The Language of Shape - The Role of Curvature in Condensed Matter, Physics, Chemistry and Biology

THE LANGUAGE OF SHAPE THE ROLE OF CURVATURE IN CONDENSED MATTER: PHYSICS, CHEMISTRY AND BIOLOGY

Cover illustration: Beyond the Euclidean desert: hyperbolic radiolaria skeletons. Adapted from SEM imoge by Roger Heady ond Michoel Ciszewski.

THE LANGUAGE OF SHAPE THE ROLE OF CURVATURE IN CONDENSED MATTER" PHYSICS, CHEMISTRY AND BIOLOGY STEPHEN HYDE DEPARTMENT OF APPLIED MATHEMATICS INSTITUTE OF ADVANCED STUDIES AUSTRALIAN NATIONAL UNIVERSITY CANBERRA, 0200, AUSTRALIA

STEN ANDERSSON SANDVIK RESEARCHINSTITUTE S. LJ~NGGATAN27 38074 L~TTORP, SWEDEN

KARE LARSSON DEPARTMENT OF FOOD TECHNOLOGY LUND UNIVERSITY BOX 124, 22100 LUND, SWEDEN

ZOLTAN BLUM DEPARTMENT OF INORGANIC CHEMISTRY LUND UNIVERSITY BOX 124, 22100 LUND, SWEDEN

TOMAS LANDH DEPARTMENT OF FOOD TECHNOLOGY LUND UNIVERSITY BOX 124, 22100 LUNDI SWEDEN

SVEN

LIDIN

DEPARTMENT OF INORGANIC CHEMISTRY LUND UNIVERSITY BOX 124, 22100 LUND, SWEDEN BARRY

W.

NINHAM

DEPARTMENT OF APPLIED MATHEMATICS INSTITUTE OF ADVANCED STUDIES AUSTRALIAN NATIONAL UNIVERSITY CANBERRA, 0200, AUSTRAUA

1997 ELSEVIER

AMSTERDAM -

LAUSANNE

-

NEW YORK -

OXFORD

- SHANNON

-

TOKYO

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

ISBN: 0 444 81538 4 91997 Elsevier Science B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the copyright owner, Elsevier Science B.V., unless otherwise specified. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper.

Printed and bound by Antony Rowe Ltd, Eastboume Transferred to digital printing ZOO5

Acknowledgments This project has gone on too long - far too long to recall all those who deserve our thanks. Nevertheless, we are very grateful to all those scientists who have kindly furnished us with data, figures and other comments. In particular, we thank Profs. Hans-Georg von Schnering (Stuttgart) and Reinhard Nesper (Ziirich), Takeji Hashimoto and Hiro Hasegawa (Kyoto) for substantial support. We have been assisted beyond the call of duty by our secretaries: Diana Wallace (Canberra) and Ingrid Mellqvist (Lund). Fiona Meldrum patiently hunted through the text, tracking down a number of typographical and language errors. Finally, we cannot forget those fresh Clyde River oysters served nightly at the Malua Bay Bowling Club, which - ably assisted by local champagne - fortified us during a week's intense work, where the form of the book was thrashed out.

Canberra, December 22, 1996.

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o,

VU

Preface

During the latest decade we have worked on periodic surfaces with zero average curvature and their significance in chemical structures, ranging from atomic and molecular arrangements in crystals to complex self-assembled colloidal aggregates. This approach has proved to be fruitful, not only in the determination of complex structures, but also in the understanding of phase behaviour and relations between structure and physical properties. Our aim has been to summarise our own understanding of this growing field, and to provide a complete description of relevant shapes and the forces behind their formation. This book deals with the role of curvature, a neglected dimension, in guiding chemical, biochemical and cellular processes. The curved surfaces that concern us might be those traced out by the head groups of phospholipid molecules that spontaneously self-assemble to form membranes and other building blocks of biology. Or they can be the surfaces of proteins involved in catalysis. They are provided in abundance par excellence by inorganic chemistry. In biology these dynamic entities have a marvellous capacity for self-organisation and selfassembly which is beginning to be understood. They transform from one shape to another under the influence of the forces of nature with an astonishing ease that allows them to manage resources, direct complex sequences of reactions, and arrange for delivery, all on time. Shape determines function, and the energetics of function dictates the optimal structure required. At least that is our thesis. The cognition and recognition of shape and form are one of the earliest tasks presented to the brain. Shape and form are so much a part of our mental processes that we tend to take them for granted. Almost any word in any language that describes objects conjures up an image that involves form. And indeed one of the deepest expressions of our sense of being is representational art. Painting and sculpture deal exclusively with colour, shape and form. Yet despite the vaunted successes of physics and mathematics that underlies m o d e m science, science remains antithetic to art because it reduces diversity to too sterile order through the imposition of Euclidean symmetry. According to conventional texts, forces act between point atoms, spheres, cylinders and planes in a kind of pythagorean and ptolemaic imperative that ignores curvature. There is nowhere an awareness that shape may have a role to play, except to please the eye. Nature ever geometrised, said somebody. True. But it has good reasons. In cell and molecular biology where mechanisms of enzyme action are not understood and attributed to some kind of Maxwell demon, all is specificity, and the lipids of membranes serve to do no more than act as a passive matrix for proteins and as a protection for the procreation of a uni-dimensional, machine like and stolidly boring DNA. There is more to it than that.

viii The thesis of this book is that two circumstances may have contributed to our present situation. The one has to do with the forces acting between chemical assemblies, and the interplay between these forces, set by the environment in which they work, and curvature. The other has to do with the absence of any language describing shapes of physically associated assemblies that are part of the subject of cellular and molecular biology. When shape is taken into account one comes to the realisation that curvature, and forces, set by constraints, are meaningful thermodynamic variables, (derived from classical thermodynamics). The key problem in the reductionist chain is how to build a statistical mechanism that uses a language of shapes. This language draws on topology and differential geometry. What we will attempt to show is that once that language is learnt, the world begins to take on a richer and more colourful unity. Through a consideration of minimal surfaces and other shapes the bewildering chaos of nature makes more sense. We are convinced that a structural description based on curvature is useful in physical and biological sciences, and the numerous examples presented here support that view. Finally, we hope that our speculations on the role of these shapes in chemical reactions and in molecular organisation in living systems will inspire new work in this field.

ix

Table of Contents Chapter The

1 Mathematics

of Curvature

1.1 ................. I n t r o d u c t o r y r e m a r k s ............................................................................................... 1 1.2 ................. C u r v a t u r e ................................................................................................................... 2 1.3 ................ D i f f e r e n t i a l g e o m e t r y of s u r f a c e s ........................................................................... 4 1.4 ................ T h e G a u s s m a p .......................................................................................................... 6 1.5 ................. G e o d e s i c c u r v a t u r e a n d g e o d e s i c s ......................................................... . ............... 7 1.6 ................ T o r s i o n ........................................................................................................................ 8 1.7 ................. T h e G a u s s - B o n n e t t h e o r e m ..................................................................................... 10 1.8 ................. T o p o l o g y .................................................................................................................... 11 1.9 ................. A p r o v i s i o n a l c a t a l o g u e o f s u r f a c e f o r m s .............................................................. 14 1.10 ............... A h i s t o r i c a l p e r s p e c t i v e ............................................................................................ 18 1.11 ............... P e r i o d i c m i n i m a l s u r f a c e s ....................................................................................... 21 1.12 ............... T h e B o n n e t t r a n s f o r m a t i o n : t h e P - s u r f a c e , t h e D - s u r f a c e a n d t h e g y r o i d ........ 27 1.13 .............. P a r a l l e l s u r f a c e s ......................................................................................................... 32 1.14 ............... F u t u r e d i r e c t i o n s ....................................................................................................... 32 Appendix: ..... A c a t a l o g u e of s o m e m i n i m a l s u r f a c e s .................................................................. 33 M a t h e m a t i c a l B i b l i o g r a p h y ............................................................................................................. 40 R e f e r e n c e s ........................................................................................................................................ 41

Chapter 2 T h e Lessons of Chemistry Inorganic C h e m i s t r y - . F r o m t h e d i s c r e t e

43

l a t t i c e of c r y s t a l s y m m e t r y to t h e c o n t i n u o u s ....... ...................... m a n i f o l d s of d i f f e r e n t i a l g e o m e t r y ......................................................................... 43 2.1 ................. T h e b a c k g r o u n d ........................................................................................................ 43 2.2 ................. T h e u n r a v e l l i n g of c o m p l e x s t r u c t u r e s .................................................................. 44 2.3 ................. D e f e c t s ........................................................................................................................ 46 2.4 ................. T h e i n t r i n s i c c u r v a t u r e of s o l i d s ............................................................................. 49 2.5 ................. H y d r o p h o b i c z e o l i t e s a n d a d s o r p t i o n ................................................................... 52 2.6 ................. P h a s e t r a n s i t i o n s , o r d e r a n d d i s o r d e r .................................................................... 55 2.7 ................. Q u a n t i t a t i v e a n a l y s i s of h y p e r b o l i c f r a m e w o r k s : s i l i c a t e d e n s i t i e s .................. 58 2.8 ................. T e t r a h e d r a l f r a m e w o r k s : T h r e e - o r t w o - d i m e n s i o n a l s t r u c t u r e s ? ..................... 63 2.9 ................. Q u a s i c r y s t a l s ............................................................................................................. 66 O r g a n i c C h e m i s t r y : T h e S h a p e of M o l e c u l e s ............................................................................. 73 2.10 ............... T h e h y p e r b o l i c n a t u r e of sp 3 o r b i t a l s ..................................................................... 3'3 2.11 ............... O r g a n i c s c u l p t u r e s : c a r c e r a n d s , c r o w n s , etc ......................................................... 75 2.12 ............... B e y o n d g r a p h i t e : f u U e r e n e s a n d s c h w a r z i t e s ....................................................... 78 Appendix: ..... T h e p r o b l e m o f q u a s i c r y s t a l s .................................................................................. 80 R e f e r e n c e s ........................................................................................................................................ 84

Chapter

3

Molecular

Forces

and

Self-Assembly

87

3.1 ................. T h e b a c k g r o u n d ........................................................................................................ 87 3.2.1 .............. T h e n a t u r e of force, s h a p e a n d s i z e ........................................................................ 88 3.2.2 .............. S e l f - e n e r g y , m o l e c u l a r s i z e a n d s h a p e ................................................................... 89 3.2.3 .............. S e l f - e n e r g y a n d a d s o r p t i o n ..................................................................................... 91 3.2.4 .............. T h e s h a p e of b o n d s ................................................................................................... 94 3.3 ................. T h e b a c k g r o u n d to s u r f a c e f o r c e s ........................................................................... 96 3.4 ................. M o l e c u l a r f o r c e s i n d e t a i l ......................................................................................... 98 3.4.1 .............. v a n d e r W a a l s f o r c e s ................................................................................................. 98 3.4.2 .............. L i f s h i t z f o r c e s ............................................................................................................. 100

3.4.3 .............. D o u b l e - l a y e r forces ................................................................................................... 103 3.5 ................. A g a l l i m a u f r y of forces ............................................................................................. 105 3.5.1 .............. Forces d u e to l i q u i d s t r u c t u r e ................................................................................. 105 3.5.2 .............. S u r f a c e - i n d u c e d l i q u i d s t r u c t u r e ............................................................................ 106 3.5.3 .............. H y d r a t i o n forces in p h o s p h o l i p i d s ........................................................................ 106 3.5.4 .............. Surface d i p o l e c o r r e l a t i o n s ...................................................................................... 107 3.5.5 .............. S e c o n d a r y h y d r a t i o n forces a n d i o n - b i n d i n g ....................................................... 108 3.5.6 .............. R a n g e of t h e d o u b l e - l a y e r force a n d i m p l i c a t i o n s ............................................... 109 3.5.7 .............. H y d r o p h o b i c i n t e r a c t i o n s ........................................................................................ 110 3.5.8 .............. N o n - i o n i c s u r f a c t a n t forces ..................................................................................... 111 3.5.9 .............. Forces of t h e r m o d y n a m i c o r i g i n ............................................................................. 111 3.5.10 ............ T h e H e l f r i c h force ..................................................................................................... 112 3.5.11 ............ Forces of v e r y l o n g r a n g e ......................................................................................... 112 3.5.12 ............ S u m m a r y .................................................................................................................... 113 3.6 ................. S e l f - o r g a n i s a t i o n in s u r f a c t a n t s o l u t i o n s ............................................................... 113 3.6.1 .............. A g g r e g a t e s t r u c t u r e in the E u c l i d e a n d e s e r t ......................................................... 116 3.6.2 .............. C u r v a t u r e as t h e d e t e r m i n a n t of m i c r o s t r u c t u r e ................................................. 117 3.6.3 .............. G e n e s i s of t h e s u r f a c t a n t p a r a m e t e r ....................................................................... 119 3.6.4 .............. T h e t y r a n n y of t h e o r y ............................................................................................... 122 Appendix A:. E v o l u t i o n of c o n c e p t s o n l o n g r a n g e m o l e c u l a r forces r e s p o n s i b l e for ........... ...................... o r g a n i s a t i o n a n d i n t e r a c t i o n s i n c o l l o i d a l s y s t e m s .............................................. 124 Append/x B: .. M o d e m c o n c e p t s of s e l f - a s s e m b l y ......................................................................... 128 Appendix C:.. R e m a r k s o n t h e n a t u r e of the h y d r o p h o b i c i n t e r a c t i o n a n d w a t e r s t r u c t u r e .. 129 R e f e r e n c e s ........................................................................................................................................ 137

Chapter

4

Beyond The

Flatland

Geometric

Forms

due to

Self-Assembly

4.1 ................ I n t r o d u c t i o n : m o l e c u l a r d i m e n s i o n s a n d c u r v a t u r e ............................................ 4.2 ................. T h e local g e o m e t r y of a g g r e g a t e s ........................................................................... 4.3 ............... T h e c o m p o s i t i o n of s u r f a c t a n t m i x t u r e s : t h e g l o b a l c o n s t r a i n t .......................... 4.4 ................. Bilayers i n s u r f a c t a n t - w a t e r m i x t u r e s .................................................................... 4.5 ................. M o n o l a y e r s i n s u r f a c t a n t - w a t e r m i x t u r e s ............................................................. 4.6 ................. G e o m e t r i c a l physics: b e n d i n g e n e r g y .................................................................... 4.7 ................. T h e m e s o p h a s e b e h a v i o u r of s u r f a c t a n t - a n d l i p i d - w a t e r m i x t u r e s ................ 4.8 ................. T h e h y p e r b o l i c r e a l m : c u b i c a n d i n t e r m e d i a t e p h a s e s ........................................ 4.9 ................. M e s o s t r u c t u r e in t e r n a r y s u r f a c t a n t - w a t e r - o i l s y s t e m s : m i c r o e m u l s i o n s ........ 4.10 ............... Block c o p o l y m e r m e l t s : a n i n t r o d u c t i o n ............................................................... 4.11 .............. C o p o l y m e r s e l f - a s s e m b l y ......................................................................................... 4.12 ............... R e l a t i o n b e t w e e n m a t e r i a l p r o p e r t i e s a n d s t r u c t u r e ........................................... 4.13 ............... P r o t e i n a s s e m b l i e s i n bacteria: a m e s h p h a s e ....................................................... 4.14 ............... S e l f - a s s e m b l y of c h i r a l m o l e c u l e s ........................................................................... R e f e r e n c e s ........................................................................................................................................

141 141 143 146 149 154 157 160 163 170 176 177 185 186 187 194

Chapter 5 Lipid Self-Assembly and Function In Biological Systems 199 Self-association of lipids in an aqueous environment ............................................................. 199 5.1.1 .............. I n t r o d u c t i o n ............................................................................................................... 199 5.1.2 5.1.3 5.1.4 5.1.5 5.1.6 5.1.7

.............. .............. .............. .............. .............. ..............

G e n e r a l b e h a v i o u r of l i p i d s in w a t e r ..................................................................... C u b i c p h a s e s .............................................................................................................. C u b i c l i p i d - p r o t e i n - w a t e r p h a s e s ........................................................................... D i s p e r s i o n s of b i c o n t i n u o u s c u b i c p h a s e s : c u b o s o m e s ....................................... L i p o s o m a l d i s p e r s i o n s ............................................................................................. Vesicles a n d m e m b r a n e s ..........................................................................................

200 203 206 207 208 209

xi

5.1.8 ............. C h o l e s t e r i c l i q u i d - c r y s t a l s a n d l o w - d e n s i t y l i p o p r o t e i n s t r u c t u r e s .................. 211 Cell membranes.............................................................................................................................. 213 5.2.1 .............. I n t r o d u c t i o n ............................................................................................................... 213 5.2.2 .............. O n i n t r i n s i c p e r i o d i c b i l a y e r c u r v a t u r e i n m e m b r a n e f u n c t i o n : ...................... A m o d e l m e m b r a n e b i l a y e r p h a s e t r a n s i t i o n i n v o l v i n g p e r i o d i c c u r v a t u r e ... 215 5.2.3 .............. L i p i d c o m p o s i t i o n c o n t r o l i n m e m b r a n e s ............................................................. 215 5.2.4 .............. T h e n e r v e m e m b r a n e , s i g n a l t r a n s m i s s i o n a n d a n a e s t h e s i a .............................. 218 ...................... B i l a y e r c o n f o r m a t i o n d u r i n g t h e a c t i o n p o t e n t i a l ................................................ 218 ...................... A n a e s t h e t i c effects .................................................................................................... 220 5.2.5 .............. A n a e s t h e t i c a g e n t s a n d c a n c e r , i m m u n o s u p p r e s s i o n ......................................... 222 5.2.6 .............. O n t h e m e t a s t a t i c m e c h a n i s m of m a l i g n a n t ceils ................................................ 224 5.2.7 .............. M e m b r a n e s i n m i c r o - o r g a n i s m s a n d a n t i - m i c r o b i a l a g e n t s ............................... 224 5.2.8 .............. P l a n t cell m e m b r a n e s ............................................................................................... 226 5.2.9 .............. T h e C 2D c o n f o r m a t i o n a n d m e m b r a n e f u s i o n ...................................................... 226 5.2.10 ............ M e m b r a n e s e n c a p s u l a t i n g o i l / f a t s a n d b i l i q u i d f o a m s ...................................... 227 5.2.11 ............ S u g a r g r o u p s , r e c e p t o r - l i g a n d b i n d i n g a n d c o o p e r a t i v i t y ................................. 229 5.2.12 ............ A C 2D m e m b r a n e s t r u c t u r e i n a Streptomyces s t r a i n ............................................ 230 R e f e r e n c e s ........................................................................................................................................ 232

Chapter

6

Folding and Function In Proteins and

DNA

237

6.1 ................. O v e r a l l f e a t u r e s of p r o t e i n s t r u c t u r e ...................................................................... 237 6.2 ................. a - h e l i x d o m a i n s ......................................................................................................... 239 6.3 .............. a - h e l i x / ~ - s h e e t d o m a i n s ....................................................................................... 239 6.4 ............... [3-sheet d o m a i n s ........................................................................................................ 241 6.5 ................ M e m b r a n e p r o t e i n s ................................................................................................... 242 6.6 ................. E n z y m a t i c a c t i o n ....................................................................................................... 243 6.7 ................. P r o t e i n f u n c t i o n a n d d i o x i n p o i s o n i n g .................................................................. 2 4 7 6.8 ................. G e o m e t r y i n h o r m o n e - r e c e p t o r i n t e r a c t i o n s ......................................................... 248 6.9 ................. Self / n o n - s e l f r e c o g n i t i o n ....................................................................................... 250 6.10 ............... D N A f o l d i n g .............................................................................................................. 251 6.11 ............... S e l f - a s s e m b l y a n d c r y s t a l l i s a t i o n of p r o t e i n s ....................................................... 253 R e f e r e n c e s ........................................................................................................................................ 256

Chapter 7 Cytomembranes and Cubic Membrane Systems Revisited

257

7.1 ................. M e m b r a n e o r g a n i s a t i o n ........................................................................................... 257 7.2 ................ R e c o g n i t i o n o f h y p e r b o l i c p e r i o d i c c y t o m e m b r a n e m o r p h o l o g i e s i n e l e c t r o n ...................... m i c r o s c o p i c s e c t i o n s ................................................................................................. 259 7.3 ................ T h e s t r u c t u r e a n d o c c u r r e n c e of c u b i c m e m b r a n e s ............................................ 266 7.4 ................ C u b i c m e m b r a n e s in u n i c e l l u l a r o r g a n i s m s : p r o k a r y o t e s a n d p r o t o z o a ......... 272 7.5 ................ C u b i c m e m b r a n e s i n p l a n t s ..................................................................................... 275 7.6 ................ C u b i c m e m b r a n e s i n f u n g i ...................................................................................... 284 7.7 ................. C u b i c m e m b r a n e s i n m e t a z o a ................................................................................. 286 7.8 ................. R e l a t i o n s h i p s b e t w e e n t u b u l o r e t i c u l a r s t r u c t u r e s , ...................... a n n u l a t e l a m e i l a e , a n d c u b i c m e m b r a n e s ............................................................. 314 7.9 ................. B i o g e n e s i s of c u b i c m e m b r a n e s .............................................................................. 317 7.10 ............... R e l a t i o n s h i p s b e t w e e n c u b i c m e m b r a n e s a n d c u b i c p h a s e s .............................. 321 7.11 ............... F u n c t i o n a l i t i e s of c u b i c m e m b r a n e s ........ ~.............................................................. 323 7.12 ............... C e l l s p a c e o r g a n i s a t i o n a n d t o p o l o g y ................................................................... 324 7.13 .............. S p e c i f i c s t r u c t u r e - f u n c t i o n r e l a t i o n s ...................................................................... 327 A b b r e v i a t i o n s .................................................................................................................................... 330 R e f e r e n c e s ........................................................................................................................................ 331

Chapter

8

Some Miscellaneous Speculations

339

Templating 8.1.1 .............. T e m p l a f i n g a n d c u r v a t u r e : D N A t e m p l a t i n g ....................................................... 339 8.1.2 .............. T e m p l a t i n g b y electric fields: e q u i p o t e n t i a l a n d t a n g e n t i a l field surfaces ....... 339 8.1.3 .............. Diffusion w i t h i n fast-ion c o n d u c t o r s ..................................................................... 340 8.1.4 .............. T h e t e m p l a t i n g of zeolites ....................................................................................... 342 8.1.5 .............. T e m p l a t i n g o r g a n i c m o l e c u l e s : the c a e s i u m effect .............................................. 344 8.1.6 .............. T e m p l a t i n g of the m o r p h o l o g y of a calcite c r y s t a l ............................................... 344 Supra s e l f - a s s e m b l y 8.2.1 .............. Biological s u p e r s t r u c t u r e s b a s e d o n s e l f - a s s e m b l y .............................................. 348 8.2.2 .............. C o l l a g e n a n d p l a n t cell w a l l s .................................................................................. 349 8.2.3 .............. The m o l e c u l a r p a c k i n g in n a t i v e s t a r c h ................................................................. 350 8.2.4 .............. S a d d l e s in the kitchen: b r e a d f r o m w h e a t flour ................................................... 352 8.2.5 .............. M u s c l e c o n t r a c t i o n .................................................................................................... 355 8.3 ................. The origin of life: a role for c u b o s o m e s ? ................................................................ 359 8.4 ................. A final w o r d ............................................................................................................... 362 References ........................................................................................................................................ 363 I n d e x .................................................................................................................................................. 3 6 5

Chapter 1

The Mathematics of Curvature

1.1. Introductory remarks his book deals with shape and form, and especially the role of curvature in the natural sciences. Our search is for a connection between structure and function posed by D'Arcy Thompson in his famous book "On Growth and Form" [1] almost a century ago. Our theme will be that curvature, a neglected dimension, is central. Some of the curved surfaces that will preoccupy us and recur are shown in the Appendix to this Chapter. The reader is invited to pursue them at once. They are not just computer generated art or mathematical abstractions, and will be seen later to be ubiquitous in nature. They represent situations as diverse as:

T

*equipotential surfaces dividing space between the atoms of a crystal *real structures formed spontaneously by the constituent molecules of biological membranes *the shapes of bio-macromolecules, from proteins to starch

Euclidean geometry underlies practically all of science and our intuition has depended on it. The shapes provided: planes, cylinders, spheres, polyhedra, all have constant or even zero curvature. Only in theoretical physics, in subjects like general relativity where the curvature of space-time is essential, has non-Euclidean geometry and especially so-called hyperbolic geometry played any part in the scheme of things. The scientific community has been prepared to leave such matters to physicists alone. It can do so no longer, and the idea of curvature is becoming an essential tool to the understanding of many phenomena. This Chapter is concerned with some of the mathematical tools required to describe special properties of curved surfaces. The tools are to be found in differential geometry, analytical function theory, and topology. General references can be found at the end of the Chapter. The reader uninterested in the mathematics can skip the equations and their development. The ideas we want to focus on will be clear enough in the text. A particular class of saddleshaped (hyperbolic) surfaces called minimal surfaces will be treated with special attention since they are relatively s t r a i g h t f o r w a r d to treat mathematically and do form good approximate representations of actual physical and chemical structures.

2

1.2

Chapter I

Curvature

The concept of curvature was d e v e l o p e d by Isaac N e w t o n in the m i d d l e of the 17th century, as a natural extension to his w o r k on the calculus. At that time, the d e t e r m i n a t i o n of the p e r i m e t e r of p l a n a r c u r v e s a n d the area u n d e r curves w e r e major p r o b l e m s . In particular, N e w t o n ' s n e w analytical tools a l l o w e d h i m to d e t e r m i n e the "quadrature" (area) of a circle. It o c c u r r e d to N e w t o n that the radius of the circle of best fit to an arbitrary planar curve at all points on the curve w a s a useful measure, for w h i c h he coined the term "crookedness"[2]. This is curvature (Fig. 1.1).

Figure 1.1: The curvature of a planar curve at a point (P) is equal to the reciprocal of the radius of the circle of best fit to the curve at P, r. The c u r v a t u r e of a planar curve relates arc length along the curve to changes of tangent vector (Fig. 1.2).

~s

.x

Figure 1.2: Tangents TP and QT at two points, P and Q, on a planar curve.

Curvature

3

The tangents TP and QT in Fig. 1.2 subtend angles ~, p+Sp with the x-axis, so that 8 ~ is the angle between the two tangents. If 5s is the length of the arc PQ along the curve, then 8_E is the average curvature of the planar curve along 8s the arc PQ. The curvature at the point P is defined to be the limit of this expression as Q approaches P, i.e. 8-E. Ss

Figure 1.3: The radii of curvature, rp and rQ at two points, P and Q, on a planar curve. The centres of the circles of best fit to the curve at P and Q lie on opposite sides of the curve and the curvature changes sign at the point of inflection on the curve between these points. The curvature at Q is positive and at P it is negative.

If PQ is the arc of a circle of radius r, the angle ~ , between the tangents at P and Q is equal to the angle subtended at the centre of the circle by the arc PQ, d~ = ~. 1 The curvature is constant at all points of a so that 8s=r5 ~, whence ~ circle, and the radius is equal to the reciprocal of the curvature (Fig. 1.1). If the curve is described in cartesian coordinates by a function y=y(x): r = ds _

dl//

ds dx _ s e c ( I g ) ~--~-" tan(I//) = d)_.[ S O dx dlg dtp dx

SCC2(Vr )d~ _d~" dx

that:

and r -

dx=

d2y

dx2 The curvature, ic, is thus given by the expression:

K'=

dx2

[' ,~,j

(1.1)

4

Chapter I

If the positive value of the root of the denominator is taken, the sign of the curvature will be the same as that of d2y" i.e. positive if the curve lies above dx--5-, the tangent, and negative below it. At a point of inflection (or a straight line), d2~---2 ' is zero and therefore the curvature is zero in these cases. The sign of the dx2 curvature signifies the convex or concave nature of the curve. It is also related to the side of the curve at which the centre of the circle of best fit is located (cf. Fig. 1.3).

1.3

The differential geometry of surfaces

The curvatures of a surface are more complex entities, but can be understood as a generalisation of the c u r v a t u r e of planar curves. Imagine a plane containing a point P on the (smooth) surface, which contains the vector (n) passing through P, normal to the surface (Fig. 1.4).

Figure 1.4: The intersection of a surface with the plane containing the normal vector (n) to the surface at the point P. The intersection of this plane with the surface is clearly a planar curve, whose curvature at P can be evaluated as described above. This curvature is equal to the value of the normal curvature, ~:n, at P in the direction prescribed by the orientation of the plane. N o w let the plane rotate about an axis coincident with the normal vector, n. The n o r m a l curvature will vary periodically, so it must attain m a x i m u m and m i n i m u m values. These values are defined to be the principal curvatures, K1 and Ic2, of the surface at P (Fig. 1.5). The directions

Differential geometry

5

at which these extremes occur are referred to as the principal directions at P. In special cases, all these curves of intersection are of equal curvature (e.g. a point on a sphere), the point is an umbilic, and principal directions cannot be defined. If the n o r m a l curvatures at the umbilic are zero (so that all the intersection curves are straight lines) the surface is locally p l a n a r at that point, which is then called a fiat point. At regular points (excluding umbilics) the principal directions are orthogonal. The principal curvatures can be combined to give two useful measures of the c u r v a t u r e of the surface, the Gaussian curvature (K) and the mean curvature

(H): K = ~q.K'2 and H - rl+~:2 2

(1.2)

(a) Both principal curvatures are of equal sign.

(b) One principal curvature is equal to zero.

(c) Principal curvatures of opposite sign.

Figure 1.5: The extrema of normal curvatures define the

principal curvatures of a surface.

The surfaces in Fig. 1.5 h a v e (a) positive G a u s s i a n c u r v a t u r e , (b) zero Gaussian curvature and (c) negative Gaussian curvature. The Gaussian c u r v a t u r e has the dimensions of inverse area and the m e a n curvature has dimensions of inverse length. The topology of the surface

6

Chapter I

(introduced below) is related to a (dimensionless) measure of the integral geometry of the surface, the integral curvature, which is equal to the areaweighted integral of the Gaussian curvature over the surface, /

K da.

The Gaussian curvature and integral curvature bear a fascinating relation to the normal vectors on the surface, and belong to the realm of intrinsic geometry, i.e. the geometry that can be deducedwithout reference to the space within which the surface is embedded. Some further results on the intrinsic geometry of surfaces will be needed throughout the book. We outline them briefly below.

1.4

The Gauss map

The Gaussian curvature has a n u m b e r of interesting geometrical interpretations. One of the more striking is connected with the Gauss map of a surface, which maps the surface onto the unit sphere. The image of a point P on a surface x under the mapping is a point on the unit sphere. This point is given by the intersection of the unit normal n to the surface at P with a unit sphere centred at P. The Gauss map of the surface x is the collection of all such points on the sphere, generated by sliding the surface through the centre of the (fixed) sphere (Fig. 1.6). If a closed curve on a surface is traversed in the opposite sense on the sphere under the mapping, the surface is saddleshaped, and the Gaussian curvature is negative. Clearly the spherical image under the Gauss map of a highly curved surface patch will be larger than that of less curved patches of the same area, since the divergence in direction spanned by the normal vectors is wider for the highly curved patch. An extreme example is the plane, which is mapped onto a single point, whose location depends on the orientation of the plane. The Gauss map is closely related to the Gaussian curvature of the surface. In fact, the surface area of the Gauss-mapped region on the unit sphere is equal to the integral curvature of the region, J K da. An alternative definition of J~ufface the Gaussian curvature follows from this result. Imagine shrinking the region progressively to an infinitesimal area about a point. In the limit the quotient of the area of the surface element and its spherical image is 1/K. If the Gauss map of a surface comprises only a single point (e.g. the plane) or a curve (e.g. the cylinder), the Gaussian curvature is zero at all points on the surface.

The Gauss map

7

Figure 1.6: The Gauss map of a surface. The normal vectors in the triangular ABC region of the

saddle-shaped surface define a region on the unit sphere, A'B'C', given by the intersection of the unit sphere with the collection of normal vectors (each placed at the centre of a unit sphere) within the ABC region. Notice that for the example illustrated the bounding curve on the surface and on the unit sphere are traversed in opposite senses. This is a necessary feature of saddle-shaped surfaces, with negative Gaussian curvature. The G a u s s i a n curvature, K, is a b e n d i n g invariant. This m e a n s that if we can b e n d a s i m p l y connected surface x into a n o t h e r s i m p l y connected surface y w i t h o u t stretching or tearing, there exists a c o n t i n u o u s t r a n s f o r m a t i o n from x to y that preserves the Gaussian curvature at every point. For example, the plane, w i t h G a u s s i a n c u r v a t u r e , K = 0, is easily rolled into a cylinder for w h i c h also K - 0. On the other h a n d there is no w a y to form a sphere (K c o n s t a n t , b u t strictly p o s i t i v e ) f r o m e i t h e r c y l i n d e r or p l a n e w i t h o u t s t r e t c h i n g , t e a r i n g or gluing. Surfaces t h a t are related b y a c u r v a t u r e p r e s e r v i n g t r a n s f o r m a t i o n (like the p l a n e a n d the c y l i n d e r ) are called

isometric.

1.5

Geodesic curvature and geodesics

A n o t h e r entit3, that we shall need belongs to the realm of intrinsic geometry:

geodesic curvature. Consider a surface x, a point P on x and a curve ~ on x p a s s i n g t h r o u g h P. The c u r v a t u r e vector of ~ at P joins P to the centre of c u r v a t u r e of ~. This c u r v a t u r e vector m a y be d e c o m p o s e d into m u t u a l l y o r t h o g o n a l c o m p o n e n t s . These c o m p o n e n t s are given by projection of the

8

Chapter1

curve ~ onto two orthogonal planes: (i) the tangent plane to x at P and, (ii) the plane containing the normal vector to x at P and the tangent vector of ~ at P. The curvature of the latter projection is the normal curvature, lCn, introduced in section 1.3. The geodesic curvature, ~Cg of ~ at P on x is equal to the curvature of the projection of ~ onto the tangent plane to x at P (Fig. 1.7). If the geodesic curvature is zero, the curvature of ~ is identical to the normal curvature. A curve whose geodesic curvature is zero everywhere is called a geodesic, and it is (locally) the shortest distance between two points on the surface. Along geodesic curves, the normal vectors to the geodesic coincide with the normal vectors to the surfaces. An infinite number of geodesics passes through any point, one for every direction emanating from the point. Geodesics on curved surfaces are rarely straight lines. Geodesics on a cur~ed surface linking two points can be constructed by stretching a string (constrained to lie on the surface) between the p o i n t s - the path taken up by the string will always follow a geodesic.

Figure 1.7: Decomposition of a curve in a surface (left) into orthogonal geodesic and normal curvatures (right).

1.6

Torsion

One further measure of the bending of curves needs mention. A nonplanar space curve exhibits curvature (which is measured by the radius of the circle of best fit to the curve) and torsion.

Torsion

9

The torsion of a curve describes its pitch: a helix exhibits both c o n s t a n t c u r v a t u r e and torsion. Its c u r v a t u r e is m e a s u r e d by its projection in the t a n g e n t plane to the curve - which is a circle for a helix - while its torsion describes the degree of non-planarity of the curve. Thus a curve on a surface (even a geodesic), generally displays both curvature and torsion. The m e a s u r e of torsion is u n a m b i g u o u s for an isolated curve p i c t u r e d in three-dimensional space. H o w e v e r , the torsion of a curve lying in a surface has a more complicated p r o p e r t y related to the g e o m e t r y of the surface. The geodesic torsion (~g) is a further m e a s u r e of the local b e n d i n g of a surface curve c o m p l e m e n t a r y to the n o r m a l and geodesic curvatures. The geodesic torsion at a point on a surface in a certain direction is equal to the torsion of the geodesic on the surface t h r o u g h that point in that direction. This can be stated more formally as follows. A triple of orthogonal vectors can be defined at a n y point on a curve lying in a surface. This triple contains the n o r m a l vector to the surface at that point, n, the tangent vector to the curve at that point, t, and a vector orthogonal to both of those vectors, k n o w n as the geodesic normal vector, u = n x t. The rate of change of n w i t h arc l e n g t h along the curve projected onto t is equal to the normal curvature. The rate of change of n projected onto u is equal to the geodesic torsion, tg. The geodesic torsion thus c o m p l e m e n t s both the geodesic and n o r m a l c u r v a t u r e s in a natural way, although (along with the normal curvature) the geodesic torsion is not a concept of intrinsic geometry. All three m e a s u r e s of b e n d i n g of curves on surfaces can be unified by the Bonnet-Kovalevsky formulae: g~- =

ds

~gU +

ICnn

du =-g~gt+ tgn

ds

da

ds

=

-~nt - tgU

Both the normal curvature and the geodesic torsion of a curve on a surface d e p e n d on the variation of the normal vectors to the surface along the curve, w h i c h implies some unexpected results. For example, a straight line certainly displays neither curvature nor torsion. H o w e v e r , the geodesic torsion of a s t r a i g h t line vanishes only if the line is e m b e d d e d in a surface of z e r o G a u s s i a n curvature. The normal c u r v a t u r e a n d geodesic torsion of a c u r v e on a surface are related to the principal curvatures of the surface (ICl and Ic2) and the angle the curve subtends with a principal direction, r tcn : Ic, cos2(+) + to2 sin2(r

(1.3)

~:s = ( K',- K'2)r

(1.4)

~) sin(~)

cItapter I

10

For a surface characterised by ~c1=-~2, the Gaussian curvature is simply related to the normal curvature and geodesic torsion: K-/('n 2 + ~g2

(1.5)

In this case, the magnitude of the geodesic torsion at a point on a straight line lying in the surface is equal to the m a g n i t u d e of the principal curvatures of the surface at that point.

1.7

The Gauss-Bonnet theorem

The Gauss-Bonnet theorem is a p r o f o u n d theorem of differential geometry, linking global and local geometry. Consider a surface patch R, b o u n d e d by a set of m curves ~i- If the edges ~i meet at exterior angles Oi and they have geodesic curvature ~g(Si) where si labels a point on ~i then the theorem says

/('g(Si) dsi + ~-' boundaD"

R

K da + ~ Oi = 211: i=1

(1.6)

Figure 1.8 illustrates the case for a surface patch consisting of four b o u n d a r y arcs, ~I=AB, ~2=BC, ~3=CD and ~4=DA.

Figure 1.8: Four arcs belonging to a surface. From the Gauss-Bonnet theorem, the integral curvature within the region of the surface bounded by the arcs (ABCD) is determined by the vertex angles (0/) and the geodesic curvature along the arcs AB, BC, CD and DA.

The Gauss-Bonnet theorem

11

Choose a triangle traced on a surface, whose three edges are geodesics. From the theorem, we have 3 [ Kda + ~_~ 0i= 2~: R i=1

(1.7)

The external angles, e i, are related to the internal angles, ai, by e i = ~ - a l so that the area of the geodesic triangle is: 3

~_~ a i = ~ + ~ Kda i=1 R

(1.8)

Thus, for a Euclidean triangle (which is located on a surface of zero Gaussian curvature, such as the plane), the s u m of the vertex angles of a triangle is indeed ~. However, if the triangle decorates a surface of negative Gaussian curvature, the s u m of the angles is less than ~, if the integral surface has positive integral curvature, the s u m of the angles exceeds ~. The angle "excess" (i.e. its difference from ~) is thus a measure of the integral curvature within the region b o u n d e d by the geodesic edges.

1.8

Topology

G e o m e t r y , differential or otherwise, deals w i t h the metric relationships of rigid objects. There are s o m e f u n d a m e n t a l aspects of shapes that are preserved if the objects studied consist of stretchable rubber sheets. A rubber sphere m a y be deformed into an ellipsoid, or a long, n a r r o w cylinder with caps, or indeed any globular object (Fig 1.9).

Figure 1.9: A sphere can be stretched and bent (without any rupture or fusion of the surface) into an infinite variety of globular surfaces, all topologically equivalent.

ChapterI

12

Similarly, a torus can be d e f o r m e d into any two-sided surface containing a single handle, or a single hole (Fig 1.10).

Figure 1.10: A donut-shaped torus (with a single hole) can be deformed into any two-sided surface containing a single handle, such as a cup. Surface topology is "rubber sheet" geometry, since only those geometrical characteristics of the surface that are maintained u p o n stretching or squeezing are relevant. The usual geometrical notions of area, length, etc., are excluded from topological analysis. Suppose a surface, x, is facetted, subdividing x into a n u m b e r of faces, edges ( b o u n d i n g the faces) and vertices. Denote the n u m b e r of faces by F, the n u m b e r of their edges by E and the n u m b e r of vertices by V. Descartes, a n d later Euler, discovered that (F-E+VO =2 for all polyhedra (Table 1.1). Table 1.1: Relation between numbers of faces (F), edges (E) and vertices (lO of conventional polyhedra. F

E

V

4 6 8 12 20

6 12 12 30 30

4 8 6 20 12

(F-E+V) 2 2 2 2 2

polyhedron tetrahedron cube octahedron dodecahedron icosahedron

Topology

13

This p r o p e r t y holds because (F-E+V) is a topological characteristic, d e p e n d e n t o n l y o n the t o p o l o g y of the f a c e t t e d surface. Since all p o l y h e d r a are topologically e q u i v a l e n t to the s p h e r e (Fig 1.9), (F-E+V) is c o n s e r v e d . The value of this integer is k n o w n as the Euler-Poincar6 characteristic Z(x): F - E + V = Z (x)

(1.9)

N o w imagine a net lying in a s m o o t h surface. This net also facets the surface into c u r v e d faces, c u r v e d edges a n d vertices. If each n o d e o n the n e t has z edges (so that the net is z-connected), a n d the ring size of each ring in the net is n, Euler's relation for the net is: F = Vz.

E =

(1.!0)

so that -- = n + z 1

Table 1.2: Relation between average ring size, n, connectivity, z, and Euler-Poincar~ characteristic per vertex, x/V, for a range of networks that are regular tessellations of surfaces 9 The nature of the network is set by the value of z/V: cages, planar networks and threedimensional frameworks are characterised by positive, zero and negative z/V respectively.

z

n

3 3 3 4 5

3 4 5 3 3

3 4 6 3 3 9

network

type

3/2 1 1/2 1 1/2

tetrahedron cube dodecahedron octahedron icosahedron

cage " " " "

6 4 3

0 0 0

hexagonal tiling square thing triangular thing

sheet " "

7 8

-1/2 -1

"skew" polyhedra

9

4 9

5 ~

X/V

framework ,I

o

-1 o

This result implies that the ring size a n d connectivity of a n e t w o r k d e t e r m i n e the t o p o l o g y of the surface w h i c h c o n t a i n s t h a t n e t w o r k . This a l l o w s for simple characterisation of cage, sheet a n d f r a m e w o r k nets, distinguishable b y the value of their Euler-Poincar~ characteristic (Table 1.2).

A n o t h e r topological characteristic, the genus, g(x), of a surface, is a m e a s u r e of its connectedness. It is equal to the n u m b e r of holes or handles in the surface a n d simply related to the Euler-Poincar~ characteristic by

14

Chapter I

g(x) = [2- X'(x)]/2

(1.11)

(The equation applies only to 'orientable' surfaces, those with distinct sides. This excludes one-sided surfaces, such as the M6bius strip.) Thus, a sphere has genus zero. A torus (or a sphere with one handle) has genus one, and so on.

Remarkably, the topology is linked to the integral curvature of a surface by the simple equation: K da = 2~X

(1.12)

lurfac~ This means that all surfaces with the same number of handles or holes have the same integral curvature! In other words, no amount of bending or squeezing of a surface can add any net integral Gaussian curvature to that displayed by the simplest topologically equivalent surface. For example, although regions of negative Gaussian curvature can be formed in a sphere by squeezing the surface to produce a region which is saddle-shaped, this contribution to the integral curvature will always be compensated by a corresponding positive integral curvature in other regions. A more general, global version of the Gauss-Bonnet theorem can now be stated: Let x be an oriented surface and R be a bounded region of x. As before, let the boundary of R be the union of m simple curves ~i that do not selfintersect, and let 0i be the external angles at the m vertices. Then we have

E

rg d~i +

i= !

naary

f K da + ~,~m Oi = 2~z(R)

(1.13)

i= I

1.9 A provisional catalogue of surface forms It is obviously impossible to offer a complete catalogue of curves. Similarly, no comprehensive list of surface forms can be drawn up. The language of surface shape is a rich one: some familiar forms like the sphere and the plane, are deeply imbued in our consciousness, while others remain difficult to describe and visualise in terms that are intuitively reasonable to all of us raised on the limited vocabulary afforded by the simpler forms. But some attempt at exhaustive classification is necessary: the wealth of form in natural structures draws on the richness of abstract form, so that if we are to

A catalogue of surfaces

15

understand natural structures, it is necessary to obtain as full an intuition about surface forms as possible. To do this we must draw on non-Euclidean geometries. These different geometries emerge if the "parallel postulate" of Euclid's "Elements" is no longer taken as axiomatic. Following the work of Bolyai, Lobachevsky, Gauss and Riemann early last century it became clear that if the parallel postulate is relaxed, three quite distinct geometric classes exist. The three classes are elliptic (single and double), parabolic (Euclidean) and hyperbolic. Riemann's great contribution to geometry lay with his program of abstracting and relating the notion of form to the concept of differential form. The Riemannian approach classifies a geometry by its functional (pointwise) structure - characterised by the curvatures and metric. The triad of local surface geometries is characterised by Gaussian curvature: elliptic shapes have positive Gaussian curvature, Euclidean shapes have zero Gaussian curvature and hyperbolic shapes have negative Gaussian curvature. The vocabulary of surface forms can be developed by marrying this local geometry to (global) topology. Any surface form can be characterised by its intrinsic shape (its metric and curvature) and its global (extrinsic) embedding in space. Since natural structures are embedded in our particular space, we are concerned in this book with two-dimensional (surface) embeddings in threedimensional Euclidean space. (We leave the cosmologists to debate the nature of the deviations from Euclidean space due to matter. However, it is certain that at the length scales we deal with here, the space we live in can be safely approximated by Euclidean space.) Thus, although the embedding space is always approximately Euclidean, its intrinsic geometry may be locally Euclidean (parabolic) or non-Euclidean, viz. elliptic or hyperbolic. In the vicinity of elliptic points, the surface can be fitted to an ellipsoid, whose radii of curvature are equal to those at that point (Fig. 1.5a). The surface lies entirely to one side of its tangent plane, it is "synclastic" and both curvatures have the same sign. About a parabolic point, the surface resembles a cylinder, of radius equal to the inverse of the single nonzero principal curvature (Fig. 1.5b). Hyperbolic ("anticlastic") points can be fitted to a saddle, which is concave in some directions, fiat in others, and convex in others (Fig. 1.5c). At hyperbolic points, the surface lies both above and below its tangent plane. The most familiar surfaces have constant Gaussian curvature over the surface. In our space this is only possible in elliptic and parabolic cases: the sphere and the cylinder and plane respectively. It is impossible to form hyperbolic surfaces of constant (negative) Gaussian curvature without singularities. In general surfaces contain elliptic, parabolic and hyperbolic regions. The "average geometry" of a surface can be characterised by the average value of its Gaussian curvature, . This is equal to the integral curvature divided by the surface area A:

16

C/u~ter 1

L (K>=

Kda rfac~

(1.14)

m'l'~:e The integral curvature of a surface is linked to the Euler-Poincar~ characteristic of that surface (X) by eq. (1.12). This allows the average geometry of orientable surfaces to be related to the number of holes or handles, characterised by the surface genus, g, and the area of the surface, A, by the relation: 2-2g {K}-- A

(1.15)

Figure 1.11: A random sponge-like surface. (Picture courtesy of Peter Pieruschka.)

Thus, surfaces that are free of holes or handles like the sphere have positive and X; they are elliptic. Surfaces with a single handle or a hole (a donutshaped torus, or - plus or minus a single p o i n t - a cylinder or a plane) are on average parabolic, or two-dimensionally Euclidean (=x=O).All surfaces with more than one hole (or handle), are hyperbolic (negative , X)-

A catalogue of surfaces

17

Within a topological context, most surfaces are hyperbolic, yet our knowledge of this geometric class is less developed than for the other two classes. This book is entirely concerned with these hyperbolic surface geometries. In particular, we focus on the most exotic topological species within the hyperbolic realm, namely surfaces of infinite genus. The most general surface within this class is a boundless sponge structure (Fig. 1.11). The mathematics of such random surfaces is at present poorly developed. We confine our treatment to the more tractable cases of periodic surfaces of unbounded genus.

Figure 1.12: A square mesh surface, which is two-periodic.

Periodic hyperbolic surfaces of infinite genus can be further subdivided into those which exhibit a lattice in one-dimension (one-periodic surfaces), twoperiodic and three-periodic surfaces. These periodic surfaces can be divided into equivalent regions, bounded by a unit cell of space that contains the smallest region of the surface that reproduces the complete surface upon translation of the unit cell alone. For geometric purposes, the surface contained within the unit cell (also called a "lattice fundamental region", to distinguish it from a unit cell of the space group to which the surface belongs) is characteristic of the complete surface. To form connected surfaces of unbounded genus, one-periodic surfaces must have genus per unit cell at least equal to unity, two-periodic surfaces must contain genus two (or higher) surfaces within the unit cell and three-periodic surfaces must have a genus per unit cell greater than or equal to three. One-periodic surfaces are relatively unexplored to date. (The double strands of DNA lie on the simplest one-periodic surface, the helicoid.) Two-periodic surfaces deserve some comment. The most interesting examples of these surfaces can be visualised as confined between two parallel bounding planes, with a regular network of pores joining the two parallel sheets. We call these surfaces "mesh surfaces", due to their characteristic twodimensional porous network, which resembles a mesh. A square mesh surface is shown in Fig. 1.12. The mean curvature of these surfaces can be

18

CA~ter1

constant [3], although it cannot equal zero. The inner and outer volumes on either side of these surfaces are quite different: the exterior volume consists of two half spaces, interconnected via a lattice of tunnels and the interior is a two-dimensional tubular network. Three-periodic hyperbolic surfaces of infinite genus carve space into two intertwined sub-volumes, both resembling three-dimensional arrays of interconnected tubes. They are simple candidates for the interfaces in bicontinuous structures, consisting of two continuous subvolumes [4, 5]. As such they have attracted great interest as models for microstructured complex fluid interfaces, biological membranes, and structures of condensed atomic and molecular systems, to be explored in subsequent Chapters. The simplest three-periodic hyperbolic surfaces are "Infinite Periodic Minimal Surfaces" (IPMS, named by Alan Schoen [6]). For these surfaces, the mean curvature is constant on the surface, and everywhere identically zero. This is a defining characteristic of minimal surfaces. For these structures, the sub-volumes can be geometrically identical. This occurs if the IPMS contains straight lines. Such surfaces have been called '~alance surfaces" by Koch and Fischer [7]. We focus primarily on IPMS in this book. Some further discussion of general properties of minimal surfaces is in order here, since a number of their geometrical and topological properties will be required for later chapters.

1.10 A historical perspective The study of minimal surfaces arose naturally in the development of the calculus of variations. The problem of finding the surface forming the smallest area for a given perimeter was first posed by Lagrange in 1762, in the appendix of a famous paper that established the variational calculus [8]. He showed that a necessary condition for the existence of such a surface is the equation __~

Zx

+

ax 41 + z~ + z,? .

-

=0

(1.16)

I+Z?+

Here Z---Z(x,y) is the equation of any surface bounded by the perimeter; Z x, Zy denote partial derivatives. The partial differential equation for a surface is then: (1 + Z ~ Z x x - 2 ZxZvZxy + (1 + Z 2 ~ y y = O .

(1.17)

Lagrange pointed out that the plane would be a trivial solution to the equation but made no further investigations to see what other possibilities existed.

History of minimal surfaces

19

In 1744 Euler discovered the catenoid, the first non-planar minimal surface. This surface is readily realised by a soap film, spanning coaxial circular b o u n d i n g wires. The film shrinks under the action of its surface tension, forming the minimal surface (Fig. 1.13).

Figure 1.13. (Top): A region of the catenoid formed by a soap film. (Bottom left): Computergraphics image of a portion of a catenoid. (Bottom right): larger view showing "trumpet" ends.

The link between curvature and minimal surfaces was made by Meusnier in 1776 [9] . He proved that eq. (1.16) implies that the mean curvature is zero everywhere on a minimal surface. In his own words: "la surface de moindre ~tendue entre ses limites a cette propri~td, que chaque element a ses deux rayons de Courbure de signe contraire & egaux". This is the defining property of a minimal surface;

20

Chapter I

For a minimal surface, the principal curvatures are equal, but opposite in sign at every point. The Gaussian curvature is then always non-positive, and the mean curvature is zero. In addition to the catenoid, Meusnier also found a further non-trivial solution to eq. (1.16), the helicoid, shown below.

Figure 1.14: The helicoid, one of the earliest discovered examples of minimal surfaces.

The helicoid is the only minimal surface built up entirely of straight lines (a ruled surface) and the catenoid is the only minimal surface of revolution. These surfaces are related through the Bonnet transformation that will be discussed later. The term "minimal" is misleading. (The terminology is however timehonoured and we shall live with it.) In fact, the differential equation (16) is satisfied by any surface that constitutes a local critical point to the area function (i.e. a minimum or a maximum). The minimal area property of minimal surfaces is characteristic only of a finite patch of the surface with prescribed boundary. For the systems that concern us in subsequent chapters, this area property is irrelevant. It is the curvature characteristic of minimal surfaces that is important. On any surface, the principal directions are mutually orthogonal at regular points (recall section 1.3). On minimal surfaces, this is true for asymptotic directions as well. (An asymptotic direction is that along which the normal curvature vanishes.) Orthogonality of the asymptotic directions can be shown

History of minimal surfaces

21

to be a requirement that is equivalent to that of zero mean curvature. Hence the orthogonality property can be used to define minimal surfaces. Gauss' paper of 1827 "Disquisiones generales circa superficies' curvas" [10] marks the birth of differential geometry. Following the advances of Gauss, it became possible to deal with surfaces by their intrinsic geometry, which includes those surface features that can be determined without reference to the external space containing the surface. A central tool introduced by Gauss was the Gauss map, discussed in section 1.4. The Gauss map of a surface is conformal (angle-preserving and representable by a complex analytic function) if and only if the surface is a sphere or a minimal surface. This property is a very useful one, since it allows minimal surfaces to be analysed from their Gauss map. The Gauss map of a surface exhibits singularities at special points on a surface, k n o w n as umbilics. On minimal surfaces, the umbilics are easily recognised: they are the points on the surface where its Gaussian curvature is equal to zero, and the surface is locally planar. The nature of these flat points can vary, leading to distinct classes of singularities in the Gauss map. It turns out that the distribution of these singularities in the Gauss map uniquely determines the (intrinsic) geometry of minimal surfaces.

1.11 Periodic minimal

surfaces

Following the discovery of the helicoid by Meusnier, sixty years elapsed before additional examples of minimal surfaces were given in explicit form. In 1835 Scherk published five more examples of minimal surfaces [11]. Of these, two are periodic. They are often referred to simply as Scherk's first and second surfaces (Fig. 1.15).* Two of the leading mathematicians of the nineteenth century, Riemann and Weierstrass, played crucial roles in the further d e v e l o p m e n t of minimal surface theory. Their most i m p o r t a n t result, for which Weierstrass is accredited, is the triplet of integrals k n o w n as the Weierstrass equations. These e q u a t i o n s offer a useful route to periodic minimal surfaces. A summary of the theory follows. A minimal surface can be represented (locally) by a set of three integrals. They represent the inverse of a m a p p i n g from the minimal surface to a Riemann surface. The mapping is a composite one; first the minimal surface is mapped onto the unit sphere (the Gauss map), then the sphere itself is mapped onto the complex plane by stereographic projection. Under these operations, the minimal surface is transformed into a multi-sheeted covering of the complex plane. Any point on the minimal surface (except fiat points), characterised by cartesian coordinates (x,y,z) is described by the complex n u m b e r co, which

*According to Nitsche [28,29], this terminology is grossly misleading, and has occasioned much debate among those concerned with such matters. Such is life.

22

Chapter I

defines its m a p p e d location in the complex plane. The value of ca is dependent only on the surface orientation of (x,y,z), via the Gauss map (Fig. 1.16).

Figure 1.15: Scherk's first and fifth minimal surfaces, d i ~ o v e r e d in 1831. The first is twoperiodic, the second one-periodic.

The Weierstrass equations allow calculation of the cartesian coordinates ((x,y,z) with respect to an origin (xo,Yo,zo)) of the minimal surface at all points on the surface - except flat points - in terms of a complex analytic function R(60). The Weierstrass equations are:

x = x0 + R e

e ie (1 - 0 2) R(60) do)

Y = Yo + Im

e i8 (1 + 602) R(60) d60

z = zo- Re

e ia (260)R(60)d60

(1.18)

Integration is carried out on an arbitrary path from 60o to 601 in the complex plane, for a fixed value of 0 between 0 and n/2. Any analytic function R(60) can be plugged into the equations, to give a minimal surface. Further, the function R(60) uniquely determines a family of surfaces related by the Bonnet transformation as 0 varies. The converse does not hold; any surface will support an infinity of functions R(60) corresponding to different orientations of the original surface. The essence of the problem in elucidating interesting minimal surfaces lies in the choice of suitable functions, R(60).

Periodic minimal surfaces

23

Figure 1.16: Mapping of a minimal surface from real space to the complex plane. A point P on the surface, whose normal vector at P is n, is transformed to a point P' by the Gauss map, given by the intersection of n (placed at the origin of the unit sphere O) with the sphere. P' is mapped into a point P" on the complex plane (real and imaginary axes o"and i"resp.) by stereographic projection from the north pole of the sphere, N, onto the complex plane, which intersects the sphere in its equator.

Figure 1.17: (a) The minimal surface spanning four straight edges of equal length, subtending vertex angles of 60". (b) "Monkey saddle", formed by six copies of the saddle shown in (a). Each copy is related to its adjacent one by rotation of 180" about the common straight edge. These engravings are from Schwarz' original paper. The w o r k of W e i e r s t r a s s a n d R i e m a n n on a n a l y t i c f u n c t i o n s a n d s u r f a c e g e o m e t r y p r o v i d e d the setting for the w o r k of S c h w a r z w h o p i o n e e r e d the s t u d y of three-periodic m i n i m a l surfaces (IPMS). Schwarz, a s t u d e n t of

24

Chapter 1

Weierstrass, worked out the Weierstrass representation (eqs. 18) for two of the simplest IPMS, now called the P- and D-surfaces. (The latter is sometimes called the F-surface.) He also described three further three-periodic minimal surfaces, the CLP-(Crossed Layers of Parallels), H-, and T- surfaces. (The last is also called "Gergonnes surface"). These IPMS are illustrated in the Appendix. For the first time, the extraordinary complexity of these surfaces was revealed [12]. For the next half-century, his work was extended by others.

Figure 1.17(c): A single node of tile three-periodlc [)-'~urface. Four tunnels Ion one side ~f the surface) meet at each node, at angle~ of 10~.5 . Image courtes\' of David Anderson. (d): A model of a portion of the D-surface. The surface partitions space into two interpenetrating open labyrinths, each lying on a d i a m o n d lattice.

Schwarz' mastery of complex analysis was certainly responsible for his prizewinning work. (He was awarded a medal from the Prussian Academy for his original essay outlining the analysis and geometry of these surfaces.) However, much of the analytical groundwork had already been laid out by his predecessors. In particular, Riemann had derived representations for a number of minimal surface patches of IPMS, apparently without realising the extraordinary complexity of the complete IPMS. Schwarz established that the extended surface can be generated from a small surface patch by a simple procedure, at least in special cases. In particular, if the surface patch is bounded by straight lines, the patch can be extended beyond the straight edge by rotation about that edge through 180 ~. Thus, for example, the saddle spanning four straight edges (each subtending and angle of 60") (Fig. 1.17(a)) leads to a "monkey saddle" (Fig. 1.17(b)) with three peaks and three valleys (one for each leg of the monkey and one for the tail.) Further rotation of the monkey saddle leads to the formation of a continuous hyperbolic surface that partitions space into two interpenetrating networks of tunnels. Each network consists of nodes connecting four tunnels meeting at tetrahedral angles (109.5 ~ (Fig. 1.17(c)). The nodes are arranged on

Periodic minimal surfaces

25

a cubic lattice, which is equivalent to the diamond network (Fig. 1.17(d)). An intuitive picture of the formation of the surface can be seen as follows. Imagine a network model of the diamond structure (Fig. 1.17(e)), blue lattice), constructed from rubber tubes. Now inflate the network, swelling the hollow tubes. The resulting structure is a curved continuous network, enclosing the tunnels in the diamond network. If the inflation procedure is continued, the surface closes up around a complementary diamond network. The D-surface is the "half-way point" during the procedure.

Figure 1.17: (e) Computer image of a surface produced by partial inflations of a diamond network (blue). The "outside" of the surface wraps around a complementary (red) diamond lattice. Picture courtesy of David Anderson.

The P-surface can be found by inflating a simple cubic network, where the tubes connect nearest neighbours, giving a cubic array of nodes, each linking six tunnels (Fig. 1.18(a,b,c)).

Figure 1.18. Portions of the P-surface. (a): a single "unit cell", (b): four unit cells.

26

Chapter I

Figure 1.18(c): Computer image of 6x6x6 conventional unit cells of the P-surface (courtesy P. Pieruschka).

Other IPMS which are free of self-intersections can likewise be described by their tunnel geometries. In general, minimal surfaces display self-intersections. The most usual cases are surfaces that intersect themselves everywhere, and the "surface" wraps onto itself repeatedly, eventually densely filling the embedding space. We are only interested in translationally (or orientationally) periodic minimal surfaces, which are free of self-intersections (thereby generating a bicontinuous geometry) or periodic surfaces with limited self-intersections. Elucidation of these cases of interest requires judicious choice of the complex function R(r in the Weierstrass equations (1.18). Remarkably, the D- and P-surfaces are related by the Bonnet transformation. They are both described by the same function in the Weierstrass equations, shown by Schwarz to be:

R(ca)= 1/~/(1-14r162 The "Bonnet angle", 0, is equal to 0 for the D-surface and n/2 for the P. The function R(r used by Schwarz for the D- and P- surfaces is simply the inverse of the square root of the product over the images of the fiat points under the map from the minimal surface to the complex plane. Using r i to denote the fiat point images, the representation can be written: 8

= [I

(1.19)

Periodic minimal surfaces

27

This product form for the Weierstrass polynomial is readily generalised, and offers a useful route to the discovery and parametrisation of three- periodic minimal surfaces (IPMS). It turns out that for all "regular surfaces" (which are the topologically simplest IPMS), the distribution and character of the fiat point images (the location and type of the branch points (of R(~o)) in the complex plane) alone suffice to construct the Weierstrass polynomial, and thus the complete IPMS, using the Weierstrass equations. The Gauss map of an IPMS (which is a function of the surface orientation only through the normal vectors) must be periodic, since a translationally periodic surface is necessarily orientationally periodic. (The converse, however, is not true.) Consequently, the Gauss map of IPMS must lead to periodic tilings of the sphere. This principle has been used to construct all the simpler IPMS, and has recently been generalised to allow explicit parametrisation of more complex "irregular" IPMS [13-24]. Some of these examples are illustrated in the Appendix to this chapter.

1.12

The Bonnet transformation: the P-surface, the D-surface and the gyroid.

The variety of curves that can be traced out by a length of string is endless. This freedom of movement does not extend to surfaces. Most surface patches can only be moulded into each other with some accompanying change in the area of the sheet. However, anyone who rolls up a sheet of paper knows that a plane may be deformed into a cylinder, and it may also be rolled into a cone. This can be done without any stretching or squashing of the sheet. Thus all angles and lengths on the surface are preserved under this operation, which is known in mathematical jargon as an isometry. All those properties that belong to the intrinsic geometry of the surface remain unchanged, including the Gaussian curvature. While the plane can be "developed" into a cylinder or a cone, a sphere cannot be isometrically deformed at all; a property called "the rigidity of the sphere". Among those surfaces that are able to be deformed isometrically are minimal surfaces and surfaces of constant mean curvature. In 1853 Ossian Bonnet discovered the isometric transformation between minimal surfaces that bears his name. He wrote: "Chaque surface minima a une conjugu~e qui est d~veloppable sur elle" (every minimal surface has a "conjugate" minimal surface into which it can be developed). The distinct sequence of deformations is characterised mathematically by the variation in the Bonnet angle 0 in eqs. (18). The simplest example of this Bonnet transformation is the bending of the catenoid into the helicoid. (Fig. 1.19).

28

Chapter I

Figure 1.19: The isometric Bonnet transformation acting on a helicoid (top left), produces a catenoid (bottom right). This transformation leaves the intrinsic geometry intact (e.g. Gaussian curvature, lengths and angles defined at corresponding points) and preserves the zero mean curvature (characteristic of minimal surfaces) throughout, although the global embedding of the minimal surfaces changes dramatically. If the process is continued beyond the catenoid, a helicoid of opposite handedness is generated.

The identification of Bonnet-related minimal surfaces is facilitated using the Gauss map of the surface. A minimal surface is characterised by its fiat points, so any two surfaces with the same collection of flat point images (and singularity types) under the Gauss map (within a rigid body reorientation of the surface) must be associated by the Bonnet transformation. Thus, for example, the P- and D- surfaces, which have an infinite number of flat points, display only eight distinct fiat points under the Gauss map due to the eight distinct surface normal vectors at the fiat points. If the origin of each normal vector is placed at the centre of a cube, the endpoints of the vectors point to the eight vertices of a cube in both the P- and D-surfaces. This feature is clearest in the P-surface, shown in Fig. 1.20. Due to their identical distribution of normal vectors at the flat points (and type of fiat points), the P- and D-surfaces are "adjoint" (or, to quote Bonnet, "conjugate"), their associated Bonnet angles differ by 7t/2 (Figs. 1.22). In the 1960's Schoen [6] made the remarkable discovery that a third intersection-free IPMS- the gyroid - is generated during the transformation (at a Bonnet angle of approximately 38~ Generic minimal surfaces formed during this transformation are self-intersecting, and aperiodic.

The Bonnet transformation

29

Figure 1.20: A unit cell of the P-surface, embedded in a cube. The normal vectors to the P-surface at its eight fiat points (one obscured) are indicated by the arrowed vectors. These vectors point towards the eight vertices of the cube.

Figure 1.21: The gyroid surface discovered by Schoen in the 1960's. (Top: view down [111] axis of a larger partion of the surface; bottom: solid model.)

30

Chapter 1

The general features of the Bonnet t r a n s f o r m a t i o n can be seen in the simplest e x a m p l e , n a m e l y the i s o m e t r y b e t w e e n the catenoid a n d the helicoid (Fig. 1.19). U n d e r the action of the transformation, each point on the surface traces an ellipse in space, c e n t r e d at the origin. If the C a r t e s i a n c o o r d i n a t e s of identical points on adjoint surfaces are (x,y,z) and (x",y",z"), the coordinates of an associate surface, characterised by a Bonnet angle of e are: (x',y',z') = (x,y,z) cos(e) + (x",y",z") sin(0)

Figure 1.22: (a) The Bonnet transformation relating the D-surface, the gyroid and the P-surface. A simple saddle of the D-surface (bounded by an alternative set of straight lines to those in Fig. 1.17(a)) is twisted, ultimately forming a unit bounded by planar curves, which is a portion of the P-surface.

The Bonnet transformation

31

The effect of the Bonnet t r a n s f o r m a t i o n on IPMS' is to t r a n s f o r m the lattice of catenoidal channels - characteristic of IPMS - into helicoidal strips, t h r o u g h a screw operation on the w h o l e surface. For example, the channels in b o t h the P- a n d D-surfaces are t r a n s f o r m e d into spiral tunnels in the gyroid. Due to its intermediate Bonnet angle w i t h respect to the P- and D-surfaces, it lacks straight lines (2-fold axes) and m i r r o r planes. The labyrinths on both sides are e n a n t i o m o r p h i c : one l a b y r i n t h is l e f t - h a n d e d a n d the other r i g h t - h a n d e d (Fig. 1.21). The P-surface, the D-surface and the gyroid, are the simplest m e m b e r s of a large family of structures w h o s e m e m b e r s are still being identified. In m a n y w a y s these three surfaces are the m o s t important: they have b e e n identified in a variety of physical systems, from silicates to cells. Their i m p o r t a n c e can be traced to the fact that they are the most h o m o g e n e o u s IPMS, w i t h the s m o o t h e s t and s m a l l e s t variations (suitably scaled) of G a u s s i a n c u r v a t u r e over the surface. This feature is d u e to their relative paucity of fiat points: they h a v e the m i n i m u m n u m b e r of fiat points per unit cell of all IPMS. (In a topological sense they are the least complex IPMS, displaying the m i n i m u m genus per unit cell of all IPMS - three.)

Figure 1.22(b): Bonnet transformation acting on a number of unit cells of the D-surface (top), viewed along a <100> axis. (The tunnels of the D-surface are obscured in this orientation.) Successive images are for Bonnet angles of 10~ 20~ 30~ 38.015~ and 90~ The transformation "locks in" to the gyroid at a Bonnet angle of 38.015~ Continuation of the transformation beyond the gyroid leads to the P-surface, with tunnels along the <100> directions (lower image). The images are all scaled equivalently.

Chapter I

32

1.13

Parallel surfaces

Two surfaces x and y are parallel if they have an identical distribution of normal vectors; i.e. their Gauss maps are indistinguishable. Thus, a family of parallel surfaces can be produced by translating a surface in the direction of its normal vectors by an equal amount everywhere on the surface. If x and y are parallel surfaces separated by a distance c, it can be shown that their Gaussian and mean curvatures are related by: Ky = Kx/(1 + 2H x + C2Kx)

(1.20)

Hy = (H x - cKx)/(1 + 2H x + c2Kx)

Two interesting conclusions can be drawn from these formulae. First, the Gaussian and mean curvatures of surfaces parallel to minimal surfaces related by a Bonnet transformation remain unchanged. Secondly, the Gaussian curvature of a surface parallel to a minimal surface increases with c. This means that the minimal surface has a larger area than related parallel surfaces.

1.14

Future directions

There has been a resurgence of mathematical research into minimal surfaces in recent years. A number of new complete minimal surfaces [25, 26] have been elucidated, which will surely lead to novel classes of IPMS. Developments have made the parametrisation of more complicated periodic minimal surfaces possible (see Appendix), and the beautiful work of Karcher and colleagues at Bonn [14, 27] and Fischer and Koch [7] at Marburg has led to a plethora of new minimal surfaces containing straight lines free of selfintersections. Clearly, many new hyperbolic surfaces of infinite genus remain to be discovered, and the program initiated by Meusnier, Gauss, Riemann and Schwarz is now well underway.

33

Appendix: A catalogue of some minimal surfaces Let us finish this chapter with a catalogue of some minimal surfaces that can be generated using the Weierstrass parametrisation (eqs. (1.18)). We start by imposing some extra conditions on the function R(ta), which defines the geometry of the minimal surfaces via the Weierstrass equations (1.18). We will consider only those surfaces whose Gaussian curvature is everywhere finite and whose fiat points are isolated. Since the orientation of the surface in space is arbitrary, the north pole on the Gauss sphere (which maps out the point at infinity in the complex plane u n d e r stereographic projection) may always be chosen to have nonzero Gaussian curvature. The Gaussian curvature within the Weierstrass representation is given by [28, 29]: K=-41 R(0))1-2(1 +1 r J2). This imposes an asymptotic behaviour of the function R(t0). lim(K) = lim {- 4 IR(t0~ (1 + ]o~)-4} = C ta-.r

where C is some negative, finite number. This implies lim IR(ta~ =lim (1+]a~)2 = to4 tO--r

ta--r

Then we can consider a general form

R(co) = ~ I a (ca- r -a /b

(1.21)

/=1

This representation enables a catalogue of minimal surfaces to be built. The least n u m b e r of flat points a minimal surface may possess is one. An example of such a surface is Enneper's surface, which is asymptotically flat (Fig 1.23). In this case the fiat point is not an isolated point on the surface. However, only a single surface orientation is displayed by the asymptotically flat boundary. The surface is unique in that the Bonnet transformation applied to this surface does not produce distinct surfaces. All the members of the isometric family related to Enneper's surface are equivalent - they differ only in their relative orientation in space. The Weierstrass parametrisation for this surface is given by R ( tO)Enneper = ( to- too)-4

( 1.22)

34

Chapter I

where too is an arbitrary complex constant.

Figure 1.23: Enneper's minimal surface, the simplest minimal surface. (Note that this surface is self-intersecting.)

The next possibility is a Gauss map containing two singularities due to fiat points. Examples of this case are the helicoid and the catenoid (Figs. 1.13 and 1.14). The normals of the fiat points on these surfaces (at the asymptotic ends of the surfaces) are antiparallel, and hence the Weierstrass parametrisation is given by R(tO)hel,cat= (to- too)-2(to+ l/too) -2

(1.23)

Three fiat points alone cannot satisfy eq. 1.21. Proceeding to four fiat points we may create the Scherk surfaces (Fig. 1.15) by distributing these evenly along a great circle on the Gauss sphere. The standard Weierstrass parametrisation of these surfaces is R(tO)Scher k = (tt~ - 1)-1

(1.24)

All these surfaces have one important characteristic in common. The fiat "points" are not located within any finite portion of the surface. Rather, the surfaces become asymptotically fiat (e.g. the trumpet-shaped "ends" in the catenoid). As the number of fiat points increases beyond four, the fiat points are located at fixed identifiable sites and the surface closes up to become periodic in three dimensions. This distinction between one- or two-periodic and three-periodic minimal surfaces is a crucial one, since it implies that the average Gaussian curvature () of one-, and two-periodic minimal surfaces is usually zero, due to the overwhelming contribution from the

Appendix

35

asymptotically flat ends. On the other hand, the average Gaussian curvature of three-periodic minimal surfaces is negative. This property is important, and may be responsible for the frequent occurrence in nature of threeperiodic surfaces as compared with topologically less complex surfaces. (This issue will be discussed further in Chapter 4.) The simplest, non self-intersecting three-periodic minimal surfaces have a genus of three per unit cell. Six such surfaces are known (plus lower symmetry cases for some). They belong to three distinct isometric families. The first family comprises the P- and D- surfaces and the gyroid discussed above. The second group comprises the H-surface and the h-CLP surface (Fig. 1.24). The H-surface was discovered by Schwarz last century, the h-CLP IPMS was first described by Lidin [30]. Computer studies of the isometric family of IPMS related to the h-CLP surface have revealed a further IPMS associate to the h-CLP, and free of selfintersections [18] (analogous to the gyroid in the D-P family).

Figure 1.24(a): The H-surface, discovered by Schwarz.

36

Chapter I

Figure 1.24(b): The h-CLP surface. The third family consists solely of the CLP-surface (Fig. 1.25).

Figure 1.25: The CLP surface All these IPMS have eight distinct normal vectors due to flat points. The relative distribution of these normal vectors determines the IPMS. They are described by Weierstrass parametrisations of the form (eqs. 18): 8

R(o) = l-I (w- a~)-~/2 i=1

Appendix

37

Surfaces of higher genera normally have a much more complicated structure, with the exception of the I-WP surface (genus 4) which has a fairly simple representation, namely: 6

n(co) = FI (co- c~)-2/3 i=1

Figure 1.26: The I-WP surface, unit cell only shown. A small selection of more complex cubic and tetragonal IMPS is s h o w n below. Many other IPMS can be found in the publications of Karcher, Fischer and Koch and others [14-18, 20, 22-24, 31].

Figure 1.27: A conventional unit cell of the Neovius surface.

38

Chapter 1

Figure 1.28. Left: one node of the F-RD surface. Right: The other node of the F-RD surface.

Figure 1.29: A unit ceil of the tetragonal IPMS, the H-T surface. P r o p e r t i e s of t h e s i m p l e r IPMS are t a b u l a t e d o v e r l e a f ( a d a p t e d f r o m [21]).

Appendix

39

Table 1.3: List of the simpler three-periodic minimal surfaces (IPMS), together with their crystallographic symmetries. Those surfaces that carve space into two interpenetrating open labyrinths are m a r k e d with a tick, a cross denotes IPMS that are self-intersecting. In most cases, two space groups are listed for each IPMS, the first is that of the surface assuming both sides are equivalent, the second is the symmetry displayed by the surface assuming inequivalent sides.

M i n i m a l surface /

Space Group

Gamus

Bicontinuous

Adjoint surface D /P

Pn3m - F d 3 m / I m a m - Pm~m

3

~4

rPD

R~.m- R~.m (c'=2c)

3

~/

tD/tP

P 4 2 / n n m - I41/amd /

3

~/

(Intersection-free)

I 4 / m m m - P4/mmm oPa / oDa

Immm - Pmmm / Pnnn - Fddd

3

oDb/oPb

Cmma - Imma / Fmmm - Cmmm

3

"~ ~/

mPD

C 1 2 / m l - C 1 2 / m l (c'=2c)

3

CLP / CLP

P 4 2 / m c m - P42/mmc (v)

3

oCLP'/oCLP

Cmmm - Pmmm / Pccm - Ccx~

3

~/

mCLP

P 1 2 / m l - P 1 2 / m l (c'=2c)

3

q

mPCLP/mDCLP

C 1 2 / m l - P12/m1 /

3

~/

P12/cl

-

C 1 2 / c l (a'=2a,b'=2b)

I-WP / S t e s s m a n

Im3m- Im3m/-

4

~//x

VAL/VAL

Cmma - Cmma (c'=2c)

5

~/

H'/H

- / P63/mmc - Prim2

3

x / ~/

40

ChapterI

Mathematical Bibliography: S. Hildebrandt and A. Tromba, "Mathematics and Optimal Form". Scientific American Library, (1985), New York: W.H. Freeman and Co. This book assumes no mathematical competence on the part of the reader (although the authors are distinguished geometers) and offers a good entree into the field of surface geometry and topology for those who find standard mathematical texts forbidding. Filled with beautiful pictures! D. Hilbert and S. Cohn-Vossen, "Geometry and the Imagination". (1952), New York: Chelsea Publishers. This is a very good general introduction to geometry and topology, accessible to the non-specialist. M. do Carmo, "Differential geometry of curves and surfaces". (1976), Eaglewood Cliffs, N.J.: Prentice-HaU Inc. A. Goetz, "Introduction to Differential Geometry". (1970), Reading, Massachusetts: Addison Wesley Publishing Company. A. Gray, "Modern Differential Geometry of Curves and Surfaces". Studies in Advanced Mathematics, ed. S. Krantz. (1993), Boca Raton, FLA.: CRC Press. T.J. Willmore, "Differential Geometry". (1985), Oxford University Press: Delhi. p. 137. All of these are recommended introductions to differential geometry. M. Spivak, "A Comprehensive Introduction to Differential Geometry". Vol. IV, chapter 9. (1979), Berkeley: Publish or Perish, Inc. Spivak gives a modern technical account of all aspects of differential geometry in five volumes, including a good historical section, covering in some detail the original work of Gauss and Riemann (vol 2). J.C.C. Nitsche, "Vorlesungen fiber Minimalfli~chen". (1975), Berlin: Springer Verlag. J.C.C. Nitsche, "Lectures on Minimal Surfaces". Vol. 1. (1989), Cambridge: Cambridge University Press. The gospel according to Nitsche. Everything you ever wanted to know about the classical theory of minimal surfaces . A picture-filled comprehensive alternative is U. Dierkes, S. Hildebrandt, A. Kfister and O. Wohlrab, "Minimal Surfaces", 2 volumes, (1992), Berlin: Springer Verlag. P.A. Firby and C.F. Gardiner, "Surface Topology". 2nd. edn. Ellis-Horwood series in mathematics and its applications, ed. G.M. Bell. (1991), Chichester: Ellis Horwood Limited. This offers a simple introduction to surface topology.

References

41

References 1. D. Thompson, "On Growth and Form". 2nd ed. (1968), Cambridge: Cambridge University Press. 2. R.S. Westfall, "Never At Rest". University Press.

(1980), Cambridge:

Cambridge

H.B. Lawson, Ann. of Math., (1970). 92: pp. 335-374.

~

L.E. Scriven, Nature, (1976). 263: pp. 123-125.

.

5. L.E. Scriven, in "Micellization, solubilization and microemulsions.", K.L. Mittal, Editor. (1977), Plenum Press: New York. pp. 877-893. A.H. Schoen, Infinite periodic minimal surfaces intersections(1970), N.A.S.A.: Technical Note # D5541.

6.

self-

E. Koch and W. Fischer, Acta Cryst., (1990). A46: pp. 33-40.

.

0

without

J.L. Lagrange, Miscellanea Taurinensia, (1760-1761). 2: pp. 173-195.

J.B.M.C. Meusnier, M~m. Math~m. Phys. Acad. Sci. Paris, pr~s. par. div. Savans, (1785). 10: pp. 477-510.

9.

10. K.F. Gauss, "Disquisitiones generales circa superficies curvas (General investigations of curved surfaces)". (1827), reprinted New York: Raven Press (1965). 11.

H.F. Scherk, Crelles Journal, (1835). 13: pp. 185-208.

12. H.A. Schwarz, in "Gesammelte Mathematische Abhandlungen. (1890), Springer: Berlin. 13.

K. Kenmotsu, Math. Ann., (1979). 245: pp. 89-99.

14.

H. Karcher, Manuscripta Math., (1989). 64: pp. 291-337.

15.

S. Lidin and S.T. Hyde, J. Phys. (France), (1987). 48: pp. 1585-1590.

16.

S. Lidin, J. Phys. France, (1988). 49: pp. 421-427.

17. S. Lidin, S.T. Hyde, and B.W. Ninham, J. Phys. France, (1990). 51" pp. 801-813. 18.

S. Lidin and S. Larsson, J. Chem. Soc. Faraday Trans., (1990). 86.

19.

A. Fogden, Acta Cryst., (1993). A49, pp. 409-421.

20.

A. Fogden and S.T. Hyde, Acta Cryst., (1992). A48: pp. 442-451.

21.

A. Fogden and S.T. Hyde, Acta Cryst., (1992). A48: pp. 575-591.

42

Chapter I

22. A. Fogden, M. Haeberlein, and S. Lidin, J. Phys. I (France), (1993). 3: pp. 2371-2385. 23.

A. Fogden and M. Haeberlein, J. Chem. Soc. Faraday Trans., (1994).

90(2): pp. 263-270. 24.

A. Fogden, Z. Kristallogr., (1994). 209: pp. 22-31.

25. D. Hoffman and W.H. Meeks III, J. Differential Geom., (1985). 21: pp. 109-127. 26.

D. Hoffman, Math. Intelt., (1987). 9(3): pp. 8-21.

27.

H. Karcher, Manuscripta Math., (1988). 62: pp. 83-114.

28. J.C.C. Nitsche, "Vorlesungen Springer Verlag.

iiber Minimalfltichen'. (1975), Berlin:

29. J.C.C. Nitsche, "Lectures on Minimal Cambridge: Cambridge University Press.

Surfaces". Vol. 1. (1989),

30. S. Lidin, Periodiska Minimalytor och Kristallstrukturer(1986), Thesis, Lurid University. 31.

W. Fischer and E. Koch, Acta Cryst., (1989). A45: pp. 726-732.

43

Chapter 2

The Lessons of Chemistry

Inorganic Chemistry: From the discrete lattice of crystal symmetry to the continuous manifolds of differential geometry 2.1

The background

n the beginning there was geometry, shape and form, perfect symmetry represented by the crystals of inorganic chemistry. All definitions are tautological, but if such be needed, inorganic chemistry is the study of compounds formed from the elements, excluding a few exceptions, like carbon. The distinction between inorganic and organic chemistry is artificial, and not altogether harmless. But classification is necessary, indeed unavoidable, since the demarcation of boundaries between disciplines does allow a mastery of detail. That is acceptable until new insights are drawn in to go further. This is the stage that inorganic chemistry has reached and its new insights have much to offer other disciplines.

I

The inorganic compounds embrace familiar terms like metals and alloys, salts and hydrides. In an earlier stage of development of the subject, properties of these solids were studied with a main emphasis on how best to prepare them. Pre-eminent among these properties is structure- where the atoms reside in space. Once that is known, our understanding of reactivity, diffusion, strength, catalytic mechanisms, phase behaviour and interrelationships between compounds is possible. A knowledge of structure gives insights into how compounds form and why they self-assemble, and how to predict the properties of unknown materials. The reduction to order of the complexity of nature is relatively easily accomplished in solid state inorganic chemistry compared with say biology. Many inorganic materials are characterised by the simplest possible building scheme, classical crystals. The atoms are repeated in an ordered array which is termed "isometric". By this we mean that a basic unit, the unit cell, is occupied by atoms in a fixed arrangement, and the unit cell is repeated with simple translation. In the great majority of crystals, the unit cell itself possesses certain symmetries, leaving even fewer degrees of freedom in the system. This is why a one gram crystal of NaC1 containing about 2x1022 atoms can be described using only a single distance! Translation and symmetries of the unit cell, examples of which are reflections and rotations, all belong to the group of operations known as isometries. These preserve all distances and angles of the original motif when creating a copy. The unit cells must fill space and this imposes constraints on their shape, and thereby also on allowed interior symmetries. Naturally all these operations are idealisations. Often we find small deviations from this ideal behaviour in a natural crystal. The properties of a crystalline material are heavily dependent on those deviations, which operate to produce non-ideal behaviour.

44

Chapter 2

Let us start by analysing the plane. There is an infinite number of ways to fill the plane with irregular tiles, but if we restrict ourselves to regular polygons, whose edge lengths and angles are all equal we find that only triangles, squares and hexagons will do the job. This is reflected in the fact that the only rotational symmetries in the plane compatible with translational symmetry are two-, three-, four- and six-fold. Pentagons (with five-fold symmetry) fail, and th~ riddle has bothered man since ancient times. The Moors in Granada created the famous palace of Alhambra, and decorated it with intricate geometrical patterns. (Religious considerations forbad the picturing of humans.) In the Alhambra one may find examples of each of the possible 17 discrete planar groups. However, the Arabs went further and tried to create pentagonal tilings [1]. Pentagonal structures are common in nature, but never exhibit translational symmetry. Earlier the Greeks had discovered something even more surprising. While the pentagon does not tile the plane, it does indeed tile the sphere. Twelve pentagons make up the pentagonal dodecahedron whose vertices lie on the sphere. Similarly, the icosahedron is a triangular tiling of the sphere with fivefold rotational symmetry. All this was known to modern crystallographers, and an article of faith enunciated and firmly enshi-ined in an eleventh commandment that said '"Fhou shalt not have five-fold symmetry". Despite this, for decades X-ray crystallography has revealed five-fold symmetries in the atomic arrangements in alloys, but crystaUographers invariably discarded such samples. What was observed was dismissed as nothing but complicated twin structures. (Twinning means that sub-units of a crystal are assembled by reflection or rotation.) It was only recently that Shechtman and Blecht [2, 3] made the bold claim that some rapidly quenched aluminium-manganese alloys exhibited five-fold symmetry within the untwinned "crystal". X-ray diffraction patterns of these alloys exhibit perfect icosahedral symmetry. The fact that the atomic arrangement was clearly not ordered in the classical sense, but still exhibited a perfectly regular diffraction pattern could be explained as an ordering in higherdimensional space, as for the so-called "incommensurate structures". Ordering in "higher" space means that the positions of the atoms in space cannot be labelled by the three cartesian indices, but require extra labels. What is learnt from this goes beyond the statement that geometry is important. Crystallography is by necessity ruled by geometry and its rules are universally valid. But this fact should not induce a state of mind where we think that we can predict all unknown structures from considering how the old ones are built. We must remain open to new ways of looking at old knowledge. 2.2

The unravelling of complex structures

Over the years a plethora of inorganic compounds has been prepared and their structures determined. Making use of the knowledge so gained, inorganic chemists try to predict new structures and how to prepare them, sometimes

Complex structures

45

successfully. Slowly a pattern has emerged, and this pattern has evolved into a new way of looking at complex structures [4, 5].

Figure 2.1: The structure of the zeolite paulingite, cubic, with a lattice parameter of 35.1 ./~, composed of (different coloured) gismondine units.

As a rule almost all complex structures, even those with horrible stoichiometries (like Nb31077F) can be described using simpler structures. There is a natural way to generalise these structures to polyhedral descriptions. Atomic coordinates are grouped to form polyhedra, which in turn form parts of larger structures. This process reduces increasingly complex structures to simpler forms by regarding them as composed of well known and understood packets. The model structures (packets) are put together using the classic crystallographic operations of translation, reflection and rotation. An excellent example of this technique is the unravelling of the structure of the complex zeolite, paulingite (Fig. 2.1). (Zeolites are open alumino-silicate frameworks, widely used for catalysis, discussed in more detail later in this chapter.) When viewed as a set of atomic coordinates in three-dimensional space, the structure of paulingite is monstrous to behold. Once it is realised that the whole can be decomposed into units of the (geometrically much simpler) zeolite gismondine, the picture becomes clearer. If the gismondine is then described as a composite of parts of the even simpler cristobalite structure, we can get a good understanding of the whole spatial arrangement. The introduction of a hierarchical system of ever increasing complexity makes larger packets of information more comprehensible. Another example is the zeolite N - the most complex of all known zeolite structures [6]. In Fig. 2.2 it is shown how this giant structure is decomposed into two much simpler zeolite structures, those of zeolite ZK5 (blue) and sodalite (yellow).

46

Chapter 2

Figure 2.2: The structure of zeolite N, a cubic framework with a lattice parameter of 36.9,/k. The framework consists of two different, interpenetrating zeolite structures, sodalite (yellow) and ZK5 (blue) structure.

To illustrate further, consider giant alloy structures, such as Cu4Cd 3, NaCd 2 and Rh7Mg44. These lend themselves very well to similar structural decompositions and descriptions as those described above. They may all be dissected into simpler structures, joined by reflections, rotations and translations to give a complete description of the atomic positions in the whole structure. The matrices describing these operations are identical to those used by metallurgists to describe the larger scale structure, characterised by the orientation of the grains (crystallites) in ordinary metals and alloys [7]. When small particles are sintered together to form a continuous material, they are on the way to forming a complex structure. Similarly, a complex alloy structure can be viewed as polycrystalline material of simpler structures.

2.3

Defects

To go further towards understanding the concepts of inorganic chemistry we need to consider the subject of defects in solids. They are a key to the behaviour of many materials. They are of central importance to diffusion, phase transformations and reactivity of solid compounds. Defect structures show up as the local occurrence of a grain boundary or sites of a structure building operation. The formation of a structure using translational defects is easily understood by a planar example, illustrated in Figs. 2.3. Modem high resolution electron microscopy (HREM) reveals how frequently these defects occur, and how they provide a setting and a mechanism for intergrowth between related structures.

Defects

47

Figure 2.3(a): Translation by half a unit step in a square lattice creates a linear defect in a two dimensional lattice between the open and filled circles. A defect of this kind may continue to

grow sideways, leading to a whole family of new structures.

Figure 2.3(b): The transformation from a simple square lattice to a centred square lattice through the propagation of a linear defect. In Figs. 2.4 some typical HREM of alloy structures are compared to a freezefracture electron micrograph of a liquid crystal. In the alloy pictures, planar defects, i n t e r g r o w t h of different structures and small f r a g m e n t s of new structures appearing at phase boundaries are clearly seen. The similarities with the lyotropic liquid crystal image are striking. The latter image contains extensive regions of well-ordered material, the boundaries of which are now known to correspond closely to the P- and D- surfaces, described in Chapter 1. Between these regions we find other structures, in some places confined to an interface, but in others extended regions of a new structure can be seen. Several new structures can be inferred to exist from this picture, and these structures are expected to be simply related to known ones. The energy difference between the different phases must be small to enable coexistence or long term metastability. In alloys structural reorientation is sluggish, while in liquid crystalline phases the changes are more rapidly accomplished.

48

Chapter 2

Figures 2.4(a),(b),(c): Typical defects occurring in different alloy structures (similar to steel).

Figure 2.4(d): EM image of a freeze-fractured lipid bilayer folded into a bicontinuous cubic structure. (For details see Chapter 5.) Note the similarity of defe~ts with those in Figs. 2.4(a),(b),(c) although the length scales are very different. (Image courtesy of T. Gulik.)

Intrinsic curvatureof solids

2.4

49

The i n t r i n s i c c u r v a t u r e o f s o l i d s

W e c o m e n o w to a f u n d a m e n t a l p r o p e r t y of solids, t h a t of intrinsic c u r v a t u r e [8]. C o n s i d e r first a salt like N a C l (Fig. 2. 5). T h e crystal can b e c o n s i d e r e d as a c o l l e c t i o n of p o s i t i v e a n d n e g a t i v e c h a r g e s p l a c e d at t h e o r a n g e a n d p i n k lattice sites i l l u s t r a t e d . T h e e n e r g y of f o r m a t i o n p e r u n i t cell is c a l c u l a t e d to first a p p r o x i m a t i o n b y a d d i t i o n of t h e p a i r - w i s e C o u l o m b p o t e n t i a l e n e r g y of i n t e r a c t i o n b e t w e e n t h e c h a r g e s [9]. If w e a s k w h e r e this p o t e n t i a l is z e r o w e h a v e for this p a r t i c u l a r s y m m e t r y t h e set of i n t e r s e c t i n g p l a n e s i l l u s t r a t e d . The m a t t e r is a p p a r e n t l y d e v o i d of interest. If w e c o n s i d e r i n s t e a d CsCl, a v e r y d i f f e r e n t p i c t u r e e m e r g e s . T h e s u r f a c e s of z e r o p o t e n t i a l a r e i l l u s t r a t e d , a n d d i v i d e s p a c e into i n t e r p e n e t r a t i n g l a b y r i n t h s (Fig. 2.6).

l'i~urc 2.~(left): The NaCI structure. The intersecting planes define the "i~erc~" equipotential

-,urface.,-,, between the Na + (red)and CI- (orange) ions. t:igure 2.(~ (right): The CsCI structure with its zero potential surfaces. The vello,,v Cs + ions lie at the centre of the body-centred cubic unit cell. T h e s e s u r f a c e s of z e r o p o t e n t i a l f o r m e d in d i f f e r e n t salts are v e r y close to p e r i o d i c m i n i m a l s u r f a c e s [ 9 ] , w h o s e m e a n c u r v a t u r e , d e f i n e d as t h e a r i t h m e t i c m e a n of the m a i n c u r v a t u r e s , is e v e r y w h e r e z e r o (see C h a p t e r 1 ) ' . On these m i n i m a l s u r f a c e s the G a u s s i a n c u r v a t u r e is e v e r y w h e r e n e g a t i v e or In our first simple example the electrostatic potential set up by CsCl is almost but not quite a minimal surface [10]. The reason is that the Coulomb electrostatic energy is only a part of the whole electromagnetic field. Two body, three and higher order, non-additive van der Waals interactions contribute to the complete field, distributed within the crystal. This leads one to expect that the condition that the stress tensor of the field is zero, as for soap films, yields the condition for equilibrium of the crystal. Precisely that condition is that for the existence of a minimal surface. Strictly speaking the minimal surface might be defined by the condition that the electromagnetic stress tensor is zero. But in any event, we see in this manner that the occurrence of minimal surfaces, should be a consequence of equilibrium (cf. Chapter 3, 3.2.4). Indeed a statement of equilibrium may well be equivalent to quantum statistical mechanics.

50

Chaljter 2

zero, and varies continuously over the surface. Immediately we have entered the field of non-Euclidean hyperbolic geometry of Lobachevski, Bolyai and Gauss. The simplest periodic minimal surfaces like the P- and D- surfaces, divide space into two congruent labyrinths. More complicated surfaces (such as the IWP surface) divide space unequally. An analysis of a structure based on interpenetration using these partitioning surfaces has proved very useful in inorganic chemistry. An example is the W3Fe3C structure (cutting steel). The tungsten atoms form an octahedral network, with carbon in some of the octahedra. The iron forms units of stellar tetrangulae, which are polyhedra made up of five tetrahedra, with one central tetrahedron sharing faces with the four others. The stellar units form a three-dimensional network through sharing of corners (Fig. 2.7(a)). It is reasonable to expect that these two entirely different structures would react with each other. Indeed they do. They interpenetrate to form a composite structure, separated by the D-surface (Fig. 2.7(b)). This surface, which is at the iron-tungsten boundary, is the location of maximal collective interaction of electrons. (There is a neat correspondence here with the bicontinuous microemulsion structures to be explained in Chapter 4. These form spontaneously from mixtures of oil, water, and surfactant that sits at the interface between phases. Here tungsten carbide plays the part of water, iron that of oil, and the electrons fill the role of the amphiphilic surfactant, able to interact with both phases.)

Figure 2.7: (a) The W3Fe3C structure (cutting steel). The red balls denote iron atoms. (b) Dsurface separating the Fe structure (red stellar tetrangulae) and the W3C structure (yellow octahedra).

Intrinsic curvature of solids

51

The reason that this structure forms is as follows: units of octahedral W3C crystallise on one side of an interface, that separates solid W3C from Fe. The Fe atoms are drawn into the interstices of the W3C elements, with a driving force dependent on the Gaussian curvature of the interface. This is one way of looking at intrinsic curvature in solids. As will become apparent, there are other ways that link the interactions between atoms, molecules and larger aggregates to local curvature. Many more examples of interpenetration in inorganic chemistry lead to a recognition of the ubiquity of hyperbolic surfaces of infinite genus exemplified by three-periodic minimal surfaces - that demands consideration. In the giant structure of C u 4 C d 3 the Cu atoms are separated from the Cd atoms by a surface that resembles a minimal surface. In diamond, cubic ice and cristobalite, all the atoms are located on one side of the surface and the space on the other side is empty. If ice is subjected to very high pressure, the same structure appears on both sides of a minimal surface (double ice or ice IX), with almost double the density of o r d i n a r y ice (Fig. 2.8). Similarly, d i a m o n d is expected to transform to a d o u b l e - d i a m o n d structure with metallic properties at sufficiently high pressure.

Figure 2.8: Cubic ice, d i a m o n d or cristobalite (yellow) on one side of the D-surface. In ice IX or double ice, a t o m s are on both sides of the D-surface, yellow and red.

The concept of intrinsic curvature is particularly useful when dealing with the intricate and beautiful structures formed by zeolites. Zeolites are commonly used as technical materials. They exhibit many special properties, due to their extraordinary ability to selectively absorb a large range of molecules. The forces that act are weak, physical not chemical, and we characterise them by invoking the idea of intrinsic curvature. Zeolites are built up from a negatively charged three-dimensional network of tetrahedra of (Si,Al)O 2 connected by sharing of the tetrahedral vertices with

52

Chapter 2

interstitial positively charged counterions, often alkali metal ions. The zeolites crystallise from alkaline, aqueous solutions at temperatures around 50-200 oc. From similar solutions but at different (higher) temperatures and pressures, feldspars, quartz and cristobalite form. Examples of zeolites sitting close to or on one side of periodic minimal surfaces are shown in Figs. 2.9(a),(b): the zeolite known as Linde-A on the Psurface and faujasite on the D-surface. Other examples are zeolite N in which the D-surface partitions the ZK5 and sodalite structures, and also paulingite which is described by the P-surface.

Figure 2.9(a): The Linde A zeolite on one side of the P-surface. The water-alkali structure is on the other side. (b): The faujasite structure on one side of the D-surface.

2.5

Hydrophobic zeolites and adsorption

In the ensuing discussion we deal exclusively with hydrophobic (dealuminated) zeolites. In these zeolites nearly all Al 3+ ions have been substituted by Si4+ ions, so that the stoichiometries of these solids are close to those of pure silica, SiO2. The diameters of the pores in the framework are typically of the order of 5~, up to 7.5~ (zeolite Y). (Recently, wider pore hydrophobic "zeolites" have been synthesised. These are discussed later in this chapter; for now we restrict our attention to the "classical" zeolites.) When these compounds absorb molecules, the heat developed is dependent on the pore size. This heat comes mainly from the non-bonding interaction between the silica framework and the absorbed molecules.

Adsorption in zeolites

53

What is it that makes these weak forces strong enough to crack hydrocarbons, breaking the carbon-carbon bonds? It is clear that it must be a collective effect. The (interior) surface of the zeolite imparts a cooperative effect to the adsorbate. Some conclusions can be d r a w n from a simple analysis of adsorption phenomena as a function of the substrate curvature. It is easy to see that adsorption energies are dependent on the curvature of the interface. Consider first adsorption on a planar interface. At low pressures, p, a sub-monolayer, gas-like, and eventually a two-dimensional liquid described by a Langmuir isotherm (or decorations thereof) forms. At higher pressures still (p/ps>0.35, where ps is the saturated vapour pressure) multilayer adsorption isotherms can occur depending on adsorbate, molecular size and adsorbate-substrate interactions. This regime is usually described by the theory of Brunauer-Emmet-Teller (BET). In this domain, l n ( p / p s ) _--l/t, where t is the thickness of the film. The BET theory, central to surface area determination, is a simple model that nonetheless captures the essentials of adsorption behaviour. It assigns a free energy of adsorption to the first layer, and a different free energy to subsequent molecular layers, equal to the latent heat of vaporisation of the bulk adsorbate liquid. (There is a continuum theory, valid asymptotically for thick films at higher pressures, the Lifshitz isotherm for which the form l n ( p / p s ) ___-t/13, takes over, cf. section 3.4.2) The BET theory is deficient, as are all extant molecular theories, in an incorrect treatment of the entropy of the adsorbed film. In fact, when molecular size and interaction pressure with these parameters are taken into account, a unified isotherm, which also predicts step-like isotherms in certain circumstances emerges. If we calculate the van der Waals energy of adsorption of a molecule on to a planar substrate its form is E ___--A/d3, where d is a cut-off distance of the order of the molecular dimensions. This energy is acutely sensitive to exactly where the molecule sits on the surface, which is rough at the atomic scale. If we imagine that the surface is corrugated at the length scale of the surface atoms, an adsorbed molecule, visualised as a sphere, has several options. If it is too large to settle into the corrugations formed by the close packed spheres on the surface its adsorption energy is typical of physisorption. If the surface atomic distances are stretched slightly so that the adsorbate molecule can sit closer, the adsorption energies turn out to be of the order of chemical bond energies. Clearly then, the geometry of the substrate plays a critical role in the adsorption energetics. Adsorption within a zeolite can occur everywhere in the crystal. The alumino-silicate framework is a convoluted curved sheet, everywhere exposed to the exterior of the crystal by way of the channels (cf. Fig. 2.9). The process described above holds for zeolites (although the adsorbed molecules hover in the tunnels, and are not bound to the zeolite), and the local geometry of the continuous alumino-silicate sheet determines the adsorption energy. This geometry varies throughout the sheet continuously, since the Gaussian curvature is not constant. It is certain then that the adsorption

54

Chapter 2

energy of an adsorbed species is dependent on the Gaussian curvature. A simple model calculation makes this clear. Consider a spherical adsorbate species hovering over five substrate atoms at a distance d from the central atom. We can calculate the "adsorption energy" for this system as a function of the Gaussian curvature of the substrate of the substrate. If the substrate atoms form an ellipsoidal surface, a sphere of radius d (Gaussian curvature 1/d2), the energy is equal to -5/d 4 (assuming, for simplicity the interaction energy scales as 1/d4). If the substrate is fiat (zero Gaussian curvature), the adsorption energy is equal to -2/d 4. For a symmetric saddle (zero mean curvature) of Gaussian curvature -1/d 2, the adsorbate experiences an adsorption energy equal t o - 3 / d 4. We see then that the heat of adsorption is maximised for an adsorbate molecule sitting in a half-sphere. However, that local arrangement cannot be globally realised. All elliptical interfaces (positive Gaussian curvature) will close up, rather than form a continuous structure like a zeolite. The simple analysis described above suggests that a saddle (negative Gaussian curvature) leads to greater adsorption power than a plane (Gaussian curvature zero). And as long as the adsorbate will fit, the tighter the saddle, the better. Further, a smooth change in the Gaussian curvature, as for a minimal surface will funnel adsorbates to preferred sites, increasing the effective catchment area for adsorption. Certainly this simple calculation is an idealisation of the actual physics of adsorption; the energy depends on the adsorbate size, shape, etc. Nevertheless, the lesson is clear; the adsorption energy depends on the curvature of the substrate. In the case of zeolite adsorption, where the mean curvature of the substrate is everywhere close to zero, the adsorption energy must depend on the variations of Gaussian curvature throughout the alumino-silicate network. Our example suggests that for minimal surfaces, the higher the magnitude of the (negative) Gaussian curvature, the stronger the adsorption energy (cf. the plane and the saddle). Other effects, such as steric phenomena, may alter this trend. However, in general we expect adsorption to occur firstly in the vicinity of saddle points within the zeolite structure. This argument suggests there should be a correlation between the integral heat of adsorption for various adsorbates and the Gaussian curvature of a substrate. Adsorption data is available for de-aluminated faujasite and silicalite. The integral heats of adsorption for various adsorbates can be compared to the integral Gaussian curvature per unit area of the zeolites [11]. This is an entity easily computed from the Gauss-Bonnet theorem once the surface areas of these surfaces are known (see Chapter 1). These arguments can be quantified, and put on a firm basis by the theory developed in section 3.2.3. The combination of the curvature-funneling effect and the wide distribution of adsorption sites within zeolites - some of which will be optimal for a

Adsorption in zeolites

55

particular adsorbate - explains how zeolites can catalyse reactions like cracking and polymerisation. The high intrinsic heat developed at the optimal sites supplies the driving force for the reactions in question. Zeolites combine the features of a general catalyst/adsorbent with those of a very specific one. This is u n d e r s t a n d a b l e in terms of curvature. The fact that a continuous distribution of different sites of varying Gaussian curvature is available vouches for the generality of the adsorption phenomenon, and the fact that the efficacy is governed by shape warrants the selectivity. Small changes in the structure of these compounds will affect the selectivity to a large extent without affecting the generality. It will be seen later in the book that the same principles that underlie their catalysis and reactive properties are exploited to a great advantage by enzymatic proteins in biology.

2.6

Phase transitions, order and disorder

One measure for the level of lack of understanding in an established field is the volume of papers and weight of the books devoted to it. The subject of phase transitions in solid state chemistry constitutes an immense field by weight. (The phenomenon of melting - even for the simplest solid - has hardly yielded to theoretical onslaughts at all since the formulation of Lindemann's rule 80 years ago. This criterion, obeyed for real or model solids which range from a one-component electron gas subject to repulsive Coulomb interactions in a positive neutralising continuum background, to a hard-sphere solid with an attractive potential, long or short range - is universal. The rule has it that a solid melts whenever the root mean square deviation of a particle moving in the mean field of its neighbours is about 1/10 of the lattice spacing. The reasons, presumably, are twofold: If one imagines the electrons of an atom to be smeared into an exponential cloud of probability density, the Rayleigh criterion for resolvability tells us that it is impossible to distinguish between one atom and its neighbour at this level of excitation of the solid. On the other hand, if we imagine the atoms as hard spheres, this mean square displacement just permits rapid diffusion from one lattice site to another. The failure of statistical mechanical theories has to do with the neglect of the anisometric structure that occurs in any real liquid, and indeed the non-existence of any serious theory of liquids.) We will concentrate our attention on a small number of special cases of solidsolid transitions, without presuming to infer too much in generality. The approach taken is different to the usual view, and the examples which yield to this attack are inaccessible to standard ways of thinking. The utility of our approach will be apparent from the examples chosen. Earlier we have given an example of a transition built on the concept of symmetry operations. Groups of atoms move by translation, rotation or reflection. Transitions that operate along these lines tend to be sluggish and have large activation energies. The driving force is temperature or pressure. At high temperature the coordination number has a tendency to decrease and

56

Chapter 2

at a critical strain, this will trigger a transition. Very often, high pressure will trigger transitions that give rise to increased coordination. The graphite to diamond transition is a classic example. The multitude of modifications of ice is another. These examples are easily understood from packing arguments. The rutile (TiO2) to the high pressure TiO2(II) (a-PbO 2type) transition used for the lubrication of guns is easily understood from electrostatic considerations. This transition, schematically shown below, leaves the oxygen atoms virtually untouched, while the titanium atoms move according to the Fig. 2.10

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

~

9

9

9

9

9

9

9

9

Figure 2.10: Schematic view of Ti02(I) --~ Ti02(II) transformation.

When the solid is compressed (e.g. upon firing), the enhanced repulsion distorts the linear arrangement of positive cations. They move sideways, and the structure becomes more elastic. The only coordination feature that changes is the cation-cation packing, which is enhanced. The new structure is relaxed and this is reflected in the cation - cation distance, which is larger in the more compressed form. Perhaps the most technologically important phase transition in solids is the martensitic transition. The classical martensitic transition occurs in steel where the face-centred cubic phase (fcc, austenite) transforms to the bodycentred phase (bcc, martensite). The hardness of the material depends on the amount of martensite formed, and this in turn is a function of initial temperature and cooling rate. The transition is characterised by a drastic and complex diffusionless r e a r r a n g e m e n t of atoms, adiabaticity, and a propagation of structural change at sonic velocity. This is very fast for a transformation that involves the making and breaking of bonds, and it is clear that bond breaking and making is minimised. Few phenomena in chemistry involving actual movement of atoms occur at comparable rates. The transportation kinetics are independent of temperature over a wide range. This phenomenon, which is widespread in metals, is triggered by a cooperative phase transformation associated with the ensuing change in size of the atoms (or equivalently the position of the potential minimum in their

Phase transitions

57

i n t e r a c t i o n p o t e n t i a l ) . It is a c c o m m o d a t e d b y t h e b c c lattice. T h e w h o l e t r a n s f o r m a t i o n o c c u r s c o o p e r a t i v e l y o v e r l a r g e v o l u m e s of the crystal.

Figure 2.1 l(a): Schematic view of the effect of the Bonnet transformation on a crystal. The elliptical trajectories of all atoms result in a bulk region abcd being transformed into the related sheared and rotated regions 0~u and a'~'7'8' for increasing and decreasing Bonnet angle, 0, respectively. The Bonnet angle 0, determines the degree of rotation, @.

Figure 2.1 l(b): Monkey-saddles of the gyroid (left) and the D surface (right). These saddles are related by the Bonnet transformation. Atoms, fixed to the fiat points of the saddles, are transformed from the face-centred cubic to a body-centred cubic array. T h e s t a n d a r d ( p o s t u l a t e d ) m e c h a n i s m for t h e t r a n s i t i o n is t h e B a i n d e f o r m a t i o n , w h i c h i n v o l v e s a 20% c o n t r a c t i o n in o n e d i r e c t i o n a n d a 12% e x p a n s i o n in t w o p e r p e n d i c u l a r directions. This w a s d e s c r i b e d by Bowles a n d

58

Chapter2

Mackenzie as a three-step process, with rotation, reflection and translation operations (see, for example, [12]). However there is another more credible and entirely different mechanism [13]. By viewing the atoms of the (facecentred cubic) austenite as placed on all the fiat points of the D-surface to which indeed they fit, and applying a Bonnet transformation to this surface to deform the D-surface into the gyroid (Chapter 1), we arrive at a body-centred cubic arrangement of flat points, characteristic of the martensite lattice. This mapping of the atoms onto the flat points of the IPMS, which then transform subject to the Bonnet transformation, fixes uniquely the relative lattice spacings of parent and product phases, as well as their relative crystallographic orientations. The ratio of lattice parameters is 1.249 under the Bonnet transformation, close to the measured value, 1.269. The measured orientation relation (known to metallurgists as the Kurdjomov-Sachs plane correspondence) is exactly that expected by the Bonnet transformation. Further, the occurence of twinning during the transformation ('tweed") is also consistent with the Bonnet transformation (since Bonnet angles of +0 gives twinned regions, cf. Figs. 2.11(a)). This model may also explain the habit plane, which invariably defines the orientation of the interface between parent and product phases in the martensite transformation [14]. Our alternative mechanism to that of Bowles et al. would cost very little energy, since the Bonnet transformation is isometric (no bond stretching along the surface) and this single operation combines all three features of the older model in one simultaneous step, involving the entire crystal. All atoms move along ellipses as shown in Fig. 2.11(a),(b).

2.7

Quantitative analysis of hyperbolic frameworks: silicate densities

A central problem in the zeolite business is to synthesise novel zeolites containing wide pores. These materials are sought after for a variety of scientific and technological reasons, ranging from fractionation of proteins to the storage of methane. This program has recently leapt ahead following the announcement of MCM-41 "zeolites" by Mobil Oil, with pore radii from 15100]k [14]. Until then, the widest pore aperture in hydrophobic zeolites realised in zeolite Y (isostructural to faujasite) - was a mere 7.5~ in diameter. Debate is continuing about the structure of the Mobil materials. The MCM series are structurally very different to conventional zeolites: they do not exhibit crystallinity at the atomic length scale (wide-angle X-ray diffraction patterns consist only of diffuse bands), and microscopic images indicate the zeolite walls have a thickness larger than 5/~. (We remark in passing that these zeolites are synthesised using ionic lyotropic liquid crystals as templates. In these cases, the templating mechanism appears to be a steric one and the alumino-silica framework geometry follows that of the hydrophobiclipophilic interface in the liquid crystalline mesophase. The formation of hyperbolic MCM frameworks is possible, using bicontinuous "cubic" mesophases of the liquid crystals, discussed in Chapters 4 and 5).

Tetrahedral frameworks

59

Wide-pore zeolites must have a low "framework density" (FD), equal to the number of tetrahedral "T" atoms (silicon or aluminium) per 1000A 3. Two questions arise naturally in this context. What sets the framework density? What is the minimum FD achievable for a hydrophobic zeolite? These are difficult questions to answer within the standard three-dimensional Euclidean view of these frameworks. However, within a two-dimensional hyperbolic perspective, they are readily answered. The latter view recognises that infinite crystalline frameworks are tesselations of periodic hyperbolic surface, as in Figs. 2.9. To many, this is an excessively complex description. However, this description does yield new insights into the nature of these covalent frameworks. The bulk density of a material depends on the surface density and the Gaussian curvature of the surface. Thus, a hyperbolic surface with large (negative) Gaussian curvature packs a greater area into a given volume than one of low curvature, since the former is more convoluted. The bulk density together with the Gaussian curvature of the framework then allows estimation of the surface density of the framework, as follows. Define the average value of the Gaussian curvature (see section 1.9) by

JJ

K da

t]~'~ __-- u n i t ce]]

(2.1)

A

where A is the area of the surface per unit cell. From the Gauss-Bonnet theorem: A = 2=g ~<)

(2.2)

so the framework density is given by: FD = 103 2~X

(K)nV

(2.3)

Here f~ is the area (in 2i2) per vertex of the framework (e.g. per SiO2 group for silicates) on the surface (surface density f~-: ) and V is the volume per unit cell. A dimensionless index that we call the "homogeneity index", H, (Chapter 4) can be defined as: HWhence:

~/

A3

-2~xV 2

(2.4)

60

Chapter 2

FD=10 3H ~,

or

FD= 103-]~--. '~/2~(-X)-~ ,G3/zV N

(2.5)

where N is the number of atoms per unit cell. Euler's relation (described in section 1.8) allows a relation to be drawn between the Euler characteristic (-x/N) per vertex to the average ring size (n) of rings on the surface, and the connectivity (z) of the atomic net on the surface. (A ring is defined to be the shortest circuit about a vertex along edges in the framework.) This relation is:

N -

n

(2.6)

Thus, if z = 4 as for tetrahedral silicate frameworks,

FD=10 3 H ~/2~(n-4)

~-~3/2

(2.7)

The framework density is dependent only on the average ring size and the area per vertex. From these equations, it can be seen immediately that the formation of wide-pore zeolites (of low FD) requires networks whose average ring size on the surface is as close to four as possible. It is shown in Chapter 4 that the value of the homogeneity index, H, is close to 3/4 for hyperbolic surfaces of infinite genus. Thus, from the framework density, the area per vertex, fl, can be deduced using eq. 2.7. It turns out that for a range of silicates, from open zeolites to dense silicates (such as quartz), the value of t'l is approximately constant, regardless of the intrinsic curvature of the silicate (Fig. 2.12), despite the range of areas that are geometrically realisable by the networks (assuming their T-T distances are equal to 3.05/~) [16, 17]. This is a striking finding, reminiscent - but not a consequence - of the more usual notion of preferred bond lengths and angles in covalent frameworks. This result suggests a remarkably simple picture of these frameworks as dense packings of flexible discs (each containing a single SiO2 group) on periodic hyperbolic surfaces close to three-periodic minimal surfaces of infinite genus. For a minimal surface, the average value of the Gaussian curvature, , is simply related to the normal curvature, *:n and the geodesic torsion, Zg (section 1.6): - = , 2 + zg2

(2.8)

Tetrahedral frameworks

61

Figure 2.12: Plot of the area per T-atom vertex (s versus the average ring size, n, for a variety of zeolites, silica clathrasils and dense silicates. (All zeolites have a silicon: aluminium ratio exceeding three, so that the approximate stoichiometry of all these frameworks is SiO2). Zeolite and clathrasil frameworks are labelled by the code adopted by the International Zeolite Association [18]). The shaded domain indicates the window of geometrically accessible values of ~ as a function of the ring size. Despite the allowed geometric variability, the value of ~ is close to 12.2~ for all these "silicates", regardless of the ring size and consequent intrinsic curvature. The n o r m a l c u r v a t u r e of curves on the surface linking adjacent T a t o m s is set b y the E u c l i d e a n d i s t a n c e s (1) b e t w e e n these a t o m s a n d the angles (A) s u b t e n d e d b y the straight lines joining adjacent T atoms. The m a g n i t u d e of this c u r v a t u r e is equal to the reciprocal of the r a d i u s of a circle that contains t h r e e vertices s p a n n e d b y t w o a d j a c e n t b o n d s . The a v e r a g e v a l u e of the n o r m a l c u r v a t u r e of the surface o v e r the region of surface s a m p l e d b y the t w o b o n d s is equal to:

~r =

(2.9)

l

The f r a m e w o r k d e n s i t y of a f o u r - c o n n e c t e d n e t w o r k on a m i n i m a l surface can t h e n be estimated with the help of eqs. (2.5), (2.8) and (2.9) as [19]: FD = 103/-/ n 2~ (4-n)

cos

+ xg

2}3/2

(2.10)

62

Chapter 2

If the (curved) edges lie along the principal directions on the surface the geodesic torsion of the network vanishes (Zg=0). In this case, the density is: FD = 2.103H co~A/2) ~2 I

(2.11)

At the other extreme, if the edges lie along the asymptotic directions, they are straight (Kn=0), and the framework is torsional only. This analysis can be recast in terms of more familiar three-dimensional Euclidean bonding dimensions within the framework. In alumino-silicate frameworks, the T-T distance, 1, is related to the T-O bonding parameters by the equation: l= 2d.sin(o~/2)

where d is the T-O bond length, and ao is the T-O-T bond angle. Recall that A denotes the (vertex) angle of collar rings about a T atom formed by connecting adjacent T atoms with straight lines. A is related to the O-T-O angle (aT) and the T-O-T angle (aox) by: O~T-/l:+(~0x -< A _< OtT+/I;-O~0x.

The bounds are achieved for torsionless frameworks (Zg=0), in which case bound reached depends on the location of the two O atoms bonded to three T atoms (which subtend the angle A) relative to the ring containing T atoms. The right-hand limit is realised if the O atom lies inside the ring; left-hand bound requires all O atoms to lie outside the ring.

the the the the

Eq. (2.10) reveals that the density is minimised - allowing the widest p o r e s when the geodesic torsion of the framework vanishes. Th~ occurs when the net edges are parallel to the principal directions. If all O atoms lie inside rings formed by connecting adjacent T atoms (A = aT+~r-a0x) the density is further reduced. If the bond angles and lengths are constrained, eq. (2.11) provides an estimate of the most open framework that can be realised in hydrophobic zeolites whose stoichiometry approaches SiO2. The Si-O bond length is typically equal to 1.61A, the O-T-O angle is 109.5 ~ and the T-O-T angle is 140 ~ [20]. These data yield a m i n i m u m framework density for silicon-rich zeolites of 10.7 T atoms per 1000.~3 (average ring size 4.2). In contrast, zeolite Y has a FD of 12.4. An upper bound on the density of four-connected silicate frameworks can also be found, which is close to that of the densest tetrahedral silicate, coesite (ca. 30 T-atoms per 1000A3). It follows from this analysis that zeolites of slightly lower density than that of zeolite Y are theoretically realisable, although significantly wider-pore materials require bonding dimensions that cannot be achieved in silicates.

Tetrahedral frameworks

63

The successful synthesis of wide-pore "interrupted" frameworks (which contain unconnected edges satisfied by hydroxyl groups) exhibiting very low FD's can also be understood within this approach. In these cases, lower FD's can be realised without compromising the preferred bonding geometry, since the average connectivity of the framework is less than four. From eq. 2.7(b), the lower connectivity (z) leads to higher FD's. The new large-aperture MCM-41 family of zeolites mentioned earlier have FD's far below those expected from this analysis. This is due to the novel structural type of these frameworks. They consist of a silica bilayer wrapped onto hyperbolic surfaces (or cylinders), rather than the hyperbolic silica monolayers characteristic of conventional zeolites. The structures of MCM-41 zeolites are closer to those of the double-layer sheet silicates [20] than zeolites. In these cases, the connectivity within each layer is three (the fourth link fuses the two layers), and pores of infinite radius are apparently realisable, e.g. CaA12Si208, which is a fiat bilayer! To sum up this rather technical analysis, it seems indisputable that the fitting of classical "monolayer" zeolites to IPMS is not just an elegant mathematical curiosity. The quantitative analysis that follows from this description allows predictions of framework densities as a function of the network topology. This u n d e r s t a n d i n g of structure, which views the (Euclidean) threedimensional structure in terms of its intrinsic two-dimensional hyperbolic geometry opens up a predictive understanding of structure.

2.8

Tetrahedral f r a m e w o r k s : Three- or t w o - d i m e n s i o n a l structures?

The density analysis of silicates described above suggests a deeper understanding of these (Euclidean) three-dimensional frameworks in terms of (non-Euclidean) two-dimensional manifolds. It appears that silicon-rich zeolites (as well as the clathrate, melanophlogite) are well described by twodimensional hyperbolic tessellations of fixed surface density, irrespective of the ring size. The ring size of the frameworks is, by eq. (2.6), a measure of the Gaussian curvature - the intrinsic curvature - of the network. This curvature is really a function of the bonding geometry, expressed by eq. (2.9). Conventional measures of the bonding geometry, i.e. bond lengths, angles and torsion, assume a three-dimensional Euclidean e m b e d d i n g of the framework. However, the non-Euclidean picture allows a structural principle comparable to the standard one of solid state chemistry. Since the application of q u a n t u m mechanics to solid state chemistry, championed by Linus Pauling, the notion of preferred bond lengths, angles and (implicitly) torsion all measured in three-dimensional Euclidean space - has been accepted as a guiding principle. The hyperbolic two-dimensional picture admits a complementary picture, namely preferred intrinsic curvature and surface density.

64

Chapter2

Quantitative comparison of these two viewpoints is possible using eqs. (2.7) and (2.11), which relate the three-dimensional (bulk) density to the ring size within the framework. According to eq. (2.11), if the bond length, angle and torsion are set, the bulk density should decrease with average ring size. On the other hand, if the area per network vertex is conserved, the density increases with average ring size (eq. 2.7). The two expectations from the different approaches are in conflict. To resolve the issue we compare open and dense atomic frameworks of silica (or silicon-rich zeolites), water, silicon, and germanium. All of these materials are characterised by tetrahedral bond angles about the four-connected vertices. We need first to make some preliminary remarks: Vertex angles (A) defined above of 109.5 ~ (the "tetrahedrar' angle) are expected for the Si and Ge frameworks. In the case of H20 and SiO2 frameworks, the vertex angle includes the bond angle about the two-connected atoms (H for H20, O for SiO2) lying between the four-connected vertex atoms. Linear H - O - H bonds in ice frameworks are believed to be energetically the most favourable, so that the mean vertex angle in these frameworks remains 109.5 ~. In silicates, statistical studies suggest that bond angles of about 140 ~ are most relaxed [20]. If all O atoms lie within the ring formed by connecting neighbouring Si vertices by straight lines, the resulting vertex angle (for torsion free bonds) is approximately 70~ if all O atoms lie outside the Si-rings, a vertex angle of 150 ~ is expected. Thus a range of vertex angles is accessible to silicate frameworks. The bond lengths (1) for silicon and germanium frameworks are taken to be equal to twice the sp 3 radii for Si and Ge, i.e. 2.34~ and 2.44A respectively. We assume bond lengths of 2.7tt~ between adjoining four-connected vertices in ice frameworks, and 3.1,/~ - a typical T-T distance - between Si vertices in the SiO2 frameworks. To compare densities inferred from these three-dimensional data with those based on the two-dimensional hyperbolic analysis, estimates of f~ are required. A priori estimates are not available but we can take "best fit" (average) values from eq. (2.7). These are: 12.2A 2 for SiO2 frameworks, 10.2~ 2 for H20 frameworks, 8.0,/~2 for Ge, and 7.2~ 2 for Si. These values hold for frameworks tesselating periodic hyperbolic surfaces whose homogeneity index, H, is close to 3/4 for open frameworks. For dense frameworks, which tessellate self-intersecting IPMS, an estimate of H is 0.65. Dense frameworks are characteristic of the silicates quartz, cristobalite, tridymite, keatite and coesite, the diamond forms of silicon and germanium, and the ambient pressure hexagonal modification of ice (ice Ih). Open frameworks included are the high silica zeolites zeolite -Y, hexagonal faujasite and ZSM-5, the silica clathrate melanophlogite and the two clathrate modifications of ice, silicon and germanium. (The details are described elsewhere [15]). Fig. 2.13 compares the theoretical curves that relate (i) the framework density to the ring size assuming fixed Euclidean threedimensional bonding architecture within each framework type

Tetrahedralframeworks

65

(ii) the framework density and ring size assuming constant area per vertex with actual density data for these solids. Clearly, the hyperbolic two-dimensional picture, which assumes constant surface density (area per SiO2, H20, Si or Ge group) is more realistic than the "classical" Euclidean three-dimensional model, which supposes fixed bond angles, lengths and torsion. In all the cases analysed here, those frameworks realised in the laboratory (or in nature) do lie near to the dotted curves deduced from reasonable bond lengths and angles. Remarkably though, the variation in bulk density follows the locus of the curves assuming constant area. The intersection of the two curves determines the preferred intrinsic curvature, characterised by the ring size. If then we abandon the standard three-dimensional Euclidean perspective and adopt this non-Euclidean two-dimensional view, it can be seen that stable polymorphs are characterised by a global geometric constraint: surface density ~2-1, and a local constraint: Gaussian curvature, . We shall see in Chapter 4 that this description is identical to one that accounts for the mesophase behaviour of lyotropic liquid crystals in amphiphile-water mixtures. How do these unconventional ideas link with the standard view of a solid as a close packed array of atoms? Evidently most of the frameworks discussed above cannot be so characterised. The two-dimensional hyperbolic picture does break down for very dense structures. Thus the densest four-coordinated silicate, coesite, violates this universality (see Fig. 2.12). (Its ring size is less than that of tridymite, cristobalite, keatite or quartz, in spite of its higher density.) This polymorph is too dense for a two-dimensional description to be useful and the three-dimensional description takes over. The notion of intrinsic curvature is less rigid for silicates than for the other frameworks, because the Si-O-Si angle usually differs from 180 ~ The hyperbolic description implies that to a reasonable approximation, tetrahedral water, silicate, silicon and g e r m a n i u m frameworks are characterised by a preferred area per vertex group and a preferred Gaussian curvature. Thus, identical tessellations of isometric surfaces, with equal areas and curvatures at corresponding points on the surface, should offer alternative possibilities for stable frameworks. Indeed this is the case for the zeolite frameworks, faujasite and analcime, which are related to each other through the Bonnet transformation. Within an intrinsic two-dimensional description, these two frameworks are indistinguishable. We have seen in section 2.6 that the Bonnet transformation describes well a number of characteristics of the fcc-> bcc martensitic phase transformation in metals and alloys. The success of this model suggests that the hyperbolic picture, intuitive and obvious for zeolites, is also valid for other atomic structures.

66

Chapter2

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(a)

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it

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.

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i .... i .... i .... , .... i..-.....5 6 7 average surface ring-size

.

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(d)

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(c)

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'=

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Figure 2.13(a)-(d): Density/ring size data for (a) SiO2 frameworks (some zeolites, clathrasils and dense silicates), (b) H20 frameworks (hydrates and ambient pressure iceI), (c) Ge (clathrates and diamond forms) and (d) Si frameworks (clathrates and diamond). Empty circles denote open frameworks (zeolites, clathrasils/clathrates/hydrates), dots denote dense frameworks (diamond/ice etc.). The full curves are plotted using eq. 2.7 assuming a homogeneity index of 3/4 (open frameworks). The upper dashed curve assumes a value of H=3/4, the lower dashed curve H= 0.65 (dense frameworks). These curves assume no preferred bond angles or lengths, only fixed film density (t~'l). The value of ~ is the average value calculated from eq. 2.7 for all frameworks of a single stoichiometry (12.2-A2 for SIO2, 10.2A 2 for H20, 8.0A 2 for Ge and 7.2/~2 for Si). The dotted curves are calculated from eq. (2.11); they assume preferred bond angles and lengths listed in the text and zero torsion. (Nonzero but constant torsion yields similar curves, displaced upwards.) Two sets of curves are plotted for SiO2 frameworks. The left curves are for a minimttm bond angle of 70~ (all O atoms inside Si rings), the right for 107.5 ~ (half O's inside, half outside Si rings). Curves for vertex angles of 150 ~ (all O's outside) lie beyond the plotted region.

2.9

Quasicrystals

The triacontahedron was discovered by Kepler; Kowalewski [21] t h a t it c a n b e b u i l t f r o m o n e r h o m b o h e d r o n (Fig 2 1 4 )

a n d it w a s s h o w n b y acute and one obtuse

Quasicrystals

67

Figure 2.14: The triacontahedron (left), with the oblate (centre) and prolate (right) rhombohedra. Ten of these rhombohedra form the triacontahedron. The faces of the polyhedra are golden rhombuses, i.e. the quotient between the lengths of the rhombus diagonals is the golden section, ( ~ + 1 ) / 2 . In the language used to describe quasicrystals, the two rhombohedra are called the "oblate" and the "prolate". They have the fascinating property that they fill space, without exhibiting translational symmetry, probably already known by Kepler. Mackay [22] showed that a projection along the five-fold axes gives a pattern equivalent to a Penrose tiling. He also showed, with an optical experiment, that this tiling gives perfectly ordered diffraction patterns, even though the tiling is not an ordered lattice in the classical sense. A diffraction pattern of a quasicrystal of five-fold symmetry, is shown in Fig. 2.15. (This is discussed in more detail in the Appendix.)

9

+

~

9

,

9

9

,

9

.

II "

9

9 . "

9

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+

.,

9

o

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-;:

9 9

9

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+

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Figure 2.15: Electron diffraction pattern of a quasicrystal.

68

Chapter2

One w a y that this can be pictured in terms of a projection from higher dimensional space is as follows. Take an ordinary square lattice and draw a line across it at 36~ to one of the fundamental lattice directions, and project a part of the lattice onto this line (Fig. 2.16). The projected points form a set of golden Fibonacci sequences. Every point is in principle the start of a new sequence. Were the entire lattice projected onto the line, a dense covering of points w o u l d ensue. Because we only project the part of the lattice that is contained in a window defined by two lines parallel to the line of projection, we get a discrete one-dimensional quasi-lattice (Fig. 2.16). A series of points constructed according to these rules leads to perfect diffraction. We explain that pattern by saying that the one- dimensional structure has translational ordering in a higher dimension (two). A two-dimensional Penrose tiling is a projection from a f o u r - d i m e n s i o n a l translational lattice, and a real quasicrystal is a projection from six-dimensional space.

Figure 2.16: A one-dimensional quasicrystal obtained by projection from a higher-dimensional isometric space. What kind of symmetry is this? It is clearly not translational. (Is it fivefold?) An isometry, or an isometric m a p p i n g , is a transformation that preserves lengths and angles, as in the Bonnet transformation. Certain isometries transform a figure into itself or its mirror image. These are the ordinary symmetry operations that classical crystallography deals with. By combining these we get the discrete groups or transformations allowed for in spaces of different dimensions; 7 groups in one dimension, 17 in two, 230 in three, about 5000 in four and a huge number in six dimensions. Another class of transformations, k n o w n as conformal transformations, preserves angles but not lengths. One example is dilatation, which is repetition by scaling. The golden Fibonnaci sequence is an example of dilatation, as is the pattern of pentagons in Fig. 2.17.

Quasicrystals

69

Figure 2.17- An example of pattern exhibiting dilatational and rotational symmetry based on a regular pentagon. The scaling ratios are golden Fibonacci ratios: 3/2, 5/3, 8/5, 13/8 ..... and the rotations are 36".

Dilatational s y m m e t r y involves expansion of a motif by geometric ratios, while classical (translational) symmetry is a result of arithmetic progression. If space is filled by the packing of the oblate and prolate r h o m b o h e d r a following certain matching rules so that space becomes conformal, a structure is obtained that will give diffraction of icosahedral point symmetry. We assume then that the decoration of the r h o m b o h e d r a with atoms fulfils a Fibonacci sequence. Such a structure has five-fold translation s y m m e t r y in six-dimensional Euclidean space. While the global structure has five-fold s y m m e t r y in three-dimensional space, the local structure will necessarily show deviations. The question of order becomes very complex in systems with dilatational symmetry. Earlier descriptions of order implied isometric repetition. A criterion widely accepted by crystallographers is that a sharp diffraction pattern indicates a well-ordered structure. In that case both Penrose tilings [22] and quasicrystals can be considered to be ordered structures. Conformal s y m m e t r y is very c o m m o n in nature; e.g. we can find it in the nautilus shell and the sunflower. These structures are clearly ordered, even if they do not give sharp diffraction patterns. Here the repetition is nonEuclidean, on a logarithmic spiral (nautilus), or on a torus (sunflower). We are inclined to say that any kind of repetition, conformal or isometric, even in non-Euclidean space, is ordered. However, classification of these more chaotic structures, as for liquids, is less certain. It may be that a liquid can be described as a structure with some of the characteristics of conformal symmetry or perhaps by a representation even more exotic, like a manifold of constant negative curvature. Let us return to quasicrystals and investigate the nature of their intrinsic ordering. It is possible to solve their structure by (almost) classical crystallographic techniques if one a d m i t s structural solutions in sixdimensions. Clearly however this is of little help when we try to grasp the real structure of the crystals. To arrive at the true three-dimensional structure

70

Chapter 2

we have to project the six-dimensional structure down to three dimensions, and the problem is that we do not know what" sort of section should be used to make that projection. Nothing suggests that the section even has to be planar. The problem of elucidating a structure from crystallographic techniques is evidently profoundly difficult.

Figure 2.18: Quasicrystal built by the interpenetration principle. Top (a): The interpenetration shown with only two icosahedra. Bottom (b): Unit containing a larger number of interpenetrating icosahedra. One attempt at full elucidation from six dimensions has been made by Janot et al. [23]. While this attempt is of great elegance, it remains deficient in that the distance between some atoms is too short and partial occupancy must be introduced. The structure can also be described directly using oblate and prolate polyhedra, and combinations of these. Such a structure must be

Quasicrystals

71

constructed from high resolution electron microscopy (HREM) pictures together with the full armoury of intuition and knowledge d r a w n from common structures, and its validity can be checked by comparing diffraction patterns. This has been attempted, starting from HREM images. The proposed structure [24] contains units consisting of an icosahedron surrounded by an i c o s i d o d e c a h e d r o n and this in t u r n is s u r r o u n d e d by a small rhombicosidodecahedron studded with icosahedra: a unit containing in total 174 atoms. These units occupy the corners of the oblate and the prolate rhombohedra. These large units are interconnected by forming parts of wellknown structures like WAll2 and pyrochlore. This would be consistent with a prehistory in the melt of these large units floating around in a loose association. As the temperature drops, these units aggregate to form the quasicrystal. The units are glued together by well-known structures. This model is a member of a series of structures, all built on a simple principle of interpenetration of polyhedra of five-fold point symmetry. The simplest member of the family is shown in Fig. (2.18(a)). Higher members, formed by successive interpenetrations of icosahedra fill space (Fig. 2.18(b)) and a perfectly ordered quasicrystal structure is obtained. There is no need to use oblate or prolate polyhedra; they are generated by a single structure building principle [24]. Whether quasicrystalline structures are limited to alloys remains an open question. It is possible that their occurrence is much more widespread than had been previously thought. Indeed there is evidence for quasicrystallinity in both thermotropic and lyotropic liquid crystals. Diffraction patterns of decagonal symmetry have been recorded in lyotropic liquid crystals [K. FonteU, private communication], (Fig. 2.19), and there is theoretical evidence for the existence of a quasicrystalline structure within the blue phase of cholesterol (Chapters 4, 5). (The decagonal structure has quasisymmetry perpendicular to the tenfold axes, and translation symmetry along them.) Viruses crystallise in icosahedral clusters and the list continues to grow. In addition to five-fold symmetry, it has been shown that eight and ten- fold quasisymmetry is possible.

Figure 2.19: Pseudo-five fold diffraction pattern of a decagonal phase (65~ from the ten-fold axes Cu - K~ radiation, sample to film distance of 200 mm) of a lyotropic liquid crystal (courtesy of the late Krister FonteU). The original image has been traced over to make the image more visible. Compare with Bendersky's study of MnA14[25].

72

~ter 2

The recognition of quasisymmetry as a means of structure building is recent, and has still not made its full impact on the scientific community. In time it will seep through to different branches of chemistry as new examples are identified in different fields. We have already touched on the subject of disorder in solids. One further point deserves comment. Having quasicrystalline materials in mind, it is conceivable that the transition from the liquid to the solid state proceeds via a phase transition exhibiting conformal symmetry. This might be a typical liquid structure, or the structure occurring at the point of solidification. The melt might contain fragments of the solid state, and these would first solidify into a conformal (quasicrystalline) structure if the temperature drops quickly enough. We have already remarked on the observation of a decagonal phase in a liquid crystalline phase. Diffraction data from liquid crystals are often insufficient to determine the symmetry with certainty. It is possible that they could be interpreted in a different way, if the five, eight and tenfold symmetries of the quasicrystalline phases are considered. The expression used for indexing five fold symmetries is Q = 2x/a [(N+zM)/(2(2+z)l 1/2, Q = 4x/~,sinO, (~F5+ 1) / 2 where N is a nonzero integer, M is an integer and the values of M and N are constrained by -0.618N < M < 0.618 N. For the quasicrystal sample of MnA14 type considered by Dubois, a was determined by Janot et al. [23] to be 6.497 A, the lattice parameter of a primitive cubic lattice in six dimensions. It is quite previously dismissed. previously

clear that this possibility permits re-evaluation of some systems, thought to be cubic. This extraordinary possibility cannot easily be Were this to be so, the lessons of inorganic chemistry, a field thought to be relatively complete, would have hardly begun.

73

Organic Chemistry- The Shape of Molecules

2.10

The hyperbolic nature of sp 3 orbitals

We turn now to another area of chemistry where until recently, the notion of shape has been ignored. Organic molecules are constructed from three basic building blocks; sp 3-, sp 2- and sp-hybridised carbons. The latter two hybridisations impart a planar arrangement of substituents around the atom in question while a tetrahedral arrangement will be found around a sp 3 carbon. In geometrical terms, arrays of sp 3 carbons, alone or in combination with other types, will lead to complicated geometrical figures. The sp 3 hybridisation can be described as an idealised saddle configuration and hence an array of such atoms will induce a kind of corrugation in the tessellation, the extent of which is determined by the actual substitution pattern. The resulting surface, necessarily hyperbolic since the ring size normally exceeds four, can be approximated with a portion of a minimal surface. Thus, it should be possible to describe m a n y organic molecules with a suitable minimal surface. Unfortunately, at the present level of m a t h e m a t i c a l dexterity, such a delineation is not readily achievable. Nevertheless some indications as to the minimal surface nature of organic molecules can still be discerned and rationalised. If we start with the simplest possible case, aliphatic hydrocarbons, we can envisage an array of sp 3 carbons, i.e. saddles, joined in some suitable fashion. If we regard each carbon atom in an array as a point belonging to the surface, and consider the fact that a minimal surface is a surface where all such points are saddle points, a minimal surface is formed. For a linear hydrocarbon chain this could be illustrated with a helicoid, a classical analytical minimal surface. This striking resemblance is somewhat stronger if it be recognised that the bonding electrons, rather then the atoms themselves, tessellate the m i n i m a l surface. Bearing that in mind, it is possible to describe conformational movements of hydrocarbon chains in terms of the Bonnet transformation, Fig. 2.20. Not only is it practicable to pinpoint the positions of, and angles between, individual atoms during chain movements, but also an explanation of the concomitant chirality interchange is provided for [26]. The conjecture that bonding electrons rather than atoms should constitute the minimal surface is pleasing, as the notion of bent bonds, a necessity if real molecules are placed on minimal surfaces, is a somewhat ambiguous and a by no means settled issue. If aligned to a minimal surface the "bonds", between atoms in a molecule, could be kept straight, i.e. representing the distances between atoms only, while the actual bonding is along the surface, i.e. curved or bent. This can be interpreted as a situation where the electrons responsible for bonding distend the minimal surface and in so doing secure the over-all geometry. In terms of geometry vs hybridisation one can state that for sp 2 and

74

Chapter 2

sp carbons, substitution geometry and bonding geometry coincide, i.e. the minimal surface in question is a plane. For sp 3 carbons on the other hand substitution geometry is tetrahedral but bonding geometry is saddle shaped, i.e. the bonding electrons are smeared out along a non-planar minimal surface and the appropriate substituents are protruding from the surface.

Figure 2.20: Conformational changes in a C4 fragment of a hydrocarbon chain (large circles denote the C atoms, smaller circles the electrons located between the atoms). These images are produced by fitting bonding electrons to sites on a series of minimal surfaces 0aelicoid-catenoid) related by the Bonnet transformation. The top left image is planar ring fragment, with electrons located around the waist of the catenoid. Successive images show the result of transforming the catenoid by the Bonnet transformation, eventually forming the most extended fragment, with electrons located along the central axis of the helicoid (cf. fig. 1.19). The obvious question to ask w h e n e v e r minimal surfaces are involved is: w h a t is minimised with respect to what? In the case of organic molecules one can apply the same reasoning as for inorganic structures; formation and m a i n t e n a n c e of different c o n t r a p t i o n s with m i n i m a l e x p e n d i t u r e and dissipation of energy. In the construction of molecules where the bonding geometry, be it due to covalent, ionic or van der Waals bonding, is modelled on a minimal surface the required flow of energy is minimised on both enthalpic and entropic grounds. Both actual bond formation and buildingblock re-organisation or relocation is facilitated by the fact that the growing structure tessellates a minimal surface. A feature of curved surfaces is their capacity, loosely speaking, to contain more surface area in a given region than the plane. In situations w h e r e a v a i l a b l e surface area is e v e n t - c o n t r o l l i n g , such as in m o l e c u l a r c o m m u n i c a t i o n , a c u r v e d surface will have the u p p e r edge. Molecules communicate by means of weak forces originating from bonding electron polarisation, either p e r m a n e n t or inducible. The efficacy of the forces is roughly proportional to the n u m b e r of electrons in a molecule and as this in turn is a function of the n u m b e r of atoms, molecular communication is a result of available surface area presented. If the b o n d i n g electrons in a

sp 3 orbitals

75

molecule are situated on a hyperbolic surface their interactive prosperity will capitalise from the area compression mentioned above, i.e. upon each molecular encounter the interactive energy involved is not limited by the actual target area but will be a composite with contributions from the entirety of the molecules. In other words, if the force-field geometries are saddle shaped, molecules are compelled, via the field gradient (including information on both magnitude and direction), to adjust themselves to a "best match" engagement. This is in sharp contrast to a situation with spherical force-field geometry where directional information is essentially useless. Another salient feature of saddle-shaped force fields lies in longrange communication. A solute molecule influences its s u r r o u n d i n g s through the impact of its force field; solvent molecules do not dart around at random but will rather be distributed and oriented according to the forces working on them. Saddle-shaped molecules in solution would be prone to induce saddle-shaped solvation shells, functioning as telltales of low-energy (relatively speaking) interaction routes. In a small molecule it is usually difficult to perceive the saddle surface. This is most certainly due to the conditioning provided by various educational agencies; students are led to believe that the H-C-H angle in, e.g. methane is 109.5 ~ the famous sp 3 angle, illustrated by the hard, pointy geometrical figure of the tetrahedron. Technically speaking it is true; if we replace the atoms with points, connect them with straight lines and measure the angle we will get something like 109.5. The problem is that by doing so we tend to assume that the actual bonding phenomenon is concentrated along those very lines. This delusion is further amplified by the usage of dangerously misleading ball-and-stick models. A methane molecule is not a hard and rigid object, but should be regarded as dense bundle of energy, characterised by smoothness and dynamics. Addition of carbon atoms allows these overall properties to be maintained although some freedom of mobility is lost.

2.11

Organic sculptures: carcerands, crowns, etc.

Organic chemistry is usually presented as a kind of chemical chess; first the chessmen are identified, i.e. the functional groups, second the rules are established, i.e. the possible reactions. Given these precepts the game is ready to begin. Note that this game is confined to single atoms or narrowly restricted regions of a molecule. The rest of the molecule, the part commonly abbreviated as R-, is treated as proud flesh. Interestingly, as the size of organic molecules increases, and structures become more complicated, a striking shift of perspective ensues. As in a chemical glissando the relevance of particular atoms moves out of focus, instead, the shape of the complex structure becomes increasingly germane. Hence the literature is abundant in, e.g. crowns, cavitands, spherands, cryptands, calixarenes, carcerands etc, all with names carrying information on both molecular organisation and chemical capacity. It is noteworthy that in many instances the names given are unconcerned with the identity of the individual building blocks.

76

Chapter2

Now this leap from particularity to collectivity is not yet another educational deception but rather an appreciation of the impact of shape on chemical phenomena. Organic chemists disclose an impressive imagination when naming new complicated compounds. An imagination obviously inspired by the shapes and properties of the molecules in question. Delectable as it may be that chemists esteem shape as a pertinent constituent, the geometrical interpretation is somewhat erroneous. Chemical symbolism is traditionally based on spherical or partly spherical entities. Accordingly construction of illustrative images is in complicity with tradition. Spheres or objects with spherical geometry can however not be reconciled with saddle geometry. Spherical geometry requires positive Gaussian curvature (K) at each point whereas saddle geometry demands quite the opposite; IC~0. As before the geometry discussed is the interactive geometry, i.e. emanating from the bonding electrons taken as an ensemble. Something of a trend in modem synthetic organic chemistry is what one might somewhat blasphemously call "designer-molecules". In an effort to mimic the performance of biological systems in terms of both specificity and efficacy, chemists have devised synthetic protocols dedicated to display the influence of organisation, i.e. aggregational shape. Relaying on brute force, i.e. strong chemical interactions, rarely accomplish specificity; weak interactions on the other hand are usually inefficient. Nature regularly avoids highly potent reagents but depends rather on the collaborative action of weak yet numerous interactions. To achieve the latter end, some kind of focussing of interactions is imminent. An aggregate with spherical geometry offers excellent opportunities to realise an appropriate organisation but at the same time is seriously hampered when considering range and availability. Maximal interaction focussing will be reached for objects with spherical geometry when the sphere is closed, annulling both interactive range and availability. (Any reader familiar with microemulsions might at this point raise an offended eyebrow but should bear in mind that a micelle, clearly spherical, is essentially non-interactive. The dynamics of microemulsions are catered for by non-spherical intermediates.) Opening up a sphere increases the accessibility and the range, although some focussing is lost, until a hemisphere is formed. At this point the operative range of the system is a cylinder of indefinite extension and with a diameter proportional to the diameter of the hemisphere. Any plausible interactive partner has to enter into that rather narrow cylinder to successfully consummate the interaction. Beyond the hemispherical level, cooperative effects rapidly deteriorate. A superior alternative is an aggregate with saddle geometry, e.g. a minimal surface, where highly efficient cooperativity is combined with, in principle, infinite range- and availability properties. The late 60's and the early 70's saw the advent of the first artificial molecules where these concepts were fully utilised, the crown ethers and their sequels the cryptands. Crown ethers or coronands are hetero-cyclic structures where the hetero-atoms are separated with C2-units; cryptands are bicyclic analogues usually with nitrogens at the bridgeheads. These molecules are used in so called "host-guest" chemistry to confine and solubilise positively charged

Futlerenes

77

metal ions of various kinds, the guests, by means of cooperative interactions via the hetero-atoms and hydrophobic embeddment respectively. The impact of these features is quite impressive; 18-crown-6, a cyclic polyether with six oxygen atoms, can solubilise potassium permanganate in benzene, a solvent in which normally the well-known oxidant is essentially insoluble. In coronand/metal ion complexes, as long as the number of hetero-atoms is low a n d / o r the metal ion in question has a reasonably snug fit, the geometry of the assemblage will be planar, or almost nearly so. An altogether different situation arises when the coronand is larger or when non-optimal metal ions are employed. In those instances planar geometry is abandoned and is replaced with saddle-shaped arrangements. This is the case both when a large coronand is folded around a small metal ion or when a metal ion interacts with more than one coronand molecule. Cryptands, being bicyclic molecules, self-evidently cannot form p l a n a r complexes, hence s a d d l e - s h a p e d arrangements abound. The general trend of forming minimal surfaces abides as molecular complexity increases and is inevitable when it comes to chiral compounds.

Figure 2.21: The carcerand fitted to a unit cell of the P-surface.

78

Chapter2

The concept of coronands and cryptands has been further elaborated throughout the years, resulting in a distinct displacement of the characteristics of both the hosts and the guests. Henceforth hosts are designed to accommodate, e.g. negatively charged ions, neutral organic molecules of differing size, etc. Multi-site hosts, both mixed and isotropic, hosts with integrated reactive groups such as push-pull acid-base systems, hosts bonded to polymeric matrixes, have all become available. The pervading minimal surface nature of these different compounds is especially accentuated in structures providing large surface-carrying units. Spherands(sic!), structures based on cyclic oligomers of p-cresol, upon X-ray analysis disclose explicitly non-planar benzene rings indicating that the so called "enforced cavity" indeed follows the geometry of a minimal surface. Some very recent synthetic achievements are equally suggestive; dodecahedrane, C20H20, a truly spherical molecule yielded to synthetic efforts as late as 1984. Its preparation was regarded as a corner-stone of synthetic organic chemistry- and was preceded by an extensive amount of manpower. On the other hand the synthesis of the so-called carcerand, an equally closed molecule, the structure of which bears a casual resemblance to a sphere, was synthesised without difficulties on an apparently first-attempt basis. This remarkable difference is a reflection of the contrasting geometries involved. The spherical geometry of dodecahedrane is compelling, creating a state of opposition, and indeed congestion, when units preferring saddle geometry are forced together. Attempts to link two hemispherical units in order to complete the dodecahedrane skeleton have been consistently futile. The carcerand structure however is one of saddle geometry. It can be shown that it is an almost perfect fit to the repetitive cell of the surface (Fig. 2.21). Hence in the final synthetic step, when two preformed units are joined, no congestive or other problems are encountered.

2.12 Beyond graphite: fullerenes and schwarzites Organic chemistry is now moving beyond its traditional preoccupation with small molecules. The most spectacular examples of this shift are the fullerenes, which are sp 2 carbon clusters forming giant ball-like molecules: C60, C70 etc [27]. Until their discovery, it was believed that planar graphite consisting exclusively of six-rings - was the sole sp 2 carbon polymer. By Euler's theorem (section 1.8), the formation of carbon shells, whose geometry is elliptic, requires the presence of five-rings in the carbon network. If five-rings are possible, why not seven-rings? The presence of seven-rings in the (three-connected) network must lead to hyperbolic frameworks. Once the average ring-six exceeds six, hyperbolic carbon structures must result, which lie on periodic hyperbolic surfaces, and form three-dimensional extended frameworks. A number of theoretical studies have indicated that these hyperbolic structures, called "schwarzites" in honour of the mathematician Schwarz, should be more stable than the fullerenes [28]. In

79

Beyond graphite

geometric terms, a unique hyperbolic surface can be traced through the threeconnected carbon atoms. Just as for the tetrahedral networks analysed in section 2.8, the curvature of this surface is related to the bonding geometry. Here too, the most stable p o l y m o r p h s are expected to form tessellations of surfaces similar to IPMS [29]. So far, theoretically predicted schwarzites do not display this universality, although a number of predictions give an area per C atom close to that found in graphite and C60 (Fig. 2.22). These theoretical frameworks are the result of c o m p l e x n u m e r i c a l q u a n t u m mechanical calculations. The a p p a r e n t conservation of surface density, irrespective of the curvatures of the surface, is clearly not a direct consequence of standard physics. It will be very interesting to compare the surface densities of actual schwarzites (although they have yet to be prepared in the laboratory) with those of fullerenes and graphite. Given the usefulness of this principle in the study of tetrahedral frameworks, our bet is that they too will lie on the dotted line in Fig. 2.22.

9

I

"

I

"

I

"

lO 8

6

~ _

.~ i

5.5

l

6.0

.~ .

.

I

I

6.5

.

.

~_

9

I

7.0

.

9

7.5

a v e r a g e ring-size

Figure 2.22: Area data calculated as described in Fig. 10 (using the same vertical scale) for a range of hyperbolic "schwarzite" sp 2 carbon networks predicted by various theoreticians. The filled diamonds denote the corresponding areas per vertex for (planar) graphite (n=6) and C60 fullerene (n=5.62). (The latter is calculated from standard crystallographic data assuming a spherical network.)

80

Appendix: The Problem of Quasicrystals The problem posed by the discovery of quasicrystals has occasioned much angst among chemists. Indeed some, like Linus Pauling [30] dispute their existence, and claim that they are an artefact which can easily be accommodated within the framework of conventional theories of space filling structures. The issue remains open. Projections from hyper-Euclidean spaces to three-dimensional and group theory [4, 22, 31], geometric packing arguments [4, 31] are some of the mathematical tools that have been invoked, and some real space models that capture the main features of particular systems have been constructed [24]. The problem is to relate observed diffraction patterns with non standard, supposedly disallowed, crystallographic symmetries, to the atomic distributions that cause them. That problem remains. Because while a physicist living in world made up of equations and group theory has no difficulty in constructing the universe, its scaling laws, and singularities like black holes, as a realisation of a sixteen-dimensional group say, the chemist is more narrowly constrained. A three- dimensional atom has a certain pedestrian reality that does not so easily lend itself to a mapping into six dimensions. The question raised by the quasicrystal debate is much deeper than whether they exist or not. To see this, we recall that the interpretation of diffraction experiments on all known translationally invariant crystals, however complicated, depends ultimately on the existence of the Poisson summation formula. This relation asserts that the Fourier transform of the periodic delta function is itself a periodic delta function, whence the term reciprocal space. Explicitly, the Poisson summation formula is oo

f(x) = ~

~

oo

8(x-m)--- ~

n ~

exp (2~mfix)= 1 + 2 ~

rr~.o

cos(2~mx)

m=l

so that the Fourier transform of a translationally invariant array of atoms represented by the periodic delta function is

f(k) =

dx exp(-2nikx)

8(x-m) ..oo

= ~exp(-27tikx)= ~ -oo

m=-oo

8(k-m)

Appendix: Quasicrystals

81

The determination of crystal structure is 9 immediate, in principle, since any standard diffraction pattern will be related to, e.g., the product of an appropriate combination of three such delta functions (periodic in x,y,z directions), with atomic form factors. Inversion to get the real space atomic positions from the diffraction pattern is then possible via the convolution theorem for Fourier transforms, provided the purely technical problem of the undetermined phase can be solved. Now the Poisson summation formula is at the core of all mathematical analysis [33]. It is equivalent in fact to the calculus, the Jacobi theta function transformations, and to a statement of the Riemann relation connecting the oo

zeta function ~(s)m ~ n!s, Re(s)> I with ~(1-s) so linking up with number n=l

theory itself. (The distribution of zeros of the ~ function in the complex splane is one of the major unsolved problem of analysis.) Fundamental solutions of the time-dependent Schr6dinger equation and the diffusion equation are the theta functions. There has been no basic formula analogous to the Poisson summation formula, characteristic of translational invariance, on which to base an analysis of quasicrystal diffraction patterns. Here successive values of 'reciprocal' space have geometric ratios instead of the arithmetic spacing of the peaked functions observed with ordinary crystalline diffraction. Fig. 2.15 illustrates a two-dimensional section in reciprocal space of a diffraction pattern. The five-fold symmetry is exact, and typically six indices instead of three are required to index each point, with the choice of origin arbitrary, and for assignment of indices, ambiguous. The features of interest are: (1).Along a given five-fold axis, the spacing of the main peaks kn is in the (geometric) ratio kn = 1 where z is the golden ratio (~5 + 1)/2. kn+l (2).Between two such peaks is a sequence of further peaks of lower intensity that all lie in an infinite set of coincident interpenetrating Fibonacci sequences of arbitrary origin. (They can be labelled through a projection from Euclidean two-dimensional space as described in the main text.) (3).Surrounding each five-fold axis are other sequences of lesser intensity that can be connected to their neighbours to form regular self-similar pentagons (icosahedra in three dimensions). (4).Along the five-fold axes the density of points as one approaches the origin becomes infinite. (5).Increasing time of exposure of the recording film results in the appearance of more and more points throughout space in an eventually dense spacefilling array. All points satisfy the same symmetry and self-similarity properties.

82

Chapter2

(6).Through a set of initial points on adjoining five-fold axes a set of 20 intersecting equiangular spirals can be drawn emanating from any chosen origin.

If (1) is taken as the main feature of the structure, the appropriate representation along any axis is [34]: oo

f(k)=~-J ~

8(~_~-m)

(where the factor I kl-j, j=0,1,2 in 1,2,3 dimensions is necessary to preserve self-similarity). This is a very different function from the periodic delta function (arithmetic spacing) that underlies the spectra with which we are most familiar. The Fourier transform of such a function, the "skeleton" of the spectrum, has been investigated [34]. It gives out the equiangular spiral as a fingerprint - the self-similar properties of figures inscribed into such spirals have been a source of much mystery for centuries [35]. The analysis gives more, and the zeros of the Fourier transform give an infinity of other real space points that also satisfy self-similarity and fill up non-occupied space. The intersecting spirals that emerge in two dimensions (or helices in three dimensions) give the Penrose tilings naturally a;td the constraints of self-similarity, scaling, five- fold symmetry and independence of origin also emerge. There is a hint here that new kinds of basis functions other than the usual periodic delta function of ordinary mathematical analysis may provide an extraordinarily rich range of new structures. The emergence of the logarithmic spiral is not surprising, and D'Arcy Thompson [35], James Bernoulli [35], (who was so fascinated by its properties that he had it engraved in his tombstone) and Kepler would have taken it as self-evident. In T h o m p s o n ' s words: "In the growth of the shell, we can conceive of no simpler law than this, namely, that it shall widen and lengthen in the same proportions; and this simplest of laws is that which Nature tends to follow. The shell, like the creature within it, grows in size, but it does not change its shape, and the existence of this constant relativity of growth, or constant similarity of form, is the essence of the equiangular spiral". The maintenance of shape, or the constant change of curvature is indeed of the essence. Quasicrystal structures have been known for a long time to occur in condensed matter and rejected as inexplicable curiosities. They may emerge naturally too in mathematical descriptions of surfaces. (The decagonal variant certainly arises, cf. [36]). It is not an entirely idle speculation to conjecture, e.g. that the principles exploited in construction of quasi-crystals may be precisely those used by nature to build proteins that solve the problem

Appendix: Quasicrystals

83

of self-recognition, the fundamental problem of immunology. Nor, that since the associated infinite incommensurate packings are missing from the usual Hilbert space description of quantum mechanics, that the problem may be related to the difficulties that confront particle physics.

84

Chapter2

Rs163163163 1.

H. Weyl, "Symmetry". (1952), Princeton University Press.

2. D. Shechtman, J.A. Blech, D. Gratias, and J.W. Cahn, Phys. Rev. Lett., (1984). 53: pp. 1951-1953. 3.

D. Shechtman and J.A. Blech, Met. Trans., (1985). A16: pp. 1011-1065.

4. M.V. Jaric, "Introduction to Quasi-Crystals'. Academic Press, (1988 and 1989), New York: Academic Press. 5.

S. Andersson and B.G. Hyde, Z. Kristallogr., (1982). 168: pp. 119-131.

6.

L. F~ilth and S. Andersson, Z. Kristallogr., (1982). 160: pp. 313-316.

7.

Q.B. Yang and S. Andersson, Acta Cryst., (1987). B43: pp. 1-14.

8. S. Andersson, S.T. Hyde, and H.-G. von Schnering, Z. Kristallogr., (1984). 168: pp. 1-17. 9.

R. Nesper and H.G. von Schnering, Z. Kristallogr., (1985). 98: p. 111.

10. I.S. Barnes, S.T. Hyde, and B.W. Ninham, J. Phys. (France), Colloque., (1990). C-7: pp. 19-24. 11. R. Thomasson, S. Lidin and S. Andersson, Angew. Chem. Int. Ed. Engl., (1987)). 26: pp. 1017-1018. 12. J.W. Christian, "The Theory of Transformations in Metals and Alloys.". (1965), New York: Pergamon Press. 13.

S.T. Hyde and S. Andersson, Z. Kristallogr., (1986). 174: pp. 225-236.

14. S.T. Hyde, Infinite periodic minimal surfaces and crystal structures (1986), Ph.D. Thesis, Monash University. 15.

C.T. Kresge, M.E. Leonowicz, W.J. Roth, J.C. Vartuli, and J.S. Beck,

Nature, (1992). 359: p. 710. 16. S.T. Hyde, in "Defects and processes in the solid state. Some examples in earth sciences.", J.N.Boland and J. D. FitzGerald,Editors. (1993), Elsevier: Amsterdam. 17.

S.T. Hyde, Acta Cryst., (1993). AS0: pp. 753-759.

18. W.M. Meier and D.H. Olson, "Atlas of Zeolite Structure Types". 3rd. ed. (1992), London: Butterworth-Heinemann.

References

85

19. S.T. Hyde, Z. Blum, and B.W. Ninham, Acta Cryst., (1993). A49: pp. 596-589. 20. F. Liebau, "Structural Chemistry of Silicates". (1985), Berlin: SpringerVerlag. 21. G. Kowalewski, "Der Keplersche K6rper und andere Bauspiele". Vol. 3. (1938), Leipzig: K.F. Koelers Antiquarium. 2.

A. Mackay, Physica (1982). 114A: pp. 609-613.

23. C. Janot, M. de Boissieu, J.M. Dubois, and J. Pannetier, J. Phys. : Cond. Matter.(1989). 1: pp. 1029-1048. 24. S. Lidin, S. Andersson, J.-O. Bovin, J.-O. Malm, and O. Terasaki, Acta Cryst., (1989). A45: pp. 33-36; M. Jacob, S. Lidin and S. Andersson, Z. Anorg. Allg. Chem. (1993), 619: pp. 1721-1724. 25.

L. Bendersky, J. Phys. (France), (1986). 47(C3): p. 457.

26.

Z. Blum and S.T. Hyde, J. Chem. Res. (S), (1989). (6): pp. 174-175.

27. H.W. Kroto, A.W. Allaf, and S.P. Balm, Chem. Rev., (1991). 91: pp. 1213-1235. 28.

A.L. Mackay and H. Terrones, Nature, (1991). 352: p. 762.

29.

H. Terrones and A.L. Mackay, Carbon, (1992). 30(8): pp. 1251-1260.

30. L. Pauling, in "The Chemical Bond: Structure and Dynamics", A. Zewail, Editor. (1992), Academic Press: New York. 31.

G. Gratias and L. Michel, J. Phys. (France), (1986). C3: p. 7.

32.

R. Penrose, Bull. Inst. Math. Appl., (1974). 10: p. 266.

33. B.W. Ninham, N.E. Frankel, B. Hughes, and L. Glasser, Physica A , (1992). 186: pp. 441-481. 40

B.W. Ninham and S. Lidin, Acta Cryst., (1992). A48: p. 640.

35. D.W. Thompson, "On Growth and Form". 2nd. ed. (1968), Cambridge University Press. 36.

A. Fogden and S.T. Hyde, Acta Cryst., (1992). A48: pp. 442-451.

This Page Intentionally Left Blank

87

Chapter 3 3.1

Molecular Forces and Self-Assembly

The background

hape and form imply the notion of a surface. In biology the shape taken up by an enzyme or DNA is crucial to its catalytic or templating properties. This shape, sometimes a fluctuating structure, is dictated by molecular forces. Intramolecular forces determine shape, and intermolecular forces guide recognition processes. Again, in cell biology, precise recycling patterns occur that require transformations between self assembled aggregates like micelles, vesicles and bilayer membranes. Indeed lipids, the surfactants from which these objects form, provide the most primitive of prebiotic assemblies. These are dynamic, equilibrium structures and even within a given class and order, transmute and reassemble into different structures in response to quite delicate changes in environment- such as temperature, salt and other factors that determine the driving molecular forces. These determine curvature. It is known and understood that lipid membranes provide a matrix for enzymatic catalysis for structural proteins, and for transport, and a protection to cells against the exterior environment.

S

We can make these remarks explicit by considering the mechanism [1, 2] of endocytosis as illustrated in Fig. 3.1. In terms of specific biochemical processes, one focusses on the selectivity of the binding of a ligand to the cell membrane protein, the dissociation of the ligand complex induced by the decrease in pH within the endocytosome, the separation of the ligand and protein in the curl, and the eventual processing of the ligand. There is an alternative way of thinking in terms of the physical forces which are a consequence of binding and dissociation processes. These forces cause changes in curvature that result in the transformation of induced cell membrane pits to protein-coated vesicles, the new cubosomal structures described in Chapters 5 and 6 and the fusion of those vesicles. Further changes in curvature are evident in the formation and separation of the curl, and in the subsequent fusion of the tubular assembly into the cell membrane. Chemical and physical processes are coupled. The bewildering diversity of assemblies that biology confronts us with is so vast that it seems impossible to make sense of the matter. Yet the patterns are so repeatable and precise that there must be an underlying unity that derives from the forces that determine curvature and guide the necessary chemical reactions. It is only recently that some of that unity has been revealed. This Chapter outlines some of the nature, delicacy and specificity of molecular forces, and how it is that these forces conspire with the geometry of molecules to organise self-assembled molecular aggregates. The shapes and topologies that set the physico-chemical environments for biochemistry are the subject of following chapters. The references provide a sufficient guide to the literature for the reader interested in exploring further complex technical issues.

88

Chapter3

Figure 3.1: (Top:) Schematicview of processes controlled by curvature in endocytosis. Adapted from [2]. (Bottom:) Representative of the simplest surfactant aggregates: spherical miceUes (v/al between 1/3 (spheres) and 1/2 (cylinders) ; planar bilayers (v/al = 1); inverted micelles (v/al > 1).

3.2.1

The nature of force, shape and size

Geometry deals with the disposition of objects, originally idealised as points, and time labels the evolution in position of those objects. The two idealisations, distance and time were linked by Newton through the separate concept of mass to give us the concepts of momentum and force. Matter, and later charge, both the measure of inertia and the source and sink of force, were once seen as immutable properties of matter. They became inextricably part of the idealised, matterless space-time continuum after Gauss developed the concept of potential energy, after Maxwell's theory of the electromagnetic field, Lebedev's proof of radiation pressure exerted by the field, and finally Einstein's synthesis in the theory of relativity. Boltzmann's statistical mechanics gave us entropy, temperature and chemical potential in a large collection of interacting molecules. Planck imposed the condition of discreteness on the electromagnetic field that led to the wave-particle duality and the muddled grandeur that is modern quantum mechanics. The rest is history, and the soaring flights of imagination that have taken us to m o d e m theories of the universe and the search for a unified field theory embracing

Force, size and shape

89

gravitational, weak and strong interactions appear to far transcend the more m u n d a n e concerns of chemists. But the point particles of physics ignore shape and size that are the axiomatic attributes of the subject of chemistry, be they atoms, molecules, proteins joined in a supposedly particular configuration by "chemical bonds", or transient lipid vesicles or micelles. And where one object ends and another begins is not so self-evident. The notion of a bond that emerges from a quantum mechanical theory of two interacting atoms is not so obvious if those objects are immersed in a sea of their neighbours, forming a solid or liquid. To make sense of a very complex matter which can easily bog down in either recondite mathematical theories, or be relegated and s u b s u m e d u n d e r qualitative, non-predictive characterisations embraced by terms like "ionbinding" to surfaces, we dissect the problem in stages. The ensuing discussion attempts first to come to grips with what we mean by the size and shape of a molecule. Thereafter we explore in general, historical, terms the conceptual background to what is known about forces between surfaces in solution to set the b a c k g r o u n d for the specific technical discussion that follows. Subsequently we show how a knowledge of these forces enables us to change the curvature of aggregates of self assembled molecules. The different sections are self contained and can be read independently.

3.2.2 Self-energy, molecular size and shape In most discussions of molecular forces [3-10], atoms have been treated as if they are point particles, and size or shape has been invoked as a separate concept. For example, the van der Waals energy of interaction V(r) between two atoms a distance r apart is V(r) or - 1 / r 6. It becomes infinite on contact, unless some preassigned hard core contact size is invoked, and so too for physisorption of an atom to a surface. This convention is necessary, but its origins lie deep. In fact no distinction can be made between energy, size and shape, which are inextricably linked. The linking concept of self-energy takes on significance whenever an object is considered to have a finite extent or is delocalised: for then the abstraction that the object can be considered separately from its surroundings becomes philosophically tenuous, as one part of the object can consider its other parts to belong to the rest of the world; hence, perhaps, the uncertainty principle. No difficulty occurs if the environment or object are immutable. If the opposite obtains, as indeed it always does at some level, the reaction of the (changed) environment to the object will be different, and the self-energy due to this reaction field will be different. The shift in self-energy due to radiative corrections to energy levels is a central problem of quantum electrodynamics. The Born electrostatic selfenergy of an ion is important in electrolyte theory, physical adsorption and in the migration of ions through membranes. The Debye-Htickel theory of electrolytes results from the change in Born self-energy of each ion due to all

90

Chapter 3

other ions in the solution. The self-energy of a dipole e m b e d d e d in a dielectric sphere is the key to Onsager's theory of the dielectric constant of dipolar fluids. Equally, in any theory for, say, the surface energy of water, or adsorption of molecule, the self-energy of a molecule as a function of its distance from an interface is involved. In adsorption proper, the same selfenergy for a molecule appears in the partition function of statistical mechanics from which the adsorption isotherm is derived. The idea is clear, but the details are complicated [3-12]. We omit details and focus on the results. For a molecule of finite extent its dispersion self-energy can be defined as the change in its energy due to its coupling with the electromagnetic field, or equivalently, as the change in q u a n t u m mechanical zero point energy of the field due to its coupling with the oscillating dipole moment it induces on the molecule. If we consider a single atom centred at position R interacting with the radiation field, we can write d o w n Maxwell's equations to describe the process. Their solution requires a closure relation, between the induced dipole moment density p(r, ca) and the electric field E(r,r at all points in space r, and at all frequencies of oscillation co of the field. For point molecules within the framework of linear response theory, this relation is

p(r, co) = a(co) E(R,o0) 8(r- R)

where ~ is the polarisability tensor of the atom and 5(r) the Dirac delta function. This relation is strictly incorrect, because the dipole is spread out over a region of space (of the order of the "volume" of the molecule), and in a semi-classical formulation of q u a n t u m mechanics it can be shown that the actual relation is p(r, co) = cx(r- R, co) E(R, ca) where the components of the tensor a ( r - R, co) are peaked around R with a range which defines the size of the molecule. It can be given an explicit expression in terms of a sum over matrix elements of atomic wavefunctions [11, 12]. That is the end of the matter, and the solution can then be effected in principle. We can assume for computational convenience that the atom is isotropic and that ~(r, ca)=a (co)f( r ) / w h e r e / is the unit tensor, and the form factor f(r) is a distribution function for the electrons peaked at the centre of the atom. A gaussian is a good approximation, and we take

/ r2) , ff(r)

f(r)= n3/21 ag.exp - ~ -

d3r = l

Self-energy

91

where a can be taken as the size (radius) of the atom. In the n o n - r e t a r d e d limit i.e. i g n o r i n g the finite velocity of light, the dispersion self-energy computed with this form factor can be shown to be [11, 12]*

Es _ ~ 3 /2h2 a 3 Jo d~a(i~) _=~ Rydbergs (Hydrogen atom)

(3.~)

The dispersion self-energy is of the same order of m a g n i t u d e as the binding energy for a hydrogen atom, but of opposite sign. The same kind of semi-classical formalism permits extension to two or more atoms. The interaction energy, the difference between the complete energy of the coupled system and the s u m of the dispersion self-energies of two isolated atoms, reduces to the van der Waals interaction energy at large distances, but n o w remains finite at zero separation. For like atoms this energy is of the order of the binding energy of the molecule that w o u l d be formed by them. (For point atoms, the van der Waals interaction energy is infinite.) The same concepts can be used to develop a simple semi-classical estimate [3-10, 11, 12] of the Lamb shift in hydrogen, and to explain the differences in binding energies [3-10, 13] (e.g. face-centred cubic versus hexagonal close packed) of rare gas crystals. Extension to include quadruple and octopole effects explains the shape of the interaction potential necessary to fit the thermodynamic properties of simple liquids like argon, calculated by Monte Carlo methods. Further extensions to include surrounding media [310], polymers or the presence of surfaces [3-10], or to physisorption [3-10], reconcile a number of difficulties.

3.2.3

Self-energy and adsorption

As an illustration consider the a d s o r p t i o n of molecules into zeolites, discussed in Chapter 2. If an adsorbate molecule or atom is immersed in a m e d i u m of dielectric constant ez(rO) instead of a vacuum, the dispersion selfenergy is modified, and now takes a form corresponding to eq. (3.1)

* Here the polarisability a(co) as a function of the complex frequency variable co=TI+i~is evaluated on the imaginary frequency axis ~. The reasons are technical [3-10] and need not concern us. The simple classical form of the polarisability for a one-electron atom is Or(CO)= e2 where e is the electronic charge, in the mass of the atom, and -h-co0 is the m~0~- ~ ~) ground state energy, coo= e2, a0 the Bohr radius. 2a0

Chapter3

92

Es(medium)

_ 2h = lt3/2 a3

d~ a(i~) r

(3.2)

The expression immediately gives an estimate of the enthalpy of adsorption in taking an atom from the gaseous (vacuum) state to a liquid, or to a composite medium like a zeolite, characterised by its measured dielectric frequency dependent response &(c0). It is, exactly as for the electrostatic Born self-energy in taking an ion from vacuum to water:

AH(adsorption) = [Es(vac) - Es(medium)!

= 2/~ -11:3/2 a3

l" [

d{ a(i{) 1 - - 1 ] ez(i~ ]

The measured enthalpies of adsorption of alkanes into zeolites are very large, and can be calculated from this formula [13]. The dielectric constants for the open networks formed by the aluminium silicate frameworks of zeolites can be related to the known and measured dielectric and optical properties of bulk quartz. The calculation assumes that the zeolite framework tessellates a hyperbolic surface as described in Chapter 2. The polarisabilities of different adsorbate alkanes are also known. When the calculations are carried out for a whole range of alkanes as adsorbate molecules and for different zeolites (with differing pore structure and size) the agreement between measured and predicted heats of adsorption is excellent (cf. Fig. 3.2). The results depend explicitly on the Gaussian curvature (AH=~I~/2) , and involve only a single parameter, a, the effective radius of a single CH2/CH3 group, which turns out to be 2.9~. The approach using the self-energy concept is very different to the simpleminded two-body potential summation method of Section 2.5. It connects the Gaussian curvature of the hyperbolic surface underlying the zeolite framework quantitatively and directly to the many body interactions and excitations of all the molecules involved. These are included through the measured dielectric properties in a way not accessible to two body potentials. The approach provides a key to the mechanism of catalysis in zeolite frameworks. Thus, from a consideration of the self energies of say a single dodecane, and two hexane molecules in vacuo, we can conclude that the spontaneous decomposition of dodecane into two hexane molecules is most improbable. (The general formula for self-energy includes temperature effects also.) But inside the zeolite a different story obtains. The vibration and other excited modes of the dodecane molecule built into the self-energy expression are very different, and self-energy calculations show that it becomes

Self-energy

93

favourable for such a molecule to split into smaller parts - just as for example a salt molecule held together by an ionic bond, will dissociate into its component ions when dissolved (adsorbed) in water, thermodynamically favourable if one considers entropy and electrostatic self energies. (The water, from our viewpoint can be considered to be a dynamic self intersecting hyperbolic surface tessellated by the hydrogen-bonded network.) The selfenergy concept has been shown to be quantitatively correct in predicting the 0-temperatures of polymers in simple liquids [3,61].

aHax p (Jg")

4O

.i

o

30"

20

ill

~

U

31"

0

"

II

2OO

~

"

d

~~

~

r

II

4O0

"

o ZeoliteY

l

6OO

"

l

800

"

l

lO00

, ~ X ~ ' (J~,'s")

Figure 3.2: Theoretical (x-axis) vs. measured (y-axis) heats of adsorption in zeolite Y and silicalite for a range of alkanes. The theoretical heats are multiplies by the "volume" of a methyl group (a3).

In organic chemistry, phenomenological rules that predict reactions and reactivity that begin to break down, e.g., with increasing hydrocarbon chain length, also makes sense once the connection between dispersion self-energy, the environment, size and shape are recognised. While the notion of size and shape, the geometry of molecules or objects, emerges naturally in semiclassical theories, and is intimately related to energy, these concepts appear to disappear in full quantum mechanical descriptions of molecules. Thus the full quantum mechanical description of molecules in the gas phase gives excellent agreement with observed spectra, but fails to explain optical isomerism. Yet the question of chirality is vitally important to the understanding of biological activity of organic molecules. The much cruder Born-Oppenheimer approximation- which assumes that nuclei are fixed in space - does build in shape, the concept of a potential energy surface, and the idea of a chemical bond. Quantum mechanics in its present form misses the notion of shape [14].

94

Chapter 3

3.2.4 The shape of bonds This paradox, of physics, is more apparent than real, and the chemists have persisted with the fiction that objects exist. The concept of a chemical bond, ionic, van der Waals, covalent, is taken for granted and is essential to chemistry. The first two make no sense except in the context of an infinite crystal. (An "ion pair" in solution, or a "hydrophobic bond" in water between two methane molecules is due to complex statistical mechanical solvent mediated association behaviour, to be discussed below.) If the positions of the atoms are assigned in a regular lattice by X-ray crystallography one can calculate the electrostatic energy of formation of an ionic crystal like CsC1. This energy, per CsC1 pair in the lattice, is the bonding energy of the pair. If the Cs +, C1- ions that form the lattice structure are modelled as hard, non-interpenetrating systems (of different radii), there is no real bond; just an attractive Coulomb potential, cut off, and infinite at a "hard core" contact. In practice, the ionic electron clouds are not sharp, and the hard core can be replaced by a softer short-range repulsive potential that gives a minimum in the combined potential at the "equilibrium" radius of contact. The pairs oscillate with zero point energy, more with increasing temperature, about this minimum, or bond length. At best such a "bond" is an effective one, since the Coulomb potential is of infinite range, and the energy per bond depends on a sum of interactions of all the ions of the crystal. A similar situation holds for the van der Waals interaction or bond, in a non-ionic crystal. Here the attractive (V(r)o~-1/r6) potential is again opposed by the short-range repulsive potential due to electron cloud overlap to give a potential minimum, which if we like, we can call a bond. As already remarked, even for this case (short-range attractive forces) discrimination between the theoretically calculated bonding energies of rare-gas crystals requires that the dispersion energy be calculated to all orders of perturbation theory [3]. It is, like the ionic crystal case, a many body problem. The entire electromagnetic radiation field due to inter-oscillations of all the atoms of the crystals must be taken into account to explain observed energies of formation. So, even for the simplest conceivable cases of ionic and van der Waals crystals, global properties, many-body forces that depend on the arrangement of atoms, and local force properties are linked. Minimal surfaces appear here in a very natural way. To see this, consider an array of electrostatic point charges arranged in the different crystallographic symmetries of say NaC1, CsC1 crystals of Chapter 2. Suppose that the particular space-group symmetry is given, and for the moment admit the approximation of a hard-sphere model for the ions to set the lattice parameters. The Ewald sum of electrostatic energies provides the major contribution to the binding energy. Now consider surfaces of zero electrostatic potential traced out inside the lattice [15-18]. These can be

The shape of bonds

95

calculated, and look very much like minimal surfaces. If the crystal zero potentials are truly minimal surfaces, then phase changes can occur easily via the Bonnet transformation. Excited states of the crystal, including anharmonic states, are then included and counted as the (infinity) of different allowed minimal surfaces that we might suppose forms a complete set. But these surfaces are not exactly minimal, although appearing close to such. The reasons seem clear enough: it can be proved that no electrostatic set of charges can be in an equilibrium state. Presumably if one built in quantummechanical zero-point energies and then calculated the new equipotential surfaces, the new zero potential surfaces would be minimal surfaces of the field. With that idea, we can turn the argument around and say that since the crystal exists as an equilibrium system, it must be permeated by surfaces of zero stress of the entire electromagnetic field. Imposition of this (mechanical) requirement, together with the condition that the Poynting vector (E x H) ( m o m e n t u m transport) is zero, p r e s u m a b l y will give back q u a n t u m mechanics. The argument implies the emergence of quantum mechanics as a consequence of minimal surfaces, a necessary Pythagorean imperative that effects the bridge between geometry and arithmetic, discrete and continuum, the particle and field points of view [18]. For covalent bonds we are likewise on tenuous ground and the lone pair concept so useful to chemists has been hotly disputed. Consider only the ground state of a covalently bonded few-atomic system, and presume that an ab initio quantum mechanical calculation gives us a lowest energy state. Let us ignore the complication that different isomers of the covalently bonded system could be degenerate. There will be a shape associated with this lowest energy configuration, set by the electron cloud distributions about the fixed positions of the nuclei. The squared amplitude of the calculated Schr6dinger wave function gives this electronic distribution. A frustration for chemists has been that this probability distribution, while appearing to mimic vaguely what one thinks a bond should look like, is too smeared out to give any real confidence beyond its mathematical expression. Some remarkable recent work [19, 20] has taken up ideas that were already a part of quantum mechanics, accessible at least thereto since the discovery of the Pauli principle, itself a fundamental building block in chemistry through Hundt's rule. By calculating an electron localisation function from the wave function, the probability that if an electron is in a given spin state another will be a prescribed distance away can be calculated, and the positions of the electron in a bonded system can indeed be pinned down. Various calculations confirm that the intuitive covalent bond picture of chemistry, and the lone pair concept, are very precise and sharp. What is more the 'shape' of the electron localisation distribution functions turns out invariably to give precisely the shape traced out by sections of periodic minimal surfaces [21].

96 3.3

Chapter3

The background to surface forces

We turn now to the nature of molecular forces between surfaces separated by a liquid. The main question is the nature of forces at distances greater than molecular dimensions, and less than those at which matter can be treated as a continuum. Very clearly, its answer must be intimately connected with the meaning assigned to the word surface. The (molecularly smooth) mica, or mercury - water interfaces are one thing, the liquid vapour or immiscible liquid liquid interface is another. And it might reasonably be imagined that the thesis that the world is fiat would be considered extravagant madness by an anthropomorphic huddle of water molecules near a biological membrane or surfactant interface. However, we persist with the idealisation of a surface. Consider first planar solid surfaces. -

Two themes emerge in dealing with the force between two such surfaces:(see Appendix 3): (i) an intervening liquid can be thought of as a structureless continuum with bulk liquid properties, up to a molecular distance from the surface; (ii) any object (surface) must perturb proximal liquid structure (density, dipolar orientation, hydrogen bonding) so that the transmission of force is propagated via a stress field passing from molecule to molecule in much the same way that the electromagnetic field is carried through the vacuum or a dielectric in MaxweU's theory of the electromagnetic field. Once the forces between rigid surfaces are understood, we can proceed to the more complex question of how an object reorganises its shape in response to the change in chemical potential, of force induced by a neighbouring object. The classical intuition on molecular forces is embodied in the famous Derjaguin-Landau-Verwey-Overbeek theory of colloid stability. It blends themes (i) and (ii) above in a contradictory way that has taken until the last few years to sort out. The best elementary account of the theory is contained in the book by Israelachvili [4]. But there have been very many developments since, as outlined below. These developments are very complicated, but cannot be ignored, and the subtleties have to be recognised to make sense of many phenomena in colloid science and biology. Imagine a suspension of colloidal particles in water. What causes stability, and what, under changing solution conditions like addition of salt causes flocculation (precipitation of the suspension)? Two opposing forces were considered to operate between two such particles. The one, attractive, is the quantum mechanical van der Waals force and treats an intervening liquid as if it has bulk liquid properties up to the interfaces of the particles (theme (i)). The other, repulsive, due to charges formed by dissociation of ionisable surface groups, is electrostatic in origin, and depends on salt concentration.

Surfaceforces: background

97

Here by contrast, it is the overlap of the inhomogeneous profiles of electrolyte concentrations induced by the charged surface of the particle that gives rise to the osmotic (double-layer) force (theme (ii)). When the repulsive double-layer forces win out, the suspension is stable. On addition of sufficient salt, the range of these forces decreases, the attractive forces take over, and the system of particles flocculates. While the picture is (usually) qualitatively correct, it is deficient in very many respects, and an understanding of how these deficiencies show up is essential to our thesis. Before launching into detail we first outline how this theory [3-10] has changed. We remark in passing that the more sophisticated m o d e m theories have been confirmed by direct measurement, or so the story goes. Note first that in this older picture, for both the attractive (van der Waals) forces and for the repulsive double-layer forces, the water separating two surfaces is treated as a continuum (theme (i) again). Extensions of the theory within that restricted assumption are these: van der Waals forces were presumed to be due solely to electronic correlations in the ultra-violet frequency range (dispersion forces). The later theory of Lifshitz [3-10] includes all frequencies, microwave, infra-red, ultra and far ultra-violet correlations accessible through dielectric data for the interacting materials. All many-body effects are included, as is the contribution of temperature-dependent forces (cooperative permanent dipole-dipole interactions) which are important or dominant in oil-water and biological systems. Further, the inclusion of socalled retardation effects, shows that different frequency responses lock in at different distances, already a clue to the specificity of interactions. The effects of different geometries of the particles, or multiple layered structures can all be taken care of in the complete theory [3-10]. For the double-layer (electrostatic) forces, extensions beyond the point ion approximation [22] to allow a hard core size to the "hydrated" cations and anions, inclusion of image interactions and higher order correlations in the inhomogeneous electrolyte profile show that these forces cannot be treated independently of the van der Waals forces. For multivalent counter-ions, like Ca 2+, the "new double-layer" forces can actually be strongly attractive at small distances ~ <20A and high concentrations [23-25]. Further, at large distances, the range of these (exponentially screened) forces is very different from the older theory for asymmetric electrolytes, a fact that again removes some puzzling experimental anomalies [26]. Beyond this, the inclusion of the competition for surface sites of different competing species (e.g. H + vs. Na +) gives rise to the further problem of surface charge regulation [22, 27-30], with a concomitant appearance of a socalled "secondary h y d r a t i o n force". Surface localised dipole-dipole correlations give rise to a further force [31, 32], and much of what was confused falls into place. These developments represent a first conceptual step forward on the way to a more complete and necessary stage of

98

Chapter3

understanding. The physical meaning of "ion-binding" so important and specific to many biochemical processes becomes clearer. The next stage in the development of molecular forces relaxes the continuum approximation for an intervening liquid, and revisits theme (ii). At smaller distances the granularity of the liquid begins to show up. Oscillations in surface forces with a periodicity of a molecular size show up to surprisingly large separations [4, 33-35]. Further, surface-induced liquid structure, termed by Derjaguin [36] the "structural component of the disjoining pressure", or by others "solvation" or "hydration forces" play an important role in the scheme of things [37-41]. There are also some extremely important strong new forces between hydrophobic surfaces hitherto unrecognised, now quantified, but with a very different origin [42-45], and some forces between molecules like DNA of extraordinarily long range [3-10]. The hydrophobic force arises from a completely different mechanism from others mentioned. Its origin has to do with gas adsorption, and or fluctuations in the thin water film between hydrophobic interfaces and it is not fully understood. It is of extreme importance and will be discussed later. We now describe these forces in more detail before showing how they can be put to work to prescribe curvature. A brief schematic outline of the evolution of present ideas on molecular forces relevant to self-assembly is given in Appendix 3A.

3.4

Molecular forces in detail

3.4.1 van der Waals forces For two like electrically neutral atoms a distance r apart, quantummechanical perturbation theory gives an attractive energy of interaction where V(r)= _3 hcao o~2(0), where ca0 is the principal absorption frequency and 2 r6 0~(0) the static polarisability of the atoms. This energy, measured with respect to a zero at infinite separation, is called the "van der Waals" or "dispersion" energy. (The van der Waals interaction energy is the change in self-energy of the two atoms as a function of their separation [3-10].) At very large distances r>>c/OOo, where c is the velocity of light, this energy goes over to the weaker (retarded) form V(r)~B/rT). These results are obtained by treating the atoms as if they have zero size. The form of the interaction due to induced dipoledipole correlations is the leading term in a power series in 1/r 6 and higher order terms (induced d i p o l e - q u a d r u p o l e interactions, q u a d r u p o l e quadrupole ....) contribute at smaller distances. At distances of the order of an atomic size, where the electron clouds of the atoms begin to overlap, the expansion breaks down, and a different form obtains. Solution of that problem is the province of quantum chemistry. For species that do not form a 'chemical bond', the problem is avoided by invoking the approximation of a

van der Waals forces

99

phenomenological, sharply repulsive, soft core or hard core, with the potential V(r) arbitrarily set to infinity for distances less than the atomic size. If both atoms have a permanent dipole m o m e n t ~t, the thermally averaged rotating

dipole-dipole

e n e r g y is V(r)Keeson = -

correlation

2/~4 9 In 3kT r 6 condensed media, the form for the interaction energy is quite wrong, as the interactions are strictly non-additive and must be dealt with differently as below. If one atom has a p e r m a n e n t dipole moment and the other is polarisable, the dipole-induced dipole (Debye) correlation energy has a similar form, again proportional to 1 / r 6. Three- and higher-body interaction energies [3-10] are much more complicated. Then if only two-body forces act between the atoms of an assembly, the energy of formation at zero temperature is simply

V(rij)--

1 Jri- r]6

where i,j run over all lattice sites of, say, a crystal. If such an idealised material is split, and the two halves separated by an infinite distance, thus creating surface area (cf. Fig. 3.3) then the difference in energy between the two states is twice the surface energy.

000000 000000 000000 000000 000000 000000 -I~I~I~

0000 0000 0000! 0000 0000

0000 0000 0000 0000 0000

--~

Figure 3.3: Surface energy related to interaction energy

If only two-body dispersion forces operate, a rough estimate of the surface energy can be found by adding up the energies of interaction on contact. The energy of interaction per unit area at a distance l has the form E=

A 12tc [2

(3.3)

where A is called the Hamaker constant, and the factor 12~ survives as a convention for historical reasons of no importance. At "contact" d, a separation of the order of a molecular diameter,

lOO

Chapter 3

2Es =

A

12v~d2

(3.4)

where E s is the surface energy per unit area. If we consider an atom interacting with a half space, the energy of interaction is proportional t o A / l 4.

The Hamaker constant is obtained by elementary integration and is related to the molecular polarisability through A - 372 ~'r p20~2(0) , p~_ number density of atoms 4 For hydrocarbons, a typical ultraviolet ionisation potential is h~a= l016 radians/sec and with a choice of 2~ for the size of the CH 2 group one obtains E s ~ 20 d y n e s / c m which is about what is obtained. The corresponding absorption energy depends on the inverse fourth power of the cut-off distance, d. If we deal with the absorption of a molecule to the surface of a crystalline inorganic material, these phenomenological estimates vary enormously depending on precisely where the adsorbed molecule sits; If the adsorbed molecule can nestle into and between the surface atoms, the physisorption energy is as large as a chemical bond. As we have already seen for the example of adsorption in zeolites, the distinction between a bond and physical interactions becomes tenuous.

3.4.2

Lifshitz forces

The idealised calculations described above presume that interacting surfaces remain unperturbed as they come into contact, i.e. that surface energies are infinite (or for finite sized atoms very large) as compared with interaction energies. If a surface is in contact with a liquid, or two surfaces are separated by a liquid, we immediately run into a conceptual difficulty. Thus if the liquid is, say, water, at a single solid surface we might expect the surface to order proximal molecules, which evidently will have a different arrangement of dipole moment, hydrogen bonding, density, etc. from that of bulk water. The range and nature of this ordering depends on both the solid (which induces changes in liquid structure) and the liquid. We return to this central question later. For the moment, suppose that the intervening liquid has bulk properties up to the surface. The solution to this interaction problem, valid at "large" distances, was obtained by Lifshitz [3-10]. Instead of adding two-body atomic forces as above, the method used is as follows. Consider the totality of electromagnetic fluctuations due to all the atoms and molecules of the interacting materials. These fluctuations satisfy Maxwell's equations with boundary conditions set by the dielectric properties (as a function of frequency) of the materials. Fourier analysis of these equations and their

Lifshitz forces

101

solution leads to a set of allowed normal modes of frequencies co and wave vector k. Then assignment of a harmonic oscillator free energy

Fto= l kT ln{2sinh~k T} 2 to each mode, and addition of these free energies gives an expression that includes the retardation of fluctuation correlation contributions due to the finite velocity of light. The expression for the resulting free energy of interaction per unit area is extremely complicated and analysed completely in

[3]. It is, for media 1 and 3 interacting across a medium, m oo

F([,T)= k T

~t~n,

8if, ]2 n=O where

R

=

l" R

R

with

A mR=Slem - pc1 Ama 1= sl - p sl C.m + pel " sl + p s1=~/p2-1+el/Sm

,

e=e(i~n)

and the prime on the summation indicates that the term in n=O to be taken with a factor 1/2, and the sum is over imaginary frequencies ton = i~n,

~,~ = 2~rn kT/h, and c is the velocity of light. This formidable formula is not so forbidding as it seems, and has been analysed in great detail, reduced to tractable forms that make sense in [3]. Pairwise summation emerges as a very special case, valid for gases only, and even then is a bad approximation to the full m a n y - b o d y interaction. Calculation of the interaction free energy for particular cases is not difficult [310] and requires a k n o w l e d g e of m e a s u r e d dielectric properties and adsorption frequencies in the infrared, visible and ultraviolet, all known, in principle.

lO2

Chapter3

If the dielectric properties are known, the expression represents a complete solution to the interaction problem, provided that a liquid separating the two interacting solid surfaces is itself not perturbed in structure by the surfaces. These forces can be calculated and measured. We make the following remarks. The assumption of two-body forces is completely misleading and qualitatively erroneous for condensed media interactions. The sum in the full expression above includes contributions from all frequencies. A contribution to the sum from a frequency ran is retarded, in effect behaves as the non-retarded component of the total interaction-frequency multiplied by an exponential factor ex~-~n/~s), ~s = c/2! C~- which has the nature of a cut-off factor. Different frequencies lock in at different distances. The zero frequency, temperature dependent contributions are extremely important in oil-water or biological systems, about half or more of the total interacting free energy. The behaviour of the interaction free energy depends strongly on geometry. Sometimes, the geometric factor can be evaluated by taking a "Hamaker function" [3-10] calculated by Lifshitz theory, multiplied by an appropriate geometric distance factor evaluated from pairwise summation. (This assumption fails for the temperature dependent contribution, i.e. in water, the most interesting liquid!) Loosely speaking, the theory shows that two bodies (or macromolecules) sense and feel their different frequencies of inter-oscillations at different distances. The zero frequency (microwave, non-retarded) classical or non q u a n t u m mechanical coupled p e r m a n e n t dipole fluctuations ~ u i d e orientation and interactions at large distances. At distances -- 200-500A, the infra-red locks in and at distances below about 50A, the strong ultraviolet correlations take over. At smaller distances still chemistry takes over, as the far ultraviolet correlations lock in. At one level the very existence of such an aesthetically unpleasing formula for the attractive interaction energy between surfaces is discouraging. It describes the interaction problem in the simplest possible idealised case, and the reader is advised that matters are going to get worse as we deal with more realistic situations. There is not much one can do about it. At another level it is encouraging, because it indicates that nature has available to it some very subtle forces to steer self-assembly. The simpler intuition concerning molecular forces in inorganic materials that are the preoccupation of colloid scientists does not map over to biological systems. (The strength, range, and subtlety of these forces in biological materials are more delicate, because of their different dielectric properties.) For unlike materials interacting across a liquid, the forces can be repulsive as well as attractive, depending on material dielectric properties and distance, a situation which also obtains for like anisotropic media, depending on orientation of dielectric axes [3-10].

Lifshitz forces

103

For inorganic materials in vacuum or in a liquid such as water, the van der Waals interactions are dominated by ultraviolet contributions. Typically, below the retardation cut-off we have A--10-20 Joules. For organic materials in water, the interaction is qualitatively different. All frequencies contribute to the strength of the interaction, about a factor of 10 or more lower in magnitude than for inorganic materials. The temperature dependent (n = 0 term of the sum) dominates in biological or oil-water systems, being at least half the total interaction. This particular contribution has the form F,,=o (/,T) = kT I.el(0)- e.m(0) 'e2(0) - em(0) 16~ ~el(0) + e.m(0) ~2(0) - era(0) where the dielectric constants are evaluated at zero frequency (static values). It is a cooperative many-body dipole-dipole correlation force that reduces to a sum of individual dipole-dipole forces only for dilute media (gases).

Salt effects

A further complication appears when the intervening liquid contains salt. The temperature dependent contribution above is then modified [3-10] and at "large" distances is damped by a factor exp(-2~/) where, for a 1:1 electrolyte ~= ~ ek T

e= era(O)

and w is the inverse of the Debye length A D of the solution. We shall return to this fundamental observation later. Salt solutions can also change the form of the long range interaction dramatically [3-10] within coiled polymers or macromolecules.

3.4.3

Double-layerforces

We turn now to the other side of the colloidal particle interaction problem idealised to the case of two half spaces separated by salt water [3-10, 22-24]. Typically such particles will contain ionisable groups at their surfaces, so that the surfaces are charged. Imagine that the water, as before, retains its bulk properties up to the surface of the half spaces. The charged surfaces create an inhomogeneous profile of cationic and anionic density. For an isolated surface at the simplest level of approximation and schematically only, this distribution follows from the equation: V. E = V 2 0(x) =

4~p(x)

104

Chapter 3

where E is the electric field, ~ the potential, p the charge density and the boundary conditions are elE 1 -e2E 2 = 4~cr, where cr is the surface charge density, and el and e2 are the static dielectric constants of adjoining media. The charge density profile p(x) is described by the Boltzmann distribution p(x)+ = p o e

-e~

-e~p

where e denotes unit ionic charge and Po is the salt concentration in the bathing medium far from the surface. Subject to the surface charge (or potential) boundary condition assumed, the potential follows from the non-linear Poisson Boltzmann equation:

V2tp= (87rP0)sinn[k-~-]

9

(The ions of the (symmetric) electrolyte are assumed to be point ions.) It is the overlap of the profiles that gives rise to a repulsive "osmotic pressure" which gives a form: Force ~ f(o)e -~, where f(6) is a function of surface charge (or potential) and ~ is the inverse Debye length, defined above. Combination of the attractive van der Waals potential --- l//z and the repulsive double-layer potential (~ e'r~, gives rise to the famous DLVO theory of colloid stability. Depending on salt, the net potential can be such as to induce flocculation directly, pose a barrier to flocculation, or lead to a stable suspension. The theory is deficient in many respects, e.g. in any real case the degree of surface charge dissociation will be affected by the presence of a neighbouring surface. This can be taken into account in a self-consistent manner [22, 27]. A further inconsistency is that the treatment of temperature-dependent

attractive fluctuation forces given by the extension of Lifshitz theory outlined above ignores the inhomogeneous profile of ionic charge distribution at the (charged) surface, whereas the double-layer force depends on that profile. In a more complete theory these must be treated at the same level, and the inhomogeneous profile is essential in treating fluctuation forces correctly. A full theoretical description must allow ionic density fluctuations about the inhomogeneous profile, a difficult problem of statistical mechanics only recently solved [3-10, 23-29]. When this is done, it turns out that no distinction can be made between the van der Waals and double-layer forces. Indeed, in solutions of di- or trivalent salts at sufficiently high concentration the (formerly repulsive) forces have a deep attractive well due to ionic correlations [23-25].

Double-layer forces

105

These forces have been measured. A further consequence of the new theories is that inferences of surface charge, "ion-binding" to surfaces as adduced from force measurements a n d / o r electrophoretic mobility studies or NMR which involve the crude Poisson-Boltzmann theory are wrong. If the simple analytic Poisson-Boltzmann theory is used to interpret direct force measurements, experimental results can be forced into that analytic form, but the real binding constants are quite different.

3.5

A gallimaufry offorces

Our brief outline so far calls in the full a p p a r a t u s of q u a n t u m electrodynamics, q u a n t u m and classical statistical mechanics, applied to inhomogeneous liquids at interfaces. Generally "agreement" between theory and experiment has been confirmed at "large" distances [3-10], and, even at small distances [23-29] - provided a hydrated size for the ions of electrolyte solutions is allowed. But the "agreement" between theory and experiment claimed in even the most recent literature is often more apparent than real. An open question is: at what distances does the molecular granularity of matter, like a liquid, show up? The answer depends on the surfaces, the materials, their interaction strength, and the liquid. It is not too surprising then that its resolution presents difficulties. Nonetheless some real progress has been made.

3.5.1 Forces due to liquid structure Such a force has been called the "structural component of the disjoining pressure" by Derjaguin [36]. These forces have usually been considered as additional to those deduced from continuum approximations. (In reality the continuum theories should be considered as asymptotic approximations to the forces, which hold only at "large" distances.) Two kinds of structural forces can be considered. If the surface does not perturb bulk liquid structure, we should still expect continuum theories to break down at some point. For example, between molecularly smooth, rigid mica surfaces separated by simple liquids, hard sphere van der Waals liquids like octamethyltetrasiloxane, (a r o u g h l y spherical molecule of d i a m e t e r ~8A),forces with large oscillations, of period equal to the molecular size, have been measured up to surprisingly large distances. Beyond about 8 molecular diameters, the measured force merges into the van der Waals-Lifshitz forces. The same occurs for hard-sphere-dipole molecular liquids, which are not hydrogen-bonded like propylene carbonate [33-35]. For soft surfaces, or surfaces rough on a molecular scale (like lipid bilayers), the oscillations tend to be smoothed and can be ignored.

lO6

Chapter3

3.5.2 Surface-induced liquid structure The second type, of more importance, is a force due to surface-induced liquid structure. At a conceptual level we have already encountered such a force in the double-layer. There the electrolyte can be regarded as the "liquid", with the suspending water a continuum background that affects the problem only through its dielectric constant. The bulk "liquid" electrolyte has a uniform distribution of cations and anions. In the presence of the charged surface, that uniform distribution changes. The overlap in profile of the surface induced liquid structure causes the force. In general, a surface will always induce re-ordering (e.g. of hydrogen bonds) resulting in dipole orientations which extend over a number of layers, as for the double-layer interactions. These are called "hydration", or "solvation forces" [37-41] and can dominate interactions in precisely the distance regimes of most interest, below about 50tL Between lipid bilayers with large hydrated zwitterionic head-groups, the forces become extraordinarily large and repulsive. These forces ought properly to be called dehydration forces since work must be done to remove proximal bound water from the head-groups. They dominate electrostatic forces below 30,/L

3.5.3 Hydration forces in

phospholipids

A vast amount of pioneering work has been devoted to measurements of forces between lipid bilayers [37-39]. Quantification of these forces represented the first real advance in understanding the subtleties of biological systems, which do not fit really into the scheme of things provided by classical colloid and surface chemistry. The repulsive forces are very large; they dominate double-layer and van der Waals interactions below distances of about 30./~. The forces prevent fusion of bilayers, an eminently satisfactory result, as otherwise cells would not exist. There is no theory of such forces beyond the simple statement which goes back to Poisson 150 years ago, and was quantified by J. Clerk Maxwell in 1876 using mean field theory [41] - that the overlap of profiles of surface-induced liquid structure gives rise to the force. The theory and exponential decay length of approximately 3/~, (the size of a water molecule), were predicted by Maxwell, rediscovered by Marcelja 100 years later [40], following earlier measurements by Parsegian and co-workers [37-39] in a set of pioneering investigations. Oscillations with a periodicity of 3/~, are smoothed because the surfaces provided by phospholipid head-groups are bumpy, at least on a molecular scale. -

For multilamellar liposomes, the hydration forces should balance the predicted longer range attractive van der Waals forces, to give an equilibrium lamellar phase spacing of about 30,/~ in water. They do not, but once surface dipole-dipole correlations are taken into account [31, 32], theory and experiment do agree.

Other forces

107

3.5.4 Surface dipole correlations Recognition that a further correlation contribution exists, due to twodimensional p e r m a n e n t dipole (zwitterionic) head-group fluctuations confined to a surface accounts for the discrepancy in equilibrium lamellar spacings [31, 32]. Elegant force measurements have been made between adsorbed monolayers of the protein cytochrome-c, and insulin on mica, immersed in water [26]. Hydration forces here play no role. If the full armoury of theoretical predictions is invoked, the complicated force curves measured all seem to fall into place. It is possible, indeed probable that with real biological membranes that contain up to 50% proteins, the hydration forces that prevent fusion of pure phospholipid membranes do not always operate. Agreement between theory and experiment is often illusory. Consider the double-layer forces measured between mica or silica in an electrolyte, under conditions where hydration forces are absent. The curves certainly decay with the predicted exponential forces (e-Kl) of the double-layer theory, at least for 1:1 or 2:2 electrolytes. But agreement with the basic Poisson-Boltzmann theory has been achieved by fitting theory with an assumed surface charge density, from which an "ion-binding" constant [46] can be inferred. The full theory [22-30] which includes ionic correlations has the form of the PoissonBoltzmann theory, it will give a different degree of binding. It is impossible to infer ion-binding constants which have any meaning, important to considerations of specific ion effects in biology, unless the full theory is used. Usually this is argued away on the grounds that if the simpler (wrong) theory gives a consistent set of parameters for a series of ions, e.g. Li +, Na +, K*, that should be sufficient. This is not so, and even for those situations where classical theory appears to work well there are problems. A further example can be seen in force measurements between ionic bilayers. The double-chained cationic surfactants didodecyl, dehexadecyl, dioctadecyl dimethyl ammonium, bromides or acetate, can be adsorbed, or deposited by Langmuir trough techniques onto molecularly smooth mica. For bromide as counter-ion, with or without added NaBr, the force curves fit nicely to Poisson-Boltzmann theory provided one postulates that 80% of the headgroups are neutralised by bound Br- [46]. This agrees with the binding deduced from NMR and other studies on miceUes with the same head-group. There is a fitted phenomenological parameter, the assumed surface charge necessary to secure agreement between theory and experiment. On the other hand, with acetate as counter-ion, the forces are an order of magnitude larger, the fit to theory is perfect, with no free parameters [46]. There are no bound counter-ions. The Poisson-Boltzmann theory here can be shown to provide an upper bound to the magnitude of the double-layer interaction. If the more refined theory [23-25] is used, the predicted result is somewhat less than the measured curve, less still if there exists any real ionbinding. (How much of a difference exists depends on the presumed hydrated

lO8

Chapter3

ion size invoked, i.e. again involving water structure, which is not taken account of in the theory. With large counter-ions like acetate, the deviations from Poisson-Boltzmann theory are less than those with smaller counterions like bromide.) The differences must be attributed to additional hydration forces, hidden if one insists on a simple Poisson-Boltzmann description [47]. Specific counter-ion effects are critical to biological function, in determining forces between individual sub-units of macromolecules and in the consequent shapes they take up. How much these effects can be attributed to physics and how much to specific chemistry can only be revealed by a reanalysis of all data in the light of the new theories of molecular forces. Until that reanalysis is done, present experimental inferences on binding surface potential and charge remain phenomenological curve fitting. A very large amount of sophisticated NMR studies and other work on specific ion-binding to proteins and membranes has been interpreted in terms of a phenomenological ion-binding model equivalent to the ionbinding model for micellisation discussed below. The model "works" and is equivalent to the Poisson-Boltzmann theory of the double-layer as we have already mentioned, but only in the special limit of strong binding [48]! Why then give up a good simple theory for a more sophisticated one? The simple theory based on the Poisson-Boltzmann description does provide some useful qualitative predictability as we shall see. The ever-present proviso is that it must not be taken too far. It is an old story, and ultimately comes back to the fact that we have no real molecular theory of water. Langmuir's view, that water is itself a giant connected molecule is probably closer to reality than a molecular picture.

3.5.5 Secondary hydration forces and

ion-binding

A different kind of hydration force occurs between hard surfaces bearing dissociable groups, like mica in water. Competition between two competing ionic species of different hydration - (e.g. H +, Na +) for the surface (and for the hydronium ion sub-surface sites) can lead to equally large, indeed ubiquitous "secondary" hydration forces. At low salt concentrations, the classical DLVO theory appears to work well. But at higher salt content, the law of mass action demands that the less easily bound Na § ions must win out over H § in competition for the surface. Work must be done to dehydrate these ions as two surfaces approach. The onset of secondary hydration forces occurs for a given pH at a critical salt concentration, and these forces appeared to dominate double-layer forces, again below about 30-40/~, separation [27, 28]. The overall effective decay length of secondary hydration forces is 10]~ (univalent), 20]~ (di-valent), 30A (tri-valent), as opposed to lipids, where the decay length is 3~. Later work has shown that while these secondary hydration forces exist, their range is probably the same as that for lipid interactions, i.e. characteristic of the size of a water molecule. The apparent long-range of secondary ionic hydration forces is an artefact of earlier

Other forces

109

theoretical analysis [29, 30]. (This analysis assigns an area per ionic adsorption site in the lateral direction, and, inconsistently, treats the completing ionic species as of zero extent in the perpendicular direction. But the "apparent" phenomenological force remains real.)

3.5.6 Range of the double-layer force and implications Just to complicate matters further, we remark on the range of double-layer forces. All theories take it as axiomatic that the force between surfaces or between charged molecules should decay as exp(-Ic/) where K"1 is the Debye length defined by

~o = (4~.(vlz12+ v2z22)e2c) kT where c is the concentration of a salt, CZ] A v2z2(e.g. Ca 2+ C~). In fact, it can be shown, and has been confirmed experimentally, that the actual decay length is given by

Ir

-l

A D = K0

( 1 + 71n (3) 4c (vl z~ + v2 z ~ + O(c2/31n (c)) + ... ) (24)~r2

(vl zl + v2 z 2)

where c is electrolyte concentration in moles per litre [26]. This result affects profoundly our intuition on double-layer forces, especially in mixed electrolytes [26]. (It has been confirmed by direct m e a s u r e m e n t of the forces acting between molecularly smooth mica surfaces in solutions of the p r o t e i n c y t o c h r o m e - c [48]. This protein has a charge of 12 + or 8 + d e p e n d i n g on pH, so the effects show up very strongly. The experiments have an interesting consequence. In many applications a trace a m o u n t of a highly charged ion like V s+, Cr 3~ added to a 1:1 salt solution like NaC1 can have a quite dramatic effect on the stability of a colloidal dispersion. The reason is due to the surface concentration, n s, of a species of charge z near a charged surface, which is determined by the formula ns = ns exp(+ze~0/kT) where n B is the bulk solution concentration far from the surface, ~ is the surface potential and z the charge of the counter-ion species. The exponential factor means that near the surface such a system becomes more like a z:l electrolyte. Hence the double-layer force b e t w e e n two such surfaces, determined by the overlap of the surface-induced ionic profiles, resembles a z:l electrolyte more than a 1:1, and can be of much shorter range than expected. The same effect occurs in many biological situations, where a m e d i u m between two interacting membranes often contains a small amount

110

Chapter3

of highly charged species of proteins or RNA that affect interactions strongly by this mechanism. The Boltzmann factor, depending on charge, also explains why it is that trace amounts of charges Se4+,V 5+, etc., are of such importance in agriculture and to animal metabolism. At the surface of a (charged) membrane, the concentration of such ions - which determine the local ionic concentrations, and hence the conformation of a membrane bound protein involved in recognition is orders of magnitude higher, than in the bulk solution far from the membrane. Biological activity will be critically dependent on trace element concentration. If the solution between two surfaces contains surfactants that form highly charged micelles, a different effect occurs. Theory, confirmed by measurement [49-51], shows that the Debye length is then to be calculated as if the micelles and their "bound" counter-ions are simply ignored. The doublelayer force is here much longer ranged than it would be on the basis of standard theory. These forces are sometimes called "depletion" forces. The mechanism is the same as that for the oscillatory forces discussed in section 3.5.1.

3.5.7 Hydrophobic interactions The hydrophobic interaction between oil-like surfaces or between oil-like molecules in water is the driving force for self-assembly. In biological systems it is the most important force of all. While the word "hydrophobic" is ubiquitous in the chemical and biological literature, its use disguises ignorance. That ignorance occurs because we have no real quantitative knowledge of water structure and the influence of solutes, be they hydrocarbons, gases or ions, on water structure, let alone at surfaces. In the brief account below we summarise what we knew of hydrophobic interactions [42-45] in the context of this overview of complex molecular forces until 1993, so as not to deviate too far from our main theme. Recent work to be discussed in Appendix 3C, has opened up a whole new perspective. The consequences of that work affect our understanding in profound ways not yet fully comprehended. The hydrophobic interaction between two small molecules (e.g. methane in water), is generally considered to be short ranged, virtually a contact force. With systems of large surface area, as opposed to small molecules, a completely different situation occurs. Measurements of forces between modified hydrophobic mica, silica, or between surfaces on which monolayers of surfactants have been deposited show that the hydrophobic interaction is extremely long ranged [4245]. These forces are 10 to 100 times larger than any conceivable van der Waals force. The contact free energy (adhesive energy) is what one expects from hydrocarbon-water interfacial energies. If fully hydrophobic surfaces (contact angle > 90") brought to contact are pulled apart, spontaneous cavitation occurs. A hydrophobic surface prefers contact with a

Other forces

111

vacuum or dissolved gas and water vapour over contact with water, and a thin film of water between a hydrophobic surface is metastable. With fluorocarbon surfaces, cavitation occurs before contact. Although well documented, and repeatable, the mechanism of this hydrophobic interaction has not been fully explained at this time. Addition of salt changes the range of the forces, and this provides a clue to the mechanism. Electrolyte correlation effects can be ruled out, as any force would depend exponentially on the Debye decay length, and this is not observed. The most recent experiments suggest that the phenomenon, along with observed cavitation effects, is related to specific ion adsorption and nucleation of gas at the hydrophobic surfaces, which induce fluctuation states in the metastable water film. The fluctuations, and hence the consequent force, are dependent on specific ions and the dampening of the forces with salt has the same kind of critical switching property present in gas bubble-bubble interactions discussed in Appendix 3C. This force, acting between biological macromolecules (with varying degrees of hydrophobicity over their surfaces, such as proteins, and hence varying gas and ion adsorption) is of extreme importance in biology. Changes in the amounts of dissolved gases like carbon dioxide and oxygen materially affect these processes and the interactions described in detail in the Appendix.

3.5.8 Non-ionic surfactant forces The forces [52] between two monolayers of non-ionic polyoxyethylene (PEO) head-groups deposited on hydrophobed mica exhibit peculiar properties. At temperatures below the so-called cloud point, the forces are repulsive. Above the cloud point they become attractive. Simultaneously with the onset of attractive forces, the measured head-group area decreases, in agreement with NMR measurements on micelles formed by these surfactants above and below the cloud point. (The head-groups give up two molecules of hydration per PEO group.) Such changes at the molecular level between head-group interactions produce large changes in the self-organisational properties of solutions of these surfactants. Therefore it ought not to be too surprising to see, as we do, dramatically sensitive changes in biological activity of enzymes and other macromolecules over small temperature ranges. The fact that such effects have been made explicit and quantified underlines the real dangers of modelling structure of macromolecules with universal potential parameters currently in vogue [53].

3.5.9 Forces of thermodynamic origin Two further forces operate in any colloidal suspension or self-organised system. The first is due to Onsager and to Langmuir [54-56] who explained colloidal stability of clays and cylindrical particles in terms of purely repulsive

112

Chapter 3

forces, due to either double-layer effects, or hard rod or hard plate repulsion. In a many-particle system, an ordered phase will always be in equilibrium with the disordered phase of plates or cylinders. The equilibrium between the two coexisting phases is set by the volume fractions of each. The "force" that causes ordering is entropic. Recent work of Lekkerkerker and colleagues has explored such effects extensively.

3.5.10 The Helfrich force Another force [57, 58] occurs in a multilayered system, like a swollen lamellar phase of surfactant bilayers or phospholipid vesicles. Shape fluctuations in the bilayers can give rise to steric effects that are supposed to stabilise such systems where the van der Waals and double-layer forces are very weak, as they often are. The magnitude of such fluctuations depends on the "stiffness" of the bilayer. The status of these forces is the subject of an active debate and unclear. This potential force occurs in microstructured fluids like microemulsions, in cubic phases, in vesicle suspensions and in lamellar phases, anywhere where an elastic or fluid boundary exists. Real spontaneous fluctuations in curvature exist, and in liposomes they can be visualised in video-enhanced microscopy [59]. Such membrane fluctuations have been invoked as a mechanism to account for the existence of oil" or water-swollen lamellar phases. Depending on the natural mean curvature of the monolayers bounding an oil region - set by a mixture of surfactant and alcohol at zero these swollen periodic phases can have oil regions up to 5000A thick! With large fluctuations the monolayers are supposed to be stabilised by steric hindrance. Such fluctuations and consequent steric hindrance play some role in these systems and in a complete theory of microemulsion formation. The status of such forces is an open and unclear situation. They exist, but cannot be too specific in nature.

3.5.11 Forces of very long range A further curious force operates when charged polymeric molecules are in a stretched out state, or indeed with any thin cylindrical structure. Fluctuations of the ions in the associated double-layer can lead to forces of extremely long range. Their existence is not in dispute, and they probably play a large role in the organisation of DNA in cell biology [3-10]. A schematic representation of the evolution of concepts of molecular forces is given in Appendices 3A,3B.

Other forces

113

3.5.12 Summary From a historical perspective, our understanding of molecular forces derives from colloid science, concerned with interactions between solid particles. The opposing forces between particles, one attractive, the other repulsive, which form the core of the DLVO theory which has been the mainstay of that subject, sit in uneasy juxtaposition. The attractive van der Waals forces were derived assuming the liquid between two interacting surfaces is a bulk continuum medium. On the other hand, if we regard an electrolyte as the operative liquid, the double-layer force is the first example of a structural force and surface-mediated interaction. We have seen h o w m o d e r n refinements have changed that simple view, especially in oil-water and biological systems of low Hamaker constant; and how in the new theories the conceptual distinction made between the opposing forces disappears. At one level, the litany above presents us with an appalling situation. The reader will observe that except for hydration forces, for which there is no real theory, we have treated water as if it is a continuum, and the ions of electrolytes as hydrated hard spheres. That approximation is not always valid [60], and one simply has to be aware of the limitations of present physical theories. Again the hydrophobic "surfaces", as discussed in section 3.5.7, the forces, and their salt dependence, depend critically on the amount of dissolved gas, and nature of the charged, or polar species. Behind that litany, at even the most elementary model approximation sits the full apparatus and complexity of quantum electrodynamics, statistical mechanics of inhomogeneous fluids which lie at the foundations of modern physics. That is comforting in a way at least in that the direct traces of a unifying thread between physics through chemistry to biochemistry can be discerned. In another way it is disconcerting. The biologist and biochemist have enough concerns of their own not to be bothered with the subtleties of physical chemistry. It is evident that one can go a long way towards understanding phenomena at the exquisitely refined level of molecular biology knowing nothing of molecular forces. Indeed too facile application of the older simpler theories of colloid and surface science fails in complex situations. Nonetheless, awareness of the new forces does enable us to make sense of a variety of issues not comprehensible without that awareness, and provides mechanisms for the extensive geometric rearrangements needed in biology.

3.6

Self-organisation in surfactant solutions: the Euclidean desert

We now move away from rigid surfaces and their interactions, and allow the interactions to dictate the formation of bodies and their surface shapes. In general the role of molecular forces in self-assembly is extremely subtle. This can be seen from the following simple examples. The simplest example

ll4

Chapter3

is a polyelectrolyte. At very low ionic strength the Debye length is large and the screened electrostatic repulsion between individual charged units along the backbone is very large. The polyelectrolyte takes up an extended linear conformation, resulting in a stiff viscous solution. With addition of salt the segment-segment interaction is reduced and the polyelectrolyte collapses to a random coiled state, with the solution becoming fully flowing. Another example is the 0-point of polymers in solvents, characterised by an abrupt change to an effectively non-interacting polymer at a particular temperature, well explained by calculations involving the dispersion self-energy and for polyelectrolytes, using the Poisson-Boltzmann equation [61]. More subtle are specific ion effects on conformation. The repulsive forces between bilayers of quaternary ammonium surfactants adsorbed onto mica differ markedly depending on counter-ion, as we have seen. With Br-as counter-ion the forces are weak, with acetate and a range of carboxylates, Fand OH- the forces are strong. In the language of the Poisson-Boltzmann theory about 80% Br- is "bound", no acetate is "bound". In a totally insoluble (hydrophobic) material like glycosamine (e.g. chitosan, the essential ingredient of crustacean shells and the most common polymer on earth), the chloride counter-ion is strongly bound. After immersion in weak acetic acid, the acetate ions exchange with chloride ions; the electrostatic interactions between polymer units increase, the polyelectrolyte dissolves easily to form a clear solution. Knowledge of the counter-ion effect is sufficient to understand this phenomenon, but further effects remain unclear: addition of further acetic acid ought to screen interactions and collapse the polymer again. It does not, so that extra interactions such as hydration effects and dissolved gas are involved (cf Appendix 3C). (Addition of NaC1 will collapse the polymer.) The acute sensitivity of many biological phenomena to temperature can in some cases be correlated with forces. As we have already discussed, the cloud point behaviour of polyoxyethylene surfactants can be understood in terms of forces. If we take a cellulose acetate polymer, ethoxylate some of the groups, then dissolve the polymer in water and add an ionic single-chained surfactant above the critical micelle concentration an extraordinary phenomenon is observed. Hydrophobic portions of the polymer act as "nucleation sites" for micelles, leading to a complex assembly, resulting in a string of charged beads. Below the cloud point both the charged micelles along the string and repulsive interactions between ethoxylated groups conspire to give a clear freely flowing solution. Above the cloud point, the solution becomes immediately extremely rigid. (The effect is reversible.) Evidently, the repulsive and (now) attractive forces are in tension, with the ethoxylated groups coiling up. This kind of behaviour is a simple system which can be understood in terms of the molecular forces operating. It allows us to understand why and how it is that, e.g. enzymes can be so sensitive to small variations in temperature. The present section gives a brief overview of ideas and the physical notions behind self-assembly of surfactant-water systems.

Self-assembly

115

A schematic listing of ideas in self-assembly is also indicated in Appendix 3B. These theoretical developments were until recently confined to spheres, cylinders and planes, a Euclidean desert. New developments that allow access to the full variety of shapes used by nature are the subject of other chapters. The word "surfactant" (sometimes "tenside") is a mnemonic for surfaceactive-agent. In a less limited sense we should take it to mean any molecule, one part of which is hydrophobic (i.e. water-hating) with one or more flexible hydrocarbon (or other oil soluble) moiety. This part of the molecule is chemically bound to a hydrophilic (water-loving) head-group, e.g. a salt, or other predominantly water-soluble group, like a polyoxyethylene. This dual or schizophrenic character is responsible for the self organisational capacity. Depending on a loosely defined hydrophilic-lipophilic balance ("HLB", lipophilic - f a t - or oil-loving) the solubility in water of m o n o m e r i c surfactants can range from a maximum of typically 10-2M for ionic singlechained chemical soaps, to the extremely low value of 10-16 M for the doublechained phospholipids that form biological membranes. Beyond the solubility limit, the surfactant molecules self-assemble into aggregates as follows. At first, as surfactant is added to water, the free energy per molecule is dictated by entropy, F = E - T S , - T S = -kTln X1 --~ + ~, X1 = mole fraction. With increasing concentration, the entropy decreases, and the solution can minimise its free energy if some of the molecules migrate to the air (or oil)water interface to form a monolayer. (The hydrophobic tails prefer association with each other or with any hydrophobic interface due to the "hydrophobic interaction".) The surface tension drops continuously until a complete monolayer is formed. Thereafter, at the so-called critical micelle concentration (cmc) the surface tension curve (and the electric conductivity for ionic surfactants) exhibits a sharp break, indicating the onset of selfassembly. The interface no longer being available, the molecules satisfy their dual requirements, and that of entropy, protected as far as possible from water by the hydrophilic head-groups by arranging their tails together into monodisperse micelles, i.e. spherical or globular aggregates. Double-chained surfactants cannot usually pack into spheres, and are forced immediately into closed bilayers which then associate under the influence of attractive van der Waals forces to form e.g. multi-walled vesicles (liposomes), or more complicated phases. (For single-chained surfactants typical micellar aggregation numbers are of the order of 25-50, set by the minimum geometric size that packs the molecules favourably.) If the chains are sufficiently flexible, and repulsive (hydration or other) forces between aggregates sufficiently large and the chains sufficiently hydrophobic, as for double-chained surfactants like phospholipids, single-walled vesicles can occur. The process of self-assembly is a dynamic one. With some single-chained ionic surfactants, (e.g. sodium dodecyl sulphate), the residence time of a monomer in a micelle is of the order of 10-6 sec, and the lifetime of the micelle as a whole is of the order of a millisecond. At the other extreme, for highly insoluble phospholipids that form the bulk of bilayer membranes, the

116

Chapter 3

time taken even for flip-flop of a molecule from one side of a bilayer to the other can vary from seconds to months, depending on chain length. That stability is necessary to preserve the integrity of biological membranes. Alcohols, at least those of short chain length, have moieties that are neither extremely hydrophilic, nor extremely hydrophobic, and partition in a complex way between oil and water, or between the oil-like interior of a micelle or membrane, and water. This property, together with their small head-group area, enables them to be used with surfactant-water, or surfactantoil-water systems to produce a rich diversity of microstructured solutions through changing curvature of the interface, as does cholesterol, for the same reasons. The self-assembly of biological aggregates is further complicated by the presence of amphiphilic proteins.

Figure 3.4: Schematic representation of surfactant-water phase diagrams, for non-ionic surfactants (left) and ionic surfactants (right).

Typical surfactant-water-phase diagrams are shown in Fig. 3.4 for singlechained ionic, and non-ionic surfactants respectively. Below a "Krafft" temperature characteristic of each surfactant, the chains are crystalline and the surfactant precipitates as a solid. Increased surfactant concentration (Fig. 3.4) results in sharp phase boundaries between micellar rod-shaped (hexagonal), bilayer (lamellar) and reversed hexagonal and reversed micellar phases. (The "cubic" phases, bicontinuous, will be ignored in this section and dealt with in Chapters 4, 5 and 7.) 3.6.1 Aggregate structure in the Euclidean desert

If we confine ourselves to simple geometries, the "explanation" of the gross phase behaviour indicated is immediate: given the maximisafion of entropy

Self-assembly

117

that accompanies the formation of the smallest possible aggregates and given that this free energy minimisation accrues to minimal exposure of the hydrocarbon chains, then the implication is that for a given volume and head-group area, micelles of smallest size satisfying these requirements will form first. When these spherical globules can no longer pack (maximum volume fraction 0.74 for a face-centred-cubic array) they have no alternative except to form a hexagonal array of cylinders (maximum volume fraction 0.91), thereafter a lamellar phase, and so on. That is, global packing, volume fractions of the several components, are and must be one requirement that accounts for the features of phase changes that occur with increased concentration. And if inter-aggregate forces, e.g. between ionic micelles, are operating, they might reasonably be accommodated by invoking an "effective" micellar radius that includes the Debye length, the range of the double-layer repulsion between aggregates, as an adjustable parameter. At first sight the multi-molecular, multi-component assemblies of biology occur in dilute solutions where global packing constraints are not a problem. In fact that is not so. Compartmentalisation of components occurs naturally, and within those compartments global packing is a main issue. Even so their transformations in response to biochemical processes require more detailed accounting of the interplay between intra- and inter-molecular forces. A clear and delicate example can be seen for the case of non-ionic surfactants. On increasing temperature the dilute micellar phase shows a sharp cloud point (section 3.5.8, cf. Fig. 3.4). Two factors are involved. At and above this temperature, the surfactant head-group gives up water of hydration. The head-group area is reduced and the surfactants can no longer pack geometrically into spheres. They increase in size to become rod shaped objects. Simultaneously the forces between aggregates change from repulsive to attractive. However weak these long range forces, they will induce a phase separation as long as the micelles are sufficiently long (rod-like). (No phase separation is possible if the attractive forces acted between small spherical micelles- the solution is too dilute for the forces to have such an influence.)

3.6.2

Curvature as the determinant of microstructure

These ideas can be formalised in terms of statistical mechanics to some extent, and an outline of the main ideas is given in the following section. We remark parenthetically that there are profound difficulties confronting the definition of an aggregate. The nature of the hydrophobic free energy of transfer of a hydrocarbon from water to the hydrophobic core of a micelle can be measured, but its temperature dependence is not understood because it depends on water, an u n k n o w n quantity. For the same reasons, solution theory, does not even tell us whether mole fractions or mole volumes are the correct ratios to use to determine entropy. However, provided certain assumptions are allowed [62-65], then simple rules emerge. The rules are: if v is the hydrocarbon chain volume, a the head-group area, and I of an optimal

Chapter 3

118

chain length, a "surfactant parameter" v/al characterises and determines the particular aggregation states which form. This molecular parameter is a function of the curvature at the surfactant-water interface. For a single component system, the volume v is fixed, and the area per head-group can change as solution conditions change intermolecular forces. The rules, in the limited class of Euclidean shapes are: v = 1_ micelles; 1 < v < 1_ rod shaped al 3 3 al 2 micelles: 12__< val < 1 vesicles, (closed single walled bilayers) or lamellar phases, v > 1 reversed phases. Indeed that progression is readily observed by adding al salt to a solution of ionic micelles, the head-groups of which interact predominantly by electrostatic rather than steric forces. Addition of salt screens the electrostatic repulsion, reducing head-group area appropriately to produce the corresponding phase. Again, simultaneously, inter-aggregate forces are reduced, and assist in causing the phase changes to be abrupt. The same progression can be observed if one adds progressively a singlechained surfactant (S) (v/al = 1/3) to a lamellar phase of a double-chained surfactant D (v/al=l) of the same head-group area and chain length. Here it is the effective volume which changes according to,

{•-}

Xs + I(XD)

ff

Xs +

9(Xs, XD mole fractions)

If curvature so prescribed is the major determinant of self-assembly, then phase diagrams ought to exhibit universality, yet ionic and non-ionic phase diagrams are different (Fig. 3.4). The latter exhibit an upper consolute point (phase separation beyond the cloud point). That issue can be resolved if we recognise that for the non-ionic surfactants we might equally replace the temperature axis by the curvature v/al. It can be shown that for ionic surfactants v/al decreases with increasing temperature. To achieve universality one need then only to reverse this direction. This can be done by taking a combined surfactant system (as above), admixing double-chained v/al = 1 to a single-chained v/al = 1/3 surfactant, both with the same area and chain length, and replacing the temperature axis by (v/al)eff. The resulting phase diagram, with the x-axis now the volume fraction of the combined surfactant in water exhibits the cloud point phenomenon. (Some unusual single-chained ionic surfactants do in fact show cloud point phenomena in any event and this can be understood only in terms of as yet unknown surfactant-induced water structure.) These arguments suggest that since the x-axis is some measure of interactions, and the y-axis in the phase diagram a measure of curvature, that it is curvature vs. interactions that provide a better characterisation of phase behaviour for such systems than the usual Gibbsian parameters (T,V,N). These ideas and observations have been put on a sound basis linking molecular forces quantitatively to the prediction of cmc's as a function of salt

Self-assembly

119

and temperature ion-binding parameters, and chain length [47], and the universality so found gives confidence that curvature, as set by molecular forces can be used to elucidate structure.

3.6.3

Genesis of the surfactant parameter

An outline of the ideas is as follows [62-65]. It suffices to assert that a dilute solution of surfactant molecules can be considered to consist of water plus monomers, dimers, trimers, and larger allowed aggregates (micelles, vesicles, liposomes .... ). The concentration is assumed to be so low that aggregates can be considered to be non-interacting. The probability distribution of aggregates, is then determined from the law of mass action

p~ + ~-Z In (-~)=/zl~ + k T l n X 1

(3.5)

where the chemical potential of an aggregate of size N has been written as NIZN 0 + kTIn(XN/N) and X N is the concentration (mole fraction, volume fraction...) of surfactant molecules in the N-aggregate. (The theory also allows for the possible formation of infinite aggregates, i.e. separate phases.) The glib assertion eq. (3.5) represents a beginning to a chemist and an essential stumbling block to a physicist who can go no further without questioning foundations. If aggregates of given size N were distinct, identical well-defined chemical species there would be no problem, apart from the vexed question of concentration units. They are not: even within a given N-aggregate, if such can be defined, there exist an infinite diversity of shapes or configurations which the association of surfactant molecules could take up. Implicit in eq. (3.5) is the understanding that for any N, there is a shape of optimal energy that exists, e.g. a sphere, cylinder, bilayer, and is overwhelmingly more probable than its fellows. These problems are profound, and unresolved. But under certain conditions and assumptions that can be spelt out [62-64] an aggregate can be defined, and an equilibrium partition function of statistical mechanics written down, from which it is possible to extract the thermodynamics eq. (3.5). The difficult question of how even to write down a partition function that averages over size and shape of aggregates, rather than a restricted set like spheres, cylinders.., is identical to that of the general theory of phase transitions. The same problem occurs in the problem of nucleation theory, e.g. gas-liquid condensation. The usual models are physically comprehensible, but give erroneous critical behaviour. The more sophisticated renormalisation group methods of condensed matter physics gives correct critical behaviour, but completely avoid shape, which remains a hidden variable. In any event, extant theories fail to provide a quantitative description of first-order gas-liquid transitions, or of melting of a solid.

120

Chapter3

If we accept these limitations, eq. (3.5) can be rewritten as

=

~ _.o )/kTl

aggregates are allowed to occur at any concentration, albeit with infinitesimal probability, even below the cmc. Above the cmc, defined by

All

oo

xl=

xN, N>I

X1 increases slowly with concentration. It can be shown that if ]~N0, the chemical potential of a monomer in an aggregate is sharply distributed about some N, then the distribution of aggregates peaks at a value of N just less than the N with minimum #N 0 and is also sharply distributed. Otherwise, pronounced polydispersity may occur (e.g. for long cylindrical micelles). Thus reduced to bare bones it can be seen that the use of the word "theory" is dubious. We have simply characterised the observation of miceUes, and claim that it can be shown that the law of mass action is an appropriate vehicle for this characterisation. The entropic term ( k T / N ) In ( X N / N ) has considerable nuisance value. In the pseudophase approximation- valid for finite disconnected aggregates- it can be dropped. Then in this approximation no micelles occur below the cmc. This is now the value X 1 of monomer concentration for which ]~N0 = ]al 0 + k T lnX 1. Above this cmc, all additional surfactant molecules form micelles or whatever aggregate has the minimum lUN0. No other aggregates form until activity coefficients, i.e. interactions between aggregates, become significant. Keeping in mind possible complications due to phase transitions and interactions, the strategy is then to compare the chemical potential of different aggregates to see which has the minimum free energy ]~N0. We can write for the free energy difference f between a monomer in a miceUe and in solution #N - #1 - f = fB + fs + "packing term"

Here fB is the bulk (hydrophobic) free energy of transfer of monomer hydrocarbon tails from water to the oil-like interior of a micelle. It is the same for all aggregates, f s is the surface free energy per monomer in the aggregate. It depends on the size of the aggregate and the curvature of the interface. Then there is a packing term: the assumption that the interior of our micelle is fluid-like, and is to first approximation incompressible, has an immediate consequence, provided we require that aggregates can obtain no holes. (The occurrence of an interior vacuum or a water-filled region inside the (oil-like) interior of an aggregate would result in a large unfavourable increase in free energy, which possibility must be excluded from

Self-assembly

121

consideration.) This can be taken into account if we assume ~N 0 = oo when the packing criterion is violated. For spherical and cylindrical micelles this criterion is usually R < Ic where R is the radius of the micelle and l c is a critical chain length which is usually about 80% of the fully extended chain length for bilayers. The packing criteria is an extreme simplification. Sometimes, e.g. for vesicles, it needs to be relaxed or extended. The welding of two notions, of a fluid-like interior for the micelle, and of "packing" is at first sight contradictory. However the two notions can be shown to be compatible in a first-order theory [48, 62]. Possible candidates for aggregates can now be examined. For surfactant-water systems these have been restricted in the past to spherical micelles, nonspherical micelles (globular, cylindrical), vesicles, liposomes, bilayers, and for oil-water-surfactant systems spherical drops, normal or inverted (water in oil) or (oil in water). The analysis requires some assumptions concerning the surface free energy of the aggregate. Under quite general assumptions, the story as outlined above does emerge. (Vesicles, single-walled bilayers represent more complicated structures [63-65], because the chains on the inner surface are subject to compression, rather than extension, as for the outer layer. Further, all closed aggregates including liposomes, have different interior solution conditions to those outside. This difference is especially marked for ionic surfactants. This effect leads to supra-self-assembly, a higher class of structures to be discussed in (Chapters 5 and 6). In the case of insoluble double-chained surfactants, the interior of a liposome can collapse to a cubic phase or micelles because the interior solution conditions that set c u r v a t u r e t h r o u g h molecular interactions are different inside [63]. Nonetheless vesicles can be shown to be thermodynamically stable structures sometimes, with aggregation numbers, size, depending on chain compressibility) [63-65]. A crucial parameter-free test of the theory is provided by its application to

micelle formation from ionic surfactants in dilute solution [47]. There, if we accept that the Poisson-Boltzmann equation provides a sufficiently reasonable description of electrostatic interactions, the surface free energy of an aggregate of radius R and aggregation number N can be calculated from the electrostatic free energy analytically. The whole surface free energy can be decomposed into two terms, one electrostatic, and another due to short-range molecular interactions that, from dimensional considerations, must be proportional to area per surfactant molecule, i.e. fs = f e s ( N , R , a ) + y a

where ~, is a constant that subsumes all other intramolecular interactions like head-group steric repulsion, hydration, and chain interactions. The optimal micelle can be shown to be given by

122

Chapter 3

Ofs = o = Ores +7' , N v = 41r Oc ~c 3

,Na=4~R 2

Hence ~ris determined - electrostatic forces must balance all others. The older pseudophase, or ion-binding model had characterised ionic micellisation through a "chemical" equilibrium reaction XM ~ X~ X~ where X1 represents the monomer surfactant concentration, X2 the counterion concentration s u p p o s e d to be in equilibrium with micelles at concentration XM. N is the aggregation number of micelle, and Q the number of "bound" counter-ions. As such it is a phenomenological description with no predictability. This model emerges as a special case of the more general theory, and the number of "bound" counter-ions emerges as the calculated physisorption excess of counter-ions at the micellar surface. In this theory, cmc and aggregation numbers all are given correctly as a function of chain length, added salt, and temperature-up to 160~ [65]. The term ~, is constant for a given chain length with salt variation, and the predicted value agrees with independent statistical mechanical calculations of chain interactions in a micellar configuration. More refined calculations that evaluate the statistical mechanics of chain packing give essentially the same result. Of more interest perhaps is the fact that from the analysis emerges a reconciliation of the older pseudophase chemical equilibrium model and the more general statistical mechanical model.

3.6.4

The tyranny of theory

The importance of advances in basic science is not so much in the advances themselves. It lies rather in the breakdown of dogma, the removal of the tyranny of theory which tends to become set in stone, and the tyranny of disciplinary boundaries so inhibiting to the opening of new vistas. That is so equally for the Ptolemaic planetary system, for Darwinism, for q u a n t u m mechanics, or even the new molecular biology. Darwinism in retrospect is in part tautological, and q u a n t u m mechanics shot through with logical inconsistencies despite its astonishing successes. Even the new molecular biology, focused on linear sequences of DNA, and beset by X-ray structure determination of the individual atoms of proteins to the neglect of the shapes of the hydrophobic regions (of hyperbolic geometry) that guide and drive RNA replication, is so beset. The neglect of the environment, and its interaction with shape, i.e. forces, has inhibited the joining to advantage of what mathematics and physics and chemistry have to offer biochemistry and life processes. D'Arcy Thompson had argued ever so gently in his book "On Growth and Form" [66] that classification and evolutionary lineage through

Self-assembly

123

morphology alone might be fraught with danger. The equiangular spiral that underlies similarity in morphology of shells and the shape of plants and other things derives from processes driven by constant rates of growth. They are not necessarily an indication of relatedness. And the present vogue of assigning evolutionary connectedness depending on statistically correlated amounts of conserved partial DNA sequences is as dangerous as phrenology. On the other side, in colloid and surface chemistry, which borders most closely from the physical sciences on biology, the DLVO theory of stability that held sway so successfully for 50 years and more, has been equally or more inhibiting to the joining of structure and function. The gallimaufry of forces outlined above removes that disjunction, somewhat, because it is now clear that nature has at her disposal a richer, more specific and stronger diversity of forces than we had imagined. And we now understand better how they can be turned on or off to advantage through shape and topology triggered by biochemistry in a changing environment engineered itself by those shapes. Awareness of the new forces, and how they can be called in to play via hydration, specific ion effects, the Debye length in mixed electrolytes or micellar systems, the new hydrophobic forces, dipole- dipole and ionic fluctuation effects (and others) to affect the state of phospholipid curvature is necessary. The older inhibitions have been removed. Experiment and theory now begin to be consistent, at least for model systems, and the rich diversity of shapes and structures available, and controllable t h r o u g h c u r v a t u r e set by molecular forces becomes comprehensible, perhaps even accessible. We move on now out of the Euclidean desert to explore these matters further. In Chapter 4 we shall see how curvature together with global packing constraints conspires to p r o d u c e and predict the rich diversity of bicontinuous cubic phases and others of constant mean curvature which can be prescribed by variations of solution conditions or temperatures.

124 Appendix 3A: Evolution of concepts on long range molecular forces responsible for organisation and interactions in colloidal systems

(A) Original theories (1930's) involved (1) steric repulsion, "hydration shells" of water to stabilise (i.e. prevent coagulation) biological aggregates and emulsions, and later invoked; (2) entropy (two-phase equilibrium) opposed to repulsive electrostatic (double-layer) and hard-core interaction between anisotropic particles, clay plates, tobacco mosaic virus ordering (Langmuir, Onsager). These ideas are coming back into vogue and rediscovered in the steric fluctuation forces of Helfrich between biological membranes and in surfactant-water surfactant-water-oil systems. The current status of these ideas, presently under intense investigation by new low angle x-ray scattering and neutron scattering techniques, is unclear. Hydrodynamic forces also play a role, but depend on short range forces at an interface (boundary layers). (B)

Forces between solid particles.

Two Themes: (1) YOUNG-LAPLACE

(2) POISSON

4--1~ water

action at a distance Intervening liquid has bulk liquid properties to atomic dimensions from an interface

surface induced liquid structure Forces due to overlap of profiles of surface induced liquid structure

odispute reviewed by Challis in report to British Association (1833, 1834) "On the Present State of Theories of Capillary Action". The dispute between Laplace and Poisson there reviewed taken up by J. Clerk Maxwell (1876) in "Theory of Capillary Action" [41].

Appendix 3A

125

1930"s

9double-layer theory and theory of van der Waals forces developed. 9attractive Van der Waals Forces by pairwise summation plus double-layer theory gives potential of interaction.

vl(I) =

r6

l2

+

v2(I)

kl ;

Ir = Debye length of salt solution yields

1940"s

9Derjaguin Landau Vervey Overbeek (DLVO Theory): explains salt dependence of lyophilic colloid stability, fails to account for biological interactions oil-water systems. 1960's

9Lifshitz theory of attractive forces across vacuum, measurements of long range forces.

9measurement of repulsive forces via soap film thickness.

1970"s

9van der Waals forces, effects of interacting media, non-additive, infrared, ultravolet, microwave, temperature dependent, salt d e p e n d e n t , r e t a r d a t i o n show extreme subtlety of forces even in continuum approximation.

9surface charge regulation, surface charge responds to presence of neighbouring surfaces to regulate interactions and re-organise structures.

9very long range forces discovered. 9awareness of recognition. 9forces even at primitive level guide recognition processes. 9van der Waals self-energy concept parallels self-energy for charged ions. 9interactions and surface energies intimately coupled.

126

Chapter 3

9measurements of long-range forces (Israelachvili et al.).

9measurements and "theories" of hydration forces between phospholipids (Parsegian and

Rand). 9measurements of short-range forces (Maxwell rediscovered).

9charge regulation, ionic hydation-~ 9"secondary hydration" forces. 9oscillatory forces of surprisingly long range in simple liquids and ionic solutions.

1980"s

9beyond Poisson-Boltzmann theories. 9ion-fluctuation correlations explain failure of DLVO theory for divalent salts. 9ion-binding inferences need reinterpretation. 9van der Waals and double-layer forces now inextricably entangled. Distinction once made disappears, especially for oil-water and biological systems. 9measurements of specific ion effects show extreme subtlety of biological interactions e.g. acetate vs, Br-. 9temperature dependence of polyoxyethylene surface interactions from repulsive to attractive in narrow temperature range show further subtlety, in biological situations. 9very long range hydrophobic interactions discovered.

1990

9surface dipole-dipole forces explain discrepancies between long-range forces predicted and measured for biomembranes, phospholipids and proteins. 9oscillatory forces in micellar and lamellar phase systems provide further mechanisms for organisation and recognition.

9forces in asymmetric electrolytes explain failure of theories in many situations.

Appendix 3A

127

1991

9awareness that measurements of short-range (< 20.~) forces may be erroneous due to experimental limitations, as for lubrication, viscosity, adhesion, friction. Similar conclusions for scanning tunnelling and atomic force microscopy "imaging" of macromolecules. New techniques available. .awareness of how to control forces and their subtlety with temperature, salt, hydration removes disjunction between colloid science and membrane science.

1993

*hydrophobic interactions shown to be critically dependent on amount of dissolved gas, salt and salt types. This completely changes our understanding of the role of "hydrophobicity" in self-assembly.

1993

9It now turns out that there is not one, but many hydrophobic interactions. Developing theories recognise that earlier ideas have ignored the role of dispersion forces acting on ions, that drive ionic adsorption. Evidence is mounting for ultrastructure in water, perhaps due to dissolved gas; recognition of the interplay between dissolved gas organisation and electrolyte concentration and type. Free radical production due to formation of nanometric sized cavities connects hydrophobicity, reactivity and recognition in fundamentally new ways. The shape of the bridge linking the molecular forces of surface and colloid science and biochemistry and selforganisation begins to emerge.

128

Appendix 3B: Modern concepts of self-assembly 1936

oconcept of micellar and lameUar phases oself organisation of surfactants (Hartley). 1945 omicroemulsions (oil-water-surfactant-cosurfactant) oequilibrium phases. 1960"s

obeginnings of microscopic theories (Tanford). oprinciple of opposing forces at surfactant-water interface + hydrophobic free energy of transfer). ocubic phases discovered (Luzzati, Fontell). 1970's osimple unified theories give good description of phase diagrams of surfactant-water systems. Restricted to simple geometries. Spontaneous vesicles discovered. 1980"s

obicontinuous cubic phases shown to be ubiquitious. ,structure of microemulsions explored and put on firm basis theoretically using concepts of curvature set by molecular forces (inter- and intra-aggregate interactions) and global packing constraints. *discovery of rich diversity of cubic and intermediate phases in phospholipids and other systems, easily transformed from one state to another. orecognition that lipids and membranes and the topology and shapes taken up play a crucial role in guiding biochemical processes. 1990"s ocubosomes, supra self-assembly and quasi-crystalline structures.

129

Appendix 3C

Remarks on the nature o f the hydrophobic interaction and water structure

For the all-important hydrophobic interaction, the problem of water structure cannot be easily swept away. Recent work raises the spectre that most of our progress has been illusory. This can be seen from studies of the effect of salts and other solutes in reducing bubble coalescence in aqueous solutions. We refer to the original papers for more detail [67, 68]. The reasons for the effects of salt (or other solutes) are not understood. Salts generally increase the surface tension of water and are desorbed from the airwater interface. These factors might be expected to destabilise bubbles. Salts would also be expected to reduce any electrostatic repulsion produced by charge build up on bubble surfaces. The mechanisms involved in the enhanced bubble stability produced by salts have yet to be elucidated. The phenomenon of interest is about as simple to demonstrate as it is dramatic. If we imagine bubbles formed by passing a gas through a glass frit at the base of a column of water, the bubbles fuse on collision and grow in size as they ascend the column. However, on addition of an alkali halide, beyond a certain critical salt concentration the bubbles will not fuse and remain the same size. That phenomena has been explored by several authors. On the other hand, increasing concentrations of HCI do not have any effect on bubble coalescence (cf. Fig. 3.5) A whole range of cations and anions in different combinations have been explored. The results are surprising. Measurements of coalescence rates for a range of typical electrolytes as a function of electrolyte concentration are shown in Fig. (3.5). There is a correlation between valency of the salt and transition concentration, defined as 50% bubble coalescence, with more highly charged salt effective at lower concentration. The effect is independent of gas flow rate. All the results scale with Debye length (ionic strength). Some salts and acids have no effect at all on bubble coalescence, a situation summarised in Table 3.1. There is no known mechanism that can account for these effects. Water structure has to be implicated. But there is clearly a remarkable correlation between the ions present in a salt and their effect on the coalescence phenomenon. A property a or I~ can be assigned to each anion or cation. The combination a a or [~1~results in inhibition of bubble coalescence at a critical salt concentration, whereas the combinations a13 or [3a produces no effect at all. Different gases of widely different molecular size, from helium to sulphur hexafluoride, affect the transition concentration a little, but do not change the phenomenon.

130

Chapter 3

Figure 3.5: Plot of bubble coalescence as a function of electrolyte concentration for a range of common salts. Beyond a fairly sharply defined concentration range, it is impossible for bubbles to fuse. The sugars sucrose, fructose and glucose have also been found to affect bubble coalescence. On addition to water these sugars raise the surface tension and are d e s o r b e d from the air-water interface. Thus their effect on bubble coalescence equally cannot be described in terms of surfactant-like behaviour and certainly no charge effects are involved. Hence, even if an "explanation" could be found within the confines of the primitive model of electrolytes, that explanation could not accommodate this observation. The reduction in bubble coalescence achieved with increasing concentration is s h o w n in Fig. 3.7. Although earlier work had focussed on those electrolytes which exhibited coalescence inhibition, it has now been s h o w n that some other salts and mineral acids have no effect whatsoever. For those electrolytes inhibiting coalescence there does appear to be a correlation with the ionic strength, which brings the results into a relatively narrow band. However, and to repeat, as yet there is no obvious explanation why some electrolytes produce no effect on coalescence.

Appendix 3C

131

Table 3.1: Effect of added ions on bubble coalescence in water.

CATION

H+ ~Mg2+ Na+

ANION

OH- , X

0~ 9

I. Sol

9

9

c,-

X

J

J

9

Br-

0~

X

NO39

9

) CIO3-

J

X

9

m

Ca2+

K+

I. Sol

J

J m

C104- j C !-I.~00-

of.

J

oxanate2- X

X

X

X

I. Sol I. Sol

Combining Rules: ~

9

9

9

9

9

9

9

ji

Li+

X

J

Unavail 9

9

I. Sol

J m

Unavail

9

Unavail

9

J

l

=

X

Unavail I. Sol

I. Sol

9

X I. Sol

J

or 1313gives J

!. Sol=Insufficiently soluble

9

1

9

X

9

UnavailI I. Sol Unavail I. Sol

!J 9

9

J,J

9

X

9

J

Unavail UnavaiI X

SO42-

Me4N+

Cs+

j=jij j

9

J

NH4+

~.

X

Unavail =Unavail ,,

X

9

J

I. Sol Unavail Unavail 0t13or 13orgives X Unavail=Salt unavailable

Addition of salt: Prevents coalescence JI-las no effect on coalescence

X

In seeking an explanation we can definitely rule out any correlation with changes in surface tension at the air-water interface that are responsible for the foaming of surfactant solutions. The electrolytes studied increase the surface tension of water. Changes in the h y d r o d y n a m i c force caused by viscosity can also be excluded because some electrolytes decrease the viscosity, yet inhibit coalescence. The t e m p e r a t u r e d e p e n d e n c e if any is weak. Predictions of conventional DLVO theory go in the wrong direction - added salt reduces repulsive double-layer forces, to suggest enhanced coalescence. There is no evidence of significant bubble charging in salt solutions. A

132

Chapter 3

different kind of double-layer force that increases with salt concentration can be considered. This can come about through the differing cation and anion hydrated sizes that affect the electrostatic potential at the water surface. Detailed statistical mechanical calculations of this repulsion force have been carried out assuming a range of ion sizes and surface charge. However, the predicted short-range (<2nm) and weak magnitude of this force has ruled it out as a possible explanation. Fluctuations in the thin aqueous membrane between two approaching bubbles cannot play a role, because the theory predicts that fluctuations increase in magnitude with added salt and favour rather than oppose coalescence.

Figure 3.6: While some salts inhibit bubble fusion beyond a 'critical' concentration, others have no effect whatever. As two bubbles approach at any reasonable rate, there must arise a hydrodynamic repulsive force, due to the need to expel water molecules from the film between bubbles. In water, coalescence is observed. Thus, there must exist an attractive force that overcomes the hydrodynamic repulsion. The attractive van der Waals force calculated between two bubbles in water is found to be orders of magnitude smaller than the hydrodynamic repulsion present at reasonable approach rates. As bubbles are highly hydrophobic ( ~ i r - w a t e r = 72mJm 2) it is reasonable to assume that the "hydrophobic force" is present, and acts to produce coalescence. Available force measurements are found to give an attraction of sufficient m a g n i t u d e to overcome the hydrodynamic repulsion. This implies that for salts and sugars to reduce bubble coalescence, the attractive hydrophobic force is reduced in their

Appendix 3C

133

presence. Indeed, the limited investigations of the effect of salt on the hydrophobic attraction between solid surfaces and between bubbles show this.

Figure 3.7: Bubble coalescence as a function of concentration for some sugars.

The measured long-range hydrophobic attraction between hydrocarbon or fluorocarbon surfaces in water is 10 to 100 times larger than any conceivable van der Waals force. The range of this (roughly exponential) force is between 10-100 nm, depending on the surface. It has been suggested that the longrange (>10nm) interaction involves some type of micro-bubble cavitation between the approaching hydrophobic surfaces; the formation of such cavities between hydrophobic surfaces is energetically favourable. Further, spontaneous cavitation has been observed between hydrophobic surfaces at close separations. However, the formation of a small vapour cavity is a high energy process, a circumstance that poses a problem for the cavitation mechanism. A modification of this approach might eliminate this difficulty. In all cases of long-range (>10nm) hydrophobic force measurements, reported so far, the aqueous solutions were apparently in equilibrium with the atmosphere. Hence, they contain about 25ml per litre of dissolved nitrogen and oxygen gases, close to saturation. These dissolved gases might accumulate in the vicinity of hydrophobic surfaces, so releasing high energy water surrounding the gas molecules back into the bulk state. Fluctuations in

134

Chapter 3

density near hydrophobic surfaces due to the "adsorption" of gas could give rise to the long-range attraction. For hydrophobic surfaces in close proximity, this accumulation appears to develop into bubble nucleation sub-critical fluctuations in density of the liquid film. Support for this notion comes from the simple observation of the effect of dissolved gas on emulsion stability. It can be shown that de-gassing a mixture of dodecane and water increases the stability of an oil-in-water emulsion produced by shaking. This simple observation suggests that the hydrophobic interaction responsible for oil droplet fusion is reduced by the removal of dissolved gases. Measurements of the hydrophobic attraction between solid polypropylene surfaces show that on degassing the water the force drops back to van der Waals forces predicted by Lifshitz theory - by one or two order of magnitude. Measurements of the hydrophobic attraction between solid surfaces using an atomic force microscope have also been performed in solutions of NaCl and NaC103. The attraction was found to be equivalent to that in water in 0.2M NaCIO3, whilst in 0.2M NaCl the attractive force was much reduced, supporting expectations. (NaCIO3, has no effect on bubble coalescence, NaC! does, cf. Table 3.1.) It has also been observed previously that the range of the attractive force acting between bubbles is substantially reduced from about 100nm to 40nm on addition of KCI above the transition concentration. The effect of NaCl on bubble nucleation in the presence of hydrophobic surfaces has also been examined. Excess nitrogen gas was dissolved in solution by equilibration under 25 atmospheres of pressure. Immediately following decompression the solution was supersaturated with nitrogen gas. In water and 0.02M NaCl, it was found that bubbles nucleated quickly (<25 sec) at a (hydrophobic) teflon surface. However, a 0.20M solution of NaCl was found to inhibit bubble formation. In repeat experiments, bubbles were found to form at the same sites on the hydrophobic surface. It would appear that the microstructure of the surface is important for the nucleation of bubbles. Microscopic surface cracks would present hydrophobic surfaces at very close separations, enabling nucleation to occur more readily.

Further consequences

These observations can be linked in a remarkable way. Salt in the human body is at the minimum level for which maximum bubble coalescence stability is achieved (about 0.15 M for NaCl). This level of salt might then act to prevent bubble nucleation at the interface between microscopic hydrophobic organelles in the human body. That is, salt prevents the effects of the "bends" that would otherwise occur even under atmospheric conditions. (Spontaneous cavitation between highly hydrophobic surfaces in water at close separations, without decompression is established.) Examination of Table 3.1 shows us that some salts give no protection and would therefore be inhibiting to organised organic structures. Decompression

Appendix 3C

135

tests show that bubble nucleation at hydrophobic surfaces is indeed influenced by the salt level in the aqueous solution. Evidently, protection is afforded against nucleation at salt concentrations where bubble coalescence is prevented. These tests show further that neither the type of gas nor its pressure is critical. The results then lead to the remarkable suggestion that eukaryotic life could only evolve and survive in sea water containing a minimum level of protecting salts like NaC1. It is possible that the extinctions of the P e r m i a n era (230 million years ago), when 95% of extant species disappeared, correlates with a known drop in ocean salinity by a factor of 2 or 3. The Burgess shales enigma (ca. 530 million years ago) and earlier Ediacara extinctions (570 million years ago) may also be related to a drop in salinity levels. Removal of water to the polar caps, in a glaciation, paradoxically, results in massive salt deposits, so that after melting, the oceans have reduced salinity. The observations discussed above may have implications for any system where water, dissolved salts and hydrophobic entities are present; and there must be many. Currently the separation of hydrophobic proteins can be achieved using a hydrophobic chromatography column, by elution with salt solutions. There is no adequate theory for this process and present understanding is purely empirical. Suppose then, that while all salts reduce electrostatic forces, only those salts that reduce bubble coalescence also reduce the hydrophobic attraction. Further, these salts have a significant effect on the hydrophobic attraction only above their transition concentration. With this notion in mind the experimental results are explained. This then enables separations to be simplified, as the salt type and concentration gradient required are easily determined. The effect of salt on the equilibrium and kinetics of protein adsorption on (hydrophobic) butylated surfaces has also been investigated. It was found that increasing the concentration from 0.1M to 1.0M results in a drop in the rate constant of an order of magnitude and a drop in the degree of adsorption. This again may be explained in terms of a reduction in the hydrophobic attraction between the surface and hydrophobic sites on the proteins. Sucrose (which consists of the monosaccharides glucose and fructose) exhibits a reduction in coalescence equal to the individual reductions in coalescence achieved by fructose and glucose separately. This is remarkable, but moreover, a mixture of fructose and glucose exhibits a far smaller effect on bubble coalescence (cf. Fig 3.7). This suggests that the presence of one sugar is antagonistic to the effect produced by the other. An analogy can be drawn here to the effect of an ((x,[3) salt where it appears one ion is antagonistic to the other ion inhibiting coalescence, and vice-versa. It is assumed that with sucrose the sub-units are locked in a conformation that precludes antagonistic effects between the sub-units. All these findings indicate that the yet u n k n o w n "water structure" induced by solutes is central to the phenomenon. Thus in the case of salts, ions may only be important in their effect on water structure rather than for any further electrostatic effect.

136

Chapter 3

The polar, ionic and even non-ionic head-group interactions of lipid membranes and other surfactants, (as indeed for many polymers and polyelectrolyte intra-molecular interactions) and the associated curvature at interfaces formed by such assemblies will be dependent on dissolved gas in quite complicated ways. Fluctuating nanometric sized cavities at such surfaces will be at extremely high pressure, (P = 2 y / r , ~,= surface tension, and r the radius) resulting in formation of I~I and OH radicals. The immediate formation of (~1 radicals and consequent damage to phospholipids offers an explanation of exercise-induced immunosuppression (through excess lactic acid CO2 production), perhaps a clue to the aging process. A mechanism for the long-range hydrophobic interaction is evidently hidden in the complicated matter of surface induced gas a d s o r p t i o n - affected by solute, electrolyte, and dissolved gas. That seems now clear from the complex behaviour of solutes on bubble coalescence, and indeed from direct force measurements and other observations mentioned above. In retrospect, that and the resulting added subtlety of molecular forces is not a surprise. After all, nature did not put these forces to work to assemble biological entities in an aqueous environment without an atmosphere.

137

References: 1. 151. g

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0

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B.W. Ninham and V.A. Parsegian, J. Chem. Phys., (1970). 52: p. 4578. B.W. Ninham and V.A. Parsegian, Biophys. J. , (1970). 10: p. 647. B.W. Ninham and V.A. Parsegian, J. Chem. Phys., (1970). 53: p. 3398. V.A. Parsegian and B.W. Ninham, Biophys. J., (1970). 10: p. 646. B.W. Ninham, Chemica Scripta, (1985). 25: p. 3.

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J. Mahanty and B.W. Ninham, J. Chem. Phys., (1973). 59: p. 6157.

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J. Mahanty and B.W. Ninham, J. Chem. Soc., Faraday Trans. II, (1974).

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138

Chapter 3

21. A. Savin, O. Jepsen, J. Flad, O.-K. Andersen, H. Preuss, and H.-G. yon Schnering, Angew. Chem. Int. Ed. Engl., (1992). 31: p. 187-188. 22.

B.W. Ninham and V.A. Parsegian, J. Theor. Biol., (1971). 31: p. 405.

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R. Kjellander and S. Marcelja, J. Phys. (France), (1988). 49: pp. 1009-1015.

24. R. Kjellander, S. Marcelja, R.M. Pashley, and J.P. Quirk, J. Chem. Phys., (1990). 92: pp. 4399-4407. 25. P. Attard, D.J. Mitchell, and B.W. Ninham, J. Chem. Phys., (1988). 89" p. 4358. 26. P. K~kicheff and B.W. Ninham, Europhys. Lett., (1990). 12: pp. 471-477. see also: T. Nylander, P. K~kicheff and B.W. Ninham, J. Colloid and Interface Sci., (1994). 164, pp. 136-150. 27.

R.M. Pashley, J. Coll. Interface Sci., (1981). 83: p. 531.

28.

R.M. Pashley, J. Coll. Interface Sci., (1981). 80: p. 153.

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V. Shubin and P. K~kicheff, J. Coll. Interface Sci., (1993).155: p. 108.

30. P. K~kicheff, S. Marcelja, T. J. Senden and V. Shubin, J. Chem. Phys., (1993). 99: p. 6098. 31.

P. Attard, D.J. Mitchell, and B.W. Ninham, Biophys. J., (1988). 53: p. 457.

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P. Attard and D.J. Mitchell, Chem. Phys. Left, (1987). 133: p. 347.

33. D.J. Mitchell, B.W. Ninham, and B. Pailthorpe, Chem. Phys. Lett., (1977). 51: p. 257. 4.

R.G. Horn and J.N. Israelachvili, Chem. Phys. Lett., (1980). 71: p. 192.

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H.K. Christenson, J. Dispersion Sci. & Tech., (1988). 9: p. 171.

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B.W. Ninham, "B.V. Derjaguin', in Prog. Surf. Sci., (1992). 40: pp. 1-4.

37. S. Leikin, V.A. Parsegian, and D.C. Raw, Ann. Rev. Phys. Chem., (1993). 44: p. 369. 38. 601.

D.M. LeNeveu, R.P. Rand, and V.A. Parsegian, Nature (1976). 259: p.

39. 351.

R.P. Rand and V.A. Parsegian, Biochem. Biophys. Acta, (1989). 988: p.

40. S. Marcelja, Croatia Chemica Acta, (1977). 49: p. 347; Che, m. Phys. Lett., (1976). 42: p. 129.

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41. J. Clerk Maxwell, "Capillary Action". 9th ed., E. Brittanica. Vol. 56. (1875). 42. R.M. Pashley, P.M. McGuiggan, B.W. Ninham, and D.F. Evans, Science, (1985). 2239: p. 1088. 43. H. Christenson, "Modern Approaches to Wettability", eds. M.E. Schraeder and G. Loeb. (1992), New York: Plenum Press. 4.

45.

V. Yaminski and B.W. Ninham, Langmuir, (1993). 9: p. 3618. V. Yaminski, B.W. Ninham, H. Christenson and R. M. Pashley,

Langmuir, (1996), in press. 46. R.M. Pashley, P.M. McGuiggan, B.W. Ninham, D.F. Evans, and J. Brady, J. Phys. Chem., (1986). 90: p. 1637. 47. V. Parsegian and R.P. Rand, Proc. Natl. Acad. Sci. USA, (1991). 95: p. 4779. 48. D.J. Mitchell, B.W. Ninham, and D.F. Evans, J. Phys. Chem., (1985). 88: p. 6'344. see also: J.B. Hayter, Langmuir, (1992). 8: p. 2873. 90

50.

R.M. Pashley and B.W. Ninham, J. Phys. Chem., (1987). 91: p. 2902. P. Richetti and P. K~kicheff, Phys. Rev. Lett., (1992). 68: p. 1951.

51. J.L. Parker, P. K~kicheff, P. Richetti, and S. Sarman, Phys. Rev. Lett., (1992). 68: p. 162. 52.

P. Claesson, R. Kjellander, H.K. Christenson, and P. Stenius, J. Chem.

Soc. Faraday Trans. II, (1986). 82: p. 2735. 53.

B.W. Ninham, Chemica Scripta, (1989). 29A :p. 15.

54.

I. Langmuir, J. Chem. Physi, (1938). 6: p. 873.

55.

L. Onsager, Ann. N.Y. Acad. Sci., (1949). 51,: p. 627.

56.

P.A. Forsyth, S. Marcelja, D.J. Mitchell, and B.W. Ninham, Adv.

Colloid Interface Sci, (1978). 9: p. 37. 57.

W. Helfrich, Z. Naturforsch., (1973). 28c: p. 693.

58.

A. Fogden and B.W. Ninham, Langmuir, (1991). 7: p. 590.

59. B.M. Kachar, D.F. Evans, and B.W. Ninham, J. Coll. Interface Sci., (1984). 100: p. 284. 60.

B.A. Pailthorpe, D.J. Mitchell and B.W. Ninham, J. Chem. Soc. Faraday

Trans. II, (1983). 80: p. 115.

140

61.

Chapter 3

D.Y.C. Chan and B.W. Ninham, J. Chem. Soc. Faraday Trans. II, (1975).

71: p. 119; V. Mohanty, B.W. Ninham and I. Oppenheim, Proc. Natl. Acad.

Sci. USA, (1996), in press. 62. J.N. Israelachvili, D.J. Mitchell and B.W. Ninham, J. Chem. Soc. Faraday Trans. II, (1976). 72: p. 1525. 63. D.J. Mitchell and B.W. Ninham, J. Chem. Soc. Faraday Trans. II, (1981). 74: p. 1472 4.

B.W. Ninham and D.F. Evans, Disc. Faraday Soc., (1986). 81: p. 1.

65. B.W. Ninham, "In Search of Microstructure'. Structure and Reactivity of Reverse Micelle, ed. M.-P. Pileni. (1989), Elsevier. 66. D.F. Evans, M. Allen, B.W. Ninham, and A. Fouda, J. Solution Chem., (1984). 13: p. 68. 67. D. Thompson, "On Growth and Form". 2nd ed. (1968), Cambridge University Press. 68.

V.S. Craig, B.W. Ninham, and R.M. Pashley. Nature. (1993),364: p. 317.

69. V.S. Craig, B.W. Ninham, and R.M. Pashley. J. Phys. Chem., (1993), 97: p. 10192.

141

Chapter 4 4.1

Beyond Flatland: The Geometric Forms due to S e l f - A s s e m b l y

Introduction: molecular dimensions and curvature

F

latland is the title of a pretty little book by E.A. Abbott published late last century [1]. It tells the story of an imaginary society of fiat polygons, who live in a fiat space of two dimensions. To two-dimensional people, those who live in one dimension are really a rather limited tribe devoid of imagination. By the same token the two-dimensional civilisation found the notion of a three-dimensional world utterly absurd and unacceptable. Probably Abbott's story was motivated by ideas like multi-dimensional geometry and Riemannian geometry that were beginning to get about. The impact of those new counter-intuitive non-euclidean geometries has been immense. As yet they have spilt over little to chemistry and biology, and relative to the story one might tell in fifty years time we are probably still in Flatland. So far we have constructed a set of tools and put them to work in building and understanding some of the elementary self-organising structures, and how it is that forces conspire with molecular geometry to allow these entities to transform from one form to another. But the forms allowed have been too limited by far, and we now call in the language of curvature to embrace a wider range of shapes that nature employs. In this section we shall describe in some detail surface structures whose curved forms are s p o n t a n e o u s l y a d o p t e d by aggregates of simple macromolecules. These can range from synthetic surfactants and natural lipidso(typically about 20,~ in length) to synthetic copolymers and proteins (1000A). The analysis includes both natural and synthetic molecules, since lipids and proteins can be considered as more complex counterparts of surfactants and block co-polymers. Two characteristics determine the shape of molecular aggregates. The first is the shape of the constituent molecules, which sets the curvature of the aggregate. The second is coupled to the chirality of the molecules, which also determines the curvature of the aggregate, via the geodesic torsion. The bulk of this chapter is devoted to an exploration of the effect of molecular shape on aggregation geometry. An account of the theory of self-assembly of chiral molecules is briefly discussed at the end of this chapter. For surfactants, the hydrophobic free energy of transfer of the lipophilic hydrocarbon tail from water to oil provides the driving force for aggregation. But the hydrophilic head-groups prefer an aqueous environment and an interface between the polar region and the lipophilic domains results. With hydrocarbon tails, like alkanes, there are a large number of accessible tail conformations, so that the hydrophobic region is usually fluid-like around room temperature. The interface can be a dynamic one of a well-defined

142

Chapter4

time-averaged geometry, with rapid interchange between the aggregate and individual surfactants dispersed in water. Or the exchange process can be very slow for very hydrophobic chains (like those in phospholipids) so that the interface is real. We shall see that the geometry of the interface is often set by the molecular dimensions of the surfactants. Just as surfactants and lipids self-assemble to segregate the lipophilic and hydrophilic domains, much larger macromolecules built from immiscible species also aggregate to reduce the extent of mixing of the blocks. For example, copolymers made up of polystyrene and polyisoprene "blocks" containing thousands of polymerised monomers of each block, or structural proteins containing hydrophobic and hydrophilic patches aggregate to form assemblies of well-defined form under suitable thermodynamic conditions. In these cases too, an interface can be traced between the blocks or patches, whose thickness depends on the degree of immiscibility. In copolymers, the interfacial geometry is dictated principally by the fact that the very flexible chains in the macromolecule prefer those conformations that maximise their entropy. Aggregates of structural proteins form a variety of complex structures, whose geometry probably depends on the molecular shape and the chirality. In Chapter 2 we have seen many examples of structures at the microscopic level, defined by the positions of atoms in atomic crystals and organic molecules. The structures that interest us in this Chapter are visible at the "mesoscopic" (intermediate) scale. While the individual molecules are molten and lacking in crystalline order, the self-assemblies of these molecular units may be translationaUy symmetric, yielding liquid crystals. In other cases, the aggregates lack any mesoscopic translational order, and a random fluid results The shapes of these self-assemblies are as varied as the capacity of the molecules to weave through space will allow. The accessible interfacial geometries span a rich range of structures: from spheres and planes to highly interconnected bicontinuous honeycombs. So how are the structures of these complex liquid crystalline and disordered assemblies best described and understood? Typically, the problem is tackled by recourse to thermodynamic principles. A complete statistical mechanical treatment is out of the question. (The difficulty is a fundamental one. We do not yet know how to write down a partition function that describes the full ensemble of possible aggregate shapes and their associated free energies.) Assuming complete segregation between the domains - be they hydrophilic and lipophilic domains in surfactant-water systems or immiscible polymeric domains in block copolymer melts - the geometry of the aggregate domains can be described by that of the interface between the domains. The interfacial geometry is characterised by two distinct signatures: the interfacial curvatures, which describe its local geometry and the interfacial topology, which describes global geometry, including the connectivity of the interface. It turns out that this duality is not only useful mathematically, it translates naturally into the language of physics and chemistry of self-assembly, once it

Introduction

143

is recognised that condensed molecular aggregates consist of discrete building blocks even if the molecules are melted. The shape of those blocks sets the local interfacial geometry, while the composition of the molecular system constrains the global topology of the interface. In the next few pages we shall discuss the question of local interfacial structures bounding idealised aggregates, tiled by blocks of fixed dimensions. The model represents one extreme idealisation of the molecular constituents that form the aggregate, most applicable to small surfactant molecules. At the other extreme, the block dimensions are not set a priori, they must be determined as a function of the temperature, concentration, etc. This case will be dealt with later. The welding of two concepts, a fluid-like mixture of hydrocarbons, with that of an idealised block is at first sight contradictory. However it can be shown to be consistent in a first order theory [2].

A schematic view of a spherical micelle. The hydrocarbon fraction of the surfactant molecules occupies the interior, and the sphere (by convention) divides the polar from hydrocarbon regions. If the micelle is built up from a number of equivalent building blocks, each such enti~ adopts the form of a cone.

Figure 4.1:

4.2

The local geometry of aggregates

Since the suggestion of Hartley in the 30's that surfactants can self-assemble to form globular aggregates - micelles- in which the hydrophobic chains are essentially molten, it has been clear that in order for surfactant molecules to pack into aggregates, the molecular dimensions must be compatible. For

144

Cha~ter4

example, to form spherical miceUes, the average chain length of the surfactant molecules that build the sphere must be equal to the radius of the miceUe (in the absence of extra lipophilic components). Further, the surface area spanned by the polar fraction of the surfactant molecules must be sufficiently large to accommodate the average shape of a molecule, which, for a spherical aggregate is a cone, whose base is located at the surface of the miceUe. This is illustrated in Fig. 4.1. These constraints upon the molecular geometry of the surfactant molecules can be quantified neatly by the surfactant parameter introduced in the previous chapter. This parameter describes the shape of a surfactant molecule in terms of the volume of the hydrophobic region per molecule (the chain volume), v, the preferred length of the chains, / and the polar head-group cross-sectional area, a. The magnitude of the dimensionless surfactant parameter is given by (v/al). It defines the preferred direction of (mean) curvature; if v/al=l, the hydrophobic chain region of each molecule has, on average, a cylindrical hull; if v/all it is tapered in the opposite sense. Close packing of these molecules leads to curved interfaces, and the direction of the curvature (towards the polar or hydrophobic fractions) depends upon the value of the parameter, as shown in Fig. 4.2.

Figure 4.2: View of the curvature of aggregates formed by surfactant molecules of various surfactant parameters, v/al. (Left:) If the surfactant parameter is less than one, the interface between the polar and hydrophobic regions curves towards the chain region, and the average molecular shape is tapered towards the hydrophobic end of the molecule. (Middle:) If the surfactant parameter is exactly equal to one, the interface exhibits no preferential CUl~,ature, and the molecules are on average cylindrical. (Right:) If the surfactant parameter exceeds one, the interface curves towards the polar regions, and the molecule tapers towards the headgroup. Since the volume, V, of a cone is: v=A/ 3 where A is the area of the base and l is the height, the surfactant parameter for cone-shaped molecules is equal to a third. Consequently, spherical miceUes are formed by surfactant molecules whose surfactant parameter,

Local geometry

145

v/al=l/3 (discussed already in section 3.6). Similarly, cylindrical micelles are formed by surfactants characterised by v / a l = l / 2 and planes require v/al=l. (Vesicles, whose surfactant parameters for the inner and outer monolayers differ, are a different story, discussed in section 5.1.7.) These examples allow us to describe the structure of surfactant aggregates in terms of the value of the surfactant parameter. Indeed, this is the case for simple closed surfaces, where the interior contains the hydrophobic fraction (v/al
(4.1)

where H and K are the mean and Gaussian curvatures of the interface. The sign of H (alternatively, the sign of d) is dependent upon which side of the interface the parallel surface sits. We adopt the convention that the mean curvature is positive if the interface is curved towards the polar regions, and negative otherwise, which implies that the value of I is always positive. The volume of space occupied by a "foliation" of parallel surfaces up to a distance I is obtained by integrating eq. (4.1) with respect to d:

v(l) = a(O) l (1 + HI + ' ~ )

(4.2)

Setting v(l) equal to the chain volume, v, l to the chain length and a(0) to the head-group area, a, leads to a simple general expression for the surfactant parameter in terms of the curvatures of the interface between hydrophilic and hydrophobic regions, scaled by the characteristic distance, l: Y-- = 1 + HI + K l 2

al

3

(4.3)

The assumption that l is equal to the chain length (assumed to be about 80% of the fully extended chain) is valid provided the molecular chains lie normal to the interface. In general, this assumption is known to be valid for typical surfactant molecules [3].

Consider, for example a spherical micelle. By our convention, if the interface encloses hydrophobic regions, the mean curvature is negative. Consequently, the surfactant parameter for a spherical miceUe of radius R is given by:

146

Chapter4

v__- l - Rz + 12 d 3R 2

(4.4)

(since the mean and Gaussian curvatures of a sphere are equal to 1/R and 1/R 2 respectively). Clearly, the chain length, I, must be equal to the radius of the miceUe, R, thus v/al =1/3. Equation (4.3) shows that the magnitude of the surfactant parameter fixes only the function of the interfacial curvatures, (1 + HI + K/2/3), rather than the curvatures of the interface themselves. In other words, the interfacial geometry - the structure of the surfactant aggregate - is not fixed by the surfactant parameter alone. Both the mean and Gaussian curvatures can be varied cooperatively without altering the value of the surfactant parameter. Nevertheless, the surfactant parameter does furnish a local constraint upon the curvatures of the interface.

4.3

The composition of surfactant mixtures: the global constraint

A large number of different surface shapes satisfy eq. (4.3). These shapes are distinguishable by their topologies, characteristic of the degree of interconnectivity of the structure. A feature of high topology surfaces is their high surface area. Indeed, it follows from the Gauss-Bonnet theorem (described in section 1.7) that the surface area per unit volume increases with the surface genus, provided the average value of the Gaussian curvature, , is conserved. This means that the various shapes that can be adopted by close packed surfactant molecules assuming a fixed surfactant parameter have different surface areas. The area per surfactant molecule at the hydrophobic-hydrophilic interface the head-group area - is prescribed by the temperature, water content, steric effects and ionic concentration for ionic surfactants. Assume for now that the area per each surfactant "block" making up the assembly is set a priori. This assumption implies that the surface to volume ratio of the mixture (assumed to be homogeneous) is set by the concentration of the surfactant. So the interfacial topology is predetermined by this global constraint, the surface to volume ratio. We have seen that the local constraint on the surface curvatures, set by the surfactant parameter, can be treated within the context of differential geometry, which deals with the intrinsic geometry of the surface. In contrast, the global constraint, set by the composition of the mixture, is dependent upon the extrinsic properties of the surface, which need not be related to its intrinsic characteristics. (For example, the surface to volume ratio of a set of parallel planes can assume any value by suitably tuning the spacing behveen the planes. Similarly, the ratio of surface area to external volume (i.e. the volume of space outside each sphere closer to that sphere than any other) of a lattice of spheres depends upon the separation between the spheres.)

Globalgeometry

147

However, in other situations a connection can be m a d e b e t w e e n the surface to volume ratio of a surface and the intrinsic g e o m e t r y of that surface. For example, the ratio of surface area to internal v o l u m e of a sphere or cylinder depends only u p o n the curvatures, since it is a function only of the radii. It turns out that this connection can be extended to certain hyperbolic surfaces, leading to accurate estimates of the relation b e t w e e n the global and local geometric characteristics of these surfaces. To see this w e need to make an approximation. The a p p r o x i m a t i o n hinges on the g e o m e t r y of "focal surfaces" to an interface [4]. These are the two surfaces traced out by the centres of curvature (the foci) on an interface. The centres of curvature of a hyperbolic interface lie on both sides of the surface, so that the focal surfaces are on both sides of the surface (Fig. 4.3).

Figure 4.3. (Left:) Two-dimensional view of a focal surface to an interface. This focal surface describes the location of centres of curvature of the interface. (Right:) Three-dimensional view of the focal surfaces F1 and F2 to a hyperbolic surface, S. (Adapted from [5].)

The geometry of focal surfaces depends on the variation of curvatures along the interface. If the curvatures of the interface do not vary, e.g. spherical and cylindrical interfaces, the focal surface degenerates to a surface of vanishing area and the surface is " h o m o g e n e o u s " . Recall f r o m section 1.12 that h o m o g e n e o u s surfaces are characterised by constant Gaussian curvature. For example, the focal surface of a sphere is just the point at the centre of the sphere; the focal surface of a cylinder is the axis of s y m m e t r y along the centre of the cylinder. By analogy, the focal surfaces of a hyperbolic surface of

148

Chapter4

c o n s t a n t G a u s s i a n c u r v a t u r e a r e t w o c u r v e s of v a n i s h i n g area on e i t h e r s i d e of t h e surface.

Fig. 4.4(a): A series of ellipses of increasing "homogeneity" (variation of curvature), and a circle, together with their respective focal curves (also called "evolutes"). The focal curve for the homogeneous case, the circle, degenerates to a single point at the circle's centre.

Figure 4.4(b): Three parallel surfaces, all hyperbolic. The focal surfaces to homogeneous hyperbolic surfaces degenerate to two curves (AB and CD) on either side of the surface. T h i s s i t u a t i o n is n e v e r r e a l i s e d , since a h y p e r b o l i c s u r f a c e of c o n s t a n t G a u s s i a n c u r v a t u r e c a n n o t b e i m m e r s e d in t h r e e - d i m e n s i o n a l e u c l i d e a n s p a c e w i t h o u t s i n g u l a r i t i e s [6]. All h y p e r b o l i c s u r f a c e s in e u c l i d e a n s p a c e

Global geometry

149

have variations of Gaussian curvature over the surface. However, as long as the variations of curvature along a hyperbolic interface are small the foliation of space by parallel surfaces does nearly tile the volume (Fig. 4.4(a),(b)). We shall see that this approximation is a good one for threeperiodic minimal surfaces (IPMS). In terms of the constituent aggregated molecules, the "quasi-homogeneity" condition - viz. small curvature variations - imposes the proviso that the variations in molecular dimensions are small. This condition implies that the surfactant parameter, which is dependent on the molecular shape, varies little throughout the aggregate. This translates naturally into chemical language: quasi-homogeneous interfaces are expected in aggregates containing an approximately monodisperse distribution of surfactant molecules. Under this assumption, the (global) surface to volume ratio can be estimated from the (local) intrinsic geometry alone using the relations derived already for parallel surfaces. The volume is tiled without overlap or gaps by a dense foliation of parallel surfaces from the original (homogeneous) surface. For example, the (internal) volume, V, of a sphere is related to the area of the sphere, A, by the local relation (4.3), valid everywhere on the sphere,

V = AI ( I - HI + ~ ) Since the scaled mean curvature, HI, and the scaled Gaussian curvature,

K12=1, V=R A 3 This is an exact value for the sphere (V = 4nR3, A = 4~R2), due to its 3

homogeneity. Recall that the surfactant parameter for a spherical interface is equal to 31"(eq. 4.4). This number is a measure of the average block shape in the interior of a spherical aggregate, yet it is also related to a global measure, the surface to volume ratio. This example is trivial, however it is immediately able to be generalised to more complex geometries. The same technique will be used in the next sections to derive similar data for hyperbolic surfaces.

4.4

Bilayers in surfactant-water mixtures

In dealing with self-assembly of surfactants and lipids, we must consider two aggregation states: monolayers and bilayers. The latter class are ubiquitous in biological membranes, discussed in detail in Chapters 5 and 7. If the constituent monolayers in the bilayer are made up of identical molecules, the local geometry of both monolayers must be identical. In this case the mean

150

Chafer 4

curvature of the imaginary surface drawn through t h e mid-surface of the bilayer (Fig. 4.5) m u s t vanish, since the surfactant parameters in each monolayer are then equal. The molecular shape is accommodated within the bilayer by tuning the Gaussian curvature, following eq. 4.3.

Figure 4.5: Local view of the hydrophilic-hydrophobic interfaces (parallel surfaces) and surfactant packing for a bilayer interface. If both monolayers are identically constituted, the mid-surface of the bilayer (at the free chain-ends) is a minimal surface. (For an interface consisting of a reversed bilayer the surfactant molecules are inverted so that the head groups lie closest to the mid-surface, and the volume between the minimal surface and the two parallel surfaces contains the polar matter, i.e. water and surfactant head-groups.) This local constraint on the bilayer geometry due to the equivalence of the constituent monolayers forces the mid-surface of the bilayer to lie on a minimal surface. Further, under the assumption that the bilayer contains identical surfactant "blocks" of nearly rigid molecular shape (and hence nearly fixed surfactant parameter), the mid-surface of the bilayer is expected to be a quasi-homogeneous minimal surface. Consider now the surface to volume ratio of a h o m o g e n e o u s m i n i m a l surface. The Gauss-Bonnet theorem (sections 1.7-1.8) requires that the (surface-averaged) value of the Gaussian curvature, , is related to the topology of the minimal surface per unit cell (characterised by the surface genus, g) by:

(K~= 47r(l-g) A

(4.5)

where A is the surface area of the minimal surface per unit cell. The average radius of curvature of this surface, , is thus given by

Using the parallel surface formula we then have:

151

Surfactant monolayers

/

A v 4~;(g-l)

4~;(l-g)

VII 2 = ,~/

1 + -------~.

A

4~;(g-l) "

]3}

(4.7)

w h e r e V 1 / 2 is the v o l u m e associated with one side of the surface. The dimensionless " h o m o g e n e i t y index", H, for h o m o g e n e o u s minimal surfaces is defined by the following relation between the surface area, A,, the genus per unit cell of the hyperbolic surface, g, and the v o l u m e containing that surface, V (= 2V1/2):

H---~/ A3 3_

(4.8)

V 4n(g-1) V

From eq. 4.7, for a perfectly homogenous minimal surface, H = 3. 4 Table 4.1: Geometric properties of some periodic minimal surfaces. The "genus" of each threeperiodic minimal surface (IPMS) is the genus of a unit cell of the IPMS (with symmetrically distinct sides). The "symmetry" refers to the crystallographic space group for the surface (assuming equivalent sides).The surfaces are tabulated in order of deviation of the homogeneity index from the "ideal" value of 3/4.

II~.q

2maua

D-surface

3

I-WP surface

4

lnnnmetrv cubic (Pr~m)

/-/.lmmoa~neitvindex .7498

"

(Imam)

.7425

"

(la~)

.766

gyroid

3

CLP surface (c/a=~/2)

3

P-surface

3

Neovius surface

9

....

F-RD surface

6

"

tetragonal (C42/mmc) .7751 cubic (In,m)

.7163 .6640

(Fcl~rr0

.6577

In fact, the value of this index varies for different m i n i m a l surfaces, due to their inhomogeneity in euclidean space. The simplest m i n i m a l surfaces have finite integral c u r v a t u r e and u n b o u n d e d surface area. These include the helicoid, the catenoid and all one- and two-periodic surfaces. In these cases, H is either 0 or its value is unbounded. Clearly then, these minimal surfaces are far from h o m o g e n e o u s . The remaining m i n i m a l surfaces have u n b o u n d e d integral curvature, and a correspondingly complex topology (infinite genus). The simplest e x a m p l e s of these m i n i m a l surfaces are the triply-periodic IPMS. The values of the homogeneity index for various IPMS (described in the Appendix to chapter 1) are tabulated above (Table 4.1). Notice that the h o m o g e n e i t y indices of these IPMS are close to those of p e r f e c t l y h o m o g e n e o u s m i n i m a l surfaces. Clearly then, three-periodic m i n i m a l surfaces are quasi-homogenous hyperbolic surfaces, in contrast to

152

Chapter4

simpler m i n i m a l surfaces. A l t h o u g h the list is far from complete (particularly for high-genus surfaces), there is a trend of enhanced inhomogeneity as the genus of the IPMS increases. IPMS of genus three or four per unit cell most nearly approach ideal homogeneity. We have already argued that a self-assembled bilayer composed of equivalent surfactant "blocks" should form a homogeneous minimal surface, tracing the mid-surface of the bilayer. Within the constraints of this simple geometric model, we thus expect the formation of hyperbolic bilayers, wrapped onto three-periodic minimal surfaces of genus three or four per unit cell. The curvature of the bilayer, characterised by at the mid-surface, is set by the block shape. In terms of the surfactant parameter, the approximate form (valid if v/al is close to unity [7]) is:

Figure 4.6: Schematic views of bilayer configurations as the value of the surfactant parameter, v/al, varies for a double-chain surfactant or lipid. The stippled regions denote polar regions (water plus head-groups). (Left:) v/al > 1, cross-section through a pore of a saddle-shaped bilayer, whose mid-surface is a minimal surface; (centre:) v/al = 1, a planar bilayer; (right:) v/al < 1, a "blistered" bilayer, containing a vacuous region. In the last case, a reversed bilayer (Fig. 4.7) is favoured over the bilayer configuration illustrated. We have seen that a symmetric bilayer composed of chemically identical monolayers will spontaneously buckle so that the mid-surface of the bilayer wraps onto a minimal surface. Since the average Gaussian curvature, , of a minimal surface must be non-positive, eq. 4.9 implies the constraint on the surfactant parameter of the molecules which make up the bilayer: ~ > I. This is a severe constraint on the molecular shape of bilayer-forming surfactants: their chains must be bulky compared with their head-groups (e.g. doublechain surfactants or lipids). If the surfactant parameter is exactly unity, a planar bilayer results. However, once it exceeds unity, the bilayer must buckle, resulting in some negative Gaussian curvature. The data for threeperiodic minimal surfaces suggest that this buckling results in a bilayer whose global structure is similar to those of IPMS, with an array of catenoidal pores. The density of these pores depends intimately on the molecular shape, as the molecule becomes less cylindrical (Y-- = l) and increasingly wedged

5 urfactan t monolayers

153

shaped (~ > 1), this density increases. The possibility of tuning the pore density and consequent transport properties through adjustment of the molecular architecture (e.g. varying the head-group area, a, by varying the electrolyte concentration at the head-groups) is a real one, with obvious ramifications for the functioning of biological membranes. If the surfactant parameter is less than unity, a reversed bilayer can form, where the constituent surfactant molecules are placed head-to-head, rather than the chain-to-chain configuration characteristic of normal bilayers. If a chain-to-chain configuration occurs, the bilayer must be "blistered", to accommodate the bulkier head-groups (Fig. 4.6).

Figure 4.7: Images of (left) a portion of a surfactant bilayer wrapped onto the P-surface, a triply-periodic minimal surface, with two interwoven polar labyrinths and (right) a reversed bilayer on the P-surface, with interwoven lipophilic labyrinths.

The molecular shape is not the sole determinant of the structure of the aggregate. If the surfactant-water mixture is to form a single phase, the surface and volume requirements set by the composition of the mixture must be satisfied. Introducing the global constraint set by the composition leads to an estimate of the relation between the local geometry (expressed by the surfactant parameter) and the composition at which the surfactant mixture is expected to form a bilayer - or reversed bilayer - wrapped onto an IPMS (illustrated in Fig. 4.7). The details of the calculations for both reversed and normal bilayers (for which the tunnels are filled with water and surfactant respectively) are given elsewhere [7-9]. We characterise the concentration of the surfactant by the The relation between volume fraction of the hydrophobic region, r composition and molecular shape for hyperbolic bilayers is:

154

Chapter 4

v {3(~/"1))1'2 Ochains = 4~[~H-~ (3~-1)3/2

(4.10)

A similar relation can be derived between the local and global geometry of hyperbolic reversed bilayers, for which v/al varies between I and 2. x

3

The regions within the local/global "phase diagram" for which these hyperbolic bilayer structures can be realised within a surfactant-water mixture are plotted in Fig. 4.8.

Figure 4.8: Plot of the relations between the surfactant-water composition (characterised by (hint) and the surfactant parameter for normal (v/aiM) and reversed (v/al
So far, we have assumed the ideal value of ~ (derived above) for homogeneity index, H. In reality, this index varies according to symmetry and topology of the IPMS (Table 4.1). If the true values of H inserted into eq. 4.4, more accurate estimates of the local/global relations be made for various IPMS. These estimates are plotted in Fig. 4.9 below. "IF

4.5

the the are can

Monolayers in surf actant-water mixtures

In the previous section, it has been shown that a surfactant bilayer is constrained to adopt a hyperbolic (or planar) geometry if the constituent monolayers have identical molecular shape (characterised by the surfactant parameter). In the case of monolayers, all three geometries - elliptic,

Surfactant monolayers

155

parabolic and hyperbolic - are accessible, since the mean curvature of the monolayer is free to vary. We now calculate local/global relations for the most homogeneous examples of these geometrical classes, consistent with the "block" model of monodisperse surfactant-water mixtures. In contrast to the hyperbolic case, truly homogeneous examples exist in the other geometrical classes: spheres are the unique homogeneous elliptic surfaces; cylinders and planes are homogeneous parabolic surfaces.

surface

0

~

,$11rface ~ce ~Tface '~Ssurface

d

N 0 > .U

0

1.2

1.4

surfactant parameter

Figure 4.9: Plot of the expected local/global relations for a range of IPMS. An immediate consequence of eq. 4.3 is that the surfactant parameter must equal i3 and l for surfactant solutions to form ordinary micelles (called the 2 "L 1 phase" in the trade) and cylinders (H 1 phase) respectively - regardless of the surfactant concentration. The local/global relations for reversed spherical and cylindrical micelles ( v / a / > 1 ) are determined by the condition that the interfacial radii of curvature and the exterior volume (which is occupied by chains, and is equal to the integrated area of all parallel surfaces from the interface out to the chain length, l) must be consistent with the interior and exterior volume fractions. Clearly, as the surfactant parameter approaches unity, the spheres and cylinders grow, so the water volume fraction in the mixture increases. Once this internal volume fraction becomes too large, the spheres and cylinders must order into a crystalline lattice to fit in the available space. At still higher water fractions, set by upper limits on the packing densities of spheres and cylinders, these structures can no longer form. The formation of crystalline arrays of spheres and cylinders is also favoured in reversed micelles to achieve nearly uniform molecular dimensions. Although the curvatures of these surfaces are constant, it is not possible to pack the volumes between the spheres or cylinders with units of equal surfactant parameter. This inhomogeneity is due to variations in the spacing

156

Chapter4

between opposing spherical or cylindrical surfaces. These variations impose a variation in the tail lengths, I. In order to minimise these variations, body centred cubic arrays of reverse spherical miceUes and hexagonal arrays of reverse cylindrical micelles are favoured. It has been argued in the previous section that IPMS are the most nearly homogeneous minimal surfaces. Since hyperbolic monolayers can adopt non-zero mean curvature, other nearly-homogeneous hyperbolic surfaces represent possible aggregate geometries. Unfortunately, little is yet known about hyperbolic surfaces other than minimal surfaces. Calculations indicate that triply-periodic constant (and nonzero) mean curvature surfaces are less homogeneous than related IPMS [10]. However, one class of hyperbolic surfaces of genus > 2 per unit cell (discovered by Lawson [11]) are veDr different from IPMS. These surfaces lie between two planes; we call them "mesh" surfaces (cf. section 1.9). They form a two-dimensional array of tunnels on the inside of the surface, and their mean curvature necessarily differs from zero (so that they are not minimal surfaces). The outer volume created by these mesh surfaces consists of two half-spaces (above and below the surface), linked by an array of catenoid-like tunnels. The simplest mesh surfaces are of tetragonal, rhombohedral and monoclinic symmetries, corresponding to square (Fig. 4.10), hexagonal and oblique lattices of holes in each sheet. These mesh surfaces are also nearly homogeneous and offer an alternative to IPMS bilayers for hyperbolic molecular assemblies. We persist with the homogeneity approximation for hyperbolic surfaces, to calculate the local/global relation. The local constraint for a binary surfactant/water mixture is given by eq. 4.3. Further, if the interface is hyperbolic, one of the focal surfaces to the interface must lie at a distance equal to the chain length (l) from the interface, where the head-groups are located. (This condition ensures that no gaps between chains or chain overlap occur.) Consequently, the cross-sectional area for this parallel surface must vanish: l + 2HI + K l 2 --- 0

(4.11)

These equations (4.3 and 4.11) uniquely determine the interfacial curvatures v____.2_2 of the monolayer. It turns out that when a/ 3 , the monolayer must lie on a minimal surface, at which point it is curved equally towards both the lipophilic and the hydrophilic regions. If v/al lies between s and ~, 3 hyperbolic monolayers (of nonzero mean curvature, e.g. mesh surfaces) result, with the interface enclosing the lipophilic (chain) regions. If the interface consists of stacks of mesh surfaces, the global geometry (set by the composition) is independent of the local geometry (v/at), since the spacing between the mesh sheets can be altered without affecting the interfacial curvatures. However, if the hyperbolic monolayer is a single-sheeted hyperbolic sponge, leading to a

Surfactantmonolayers

157

structure where both hydrophobic and hydrophilic domains are continuous (a "bicontinuous" structure), the global geometry is dependent on the magnitude of the surfactant parameter. If it exceeds 2_ a reversed hyperbolic 3' monolayer results, which encloses the polar (hydrophilic) domain.

Fig. 4.10: Some square "mesh" surfaces, of differing (nonzero) mean curvature. LikeIPMS,these surfaces are hyperbolic, however, they are confined between parallel planes and form twodimensional tunnel networks on one side only of the surface. The resulting local/global "phase diagram" for elliptic, parabolic and hyperbolic monolayers, together with bilayers (fiat, as in the classical lamellar La phase, and hyperbolic) is shown in fig. 4.11. To generate specific data, the value of the tail length, l, is set to 14A, which is characteristic of a molten 12carbon tail, for which the fully stretched length is about 16J~. 4.6

Geometrical physics: bending energy

So far, the analysis has led to estimates of the average molecular shape (characterised by the value of the surfactant parameter) as a function of (i) the concentration of the surfactant and (ii) the geometry of the interface. The calculations summarised in Fig. 4.11 have been limited to homogeneous, or nearly-homogeneous, surface shapes. In the process, we have seen how the interfacial topology apparently affects the homogeneity: higher genera hyperbolic surfaces are generally less homogeneous than genus-two mesh surfaces and genus-three (or four) IPMS. Due to their inhomogeneity, hyperbolic and reversed micellar aggregates require variations in the

158

Chapter4

molecular shape in three-dimensional euclidean space. If a molecule has a preferred average geometry, departures from this preferred shape will necessarily incur some energy cost associated with the inhomogeneity.

Figure 4.11: Plot of the approximate compositions for which suz~ctant/water mixtures can form monola~rers versus the surfactant parameter of the surfactant. This plot is for chain lengths of 14A, which corresponds to hydrocarbons made up of about 12 carbon atoms. The notation for various mesophases is as follows: V1, V2 are bicontinuous cubic phases (the former containing two interpenetrating hydrophobic chain networks in a polar continuum, the latter polar networks in a hydrophobic continuum), H1 and H2 denote normal and reversed hexagonal phases, Lot denotes the lameUar phase, and L1 and L2 denote isotropic micellar and reversed micellar phases (made up of spherical miceUes). An estimate of this energy can be gained from the local model outlined above. If (v/al)o is the preferred surfactant parameter for a given surfactant species (which value depends on temperature, dilution, etc.), an elastic approximation implies that deviations from this preferred value incur a "bending energy" cost, analysed by Canham and Helfrich [12]. We adopt the following form for the bending energy per molecule:

Fbe~= Ic(~)0 - ~

(4.12)

where v/al is the actual surfactant parameter a d o p t e d b y the surfactant molecule w i t h i n the aggregate and K is an elastic b e n d i n g m o d u l u s . Molecular dynamics calculations show that this simple expression is a

159

Beratingenergy

reasonable one for a variety of surfactants, although the assumption of elasticity is valid only for small deviations about the preferred state [13]. For a lipid or surfactant bilayer, this expression can be written in terms of the mean and Gaussian curvatures (both scaled by the magnitude of the tail length, l) of the interface at the mid-surface of the bilayer: Fbend = tel (H 1- Hol~

+

/c2(K 12- KOI2~

(4.13)

where the material elastic moduli z l a n d ~:2 are coupled [14]. For an inhomogeneous aggregate, the bending energy density is obtained by integrating over the interface: I,, Ebend =

[~l (H I-Hol~ + ~2(K 12- K012)2] da "r

l,,,,~.,da

(4.14)

If the aggregate is a bilayer lying on an IPMS, scaled so that the average value of its Gaussian curvature, (determined at the mid-surface of the bilayer) is equal to K0, the elastic bending energy of the bilayer is a function of ~K2 da only. If different IPMS are scaled to give the same value of , the relative magnitudes of ~K2 da offer an estimate of the relative homogeneities of these IPMS and hence the relative stability of bilayers (containing a monodisperse distribution of surfactant molecules) folded onto these surfaces. Numerical calculations support the trends derived in section 4.4: lower genus IPMS are the most homogeneous candidates. Further, the most symmetric IPMS are also the most homogeneous. Cubic genus-three IPMS (the P-surface, D-surface, gyroid) can be formed with the least bending energy cost. Rhombohedral distortions of the P- or D-surfaces - leading to the rPD oneparameter family of IPMS - can also occur with only marginally greater bending energy cost than the cubic IPMS. This approach to bilayer stability is deficient in that it is dependent only on the local geometry of the bilayer. Hence, for example, the P-surface, D-surface and gyroid IPMS' are expected to be equally favourable (at slightly different concentrations, see Fig. 4.9), due to their identical local geometry. Where longer-range interactions are at work in the system, global measures of stability are in order. According to the simplest global forms (which favour equal distances between facing surface elements), the D-surface and gyroid are the most homogeneous in this global sense, with the narrowest distribution of pore radii.

160

Chapter4

Figure 4.12. (Top:) The binary phase diagram of didodecyl phosphatidylethanolamine-water mixtures. (Adapted from [15].) Single-phase regions are white, two-phase regions shaded. The thermotropic behaviour at about 20% w/w water is illustrated by the line ABC. (Bottom:) The trajectory of the line ABC in the local/global domain (see previous figure), showing the variation of molecular shape as a function of temperature for this lipid. The phase diagram can be reconciled with the local/global behaviour if the "lamellar" (L) phase is in fact a mesh structure, i.e. porous lameUae.

4.7

The mesophase behaviour of surfactant- and lipid-water mixtures

The free energy o f surfactant interfaces is due to interactions between water and the surfactant head-groups, as well as interactions between the surfactant chains, b o t h of w h i c h c o m p e t e to set the c u r v a t u r e s of the interface. Consequently, all else being equal, h o m o g e n e o u s interfaces are preferred over o t h e r g e o m e t r i e s for a m o n o d i s p e r s e d i s t r i b u t i o n of s u r f a c t a n t

S urfactan t-water mesophases

161

molecules in water. This explains the apparent presence of the "classical" mesophases, consisting of spheres (normal or reversed micelles, L1 and L2 phases, Fig. 4.1), cylinders (normal or reversed hexagonal, H1 and H2 phases) and planes (lamellar La phases) in surfactant solutions. In most cases, the complex array of interactions at work within an actual surfactant-water mixture leads to variations of the surfactant parameter with surfactant dilution and temperature. In general then, the phase behaviour of a binary surfactant-water mixture follows a curved trajectory through the local/global domain plotted in Fig. 4.11. If these variations in molecular conformation are small, the phase progression with water dilution is expected to follow a nearly-vertical line in the plot; if the molecular architecture is sensitive to these external parameters, the succession of phases with water dilution is more nearly horizontal. In the case of many zwitterionic double-chain surfactants or non-ionic diacyl (double-chain) lipids and surfactants (for which v/al >1) the addition of water has little effect on the mesophase. By contrast, temperature has a strong effect. Thus these systems are predominantly thermotropic, and a variety of mesophases result [15]. This feature can be traced to an increase in the wedge shape (increased v/al) due to enhanced fluctuations of the chains at higher temperatures, decreasing the effective tail length I. The behaviour can be explained with the help of the local/global relations plotted in Fig. 4.11. If v/al is close to, but exceeds, unity, the H2 (hexagonal) and L2 (lamellar) mesophases can form in very dilute mixtures only. At higher concentrations only the hyperbolic interfacial geometries remain accessible. However, for mesh or sponge-like monolayers and bicontinuous bilayers, the curvatures of the interface must be very low, since v/al=l. In all these cases, the aggregate geometry can be pictured as an array of lameUae, containing disclinations in the form of catenoidal tunnels. Since the (Gaussian) curvature, , is low, the density of these disclinations is also low, and there is no compelling energetic reason why the disclinations need order (to produce a crystalline mesophase exhibiting two- or three-dimensional order). Thus, these mesostructures are predominantly lamellar, and identified as conventional (parabolic) lamellar phases, although they may in fact be hyperbolic. Indeed, unless v/al is exactly unity, a planar interface (lameUar mesophase) incurs a bending energy cost; hyperbolic sponge monolayers or bilayers or mesh monolayer mesophases are favoured if v/al differs from unity. It is likely then that many "lamellar"" phases in fact adopt a hyperbolic geometry. Careful neutron-scattering studies of a lamellar phase have revealed the presence of a large number of hyperbolic "defects" (pores within the bilayers) in one case [16]. (An example of this mis-identification of hyperbolic phases in block copolymers is discussed in section 4.10.) The effect of temperature on double-chain zwitterionic surfactants and uncharged lipids is to increase v/al further beyond unity, allowing a range of a g g r e g a t i o n states. For example, the diacyl lipid d i d o d e c y l phosphatidylethanolamine mixed with about 20% water exhibits "lamellar", cubic (V2) and reversed hexagonal (H2) phases upon heating [15]. This

162

Chapter4

progression is consistent with a gradual increase in the value of v/al from a value marginally larger than unity to about 1.5 (Fig. 4.12).

Figure 4.13. (Left:) The binary phase diagram of AOT-water mixtures (after [17, 18]). The lyotropic behaviour at room temperature is illustrated by the line ABC. (Right:) The trajectory of the line ABC in the local/global domain. The mesomorphism of ionic surfactants and lipids is only weakly temperature dependent, in contrast to non-ionic and zwitterionic systems. However, a variety of mesophases are formed as the concentration of these charged molecules is varied. This effect is due to the dependence of the molecular shape on the electrostatic conditions. The addition of water to ionic surfactants generally increases the lateral electrostatic repulsion between the head-groups, due to enhanced dissociation within the lipid salt. This leads to a decrease in the magnitude of v/al, so that a variety of lyotropic mesophases can result. For example, the double-chain ionic surfactant sodium di-2-ethylhexyl sulfosuccinate (AOT) forms reversed hexagonal (H2), bicontinuous cubic (V2) and "lamellar" (La,, denoted "L" in Figs. 12-15), mesophases on water dilution at room temperature [17, 18]. This behaviour can be traced to a gradual decrease in the value of the surfactant parameter upon water dilution (Fig. 4.13). Lyotropic mesomorphism is also exhibited by charged single-chain surfactants, for which v/al is usually less than unity. The remarkable variety of mesophases detected in sodium dodecyl sulfate (SDS)-water mixtures [19, 20] can be understood by invoking a similar trend of decreasing v/al with water content as that found in the AOT-water system (Fig. 4.13), although the magnitude of v/al is smaller than that of AOT, due to the presence of a single chain only (Figure 4.14). The geometrical analysis supports the contention that the rhombohedral and tetragonal intermediate phases contain hyperbolic interfaces [9]. In the next section we will discuss these hyperbolic mesophases in detail.

Surfactant-watermesophases

163

Figure 4.14. (Left:) The binary phase diagram of SDS-water mixtures (after [21]). The tetragonal mesophase is denoted T1, rhombohedral, R1 and monoclinic, M1. The lyotropic behaviour at room temperature is illustrated by the line ABC. (Right:) The trajectory of the phase progression marked by the line ABCD in the local/global domain. The three examples cited above illustrate the power of the simple block developed here. While complex physics governs the variation of the surfactant molecular shape with temperature and concentration, clear trends can be seen for the various classes of surfactants and lipids.

4.8

The hyperbolic realm: cubic and intermediate phases

A central issue in the field of surfactant self-assembly is the structure of the liquid c r y s t a l l i n e m e s o p h a s e s d e n o t e d b i c o n t i n u o u s cubic*, and "intermediate" phases (i.e. r h o m b o h e d r a l , monoclinic and tetragonal phases). Cubic phases were detected by Luzzati et al. and FonteU in the 1960's, although they were believed to be rare in comparison with the classical lamellar, hexagonal and micellar mesophases. It is n o w clear that these phases are ubiquitous in surfactant and lipid systems. Further, a number of cubic phases can occur within the same system, as the temperature or concentration is varied. Luzzati's group also discovered a n u m b e r of crystalline mesophases in soaps and lipids, of tetragonal and rhombohedral symmetries (the so-called "T" and "R" phases). More recently, Tiddy et al. have detected systematic replacement of cubic mesophases by "intermediate" T and R phases as the surfactant architecture is varied [22-24]. The most detailed mesophase study to date has revealed the presence of monoclinic, *This phase should not be confused with the 'T' discontinuous cubic phase formed in some systems, which consists of a cubic packing of closed aggregates.

164

Chapter4

rhombohedral and tetragonal phases, as well as the better-known hexagonal, cubic and lamellar phases, in aqueous "solutions" of SDS (Fig. 4.14). It is surely no coincidence that the symmetries of these exotic mesophases are precisely those of the most nearly-homogeneous hyperbolic surfaces, which are expected to lead to aggregates of relatively low bending energy (section 4.6). Among possible bilayer geometries, low genus three-periodic minimal surfaces of cubic and rhombohedral symmetry are the most homogeneous. Similarly, the most homogeneous monolayer geometries are mesh surfaces of rhombohedral, tetragonal and monoclinic symmetry. Detailed data are available for some cubic phases, which offer a good test of the model of hyperbolic bilayers wrapped onto IPMS. In these cases, the results suggest that these phases consist of bilayers of cubic symmetry, whose mid-surfaces closely follow IPMS [25, 26]. The equations developed earlier in this chapter offer a useful route to decoding the mysteries of these cubic phases. For example, in a cubic mesophase, the surface averaged value of the Gaussian curvature at the midsurface of the bilayer, , is related to the lattice parameter of the cubic unit cell (o0 via the equations: c~= VIt3 ~ ~ =

(4.15)

using eq. 4.8 where H is the homogeneity index and X is the Euler-Poincar6 characteristic per (cubic) unit cell of the bilayer. Eq. 4.10 allows the lattice parameter to be related to the surfactant parameter, the homogeneity index (=~) and the Euler-Poincar(~ characteristic alone. Typically, the surfactant parameter varies between about 1.05 and 1.5 for cubic phases (Fig. 4.11) and the cubic bilayers wrap onto IPMS whose genera lie between three (Z=-4) and five (Z=-8) per unit cell. This implies that the lattice parameter, c~, varies between about 80~ and 250~ in homogeneous cubic phases - consistent with measured values [27]. These geometrical constraints are not sufficient to predict the occurrence of a mesophase, since the temperature of the surfactant-water mixture must be sufficiently high to overcome the bending energy cost associated with a particular bilayer geometry and topology, but not too high to "melt" the bilayer crystalline mesophase. However, we are able to know something about the expected relative location of mesophases - if they occur - within the phase diagram. Some information is available that complies with the progression of IPMS theoretically expected upon water dilution. There is here a critical test of theory: the expected progression of symmetries of crystalline mesophases within the phase diagram. Including only the low genus cubic minimal surfaces, the relative location of these phases is expected to be (in order of increasing water content, cf. Fig. 4.9):

Cubic & intermediate phases

165

9the gyroid ("CG" phase, space group Ia3d, genus three) 9the D-surface ("CD" phase, Pn3m, genus three) sthe I-WP surface (Imam, genus five) 9the P-surface ("Cp" phase, Im3m, genus three) This progression fits well experimental data collected so far. In the binary glycerol monooleate-water system there is conclusive experimental evidence for the presence of a body-centred structure of symmetry Ia3d at lower water contents and a primitive cubic bilayer (symmetry Pn3m) at higher water contents [28] (Fig. 4.15).

Figure 4.15: Partial phase diagram of glycerol monooleate-water mixtures, after [29]. This finding fits with the expected relative phase locations well; the bilayer transforms from the C G phase to the C D phase upon dilution [29]. The details of this transformation are discussed in more detail in section 5.1.3. Further evidence that supports these calculations derives from studies of the ternary mixtures of the cationic double-chain surfactant DDAB (didodecyl dimethyl ammonium bromide), cyclohexane and water. Within the cubic mesophase region of this surfactant-water-oil mixture, all the cyclohexane is adsorbed between the surfactant chains, so that the system is a pseudo-binary one, for which our theoretical analysis ought to hold. (The effective surfactant parameter for this surfactant in the presence of cyclohexane is slightly larger than unity.) Close scrutiny of the cubic phase region within this ternary phase diagram has revealed the presence of at least one - and

166

Chapter4

possibly more - first order phase transformations within this region (Fig. 4.16) [30,311. At lower water contents of the single phase region, the mixture forms a cubic phase of symmetry Pn3m, and structural analysis indicates that the structure consists of a bilayer lying on the D-surface (C D phase), as in the glycerol monooleate solution. At higher water fractions, the cubic phase spontaneously forms a phase of symmetry Im3m [32]. Within this region of the ternary phase diagram, there is some calorimetric evidence of a further transformation, although the symmetry of the mesophase is unchanged. This may indicate a symmetry-preserving topological change in the bilayer, e.g. from the P-surface to the I-WP surface (both of symmetry Imam, described in the Appendix to chapter 1).

Figure 4.16: A portion of the ternary phase diagram of DDAB/cyclohexane/water mixtures showing the cubic phase region. Within this region, the mixture forms symmetries of Pn~.mand Im3m at low and high water contents respectively. Measurements of the lattice spacing as a function of the surfactant concentration fit well expected values, assuming the Pn3m mesophase is a bilayer wrapped onto the D-surface, and the Im3m phase follows the Psurface. The analysis is sensitive to both the surface to volume ratio of the model interface and the topology of the interface, yet the data is remarkably consistent with the assumption of bilayers wrapped onto IPMS (Fig. 4.17). The earliest structural models (by Luzzati et al.) for these bicontinuous cubic phases involved interpenetrating networks of polar rods, immersed in the hydrophobic continuum. These remarkable models were consistent with the experimental data, although it was difficult to understand why the structures should be so complex compared with much simpler geometries of the hexagonal and lameUar phases. These rod models are simply related to our hyperbolic surface models: the rods lie in the interpenetrating tunnel networks defined by the surface. The genesis of these phases is now clear: they arise due to the dual requirements set by the molecular shape and the composition. These constraints force the bilayer to adopt a hyperbolic geometry whose Gaussian curvature varies as little as possible, ergo three-

167

Cubic & intermediatephases

p e r i o d i c m i n i m a l surfaces of cubic (and p o s s i b l e r h o m b o h e d r a l ) s y m m e t r y . The other family of hyperbolic surfaces that are n e a r l y h o m o g e n e o u s are the m e s h surfaces, w h i c h also a p p e a r in these s y s t e m s , classified as i n t e r m e d i a t e phases.

3.5

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.

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1.5 I

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.18

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Figure 4.17: Plots of A/20t2 (A denotes the head-group area per unit cell and c~the lattice spacing) vs. 2rd2/r 2 (I denotes the chain length) measured in the Pn~m and In~m cubic mesophases found in DDAB-cyclohexane-water mixtures for various surfactant concentrations. The curves are calculated assuming the aggregate consists of a bilayer whose mid-surface lies on the P- and D -surfaces and the DDAB density is equal to l gcm"3. The chain length is equal to 11~ (measured in the neighbouring lamellar mesophase) and the head-group area is equal to 68]k2.

The e x p e r i m e n t a l d a t a for i n t e r m e d i a t e p h a s e s is less conclusive. Luzzati et al. p r o p o s e d a p l a n a r rod structure for the R p h a s e in lecithin-water m i x t u r e s

168

Chapter 4

[33]. Just as for cubic phases, the rod description of the R phase is an approximation to a hyperbolic surface. The smooth surface that defines the hydrophobic-polar interfaces resembles mesh surfaces containing a hexagonal array of pores. The genesis of this phase can also be understood as a resolution of the requirement for quasi-homogeneous interface. An intermediate phase of tetragonal symmetry - the T phase - has also been detected in a number of systems. A rod structure related to a square mesh surface was found to agree well with X-ray and NMR data on a perfluorinated surfactant-water mixture forming the T phase [22], [34]. These examples demonstrate that surfactant or lipid monolayers lining mesh surfaces as well and bilayers wrapped onto three-periodic minimal surfaces (IPMS) are indeed found in these self-assembled systems.

Fig. 4.18: TEM images of freeze-fractured specimens of egg lecithin (5 mM) mixed with dodecyl trimethyl ammonium chloride (12 mM) in slightly saline water (12 mM NaCI). The left image shows regions of two vesicles, exhibiting fine structure within the walls. The right image shows a single bilayer (near top), riddled with perforations, following the topology of the mesh surfaces described in the text. Images courtesy of B. Gustafsson, K. Edwards, G. Karlsson and M. Almgren. Under certain conditions (e.g. high dilution) translationally disordered mesh structures can form. Striking confirmation of such a structure is to be found in transmission EM images of rapidly-cooled lipid/detergent mixtures which form "bilayers" in excess water, often folded into vesicles. Here certain regions display perforations within the bilayers of varying dimensions (fig. 4.18). The relative stability of mesh and IPMS structures is still unclear. For example, the R1 mesophase (of rhombohedral symmetry) in the SDS-water system transforms continuously into the neighbouring bicontinuous cubic phase (Fig. 4.14) [20]. This suggests that this mesophase is a hyperbolic (reversed) bilayer lying on a rhombohedral IPMS. Indeed, the rhombohedral rPD surface is only marginally less homogeneous than its cubic counterparts, the P- and D-surfaces.

Cubic& intermediatephases

169

It is clear from the universal diagram (Fig. 4.11) that a variety of bilayer phases can form only if the surfactant parameter is between about 0.5 and 1.7. For higher values of the surfactant parameter, steric constraints (e.g. headgroup crowding) preclude the formation of curved hyperbolic bilayers or monolayers. The opportunity for bilayer polymorphism exists for surfactant parameters lying between 1.0 and about 1.5. These bilayer phases are expected to adopt cubic or rhombohedral symmetries, corresponding to the most homogeneous three-periodic minimal surfaces. The geometric analysis indicates that cubic (and other curved bilayer phases) are only expected to be found in surfactant solutions that also form "lamellar" phases readily (v/al=l). This fact alone suggests that hyperbolic geometry may be present in biological systems whose lipids satisfy this constraint. In Chapters 5, 7 and 8 we shall see that these non-euclidean forms are indeed present in a variety of living systems: from cell membranes to muscle assemblies. Although a purely geometric approach to the issue of selfassembly has limitations, once the existence of hyperbolic interfaces is taken into account, the apparently baffling array of different phases (cubic, rhombohedral, tetragonal) are seen to be nothing more than a catalogue of low genus ordered monolayer and bilayer structures, whose relative locations are determined by the molecular shape and the composition. It seems that bicontinuous crystalline phases form to minimise the bending energy of the bilayers. The changes of symmetry and topology within these phases are a response to the geometric demands placed upon the interface by the local constraint (which is set by the architecture of the surfactant molecules) and the global volume requirements (imposed by the composition of the mixture). In this approach, the effect of entropy on the aggregation geometry has been ignored. Some elegant recent work of Marcelja and Pieruschka offers promise as a means of including entropic contributions to the free energy of elastic surfaces [35]. Until this promise is fulfilled, we persist with this approximation, which is valid for interfaces whose bending moduli are sufficiently high to overwhelm thermal entropic effects. It is interesting to note that similar mesostructures, although disordered, are reproduced by their analysis, which is complementary to that detailed here. A limitation of the calculations presented so far is the assumption of a single preferred molecular shape (v/alo). This confines the model to "clean" systems, with a monodisperse distribution of amphiphilic molecules. Clearly, such an assumption is violated in many cases of industrial and biological interest. As a general rule, the inclusion of a range of amphiphilic molecular shapes allows an even greater variety of interfacial forms than those considered to date, since inhomogeneous interfaces may then be preferred, allowing partitioning of different amphiphiles into domains of different curvature. For example, a variety of higher genus (per unit cell) minimal surfaces emerge as energetically favourable bilayer geometries, leading to more complex assemblies. In these cases, hyperbolic mesophases are expected to be more readily formed than in monodisperse systems. A striking example

170

Chapter 4

of this possibility has been found in the ternary DDAB-styrene-water system, where no less than five bicontinuous cubic phases have been detected [36].

4.9

Mesostructure in ternary surfactant-water-oil systems: microemulsions

The geometrical constraints for a ternary mixture of surfactant, nonpolar solvent and water are less easily calculated for more general systems, which contain more than two components. Nevertheless, geometric considerations lead to similar behaviour as has been detailed in the previous sections. To illustrate this point, we analyse the microstructure of simple ternary microemulsions, consisting of a mixture of the cationic surfactant, didodecyl dimethyl ammonium bromide (DDAB), a range of alkanes and water. If water is added to a mix consisting of surfactant, alcohol and hydrocarbons, the resulting cocktail is usually an inhomogeneous mixture of many liquid phases. However over certain composition ranges, the mixture spontaneously self-assembles into a liquid whose appearance is indistinguishable to the naked eye from pure water. This is a "microemulsion". The most common definition of a microemulsion characterises it as a thermodynamically stable, transparent, optically isotropic, freely flowing surfactant mixture, often containing co-surfactants (e.g. alcohol) and added salts [37]. We restrict the definition further to non-crystaUine (disordered) aggregates, since crystalline isotropic phases are better considered as liquid crystalline mesophases. Indeed, the most succinct description of a microemulsion would involve its microstructure. However, this has proven to be a very equivocal issue. So much so that until very recently it was widely believed that microemulsions were devoid of microstructure hence the thermodynamic definition. The structure of microemulsions can change widely with composition within the domain of existence of the single "phase", despite the fact that some microemulsions are synonymous with "micellar" or "reversed micellar" phases. We shall see that the topology of the interface as well as the geometry varies continuously as a function of composition throughout the single phase region. The geometry adopted by the monolayer interface of surfactant separating oil and water spans a wide range of hyperbolic bicontinuous interfacial geometries as well as conventional spherical and cylindrical interfaces. In general it is very difficult to pin down the microstructure of microemulsions. Scattering probes yield only a single, broad scattering maximum, which taken alone is not very informative. To further aggravate the problem, many microemulsions consist of at least four components. Since the usual co-surfactant additive can partition between oil and water and the interface between them, it is impossible to sort out the structural

171

Microemulsions

parameters of the mix: the interfacial area and volume fractions on either side of the surfactant interface. But some double-chain cationic surfactants form microemulsions w h e n mixed with only water and oil over a large region of the ternary phase triangle [38, 39]. These surfactants are virtually insoluble in both water and oil and therefore are located exclusively at the oil-water interface. This aids structural analyses significantly. We shall focus on mixtures containing DDAB. Some typical phase diagrams for these mixtures are reproduced in Fig. 4.19.

DDAB

DDAB

v i a l = 1.83

f

v / a l -- 1.53

DDAB

Figure 4.19: Single phase microemulsion regions within the ternary phase triangles of mixtures of DDAB/water and a range of hydrocarbons. (Adapted from [40]). A variety of experimental techniques are available to investigate the structure of microemulsions: small angle scattering, specific heat, viscosity and electrical c o n d u c t i v i t y m e a s u r e m e n t s . In the DDAB systems, conductivity measurements exhibit a dramatic decrease (typically eight orders of magnitude) as water is added to the mixture. Such changes occur over just a few percent variation in water content, apparently difficult to reconcile with the fact that the oil is (relative to water) non-conducting. It implies that the

172

Chapter4

aqueous regions are becoming less connected as the volume fraction of water increases, a counter-intuitive trend. Calculations of the small-angle x-ray scattering expected from a disordered array of reverse miceUes (whose dimensions can be accurately determined for this system since the interracial area and volume fractions are well known) differ markedly from measured scattering spectra, except in the most waterrich microemulsion mixtures. Only at the highest water contents which form microemulsions alone, are conductivity and X-ray spectra consistent with water-filled reverse miceUes embedded within an oil continuum. The most water-dilute boundaries of the microemulsion phase regions within the ternary phase triangle of our systems are invariably lines of constant surfactant to water fraction (see Fig. 4.19). It is easy to show that this implies the micellar radii at the upper water limit of the microemulsion region are constant, irrespective of the oil content. The implication is that the curvatures of the surfactant interface, which separates the hydrophilic from hydrophobic regions, are fixed within this region. The curvatures of this interface are related to the molecular dimensions of the surfactant (plus any oil absorbed between the surfactant chains), characterised by the effective surfactant parameter, I ~ l , by the equation: xr le8

/+ ~)eff v = l + r s 3r12 2

(4.16)

where v is the hydrophobic (chain) surfactant volume (plus any absorbed oil), a is the head-group area and I is the chain length of the surfactant. The phase diagrams shown in Fig. 4.19 indicate that the water to surfactant ratio at the upper water limit of the microemulsion region increases with the chain length of the hydrocarbon in the ternary mixture, which means the micellar radii also increase. This indicates a reduction in the value of the effective surfactant parameter as the hydrocarbon molecular length increases. The trend seems to be universally valid, and is explicable in terms of a reduction in the a m o u n t of hydrocarbon absorbed between the chains as the hydrocarbon chain length increases and becomes less polar (thereby reducing the effective volume, v). Thus, the molecular shape of a surfactant molecule can be varied by the addition of hydrophobic solvents. As a rule, if the dimensions of the solvent are less than or equal to those of the surfactant chain, the solvent resides among the surfactant chains, swelling the hydrophobic region of the surfactant relative to its hydrophilic moiety, and increasing the value of the effective surfactant parameter from the surfactant parameter of the bare surfactant. This increase is most marked for smaller hydrophobic solvent molecules. For example, DDAB has a "bare" surfactant parameter just less than one; the effective surfactant parameter is about 1.6 for DDAB with cyclohexane, 1.4 with octane, 1.2 with decane and 1.0 with tetradecane.

Microemulsions

173

Within a simple ternary DDAB-water-hydrocarbon system, it is reasonable to expect that the effective surfactant parameter remains approximately constant throughout the triangular phase diagram, just as it does along the u p p e r water limit. (Note however, that the head-group area can change at low water fractions due to the effects of hydration on the polar head.)

Figure 4.20: Artist's impression of the interfacial geometry of a ternary microemulsion made up of a double-chain cationic surfactant "dissolved" in a mixture of water and short chain hydrocarbons. The connectivity of the surfactant interface - which encloses the water network decreases as water is added to the mixture (left: average coordination number of four; right: average coordination number of two). As in binary surfactant-water systems considered previously, two constraints on the geometry of the surfactant interface are active: a local constraint, which is due to the surfactant molecular architecture, and a global constraint, set by the composition. These constraints alone are sufficient to determine the microstructure of the microemulsion. They imply that the expected microstructure must vary continuously as a function of the composition of the microemulsion. Calculations show - and small-angle X-ray and neutron scattering studies c o n f i r m - that the D D A B / w a t e r / a l k a n e microemulsions consist of a complex network of water tubes within the hydrocarbon matrix. As water is a d d e d to the mixture, the Gaussian curvature - and topology [41]. Thus the connectivity of the water networks drops! (Fig. 4.20). decreases The microstructure can be described in terms of the average coordination number of water tubes, which is defined as the average n u m b e r of tubes sharing a c o m m o n node. Contours of constant coordination n u m b e r are roughly parallel to the surfactant-oil axis of the ternary phase diagram, so that oil dilution does not alter the topology of the interface. Percolation theory suggests t h a t a r a n d o m n e t w o r k loses all continuous paths t h r o u g h the network when the average coordination number drops below about 1.1. We

174

Chapter4

thus expect the electrical conductivity of a microemulsion (which is d e p e n d e n t on continuous diffusion paths within the polar phase) to essentially vanish for lower coordination numbers. This explains the catastrophic conductivity drop in dilution described above. The expected corresponds to the contour of coordination number equal to 1.1, which is consistent with conductivity data [42].

Figure 4.21: Microstructures of self-assembled surfactant monolayersformed as a function of water content, assuming a fixed surfactant parameter. The graphs illustrate the variations of Gaussian and mean curvature as a function of water content. The lower drawings illustrate schematic cross-section through aggregates where the surfactant parameter is larger than unity (upper axis, internal shaded regions polar) and less than unity (lower axis, internal shaded regions hydrophobic). The general situation for surfactants whose surfactant parameter is larger than unity (i.e. small head-groups relative to the chain volume) is as follows. At very low internal fractions, the interface consists of a surfactant monolayer of high topology (connectivity), with a sponge-like morphology. As the internal volume fraction is increased, the topology of the monolayer decreases gradually, although the dimensions of the tubules within the sponge increase, to accommodate the extra internal volume. Eventually the internal network forms a spaghetti-like structure (topologically equivalent to an array of cylinders). At still higher internal volume fractions, a sufficient number of connections are severed to remove all connected paths through the polar domains. Finally, the internal polar volume is completely

Microemulsions

175

partitioned into isolated spheres, and the mixture consists of reversed spherical miceUes. This progression is represented schematically in Fig. 4.21. This behaviour is a good example of the counter-intuitive nature of these structural phenomena. It seems absurd to suggest that the monolayer transforms from a convoluted, highly interconnected sponge to spheres as the internal fraction increases, but this effect is a simple manifestation of the constraints imposed by the surfactant molecular architecture. If this problem is considered in terms of the curvatures of the interface rather than the global shape, it is fairly easy to understand the structural progression. At very low internal volume fractions the mean curvature of the interface must be large (to fulfill the global constraint set by the surface to volume ratio), forming tubes of small cross-section. But if this is large (and positive - convex), the Gaussian curvature must be large and negative, to maintain the surfactant parameter at its preferred value (eq. 4.1). As the internal volume fraction increases, the mean curvature of the interface decreases to satisfy the global constraint, and the Gaussian curvature slowly increases, maintaining the value of the surfactant parameter. At sufficiently large internal volume fractions, the Gaussian curvature becomes positive (Fig. 4.21). The relation between surface topology and Gaussian curvature (eq. 1.15) offers a simple explanation of the global changes in interfacial morphology. The theorem states that the topology is proportional to the (negative) magnitude of the Gaussian curvature. Since the Gaussian curvature of these monolayer structures becomes less negative as the internal volume fraction increases, the topology of the interface must also drop. Thus, the monolayer initially forms a highly inter-connected structure, and slowly transforms to cylinders (=0) then spheres (, both positive; = 2) as the internal volume fraction is raised (Fig. 4.21). The formation of spheres exhausts the range of geometrically accessible monolayer structures, since the topology of spheres is lower than that of all other shapes. (In more familiar terms, spheres minimise the surface to volume ratio - thus soap bubbles form spheres. In other words, the volume associated with a unit surface area of a surface is maximised if that surface forms a part of a sphere. So we expect the surfactant interface to become spherical as the internal volume associated with each surfactant molecule becomes large.) This critical volume fraction is not particularly large; for typical double-chain surfactants it lies between 20% and 50%. Many more common microemulsions are formed by single-chain surfactants, in which case the structural sequence on water dilution is reversed. However, the microstructures are identical. This model calculation illustrates a recurring theme in this book: the notion of an intrinsically preferred curvature implies profound consequences for structure. The analysis of mesostructure of these microemulsions is helped by the fact that DDAB resides exclusively at the interface. In m a n y microemulsions this is not the case, so that more detailed calculations are

176

Chapter 4

required to estimate the global geometry, i.e. the surface to volume ratio. Nevertheless, the results are qualitatively identical.

4.10

Block copolymer melts: an introduction

The limitations imposed on self-assembled structures by the geometry of space have led us some way towards full elucidation of the phase behaviour of simple self-assembling systems, like binary surfactant/water mixtures. The gross morphological changes that occur as the composition is changed (the topology of monolayers or bilayers) seem to be predominantly constrained by the restrictions of space. However, the finer details of the complete phase diagram remain uncertain. So far, no attempt has been made to understand the variations in the molecular dimensions of the amphiphiles as a function of the composition. It is certain that hydration effects are involved, but we really have little idea of the physics underlying these effects, which are central to the intricacies of biology. We now turn to a complementary model of self-assembly of polymeric molecules, for which the molecular shape is flexible. In the ideal limit where the self-assembly is driven exclusively by entropic demands of the polymer chains, the molecular shape can be calculated as a function of the global constraints imposed by the composition. The dimensions of small amphiphilic molecules (particularly uncharged lipids) are nearly independent of their concentration and the variations in the surfactant parameter are small, so that their progression of lyotropic phases as a function of water dilution follows a nearly vertical trajectory in the local/ global domain drawn in Fig. 4.11. By contrast, the long flexible chains characteristic of polymers continually adjust their dimensions in response to their environment (at constant density), so that the structural transitions occurring as a function of concentration are mapped into nearly vertical lines in the local/global domain (Fig. 4.11). The conformational changes as a function of concentration of ionic and zwitterionic surfactants (shown in Figs. 4.12-4.14) and lipids, polyelectrolytes and some proteins lie between these extremes. The features of self-assemblies of all of these (achiral) molecules have much in common within the perspective afforded by the languages of differential geometry and topology. A number of experimental results suggest that as rich a variety of aggregation geometries can be found in copolymer melts as has been described in amphiphilic assemblies. In particular, "interpenetrating networks", which appear to be related to the D-surface and/or the gyroid, have been discovered in star and linear block copolymers [43, 44].

Block copolymers 4.11

177

Copolymer self-assembly

Copolymers consist of at least two distinct polymer species that are grafted onto one another. The simplest molecular geometries of these copolymers are linear diblock copolymers, which are made up of two polymer moieties (A and B), chemically bonded to form a single flexible polymer chain. A number of other molecular topologies have been synthesised besides linear copolymers, including star, brush.., copolymers (Fig. 4.22). If the blocks are immiscible these copolymers self-assemble spontaneously in solution to form a variety of phases - usually non-crystalline. If the solvent is evaporated off to leave the pure melt, a variety of mesophases can result, depending upon the temperature of the system and the relative molecular weights of each component in the copolymer molecule. At sufficiently low temperatures, the mesophases are often liquid crystalline. For example, spherical micellar (often in a body-centred cubic arrangement), hexagonal and lamellar phases are well known in these systems [45]. A physical understanding of these mesophases in block copolymers is certainly not to be found in the rigid-block model, since these copolymers are very flexible molecules, able to adopt a number of different shapes in response to environmental conditions. However, we shall see that even in this extreme case, we can talk about an (average) molecular shape, which induces a similar variety of mesophases to those found in surfactant systems. The dominant contribution to the free energy of lengthy (rubbery) polymer chains is entropy. This is known to account for rubber elasticity, which can be satisfactorily modelled by the entropy of the cross-linked polymer chains alone. A simple illustrative model of copolymer self-assembly can be developed by extending rubber elasticity theory to include bending as well as stretching deformations, to calculate chain entropy as a function of interfacial curvatures in diblock aggregates. The presence of two distinct blocks in diblock molecules means that the final configuration of a copolymer molecule involves a compromise between the preferred states (of maximal entropy) of each moiety. The immiscibility between the blocks is modelled by assuming segregation between the A and B domains, with an effective surface tension acting at the AB interface. The mesostructure of the aggregates can then be characterised by the geometry and topology of the interface(s) separating the block domains. The model leads to a variation of structure as the relative fractions of each block within the copolymer molecules changes. This can be seen heuristically as follows. If the individual blocks have different relaxed radii of gyration (set by the maximum entropy constraint imposed on each block species on its own), the existence of a bond that fuses the moieties in the copolymer imposes the condition that assemblies of the copolymers cannot form without perturbation of the original configurations (Fig. 4.23(a),(b)).

178

Chapter4

Figure 4.22: Schematic drawings of various block copolymers. These long-chain molecules synthetic molecules consist of chemically distinct polymeric "blocks" (denoted by lines of different thicknesses in the figure), chemically grafted. (Left to right:.) Linear diblock copolymer molecule (AB); linear triblock (ABC);star copolymer; brush copolymer. If the blocks are mutually immiscible, under suitable conditions the molecules spontaneously clump together forming an array of mesophases. The first requirement is that the cross-sectional areas for each block be equal at the interface between the blocks. Let's assume for simplicity that the blocks are completely immiscible, so that the interface between the blocks occupies zero width and is an ideal surface. In order for this area requirement to hold, stretching a n d / o r shrinkage of the chains must occur, so that the unperturbed areas are reduced or increased respectively (Fig. 4.23(c)). This process results in a significant loss of entropy, particularly for the blocks of higher molecular weight, due to the necessary anisotropy induced into the chain configurations (i.e. a unique director normal to the interface). Some of this entropy loss can be regained by curving the interface away from the chain, thereby restoring a measure of isotropy to the chain (Fig. 4.23(d)) [46]. A different approach to the problem can be taken, wherein the free energy of the melt is determined as a function of the curvatures of the interface. This then allows us to sample all possible (local) geometries. Ignoring the effects of interdigitation and chain tangling, we can assign an average shape to each copolymer molecule as a function of the local geometry of the interface (using the results in section 4.5), and calculate the entropy of the induced shape. The entropy content is set by the stretching and bending of the relaxed chains, assumed to be rubber-like. Since the density is necessarily uniform, the molecular volumes for each shape must be constant and the chain is assumed to undergo an affine transformation (which conserves volume) to form the curved interface. The detailed calculations can be found elsewhere [47]. Although this intuitive picture is not exact (for instance, we have not allowed for interdigitation between the chains), it captures the essential physics behind the self-assembly process. Many sophisticated theories have been presented for self-assembly in polymer melts. Most suffer from the usual limitation associated with the usual geometrical problem: they cannot

Block copolymers

179

handle the full ensemble of interfacial geometries, and have been restricted to spherical, cylindrical or planar geometries. These local calculations predict the formation of planar, hyperbolic, parabolic (cylindrical) and elliptic (globular) interfaces as the v o l u m e fraction of the larger block increases from 50%. The compositional range of existence of the various interfacial geometries d e p e n d s to a limited extent on the effective surface tension acting at the interface.

Figure 4.23: The origin of curved interfaces in aggregated diblock copolymer melts. If the natural radii of gyration of each of the two moieties making up the copolymer molecules are different (a), the molecules cannot self-assemble to form uniform densities within each region without some conformational changes (b). Some extension/compression of the chains must occur so that they can close pack at the interface (c). The interface is generally curved, to increase the isotropy (and entropy) of the major component of the copolymer (d). Some typical results are plotted in Fig. 4.24, which relates the interfacial geometry, to the degree of stretching of the larger block [47]. These local results suggest that the presence of hyperbolic interfaces in diblock c o p o l y m e r assemblies is e x p e c t e d for a r a n g e of m o l e c u l a r architectures. For these structures to be realisable in space, the curvatures m u s t be compatible with global dimensions (e.g. the radii of curvature m u s t allow a tiling of space without overlap or voids). The global problem of copolymer self-assembly can be tackled as follows. For a certain composition of A-B c o m p o n e n t s (which constrains the v o l u m e

18o

Chapter 4

fractions required on each side of the interface), the surface area is coupled to the curvatures of the interface, provided the curvatures of the interface are nearly homogeneous. (This follows from the fact that the surface to volume ratio can be expressed approximately as a function of the interfacial curvatures alone, discussed in detail in earlier sections.) The equilibrium interfacial (surface-averaged) curvatures can then be determined by minimising the free energy expression with respect to the curvatures.

Figure 4.24: Schematic "phase diagram" for AB linear diblock molecules as a function of the molecular composition and the stretch factor, ~. To obviate calculations for an infinite number of particular structures, we determine the optimal surface-averaged curvatures for each of the three geometrical classes - ellipsoidal ( > 0), parabolic ( = 0) and hyperbolic ( < 0). These correspond to isolated regions of the minority fraction in a continuous matrix of the major component, cylinders of one component in the other, and bicontinuous structures respectively. The resulting energies invariably favour elliptic and parabolic geometries over hyperbolic forms in contrast to the locally favoured forms, due to the (severe) global constraints. However, with increasing surface tension, hyperbolic "bilayers" are of comparable free energy to planes and cylinders over a small composition range (Fig. 4.26). These structures consist of intertwined A and B labyrinths, both continuous. A number of different global embeddings of this hyperbolic phase appear to occur in block copolymers, just as in surfactant- and lipidwater mixtures. So far, morphologies consistent with the symmetry and topology of the D-surface and gyroid have been reasonably definitively characterised [43, 44]. (These IPMS - or closely related surfaces - define the mid-surfaces of the domain containing the bulkier block.) Beautiful images of these phases have been recorded (Fig. 4.25).

Block copolymers

181

Figure 4.25. High magnification images obtained by transmission electron microscopy of stained thin sections of a hyperbolic mesophase of a linear diblock copolymer, polystyrenepolyisoprene, whose morphology follows the D-surface (single node circled in middle picture) and possibly the gyroid. Staining produces high contrast between the two block domains. Note the very different magnifications.

182

Chapter 4

Figure 4.25 (continued). Note the long-range order evident in this image. (Images courtesy of H. Hasegawa. Reproduced with permission from [43].)

Within each geometric class, the optimal interfacial geometry will be that which has the smallest curvature variations about the optimal values of the local curvatures (i.e. the most homogeneous structure), consistent with the required interior and exterior surface to volume ratios. In the case of elliptical and parabolic interfaces, these constraints are sufficient to fully determine the interfacial structure. Table 4.2: Number of faces in Voronoi cells of some two- and three-dimensional arrays. Array

No. of faces in the Voronoi cell 2-dimensional arrays:

square

4

hexagonal

6 3-dimensional arrays:

simple cubic cubic close packing random dose-packing bcxiy-centred cubic

6 12 13.4 (spatial average) 14

For these convex surfaces, the free energy of the copolymer assembly is a function of the mutual arrangement of the (unconnected) surfaces in space.

Block copolymers

183

In order to m i n i m i s e the free energy, the p o l y m e r s e g m e n t s o u t s i d e the interface should be as close to the optimal chain length as possible. Again, the question of homogeneity is p a r a m o u n t for the global geometry. This p r o b l e m can be analysed in terms of Voronoi cells for various packings. The Voronoi cell (also k n o w n as the Dirichlet region) about a lattice site is defined to be the p o l y h e d r o n w h o s e faces bisect p e r p e n d i c u l a r l y the line segments joining a lattice point to adjacent points. The optimal Voronoi cell - which defines the region of space closer to that surface than any other, and hence sets the region of space available for the outer chains - should be a sphere, of radius equal to the p r e f e r r e d chain length. This is, h o w e v e r , i m p o s s i b l e in t h r e e dimensional euclidean space, since space cannot be tiled by spheres alone. Nevertheless, the n u m b e r of faces in the Voronoi cell should be as large as possible, thereby approaching m o s t nearly the optimal spherical geometry. Values for a range of structures are given in Table 4.2 above.

5

.

,

.

,

.

I

',K" ""

9

I

0.2

I

4

~

0.1

:8

,

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9

-

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I

i

0.3

-

--

~ 16

'

'

0.4

r

5O

9

141

0.5 I

"

,

0.1

NI

"

0.2 9

I

'

I

0.4

'

0.5

l

II

\ 45

cyhnders ~ . . . ~ _ ~ . plan~

40

O.l

,

l

0.2

i

I

0.3

9

i

0.4

.

0.5

Figure 4.26: Relative free energies per AB linear diblock copolymers as a function of aggregation geometry and molecular composition, including global constraints. In contrast to the local calculations, hyperbolic forms are rarely favoured. The two hyperbolic curves represent calculations for "monolayer" (AB, spheres, cylinders, planes) and "bilayer" (ABBA) structures. The monolayers consist of intertwined A and B labyrinths, while the bilayers are made of A labyrinths in a continuum of the B blocks. The latter geometry is of significantly lower energy than the former. The three plots illustrate the effect of increasing surface tension (i.e. immiscibility) on mesophase behaviour. (The surface tension increases from upper to lower plots.)

184

Chapter 4

These values indicate that spherical objects prefer a body-centred cubic lattice, since this lattice maximises the n u m b e r of faces in the Voronoi cell. Similarly, where the interface is cylindrical, a (two-dimensional) hexagonal network is expected. These arrays are indeed those found in practice [45].

Figure 4.27: Transmissionelectron micrographs of a mixture of a star diblock copolymer (polybutadiene-polystyrene) with a homopolymer (polystyrene). The upper EM images show mesh layers viewed end-on. The lower image shows the mesh sheets viewed from above, revealing the dense network of pores in the layers, so that the sheets are in fact a filigree of interconnected tunnels. The large-scale dark (one marked A) and bright 03) fringes are due to variations in the thickness of the specimen only. Pictures reproduced with permission from [48]. Although it is more problematic to determine the free energy of hyperbolic geometries including curvature and distance variations, some conclusions can be d r a w n without detailed calculations. In general, low topology interfaces are preferred, due to their relatively high homogeneity (discussed in sections 4.6 and 4.8). We expect crystalline structures to be preferred; the Dsurface and the gyroid are expected to be most favourable. Preliminary calculations suggest that copolymer melts in the presence of added h o m o p o l y m e r (e.g. pure A or B polymer) should form bicontinuous "sponge" and "mesh" structures between lamellar and hexagonal (cylinder) phases. These geometries have indeed been observed [48]. The example shown in Fig. 4.27 illustrates the difficulty in distinguishing these hyperbolic phases from "classical" phases, m e n t i o n e d in section 4.7. The mesh

Block copolymers

185

morphology seen here is readily mistaken for a planar lamellar morphology. Indeed, small-angle X-ray spectra from this phase are typical of the lameUar phase, due to the strong one-dimensional ordering of the mesh layers, compared with the disordered catenoidal pores. A cross-sectional view through the material shows stacks of parallel sheets, distinguishable from the standard lamellar phase only by the slight modulations of intensity along each sheet. However, if the sheets are viewed from above, a dense network of pores is visible. It is clear then that the self-assembly process in these systems can result in the full spectrum of geometries; viz. elliptic, parabolic and hyperbolic. Further, a similar range of structures is realised at both extremes of molecular selfassembly. At one end, we have seen that a rigid block model, most suited to small surfactant molecules in water, leads to mesh and strut hyperbolic forms. A striking example of a mesh structure in a single-chain surfactantwater mixture (with some added salt) can be seen in Fig. 4.18. At the other, where the molecular shape is not preset, and dependent on chain entropy only (e.g. copolymers), hyperbolic aggregates are also formed.

4.12

Relation between material properties and structure

Copolymers were developed in response to industrial needs for material properties that could not be achieved by a single polymer. For example, many applications require a material that is both thermoplastic and thermoelastic. Simple mixing of polymers exhibiting these properties invariably results in phase separation, due to the high interaction energy between the polymers. Intimate mixing - preferably on a molecular scale - is necessary to achieve the material properties of both polymeric components. In our language, material properties are optimised by forming interfaces of high topology. For example, preliminary material measurements of bicontinuous copolymer mesophases indicate vastly superior material properties (e.g. significantly larger storage moduli) over the "classical" lameUar, spherical or cylindrical mesophases. However, the formation of hyperbolic interfaces is often unfavourable, due to curvature variations associated with such topologically complex structures. To be able to accommodate the associated curvature variations, the flexibility of the chains must be enhanced. This can be achieved by (i)increasing the temperature, or (ii) using a polydisperse molecular weight distribution. In the latter case, the molecules can distribute themselves over the regions of variable curvature according to chain lengths. Ironically then, reduction in the monodispersity of copolymer molecules may enhance the strength of these materials! Recent studies by Hashimoto et al. support this suggestion. For example, admixing of polydisperse homopolymer to copolymer systems (which form lameUar mesophases on their own) leads to the formation of bicontinuous mesophases [50].

186

4.13

Chapter4

Protein assemblies in bacteria: a mesh phase

The synthetic copolymers described in the previous section are particularly simple molecules compared to the macromolecules that occur in living systems. Virtually all bio-polymers exhibit some amphiphilic character, due to the presence of polar and lipophilic patches in the single molecule. The complex molecular architecture of proteins and enzymes leads to more subtle aggregation properties than those of synthetic copolymers. In particular, the chirality of these biological molecules offers an extra degree of freedom to the self-assembly process. This is discussed in more detail in the next section; for now we neglect the effect of chirality. Nevertheless, the range of aggregation geometries introduced in this chapter is undoubtedly to be found in biological systems. Awareness of the richness of the range of accessible shapes is important, since the understanding of function must involve a broad intuition of surface form, beyond the classical shapes. A wealth of biological examples will be dealt with in Chapters 5 and 6. For now we mention only a novel example of a hyperbolic "mesh" structure in the protein coats of cells. The most spectacular examples of these structures are found in so-called "Slayers", which consist of protein assemblies which envelope the cells of many eubacteria (both Gram-positive and Gram-negative) and archaebacteria [51]. Freeze-fracture images of these protein coats show highly ordered arrays of proteins, forming mesh structures of tetragonal, hexagonal, trigonal and oblique symmetries [52] (Fig. 4.28). A l t h o u g h the self-assembly e n v i r o n m e n t in vivo is complex, the neighbouring cell plays no part in the formation of these structures. For example, proteins have been found to self-assemble to form identical lattices even when the cell is not present [51]! It has also been found that these structures rely on the amphiphilic character of the molecular components. For example, if the pH of the solution is altered, the structures break up once the pH reaches the pK values of the carboxyl and amino groups of the amino acids making up the proteins. This is to be expected, since the self-assembled structure depends critically on the interaction between antipathic polar and apolar groups in the proteins. If this interaction is altered the driving force for self-assembly, which is segregation of these groups, is removed. Similar images have been recorded for bacterial sheaths, protein coats of viruses, bacterial spore coats and cell walls of eukaryotic algae. The hexagonal network has also been detected in gap junctions and in the photosynthetic centre. Clearly, these structures serve a useful function in these disparate biological assemblies. For example, they act as molecular filters for the cells allowing useful nutrients into the cell while repelling unwanted chemicals. (In fact, these S-layers are now sold commercially as ultrafilters for separating proteins.) There is a lot of speculation as to their role in cell-cell recognition, immune responses and cell adhesion. These are difficult issues to resolve -

Chiral self-assembly

187

we shall deal with some of them later on in this book. Nevertheless, clearly there is no need to invoke complex mechanisms to explain these structures: they form spontaneously, without requiring specific bonding bridges to stitch the network together.

Figure 4.28: (Top:) Freeze-fracture electron micrographs of S-layers enveloping archae- and eubacterial cells, together with image reconstructions (bottom). (Images adapted from [53].)

4.14

Self-assembly of chiral molecules

So far, the self-assembly phenomenon has been analysed in terms of the molecular shape, which constrains the curvatures of the interface between immiscible moieties. In fact, the molecular shape sets the tilt between neighbouring molecules in the aggregate. A further possibility arises for chiral molecules that demand a preferred tilt, set by the molecular geometry and twist, which is dependent on the chirality of the molecules. The following simple analogy exposes the link between chirality and twist. Threaded screws - right- or left-handed - are chiral. Closest packing of screws with a single handedness is achieved by twisting adjacent screws relative to each other (Fig. 4.29). So a strand of close-packed screws resembles a twisted fibre, whose handedness and pitch are dependent on the handedness and pitch of the threads in the screws. In the language of differential geometry, the tilt angle is equal to the (integral) normal curvature along an arc of the surface containing the molecules, while the twist angle is the (integral) geodesic torsion along the same geodesic (cf. Chapter 1.5). Both normal curvature and geodesic torsion depend on the

188

Chapter 4

tangential direction of the geodesic, relative to the principal directions on the surface, described explicitly in eqs. 1.3-1.4. If the molecules pack in the surface such that they all lie along principal directions, the twist vanishes. Further, if the arcs joining adjacent molecules are equally inclined relative to the principal directions, the net twist (geodesic torsion integrated over the surface) also vanishes, containing equal components of positive and negative twist. Thus, for square or hexagonal ("hexatic") packings of molecules within any boundary-free surface (at all points except umbilics), no net twist is realised - provided the molecules are aligned normal to the surface. On the other hand, if the molecules are tilted relative to the surface normal, the average twist is now nonzero, and dependent on the tilt angle and the (average) tangential direction between neighbouring molecules relative to the principal directions. Alternatively, the surface occupied by aggregating molecules may be bounded in such a way as to yield a nonzero net twist.

Figure 4.29: Cartoon illustrating the twist associated with packing of threaded (chiral) screws. In this case, since the screws are not tapered, close-packing is achieved without any tilt (the screws are all perpendicular to the straight axis running through their centres). In general both tilt and twist components are present. Many amphiphiles of biological origin exhibit chiral centres (e.g. lecithins, amino acid based amphiphiles), although in practice the resulting chirality is very weak once the molecular chains are molten. However, if the lipids remain crystalline (for example lipid-water mixtures maintained below the Krafft temperature), the resulting aggregate geometries can be strongly influenced by the r e q u i r e m e n t of n o n z e r o average twist b e t w e e n neighbouring amphiphiles. For example, if the molecular shape results in a spontaneously fiat bilayer, chirality induces a helical deformation of the bilayer, resulting in long, corkscrew-like ribbons, shown below [54-58].

Chiral self-assembly

189

Fig. 4.30: Transmission electron micrographs of chemically modified acetylcholine amphiphilic chiral assemblies (diacetylenic aldonamides), containing monomers (left) and polymers (right). In some cases the helical crystalline amphiphilic assemblies wrap into

"lipid tubules". Images reproduced from [57]. The Gaussian curvature of the ribbons remains close to zero - to satisfy the molecular shape (v/al=l). However, provided the ribbons are narrow compared to their length, some net twist is realised. In these surfaces, principal curves form an orthogonal net, consisting of parallel helices running the length of the ribbons (~=0 say in equations 1.3-1.5), and latitudinal curves running along the short width of the ribbon (~=-1r/2). Thus, integrating equation 1.4 over the ribbon area results in nonzero average twist, since there is an excessive contribution of tangential directions ~ between 0 and ~/4 over those between ~/4 and ~/2. The relation between twist and chirality remain unclear. Some intriguing phenomenological rules have been developed [59], although a priori calculations of the preferred twist angle between neighbouring molecules is impossible, in contrast to the situation for the tilt, whose relation to the molecular shape is detailed in section 4.1. The arrangement of chiral molecules in thermotropic liquid crystals is more complex, since entire volumes of space - rather than the bounded twisted ribbons discussed above - must be filled subject the constraint of a preferred twist between neighbouring molecules. The simplest examples of such mesophases are the cholesteric liquid crystals, discovered last century, (cf. section 5.1.8). This class of thermotropic liquid crystals derives its generic name from chiral cholesterol derivatives (shown below), which were found a century ago to exhibit peculiar optical changes as they were heated. Cholesterics are characterised by a "single twist", which is characterised by a relative rotation of fiat layers of molecules. (These layers are illusory, and useful for illustrative purposes only, since there is no evidence of lamellar ordering perpendicular to the layers.) The optical novelties of these cholesteric phases are due to the pseudo-Bragg reflections from the helical

190

Chapter 4

arrays of molecules, whose pitches are comparable with the wavelength of visible light (so that they are strongly coloured).

or

R= CH 3

RCO 2

Cholesteric phases are thus "frustrated", in that their preferred twist is accommodated along one axis only. In some cases, further heating of the cholesteric mesophases leads to a new mesophases, called "blue phases". Two of the three known blue phases are characterised by three-dimensional pseudo-lattices induced by the molecular orientation, again of dimensions comparable with the wavelength of light (although they are more commonly red than blue) [60]! (The third "blue fog" mesophases is probably topologically similar to the other two, although spatial correlations are much weaker.)

Figure 4.31: (Left:.)Schematic view of the relative arrangement of chiral molecules (extended lozenges) in the cholesteric liquid crystalline mesophase (after [59]. The twist between layers is greatly exaggerated. In reality approximately 104 layers lie between equally inclined layers. (Right:) Helical arrangement of molecules, with a relative twist between molecules along one direction only: the axis of the helical ribbon. The most favourable relative configuration of identical chiral molecules is that where all neighbouring molecules are twisted relative to each other. This is achieved by a "double-twist" stacking, illustrated in Fig. 4.32. In threedimensional euclidean space, this double-twist cannot be realised throughout space; some "disclination" singularities must occur [61]. How then can this double twist be most closely approached? A simple model, involving nothing more than potatoes, and oven and matches, is useful. The lower-

Chiral self-assembly

191

temperature molecular crystalline phase can be represented by parallel layers of matches (directors), without any twist between the layers. At temperatures within the cholesteric phase, the layers are twisted with respect to each other. At still higher temperatures, the layers remain twisted with respect to each, but each layer is also warped into saddles, allowing a double twist: along a single axis within each layer and between adjacent layers (Fig. 4.33). This (local) double twist configuration clearly involves a hyperbolic deformation of the imaginary layers. In contrast to the hyperbolic layers found in bicontinuous bilayer lyotropic mesophases, the molecules within these chiral thermotropic mesophases are oriented parallel to the layers, to achieve nonzero average twist. The magnitude of this twist is determined by the direction along which the molecules lie (relative to the principal directions on the surface), and a function of the local curvatures of the layers (K1-~:2), cf. eq. 1.4. Just as the molecular shape of (achiral) surfactant molecules determines the membrane curvatures, the chirality of these molecules induces a preferred curvature-orientation relation, via the geodesic torsion of the layer. In the chiral mesophases however, space must be filled by a foliation of hyperbolic layers. This cannot be achieved without disclination singularity lines (since parallel planes are the only structures that can tile space without singularities), whose geometry is dependent on that of the underlying hyperbolic leaves inducing the foliation. A single type of molecule, with a well-defined chirality, will thus pack into an arrangement whose curvatureorientation relation is as close as possible to that preferred value. Ideally, a global arrangement involving parallel hyperbolic sheets, each of constant Gaussian curvature, is preferred. In other words, homogeneous hyperbolic sheets are most favourable. Thus, for related reasons to those leading to triply-periodic minimal surfaces (IPMS) in bicontinuous cubic mesophases, foliations of hyperbolic sheets, all parallel to quasi-homogeneous IPMS, describe the orientation of molecules as a function of position in space in blue mesophases. It is surely no coincidence then that the symmetries of the lower temperature blue phases ("BPI" and "BPII") are precisely those of the D-surface and the gyroid - Pn~m and Ia3d respectively. These IPMS, the D-surface and the gyroid, are the most homogeneous "leaves" upon which the foliation of space is built [60]. In the case of thermotropic liquid crystals, a surface description is mathematically useful, but physically misleading, since these surfaces are fictional: they serve only to describe the three-dimensional variation of molecular orientation. An alternative description of blue phases in terms of close-packing of chiral rods can be found in the next Chapter (section 5.1.8). Note that due to the inhomogeneities of Gaussian curvature in hyperbolic surfaces (in euclidean space), blue phases are characterised by twist numbers slightly less than two. Since the single preferred value of the Gaussian

192

Chapter4

curvature can only be achieved within a local patch, ideal double twist can only be achieved locally. We conjecture that this preferred twist configuration is satisfied then only in the pure melt, which is locally structured only. This geometric argument can be inverted, offering an interesting explanation for melting in these systems. Assume only that the twist geometry becomes increasingly important as the temperature of the system is raised. That requires a monotonic increase in effective "twist number" on heating: via a cholesteric phase (twist number equal to unity), and blue phases (twist numbers between unity and two), culminating in the most favoured twist number of two - realisable only in a locally structured f o r m - ergo the very short-range order of a melt.

Figure 4.32: The ideal (left-handed) double-twist configuration of neighbouring molecules about a central chiral molecule. For simplicity, a square arrangement of molecules is assumed; in practice, the array is more likely to be hexagonal. Some novel predictions can be made by allowing this twist number to change continuously from zero (the crystal) to two (the melt). Twist numbers less than one can be achieved by a partial tilt of the molecules within hyperbolically deformed layers; in which case the twist is of the right sense along one direction, but in the wrong sense along the orthogonal direction within the sheet. This structure should not be confused with that of blue phases, where the molecules are (maximally) tilted (by ~/2) relative to the layer normals. However, the net twist remains nonzero, and of the correct sense, so that the twist number lies between zero and unity. Thus, between twist numbers of zero (the crystal) and one (the cholesteric phase), hyperbolic geometries are also expected. These novel phases are now being found experimentally; they have been called "twisted grain boundary phases" [62,

63].

Chiral self-assembly

193

Figure 4.33: Representation of the local structure of some chiral mesophases on heating. The matches describe the relative orientations of chiral molecules in space. As the temperature is raised, the system transforms from a crystalline phase (left) to a cholesteric phase (centre): characterised by a single twist, to a double-twist "blue" phase (right).

These chiral thermotropic m e s o p h a s e s are of great i m p o r t a n c e technologically, and it is expected that a number of novel structures have yet to be found - similar to the plethora of hyperbolic mesophases now being identified in lyotropic systems. Their technological value lies in the fact that the relation between twist angle and molecular dimensions results in pseudo-Bragg spacings comparable with visible wavelengths, and their phase behaviour is often sensitive to applied electromagnetic fields - hence they have been used in active matrix display screens. Further, ferroelectric phases have also been identified in these systems, of great interest for fast-switching electro-optic couplers (see, for example, [64]). In contrast to typical chiral thermotropic constituent molecules, protein molecules have a huge number of chiral centres, and the twist between assembled proteins is typically much larger (--1/10 of a revolution). We can expect to find a range of similar liquid crystalline phases in protein aggregates, although with significantly smaller lattice parameters (compared with typical protein dimensions). Indeed, the aggregation processes of (e.g. structural) proteins may be driven by their chirality as much as by their molecular shape (and amphiphilicity). This is discussed in more detail in Chapter 6.

194

Chapter4

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29. S.T. Hyde, S. Andersson, B. Ericsson, and K. Larsson, Z. Kristallogr., (1984). 168: pp. 213-219. 30. E.Z. Radlinska, S.T. Hyde, and B.W. Ninham, Langmuir, (1989). 5: pp. 1429-1430. 31. P. Barois, S.T. Hyde, B.W. Ninham, and T. Dowling, Langmuir, (1990). 6: pp. 1136-1140. 32. P. Barois, D. Eidam, and S.T. Hyde, J. Phys. (France), (1990). 51(Colloque C-7, supplement au no. 23): pp. 25-34. 33. V. Luzzati, T. Gulik-Krzywicki, and A. Tardieu, Nature, (1968). 218: pp. 1031-1034. 34. C. Hall and G.J.T. Tiddy, Solution, NewDelhi, (1986.).

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35. P. Pieruschka and S. Marcelja, J. Phys. (France) II, (1992). 2: pp. 235-247 ; S. Marcelja, Fizika, (Dec., 1995). B4. 36.

P. Str6m and D.M. Anderson, Langmuir, (1991). 8, p. 691.

37.

I. Danielsson and B. Lindman, Coll. Surf., (1981). 3: p. 391.

38. 538.

L.R. Angel, D.F. Evans, and B.W. Ninham, J. Phys. Chem., (1983). 87: p.

39. S.J. Chen, D.F. Evans, and B.W. Ninham, J. Phys. Chem., (1984). 88: p. 1631. 40.

K. Fontell and M. Jansson, Progr. Coll. Polym. Sci., (1988). 76: p. 169.

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41. S.T. Hyde, B.W. Ninham, and T. Zemb, Journal of Physical Chemistry, (1989). 93: p. 1464. 42. I.S. Barnes, S.T. Hyde, B.W. Ninham, P.-J. Derian, M. Drifford, G.C. Warr, and T.N. Zemb, Progr. Colloid Polym. Sci., (1988). 76: pp. 1-6. see also: M.A. Knackstedt and B.W. Ninham, Phys. Rev. Left., (1995). 75: pp. 653-656. 43.

H. Hasegawa, H. Tanaka, K. Yamasaki, and T. Hashimoto,

Macromolecules, (1987). 20: pp. 1651-1662. 4.

D.M. Anderson and E.L. Thomas, Macromolecules, (1988). 21: pp. 3221-

3230. 45.

B.R.M. GaUot, Polym. Rev., (1978).

46. H. Hasegawa, H. Tanaka, T. Hashimoto, and C. Han, Macromolecules, (1987). 20: pp. 2120-2127. see also: E.L. Thomas, D.B. Alward, D.J. Kinning, D.C. Martin, D.L. Handlin and L.J. Fetters, Macromolecules, (1986). 19: pp. 2197; D.A. Hajduk, P. E. Harper, S.M. Gruner, C.C. Honeker, E.L. Thomas and L.J. Fetters, Macromolecules, (1994). 27: pp. 4063. 47. S.T. Hyde, A. Fogden, and B.W. Ninham, Macromolecules, (1993). 26: pp. 6782-6788. 48.

T. Hashimoto, S. Koizumi, H. Hasegawa, T. Izumitani, and S.T. Hyde,

Macromolecules, (1992). 25: pp. 1433-1439. 49. H.S.M. Coxeter, "Introduction To Geometry". (1969), New York: John Wiley & Sons, Inc. 50.

H. Hasegawa, T. Hashimoto, and S.T. Hyde, Polymer, (1996). in press.

51.

U.B. Sleytr and P. Messner, Ann. Rev. Microbiol., (1983). 37: p. 311.

52.

S. Hovm6Uer, A. Sj6gren, and D.N. Wang, Progress in Biophysics and

Molecular Biology, ed. D.N. Blundell (1988). 51. 53.

M. S~ra and U.B. Sleytr, J. Bacteriology, (1987). 169: pp. 4092-4098.

54. T. Kunitake, Y. Okahata, M. Shimomura, S. Yasunami, and K. Takarabe, J. Am. Chem. Soc., (1981). 103: pp. 5401-5413. 55.

W. Helfrich, J. Chem. Phys., (1986). 85(2): pp. 1085-1087.

56.

J.-H. Fuhrhop, P. Schneider, E. Boekema, and W. Helfrich,

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Chem. Soc.., (1988). 110: p. 2861. 57. D.A. Frankel and D.F. O'Brien, J. Am Chem. Soc., (1991). 113: pp. 74367437. 58.

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References 59.

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60. B. Pansu and E. Dubois-Villette, J. Physique, (1990). 51(Colloque C-7, supplement au no. 23): pp. 281-296. 61.

J. Sethna, Phys. Rev. B, (1985). 31: p. 6278.

62. K.J. Ihn, J.A.N. Zasadzinski, R. Pindak, A.J. Slaney, and J. Goodby, Science, (1992). 258: pp. 275-278. 63. H.T. Nguyen, A. Bouchta, L. Navailles, P. Barois, N. Isaert, R.J. Twieg, A. Maaroufi, and C. Destrade, J. Phys. II (France), (1992). 2: pp. 1889-1906. 64. L.A. Beresnev, S.A. Pikin, and W. Haase, (1992). 1(8): pp. 13-19.

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5

Lipid Self-Assembly and F u n c t i o n Biological Systems

In

5.1

Self-association of lipids in an aqueous environment

5.1.1

Introduction

o far, our aim has been to put together knowledge gleaned, in most part over the last decade, on periodic surfaces of zero average curvature, and surfaces of constant average curvature. Once seen as esoteric, the universal significance of these surfaces in understanding chemical structures has become startlingly apparent. This is so equally for systems as diverse as atomic and molecular arrangements in crystals, in block copolymers, or selfassembled colloidal aggregates formed from surfactants. An approach to organisation in matter through the linking concept of curvature has been seen to give insights into the determination of complex inorganic and organic structures, into phase behaviour, and into the relation between structure and physical properties. Those insights become sharper, and the emerging mosaic less blurred, once the description and catalogue of known shapes is taken together with knowledge of the forces behind their formation.

S

This and the succeeding chapters take us a stage further. They are focused on the role of curvature and its connection to function for supramolecular assemblies of biomolecules. A number of examples from biology are presented where cur~,ature and function are clearly intimately connected. These examples allow us to speculate on the central part played by shape in chemical reactions, and in molecular organisation in living systems: the beginning of a language of shape. We begin with the field of lipid-water phase behaviour. Lipids, usually double-chain (sometimes single- or triple-chain) are generally highly insoluble surfactants. Together with proteins, they are essential ingredients of all cell membranes and other organelles. They provide protection to DNA, segregated containers, and a fabric in which biochemical processes can take place. Usually thought of as passive, they self-assemble, transmute and transform in response to changes in physico-chemical environment induced by biochemical reactions or interactions into a bewildering diversity of forms. Structure, self organised, is imposed by the biochemical environment, in turn dependent on that structure. Shape and function are here closely linked. How this can be, we shall see below. Numerous reviews exist that deal with characterisation of different liquid crystalline phases formed in pure lipid-water mixtures [1]. Our concern is rather with new features of lipid-water phases as revealed by thinking in terms of curvature.

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There is strong overlap here with Chapters 3 and 4, which deal with association properties of surfactants in general, but the material is presented so that it can be read independently. All features of lipid association are common with non-biological surfactants. The main difference lies in nature's choice of particular compounds that can take part in and guide biochemical processes.

5.1.2 General behaviour of lipids in water From a biological standpoint an essential property of lipid molecules is their ability to form aqueous phases, possessing long-range order combined with disorder at molecular distances. A variety of different phases exist for a particular lipid (polymorphism), and a small change in solution conditions is sufficient to cause a transformation from one form or structure to another.

0

D

O----CH 2

0

Phosphatidyl choline: X=

(Iedt166)

Phosphatidyl ethanolamine: X=

CH2 ~CH3 3

CH 2

CH2-- N +~-~~H

It is natural to classify lipids as polar or non-polar according to their interaction with water. Non-polar lipids, for example triglyceride oils, do not form aqueous phases, whereas polar lipids do. Except for cholesterol, membrane-forming lipids form aqueous phases and have polar head groups. Within membranes there are also trace amounts of lipids in membranes that do not interact with water, for example diacylglycerols. The structural formulae of two common membrane lipids, phosphatidylcholine (PC) and phosphatidylethanolamine (PE) are shown above.

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201

The physical properties of polar lipids, such as phospholipids, glycolipids or monoglycerides, are directly related to their association behaviour. If a pure lipid crystal is heated, it either melts directly or is transformed into a liquid crystalline phase. Those liquid crystalline phases obtained by heating are termed "thermotropic". The formation of these thermotropic phases can be u n d e r s t o o d as follows: The m a g n i t u d e of thermal vibrations of the hydrocarbon chain atoms increases with temperature. In polar lipids, which are able to form liquid crystalline phases, the "bonds" in the sheets formed by the polar h e a d groups are strong compared to the weak van der Waal's interactions between adjacent chains. Hence a temperature range can exist prior to melting, in which range the chains become fluid while the polar groups are still associated into sheets. The overall structure then comprises lipid bilayers with disordered chains.

% (u/w) in u , a ~

Figure 5.1: Characteristic phase diagram of a membrane lipid, such as a digalactosyldiglyceride, with the conventional view of the structure of the dominating (Lot) "lamellar" phase illustrated above. A feature of membrane lipids, as opposed to other surfactants, is the existence of this large lamellar phase region. Liquid crystalline phases form more commonly in the presence of water (these solvated phases are called "lyotropic'). Above a critical hydrocarbon chain melting temperature, water penetrates the polar region, and a lamellar

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lipid-water structure (the "Lot" mesophase) is formed with water layers alternating with lipid bilayers (see Fig. 5.1). Two other structures, which have hexagonal symmetry are shown in Fig. 5.2. The two hexagonal structures are termed HI and HII. These hexagonal phases and the lamellar phase are the most important liquid-crystalline phases in lipid-water systems. In fact, the HI phase occurs only exceptionally, for example in lyso-PC and similar very polar lipids. (In the language introduced in the previous chapter, the surfactant parameter is v / a l > 1, for nearly all double-chain lipids, with the hexagonal phase HI requiring v/al = 1/2, which occurs only for single-chain lipids.) The HI-phase consists of infinite cylinders of lipid molecules, which expose their polar groups to the surface of water with the core of hydrocarbon chains in a disordered liquid-like conformation. These cylinders are arranged in a hexagonal array in a water medium. The HII-phase is the inverse of HI, i.e. water forms cylinders and hydrocarbon chains form the continuous m e d i u m with the polar groups at the water/hydrocarbon chain interface.

Figure 5.2: The two types of hexagonal lipid-water phases, HI and HU. HI consists of lipid rods in water arranged on a two-dimensional hexagonal lattice, whereas HII has the reversed structure. The HII phase can also be regarded as intersecting lipid bilayers (infinite in one direction) as illustrated by the corresponding HII asymmetric unit (circled), shown enlarged to the right. The first ideas on the nature of liquid crystals in lipids were derived from Xray studies by Luzzati [1]. A crucial discovery was his demonstration of the liquid character of the hydrocarbon chains, which are thus space-fiUing. This was evident after it was found that the lipid bilayer thickness decreases with temperature with a large linear thermal coefficient: about 10-3/~ Such an effect is consistent only with a highly disordered chain conformation. Also the X-ray scattering characteristics were found to be very similar to those of liquid paraffins. Lamellar and hexagonal phases are usually identified f r o m their X-ray diffraction patterns. The lamellar phase shows a series of reflections

Lipids in water

203

corresponding to (dominating) one-dimensional periodicity, and from the Xray data it is possible to determine the thickness of the lipid bilayer, the crosssection per polar head group, and the water layer thickness, assuming a certain sheet geometry. In the same way the diameter of the cylinders in the HI and HII phases can be determined, as well as the cross-sectional area per polar head group and the distance between adjacent cylinders (assuming the model structure shown in Fig. 5.2). The emerging intuition of mesostructures, discussed in detail in the preceding chapter, implies that many structures, formerly considered to consist of planar sheets, can be very much more complicated, e.g. "mesh" structures. True membrane lipids only exhibit phases that share a common feature with the infinite lipid bilayer. The average curvature is zero in the centre of the bilayer. These phases are L0~ (infinite planes), HII (selfintersecting lamellae) and the alternative cubic phases.

5.1.3 Cubic phases Cubic lipid phases have a very much more complex architecture than lamellar and hexagonal phases. Their structural characteristics have been elucidated only very recently, and it has become clear that their subtleties are the key to a variety of biological problems. We will consider those subtleties in some detail. The three fundamental cubic minimal surfaces the Psurface, the D-surface and the gyroid (or G-surface), introduced in Chapter 1, can all be found in cubic lipid-water phases. The lipid bilayer is centred on the surface with the polar heads pointing outwards. Water fills the labyrinth systems on each side of the surface. These cubic phases will be termed Cp, CD and CG, respectively. It is likely that there are other more complex IPMS morphologies in cubic phases of lipid-water mixtures, as yet uncharacterised. -

A structure for cubic phases of lipids was first proposed in 1968 by Luzzati and coworkers [2] that consisted of interpenetrating rod networks. The polar head groups were presumed to form the rods and the hydrocarbon chains a continuous matrix. A different type of structure was later proposed [3] based on studies of monoglyceride-water systems, with lamellar lipid bilayer units forming a continuous lattice with water on each side. The main evidence was derived from X-ray diffraction data and NMR diffusion measurements, which showed that the structure was both water- and lipid-continuous, with lipid molecular diffusion very similar to that of the L(x-phase. The proposed fused network of lipid bilayer discs was organised in agreement with the space group Im3m. It was then realised that such a structure needed only a minor change from planar to curved disc units to be topologically identical to Schwarz" celebrated P-surface [4] (with a lipid monolayer on each side of the surface separating the water channel systems in the x-, y- and z-directions). The glycerol monooleate (GMO)-water system shown in Fig. 4.14 is particularly interesting as it exhibits two cubic phases, CG and CD. The two

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lattices are related by the Bonnet transformation, and the Cp structure proposed originally is a "least-common-denominator" in the indexing of the lattices. Work by Longley and Mc~tosh [5] on the cubic phase in excess water showed that the lattice was primitive with space group Pm3n, corresponding to the CD structure. In view of this result, it was natural to reexamine other X-ray data on the whole cubic region [6] of the phase diagram. This showed that there were two cubic phases in this region. On swelling with water, at first CG, and then the CD structures are formed. The Cp phase was later shown to exist in monoolein-water systems mixed with proteins [7]. Thus all three phases Cp, CD and CG dan be formed within one cubic region in a temary lipid-protein-water system. The minimal surface description naturally reveals the infinite lipid bilayer nature of cubic phases, viz. the fact that a single bilayer with no selfintersections can separate two continuous water regions. If we consider models of these cubic lipid-water phases, the structures of the Cp (or CD) phases look like water globules separated by bilayers and fused in four (or three) lateral directions, respectively. Such a structure is not consistent with the earlier rod description. The differences from the earlier rod based description are most striking in the case of the CG structure. One interesting feature of this structure is the occurrence of (infinite) helical tubes oriented parallel to the four [111] directions of the cubic unit cell. We now discuss why it is that the different alternative structures, Cp, CD and CG form. A clear requirement for existence of a particular structure is that there must be enough space for the lipid bilayer to fold into that form. If we consider the Cp or the CD structures, there are narrow necks in the surface at the points where the globular units join. The closest distance corresponds to the circular cross-section of catenoids spanning the globular units. If we imagine a lipid bilayer centred on this surface, the thickness of the bilayer must be less than this circular cross-section. This requirement gives a lower limit to the water content of the Cp phase. If we compare this limit with that of the CD phase, it is shifted to allow a higher bilayer content per unit volume. This may explain why the Cp phase cannot be present in the binary system monoolein-water. The structure is not sufficiently swollen for this structure, with water content at the highest concentration for which it can exist about 40% (w/w). However it can exist when proteins are introduced, as the bilayer then takes up a smaller proportion of the unit volume. On the other hand the CG structure contains smaller circular necks. For this reason the CG structure can be formed at low water content, even in anhydrous lipids. Recall also that surface to volume requirements for these different IPMS demand this phase sequence (section 4.4). Another critical factor is the overall molecular shape, or local packing requirement, discussed in Chapters 3 and 4. The volume per unit surface (expressed as V 2 / 3 / A ) for the P-, D- and gyroid surfaces of the same local

Cubic lipid-protein-water phases

205

shape (Gaussian curvature) are 1.07, 1.02 and 1.00, respectively. Therefore, at a single lipid/water ratio, there are differences in local molecular (wedge) shape taken up by the lipids in the C D , Cp and CG phases. In the case of the monoolein system, the CG phase exists at a very low water content for reasons described above. Swelling of the CG phase necessarily means a reduction in wedge-shape and a change in molecular shape opposed to the desired increase in disorder. Thus, the swelling is accompanied by increased strain in the chain region, and the phase transition can be expected to release this strain. At the phase transition at 35% (w/w) of water in the monoolein system, the calculated wedge-shape - defined by v / a l - of 1.27 for the CG phase is increased to 1.31 in the CD phase. The strain is thus released. A Bonnet transformation (discussed in Chapter 1) from the gyroid to the Dsurface involves a change in the surface area per unit volume. An ideal Bonnet transformation maintaining the preferred Gaussian curvature constant, is therefore not possible. Some swelling or contraction must occur to accommodate the fixed bilayer volume fraction and fixed head-group area. In agreement with this expectation, a corresponding change in the lattice occurs to account for the constant lipid-water composition [8]. The a-axis of the CD phase is shorter by a factor of 0.96 than that calculated from the CG axis, assuming an ideal Bonnet transformation. It is impossible to determine the electron density distribution within the unit cell with a resolution at the atomic scale. Hence there is no straightforward way of determining the exact structure. In summary the evidence for the zero average curvature model structure of cubic lipid-water phases then rests on: 1. Space-group determination from X-ray data (although in many cases so few lines that unambiguous assignment is impossible). The space groups Ia3d, Im3m and Pm3n are the same as those of the three fundamental cubic surfaces (G-, P- and D-respectively). 2. The fact that there are two cubic phases in the same region of a binary system with unit cell sizes in agreement with the Bonnet transformation. 3.

Studies by NMR and electron microscopy consistent with the model [10]).

(cf. [9] and

4. Swelling analysis of model double-chain surfactants within the cubic phase region of a ternary phase diagram, which is in perfect agreement with the expected swelling, assuming fixed local molecular shape, and a bilayer lying on IPMS [11, 121. Among the various lipids that form cubic phases, monoglycerides have been studied most. Phosphatidylcholines exhibit cubic phases only at very low water content, whereas phosphatidylethanolamines form cubic phases in similar regions of the lipid-water system as monoglycerides (e.g. GMO). Lysophospholipids (single-chain lipids) exhibit cubic phases of quite different kinds. They will not be discussed here as they are not considered to be relevant in the formation of self-assembled structures of biological

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~ter 5

importance. Lysophospholipids form micellar solutions and cubic phases consisting of micellar units in a cubic lattice or other structures discussed in Chapter 4. Such lipids only occur in trace concentrations in biological membranes, and are harmful in higher amounts as they tend to solubilise the bilayer (by changing the effective surfactant parameter, discussed in Chapters 3 and 4).

Figure 5.3: Freeze-etchingelectron micrograph of a monoglyceride-cytochromec- water cubic Cp phase. A schematic view of the projected view of the structure is shown in the top right comer, with the tunnels shown in black. After [10].

5.1.4 Cubic lipid-protein-water phases Cubic phases are also unique in their ability to accommodate proteins as compared to other lipid-water phases. A wide range of globular proteins with molecular weights 5,000-150,000 are known to form cubic phases when mixed with lipids and water. So far few single ternary lipid-protein-water phase diagrams have been completely determined [7], [13]; one system that has been looked at is that of monoolein-water-lysozyme. Protein incorporation results in increased water swelling, and all three phases, Cp, CD and CG, occur. The protein molecules are located in the water channel systems; and retain their native structure. This has been proved by thermal analysis of the phase, and measurements of enzymatic activity [7].

Cubic lipid-protein-water phases

207

Other lipophilic molecules can be incorporated into the lipid bilayer of the cubic monoolein-water phase, for example gliadin. An electron micrograph showing the Cp structure of a lipid-protein-water phase is shown in Fig. 5.3. The body-centred structure characteristic of the P-surface is evident from the displacement of adjacent fracture planes. Other detailed structural analyses of cubic phases of systems involving monoolein have been reported [14], and space groups observed that correspond to "asymmetric" or "unbalanced" surfaces of nonzero mean curvature, related to the "balanced" IPMS. If the channel systems on each side are different, or if the lipid bilayer contains constituent monolayers of different curvature, an asymmetric (bicontinuous) cubic phase results. There are thus three other asymmetric (or unbalanced) CD, Cp or CG structures.

5.1.5 Dispersions of bicontinuous cubic phases: cubosomes When a bicontinuous cubic lipid-water phase is mechanically fragmented in the presence of a liposomal dispersion or of certain micellar solutions (e.g. bile salt solution), a dispersion can be formed with high kinetic stability. In the polarising microscope it is sometimes possible to see an outer birefringent layer with radial symmetry (showing an extinction cross like that exhibited by a liposome). However, the core of these structures is isotropic. Such dispersions are formed in ternary systems, in a region where the cubic phase coexists in equilibrium with water and the La phase. The dispersion is due to a localisation of the Lr phase outside cubic particles. The structure has been confirmed by electron microscopy by Landh and Buchheim [15], and is shown in Fig. 5.4. It is natural to term these novel structures "cubosomes". They are an example of supra self-assembly. The cubic phases can also be dispersed by amphiphilic proteins. Caseins, for example, which also are very effective as emulsifiers, can disperse the cubic phase just as do simple surfactants, such as bile salts. The mechanism is exactly the same as the solubilisation of bilayers by detergents at the c m c - the mixed system has a different surfactant parameter, i.e. local curvature, and the system takes on a different structure. The discovery of cubosomes represents an important step in our understanding of self-assembly processes in biology. Most work on selfassembled biological structures has been conceptualised in terms of the simple Euclidean geometries represented by planes (membranes, lamellar phases); spheres, micelles (lipoproteins), or vesicles (transmission of the neuron axon impulse across synaptic junctions) or cylinders (organised polymeric fibres, collagen, muscle fibre). The role of lipids has been seen as a passive one. The intuition has been drawn from studies of model (few component) phase diagrams, with the inbuilt supposition that the various phases that exist are of infinite extent (thermodynamic equilibrium). Cubic phases, with their great advantage in terms of protein uptake and

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Chapter5

localisation, and lamellar phases, multi-walled liposomes, were seen as disjoint. But only closed containers like the multi-walled liposomes to be discussed further below will have different solution conditions, e.g. ionic strength, inside and outside, and compartmentalisation of physico-chemical environment is a self-evident requirement in cell biology.

Figure 5.4: Electronmicrograph of a freeze fractured dispersion of cubosomes [15].The cubic periodicity is about the same as in Fig. 5.3. For the phase diagrams of lipid-water systems, the cubic phases exist close to the lamellar phase in terms of component ratios. Consequently, one expects that lipid-protein mixtures can spontaneously self-assemble to give small cubic phase organelles, protected and surrounded by several layers of primitive bilayers. There is an obvious advantage to a bicontinuous network containing a high concentration of proteins subject to biochemical processes that require rapid interchange of reactants, possible through a bicontinuous network. This makes sense in terms of the existence of spontaneous hyperbolic cubosomal structures.

5.1.6 Liposomal dispersions

We make some further remarks on the better known liposomal dispersions. The globular bilayer-bounded particles are called liposomes, and many

Liposomal dispersions

209

studies have utilised liposomes as membrane models. There are also many studies using liposomes as drug delivery agents. It was only 30 years ago that Bangham first described how phospholipids in dilute aqueous systems form closed aggregates of lipid bilayers alternating with water [16]. The general conditions under which dispersions of such spherical structures form were later studied in monoglyceride-water systems. They were found to be an Lot phase (with limited swelling) in equilibrium with excess water [17]. Liposomal dispersions usually have a high kinetic stability, but there is for most biological lipids a clear progression w.ith time towards larger growth of particles, with reduction in curvature of the bilayer. The dynamic properties of liposomes have been much investigated [18]. Morphological changes have been studied during formation, flow and changing osmotic pressure. Remarkable regularities in the sequential transformations in shape have been observed in these studies. Alternative pathways from biconcave shape via triangular, square or pentagonal shapes to "stable" spherical or filamentary forms occur [19].

5.1.7

Vesicles and membranes

Liposomes bounded by a single molecular bilayer are called vesicles. The production and recycling of vesicles in cell biology is ubiquitous, a universal means of transport of materials and communication. Conceptually, this process is not difficult to understand. Plasma membranes contain a variety of lipids (whose surfactant parameter, v/al --- 1) and proteins. At the crudest level we can think of the lipids as in a planar bilayer state. Depending on changes in local solution environment the lipid mix can phase separate in response to such changes to produce local regions of different surfactant parameter that allow vesicle formation. Biochemical inputs, e.g. the release of Ca 2+ [20], or the decapitation of particular lipids by a particular enzyme [21] to produce others of lower head group area, can change the surfactant parameter locally. The production and solubilisation of lipophilic compounds, such as diglycerides or cholesterol, can do the same. Likewise changes in solution conditions like local salt concentration, pH, ion binding, etc., induced by proximity of one membrane to another, can effect the necessary changes of curvature. As we have seen, macroscopic lamellar phases can be dispersed into vesicles or micelles by admixture of single-chain surfactants or by changing counterions, both of which change curvature either directly or through intra- and inter-aggregate interactions. With biological cells these processes can be induced and followed directly by video enhanced microscopy [22]. Evidently, once mixtures are involved, or biochemical processes, the problem of making and using vesicles is a trifle for nature. But to so dismiss the problem of spontaneous vesicle formation is to avoid a question of some importance. A related question is this. How does nature

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form the giant single-walled vesicles - cell membranes - crucial to the existence of cells and therefore life itself? Can these structures form spontaneously driven by physico-chemical mechanisms as an equilibrium state of matter, or must one appeal to the higher tautology of non-equilibrium thermodynamics for their existence? Let us discuss the second question. Giant single-walled vesicles, metastable, but stable for months at least, can be made from double-chain membranes mimetic surfactants in a variety of ways. One of the most dramatic is to prepare by ion exchange a solution of didodecyldimethyl ammonium hydroxide (DDAOH) vesicles from the bromide salt (DDAB) which itself forms a dispersion of liposomes. The DDAOH vesicles are apparently admixed with a high proportion of miceUes in their interior [23] and typically have diameters around 2000/~. Titration of the solution with hydrogen bromide to neutralise the hydroxide ions results in the immediate formation of micron-sized, apparently single-walled "cells". Fusion and collapse to the organised lamellar phase state are inhibited by both the large hydration and electrostatic forces, and the high insolubility of these surfactants. Often large worm-like structures (myelin figures, due to shear forces) can be observed by video microscopy. The worms eventually settle down to take on cell shapes, not spherical, but reminiscent of red blood cells [22]. The interior of these membrane-mimetic systems contains a high density of microstructures, presumably micelles, that cannot be resolved optically. Similar things happen with naturally occurring lipids. Their phase diagrams are usually much the same as those double-chain surfactants, discussed above. There are domains in the temperature-concentration diagram, where small single-walled vesicles sometimes occur. This region is imperfectly explored, and most studies have focused on the concentrated lamellar phase. With increasing temperature there have been reports of a "critical" temperature at which the system transforms to giant vesicles, which it has been said represents an equilibrium thermodynamic state [24], [25]. It has also been reported that the membrane lipids of different single-celled animals form giant vesicles at precisely the temperature at which the different cells live [25]. The implications of this discovery are profound. It suggests that changes in temperature will induce changes in production of lipids that can allow the mixture in the membrane - in such a way that the critical temperature for the new mixture preserves the (giant vesicle) membrane integrity to ensure survival in the changed conditions. In view of later studies of lipid phase properties discussed above and in the sections on biomembranes (section 5.2, Chapter 7), it seems probable that this critical temperature could correspond to the transition from lamellar to cubic structure, so that we have the possibility of spontaneous formation of cubic structures. If we come back to smaller (single-walled) vesicles, we are on more familiar ground. The experimental situation is nevertheless unresolved. Some believe that vesicles are always thermodynamically unstable, the stable state being the lamellar phase. Monodisperse vesicles of the size given by simple

Vesicles

211

theory [26] can be produced through prolonged sonication. This is generally taken to mean that energy is required for their production. When monodisperse vesicles are so produced and cooled below the gel temperature, they become unstable and grow in size, eventually forming lamellar particles. On reheating, whatever size distribution existed at the lower temperature is apparently "frozen" and seems to be kinetically stable. Again with other preparation techniques any size vesicle distribution can be made to order, but these use mixtures or solvent extraction methods. Not all these distributions of vesicles can be thermodynamically stable states. All that can be inferred is that the dispersions of vesicles are slow to equilibrate because of the low solubility of the monomer. On the other hand, on addition of excess water, the lamellar phase disperses spontaneously to form liposomes, which suggests that the lamellar phase is not stable in too dilute solution. These phenomena have been explored elsewhere. Immense variability is allowed, and this depends on the particular surfactant or surfactant mix. So the question whether single-walled vesicles are thermodynamically stable or not depends on the lipid or lipid mix. Their existence does not necessarily require us to invoke non-equilibrium thermodynamics in biology. Some vesicles are stable, some are not, and this variability is exploited by nature. When the bilayer contains defects (e.g. mesh structures (section 4.5)), the allowed structures are evidently more variable.

5.1.8 Cholesteric liquid-crystals and low-density lipoprotein structures Cholesterol esters form crystalline structures that are similar to those formed by other lipids, consisting of alternating infinite lamellae, so that the hydrocarbon chains form close-packed sheets segregated from layers of cholesterol skeletons. There are three types of such structures [6]. One such can be represented by the chiral molecule cholesterol oleate, where pairs of cholesterol skeletons are arranged in an antiparallel packing in one layer, with the hydrocarbon chains in the adjacent layer. The cross-sectional area of the cholesterol molecule is about 40 A 2 (derived from pressure-area monolayer curves), corresponding to the cross-sectional area of two hydrocarbon chains. The chains therefore form an interpenetrating layer. If we imagine the thermal changes that occur when such a structure is heated, it is natural to expect that the chains can melt, although there is a tendency to retain the segregation between the steroid skeletons and the disordered chain regions. This can be achieved in liquid crystalline phases in a way completely analogous to the thermotropic and lyotropic mesophases of lipids discussed so far. The most remarkable feature of the cholesteric liquid crystalline phases is their ability to rotate the plane of transmitted plane-polarised light. Other liquid crystalline phases (for example those formed by certain polypeptides) with such properties have also been classified as cholesteric liquid crystals.

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Chapter 5

Particular attention has been given to one such cholesterol ester phase, the so-called "blue phase", introduced in Chapter 4. There is sometimes a smectic (i.e. layered) mesophase just above the melting point of the cholesterol ester. The spacing in this phase implies a shortening by about 10% of the fully extended molecule [6], so that the structure is similar to that of the L(x phase. When the smectic phase is heated through the transition to the cholesteric phase, the position of this spacing remains unchanged but the associated X-ray peak becomes diffuse. Due to the rotation of light it is evident that the structure must consist of helically twisted molecules, and it seems clear that the cholesterol skeletons, exhibiting chirality, form such helices, as described in Chapter 4. What type of packing of helices could explain these remarkable optical properties? The space group that has been proposed on the basis of x-ray studies is I4132, according to the surfaces exhibited by single-crystal type of domains of one of the two forms of the blue phase [27]. Different models of the structure have been discussed during recent years, and the main puzzling question posed that confounds interpretation of these structures is the following, summarised by Stegemeyer et al.: "The problem of setting up a molecular distribution function for the blue phases is to construct a continuous arrangement of form-anisotropic molecules both with cubic and chiral symmetry". However, there is a structure consistent with both the required space group and the optical properties. The gyroid surface, which occurs frequently in lipid-water systems, provides such a possibility. If we assume that cholesterol skeletons form rod-like infinite helices, this structure represents an effective three-dimensional packing of such helices. Thus, the rods form a bodycentered arrangement as shown in Fig. 5.5. In this structure, there is a helical twist between the rods, in addition to the cholesteric twist within each rod. The hyperbolic structure is a consequence of the chirality of the esters, which induces torsion into the packing arrangement. A racemic mixture does not exhibit this phase; natural cholesteric esters contain a single enantiomer only. The structure of the blue phase is of some importance. Among the lipoproteins carrying lipids in the blood, low-density lipoproteins (LDL) have attracted much attention. They are the factors mainly responsible for plaque formation, which ultimately leads to atheriosclerofic changes and heart disease. The major components of the LDL-parficles are cholesterol fatty acid esters. A remarkable property is the constant size of LDL particles [28], which indicates that the interior must possess some degree of order. It seems probable that the structure proposed above for cholesterol esters in the cholesteric liquid-crystalline structure should occur also in the LDL-particle. In that case the LDL particle can be viewed as a dispersed blue phase, whose size is related to the periodicity of the liquid-crystalline phase, and the protein coat at the surface is oriented parallel to adjacent specific crystallographic planes of the blue phase. These amphiphilic proteins will expose lipophilic segments inwards and expose hydrophilic groups towards the environment.

Low-density lipoproteins

213

Figure 5.5: Model of the packing of helical rods of cholesterol skeletons according to the geometry defined by the gyroid. The three rod-directions are parallel to the [111]-directions, and one set of rods is perpendicular to the plane of the figure.

5.2

Cell membranes

5.2.1

Introduction

Membranes based on the lipid bilayer form the envelopes of cells and the organelles within the cell. Numerous functions of the living cell are directly related to the curvature of the lipid bilayer. From studies of lipid-water mixtures and isolated membranes the general functional features of the bilayer are known: barrier properties, lateral diffusion, acyl chain disorder and protein association. To understand the mechanisms behind a wide spectrum of membrane functions, a detailed picture at the level of local curvature is needed. Examples are fusion processes, cooperativity in receptor/ligand binding or transport through the bilayer of the proteins that are constantly synthesised for export from the endoplasmic reticulum. Some preliminary discussions of the possibilities of curved, rather than fiat, membranes follow. The view of the lipid bilayer in membranes as a passive barrier, with the embedded proteins assigned all functionality, has stood unchallenged for about half a century. Numerous results during the last decade have drastically changed this picture. It is now well established that the lipid molecules also participate in biochemical processes related to many basic membrane functions. An illustrative example is cell activation due to an

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increased intracellular calcium ion concentration; a common process. A consequence of this is that the phospholipids of the membrane are modified by phospholipase A2 and C, and that protein kinase C is activated. It is obvious that the induced changes in lipid composition must influence the bilayer structure, but the significance of these conformation changes of the lipid bilayer in relation to the actual membrane functions is not known.

Figure 5.6: (a) Image of l.~ 2D fiat membrane. The holes represent space occupied by proteins. (b) The C2D membrane phase.

The implications of the existence of an enormous diversity of lipid species and fatty acid patterns in different membranes within one organism as well as the variations between different organisms have posed a challenge for a long time. Any model of lipid bilayer function has to take account of these variations. If we consider erythrocyte membrane phospholipids for example, the rat has about 50% phosphatidylcholine (PC) and only about 10% sphingomyelin (SM), whereas the sheep erythrocyte membrane contains more than 50% SM and no PC. By contrast the total membrane content of phosphatidylethanolamine (PE) + PC + PM in mammalian species is fairly constant, equal to about 15-20%, and the rest are charged lipids

Cell membranes: phase transitions

215

(phosphatidylserine, PS, and phosphatidylinositol). The a s y m m e t r i c distribution of lipids is another intriguing problem. Thus in the erythrocyte membrane, cholesterol, SM and PS are enriched in the outer monolayer. In 1980 it was pointed out that the prolameUar body is a perfect example of a Cp structure [4]. (Later, more detailed, analyses have revealed that it may also be a CD structure; cf. section 7.2.) Following work on the structure of cubic phases, it was also realised that two-dimensional analogues are possible. This in turn suggested that a phase transition involving changes in the intrinsic curvature of membranes might be possible [29, 30]. Such a mechanism has far reaching implications. Clear evidence for such transitions between bilayer conformations has been reported [9]. This membrane bilayer model will be described below.

5.2.2 On intrinsic periodic bilayer curvature in membrane function: A model membrane bilayer phase transition involving periodic curvature The proposed model [29, 30] is based on the possibility of forming sections of a cubic lipid-water phase which gives a two-dimensional minimal surface with "holes" facing alternate sides of the bilayer. If these holes through the bilayer are plugged with protein molecules, we can form a bilayer that is closely related to the "planar" bilayer conformation (cf. Fig. 5.6). As this new conformation is related to the cubic phase, we will call it C 2D, whereas the "normal" conformation, which corresponds to the La phase, is called Lr 2D. The c o r r e s p o n d i n g phase transition is thus Lr 2D. The C 2D conformation is regarded as an excited functional state of the bilayer that occurs, for example, during fusion events. According to electron micrographic evidence (principally comparison with bicontinuous cubic phases), C 2D phases are an accurate representation of socalled "non-bilayer" conformations of membranes. It must be stressed, however, that this conformation is a true bilayer.

5.2.3 Lipid composition control in membranes The same driving force behind lipid polymorphism in three dimensions changes in the average molecular wedge-shape - can also operate in a membrane bilayer. It would be remarkable if polymorphism exhibited by lipid molecules is not utilised in membrane function. Further it must be expected that chemical changes of the lipid molecules that we know occur in membranes (e.g. formation of diacylglycerol) are associated with bilayer conformational changes. The lipid-protein interaction in the bilayer means that a transition to an "excited" intrinsically curved conformation can be induced by a protein

216

Chapter 5

conformational change as well as by changes in average lipid molecular wedge-shape. The two processes - lipid and protein changes - go hand in hand

(cf. Fig. 5.7). A transition induced in the lipid bilayer will result in a conformation "pressure" on the protein so that it can adapt to the changed bilayer structure. A consequence of this is the possibility of controlling lipid composition by an on/off switch of enzymes responsible for lipid modifications. For example the methyl transferase enzymes that convert phosphatidylethanolamines into phosphatidylcholines might be controlled in this way. (PE favours C 2D whereas PC favours the L0t2D conformation and C 2D should thus be expected to switch on and La 2D switch off the enzyme activity). Extensive studies of lipid-water systems show that increased chain disorder upon heating tends to increase molecular wedge-shape (i.e. the magnitude of the surfactant parameter, v/al, via a decrease in 1), resulting in a phase transition from the Lot to a cubic phase (see Chapter 4 and section 5.1). Increased amounts of cis-double bonds or decrease in the size of the polar head group all have the same effect. Changes in the polar region can be induced by variations in pH or ionic environment, or indeed of dissolved gas like CO2, cf. Chapter 3). If membrane charge is reduced (by increased pH or counter-ions) this will favour C 2D over Lot2D. Divalent ions like calcium should be most effective in switching from Lcx2D to C 2D due to the increase in wedge-shape induced by cross-linking by calcium of anionic polar heads of the bilayer. Furthermore it is k n o w n that m e m b r a n e lipids u n d e r physiological conditions are very close to a phase transition from Lot to a bicontinuous cubic phase or HII phase [9] [31]. Variations in environmental factors (like temperature) seem to be compensated by compositional changes of the membrane lipids so that the membrane adopts a conformational state delicately poised, and close to this kind of transition. The main evidence for such control of the lipid composition in membranes, monitored by a phase transition switch (Lcx2D (-->c2D), is the diversity of lipid composition and the effect of environmental factors, like temperature, pressure, and lipophilic agents on function. Related proposals [32] for "non-bilayer" lipid structures in membranes invoke the occurrence of the transition Lot---> H I I . The data from which such a proposal is deduced are also consistent with an Lr 2D ---> C 2D transition. As discussed in Chapter 4, cubic structures are intermediate between Lot and HII. The difference is that for the La 2D ---> C 2D phase transition the bilayer conformation alone is involved, whereas any membrane conformation related to the HII structure seems improbable in the extreme because of the resulting exposure of hydrophobic tails to the aqueous medium. Pressure increases the trans/gauche ratio of the chains, which is reflected in increased bilayer thickness and decreased molecular cross-section. A pressure increase of 100 atmospheres is considered to be equivalent to the effect of a temperature decrease in the range 2-8~ Reported effects of pressure as well

Lipid composition control

217

as of temperature on lipid composition of membranes are consistent with this phase transition model [33]. Any mechanism proposed to control lipid composition must take account of enzyme system activity monitoring by such diverse factors as temperature, pressure, pH, ionic strength, variations in content of cis-double bonds and trans/gauche ratio, and the presence of sterol skeletons. A phase transition can provide a direct conformation related switch, with the ability to control all these factors [34]. Asymmetry in the lipid distribution over the bilayer could also be controlled in a similar way by the lateral packing pressure, which is likely to differ between constituent monolayers, due to the distinct chemical environments inside and outside the membrane. The enzymes involved m a y also be distributed asymmetrically. A configuration with constant, but nonzero, mean curvature, shown in Fig. 5.7, reflects such a situation. A membranespanning protein can then be viewed as a sensor of the lateral packing pressure in both monolayers. This speculation has some experimental justification. In a recent study of chromaffin granules, trans-membrane lipid asymmetry was shown to be induced by an ATP-dependent "flippase" [35]. The "fixing" of the proteins that fill holes in the C 2D structure can be considered to take place in the following way. In an L0~ phase all molecules are subject to the same packing conditions. In the cubic structures, however, there is a continuous variation in packing conditions, from fiat points to the points with highest Gaussian curvature due to the necessary inhomogeneity of hyperbolic surfaces. The disorder must increase with increasing Gaussian curvature. In a membrane bilayer the lipid molecules in a L(x2D conformation are statistically distributed about a mean shape, whereas a C 2D conformation must exhibit continuous variations. A higher Gaussian curvature might correspond to a higher relative proportion of lipids with cis-double bonds, for example. It should be pointed out also that a C 2D bilayer consisting of only one lipid type (like the cubic phases discussed in Chapter 4) must have v a r y i n g molecular disorder c o r r e s p o n d i n g to the varying Gaussian curvature. The lateral lipid diffusion is almost identical to that of the La phase [3]. Studies of lipid mobility have shown that the molecules close to membrane proteins are more disordered than those lipids further removed. In a lipid bilayer with variation in lateral disorder, the proteins should therefore be expected to go into positions of highest disorder. Thus, there is a driving force for proteins to locate in the "holes" in the C 2D bilayer structure, which are surrounded by maximally curved membrane. There is also a lateral packing pressure to adopt the protein according to the average hydrocarbon chain direction. Ideally, the proteins filling the "hole" of the C 2D lipid bilayer should be wedge-shaped (see Fig. 5.7). Thus the C 2D conformation (contrary to the Lr 2D one) provides a periodic variation in the degree of lipid molecular disorder, or "fluidity", and this

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Chapter5

might have functional significance. Different studies show that the activity of membrane-bound enzymes can be directly related to the bilayer fluidity, e.g. p h o s p h o l i p a s e s , N a + / K + A T P a s e or Ca2+-ATPase. It is obvious that a transition Lot2D ---) C 2D, with the enzyme molecules in the "fixed" locations, will result in an enzyme environment of lipid molecules of higher fluidity than that of the L~ 2D conformation.

Fig. 5.7: Periodically curved membrane conformations. Proteins filling the "holes" towards the corresponding three-dimensional phase are indicated by filled units, whereas open units are free to diffuse along the bilayers. Successive changes towards asymmetric and constant (nonzero) mean curvature are shown from top to bottom. All the other protein molecules of the membrane in the C 2D conformation, which are not in the "fixed" or "hole" positions discussed above, will possess lateral diffusional freedom just as in the Lr 2D conformation.

5.2.4 The nerve membrane, signal transmission and anaesthesia The ion fluxes and the functions of ion channels and p u m p s in nerve signal transmission have been studied and are k n o w n in considerable detail. The lipid bilayer of the nerve membrane is highly specific in composition with well-controlled asymmetry. In spite of this, no other role has been assigned to the lipid bilayer except to provide a barrier and a passive carrier function for the membrane proteins involved in the p u m p s and channels. Arguments will be given below that during the propagation of the action potential there occurs a bilayer phase transition involving a periodically curved bilayer.

Bilayer conformation during the action potential Several independent studies indicate that a lipid bilayer phase transition takes place in conjunction with transmission of the action potential. Forty

219

Nerve membranes

years ago it was reported that heat was associated with the passage of the action potential, and detailed thermal data were later reported in studies of impulses through the pike olfactory nerve [36]. Two regimes were observed first a positive value (at 0~ 44.2 ~tcal/g/ impulse), followed by a negative value (48.9 ~cal/g/impulse). Calculation shows that these values are much too high to correspond to free energy changes associated with the membrane capacitance. The natural interpretation is therefore that the thermal transition is related to changes in the lipid bilayer. Studies of optical properties of the squid giant axon have shown that the pulse gives a change in the birefringence [37]. It was proposed that these changes reflect changes in membrane thickness. Also, in a fluorescence study of an olfactory nerve fibre, a reduced fluidity of the hydrocarbon chains of the membrane on the passage of the action potential was found [38], which is consistent with reduction in entropy of the lipid. Taken together the observations imply that transmission of the action potential may involve the propagation of a phase transition "wave". In that case the spike

can be d u e

to the transitions

L~D'-~C2D--~I-~D

or

C 2D .._)I~D --~ C 2D .

As periodic structures have not been reported from electron microscopy studies of freeze-fractured axon membranes, it seems probable, if this phase transition is involved, that the region of spike exhibits the C 2D conformation. The inner lipid monolayer of the axon membrane (with PE and C22:6 acyl chains dominating) is expected to favour a periodically curved structure, due to the high average wedge-shape associated with these lipids, whereas the outer monolayer (sphingomyelin and PC dominating) should favour a planar conformation. From this perspective it seems reasonable that the membrane bilayer can exhibit both an L~2D as well as a C 2D conformation. At the neural membrane region there is a requirement for regularly packed protein molecules, p r e s u m a b l y the ion channels, located at the hole positions. The regularity in packing might be related to the required narrow spread in the speed of signal transmission. The m i n i m u m time between action potential spikes is so short that the protein order is not lost due to diffusion during the period of existence of the L~ 2D conformation of the bilayer. The existence of a curved C 2D conformation associated with the action potential is supported by the fact that the ion influx at the spike will induce an increased average wedge-shape of the molecules, due to electrostatic screening of the lateral repulsion of p h o s p h a t i d y l s e r i n e molecules. Furthermore a C 2D conformation associated with the spike would directly relate action potential propagation to the mass-cooperative vesicular fusion, involved in the chemical signal transfer by transmitter molecules at the presynaptic membrane. Experimental support for this concept has been recently reported [39]. This well-controlled fusion process of numerous "vesicles" with the presynaptic membrane must take place as a phase transition. The

220

~ter 5

process is a beautiful example of cubic phase function, as will be described below. This membrane fusion process (outside the brain) is known to involve thousands of single membrane units, previously thought of as vesicles, assembled into units that have been termed "boutons". We have examined the EM texture of the boutons and found that they are in fact a cubic phase. The synaptic signal transmission can take place as frequently as hundreds of times per second. A fusion process involving a hyperbolic membrane can be well controlled, and the calcium ion i n f l u x - which induces fusion - is expected to change the conformation of the cubosome surface membrane from its planar bilayer conformation to the fusogenic saddle-saddle conformation. (It is known phase transitions of membrane lipids can occur when exposed to calcium, e.g. [40]). In this connection it can also be mentioned that the lipid composition of synaptic membranes responds to temperature changes (fish acclimatised in the range 2-37~ as well as pressure changes, expected if the membrane bilayer must balance on the edge of an La2D<-~ C 2D transition (cf. [41]).

Anaesthetic

effects

Among various membrane functions that have been assigned to lipid bilayer perturbations, the anaesthetic effect has probably been the subject of most studies. The general anaesthetic effect can be induced by a wide range of chemically "inert" molecules (from noble gases to hydrocarbons), whose lipophilic character is directly correlated to the anaesthetic potency.

Figure 5.8: Model of a hypothetical transition L(x2D---)C2D (top to bottom) near a sodium channel (cross-hatched regions) during the action potential propagation.

Nerve membranes

221

If, as discussed above, the L(z2D(--~C 2D phase transition is involved in the propagation of the action potential, a shift in the transition in any direction can be expected to induce in an anaesthetic effect. General anaesthetic agents, like ethyl ether and chloroform, have been observed to induce the L0~--~ cubic phase transition in phospholipid-water systems. The degree to which they do so has furthermore been related to their anaesthetic potency [42]. If we consider the bilayer transition, as illustrated in Fig. 5.8, it seems clear that the curved bilayer conformation can induce a conformational change of an ion channel protein. The effects have also been examined in a model system exhibiting both Lc~ and a cubic phase in excess water. In the presence of small amounts of the anaesthetic agent, the L0~ phase cannot exist [42]. The reversal of the anaesthetic effect with pressure is also consistent with the phase transition effect; a pressure increase should produce the transition C 2D L~2D (see section 5.3.1). Local anaesthetics tend to shift the phase transition in the opposite direction. Lidocain hydrochloride, for example, can transform the cubic phase of monoolein into the L0~ phase. Local anaesthetics are usually cationic, weakly amphiphilic molecules, and their effect on the lipid bilayer conformation depends on the ionic and hydrophobic character, both of which affect membrane curvature. A study of the effect of lidocain [43] on the cubic phase of monoolein in excess water shows that lidocain hydrochloride gives a transition to the Lo~ phase, whereas the free base gives a transition to the HII phase. (These changes are completely in accord with those ion-specific CI-, Br" v s . OH-, Ac- curvature changes discussed in Chapter 3 with reference to model cationic surfactant systems (section 3.6).) Keeping these variations in mind, controversial reports in the literature concerning effects of local anaesthetics on lipid phase transitions can be explained. What is not known, however, is the degree of ionisation of a charged anaesthetic agent at its site in the axon membrane. (There is an additional effect of the charge, which is probably related to the immuno-suppressive effect of cationic surfactants, discussed later.) Local anaesthetic agents are considered to either perturb the lipid bilayer or exhibit a specific effect on the sodium channels (cf. [44]). Many observations support the simple concept of a phase transition shift. Divalent ions, for example, should be expected to induce a higher average wedge-shape of the lipid molecules through cross-linking of anionic heads of phospholipids. It is therefore not surprising that they have antagonistic effects in relation to local anaesthetic agents. Reports of the effect of anaesthetic agents on enzyme systems controlling neuronal membrane compositions can be a secondary consequence of phase transition shifts, according to the proposed L~2D~--~C 2D control mechanism of lipid composition, described above. Thus inhibition of methyl transferase activity and effects of anaesthetics on sialidase degradation of gangliosides [45] can be related to an on/off switch of enzymatic activity provided by an L~2D(--~ C 2D transition.

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Cdmpter 5

The lipid theory of the anaesthetic effect has recently been questioned [46], following studies of the effect of the influence of anaesthetics on the activity of the luciferase enzyme. Thus this enzyme is inhibited by relevant concentrations of anaesthetic agents and the anaesthetic potency correlates well with the level needed for inhibition. The most important observation was a comparison of the effect of the two enantiomers of isoflurane on the potassium current in snail neurons. They had shown earlier that isoflurane activates an outwards flow of potassium ions from the neuron. It was found that one of the isoflurane isomers increased the potassium flow twice as much as the other. The basis for their conclusion is that the disordered lipid bilayer should be influenced in the same way by the two isomers. We disagree, however. It would not be surprising if membrane lipids in the bilayer interact differently with the two isomers as they are themselves enantiomeric. The isoflurane molecule is amphiphilic (an ether) and should be localised at the polar region of the bilayer. The average shape of the bilayer molecules can therefore be influenced differently by the isoflurane isomers. This can easily be demonstrated by making up microemulsions of water, the double-chain cationic surfactant DDAB, and enantiomer alkanes. The phase diagrams, i.e. curvature at the oil water interface, are completely different [47].

5.2.5

A n a e s t h e t i c agents, cancer and immunosuppression

When the Canadian Anaesthetists Society recently held their 48th annual meeting Moudgil reported that the anaesthetics halothane and isoflurane significantly increase the development of metastatic tumours in mice inoculated with melanoma cells. The effect was interpreted to be a result of suppression of the immune response. We point out the tendency towards the periodically curved bilayer conformation induced by anaesthetic agents discussed above and the evidence for hyperbolic membrane regions in metastasising cells described in the next paragraph. This indicates that the mechanism behind this observed relationship is also due to related effects on membrane curvature. An apparently related effect due to membrane curvature is the phenomenon of i m m u n o s u p p r e s s i o n induced by cationic surfactants. The cationic quaternary a m m o n i u m and pyridinium surfactants are widely used as sterilising (antibacterial) agents in an enormous variety of applications. Although the biochemical and genetic mechanism by which bacteria like streptococcus aureus develop immunity to these is not understood, it appears certain that the antiseptic effect is simply related to membrane disruption. At and above the critical miceUe concentration ("cmc", discussed in Chapter 3), 1 any single-chain ionic surfactant ( v / a l = ~) will change the curvature of a lamellar phase or phospholipid membrane ( v / a l = 1) catastrophically. This can easily be demonstrated with an in vitro suspension of cells to which is added progressively higher concentrations. At the cmc the cells explode (the cmc is that appropriate to the physiological saline, very different to that in

Anaesthetic agents and cancer

223

pure water). Optimal effects are achieved with long-chain surfactants like cetyl pyridinium chloride. The longer the chain, the larger the hydrophobic driving force for adsorption. But the chains cannot be too long, as the Krafft temperature for the surfactant then will become higher than physiological temperatures. With longer chained surfactants too the cmc is lowered substantially, so lowering the required dosage to induce membrane disruption. (It is of interest that best results are obtained with impure surfactants the impurities act as hydrophobes that assist micellisation and lower the cmc again.) -

At much lower concentrations still, below the cmc, both single- and doublechain cationic surfactants do not destroy cells. Radio-labelling studies show that the process of adsorption follows a Langmuir isotherm, i.e it is a physicochemical effect. At concentrations 10 to 100 times lower than the cmc, these surfactants are potent immunosuppressants [48, 49], as potent or more than cyclosporin A, the drug of choice in organ transplants. The required concentration is the same as that for cyclosporin, although the mechanism is very different. Cyclosporin acts on the DNA of T-cells involved in the immune system and is not reversible in its action. The surfactant-induced immunosuppression, associated with membrane-induced curvature that affects the key recognition process involving the major histocompatibilitT complex is reversible, for both in vitro and mouse thyroid allografting experiments. The measured levels of uptake in membranes at which these surfactants begin to induce immunosuppression in in vitro experiments on T-4 cells coincides roughly with known density of charge on the T-cell inner membrane surface. So the normally negatively charged surface becomes electrically neutral, and near-membrane Ca 2+ concentration, usually high because of the double-layer ionic distribution, drops substantially. This would imply [50] that along with changes of membrane curvature caused by surfactant adsorption the concentration of perimembrane Ca 2+ at the normally (negatively) charged membrane would decrease substantially, providing another mechanism for membrane bound protein conformation changes. Another curious observation[50] is that the dose required to induce immunosuppression decreases progressively with increasing hydrophobicity of the hydrocarbon tails (number of CH2 groups, single- or double-chained), as expected, but abruptly disappears for longer double-chain (C14)and (C16)surfactants (dihexadecyl dimethyl ammonium chlorides). A clue to the reason may lie in experiments on black bilayer films, which show that the flip-flop rate increases exponentially with increasing chain length (up to many days). Hence in 5-day mixed lymphocyte culture tests or mouse thyroid graft experiments (one month), the effect is not observed if, as seen to be so, it depends on effects induced on the inner side of the membrane. The principle of using agents like cationic surfactants in non-toxic dosages to alter cell membranes in a way that change recognition processes, viral attachment, etc. has not yet been widely exploited. For example large cage compounds like cyclodextrins [51], appropriately modified to avoid bacterial

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attack, can encapsulate particular hydrophobic drugs like ellipocene depending on the size and shape of their catenoid-like interiors. They can be delivered to particular sites by attaching to the ring double-chain hydrocarbon tails compatible with the membrane lipids. The tails can have attached to them specific receptors that bind to specific cells and cell sites, so reducing the general toxicity problems associated with chemotherapy.

5.2.6 On the metastatic mechanism of malignant cells The fact that metastatic potential can be transferred with the plasma membrane [52] has been the basis of studies of the role of cell membrane structure in relation to the ability of cells to metastasize [53]. Mountford and coworkers [54, 55] have shown that the lipids of the plasma membrane of malignant cells contains about 6% of neutral lipids, with triglycerides as the predominant fraction. Furthermore they concluded from the occurrence of high-resolution sharp NMR peaks that these neutral lipids are isotropicaUy distributed in membrane-associated domains. These domains were modelled as oil droplets invading the space between the two halves of the bilayer. We propose that these triglycerides-enriched domains consist of folded bilayer units in the form of a hyperbolic bilayer. Evidence for such a structure comes from lipid phase studies of transitions from the Lr phase to a cubic phase accompanying solubilisation of triglycerides into the L0~ phase [56]. A hyperbolic bilayer domain of this type is consistent with the high-resolution NMR signals. A surface structure of a malignant cell consisting of regions will tend to fuse with any other plasma membrane. This tendency will be strongly enhanced if the other cell also has the same kind of surface structure. The presence of a general anaesthetic agent will induce a phase transition from the planar bilayer (Lcx) towards an intrinsically curved bilayer as in the cubic phase. The induction of metastasis by anaesthetic is then an obvious and expected phenomenon, unavoidable, and independent of the anaesthetic used.

5.2.7 Membranes in micro-organisms and anti-microbial agents There are striking similarities between the anaesthetic effect discussed above and anti-microbial effects. Thus gases that are general anaesthetics, even the inert noble gases, exhibit an anti-microbial effect. Furthermore this is correlated to the anaesthetic potency [57]. Moreover, it is possible to reverse this effect by hydrostatic pressure. As discussed above there is an opposing effect of temperature on lipid phase transitions as compared to that of pressure, and this is also reflected in living systems. At 1,000 bar for example certain organisms can survive at 104~ [57].

Membranes in micro-organisms

225

In accordance with the observation that the anti-microbial effect of gases like nitrogen dioxide, carbon dioxide or xenon can be related to the effects of general anaesthetics, it seems natural to relate the well-known anti-microbial effect of cationic surfactants to local anaesthetic agents. The first type of perturbants shifts the actual transition in the direction Lot--> C 2D, whereas the second type of agent is expected to shift the transition in the opposite direction. It should be mentioned that quaternary ammonium surfactants such as cetyltrimethyl a m m o n i u m bromide in micellar solution in water were found to transform the cubic phase into the La phase [34]. Simple fatty acids exhibit a weak anti-microbial effect, whereas certain branched homologues exhibit quite strong effects. Isotetradecanoic acid, being the most active homologue in this respect, was observed to induce an Lot cubic transition in a phosphatidylcholine liposomal dispersion [34]. Certain poly-alcohol surfactants have also been reported to exhibit considerable anti-microbial activity. Monocaprin and monolaurin, as the most active, have been examined in detail. No effects on the Lot phase of phosphatidylcholine dispersions were observed, whereas they gave two coexisting phases - cubic (C) + Lot- when added in minor amounts to the cubic phase of monooleylglycerol [34]. This indicates that their additions favour the Lcx2D conformation over the C 2D type. Again, if the membrane utilises the phase transition in certain functions, any disturbance in the balance between the two conformations would be manifested as an anti-microbial effect. Monolaurin exhibits a minimum inhibitory concentration of about 5 ~ g / m l (studies on Vibrio paranaemolyticus (cf. [34]). The monomer concentration in distilled water in equilibrium with monolaurin crystals at room temperature is about 4-5 ~ g / m l . Thus the concentration w h e n self-assembly starts coincides with the concentration where biological effects are observed. As mentioned earlier monolaurin favours the Lcx conformation. It is interesting in this connection to consider peptide antibiotics, which act on the lipid bilayer, like gramicidins (linear pentadecapeptides) and the tyrocidines (cyclic decapeptides). Both are very hydrophobic and assumed to form ion-permeable channels. Gramicidin is known to induce the Lot---> HII phase transition in membrane lipid systems. A study of the interaction between gramicidin A and tyrocidine on a model membrane showed that they have antagonistic effects, and tyrocine-gramicidin gives a cubic structure [58]. This indicates that the antagonism is due to opposite effects on the average molecular wedge-shape, with gramicidin increasing and tyrocidine decreasing the average wedge-shape in relation to the Lcx phase.

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5.2.8 P l a n t cell membranes

Recently membrane lipids from Brassica napus root cells were examined with respect to effects from dehydration-acclimatised plants [59]. It was found that the lipids formed a cubic phase with excess water under physiological conditions. On heating, this phase transformed directly into a reversed micellar phase. The transition was also found to coincide with the temperature limit of survival of the plant. Also after repeated water-deficient stress, a cubic phase is formed in excess water, although there are differences in the phase properties compared to lipids from membranes of plants grown normally. In the ripening process of certain plants, e.g. vegetables, ethylene is produced to increase the respiration rate. The mechanism is believed to be a nonspecific membrane lipid effect. Ethylene has an anaesthetic effect comparable to nitrous oxide, and it has even been used as a general anaesthetic agent. It is therefore natural to assume that the corresponding membrane bilayer transition L(z2D~--~C2D is the mechanism behind this stimulation of mitochondrial respiration reactions. The thylakoid membranes of chloroplasts are formed from a threedimensional assembly of the membrane, the prolamellar body. There can also be cyclic transitions between thylakoid stacks and the prolameUar body between day and night, or even between summer and winter. The ultrastructures of a series of morphological changes from the threedimensional storage form to the membrane arrangement of the thylakoid have been described by Gunning [60]. In the first assignment of a minimal surface structure in a cubic lipid-water phase, it was pointed out that Gunning's interpretation of the electron micrographs is fully consistent with a three-periodic minimal surface [61]. In connection with current work on the mechanisms behind these light-induced membrane transitions, it is also becoming clear that besides the dominating Cp structure, which was seen initially, the CD and the CG structures also occur during the transformation process (cf. section 7.2). The whole prolamellar body thus has the structure of a cubic phase, as discussed in detail in Chapter 7.

5.2.9

The C2D conformation and membrane fusion

There are interesting recent reports linking membrane fusion to cubic lipid phases [9] or to reversed (or "inverted") phases [62]. So-called "inter-lamellar attachments", formed between the bilayers of liposomes on fusion, show freeze-fracture electron micrograph textures identical to those of a cubic lipidwater phase (see Fig. 5.9). The inter-lamellar attachments seem to be identical to "lipidic particles" described earlier [63]. It is also interesting to note that diacylglycerols, secondary messengers from the PI-cycle, produce fusion in

Membranefusion

227

phospholipid model systems at the levels occurring in membranes [62]. The presence of this lipid with its small polar head group will favour the C 2D conformation over the L~2D type. In a recent review n u m e r o u s examples were given of m e m b r a n e ultrastructural textures consistent with the C 2D conformation discussed here [64]. Another obvious case of a C 2D conformation will be mentioned. The brain astrocytes are rich in potassium channels, which appear to play an important role in the regulation of the ion concentrations in the brain. Freeze-fracture electron micrographs of the outer astrocyte m e m b r a n e contain patches of a periodic structure [65]. These ordered assemblies are thought to be potassium channels. In our membrane description these channels serve to plug the "holes" of a C 2D bilayer, whereas the rest of the membrane is in the L(z2D conformation.

5.2.10 Membranes encapsulating oil/fats and biliquid foams Triglycerides are stored as an energy source in living systems. Sometimes there are specialised tissues, like adipose tissue, consisting of cells containing fats or oils. There are two aspects of such storage structures where curvature plays a significant role, which we consider below. Under physiological conditions the fat is normally liquid, but even in mammals the fat can crystallise, for example in the subcutaneous adipose layer. It is therefore most important that the crystal grows in a specially adapted form in parallel with the membranes, so as not to cause mechanical damage. The triglycerides that constitute fat crystallise in three polymorphic forms, and the form obtained first, when the liquid solidifies, can adapt to the curvature of the membrane [67]. It is the so-called (z-form, with partly disordered hydrocarbon chains. This disordered nature of the chains results in plastic properties of the crystals and a tendency to grow in thin films perpendicular to the chain direction. The adipose tissue represents a close-packing arrangement of oil-containing cells with water as a continuous phase. A similar colloidal structure can also be produced in simple ternary mixtures of oil, water and surfactants, and it has been described a biliquid foam [68]. The milk-fat globule membrane has been studied extensively due to its significance in milk processing and in nutrition. It is quite different from most other membranes, as it has to form an interface towards oil as well as water. The m e m b r a n e consists of three units. One unit is formed by a "normal" membrane consisting of a lipid bilayer with embedded proteins exposed towards water. (When the milk globule is secreted it takes the plasma membrane of the gland as an envelope after endocytosis.) Then there is a so-called protein aqueous layer, consisting mainly of xantine oxidase. The third and innermost unit is the surface of the fat globule. As this globule

228

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contains free fatty acids, monoglycerides and phospholipids (mainly PC), there must be a segregation of the lipids so that the polar lipids form a monolayer between the oil phase and the aqueous exterior. Buchheim has reported translational periodicity in the "protein aqueous" layer [69], and it has been pointed out that the electron micrographs are consistent with the formation of the D-surface [64]. It is therefore natural to assume that this zone represents a periodic hyperbolic membrane. Our picture of the relation between the fat globule and the external bilayer membrane is a linking struct u r e - formed by a periodically curved bilayer- from the outside, which fuses with the fat globule to form a monolayer. This bridging structure is thus similar to the link between the L0~ layer and the cubic phase outside and inside a cubosome respectively.

Figure 5.9: Freeze-fractureelectron microscopyof phospholipid liposomes undergoing fusion, after Ellens et al. [66]. It has recently been found that the fatty acid distribution of the triglycerides forming the core of milk fat globules varies with the globule radius; the

Encapsulating membranes

229

smaller the radius the higher the relative amount of unsaturated chains [70]. This is a striking illustration of the significance of curvature, and clearly consistent with the picture of the membrane zone structure of milk fat globules presented above. Here too, differences in curvature are likely to be related to differences in intrinsic effective molecular wedge shape.

5.2.11 Sugar groups, receptor-ligand binding and cooperativity The receptor mechanism involved in cell uptake of lipids from blood in the form of LDL (low-density lipoprotein) has been revealed by the work of Brown and Goldstein [71]. When the lipoprotein is bound to the receptor, a "coated" pit deepens and buds off to form a "coated" vesicle inside the cell (endocytosis). The "coat" represents a general principle of cell transport that is known mainly from the work by Bretscher [72]. The protein forming this coat is clathrin, which aggregates into planar networks (or basket-like cages, morphologically similar to "mesh" phases of lipids or surfactants in water, cf. Chapter 4) and the lipid bilayer can span these nets, like a soap film spanning a wire net. The membrane lipid-clathrin interaction at work during the formation of vesicles from the membrane is not known. It is usually assumed that clathrin reconformation drives the endocytosis process. Alternatively these conformational changes might be induced in the attached lipid bilayer. It should also be pointed out in this connection that clathrin-coated pits are not the only route of endocytosis into eukaryotic cells. There are also other coat proteins. Studies of protein toxins have shown that some (e.g. cholera toxin) enter the cell without the aid of clathrin. Receptor-ligand binding on a membrane surface in general is usually a highly cooperative process. A possible bilayer conformation mechanism due to the presence of sugar groups will be considered here. Sugar groups are known to play a key role in surface recognition phenomena. Plant lectins or bacterial lectin-adhesins can recognise specific sugar groups on the surface of the membranes, for example in the gastro/intestinal system. There is also the opposite topological alternative. For example, the presence of mannose receptors on liver endothelial cells implies that glycoproteins with terminal mannose will be bound and taken up by endocytosis. One primary mechanism might be a conformation change of the mannose receptor, which triggers a change in curvature, leading to vesicle formation. Another effect that must be taken into account is a conformational change of the bilayer induced when the sugar groups approach the membrane surface. Recent work has shown that very low sugar concentrations in lipid-water systems induce phase transitions following the sequence L0~--~C --~ HII. If a corresponding two-dimensional lipid bilayer phase transition is involved, it

230

Chapter5

could trigger the endocytosis process and induce cooperative b i n d i n g to n e i g h b o u r i n g receptors.

5.2.12 A C 2D m e m b r a n e structure in a Streptomyces s t r a i n

M e y e r et al. have found that the L-ceUs (without cell wall) and hyphal cells of a streptomyces strain exhibit periodically curved cytoplasmic m e m b r a n e s [73]. This sort of periodicity, observed earlier in m i c r o - o r g a n i s m s , has b e e n a t t r i b u t e d to u n d e r l y i n g vesicles. This n e w w o r k [73] has s h o w n that the periodicity occurs in the absence of vesicles, and it can be s u p p o s e d that a p e r i o d i c a l l y c u r v e d b i l a y e r (C 2D s t r u c t u r e ) is the t r u e m e m b r a n e conformation. Their results are shown in Fig. 5.10.

Figure 5.10: Electron micrographs of L-cells of Streptomyces hydroscopicas [73]. The freezefractured texture shows the periodically curved lipid bilayer. The curvature is weakly expressed in a and very distinct in c. Two attached lamellar bodies and the underlying membrane are shown in b. The bar (in c) is 500 nm.

Streptomyces

231

To sum up, the ubiquity of hyperbolic geometries in biomembranes parallels the occurrence of these structures in model membrane mimetic surfactant systems. A large n u m b e r of examples have been outlined above where these structures occur spontaneou~Jy, and their transformations between flat and curved forms are evidently used ex~ensively in nature. The approach to thinking about structure and function in terms of membrane curvature presented here allows insights into biological function that are inaccessible through the conventional view of membranes, that largely focuses on the role of proteins, to the exclusion of the lipid matrix.

232

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237

Chapter 6 6.1

Folding and Function In Proteins and D N A

Overall features of protein structure

f a carboxylic acid and an amine are mixed in a suitable solvent, some catalyst is added, and the temperature is increased, an amide together with an equimolar amount of water will be formed. The reaction, a so-called "condensation", is straightforward and the resulting amido linkage is chemically resistant. If both the carboxylic acid and the amine functionalities are present in the same molecule, a large number of molecules can be linked, forming a polyamide. These unsophisticated organic reactions are the very basis of all living matter.

I

Shifting from one interdisciplinary nomenclature to another we can view the bidentate molecule as an amino acid, the amide becomes a peptide and the polyamide a polypeptide or a protein. Hence, we have abjured organic chemistry in favour of biochemistry. Proteins are built up from approximately 20-25 different (z-amino acids, the individual order of which decide the chemical and physical properties of a particular protein. Due to a combination of certain attributes of the peptide linkage, and the presence of functionalities enabling the formation of hydrogen bonds, protein strands fall into one of three geometrically different categories; random coil, cz-helix and ~-pleated sheet. Onomatopoetic as the names may be, some clarification seems appropriate. A r a n d o m coil displays no apparent geometrical o r d e r - a result of a low abundance of hydrogen bonding units. The (z-helix, the energetically most favoured arrangement, is evidently helical, allowing a maximisation of hydrogen bonding. A ~-pleated sheet is essentially a planar arrangement where polypeptide chains are cross-linked with hydrogen bonds. (z-helix and ~-pleated sheet conformations are intimately associated with protein action and hence these conformations are profoundly represented in most proteins. The random coil conformation adopts a role as a linker of helices and sheets, resulting in a lower relative abundance of that conformation. The chemical and topological complexity of proteins - their structure - are most conveniently communicated using a sequence of structural descriptors. The primary structure deals only with the protein molecule as a one-dimensional string: the succession of amino acids along that string. The peptide sequence spontaneously winds into regular clusters of sheets or helices at certain locations along its length; these are features of its secondary structure. At a still larger length scale, the twists and turns of the molecular thread fold into a tertiary structure of the molecule. Subsequent self-assembly of protein molecules gives their quaternary structure. Quaternary structure is akin to the mesostructure of lipid or surfactant selfassemblies, such as the aggregates characteristic of mesh-structures in bacterial protein coats (described in Chapter 4), or the cholesteric liquid crystals found in

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many structural proteins, such as collagen, discussed later in this chapter. Other better-known examples (although, geometrically far duller than the spectacular assemblies of structural proteins) are the dimers and hexamers of insulin. As a consequence of the individual order of the amino acids and the conformation of the ensuing polypeptide strands (the primary and secondary structures) the three-dimensional structure of each molecule (its tertiary structure) is formed. The bulk of this Chapter is focussed on this aspect of molecular structure. Some comments on quaternary structure and protein crystaUisation form a shorter afterword. A fundamental feature of proteins is their amphiphilic and chiral characteristics, which induce folding of the peptide chain that results in a mainly hydrophobic core covered by a hydrophilic surface layer. In the evolution of protein structure, which has taken place in an aqueous environment, the forces that drive selfassembly (discussed in Chapter 3) have played a central role. In order to understand tertiary and quaternary structure, we need to explore briefly the secondary structural possibilities of protein molecules. The (z-helix was predicted in 1951 by Linus Pauling. The peptide chain forms a compact core with the residues pointing outwards. The pitch is formed by 3.6 amino acid residues, with a helical period of 5.4/~. The structure is stabilised by a very effective network of hydrogen bonds. The other form is the planar peptide chain conformation, the ~-sheet, where hydrogen bonds linking all carbonyl oxygens from one chain to all amide hydrogens of an adjacent chain, link neighbouring chains into a sheet. The peptide chains can either all have the same direction: a parallel 13-sheet; or the directions can alternate, to give an antiparallel ~-sheet (sometimes hybrids of these two types can occur). The amino acid residues from each ~-strand point upwards and downwards in an alternating way in relation to the [}-sheet. The 13sheet can thus be regarded as a surface separating one set of amino acid residues from another, and the curvature of this surface is a significant factor that determines protein molecular shape. All known protein structures contain domains consisting of a few types of folding patterns, which will be discussed in detail below. The relatively strong hydrogen-bond systems in 0t-helical and 13-sheet regions, taken together with ionic bridges and disulphide bonds, can result in various deviations from the simple polar surface/non-polar core gross structure. We can regard these deviations as a "tuning" of the structure towards specific functions. Therefore hydrophobic regions exposed on the surface after a binding site for a non-polar molecule, and the changed energetic conditions after binding induce a conformational change of the peptide chain. An excellent review on protein structure containing clear diagrams of their topology is that of Br~ind~n and Tooze [1]. Our introduction of curvature in the structural description given below is based on their classification of domains.

a-helices

6.2

239

c~-helix domains

This domain consists of a bundle of (z-helices packed in pairs against each other. The most common arrangement is illustrated in Fig. 6.1. The space between the four helices is occupied by hydrophobic side chains, whereas polar side chains are directed towards the surrounding solution. The (z-helices are twisted with respect to each other, and their arrangement is similar to that of a fragment of a blue phase (cf. Figure 5.5). In both cases, the chirality of the structural units leads to hyperbolic curvature within the aggregate.

Figure 6.1: Four (x-helices of an (x-domain structure. (Adapted from [1].)

The active site in this kind of structure is a pocket located on one side of the hydrophobic core at the end region of the helices. There is another group of important (z-helix domains, which is the globin fold known from haem proteins. The haem group is located within a bundle of eight (z-helices. It is interesting to note that the globin fold has been preserved during evolution. Comparative studies have shown that the mechanism behind the preservation of this fold during evolution is conservation of the hydrophobic character of the amino acid forming the core between helices [2]. This illustrates the significance of the helices in regulating the interface between polar and nonpolar regions.

6.3

~-helix / ~-sheet domains

This is the most common domain structure, consisting of a [3-sheet surface surrounded by (z-helices. There are two types. One is shown in Fig. 6.2.

240

Chapter 6

Figure 6.2(a): Illustration of the so-called 0t/J3barrel structure for triose phosphate isomera~ after [1]. The parallel ~sheet chains are drawn in red and they form a catenoid. Outside the catenoid there are eight (z-helices (green). (Adapted from [1].)

Figure 6.2(b): Fitting of a catenoid to the core created by the [3-sheets. (Adapted from [3].)

The [3-sheet is thus deformed from the planar conformation and forms what has been described as a barrel. In fact the curvature is that of a catenoid [3]. The 13chains all have the same direction, and the "active site" is conventionally assigned to the carboxyl edge, the open end of the catenoid. The tunnel of the catenoid is formed by hydrophobic residues, and the exterior of the catenoid has hydrophobic side chains that are bonded to hydrophobic groups from each 0thelix. The ~-sheets lie along one set of asymptotic directions on the catenoid tilted with respect to the axis of the catenoid tunnel. This arrangement allows for twist between the 13-sheets, demanded by their chirality. (In the language of differential geometry their normal curvature is zero, i.e. the sheets are straight, and their relative twist is equal to the geodesic torsion along the other asymptotic directions perpendicular to the sheets, cf. section 4.14.) This twist sets the magnitude of the Gaussian curvature of the catenoid.) This structure is the most common of known domains, and it is also the most regular, involving 200 amino acid residues as the minimum number required to

~-sheets

241

form the structure. It is most remarkable that the structure has evolved independently in different classes of proteins [1].

Figure 6.3: Illustration of a catenoid with tangentially arranged ~-sheets indicated by red lines.

The second type of (x-helix/~sheet domain consists of an open ~)-sheet which is twisted. The structure of the 13-sheet resembles the helicoid, as shown in Fig. 6.4. Here again then the curvature of this surface also corresponds to that of a minimal - or closely related hyperbolic - surface. As the ~)-sheet is open it can consist of different numbers of ~)-chains; four to ten have been observed[I]. The active site is also in this case at the carboxyl end region, and it is located in a crevice formed when loops from the [3-chain to the (x-helix have opposite directions (between a set of (x-helices on one side of the ~-sheet and a set on the other side).

6.4

~-sheet domains

This group of structures consists of antiparallel ~)-chains forming two sheets packed against each other. The number of chains can vary from four to ten, and the shape of the [~-sheet also resembles a catenoid. The core consists of hydrophobic residues whereas the 13-sheet exhibits hydrophilic side chains towards the solution. This is also clearly reflected in the amino acid sequence of the peptide chain forming ~)-strands. Thus every second chain pointing towards the inside is non-polar, as opposed to the rest, which are polar chains. This 13sheet is obviously a surface separating hydrophobic and hydrophilic regions. The average curvature at this type of [3-sheet seems to tend towards zero, although the matter is more complex.

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Figure 6.4(a): Typicalexampleof an open a/~ domain in the redox protein favodoxin(from [1]). (b) The helicoid minimalsurface that traces the arrangementof [}-sheetsin favodoxin.(red arrows).

6.5

Membrane proteins

Proteins embedded in the lipid bilayer of membranes play an important role in membrane functions, involving transport across the bilayer, electron flow and energy conversion, cell recognition, receptor functions, etc. There is not much information available on structural features of these proteins due to difficulties in crystaUisation, necessary for complete structure determination. The first detailed structure of a membrane protein was made by Henderson and Unwin [4]. They determined the structure of bacteriorhodopsin from electron microscopy data from two-dimensional crystals of the protein. Seven transmembrane a-helices were identified, tilted about 20~ towards the bilayer plane. The tilt by itself indicates a catenoid-like shape of the hydrophobic region through the lipid bilayer. The best characterised membrane protein is the photosynthetic reaction centre, from which the interaction between pigment and protein in photon capture and electron flow has been revealed [5]. A general feature of most of the known membrane protein structures is the occurrence of hydrophobic segments forming a-helices, which are buried in the bilayer. To span the lipid bilayer an a-helix needs about 20 residues. From the amino acid sequence it is therefore possible to predict trans-membrane a-helices.

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243

There are also membrane proteins with extended ~-chains through the bilayer, and channel proteins with their hydrophilic inner opening must also contain polar amino acid residues within the lipid bilayer. There is also a group of membrane proteins that are covalently bonded to bilayer lipids, including the glycosyl-phosphatidylinositol anchor [6]. These proteins are exposed on the membrane surface via a spacer arm consisting of an oligoglycan, and specific phospholipases can release the protein.

6.6

Enzymatic action

Proteins are involved in a vast number of highly disparate transactions. Enzymatic processes, immunological reactions, transportation, signal transmission, regulation, structural support and maintenance are all examples of biochemical events where protein participation is indispensable. Using the cur~,ature approach, we believe the modus operandi of this plethora of delicately tuned incidents can be identified. In enzyme chemistry the protein exerts its action upon a more or less rigorously defined target molecule, the substrate. Two consecutive events precede the actual reaction, identification and interception. In fact, the sequence identification- interception- action, pursued by enzymes can serve as a prudent model for the entire ensemble of protein-based transactions. Not only is the making and breaking of particular bonds in a chemical transformation energy intensive, but also movements and relocations of reactants increase the expenditure of energy, and thus the total number of discrete events involved will seriously influence the overall flow of energy. In vitro reactions are usually conducted without limiting energy considerations, the purpose being preparation a n d / o r high yield of a specific product. Nature, however, is normally less than extravagant when energy related issues are addressed; rather it is qualified economising with energy that is the hallmark of biological systems. Hence in in vivo reactions, precautions are taken to diminish energy dissipation. This is accomplished through careful design of transaction matrices. Accordingly protein-mediated biochemical events capitalise from the considerable structural variability of polypeptides that provide an almost unlimited supply of adequate loci. Most biochemical events are chemically quite unsophisticated but can at times seem astonishing as regards their energetics. Apart from certain capacities furnished by various prosthetic auxiliaries, the true basis of protein-mediated action is a continuously debated issue. The foundation of chemical events is the non-equivalent distribution of electrons created when different atoms or groups are joined. The imbalance might be either fluctuating or permanent. Whenever this imbalance is substantial, there is a high probability that an event will occur. Irrespective of how proteins consummate their duties, they apparently manage to induce and correctly adjust and maintain the electronic imbalance of their targets, which in turn triggers the cascade of events required.

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~ter

6

The peptide linkage is chemically stable, indicating a low aptitude to participate in chemical events. Other functionalities, except various hydrocarbon chains, present in polypeptides include hydroxyl-, sulfhydryl-, carboxylic acid-, aminoand alkylamino- groups. These groups surpass the peptide linkage as regards the partaking of chemical events and their presence increases the interactive propensity of proteins. Biochemical events, like any chemical event, are compelled to follow certain dispositions in terms of location, time and energy. To accomplish an event the counterparts have to occupy the same, narrow, space for an exact time-span, and carry an adequate amount of energy. In vitro, statistics are relied on, i.e. ~ v e n enough time and an excess of energy any reasonable combination of reactants in a suitable medium eventually undergoes the anticipated event. In vivo events cannot, as a rule, afford random encounters and have to depend on limited supplies of energy. Thus, in some enzymatic reactions the concentration of the substrate is so low that ordinary "random encounter chemistry" is simply out of the question. A similar situation prevails in hormone-receptor interactions. In fact most biological systems seem to work on a first-hit basis, i.e. the incidence of events by far exceeds the level presumed from ordinary chemical expectations. How does nature achieve such efficiency? One possible explanation as to the secret of the efficacy of protein-mediated chemistry may be hidden in the very organisation of the proteins. An organisational form where low-impact interactive ability is focussed and at the same time is communicated over substantial distances would be ideal. Different geometrical solutions within the realms of fiat and elliptical (spherical) geometries can cater for either focussing or communication but never the two in combination. An amalgamation of these capacities can be achieved using a geometry where hyperbolic curvature prevails and it is clear that such structures occur in proteins (recall Figs. 6.2,3 and 4). An attempt to discern the structure of a protein by just following the extensive backbone of the polymer is an essentially futile occupation. Protein anatomists have however devised a more rewarding approach relating to the three generalised conformations of polypeptides. By combining and appropriately connecting relevant numbers of the latter a more comprehensive image of the structural diversity emerges. Of the numerous polypeptide structures two discrete classes of enzymes stand out regarding their obvious resemblance to two classical minimal surfaces, the catenoid and the helicoid discussed above. In differential geometry the catenoid and the helicoid are intimately connected through the Bonnet transformation where one surface is isometrically transformed into the other (cf. section 1.12). During the transformation the mean curvature is kept at zero indicating that local conditions remain unchanged. One feature of this close kinship is that it is impossible just by sampling the metric to decide which surface is at hand since they are identical in their intrinsic geometry. In other words an object passing through a catenoid or travelling along a helicoid would experience the same sensations, except for one thing; the catenoid is highly symmetric and hence achiral while any given helicoid is chiral. (The handedness of the helicoid is decided from the sign of the association

Enzymatic action

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parameter in the Bonnet transformation and there is no preference for either.) Drawing on the above, the following discussion should be taken as equally valid for both the catenoid and the helicoid. Now why would an enzyme, or any protein, adopt a spatial organisation bearing a close resemblance to a minimal surface? Plausible reasons have been more than hinted at in the above: focussing and communication. By assuming this approach, biological systems circumvent the need to resort to aggressive chemistry. Entities that possess weak interactive faculties on their own, can, when collected on a minimal surface, act cooperatively. When each small contribution is added up the resulting interactive power would be quite impressive. Further, this power can be delicately tuned just by slight changes of the shape of the aggregate which in turn will affect the average curvature of the particular minimal surface. Hence by appropriately connecting, from a limited number of basic building blocks interactively soft entities a continuous range of more or less reactive aggregates can be constructed. As to the problem of communication, minimal surfaces (or close hyperbolic relatives thereof) also have the upper edge. Hyperbolic surfaces like the catenoid and the helicoid are non-self-intersecting and have infinite extension. They also have regions where the Gaussian curvature attains its most negative value, e.g. the waist of the catenoid and the axis of the helicoid. Thus, when approaching a catenoid or helicoid from infinity, a curvature gradient is encountered. A protein, immersed in an aqueous solution, will be accompanied by successive solvation shells, the geometry of which will certainly be dictated by the geometry of the protein in question. If the protein comprises hyperbolic geometry the first solvation shell will attain approximately the same geometry. As shell by shell is added the influence of the templating geometry will gradually vanish until finally solvent molecules belonging to the solvation shell and the bulk solvent respectively become essentially indistinguishable. In a protein-mediated event the target, as soon as it makes contact with the outskirts of the solvent-protein complex, is intercepted and by means of the curvature gradient guided to the region of maximal negative curvature. This, if one likes, "forced diffusion" works on two levels; firstly the effective interceptive range of a protein is substantially enhanced, secondly the smooth curvature gradient perpetually secures that the most reactive part of the protein is reached. In an enzymatic reaction the substrate, after interception, is accelerated and gains increasing amounts of kinetic energy. The focussing of interactive forces depends on the curvature (quantified in section 3.2.3) and when the maximum level is arrived at the substrate is energised to an extent where bonding integrity may be regarded as sufficiently lost. At this point, highly facilitated specific manipulations of the substrate can be undertaken. Enzymes have an almost mythological reputation for infallible specificity. Surely, when acting in their pertinent biological environments enzymes are highly selective. This selectivity is however partly a consequence of substrate concentration and the absence of relevant competition. It has been shown that when working at non-natural conditions enzymes lose absolute specificity, both chiral and otherwise. Nevertheless, the minimal surface approach to enzyme

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action can accommodate selectivity issues as well. The curvature gradient and the absolute value of the curvature constitute ample selectivity parameters; if the intended substrate arrives at, the maximally curved region in a sub-optimal state, as would be expected for anything but the correct substrate, the particular transformation would simply not occur. Chiral selectivity, concerning substrate recognition as well as product formation, is a complex phenomenon and is probably a conjunctional result of the overall chirality of the polypeptide and, when applicable, the presence of a chiral hyperbolic surface, e.g. the helicoid. It should however be emphasised that all achiral minimal surfaces have chiral associates easily accessible by means of the Bonnet transformation. The association parameter can be arbitrarily small and thus a chiral surface, infinitesimally different from the starting achiral surface, can be formed. A complete transition from one enantiomer to the other is, however, generally not possible, because of topological constraints imposed by the supporting molecular framework outside the hyperbolic region of the protein, such as the random coils. As has been mentioned earlier, the overall disposition of proteins is not altered whenever differing objectives are addressed. Drawing to both this structural conservatism and the consistency of the operative sequence, and connecting to the involvement of minimal surfaces in enzymatic processes, it seems only natural to try to identify geometrical influences among other protein-mediated transactions. Of the numerous serum-resident proteins, several are employed as transporting agents. Serum albumin is extensively used as an expedient ferrier of fatty acids. In terms of selectivity such an undertaking is in many ways vexatious. Fatty acids constitute a diverse class of compounds where attributes such as hydrocarbon chain length, number and location of double bonds are subject to variation. Instead of using an individual, dedicated, protein for each discrete fatty acid (a rather impractical solution if no manipulations or transformations are attempted) a single but by necessity indiscriminate protein is relied on for transportation per se; a carrier protein should be charged with enough interactive power to successfully intercept and retain its load, but, the interaction must be soft enough to permit a non-destructive passage. Unloading should also be expeditious with respect to both energy demands and carriage integrity. A carrier protein with a saddle structure can easily adjust itself to the differing requirements of multi-target and -task objectives. Low- and high-molecular cutoff is provided by properly set curvature and steric hindrance respectively. The curvature and hence the interactive power is continuously tunable through the Bonnet transformation. This allows the protein to accommodate to a v~dde range of targets. Also, the tunability ensures minimal expenditure of energy when the intercept is released. Thus, the events in the sequence identification, interception, transport, release, are all related to curvature and can be sustained by a single basic structure.

Eh'oxin poisoning

6.7

Protein f u n c t i o n and dioxin

247

poisoning

The idea of "barrel" and twisted sheet proteins being biological equivalents of catenoids and helicoids respectively, carries some indication as to their mode of operation. To any object with the faculty to sample the metric of a system, i.e. the local properties, the catenoid and the helicoid are indistinguishable. Hence, the sensation of passing through the central hole of a catenoid would be identical to the passage along the helicoidal axis. In any case the particular target, the substrate in case of an attempted enzymatic reaction, is attracted to the protein by means of global amplification of interactive forces. Due to the curvature gradient, the attraction increases as the surface is approached. This acceleration is experienced by the substrate as a growing thermal excitation. Given that the number of interactive units acting collectively is sufficiently high and that they are arranged in an appropriately curved manner, the energy content of the excitation can be of the order of a chemical bond, i.e. high enough to cause substantial degradation of overall b o n d i n g integrity. If the chemical transformation in question is non-elaborate, like a hydrolysis or an isomerization, it could be expected that it would be completed during the passage. The level of interaction is not only dependent on the protein but also on the substrate; the higher the number of polarisable electrons in the substrate, the more intense the interaction. Hence it would be expected that the interactive propensity of a substrate would benefit from a high abundance of non-carbon and -hydrogen atoms. It could even be argued that this is the reason for the energy carrying ability of phosphate esters. N o w , if a protein encounters a non-endogenous material that nevertheless is

attracted, the result of such an encounter can be fatal. If the false substrate combines high electron polarizability and high chemical stability, a somewhat contradictory situation yet perfectly realisable, the resulting thermal tension cannot be relaxed through a reaction. The highly toxic substance 2,3,7,8tetrachlorodibenzodioxine (TCDD, or dioxin)) might provide the ultimate example of such a catastrophic encounter. Being essentially inert to any conceivable biogenous manipulative action, and at the same time both properly scaled and loaded with mobile electrons, it constitutes an ideal target for barrel and twisted sheet proteins. Upon interception, TCDD is expected to be immobilised instead of just passing through. A relatively massive amount of interactive energy is released from the protein-TCDD complex. If one assumes that the thermal energy of the complex increases at too high a rate with respect to dissipative processes, protein thermolysis will result. The disintegration of the protein eventually will release the TCDD unharmed, and thus ready to destroy further proteins. A qualitative validation of the notions presented above can be found in viral therapy. A common feature of viruses is a high abundance of "barrel" proteins within the protein coat. It has been proposed that these barrels could be employed in the struggle against viral infections. This could be accomplished either by blocking of the catenoidal holes or by protein destruction as outlined above. Interestingly in a recent publication addressing the action of the antiviral

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agent 5-[7-[4-(4,5-dihydro-4-methyl-2-oxazolyl)phenoxy]heptyl]-3methylisoxazole the mode of action was found to be in accord with the latter proposals. The antiviral compound, a hydrocarbon chain connecting two heterocyclic rings, is, by means of the heterocyclic rings, firmly embedded in two adjacent barrel coat proteins and effectively blocks them. Calculations on the prevailing situation revealed that the connecting hydrocarbon chain experiences severe thermal stress. The conclusion arrived at emphasises that the protein coat acts as a both a source and sink for kinetic energy and in doing so provides a kind of thermal bath.

6.8

Geometry in hormone-receptor interactions

Signal transmission and regulational activities are areas where protein participation is ubiquitous. The receptors involved are consistently proteinaceous whereas the donors, or hormones, are chosen from a wide range of substances, including proteins. Hormone interception by the receptor sometimes accomplishes the desired activity but sometimes constitutes only the initiation of a chain of events. The former could be exemplified with the opening of trans-membrane channels and the latter with the action of insulin. Disregarding these discrepancies it is worthwhile to examine a general and prominent feature of hormone-receptor interactions. Upon binding of the hormone the receptor commonly undergoes gross conformational changes. Additionally, a major change of conformation affects not only the particular receptor itself but also involves substantial distortion of the surroundings. Hence, by usual standards, a necessarily energy intensive process is faced. As the hormone is normally several orders of magnitude smaller than the receptor, limiting the level of impact, and the changes in the receptor embrace a huge number of atoms, the change cannot be a result of net energy transfer. A more fertile option would be that the hormone, by activating a critical site, would trigger the onset of a conformational change having the capacity to bypass energy dissipative routes. Such a scenario is perfectly imaginable if the receptor has a suitable geometry, allowing it to engage in the Bonnet transformation. Recall that this surface transformation is isometric, i.e. keeps local conditions constant, and, as a consequence, progresses adiabatically. Further, the geometrical circumstances dictated by the presence of the receptor enforces a similar zero-energy relocation of the immediate surroundings, including water of solvation and membrane components. At hand we thus have a transformation route that offers fast kinetics in both directions with minimal deployment of energy. The nicotinic acetylcholine receptor consists of four different yet similar subunits combined in a 2:1:1:1 ratio that form a 250 kD membrane resident complex. The overall disposition of the receptor could be described as five membranespanning helical bundles of helices. Upon stimulation with two molecules of acetylcholine the helical bundles move collectively and open up an ion channel through the membrane. The organisation of the open receptor is fairly well known from electron microscopy studies and the channel is colloquially referred

Hormone-receptor anteractions

249

to as "funnel-shaped". Closing of the channel is provided for by the enzymatic hydrolysis of acetylcholine. The total time for this neuronal signal transduction is about 0.2 ms. Acetylcholine is a small, 146 D, positively charged molecule and it is highly unlikely from an energetic point of view that its per se association to the receptor could effectuate the substantial transmutation of which, if the reorganisation process followed regular step-by-step procedures. Also, the narrow time span of the signal transduction indicates that an extraordinary reorganisation mechanism is at work. The only way to maintain the spatial integrity of literally millions of atoms during a very fast and at the same time extensive reorganisation, without the continuous involvement of an energy consuming means of control, is to nullify local variability. To achieve this end the entire set of moving atoms has to be localised to a geometric unit where an isometric transformation can be conducted. In the case of the nicotinic acetylcholine receptor the five helical bundles could be arranged on a helicoid, very close to the catenoid. When the two acetylcholine molecules bind to the two identical sub-units of the receptor complex the isometric transformation is triggered and the catenoid is formed, opening up the "funnel" shaped channel. Actually, the channel is there all the time, along the axis of the helicoid, but at the helicoidal stage the radially unsymmetrical bundles are rotated out of phase, disallowing the formation of a hydrophilic continuum. The open catenoidal channel induces a geometrically similar arrangement of the solvating water. This structuring of the water, which also profits from the isometricity of the protein rearrangement and hence is coincidental, extends beyond the receptor boundary and enhances the catchment area of the receptor. The combination of a fast, reversible, isometric rearrangement and the water-structure related amplification of receptor objectives explains the high velocity of the neural signal transduction. In case of the insulin-insulin receptor system a different chain of events is triggered by hormone interception, the ultimate outcome of which is the metabolism of glucose. Insulin is a proteinaceous hormone of moderate size, 5 kD, whereas the receptor is a tetramer of two 90 kD and two 130 kD sub-units where the heavy sub-units transverse the membrane and the light sub-units are located entirely extra-cellularly. When in the hormone free state neither the trans-membrane nor the external sub-units of the receptor display any apparent activity. However, upon stimulation, when two insulin entities become attached to the external sub-units, the intraceUular parts of the trans-membrane sub-units take on kinase activity. The ensuing phosphorylating activities of the enzyme so formed fires the required cascade of events involved in glucose metabolism. The transformation of the receptor does not involve any local changes, i.e. no structural units are either removed or added, implying that the expressed enzymatic potency is a latent feature accessible through global changes only. As before, the entering hormone cannot be expected to mediate the gross permutation of the receptor by means of any conceivable per se chemical process, but rather through attachment to a pivotal site that switches on the more or less physical transformation. Insulin, being a peptide, confines the possibilities to reset the activated receptor to proteolysis. Apparently the structural changes induced by the hormone prevents the selective degradation of which and thus the entire hormone-receptor complex is fed into intracellular catabolic processes.

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Cha~ter6

In conclusion, it appears that hormone-receptor regulated processes, although dependent on chemical messengers, rely primarily on unequal global dispositions of the hormone free and stimulated receptor respectively. Receptor structures are chosen in order to enable a fast presentation of the critical global isomer without having to resort to hard-core chemistry. To achieve this essentially extra-chemical end the particular structures must be able to partake in an isometric transformation and hence adopt a structural disposition in compliance with the requirements called for.

6.9

Self / non-self recognition

The immune system is characterised by an ability to distinguish between self and non-self. Exogenous materials are consistently bracketed as non-self but also endogenous entities, if recovered outside their principal habitat, can be immuno active. The immunological response originates from either of two different systems, B-cells or T-cells, where the B-cell system handles non-cellular intruders and the T-ceU system is dedicated to whole cells. Stimulation of the B-cell system triggers the production of antibodies, i.e. highly specific proteins that are designed to intercept the intrusive materials, the antigens. The antigen-antibody complexes are finally fed into catabolic processes. Generally, antibodies exhibit two major structural domains, a nonvariable backbone and a highly variable active region, the site of antigen interception. Considering the fact that each antigen induces the production of a particular antibody we are faced once again with the problem of infinite complementarity based on a limited supply of different building elements. Here too it is tempting to propose that global, i.e. shape oriented, rather than local properties are of utmost importance. Recalling the impact of saddle-shaped protein structures, especially when it comes to features such as long-range communication and interactive variability, it seems highly probable that the antibody-dependent part of the immune system functions along similar lines. Intriguingly, X-ray analyses of antibodies indicate that the variable domains indeed are of hyperbolic geometry. The T-cell system acts as a scavenger for alien, or alienated endogenous, cells. Its mode of action is based on donor-receptor concepts and is shared by processes involved in signal transmission and regulation. Eukaryotic and prokaryotic cells, and viral particles, are all identifiable by means of special surface structures. In higher eukaryots, e.g. mammals, these structures consist of trans-membrane proteins. The proteins themselves are specific to a particular individual but also carry small peptides of intracellular origin, collected when the proteins were synthesised. The extraneous peptide presentation serves as a secondary source of information, depicting the general outline of the cell interior, and is especially important in the case of infected cells where the intra-cellular protein cocktail is contaminated with foreign proteins. This protein/peptide ensemble, the transplantation antigen, is sampled by the T-cells and whenever the message is incorrect specialised killer cells are activated and the cell in question is

Self~non-self recognition

251

destroyed. Information transfer in the T-cells is handled with a proteinaceous receptor tuned to the specific donor. According to recent investigations its mode of action seems to be shape dependent. This is in accordance with the known properties of the antigen. It is a trans-membrane protein with four domains, three of which are extra-cellularly exposed. The domain closest to the external side of the membrane does not contribute to the antigenic message confining the possible variability of the latter to the remaining two domains.

6.10

D N A folding

We have investigated a plethora of systems in which the dominant shapes are minimal surfaces, or at least close approximations to them. In some of these cases the interpretation of these shapes is complicated or even still unclear. Why are minimal surfaces formed and how does this shape affect the local and global environment of the system? In the case of DNA folding the interpretations are unusually straight-forward. The fact that the uncoiled DNA of an eukaryotic organism, e.g. a human, is approximately 2 m long, albeit split up in 5 cm strands in each chromosome, represents a formidable packing problem. Not only must allowance be made for the total confinement of such a gigantic molecule, but also for the fast and easy access to the different segments thereof, coupled with the obvious necessity to maintain structural integrity. This calls for a packing mechanism that is fast, reliable and that causes as little change as possible in the interactions between the DNA molecule itself and the intracellular fluid in which it is immersed. According to common belief, DNA is packed along hierarchical levels of rising complexity. The lowest level is of course the DNA double-helix itself. This is then transformed to the nucleosome, i.e. chromatin, level where the double helix is wound around a highly specific protein cluster creating the bead-on-a-string form of chromatin. At the next level, the bead-on-a-string is condensed to a chromatin fibre, again aided by a highly specific protein, which is further compacted to the final metaphase chromosome. The question is now how to account for a more than 10000-fold compactification that takes place rapidly (in less than 0.5 ms) and yet is so gentle that the fragile DNA molecule, that will break when pipetted, is preserved intact throughout the transformation, and will survive through a large number of repetitions of this folding. The main reason for this remarkable property of the DNA molecule is its general shape. The double helix sits on a helicoid, and therefore it shares the properties of that surface. The most important of these is the way the helicoid can be deformed, via the Bonnet transformation.

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During the Bonnet transformation, the Gaussian curvature, and hence the local metric, of the surface is preserved. This means that all inter-atomic distances are preserved, no bonds are stretched or compressed measured along the surface and the local structure remains unchanged. All this is of course important, but the Bonnet transformation is more restrictive than that. It also preserves the mean curvature. This means that the surface remains a minimal surface, and that is of great importance to a highly solvated molecule like DNA, since the metric of the parallel surfaces remains the same (of Chapter 1). That is, the solvation shells will remain unperturbed during folding if this can be described as a Bonnet transformation. Nevertheless, globally, the differences are huge. 10000-fold compactification is accomplished, but only by moving bulk solvent, not by changing solvation shells. Still another important property of the Bonnet transformation is that it imposes simultaneity on the system. In a true, mathematical, Bonnet transformation, all points on the surface move in unison. In the DNA molecule, which is only a good approximation of the helicoid, the partial folding of the molecule at one location will lead to the imposition of a similar structure at nearby sites. The trigger to start folding must come from outside the molecule if the Bonnet transformation mechanism is to be used. Since the local environment is virtually unchanged, the impetus must be a global effect. This is supplied by the histones. By binding to sites that are distant from each other along the helicoid surface and bringing them close together in 3-space, they pull the DNA strand together. The mechanism is just like the coiling and super-coiling of a telephone cord.

Figure 6.5: Proposed condensation process of chromatin fibre based on scattering measurements of chicken erythrocyte chromatin from [7] (a) Maximallyextended chromatin showing the helicoidal space arrangement. (b)-(d) Condensation of chromatin with final catenoidal space arrangement shown in (d). (e) View down the central cavity only for a portion of the chromatin chain. (f) View of the condensed arrangement for longer fibre than that of (e).

Protein aggregation

253

The general scheme of the folding would then be this: The loosely curled DNA strand is swimming in a soup of intracellular fluid, containing the histones. At a critical pH, the conditions become just right for the histones to bind to the DNA strand. Once the first histone is in place, the folding will be self-catalysed, since the binding induces a Bonnet transformation upon the nearby parts of the DNA strands, creating ideal binding sites for free histones (Fig. 6.5). This auto catalysis yields an ever accelerating process that propagates through the entire DNA strand like a sonic wave, dramatic on the global scale, but gentle on the local, thereby ensuring structural integrity of the genetic material. Considering the fact that the ensuing condensation of the chromatin fibre into the metaphase chromosome is achieved by further winding of the molecule, it is fair to assume that this follows a similar mechanism, creating a self-similar sequence, a cascade of Bonnet transformations [8].

6.11 Self-assembly and crystallisation of proteins In contrast to typical chiral molecules synthesised in the laboratory to form chiral thermotropic mesophases, protein molecules have a huge number of chiral centres, and the twist between adjacent protein molecules (in the quaternary structures of proteins) is typically much larger: --1/10 of a revolution. Nevertheless, the essential feature for formation of twisted aggregates, viz. chirality, means that we can expect to find a range of similar liquid crystalline phases in protein aggregates, although with significantly smaller lattice parameters, appropriately scaled by protein molecular dimensions. A look at some quaternary structures of proteins reveals that the aggregation processes of some proteins are driven by their chirality as much as amphiphilicity. That is not surprising, given the strong structural manifestations of chirality present already in the secondary and tertiary structures of individual protein molecules. Striking evidence for chiral liquid crystallisation in vivo has been collected by Bouligand and co-workers. Optical micrographs of stained sections of crab (Carcinus maenas) and insect (Locusta migratoria) cuticle for example, reveal the single twist characteristic of the cholesteric mesophase (Figure 6). This chiral liquid crystalline arrangement is not limited to structural proteins. "Precholesteric" and cholesteric mesophases have been found in DNA aqueous "solutions" [10]. It is likely that the other more complex hyperbolic mesophases are also to be found in chiral systems: twisted grain boundary and blue phases (whose twist numbers are less than and greater than unity respectively, cf. section 4.14), are also prevalent in proteinaceous matter. (Note that a blue phase of DNA in water has recently been reported [11]).

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In this context, some comments on protein crystallisation can be made. The process of crystallisation can be viewed as one of self-assembly of the quaternary structure, although the constituent units now have a well-defined arrangement in space, in contrast to their less rigid shape in liquid crystalline mesophases. Indeed, twisted structures are very commonly found in globular protein crystals, which are reminiscent of the hyperbolic forms of micro- and mesoporous zeolites, described in Chapter 2.

Figure 6.6: (Left:)Stained section of a crab cuticle (Carcinus maenas), showing the single twist of protein fibres (some indicated by red lines), characteristic of the chiral thermotropic cholesteric mesophase. (Right:) Schematic relation between nested arc texture in the micrograph and the cholesteric mesostructure. Micrograph and drawing adapted from [9]. The similar dimensions of many globular proteins and novel mesoporous zeolites offer fascinating possibilities for bio-organic-inorganic composites. For example, the inorganic mesoporous alumino-silicates have been proposed as templates for protein crystallisation [12]. Conversely, proteins could be used to template mesostructured inorganic materials. It is known that transport and membrane proteins are difficult to crystallise, in comparison to structural proteins. This is certainly a reflection of the fact that the former class of proteins abhor self-assembly, which would inhibit their biological functions (at most they may operate in vivo as tetramers or hexamers). In these cases, nature has tailored these molecules, via their molecular shape and chirality, to avoid self-assembly! An arresting example of the importance of segregation of proteins into small quaternary units (as opposed to self-assembly into large polymeric units) lies in the structural basis of sickle-cell anaemia, discovered by Pauling in 1949. Mutation of a single ~-globin allele in the haemoglobin molecule results in the formation of a hydrophobic patch on its surface. The resulting shape change in haemoglobin allows for aggregation of the protein into polymeric fibres, precluding the usual oxygen transport function of a healthy person, which can only occur in the tetrameric form.

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255

Figure 6.7: Schematic drawing of the crystal structure of glycolate oxidase. Each disc is an octamer, and these octamers are twisted relative to each other to form an open three-dimensional lattice. The image is adapted from [1].

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References 1. C. Br/inddn and J. Tooze, "Introduction to Protein Structure". (1991), Garland Publishing Inc.: London.. A. M. Lesk and C. Chotia, Mol. Biol., (1980). 136: p. 225. 3. Z. Blum, S. Lidin and S. Andersson, Angew. Chem. Int. Ed. Engl., (1988). 27" p. 953. R. Henderson and P. N. T. Unwin, Nature, (1975). 257: p. 28. ~

6.

H. Michel, O. Epp, and J. Deisenhofer, EMBO J., (1986). 5: p. 2445.

M. G. Low, Biochim. Biophys. Acta, (1989). 989: p. 427.

7. J. Bordas, L. Perez-Grau, M.H.J. Koch, M.C. Vega and C. Nave, Eur. Biophys. l., (1986). 13: pp. 175-185. 8. .

Z. Blum and S. Lidin, Acta Chem. Scand., (1988). B42: pp. 417-422. Y. Bouligand, Tissue & Cell, (1972). 4: pp. 189-217.

10.

F. Livolant, Physica A, (1991). 176: pp. 117-137.

11.

A. Leforestier and F. Livolant, Liq. Cryst., (1994). 17: pp. 651-658.

12.

Z. Blum and S.T. Hyde, Acta Chem. Scand, (1994). 48: pp. 88-90.

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Cytomembranes and Cubic Membrane Systems Revisited

Membrane organisation

C

y t o m e m b r a n e s and their organisation have both fascinated and puzzled cell biologists for m a n y decades. There are many functions thought to be maintained by cytomembranes that are still poorly understood. One of their roles, which is generally accepted among biologists, is the discretisation of cell space into subspaces, or organelles. These subspaces are often thought of as being isolated from each other, circumscribed by a continuous membrane. However, surprisingly little has been learnt about the overall biogenesis and organisation of cytomembrane morphology. The favoured solution to the problem of organising the heterogenous matter existing in a single cell has been based upon the assumption of an excess of material, randomly distributed throughout the cytoplasm. However, other solutions to the problem of optimisation of the cell space have started to emerge. A particularly interesting solution is based on hyperbolic multicontinuous, topologically equivalent, periodic (in a crystallographic sense) surfaces (membranes), which are folded into an optimal form. Such m e m b r a n e s discretise space into a number of subspaces. For quite some time, analogous self-organisation principles have been known to exist in crystals, liquid crystals, and block copolymers - as pursued in Chapters 2, 4 and 5. But these structures have not been recognised and hardly discussed in the field of cytomembranes. In this section we will discuss - in more detail than in Chapter 5 - a particular case in which membranes are folded upon periodic minimal surfaces (IPMS) of cubic symmetry or their periodic constant mean curvature (H-surfaces) analogues; we call both cubic membranes and refer to their mathematical surfaces as periodic cubic surfaces (PCS). While the former can either have zero or constant mean curvature (equal or unequal subvolume sizes), the latter morphology is composed of a set of membranes of which one, if present, can be described by the particular IPMS and the rest are described by H-surfaces. As we will see the cubic membranes comprise a large family of morphologies that are based on either one or many parallel membranes. The evidence for translational hyperbolic membranes rests here largely on stained sections of cells, viewed under the optical microscope, published originally by cell biologists. Many readers may baulk at the hard-core biological content of this Chapter. Fear not. The jargon extends only to a description of w h e r e these extraordinary thin-sections have been found: in which organism, and in which part of their anatomy. To those unable to digest the jargon, look at the pictures! Look again, then compare with the thin-sections of inorganic, surfactant/lipid and polymer systems shown in Chapters 2 and 4.

258

Chapter7

The cell represents the most heterogeneous chemical fabric known, yet remains an astonishingly efficient machine. It is a machine with an enormously complex and heterogeneous product line. Although it is well established that the cell has adapted a high degree of sophisticated regulatory mechanisms to control many aspects of its activities, such a complicated selfassembled system requires spatial organisation. Though the cytoskeletal network is often held responsible for certain principles of cell organisation, such as location of organelles, targeted vesicular transport, formation of pseudopods, etc., cell membranes do not necessarily have to interact with such elements in order to optimise their morphology. This must be so, since membranous organelles apparently lacking cytoskeletal elements, can adopt symmetric and specific morphologies. Mitochondria exemplify such spatially organised organelles, smaller and apparently less complicated than a whole cell. They can be described in the most simple and classical form as two independent, topologically equivalent though occasionally intertwined, spaces, separated by two continuous but closed membranes. Normally, the shape of the membrane is lamellar- or tube-like, with regular points of high curvature. This indicates the possibility of maintaining spatially discretised and symmetrically organised regions within membranes, without the action of cytoskeletal elements. Apparently the rather exact morphology and topology of the mitochondrial membranes has been selected to serve the purposes of the collective assembly. Imagine such a collective unit that either requires high amounts of membranous cristae per unit volume bound by the outer membrane, or more than two hyperbolic, independent spaces, still intimately close and intertwined, which take on an optimal form to fulfil the purposes of such a unit. Independent of the specific purposes, o n e solution to this membrane packing problem is realised by multicontinuous periodic hyperbolic surfaces. These surfaces may be parallel surfaces to the corresponding IPMS, introduced in Chapter 1. Besides their beauty and harmony, these surfaces are efficient partitioners of (not necessarily) congruent spaces. They allow very high surface-to-volume ratios compared with other membrane packings. In addition, they offer regular networks that define labyrinths of easily accessible positions. Such an arrangement is well suited for repetitive tasks and masssyntheses, as those which occur in highly active mitochondria; for example in the flight or cardiac muscle cells of certain birds. The apparent existence and ubiquity of such multicontinuous cubic membrane systems, even in mitochondria, is the major topic of the following sections. It would be a surprising omission of nature if such exquisite membrane arrangements were not exploited to fulfil a functional purpose. We believe that cytomembranes, like all systems controlled or influenced by the principles of self-assembly and self-organisation, whether regulated or not, are likely to adopt an optimally organised morphology under the influence of selective processes. This morphology may sometimes be dynamic and short-lived. This is due, in large part, to the very many degrees of freedom that exist in a cell. Changes in self-organisation occur due to a changing physico-chemical environment, as well as the restrictions imposed

Membrane organisation

259

by the abundance of regulatory events. It should therefore be stressed that we do not consider cubic membranes, or for that matter any m e m b r a n e morphology, as static or constant configurations of cytomembranes, as those which exist in the basolateral membrane of endothelial cells. On the other hand, as a first approximation, it is plausible to assume that the topology of cytomembranes is a static feature that is both conserved and controlled. However, if a particular cytomembrane unit requires symmetry for its organisation a n d / o r function, we might reasonably assume that membrane structure is influenced or controlled by the cytoskeletal elements, or the symmetry and periodicity of the underlying membrane. The latter option implies that the symmetry of biomembrane systems is closely linked to the mechanisms of assembly and self-organisation. The membrane assemblies to be discussed in the chapter have generally been regarded as novelties until recently. There is a profound lack of systematic understanding of their structure, occurrence, ontogeny, and, particularly, function (if any). It is clear that cubic membrane systems have been observed by many authors - although they have not been appreciated as such. In the following sections we describe the structural principles of cubic membranes and provide a catalogue of such. From this catalogue it will emerge that they are a much more common feature of cytomembranes than had earlier been believed. We first systematise their structures and occurrence with respect to cell type, organeUes etc. With that established, we then discuss their origin and function. We stress the organisation of the bilayer making up the collective, and contrast and compare that with condensed lipid-water phases, especially reversed "cubic" phases, described in Chapters 4 and 5. This leads to an explanation of the "living" counterpart of lipid p o l y m o r p h i s m in condensed systems, which has been widely debated since the early 1960's. Finally, the uses of cubic membranes as hyperbolic space-dividers, membrane storage assemblies, and intra- and extra cellular "communication-centres" are discussed.

7.2 Recognition of hyperbolic periodic cytomembrane morphologies in electron microscopic sections The shape and function of cytomembranes are intriguing and challenging research areas, inasmuch as their heterogeneous composition and size make them technically very difficult to study. Usually, we have to rely on experimental probes that inevitably perturb the sample - such as electron microscopy (EM) - to study their structures. The preparation techniques used for biological specimens often affect the shape of the membranes. Even if these obstacles can be overcome, the result is a two-dimensional impression of a three-dimensional structure. Serial sections or scanning electron microscopy (SEM), as well as tilting and rotation of the sample, will of course improve the result.

260

Chapter 7

There are, however, some periodic 3-D structures that can, due to their symmetries, give rise to characteristic 2-D projections, allowing unequivocal structural identification. The cubic-membranes are one such group of 3-D structures that may be identified from their sections, using a combination of techniques. Once a particular morphology is recognised in an EM section, we have used mainly computer simulations to display various sections through the corresponding IPMS* or H-surface to compute images which match the experimental section. In addition, some sections have been visualised using plastic models. The computer generated projections shown here have been computed as projections of either the periodic nodal surfaces (PNS), which are excellent approximations to IPMS [1], or of the representation by Anderson and coworkers [2] (see [3-5] for computational details). We will see that many membranes have the structure of PCS's, which are based on the gyroid (G), the double diamond (D), and the primitive (P) IPMS (or their corresponding PNS). It should be stressed that the symmetries are the same for the IPMS and its H-surface associates (and thus for their projections). Generally the knowledge of the IPMS projections is sufficient to recognise a particular cubic membrane. Estimation of the mean curvature of a membrane requires careful matching of the electron micrographs with the computer generated projections. It should be pointed out that if the space on one side of the membrane occupies a large volume fraction, the appearance of the cubic membrane can easily be mistaken for a cubic membrane formed from several (parallel) bilayers. This is because one subvolume of the structure becomes very small, therefore the opposing membranes may in certain sections look like two separate, parallel membranes, rather than one. Another important feature of cubic membranes is their topology. Even though the topology of a cubic IPMS is generally described by its Euler characteristic (described in Chapter 1), we cannot usefully apply this to cubic membranes, since they are formed subject to a constraint on their integral Gaussian curvature. They often exhibit the same topological indices as the form from which they emerged. Thus, if a membrane grows from a sphere, an intersection-free invagination of its surface will not change its global topology, though the local topology is that of the invaginated membrane form. It is thus of importance to realise from which membrane system each particular cubic membrane has evolved. Cytomembrane topology is an area of research that has hardly begun, and we have, for simplicity, assumed the principal topology to be that of a sphere (see section 7.6). It is unfortunate that the only approach by which we can currently study cubic membrane structure is from electron microscopy, which often induces artefacts. On the other hand it is more likely that perturbations caused by EM preparation and fixation techniques are destructive to cubic membranes rather than constructive. Therefore it is unlikely that EM preparation techniques would induce the formation of cubic membranes. Indeed similar conclusions have been deduced by several authors dealing with the membrane assemblies that we identify as cubic membranes (see, e.g. [4] and [6]). Besides artefactual problems, our approach is inherently limited, because * All abbreviations are listed at the end of this chapter.

Membrane structure from electron microscopy

261

no special cleavage planes are revealed in transmission electron microscopy (TEM) studies. Sections are usually r a n d o m cuts through a cubic membrane, and often these lead to such complex patterns that the m e m b r a n e appears more or less disorganised. Other parameters, such as the thickness of the section, angle of view, etc. will further complicate the final appearance of the section t h r o u g h a cubic membrane. In addition it is rare that a structure can be understood by the observation of one section alone, since similar images can be p r o d u c e d by complex and c o m p o s i t e sections of other cubic morphologies. In particular, certain views of cubic m e m b r a n e s can be m i s t a k e n l y assigned w h e n TEM m i c r o g r a p h s of low magnification are analysed, due to the loss of certain important structural details, such as the sharp "saw-tooth" appearance of the D- membrane, and details determined by the electron distribution. Thus, s u b - o p t i m a l r e p r e s e n t a t i o n s of certain sections of the G- and the P-membranes m a y look similar, while the P- and the D-membranes have some projections that appear alike.

Fig 7.1(a): The diamond (D-) membrane system of the PLB in etiolated leaves. Projection of a section cut approximately normal to the {100}plane. The lower inserts show the match between the experimental micrograph and the computer generated constant mean curvature PCS projections for two different distances along the [100] direction. The upper inserts show the Fourier transform (calculated for the regions indicated) for the corresponding experimental and theoretical projections (a and b, and a' and b', respectively).

H o w e v e r , w h e n TEM m i c r o g r a p h s of optimal magnification and good resolution are available, the structural evaluation is mostly satisfactory. Thus, excepting the rather rare cases w h e n we are able to identify unique signatures

262

Chapter 7

of the corresponding PCS of the cubic membrane, or when several different sections are available, the identification of particular lattice planes remains tentative. The complexities of cubic membranes, their polymorphism, and the fact that artefactual distortions cannot be ruled out, make detailed structural determination of cubic membranes more or less uncertain. In view of the unfavourable likelihood of an obvious occurrence of cubic membranes in EM sections, the apparent ubiquity of cubic membranes is somewhat surprising. Assume that a cubic membrane assembly is approximately spherical with a diameter of one tenth of that of an approximately spherical cell. We would then expect to find even the tiniest portion of it in only 10% of the cell sections. The likelihood of finding images displaying signatures of a particular cubic membrane is thus very small. We can estimate that less than 1% of the cell sections will display such signatures. Many of the works cited are thus more or less fortuitous sections of cubic membranes, some through several directions of the lattice, which have helped us to identify it. As we will show, there are, however, certain cell types whose cytomembranes do largely conform to the structure of cubic membranes.

Figure 7.1(b): The diamond (D-) membrane system of the PLB in etiolated leaves. Projection along the [110] direction simulated. The corresponding simulated projection is overlayed (right) on two regions with different direction. The Fourier transform calculated for the region indicated is also shown (experiment: left, calculation: right).

In the following we shall use Miller indices (h, k, l) to represent symmetrical equivalent planes of a form by the symbol {100}. This implies that projections of the planes (100), (010), (001), (100), (010), and (001_), will have identical views, since the lattices we deal with are cubic (with this degree of multiplicity of form). Analogously, equivalent symmetry directions will be written as [100], while a specified orthogonal plane along such a direction will be represented by x, y, z; e.g., 1/2, 0, 0 (relative to the origin). This, of course, corresponding to the distance along [hkl] (relative to the origin). However, by representing it as a plane x, y, z the original projection plane together with its direction is emphasised. For further details regarding crystallographical and computational aspects of the projections see [4] and [5].

Membrane structurefrom electron microscopy

263

The recognition that cytomembranes can adopt the shape of IPMS with cubic symmetry, or a constant, non-zero mean curvature analogy thereof (an Hsurface), is due to Larsson et al. [7] who suggested that the structure of the prolamellar body (PLB) in etiolated (dark-adapted) leaves, as characterised by the model of Gunning [8], describes a P-surface. It should be pointed out that to date the true structure of the PLB remains unresolved. Most detailed structural analyses have involved pleomorphic descriptions. Along with such suggestions, both the P- and the D-surfaces have been considered as possible candidates for models dealing with the structure of the PLB, and it has been suggested that the "hexagonal" pattern described by Gunning [8] and others (see e.g. [9-11]) is consistent with a structure based on the D-surface, rather than on the P-surface [12]*. A similar pleomorphism of the PLB has also been explored by others (see, e.g. [13]).

Figure 7.1 (c): As for (b); here the [111]projection. Our analysis of the structure of the PLB does not confirm any cubic pleomorphism, and shows that it is clearly described by the D- rather than the P-surface [4, 5]. Of the views depicted in Fig. 7.1(a-d) of the PLB, the "square" pattern is consistent with projections along the [100] direction (Fig.7.1(a)), the "waveform" and the "hexagonal" patterns are reconciled by projections along the [110] (Fig. 7.109) see also Fig. 7.1(e) and (f)) and [111] directions, respectively. More complex projections, e.g. the [211] and the [322] (Fig. 7.1(d)) are also readily identified. The "fork" structures are consistent with projections along the [211] direction (Fig. 7.1(d)). The alternating "hexagonal" and "fork" structures are consistent with projections along, for example, the [322] * In this chapter, D- and P- (PCS) surfaces, as well as the gyroid denote surfaces which are topologically equivalent to the D, P- and G- periodic minimal surfaces (i.e. IPMS), although they need not be geometrically identical to the IPMS. In other words, the notation refers to the geometric and topological arrangement of tunnels in the convoluted structure.

264

Chapter 7

direction of the double diamond cubic membrane based on the constant mean curvature D-surface [4, 5]. It is of some importance to mention that the "waveform" pattern as defined by Gunning [8] cannot be comprehended as a projection normal to the 1221} plane of his model, which model resembles the P-PCS-structure. Among the possible candidates that we have investigated for the description of the PLB structure (the D-, P-, and G-PCS structures and four other cubic PMS's based structures), the D-PCS structure alone accounts for all structural features of the PLB that are available to us. Of particular value is the recognition of the prominent triangular-like structural elements of the D-surface, which have only been observed in the D-PCS structure. We should point out that the opposite case; i.e. the lack of such triangular elements, does not exclude the DPCS in a structural evaluation of a cubic membrane, and we have encountered cases in which the aspect of the D-cubic membrane is smoothly curved.

Figure 7.1 (d): A complex projection showing two regions corresponding to the [211]projection (lower with Fourier transforms (a (experiment) and a' (theoretical)) and the [321] projection (upper with Fourier transforms (b (experiment) and b' (theoretical)). The structure of the PLB has been related to that of cubic phases[7], discussed in Chapters 4 and 5. However, as we shall see, a description of these membrane morphologies as equilibrium phases seems to be applicable, if at all, in only a few cases that we have encountered. Independently of Larsson et al. [7], Linder and Staehelin [14] also suggested that a certain "membrane lattice" in a parasitic protozoa did indeed correspond to an infinite periodic minimal surface. However, no further structural details, such as the symmetry or form of IPMS, were deduced or discussed. Some ten additional examples of membrane assemblies displaying cubic symmetries have been pointed out [15, 16]; but no structural details were inferred. To the best of our knowledge, the above references ([7, 14-16]) are the only reports in which membrane assemblies have been related to the structure of IPMS. There are,

Membrane stnlcture from electron microscopy

265

Fig. 7.1(e): Computer generated projections of the D-PCS for various constant mean curvatures (0-1) for the indicated lattice directions. Shown is also the corresponding 3-D unit cell (right). Notice the change in sub-volume relation between the two spaces. h o w e v e r , s o m e authors, in addition to G u n n i n g [8] that have a p p a r e n t l y inferred cubic s t r u c t u r e s for m e m b r a n e assemblies [17, 18], b u t h a v e not p o i n t e d out their s y m m e t r i e s or related them to IPMS. Despite this general i g n o r a n c e of h y p e r b o l i c periodic shapes, a v e r y large n u m b e r of s u c h m e m b r a n e s t r u c t u r e s can be found in the literature, some of w h i c h are described and discussed here.

266

Chapter 7

Figure 7.1(0: Shows the influence of specimen thickness for a quarter and a half unit cell

thickness on the projections for a constant mean curvature D-PCS (0.75 in relative mean curvature). Figs. (a)-(d) from [8], Fig. (d) from [9], reproduced with permission.

7.3

The structure and occurrence of cubic membranes

The ultrastructure of biological membranes has been an intense area of research for at least the past 35 years. The apparently widespread occurrence of cubic membranes, together with their variable appearance in cross-section, makes an exhaustive literature review of these structures nigh impossible. In addition, the differing treatments that these structures have been subjected to make it even more complicated to find examples that are possible cubic membranes. Some authors do not even comment upon the existence of such membrane assemblies, though their existence is apparent in their published micrographs. Others have recognised morphologies related to that found in

Struch~re and occurrence of cubic membranes

267

the chloroplast. During the progression of this work t h o u s a n d s of micrographs have been reviewed to find possible cubic membranes, and the hundreds of cases identified and analysed demonstrate the existence of what we define as cubic membranes. Tables 7.1 and 7.2 systematise part of our findings regarding the occurrence and structure of cubic membranes in cells a n d / o r tissues. Undoubtedly, cubic membranes of all three principal structures, the G-, the D-, and the P-PCS, exist without any obvious restrictions or preferences, throughout all kingdoms, both in normal and pathological or manipulated cells. Their existence does not seem to be restricted concerning cell type. Though there are some cell types in which cubic membranes are more frequently encountered than in others, it seems likely that this is due to, firstly, our restricted review, and, secondly, to the focussed interest of the research community on the study of certain cells and tissues in more detail than others. Despite this sampling bias, the possibility cannot be excluded that certain cell types form cubic membranes more often than do others. Some examples of this are discussed below. Cubic membranes are not strictly associated with any particular organelle. Indeed, cubic membranes can apparently evolve from virtually any cytomembrane involving the plasma membrane (PM), the rough and smooth endoplasmic reticulum (RER and SER, respectively) including the nuclear envelope (NE, both inner, INE, and outer, ONE), the mitochondria (only the inner membrane), lysosomes, and the Golgi complex. The ER seems, however, to be the organelle most frequently associated with a cubic membrane. But distinctions between the PM and the ER are sometimes difficult to draw. The origin of the ER from the NE is well established. The relationship and formation of cubic membranes with respect to classical cell organelles is further pursued below. The formation of cubic membranes is not confined to the cytoplasm, they also appear at intranuclear sites. At first glance this may appear surprising, but the formation is realised by infoldings of the INE. While the structure of these cubic membranes does represent, to date, an enigma, their functional role is an even more challenging issue since virtually nothing is known of it. U n d o u b t e d l y , structure-function relationships, so successfully applied to many other areas of science, could prove to be equally fruitful when applied to cubic membranes, as briefly addressed below. In the following sections we will highlight some of our structural findings. The outline will largely follow the order of appearance in Tables 7.1 and 7.2.

268

Chapter 7

T a b l e 7.1. Membrane assemblies, whose structures are consistent with cubic membranes, observed in E M studies o f normal cells/tissues Description o f cells/tissues E~mcter/a Cyanobacteria Thylakoid lamellae in Anabaena Thylakoid lamellse in Heterocyst of Anabaena am//ae ProUm~ Zoomasg&ophorea Lepmmonas collosoma Sarcomasgsophom Mitochondria in Pelomyxa carolinensis MaWhym Phycophyta (A,I,gae) Zy&nema Chara corallin~ C. braunii Pteridophyta Oocytes of SelagineUa krauss~zna Spermatophyta PLB in Etiolated leaves of Arena sativa Nectaries of Helleborus (H. foetidus, H. niger) Ovules of Ficaria ranunculoides Plastids in bean root tipsof Phaseolus vulgaris Sieve elements of Dioscorea (D. bulbO~era, D. macroura, D. reticulata, Tamus communis) Sieve elements of Snu/ax exce/sa Sieve elements of Ubnus amer/cana Sieve elements of Phaseolus vulgaris Sieve elements of Acer S i e v e e l e m e n t s o f Arecales

Sieve elements of Nymphoides peltata Myeopkym ~ ) Myxomycetes (slime moulds) Mitochondria in D/dynuum Ascomycetes Apothelial cells of Ascobolus stercomrius * Masti&oomycetes Oedogoniomyce$ zoospores * Maazoa Arthropoda Mitochondria in slamnafids of s flavicaudis Spermatozoa of Cran#on septemspinosa Spermatids of Melanoplus diffengolts ~fifferent~b ** Mitochondria in oocytes of D e c ~ Spermatids of Dysdercusfasciatus Mitochondria in corpus Mlata of Locust

migratoria migratorioides

Cognomen

PCS

Ref.

D

[19]

honeycombed hunellae

P

[20]

Membrane laUice

-

[ 14]

P

[22]

P -

[41] [461, [511

-

[29], [30]

D

[8]

Coue de marines Cotte de mailles

P -

[26], [27] [27]

Tubular complex

-

[:53]

Lattice-like memtnne Convoluted ER Complex ~ Convoluted membranes Quasi-crystalline membranes Convoluted tubular ER Convoluted ER

D, G G G G, P G G

[31],[32] [33] [36] [37] [35] [38],[39] [40]

P

[60]

Quasi-crysudline lamellar

Pseudocrysud

Lattice bodies

G, D

[6],[61],[62]

Organized lamellar system

-

[ 142]

.

-

[63]

bnice

P

[70].['71]

Texunn Honeycombed crime Sinusoidel tubules

D

[6/]

P

[641 [68]

~sudline

[65]

Structure and occurrence of cubic membranes

269

|

Tabie 7.1. continued Description of cells/tissues Rectal epithelial ceils of Petrobius maritmus Mitochondria in intestinal cells of Petrobius maritmus Oocytes of Pyrrocoris apterus Mitochondria in flight muscle cells of Calliphora erythrocephalo Belionci organ of Sphaeroma setratum Annelida; Polychaeta Luminous cells: Acholoe astericola Logisca extetmato Harmodwe lunulala Harmotkoe imporo H. lon&isetis, H. iunulata, Gottyana cirrosa, Polynoe scolopendrina Lens/Photoreceptor cells: Same as luminous cells and Halosydna gelotinow~ Lepidonotus sqaanuaus, 5calisetosus pellucidas, $1galion methildae, Sthenelais boa, Hermione hystrix Arctonoe vittata 5yllis arnica Nereis virens Spermatids of E/sen/afoet/da Mollusca Type I p ~ ceilsof L/max Photoreceptor cellsof Helix pomatla Spermatids of Spurilta ncpolitana Spermatids of Planorbarlus corneas Chordata Urochordata Golgi in test cells in the ovary of 8tye/a

Cognomen

PCS

[113]

Cotte de mailles

[144] [72]

.

PER

[~) [143]

Regular fenestrated cristar

[73],[75],[76]

PER, Photmomes Photogenic granules PER PER

[st]

[751,[76] [73]

PER

P

[79]

PER Crystalline element PER Paraorystalline body Undulating tubules

P

[791 [83]

Cmmgated BR Crystalfine-like structure Undulating tubular body Cytoplasmic crystalIoid

P

.

P P .

.

P P

Epithelium in the olfactory organ of

Turtuous interconnected ER

[8o]

[82]

[84]

[89) [SS) [S6)

[ST)

[90)

Honeycomb, lattice-like

Pisces

So/too trutna truffa Glandular cellsin the externaldendritic orpn of the catfish Retinal pigment epithelium cellsof i~npetm fluviatalis Admnocm'dr cells of 5a/mofar/o ** Oocytes of Brachydanio rerio Retinal pigment epit~lium teaLs of lauimeria chalumnar Amphibia Mitochondria in Sertoli cells of Xenopus /a~/s Oocyte of Necmrus maculosus maculosus Intestinal epithelium cells of Alytes obstetrlcans

Ref.

[145]

.

Tubular network Undulated membrane complex Imbricated cisternae of ER Fenestrated membranes

D

[55)

G G

[gz) [571 [1151

Regular arrays of tubules

G

[93)

Regularly fcnestrated cristae Annulate lamellae

D

[231 [116]

Sinusoidal tubules

~

.

[~461

Chapter 7

270

Table 7.1. continued Description o f cells/tissues Retinal pigment epithelium cells of Notophtalamus viridescens Reptilia Spermatids of Anolis carolinensis Sertoli cells in Eumeces iatticel~ Ayes Skeletal muscle cells derived from chick

Cognomen

m

b

r

y

o

Ref.

Fenestrated lamellae

[147]

Membranous body Sertoli membrane body

[148] [149]

Tubular network Mitochondrias in cardiac muscle cells of Serinus canarius Mammalia Lymphoblast from culture No. 2117 * UMS Mitochondrias in p h o t ~ t o r cone cell of the Yio~aiaglis Concentric whorls of cristae Vomemmmal epithelium of the rat Jejunal absorptive cells of rat intestine Membranous body Mitochondria in skeletal muscle of the rat Plasma cells of the rat Meibomian gland of the rat Bright columnar celh in the vomeronasal HexaSonal crystal-like organ of the cat membrane body Endothelial cells in liver of Macaca mu/atm ** Cytoplasmic ~yualioid Retinal epithelium cells of Macaca mulana Peculiar body Endothelial cells in the giomerular capillaries of Macaca mulana Round or hexagonal bodies Interstitial cells in the antebrachial organ of the Lemur catta Crystalloid Villus absorptive cells in fetal small intestine of man Convoluted membrane e

PCS

s

.

~

.

o

P,G

[181.[52]

~

[15o] [1511

G P

[241 []52] [153]

D P

[154] []7] [17]

G

[]o6-]o81

-

[]551 ~

[158]

.

[156]

G

[171 [157]

StruchLre and occurrence of cubic membranes

271

T a b l e 7.2. Membrane assemblies, whose structures are consistent with cubic membranes, observed in E M studies o f pathological or manipulated cells/tissues Description of cells/tissues Eubacteria Cyanophyceae (blue-green algae) Anabaena variablis infected with cyanophages * Metaphyta Triticum aestivum affected by wheat spindle streak mosaic virus Metazoa Arthropoda; insecta X-ray exposed spermatids and spermatocytes of Mela~**pl~ diffentialis differentiali$ Chordata Endothelial cells in the liver of Macaca mulana with nutritional cirrhosis ** Neoplastic cells: Rous sarcoma virus induced tumour cells of 5aquinus Various cell lines *: Subdermal tumour in the hamster produced by inoculation of M-I cells, F-53, F-3, F-9. [:-24 P3J cells HeLa cells* Epithelial lung carcinoma in man * Rhabdomyoma in man (T-tubule system in skeletal muscle) Parosteal sarcoma in man Bronchiogenic carcinomas Endothelial cells in the glomerular capillaries of nephritic man Endothelial cells of a hepatoblastoma in

Cognomen

PCS

Ref.

Pl..B-likc structure

P

[21 ]

Membranous body

G

[59]

Texmm

D

[67]

Cytoplasmic crystalloid

[155]

Membranous cytoplasmic inclusion

[Ill]

Undulating tubules, UMS UMS Cotte de maillet TItS

P(Fg)

Lattice-like parac~stalls UMS

D P

.

p

.

Crystalline bodies

[151] [159]

[ll4] [1241 [160] [ll2] [104]

[156] [161]

mall

Mitochondria in epithelial cells in adenoma of the submandibular gland of man Mitochondria in metastatic melanoma T-tubule system in skeletal muscle fibers: Late onset acid maltase defiency in man Denervated skeletal muscle from albino rats Hyperkalemic periodic paralysis in man Myotonic dystrophy in man Infections: Kidney cells of Macaca mlana infected with Tana poxovirus * Hepatocyte of Pan trode&lytes post experimental hepatitis Brain tissue of mice after inoculation with St. Louis encephalitits virus HEp-2 cells infected with llheus virus *

.

Reticulate cristae

[IOt] [IOt]

Tubular network

[96]

Tubular masses Tubular aggregates

[98]

Lattice-liketubular network

197] 199]

Honeycombed crystals

[1621

UMS Convoluted membranous

[163]

m a s s

Knotted membranes

[164] [~65]

272

Chapter 7

Table 7.2. continued Description of cells/tissues Spindle and endothelialcells in Macaca mu/atta after tumor induced by sarcoma virus

Monkeykidneycells(I/X:-,MK2) infected with rubellavirus Human emb~onic kidneycells infected with HRV T Manipulated cells/tissues: UT-I cells derivedfrom Chinesehamster *

Lutein cells of the rat aftercyciobeximide treaunent Hepatocytesof the hamsterafter phenobarbitonetreaunent Mice retinalpigmentepitheliumpost mild thermal exposure Adrenocorticalcells of Sa/mofar/o **

Cognomen

Ref. [166]

Crystalline inclusion

Crystal lattice-likestructure

[il7]

Micro-TRS

[1041

Sinusoidal ER Crystallinetubular aggregates

P

Membrane complex

-

[ 104]

Lacy pmmmO ~ Imtricated cistenmeof ER

G

[69]

[1101

[1671

P

[57]

Table 7.1 describes the existence of cubic membranes in apparently normal cells and tissues, while Table 7.2 summarises our findings in pathological and manipulated cells and tissues. It is important to point out that in none of these cases has a true three-dimensional structure been previously comprehended or assigned. Some authors have recognised similarities with the PLB, and pointed out the existence of comparable membrane systems. Although the specific structural assignments given in Table 7.1 and 7.2 are based on unique signature projections, a conclusive structural determination requires diffraction measurements. * Indicates that the cell/tissue was cultured. ** Indicates that a cubic membrane can be identified in both normal and pathogenic instances of the same cell/tissue.

7.4

Cubic membranes in unicellular organisms: prokaryotes and protozoa

In this group there are quite a few reports dealing with PLB-like membrane a s s e m b l i e s of p h o t o s y n t h e t i c l a m e l l a e , h o w e v e r f e w c a n b e u n a m b i g u o u s l y i d e n t i f i e d as c u b i c m e m b r a n e s . O n e e x a m p l e i n w h i c h w e c a n m a k e t h a t i d e n t i f i c a t i o n h a s b e e n o b s e r v e d i n a n a g e d b l u e - g r e e n a l g a [19] w h o s e photosynthetic thylakoids were observed to continuously fold into a

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273

morphology resembling the PLB. Its structure was described as the same as the PLB structure suggested by Gunning [8], i.e. resembling a P-PCS [7, 16]. However, as with the PLB, we can only comprehend the appearance of the published sections [20] through a structural analysis based on the double Dsurface, rather than the P. Even t h o u g h this cubic m e m b r a n e is morphologically identical to the PLB it is, unlike the PLB, not dependent on light conditions. The structure was only seen in aged cells in which the photosynthetic apparatus is fully developed. Thus the age, i.e. stage of differentiation, rather than the light conditions, seems to determine the formation of this cubic membrane. Similar arrangements of the thylakoid membrane have been observed in the heterocyst of Anaebena azollae, [19] and in cyanophage infected vegetative cells of A.variablis [21]. We can identify both as cubic membranes. A tentative structural determination of the first favours a P-PCS membrane, though we cannot unambiguously exclude a structure based on a D-PCS. The second case cannot be reconciled with a membrane folded onto a D-PCS, rather, it is consistent with a P-PCS membrane, based on a section [21] projected approximately perpendicular to the {431} plane of P-PCS. Since quite a few projections of P-PCS' produce patterns similar to the [431], this does not allow for an unambiguous identification of the particular lattice plane. Cubic membranes formed by folding of the thylakoid membrane are, as such, associated with the PM and hence realised by invaginations of the PM. In eukaryotic unicellular organisms we identify the "membrane lattice" found in the kinetoplastid Leptomonas collosoma [14], a parasitic protozoan, and the tubular cristae in the mitochondria of the giant amoeba Pelomyxa carolinensis [22], as cubic membranes. The cubic membrane in Leptomonas seems to be formed by membrane foldings of the ER, while that of the amoeba is formed by infoldings of the inner mitochondrial membrane. Linder and Staehelin [14] suggested that the structure of the Leptomonas "membrane lattice" is related to that of infinite periodic minimal surfaces, without being more specific. The sections reported of the "membrane lattice" clearly show a cubic membrane viewed normal to the {100} plane (see e.g. Figure 2 in [14]). However, in this particular section, it is not possible to distinguish unambiguously between the [100] projection of a P-PCS and a similar pattern obtained by the same projection of a G-PCS. But it does not seem to be reconciled with any projection of the D-PCS. The other sections reported by Linder and Staehelin [14] are evidently more complex composite sections and this circumstance does not allow for any further elaboration of the particular PCS structure. A significant observation made by these authors was the presence or absence of the cubic membrane delSendent on growth conditions [14]. This indicates sensitivity in the development of the cubic membrane to environmental factors. The lattice-like membrane arrangement of the cristae of mitochondria in the giant amoeba was interpreted by Pappas and Brandt [22] to be composed of membranous tubules. If instead of discrete tubules, a continuous surface is assigned in such a way that it links the sections, a cubic membrane can be

274

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Figure 7.2: The mitochondrial P-membrane in Pelomyxa carolinensis. (a) Projection along the [110] direction at a distance of ca. 1/2, 1/2, O. (b)-(d) Projections of sections cut approximately normal to the {111} at a distance about 2, 2, 2 (b), the {211} (c), and the [553} at a distance of 5/4, 5/4, 3/4) (d) planes, respectively. (e) A complex projection along the [432] and the [433] directions. From [22], reproduced with permission.

Cubic membranes in prokaryotes and protozoa

275

envisaged, whose structure corresponds to a P-PCS. Fig. 7.2(a) shows a section viewed along the [110] direction at a distance corresponding to about 1/2, 1/2, 0, which appears as straight lines with regular nodes. Projections of sections cut approximately normal to the {111} and the {211} planes are shown in Figs. 7.2(b) and 7.2(c), respectively. The latter projection rules out the D-PCS. Two more complex sections that can be understood as projections along the [553] and the [433] directions are shown in Figs. 7.2(d) and (e). Diamond, as well as gyroid cubic membrane structures in mitochondria are evident in other works [23, 24], as analysed below. An interesting finding regarding the biogenesis of the mitochondria in the giant amoeba is the close relationship between the mitochondria and the ONE, suggested by observations of direct continuities between these organelles during the d e v e l o p m e n t of the mitochondria [22]. Since several works report similar relationships between mitochondria and ONE a n d / o r ER, this may have implications regarding the biogenesis of mitochondria. The observation of cristae adopting a cubic membrane structure supports such a relation, since it could be viewed as a topologically consistent remnant of the INE, with the ONE forming the outer mitochondrion membrane. Not only is this feasible from a structural point of view, but it also provides an efficient transport of mitochondrial DNA.

7.5

Cubic membranes in plants

Cubic membranes can apparently be found throughout the plant kingdom, and are not restricted to foldings of the photosynthetic lamellae. Though other cubic membranes are less commonly observed than in the PLB, we have found two major groups of cell types in which we frequently identify cubic membrane systems, namely germ cells [25-30], and sieve elements [3140]. The pseudo-crystalline "lamellar lattice" reported by McLean and Pessoney [41] in Zygnema was assigned a structure identical to that of the PLB as described by the model of Gunning [8]. Recall that the structure of the PLB corresponds to the D-cubic membrane rather than the P. Our analysis of the structure of the "lamellar lattice" in Zygnema shows that it is a P-membrane, and thus differs from that of the PLB. Sections viewed along the [100] direction at distances approximately between 1,0,0 to 2,0,0 are shown in Fig. 7.3(a). Fig. 7.3(b) shows a section cut approximately normal to the {211} plane, between a distance of about 4,2,2 in the upper part, to about 2,1,1 in the lower part of the figure. Figs. 7.3(c) and (d) show two complex sections that cannot be identified as single lattice planes. The upper part of the section seen in Fig. 7.3(c) is consistent with a projection approximately normal to the {752} plane, while the lower part corresponds to a higher lattice plane of the same appearance. Fig. 7.3 (d) is consistent with the [522] projection. From the serial sections reported by McLean and Pessoney ([41], Figs. 6-8), which are projections of the P-PCS along the [100] direction, the lattice parameter can be determined to be between 360-430 nm. In view of these circumstances the correct assignment by McLean and Pessoney [41] of the structure was perhaps fortuitous. Furthermore, the cubic

276

Chapter7

m e m b r a n e in Zygnema is multicontinuous, rather than the bicontinuous m e m b r a n e in the classical PLB. The membrane structure is composed of several approximately parallel bilayers, and possibly additional proteinaceous layers as seen in Fig. 7.3(e). At least 7 membranes can be distinguished. Such a construction is thus a multicontinuous cubic m e m b r a n e modelled by a number of approximately parallel PCS' differing only in the magnitude and sign of their mean curvature.

Figure 7.2(0: Computer generated projections of the P-PCS corresponding to the assignments in (a)-(e).

Cubic membranes in plants

277

Figure 7.3: The multicontinuous primitive cubic membrane system in Zygnema. (a)-(b) Projections along the [100] (a), and the [211] (b) directions, respectively. (c) Complex projection which can be understood as the [752] projection generated at a distance corresponding to 7/4, 5/4, 1/4 relative the origin. (d) Another complex projection which is best understood as the [522] projection (see also the [311] and the [411] simulations shown in (f). (e) Higher magnification of a [110] projection at a distance varying between 1, 1, 0 to 2, 2, 0, showing the multi-bilayer configuration, note the intertwined proteinaceous network seen particularly well in (a) and (c). Figures (a)-(e) are modified from [41], with permission.

278

C_Aapter7

The P-membrane of Zygnema creates 8 different spaces. One space, the principal space (electron opaque), occupies the majority of the total volume. The other 7 spaces are all of about equal size, and small compared with the principal space. As we shall see, such partitioning of space is not uncommon. Even more intriguing is the regular network of proteinaceous material interwoven with the cubic membrane, best seen in Fig. 7.3(a) and (c). Even though the exact spatial relation and biogenesis of all the layers of this Pmembrane and its interwoven network is unknown, such an arrangement indicates the importance of spatial segregation and membrane versus protein packing. The latter, which is necessarily different in the fiat lamellar membrane compared with that in the cubic membrane, may be of relevance to the regulation of lipid-protein interactions. It is noteworthy that the membrane and proteinaceous material from at least two different sources, the thylakoid and the stroma, seems to simultaneously change its morphology in register with the lamellar-to-cubic membrane folding. Apparently the choice of PCS differs between different species of Z ygnema. We have recently shown that a culture collected from Bloomington, IN, USA formed a G-PCS with both single and multiple membranes [42] rather than a P-PCS as in the Zygnema studied by McLean and Pessoney that was collected in Austin, Texas [41]. The developmental and environmental influence on this apparent species-dependent choice of PCS is currently under investigation. The genesis of this cubic membrane seems to be through invaginations of the thylakoid membrane system [41, 42]. Besides the structural differences between the cubic membranes of Zygnema and those of PLB's, there are, in addition, important physiological differences. First, the cubic membrane was only observed in the chloroplasts of Zygnema in the stationary phase of growth. Second, as in the case of A nabaena analysed above, this cubic membrane is not light sensitive. In addition to Zygnema, many reports deal with PLB-like membrane structures in unicellular green alga of both normal (see e.g. [43]) and manipulated instances (see e.g. [44, 45]). These structures are seldom regular enough to allow for any structural identification. In mature internodal cells of the algae Charae, invaginations of the cell membrane ("charasomes") have been observed to form regular structures [4651], some of which we identify as cubic membranes. Though an extensive structural analysis of this membrane configuration is yet to be done, it seems to be similar to the gyroid. However, since this membrane structure apparently exhibits polymorphic behaviour, the gyroid membrane might only represent its average shape. Similar appearance and membrane morphogenesis are apparent in other tissues, particularly muscle tissue (see, e.g. [18, 52]), analysed below. The structural identification of cubic membranes does not necessarily show that cubic membranes are physiologically active structures. But there is some evidence for this in, e.g., PLB's that arrange the molecular structure of their photosynthetically active membrane.

Cubic membranes in plants

279

Figure 7.3(0: Computer generated projections of the P-PCS corresponding to the assignments in

(a)-(e).

280

Chapter 7

An example of such a physiologically active cubic membrane can be found in the work by Newcomb [53], who describes a light insensitive membranous "tubular complex", which he recognises as a PLB-like membrane complex. It appears to form as an invagination of the inner membrane of the plastic envelope [53]. From the sections shown in [53] we cannot unequivocally determine the particular symmetry of the cubic membrane, although it appears to be a gyroid membrane. More intriguingly, this particular cubic membrane seems to serve as a "centre" for the crystallisation/concentration of a protein, as shown in Fig. 7.4. The membrane sac surrounding the protein crystals (Fig. 7.4(a)) or the amorphous proteinaceous mass (Fig. 7.4(b)) originate from the cubic membrane. Although the role of this membrane arrangement and the function of the cubic membrane itself is unknown, it is clearly involved in the transport and storage of the proteinaceous material embedded in the membrane sac evolving from the cubic membrane. A curious observation is that only one of the two spaces defined by the cubic membrane seems to be involved in this process, as indicated in Fig. 7.4(b) The most commonly described membrane system in plants, after the PLB, which we identify as a cubic membrane system, is that of the differentiating sieve elements. It was first described in Dioscorea reticulata and characterised as a PLB-like "lattice-like" membrane body [31]. Since then, many biologists have studied the aggregation and accompanying structural changes of the ER and other organelles in differentiating sieve elements ([32-40, 54] a n d references therein) but its structure has remained uncertain. Its structure is indeed sometimes that of the PLB, i.e. a D-membrane, although important structural and physiological differences exist between the two. It is clear that the cubic membrane in differentiating sieve elements is formed by foldings of the ER, rather than from the PM, from which the PLB is formed. Although the structure of the PLB in etiolated chloroplasts is static in the dark, as well as reversible upon cyclic light exposures, the cubic membrane of differentiating sieve elements is not light sensitive. Rather, it appears during the ontogeny of the sieve elements, and only in mature or nearly mature elements, where it is frequently seen to be continuously linked to fiat lamellar configurations (Fig. 7.6(a) (See also, e.g., [32] (Figs. 1, 6, 8(a) and (b)), [38] (Fig. 31), and [40] (Fig. 5)). Further, intermediate configurations between flat lamellar and cubic membrane arrangements are often observed. Less frequently, the ER seems to close-pack in a hexagonal arrangement (see for example [32] (Figs. 17(a)), and [40] (Figs. 6 and 7)) in a fashion analogous to the structure of hexagonal phases in condensed lipid- and surfactant-water systems. The apparent polymorphic ER configurations makes these cells a good prototype for a continuous, membrane folding process which we believe to be a general property of cytomembranes. Of special in : ~st is the apparent existence in differentiating sieve elements of all thr..c lbic membrane structures investigated: the gyroid, the D- and the P. Cytomembrane structures consistent with the gyroid structure (and/or IPMS) were discussed first by Bouligand [15]. Quantitative comparison between the gyroid and a number of membranes can be found in [4, 55] (see also [56]),

Cubic membranes in plants

281

some of which are described here. One such membrane, whose sections were studied by Behnke [32], is shown in Figs. 7.5 (a) and (b). These figures illustrate two serial sections approximately normal to the {211} plane of the gyroid. The images are similar to the computer generated projection of this lattice plane in the gyroid-PCS (Fig. 7.5(c)). The lattice parameter of the underlying 3D gyroid membrane is approximately 240 nm. The [211] projection of the gyroid represents one of the more frequently observed impressions of the G-PCS membrane. However, caution must be exercised in assigning this particular gyroid section, since it can sometimes be confused with the [210] projection of the P-PCS, discussed in more detail below. In this particular case, serial sections were available [32] which are consistent only with the I~roid.

Figure 7.4: A cubic membrane in leucoplasts of root tip cells that are actively involved in "protein storage". (a) Shows two protein crystals which are encompassed by a membrane which is continuous with the cubic membrane. (b) Shows the direct continuities (arrows) between one space of the two defined by the cubic membrane and the amorphous protein-sac. From [53], reproduced with permission.

Fig. 7.5(b) shows a quite unusual polymorphism, in which the gyroid cubic membrane projected along the [211] axis is seen in the middle, while the lower right shows a membrane lattice consistent with a D-membrane projected approximately along the [111] direction. The latter image bears many similarities to the cubic membrane structure of the PLB, evident by comparison with, e.g. Fig. 7.1(b). Indeed quite a few of the reported ER aggregates in differentiating sieve elements can be characterised as Dmembranes (see, e.g. [31, 32]). In addition to the clear continuity between the two cubic membranes, several other intriguing observations can be made regarding this apparent polymorphism. First, the lattice size of the two cubic

282

Chapter7

m e m b r a n e s differs b y a factor of about 10. Second, the electron-lucent space has a p p a r e n t l y increased m u c h m o r e in v o l u m e t h a n the e l e c t r o n - d e n s e space. That the electron d e n s i t y characteristics are m a i n t a i n e d even at the sharp transition b e t w e e n the t w o lattices indicates that m e m b r a n e continuity is preserved, e v e n t h r o u g h the transitional state.

Figure 7.5: A multicontinuous G-PCS (the gyroid) differentiating sieve-elements (see also Fig. 7.6). (a) A [211] projection of a double bilayer G-PCS. The match of the theoretical projection is excellent (as well as the correlation between the Fourier transforms (a (experiment) and a' (theory)) and it is easily seen that a single bilayer G-PCS does not account for the experimental projection (see (c)). (b) Serial section of (a). Note the apparent pleomorphic behaviour of the cubic membrane in (b), which shows co-existing D- and G-morphologies, related by an intersection-free, and thus, topologically constrained transformation. (c: bottom) Computer generated projections of the single G [211] and the double bilayer G2 [211] projections. Figures (a) and (b) are modified from [32], with permission. P o l y m o r p h i c b e h a v i o u r s e e m s also to be p r e s e n t in d i f f e r e n t i a t i n g sieve e l e m e n t s of A c e r [35], a l t h o u g h c o n t i n u i t y of the m e m b r a n e across the

Cubic membranes in plants

283

different polymorphs is uncertain. Both P and gyroid cubic membranes can be identified in this organelle. A section through the P cubic membrane, which exhibits an unusual complex membrane pattern is shown in (Fig. 7.6(a)). It cannot readily be described by a single lattice plane. However, due to its remarkably well developed order, it can be described in part by the [320], [433], and [553] projections of the corresponding P-PCS. The gyroid cubic membrane, seen in Fig. 7.6(b), is partially understood as a projection approximately along the [111] direction (lower right part of the section) of a rather thick specimen (at least as thick as the unit cell edge length).

Figure 7.6: The cubic membrane system of differentiating sieve elements, showing the occurrence of a P- (a) and a G-PCS (b) in the same cell. The section of the P-PCS has not been identified, while the lower right in (b) is a projection close to the [111] direction of the G-PCS. See text for further details. (c) A gyroid cubic membrane projected along the [211] direction. This section has a thickness of about one unit cell due to the appearance of the nearly zig-zag to linearly arranged "circular" profiles between the in-phase sinusoidal pattern.

A well-developed gyroid m e m b r a n e can also be identified in the differentiating sieve elements of Nymphoides peltata investigated in [40], in which planes cut approximately normal to the {211} plane are evident (Fig. 7.6(c)). These and similar sections of the gyroid (see [57] for another example) are sometimes very difficult to assign, and can be confused with certain sections of the P-PCS structure. We have found it particularly important to investigate the influence of section thickness to avoid any confusion. Cubic membranes are also apparently formed in pathogenic instances of plant cells. An example of such, which in view of our analysis is consistent with a gyroid cubic membrane, is found in the early stage of the formation of the "pinwheel" structures derived from the ER in virus-infected wheat cells [58, 59]. Fig. 7.7(a) shows a projection approximately along the [111] direction of this G-PCS. It can be better modelled as the [332] projection of the G-PCS viewed at a distance of about 3/4, 3/4, 1/2, shown in Fig. 7.7(b).

284

~ter 7

Figure 7.6(d): Computer generated projections of the G-PCS assignments in (b) and (c). Parts of the section in (c) can be understood as deviations from the [211] direction with respect to any axis, which produces alternating patterns as exemplified by the [743] projection seen in (d). Figures (a) and (b) are reproduced from [35] and (c) from [40], with permission.

7.6

Cubic membranes in fungi

We have found images in some papers dealing with membrane complexes in fungi whose morphologies match those expected for cubic membranes. Interestingly, one example is of a mitochondrion in the amoeboid stage of the slime m o u l d (Didymium) [60], in which the inner m e m b r a n e s of the mitochondria conform to a configuration consistent with a cubic membrane. The section, shown in Fig. 7.8, cannot easily be described in terms of lattice planes and directions due to its complexity. But it is consistent with a P-PCS membrane, based on its correlation with projections (roughly) along the [110] direction, at a distance of approximately 3/4, 3/4, 0, and the [100] direction (from the left to the right in the figure). The structure of so-called "lattice bodies" occurring in apothelial cells of the

fungus Ascobolus [6, 61, 62] is consistent with that of cubic membranes. It has

Cubic membranes infungi

285

been pointed out that this structure is PLB-like and that several other possible structural homologies are apparent from a number of studies of plants [61]. It was further suggested that the membrane morphology represented closed packed "minute vesicles" or "membranous tubules" arranged as a cubic lattice [61] (see also [62]).

Fig 7.7: A gyroid cubic membrane identified in a pathological instance, classified as a viral infection. (a) View of the cubic membrane, probably a [332] projection generated at a distance of about 3/4, 3/4,1/2 from the origin (cf. Fig. 7.6(b)). (b) Computer generated projection corresponding to the section assignment in (a), in which the contrast has been enhanced relative to that of Fig. 7.6(d). See Fig. 7.18(e) for the [111] projection. Compare also the projections close to [111] in Fig. 7.16(g). Figure (a) reproduced from [59] with permission. However, the structural evaluation by Zachariah [61], which was based on the comparison with the structure of the PLB proposed by Gunning [8] is less than complete, since our analysis shows that this cubic m e m b r a n e assembly exhibits polymorphic behaviour. In certain cases, the "lattice body" apparently adopts a D-PCS membrane structure, similar to that of the PLB, based on projections of sections normal to the {111} plane, seen in Fig. 5 of [61]. However, other sections shown in [6, 61, 62] cannot be matched to a D-PCS. Closer examination reveals the existence of an additional cubic membrane type based on the G-PCS. Fig. 7.9(a) shows a projection close to the [331] direction of the gyroid. The section seen in Fig. 7.9(b) is consistent with a projection normal to the {311} plane. Note that this section is easily confused with projections of sections normal to the {322} plane. Other projections of this gyroid cubic membrane that can be identified are those following any plane for which two axes are zero or close to zero (cf. [62], Figs. 4, 7, 9, and 10), and several projections close to the [100] direction. An important observation is the existence of nearly congruent volumes of the two spaces separated by the cubic membrane, which is also clear in Figs. 7.7, while the volume relations are clearly very different in the gyroid membranes in, e.g. Figs. 7.5, 7.6(a), and 7.12(b), 7.13(b), and 7.18(b). Other

286

Chapter 7

gyroid membranes with approximately equal subvolume relations are seen in Figs. 7.16, 7.19, and 7.20.

Figure 7.8: A section of a mitochondrion P-PCS membrane identified by the [110] projection, to the left in the figure, and by the [100] projection, to the right in the figure. The more complex directions are less obvious. (Scale bar indicates I l~m.) From [60], reproduced with permission.

An arresting finding, which strongly supports the structural evaluation, is that the so called "osmiophilic minute particles", which are 90-110 nm in size [61], are in fact not particles. The particulate appearance in the micrograph simply arises as an effect of the electron density distribution of that particular section through the gyroid cubic membrane! The cubic membrane in the apothecial cells is of special interest in many ways. It is not static, but reversible, and its appearance seems to be correlated with the development of the apothecium. It is believed to form de novo from an "amorphous lipoprotein matrix" [61]. This would indicate that true phase condensation could occur in some instances of the formation of cubic membranes. However, the interpretation presented [61] is, in view of our analysis, slightly inaccurate. In particular, it is incorrect to suppose that the development of the "lattice body" takes place through the formation and subsequent fusion of vesicles. The membranous "vesicles" are properly interpreted as sections normal to any lattice plane of the gyroid in which two axes are close to zero. It should be emphasised that similar images can also be produced by projections of the D- and the P-PCS's.

7.7

Cubic m e m b r a n e s in m e t a z o a

Many of the most commonly encountered phyla in the kingdom of Metazoa are represented in Table 7.1. Even though the occurrence of cubic membranes with respect to the type of tissue does not seem to be restricted, they are

Cubic membranes in metazoa

287

apparently encountered more frequently in certain cell types like germ cells and epithelial ceils than others. The significance of this is unknown.

Figure 7.9: The gyroid membrane identified in Ascobolus. (a)-(b) Projections along the [331] and the [311] direction, generated at a distance of 3/4, 3/4, 1/4, and 3/4, 1/4, 1/4, respectively. (c) Corresponding computer generated projections. Note the part of unfolded continuous membrane shown in (b), with a remaining twisted geometry. Figures (a) and (b) are reproduced from [62], with permission. Among the Arthropoda, we have identified cubic membrane structures in m i t o c h o n d r i a [63-66], in spermatids [67, 68] and in mitochondria of spermatids [63]. The latter is an interesting observation, since the occurrence of cubic membranes in the mitochondria of spermatids and of oocytes, as well as in association with the ER and the NE, suggests a close relationship between these organelles during cell differentiation, and perhaps in organelle development. Indeed, mitochondria have occasionally been observed in close relationship with, and continuously linked to, cubic membranes (see, e.g. [61]). Further, as pointed out above, mitochondria in which the cristae form cubic membranes have been seen in continuous association with the NE [22,

63,69].

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Chapter7

Figure. 7.10: The P-PCS membrane system in scaleworm photocytes (a)-(c) and lens (d). (a) A section identified as the [100] projection (mid to left regions of the section). It is best understood as cut between the {100} and the {110} plane, at a distance of about 1/2, 0-1/2, 0, relative to the origin. (b) Complex projection along the [332] and the [553] directions. The latter is generated at a distance of approximately 5/4, 5/4, 1/4. (c) Section cut normal to the {432} plane. (d) The PPCS membrane in the lens of Lagisca extenuata. Oblique section corresponding to the [210], [221], and the [432] projections of the P-PCS. Figures (a)-(c) from [76], and figure (d) from [79], reproduced with permission.

In spermatogenesis several fascinating examples of cubic membranes can be identified. One example is the structure of the "textum" found in grasshopper spermatogenesis [67] which seems to be consistent with a D-

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membrane. However, since the only EM section available to us depicts projections normal to the {111} and {100} planes of the cubic membrane, we cannot unambiguously exclude the P-PCS, which can exhibit similar profiles. Apparently, this cubic membrane is not sensitive to X-ray irradiation, since it appears both in non-irradiated as well as irradiated material [67]. It also appears both in spermatocytes and spermatids, although most frequently in the latter [67]. In most other cases of spermatogenesis that we have analysed, the cubic membrane appears only in late spermatogenesis. Another example of a cubic membrane whose development seems to be associated with spermatogenesis is found in a shrimp [70, 71], in which a "paracrystalline ER" was presented. This we identify as a cubic membrane. Besides cubic membranes in spermatogenesis, many examples of well-developed P-PCS membranes in other germ cells, specially oocytes and ova, have been found. One such case is the "paracrystalline ER" in ova of Pyrrhocoris apterus [72] which is a D-PCS membrane, based on two complex projections along the [772] and the [621] directions (cf. [72] Figs. 2 and 3, respectively). In the phylum Annelida, the class of polychaetae (bristleworms) contains several luminous scaleworms belonging to aphroditidae (polynoids) whose intriguing bioluminescence has attracted much attention. The luminous activity is intracellular and arises from the ventral epithelial cells, modified as photocytes, which in part cover the bioluminescent scales of the scale worm. At the ultrastructural level, the work of Bassot and coworkers, carried out over a period of 30 years, has led to an unusually detailed characterisation of the underlying ultrastructural, as well as physiological mechanisms for the bioluminescence (for reviews see [73, 74]). The ultrastructural source for the luminescent light was first described as microtubular photogenic grains arranged as paracrystalline ER [75] and later given the name "photosomes". These too are cubic membranes. Several details, described by Bassot and coworkers [73, 76], such as the presence of a continuous membrane, and especially its bicontinuity, are consistent with our cubic membrane model. Indeed, similarities between the structure of cubic phases in lipid-water systems (cf. Chapters 4 and 5) and that of the photosome membrane system have been discussed by Bassot and coworkers [77]. These particularly well developed cubic membranes have been exhaustively studied; surpassed only by studies of the PLB cubic membrane. Photosomes are unusual regarding their size and number of cubic membrane assemblies. Two important findings are the intracellular coupling between photosomes, and their progressive recruitment, both mediated by dyadic junctions [76, 78]. Apparently, this coupling is of importance for both the translocation of the membrane-bound nerve stimulus, that underlies the luminescent response, and the fast membrane transport and transformations which follow stimulation. The transformation takes place over surprisingly large distances and occurs in an equally surprisingly short time (seconds). A continuous m e m b r a n e coupling scheme was d e d u c e d in which three morphologies were distinguished: the dyadic junctions, the intermediate ER,

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and the paracrystaMne ER (in the terminology of Bassot and coworkers [73]). How such a continuous membrane arrangement has evolved remains an enigma and a detailed study of the ultrastructural development and ontogeny of a scaleworm has yet to be conducted. Similarly intriguing continuous membrane arrangements have also been observed in other cells, and this remarkable structure is thus not unique to scale worm photocytes.

Figure. 7.10(e): Calculated projections of the P-PCS membrane in various sections.

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Analysis of the structure of the cubic m e m b r a n e describing the "paracrystalline ER" membrane arrangement of the photosomes shows that without exception- it is accurately described by the P-PCS. Fig. 7.10(a) shows a section approximately normal to the {100} plane. It can be s h o w n to correspond to sections obtained between the {100} and the {110} planes. The section seen in Figure 7.10(b) corresponds to a projection between the [322] and the [553] directions generated at a distance of approximately 5/4, 5/4, 3/4 from the origin. The former view can sometimes be very complex, and is similar to the [221] projection of the double diamond structure. Fortunately, the two surfaces can be distinguished by studying sections normal to neighbouring directions and by the absence of "triangular" shapes. Fig. 7.10(c) shows a section approximately normal to the {432} plane. Of great significance is the observation that similar membranous paracrystalline ER are present as a manifestation of folded ER not only in the photoemitters but also in the photoreceptors of certain scaleworms [79-83]. This m e m b r a n e system apparently makes up part of the inner segment, as well as the lens, in the eyes of both nonluminous and luminous Polynods. In view of our study, this paracrystal membranous system is also consistent with a cubic membrane. In particular we can identify a P-PCS membrane in the work of Bassot and Nicholas [79], based on sections approximately normal to composite views of the {210}, the {332} and the {432} planes of a P-PCS (Fig. 7.10(d)). However, even though all the structural details of this particular section can be understood with the P-PCS as a model, we cannot unambiguously exclude polymorphic behaviour, especially between the gyroid and primitive cubic membranes, which is indicated in other sections available to us.

Figure 7.11: see overleaf.

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Figure 7.11: The P-PCS membrane system identified in Spurilla nepolitana. (a) Over~,iew of this cubic membrane system. (b)-(c) Sections approximately viewed along the [100], and the [110] directions. (d) Projection along the [510] direction at a distance of 5/4, 1/4, 0 relative the origin. (e)-(f) Projections approximately along the [553] and the [322] directions, respectively. See text for further details. Figures (a)-(d) are reproduced from [85] with permission.

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Several similar cubic membrane in particular the ER of retinal Further, the "Undulierender spermatids of Eisenia foetida [84]

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systems can be found in photoreceptor cells; pigment epithelial cells, discussed below. TubulikOrper" associated with the ER in are cubic membranes.

Figure 7.11(g): Corresponding computer generated projections according to the assignments in (d)-(f) (see figure 7.3(0 for the [100] and the [110] projections).

In mollusca cubic membranes can be found in spermatids [85-87] and in photoreceptors [88, 89]. The structure of the "TubulikOrper" [87] in early spermatids of pulmonate mollusc is consistent with a P-PCS membrane, although there are some sections that more closely resemble the gyroid morphology. The latter views may be due to polymorphism, but are more likely to result from complex oblique sections rendering the P-PCS structure virtually unrecognisable. (Distortions of the structure due to fixation also cannot be excluded.) Sections approximately normal to the {100}, the {110}, and the {111} planes of the corresponding P-PCS can be identified ([87], Figs. 2 (left and lower right), Fig. 6 and Figs. 3, 4 and 5 respectively). An identical

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membrane structure, termed "undulating arrays" of ER, was identified by Eckelbarger [86], whose semi-serial sections admit detailed structural analysis. Fig. 7.11(a) shows an overview of this P-PCS membrane. Note the proximal Golgi apparatus, and the membranes continuously folding into the cubic membrane. Fig. 7.11(b)-(c) shows sections approximately normal to the {100} Co), and the {110}. In Fig. 7.11(c) a section similar to (a) is shown. Comparison between the two sections suggests that they are slightly oblique to the [100] direction, towards the {hl0} direction. The serial sections seen in Fig. 7.11(d) and (e) are more complex. Nevertheless, the section in (d) is well modelled by the [553] projection at a distance of approximately 5/4, 5/4, 3/4, while the section in (e) is best described as the [322] projection generated at a distance of about 3/4, 1/2, 1/2 from the origin. The structural resolution in these two sections is remarkable in that they share some characteristics of those along the [111] direction though they are still clearly distinguishable (compare the computer generated projections). All sections are thus consistent with projections of the P-PCS. In view of our findings, the "electron-dense flocculum" [86] is an interpretation of a structural feature similar to the "osmiophilic bodies" discussed above [61]; i.e., a structural characteristic of the particular projection of the continuous cubic membrane. In the phylum Chordata we have identified about 100 cubic membranes with representative examples from all classes, including urchordata, in which we identify the so-called "honeycomb lattice" in the test ceils of the Styela [90] as a cubic membrane. This is an intriguing example of a cubic membrane, since it is apparently associated with the Golgi complex. Unfortunately, further structural details are uncertain, although a D-PCS membrane seems most likely. Several other membranes, which we identify as cubic membranes, have been observed in close apposition to the Golgi apparatus, although seldom described as part of it. Specifically, the origin of the photosome cubic membrane in annelids has occasionally been related to that of the Golgi complex (see, e.g. [81]). Some of the most striking examples of cubic membranes are found in tissues of certain fish. The structure, originally referred to as an "interlacing network of tubules", found in glandular cells of the dendritic organ of some marine catfishes [55], is one of the largest and most regular D-PCS membranes that we have identified so far. This m e m b r a n e structure was classified morphogenetically as an invagination of the PM [55] and therefore represents an unusually large PM-associated cubic membrane. This structure is virtually identical to that of the PLB. Accordingly, the structure of the cubic membrane is that of the D-PCS. Figs. 7.12(a) and (b) show rectangular and hexagonal patterns that are consistent with projections normal to the {100}, and the {111} planes of the D-PCS, respectively. The zig-zag pattern also represents a projection normal to the {111} planes, as for the PLB, shown in Fig. 7.1. From a structure-function perspective, it is interesting to note that mitochondriarich accessory cells in the gill epithelium, which are believed to be involved in ion excretion, in both sea- or freshwater-adapted euryhaline fish, exhibit a similar, though not prominent, "tubular network" [91]. Van Lennep and

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Lanzing [55] did relate their findings to salt transport, and recognised structural similarities with avian salt cell glands.

Figure 7.12: A D-PCS cubic membrane in glandular cells of a marine catfish, which exhibit a similar lattice to the PLB. (a) Overview showing the unusually large extension of this cubic membrane. (b) Higher magnification of a detail of (a) which shows, among more complex lattice planes, projections along the [100], the [111] and the [311] directions. See Fig. 7.1 for the corresponding computer generated projections. Reproduced from [55],with permission. The "undulating membrane complexes" composed of SER, observed in the retinal pigment epithelium of the river lamprey [92] is consistent with a gyroid-PCS membrane. It is one of the most convincing and delicate gyroid membranes yet identified. Furthermore, the structure is composed of two bilayers, rather than one [92]. It is thus consistent with a multicontinuous gyroid membrane, dividing space into five, rather than two, continuous spaces. Fig. 7.13(a) shows a section normal to the {211} and the {111} planes of the G-PCS. Fig. 7.13(b) shows a larger magnification, in which the two bilayers are distinguishable. In Fig. 7.13(c) is a simulation of the projection as the plane of section changes from along the [111] direction to that of the [211] direction, corresponding to the change seen in Fig. 7.13(a). This transition is frequently observed to take place over the six-armed structural element normal to the 3-fold axes (and the {211} plane) of the G-PCS.

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Figure 7.13: The multicontinuous G-PCS membrane system identified as a part of SER retinal pigment epithelia cells of a river lamprey. (a) Projection along the [211] (left) and the [111] (upper right) directions. The lower right shows an oblique section which can be understood as cut between the {211}and the {111}planes. Scale bar I I~m. Co)Detail of the gyroid membrane showing the four approximately parallel membranes defining 5 different spaces. Scale bar 0.5 I~m. Figs. (a) and (b) are reproduced from [92], with permission. A structure similar to this gyroid PCS m e m b r a n e w a s reported in the retinal p i g m e n t e p i t h e l i a l cells of Latimeria chalumane [93]. T h i s m e m b r a n e structure w h i c h is c o n t i n u o u s with the SER, and described as "regular arrays of t u b u l e s " [93], is in fact yet a n o t h e r gyroid m e m b r a n e . This s t r u c t u r a l assignment is b a s e d on the recognition of a section n o r m a l the {111} plane of the G-PCS ([93], fig. 4).

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Figure 7.13(c): The corresponding computer generated projections of the transition between the {211} plane and the {111}plane of the gyroid. See Fig. 7.5 for the [211] computer generated projection.

Further examples of gyroid-based structures can be found in other ultrastructural studies of fish. One example is the membrane structure composed of "imbricated cisternae" of SER of the inter-renal cells of normal and dexamethasone-treated Salmo fario [57]. Since the particular sections reported bear strong resemblances to the P-PCS membrane, it is important to point out some of the more delicate differences between the P- and the G-PCS membranes that are easily overlooked. Projections along the [211] direction of the gyroid and the [210] direction of the P-surface can appear similar, especially for thick sections. The section shown in Fig. 7.14(a) of the gyroid membrane of an inter-renal cell shows a view close to the [211] direction of the G-PCS. By comparison, sections normal to the [210] direction of the P-PCS all exhibit a m e d i u m to strong electron density in between each adjacent membrane patch of the high-amplitude sinusoidal pattern (see Fig. 7.5(c) and (e)). Changing direction slightly will cause either an increase or decrease of this electron density, generating the appearance of either an out-of-phase, identical sinusoidal pattern (fully developed in sections normal to the [110] direction), or an in-phase pattern, seen in sections normal to the {211} plane. Both patterns are produced by the "closing" of the high-amplitude sinusoidal motif seen in a view normal to the {210} plane, generating in each the appearance of spherical electron-lucent structures. Sections normal to the {211} plane of the G-PCS change very differently as the direction or distance of sectioning is altered. On the basis of these and arguments mentioned before, sections like those shown in Fig. 7.14(a) and (b) can be identified relatively easily. Several examples of cubic membranes are identified in Amphibia. In particular, the regularly fenestrated mitochondrial cristae described in m a t u r e d Sertoli cells of Xenopus laevis[23] are consistent with a cubic m e m b r a n e . This m i t o c h o n d r i a l cubic m e m b r a n e shares s e v e r a l ultrastructural characteristics with the mitochondria observed in Pelomyxa

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carolinensis [22]. However, instead of being described by a P-PCS, this mitochondrial cubic membrane is a D-PCS membrane. Sections typical for the D-PCS membrane are seen in Fig. 7.15(a)-(d).

Figure 7.14: The SER associated gyroid membrane in adrenocortical cells of Salmo fario. (a) A rather thick section (> 0.5 times the unit cell edge) cut approximately normal to the {211} plane. Note the hatched low amplitude sinusoidal pattern, in between the large amplitude sinusoids. Compare this with Fig. 7.6(c) and its corresponding computer generated projection. (b) A typical oblique section of the gyroid membrane. From [57], reproduced with permission. Cubic membranes have also been found in Aves. The study by Ishikawa [18] on the development of the transverse tubule membrane (T-tubule) system in cultured skeletal muscle cells derived from chick embryo represents one of the more arresting examples of cubic m e m b r a n e s that are p r o d u c e d as invaginations of the PM. Indeed, Ishikawa's model for the T-tubule is in many respects similar to that of cubic membranes. Although there were no comments in the original report [18] regarding the s y m m e t r y of the "tubular network", the model is in m a n y respects similar to the D-surface.

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Figure 7.15" The D-PCS membrane in mitochondria of mature Sertoli stage sustentacular cells of Xenopus laevis. (a) A projection generated along the [111] direction. (b)-(d) Sections with the typical appearance of the D-PCS membrane. Compare Fig. 7.1 and 7.12. Reproduced from [23] with permission.

However, closer analysis has shown that the mature T-tubule membranous complex is consistent with a G-PCS membrane. As will be discussed below, the structure seems to be consistent with a G-membrane in many of the published examples of mature "T-tubular networks" of mammals which can be identified as cubic membranes. The work by Ishikawa [18] allows detailed comparison with the gyroid m e m b r a n e , which makes the structural assignment of the "T-tubule network" convincing. In many respects the structural analysis is similar to that described below of interstitial cells in an antebrachial organ of a lemur [17]. Some of the results of this analysis of the T-tubule associated gyroid membrane [4, 94], are shown in Fig. 7.16.

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Figure 7.16: see overleaf.

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Figure 7.16: The gyroid membrane of the T-tubule system in skeletal muscle cells of chick embryos. (a) Overview showing the relation between the T-tubule system (tt), the sarcoplasmic reticulum (sr), and the myofibrils (mf). Note the continuous invagination of the tt (the plasma membrane) which give rise to the cubic membrane. (b) View of the gyroid cubic membrane projected along the [211] direction at a distance of about 1 / 2 , 1 / 2 , 1 / 4 (upper part), which changes direction towards the [321] direction cut approximately at a distance corresponding to 3/4, 1/2,1/4 (lower part). This corresponds to a change along the x-axis by a distance of only a quarter of a unit cell edge (103/4 nm). (c) Section that can be assigned in part to the [322] projection seen in (g). (d) Ferritin diffusion experiment, which shows the accessibility of particles (approximately 11 nm in diameter) suspended in the culture media to the cubic membrane, indicating its continuity with and origin from the plasma membrane. The projection is along the [544] direction of the gyroid at a distance of 5/4, 5/4, 4/4. (e) Complex projection described by the [953] projection generated at a distance of 9/4, 5/4, 3/4 relative the origin. (f) Section that is described almost perfectly by the [320] projection of the gyroid at a distance of 3/4, 1/2, 0 relative the origin. Figures (a)-(f) are modified from [18], with permission.

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Figure 7.16(g): Computer generated projections corresponding to the assignments in Figs. (b)-(f). It appears that the immature T-tubule membrane exhibits polymorphism. In newly developed invaginations of the PM, it is apparently able to adopt other cubic membrane ultrastructures than the gyroid. The membrane morphology seen in Figs. 7.17(a) and (b) seems to be described by a P-PCS, rather than a GPCS. These particular sections are similar to sections through a P-PCS in which two indices are zero, or close to zero. However, gyroid-like structural elements

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are also present, as seen at the left part of Fig. 7.17(b). Again this indicates a continuous p o l y m o r p h i c m e m b r a n e folding. M e m b r a n e m o r p h o l o g i e s of the T-tubule system, similar to those reported by Ishikawa [18] have been reported by m a n y authors ([95-102]; see also [103] and [104]. Clearly, the formation of this cubic m e m b r a n e is not restricted to any specific m a m m a l i a n species. It seems to occur in a variety of circumstances, t h o u g h most reports deal with m a n i p u l a t e d or pathological tissues. An i m p o r t a n t factor linking the plasma m e m b r a n e associated cubic m e m b r a n e s in skeletal muscle cells seems to be the absence or malfunction of innervation.

Figure 7.17: The cubic membrane system of T-tubules in skeletal muscle cells. (a) Section that cannot be reconciled with a gyroid-based structure as in Fig. 7.16; rather it corresponds to a section along any direction of a P-PCS in which two of the axis are zero, or close to zero. (b) Same structure as in (a) with the possible exception of the left part of the section that appears to correspond to a gyroid structure. Note the "pores" of the plasma membrane (sarcolemma (sl)) from which this cubic membrane originates and their regular size and orthogonal arrangement (see [18] for further micrographs regarding the early stages in the development of this cubic membrane). Note also the adjacent sarcoplasmic reticulum, indicating a regular interpenetration between the (sl) and (sr) (see also Fig. 9 in [18]). A m o n g the n u m e r o u s examples of cubic m e m b r a n e s in a p p a r e n t l y n o r m a l tissue of m a m m a l s , perhaps the most appealing is that of the extraordinarily large mitochondria in photoreceptor cells of Tupaia glis [24]. It was originally reported as "patterns of concentric cristae arranged in highly ordered whorls of lamellar configurations" [24]. This m e m b r a n e configuration is in fact a gyroid m e m b r a n e [4, 94]. Fig. 7.18(a) shows an overview of the photoreceptor outer segment, in which the regular array of cristae of the m i t o c h o n d r i a is identified as a section cut close to the normal of the {100} plane of the gyroidb a s e d s t r u c t u r e . This section is easily r e c o g n i s e d , since it contains a

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c h a r a c t e r i s t i c n e t w o r k of s t r a i g h t lines p e r p e n d i c u l a r clearly s e e n in Fig. 7.18(b).

to i n - p h a s e s i n u s o i d s ,

Figure 7.18: The remarkable gyroid membrane in mitochondria in photoreceptors (cones) of the tree shrew. (a) Overview showing the relation between the rod outer segment of the cone (Os) and the adjacent giant mitochondria 0VI) whose cristae are folded on a gyroid cubic membrane. (b) Magnification of the section of the cubic membrane in (a). It can be identified as a section along the [100] direction. Note the multi-bilayer arrangement, in which at least four membranes are identified; i.e., five continuous spaces. (c)-(d) Sections that are identified as the [111] and the [211] projections, respectively. Figures (a)-(d) are modified from [24], with permission.

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Figure 7.18(e): Computer generated projections corresponding to the assignments in (b)-(d).

Other sections are consistent with images from a gyroid PCS cut approximately normal to the {111} and to the {211} planes (shown in Figs. 7.18(c) and (d), respectively). This is the only example of a gyroid membrane we have identified in conjunction with mitochondria. It is multicontinuous, (four, possibly five bilayers and thus separating up to six distinct spaces), seen in Fig. 7.18(b) and (c). In addition to these remarkable features, this gyroid cubic membrane is one of the largest we have identified; its lattice parameter is about 1200 nm! A number of reports can be found in the literature dealing with certain muscle cells that often have mitochondria exhibiting regularly arranged cristae resembling gyroid-based multicontinuous profiles, though seldom with such a regularity as in Tupaia glis (see, e.g. [66, 105]). Epithelia are one of the most commonly encountered tissue-types in which cubic membranes can be identified, with well over 100 examples, documented in part earlier in this chapter. A particularly interesting case, since it is a sensory epithelium, is the vomeronasal epithelium in certain mammals. In cats, SER-associated "hexagonal crystal-like membrane differentiations" have been described [106-108]. This membrane configuration is well modelled by a gyroid membrane, shown in Fig. 7.19. Fig. 7.19(c) shows a section similar to

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t h a t s e e n in Fig. 7.6(c). It c a n b e u n d e r s t o o d as a p r o j e c t i o n of a r e l a t i v e l y thick section of the g y r o i d - P C S m e m b r a n e a l o n g the [211] direction.

Figure 7.19: The gyroid membrane in the vomeronasal epithelia of the cat. (a) Overview of the cubic membrane showing its close relation to the ONE. (b)-(c) Magnifications of the right, and left sides, respectively, in (a). The views can be understood as the [110] projection at a distance of approximately 3/4, 3/4, 0, and the [211] projection, and correspond to a section thickness of about one unit cell edge (150-160 nm). Figures (a)-(c) are reproduced from [108], with permission.

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Figure 7.19(d): Computer generated projections corresponding to the assignments in (b) and (c).

It is natural to investigate cells that exhibit prominent SER membranes, such as steroid-secreting cells, in which the SER is often described as a closemeshed network of branching and anastomosing tubules. It appears that many of these SER are at least in part more correctly described as cubic membranes, though they are rarely preserved in a manner that allows for any elaborate structural assignments. One steroidogenic cell, in which a gyroid cubic membrane can be envisioned, is that of the antebrachial organ (a scent marking cutaneous gland) of Lemur catta, in which an SER associated membrane structure has been described as a "rigidly patterned crystalloid composed of stacked layers of parallel interconnected tubules" [17]. From a series of sections, the authors constructed a three-dimensional model. Although the details are incorrect, in that their model has a connectivity of four tunnels meeting at each node, rather than three expected for the gyroid, their model is nonetheless bicontinuous. Reanalysis [4, 5, 56] shown in Fig. 7.20(a)-(e), reveals that this is a gyroid membrane, of similar appearance to those shown in Figs. 7.16 and 7.19. Other than the cases of well-developed cubic membranes in steroid secreting cells, there are several such cells that exhibit apparently hexagonal membranes (see, e.g. [109, 110]); i.e., bilayerbased tubular extensions. These can be efficiently formed by folding of any of the three principal cubic membranes, along the [111] directions. Hence, the appearance of a hexagonal membrane can often be taken as an indication of the existence of a cubic membrane. Indeed, as analysed below, such a continuous membrane transformation has been reported, t h o u g h not appreciated as such [111]. Most examples of cubic membranes in pathological tissue appear in neoplastic and infected cells (Table 7.2). In view of our results, and of the apparent ubiquity and widespread appearance of cubic membranes, a correlation between certain pathological conditions and the expression of cubic membranes deserves serious consideration. It is our opinion that the

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qualitative a p p e a r a n c e of a cubic m e m b r a n e c a n n o t be taken as an indication of a pathological condition. Rather, the state of disease alters the p a r t i c u l a r cell differentiation, a n d therefore the time of the a p p e a r a n c e a n d a m o u n t of specific cubic m e m b r a n e s . It seems, for instance, that the c o m m o n l y s t u d i e d ER of t u m o u r a n d infected cells are m o r e frequently o b s e r v e d to exhibit cubic m e m b r a n e s t h a n o t h e r p a t h o l o g i c a l cells. Of c o u r s e , it c o u l d be s i m p l y b e c a u s e the r e s e a r c h c o m m u n i t y at large has a g r e a t e r i n t e r e s t in f i n d i n g u l t r a s t r u c t u r a l m a r k e r s of cancer cells, as well as infections, t h a n for m a n y other p a t h o g e n i c instances, so biasing the statistics.

Fig. 7.20: The gyroid cubic membrane system in a steroidogenic cell of Lemur catta. (a) Overview which is close to the [211] projection. (b) Section viewed close to the [110] projection. (c) Section that can be understood as the [331] (upper left to the middle region, relative to the straight line pattern making approximately 45" with the edges of the micrograph) and the [330]] (in the middle of the section) projections, at a distance of approximately 3/4, 3/4, 1/4 and 3/4, 3/4, 0, respectively. The lower right part can be understood as cut in between these directions. (d) High magnification of a similar section as in (c). The continuous path seen just to the right of the centre corresponds to the [320] direction viewed at a distance of approximately 3/4,1/2, 0. Figure (a)-(d) are reproduced from [17] with permission.

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Fig. 7.20(e): Corresponding computer simulations to the assignments in figs. (c) and (d) (the computer generated [110] and [211] projections are seen in figs. 5.19d and 5.5c, respectively).

A typical membrane configuration, identified as a P-PCS membrane, has been described in a particular pathogenic instance, classified as a virus-induced tumour [111]. The structural analysis is shown in Fig. 7.21. More interesting membrane configurations, that we identify as cubic, are the intranuclear "undulating m e m b r a n o u s structures", originally described in cells of a human parosteal osteosarcoma [112]. This membrane is readily identified as a P-PCS membrane (Fig. 7.22). Though direct continuity between the cubic membrane and the INE was not described[113], it seems most likely to be formed from invaginations of the INE, by analogy with the cubic membranes continuously folded from the ONE (see, e.g. [113] and [110]). One quite extraordinary example of the latter has been described in UT-1 cells [110]. These cells are a compactin-resistant line derived from Chinese hamster ovary ceils (Fig. 7.23). Fig. 7.23(c) shows the relationship between the different membrane morphologies and the ONE from which the ER emerges. Since the electron density distributions in the different spaces separated by the membranes are similar across the very different membrane topologies, there is little doubt that shape transformations of the membrane occur through intersection-free m e m b r a n e folding. The work by G.W. A n d e r s o n and

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colleagues [110], revealing that compactin competitively inhibits 3-hydroxy-3methylglutaryl coenzyme A (HMG CoA), which is the rate limiting enzyme in cholesterol synthesis, allows us to conjecture a new interpretation of m e m b r a n e folding. UT-1 cells g r o w n in the presence of compactin are deprived of cholesterol and show a build-up of HMG CoA [110]. Since both the hexagonal and cubic membranes are present only in those UT-1 cells grown in the presence of compactin, this indicates the importance of either cholesterol deprivation, or HMG CoA build-up, or a combination of the two, in the formation of these membrane configurations. Cholesterol deprivation is expected to increase the membrane fluidity (and decrease its bending m o d u l u s , discussed in Chapter 4), which would allow the formation of structures that exhibit large variations of curvature. On the other hand, it is generally thought that an increase in membrane rigidity is a prerequisite for any regular structure to exhibit large unit cells. The build-up of HMG CoA could be responsible for the increased m e m b r a n e rigidity. A s o m e w h a t different theory conjectured by one of us (T.L.) is presented in Section 7.8.

Figure 7.21: Sectionsof the P-PCS membrane identified in a virus-induced tumour cell. (a) Projection along the [510] direction at a distance of about 5/4, 1/4, 0 (compare Fig. 7.11(d)). (b) and (c) Sections cut normal to the {110}and the {310}planes, respectively. Note that the latter pattern can be interpreted only by assuming a section thickness of 1/4-1/2 of a unit cell edge. (d) A projection along the [331] direction at a distance of about 3/4, 3/4,1/4. Figures (a)-(d) modified from [111], with permission.

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Figure 7.21(e): Computer generated projections corresponding to the assigmnents made in (b)-(d) (see Fig. 7.11(g) for the theoretical projection corresponding to (a)).

Figure "~.__:'~') The primitive intranuclear cubic m e m b r a n e in cells of a h u m a n bone t u m o u r . (a)-(b) Sections that can be identified w i t h a P-PCS m e m b r a n e . Modified from [112], w i t h p e r m i s s i o n .

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Figure 7.22 (c-e): The primitive intranuclear cubic membrane in cells of a human bone tumour.

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Figure 7.23: The cubic membrane system in UT-1 cells. (a) Overview showing the relation between the P-PCS and the fiat lamellae-like membrane arrangement. (b) Higher magnification of a cubic m e m b r a n e that is identified as P-PCS. Note the interesting electron dense line-segment spanning over the small subvolume. It seems as if it is membrane-bound, in which case it is possibly a protein that interpenetrates the cubic membrane in similar fashion as in Fig. 7.3. (c) Section that shows at least three regular membrane arrangements; hexagonal (H), cubic (C), and fiat lameUar (L). The hexagonal membrane arrangement is consistent with bilayer "tubes" closed-packed in a hexagonal fashion (similar to that discussed in [109]), and corresponds to the complex reversed hexagonal phase postulated in condensed systems. Note that the cubic membrane seen in this section may correspond to a G-PCS, rather than a P-PCS, as in (a) and (b). From [110], reproduced with permission.

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7.8 Relationships between tubuloreticular structures, annulate lamellae, and cubic membranes

Annulate lamellae (AL) (Fig. 7.24) are frequently observed in mainly differentiating gametes and tumour cells. AL's occur in several of the examples listed in Table 7.1 and 7.2 in close association with the occurrence and the formation of cubic membrane structures [114-117] (see also [119] for a similar structure to that described in [115]). The structure, origin, and function of AL's are mysterious. Regarding their structure, many different hypotheses have been suggested, but none accounts for all the features of AL's. Tangential sections of AL's most often exhibit a hexagonal arrangement (see, e.g. [119]), while perpendicular sections (Fig. 7.24(a)) do not reveal any obvious symmetrical arrangement, even though they always exhibit an astonishingly regular lamellar organisation, indicating an underlying periodic structure. There are, however, structural variants, such as those seen in Fig. 7.23(c), which are sometimes classified as AL's [115], but also as novel features of the ER [118]. Based on the apparent morphological similarities between the AL and the NE, it has been suggested that AL's represent a cytoplasmic NE working as a reservoir of both ER membrane and nuclear pores (see [119] and [120] for reviews). In favour of such speculations is the fact that AL's have been observed in direct continuity with the ONE. However, in a comparative study of antigenic epitopes of AL's and the NE, the antibodies directed against the NE laminae did not cross-react with constituents of the AL membrane assembly [121]. This suggests that the AL is an antigenically distinct membrane system. Other results indicate the opposite (see, e.g. [120]). In addition, other scientists have observed a direct as well as indirect continuity, via the RER, between AL's and the Golgi complex, suggesting that AL's may form from the Golgi complex, in addition to the ER [122]. The apparent complexity associated with the origin of AL's thus presents an analogy with cubic membranes. In view of our analysis, we suggest that AL's offer a further ultrastructural type available to cytomembranes in general. The AL structure is probably related to that of IPMS (analogous to cubic membranes, although not necessarily with the same symmetry). Tubuloreticular structures (TRS) are the second major membrane assembly that has been frequently associated with the occurrence, as well as the origin, of cubic membranes. Indeed membrane assemblies that are now characterised as cubic membranes, have often been described as being similar to TRS. More specifically, undulating membranous structures (UMS) have often been interpreted as TRS-like structures. This is despite the fact that TRS are irregular membranous arrangements with no obvious symmetry, having a typical appearance shown in Fig. 7.25. Unfortunately, we are not aware of any study of TRS in which serial sections have been prepared. TRS have attracted interest due to their potential use as an ultrastructural marker in pathology (for reviews see e.g. [104] and [125]). Accordingly, TRS

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Figure 7.24: Micrographs showing typical (a)-(b) and atypical (c) appearance of annulate lameUae. (a) Overview showing an AL in Xenopus laevis. (b) Detail of AL in (a). (c) A rather unusual regular membrane arrangement of what can be classified as an AL. Figures (a) and (b) are reproduced from [123],and (c) from [118] with permission.

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occur in virus-infected and cancer cells, and the structure of TRS has often been interpreted as an indication of viral infection. The current view is that it is a conformation of the ER, or the Golgi complex, which is not necessarily associated with the state of virus infection.

Figure 7.25: Micrograph showing a typical appearance of a regular tubuloreticular structure (TRS) in a tumour cell of rabbit myxosarcoma. Reproduced from [124] with permission.

More recently, TRS have attracted interest due to their presence in cells infected with the H W virus. Many of the examples listed in Table 7.2 have indeed been interpreted as TRS, although exhibiting a distinct cubic symmetry. As with AL's, the occurrence of TRS is often correlated with that of membrane structures we now know to be cubic membranes. Indeed, direct continuity has been observed by s4veral authors ([104], p.96, plate 48, Fig. 2; [103], p. 29, Figs. 1.11 and 12). This indicates yet another membrane morphology, which may exhibit translational periodicities. Even though we have been unable to identify such features in TRS to date, it can be argued that they are isotropic, and then possibly cubic. If this turns out to be the case, it is conceivable that they exhibit a higher connectivity than the simpler examples of cubic hyperbolic surfaces considered in this chapter. Not only would this render the translational symmetry difficult to comprehend, but it also presents analogies with the progression of cubic phases in simpler equilibrium condensed systems. Some TRS membrane structures exhibiting such features could then possibly correspond to IPMS other than the G-, D-, and P-PCS-based cubic membranes; e.g., they could they be based on the IWP, F-RD, Neovius (C(P)) a n d / o r the C(D) surfaces (cf. Chapter 1). However, this remains to be established.

Biogenesis of cubic membranes

7.9

317

Biogenesis of cubic membranes

This exhaustive study of cubic membranes reveals that virtually all membranes are capable of forming cubic membranes. That is not surprising to us, given the progression of such phases in equilibrium amphiphile systems described in Chapters 4 and 5. Their origin thus seems to be strongly coupled to the mechanisms of membrane biogenesis in general. From this point of view, cubic membranes are just another configuration of cytomembranes. But there must be much more involved. The formation and growth of cubic membranes may be selected to fulfil a purpose under the influence of self-organisation under changing (nonequilibrium) conditions. To verify this we ultimately have to understand the underlying rule(s) for selection between the different growth morphologies. Hence, it is of importance to establish some empirical relationship between different mechanisms of cubic membrane biogenesis, which allows elaboration of structure-functional relationships. The thesis that we shall develop introduces the concept of optimisation of cell space and membrane organisation under the influence of a selective cytomembrane topological organisation. From these concepts, a hypothesis can be advanced regarding topologically constrained membrane morphologies, which parallels our understanding of condensed lipid polymorphism in equilibrium systems. Cytomembrane morphologies are reversible structural features that depend on changing physico-chemical conditions. But, the connection between the formation of these structures (of periodic symmetry) and function is far from clear. Perhaps the most perplexing matter one has to resolve is that we generally require near-equilibrium conditions to grow a beautiful crystal. This is especially so for cubic phases in lipid-water systems. On the other hand, crystals of proteins have long been known to exist in a manifold of cells, and more recently in composite crystals of fluid membranes. Clearly, the one invariant variable is the symmetry. It thus appears that symmetry is demanded by spatio-temporal requirements of the cell. If the growth of a certain periodic membrane is considered to be caused by asymmetrical fluctuations that lead to self-organisation by some unknown parameters; how then is a particular symmetry selected, and what is the selection procedure between the different cubic membrane structures? From a topological point of view, cubic membranes are in general formed from a structural "template" (the precursor to the cubic membrane), such as the invaginations or "pores" of the PM membrane. This can lead to intersection-free invaginations or out-vaginations of the membrane (relative to the cell interior). Two exceptions seem to be out-vaginations of the PM and of the outer membrane of mitochondria. (Though we have not observed direct invaginations of the latter there are indications of such.) During the folding, the membrane must necessarily be a contiguous fluid membrane. Thus the final global topology of the cubic membrane depends on that of its precursor (cf. Fig. 7.26). If one pore of the invaginated membrane triggers the folding process, the topology must be that of a sphere. If more than one

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invagination is allowed, as indicated in Fig. 7.17, it may require points of fusion, to achieve a connected membrane within the cell. An intriguing observation is that of the periodicity of the template (Fig. 7.17) which seems to be symmetrically arranged about the normal to the PM surface. Perhaps it is this symmetry that is the driving force for fusion, since, if fusion did not occur several independent cubic membrane systems would form, which would not necessarily bear any spatio-temporal memory with respect to each other, or the template. Since there are very few systems in which we can envision the template for a cubic membrane, and thus in which membrane biogenesis can be directly observed, it is not possible at present to construct a general theory of membrane folding a n d / o r fusion that can lead to a cubic membrane. Nevertheless, we can consider some special cases. We confine our discussion to membrane systems in which the folding itself is intersection-free (i.e. ignore fusion). In that case why is a symmetric element formed in the first place, and not simply a random membrane configuration? A typical example is the red blood cell, whose PM invaginates upon certain perturbations [126] with the adoption of symmetrically arranged membrane profiles. Further, polymeric prote'ms and proteoglycans apparently can impose organisation on membranes. Some examples are the apical surface of endothelial cells, and the nuclear envelope. While such protein interactions are not necessary for the formation of cubic membranes, as is most evident in their formation in mitochondria, they can certainly have a bearing on the final symmetry. An example might be the regular network of proteinaceous material in the cubic membrane of Zygnema, analysed above. Though the relative importance of the arrangement of lipids versus the protein network is unknown, it cannot be ruled out that the membrane arrangement is influenced by the symmetry of the network, or vice versa. It is conceivable that similar interpenetrating networks can be established between, e.g. ER associated cubic membranes, and the cytoskeletal network. Indeed, it is interesting that the latter is often seen to adopt periodic structures. Perhaps the most challenging question concerning cubic membranes, or for that matter, any periodic membrane arrangement, is the nature of the rules that govern the selection of a particular structure under the influence of a set of physical and chemical constraints. The necessity for such rules should be clear, since there apparently exist structural invariants, independent of evolution. An example is the D-PCS membrane of the PLB. There are very good physical and chemical reasons for the choice of this particular geometry. As a first approximation we assume a topologically invariant structural growth caused by an asymmetrical perturbation. Two parameters are obvious, the connectivity and the symmetry. Since the perturbations are asymmetrical, we may assume that one subvolume (the cell interior) responds to the perturbation induced in the other subvolume in such a way that the induced strain is minimised. Thus a large scale communication, based solely on the geometry of the perturbation (curvature), is established. Furthermore, a cubic membrane can apparently form either by orthogonal invaginations, producing, e.g. a P-PCS membrane growing along the [100] directions, or as

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hexagonal vaginations, which fold into, e.g. a G-PCS membrane, growing along the [111] direction, etc. Similarly, a cubic membrane can grow from a pre-existing entity with different symmetry. For example, a D-PCS structure can grow without intersections from a gyroid membrane. Such intersectionfree cubic membrane transformations must then be necessarily viewed as growth (or reductive) transitions. Recall that the global topology is kept constant during this particular transition although the local connectivity changes from three (G-PCS) to four (D-PCS). Fig. 7.26 presents a scheme for the formation of cubic membranes on this basis. A gyroid membrane, formed by m e m b r a n e folding (invagination) of the plasma m e m b r a n e , which transforms into a D-PCS membrane, is shown schematically in Fig. 7.26(c). The origin of the change of curvature as a fiat membrane is folded into a cubic configuration is virtually unknown. Several plausible mechanisms for generating the balanced saddle shape of the bilayer have been pointed out by Bouligand [15]. The presence of proteins in the bilayer causes an increase in membrane rigidity, which in principal would allow for the large unit cell sizes observed, as compared to pure lipid-water systems (Chapter 5). On the other hand, it has been argued in Chapter 4 that the periodic hyperbolic structures minimise curvature variations and are the most homogeneous saddle shapes. These are obvious parallels between biomembrane and lipidwater mesostructures. Nevertheless, the presence of proteins in the membrane complicates matters. Further, the inherent bilayer a s y m m e t r y present in all cytomembranes is likely to change the volume relations between the subspaces, while its effects on curvature will depend on the specific distribution of membrane components along, as well as within, the bilayer. Note too that the fixation procedure of EM specimens may affect these volume relations. These effects are likely to be particularly marked in fixing of cubic membranes, since a change in the volume relations may lead to unfolding of the cubic membrane. A typical micrograph where such an effect is conceivable is shown in Fig. 7.27. We cannot, at this point, distinguish between this artefact and other apparent m e m b r a n e morphologies that exhibit either more complex topologies or different symmetries to the three cubic membranes focused on herein, based on the P-, D- and gyroid surfaces. The need for a concept of cell topology and the organisation of intracellular space is evident, given the ubiquity of cubic membranes. Some speculations follow. Consider the topology, as a consequence of the biogenesis, of multicontinuous cubic m e m b r a n e s associated with mitochondria. The example identified in Tupaia glis, analysed above (Fig. 7.18), may be regarded as a multicontinuous cubic membrane, composed of n multiple bilayers, separating (n+l) spaces. From the classical perspective of a mitochondrion with a continuous crista, the multicontinuous cubic membrane can separate only two spaces, since it must be formed as an intersection-free folding with multiple self-interpenetrations. On the other hand, if we allow more than one crista or intersections, more than two spaces can form. In addition, there are some indications of more complex topologies, in which the outer mitochondrial membrane periodically invaginates in phase with the cristae, providing an immense contact area between the outer mitochondrial

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Figure 7.26: Schematic cartoon of intersection-free membrane folding of PM. (a) A template in the form of a pore (note that the pore neck can be arbitrarily long and thus penetrate into the cell) which grows into a triad element of a gyroid (left) which continuously grows into larger assemblies (right) until it extends to a recognisable gyroid cubic membrane schematically shown in (b) with two pores as template. (c) The gyroid-to-D-PCS folding pattern.

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membrane and the cristae. However, the existence of direct continuities, "pores", of the outer membrane of mitochondria remains to be shown. Currently, we cannot rigorously distinguish between these alternative scenarios. The situation is equally complex when other multicontinuous cubic membranes are analysed. The primary problem that remains to be resolved is the biogenetic origin of the membrane(s) in each particular case of cubic membranes. In other words, we cannot always assume that the membranes constituting a cubic membrane system originate from the same organelle. Specifically, in multicontinuous cubic membranes, a putative hypothesis is that each continuous bilayer originates from a particular membrane organelle. For example, such a membrane arrangement can easily be imagined if for example the PM and the ER are intermingled. This hypothesis is supported by, the observation of the intertwining of the invaginated cubic plasmalemma and the sarcoplasmic reticulum in skeletal muscle cells [18]. There are many multicontinuous cubic membranes that clearly do separate different material through the partitioning of space. Perhaps the best example is that of Zygnema analysed in Fig. 7.3 in which several different (and normally independent) bilayers and a network of proteinaceous material interpenetrate.

Figure 7.27: A cubic membrane that s e e m s to unfold asymmetrically.Note such effects can in principal be introduced as an artefact d u e to fixation procedures, and the (asymmetric) osmotic pressure present under these circumstances. From [104],with permission.

7.10 Relationships between cubic membranes and cubic phases Lipid-water or surfactant-water bicontinuous cubic phases of the reversed type, treated in Chapters 4 and 5, consist of hyperbolically curved bilayers,

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where each monolayer is draped over an IPMS in such a way that the surface at the centre of the bilayer is described by an IPMS, and the apolar/polar interfaces are two identical PCS'. Thus with respect to bilayer configuration, the geometries of cubic phases are similar to those of the cubic membranes treated here. There are considerable differences between the formation of cubic phases and cubic membranes. Firstly, if cubic membranes were cubic phases, why do they not translocate throughout space? A possible explanation is that cubic membranes are formed at conditions corresponding to a highly regulated multiphase "equilibrium" process. This would be supported by the fact that they are usually closely formed in direct contact with other membrane configurations. However, as previously discussed, the composition of a cubic membrane, which necessarily must be treated as a time-averaged property, is likely to affect the constituent monolayers of the membrane asymmetrically, rather than symmetrically. A change induced by, for example, an increase in the concentration of calcium ions along the outer surface of the PM, or alteration of the physical properties of an asymmetrically distributed protein, can lead to a new coupling over the bilayer, as well as along the outer bilayer surface, thereby inducing a change of membrane curvature. One possible role of the PM glycocalyx could thus be to regulate the membrane morphology, a function that is consistent with its established role as an information-carrier in cell-cell communication, etc. Similar mechanisms can be put forward for plasma membrane bound receptors, signal transmissions, secondary messengers, etc. Besides the asymmetry between monolayers in cytomembranes, two of the more obvious differences between cubic phases and membranes are the unit cell size and the water activity. It has been argued that the latter must control the topology of the cubic membranes [15], and hence that the cubic membrane structures must be of the reversed type (in the accepted nomenclature of equilibrium phase behaviour discussed in Chapters 4 and 5: type II) rather than normal (type I). All known lipid-water and lipid-protein-water systems that exhibit phases in equilibrium with excess water are of the reversed type. Thus, water activity alone cannot determine the topology of cubic membranes. Cubic phases have recently been observed with very high water activity (75-90 wt.%), in mixtures of lipids [127], in lipid-protein systems [56], in lipid-poloxamer systems [128], and in lipid A and similar lipopolysaccharides [129, 130]. Most cubic phases in lipid-water systems exhibit unit cell parameters not larger than 20 nm, while the unit cell of cubic membranes is usually larger than 100 nm. Some exceptions have been apparently found [131, 132]; although at this stage such findings should be treated with caution, as the determination of lattice parameters is dependent on the indexing of diffraction patterns, based only on a small number of reflections. Further, in lipid-protein-water, lipid-poloxamer-water and lipid-cationic surfactant-water systems, cubic phases with cell parameters of the order of 50 nm have been observed [56, 127, 128]. Due to the small number of reports dealing with the

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existence of cubic phases under dilute conditions, it is premature to draw any conclusions regarding apparent difference between the lattice parameters of cubic phases and membrane periodicities. Furthermore, the nonequilibrium conditions under which cubic membranes are formed are likely to offer a range of different, unknown physical mechanisms for the control of unit cell size compared to condensed systems. (This view is supported by the existence of a labile cubic phase in a lecithin-water system [133]. It is tempting to speculate that this might represent an example of a cubic membrane, rather than an equilibrium phase.) It is now well established that proteins can induce phase transitions in lipid membranes, resulting in new structures not found in pure lipid-water systems (cf. section 5.1). However, this property is not peculiar to proteins; the same effect can be induced by virtually any amphiphilic molecule. Depending on the structure and nature of proteins, their interactions with lipid bilayers can be manifested in very different ways. We may further assume that the role of proteins in the biogenesis of cubic membranes is analogous to that in condensed systems, and lipids are necessary for the formation of a cubic membrane. This assumption is supported by studies of membrane oxidation, which induce a structure-less proteinaceous mass [113]. However, the existence of a lipid bilayer by itself does not guarantee the formation of a cubic membrane, as proteins may also play an essential role in setting the membrane curvature. In this context, note that the presence of chiral components (e.g. proteins) may induce saddle-shaped structures characteristic of cubic membranes. (This feature of chiral packings has been discussed briefly in section 4.14)

7.11 Functionalities of cubic membranes What are the functions, if any, of the cubic membranes? It may be that cubic membranes are but an inevitable self-assembled product of the complex molecular soup of lipids and proteins; the result of molecular packing considerations and inter-molecular interactions. This would be in analogy with known phase behaviour in equilibrium systems. Even though this is a very appealing solution to the long and unresolved debate about "nonlamellar" lipids in conjunction with cell membranes, we rather believe that these structural organisations have been chosen to fulfil a purpose (see, e.g. [134] and references therein for current theories, and [4] for a more comprehensive discussion), and the formation cannot be rationalised solely by molecular packing. Following recognition of the apparent ubiquity of cubic membranes and their biogenesis, many structure-function relationships can be discerned. In the following sections we speculate and elaborate on some of these. In particular, a hypothesis regarding cell topology [4, 55] is briefly introduced, driven by the recognition that periodic hyperbolic surfaces are in many senses efficient space partitioners. We stress that these hypotheses have been

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developed in an attempt to reconcile a variety of morphological observations and to motivate further experimental work in this area. They do not necessarily reflect or take into consideration current knowledge on cell function (see, e.g. [135]), which ultimately must be done.

7.12

Cell space organisation and

topology

Continuous hyperbolic forms - as in cubic membranes - necessarily partition space into discrete, and unconnected, subvolumes. At least two discrete spaces are to be found in a cubic membrane. The identity of each space itself is not a consequence of the existence of a cubic membrane. Rather, it is an effect of the topologically invariant membrane morphogenesis. Although an intracellular space maintains its identity as long as membrane continuity is preserved, it is only by means of a hyperbolic membrane structure that such spatial relations can be rigorously studied. Thus, solely due to the existence of cubic membranes in cytomembranes, we are bound to acknowledge that bilayer m e m b r a n e s - which ultimately emerge from the nuclear envelope can partition intracellular space. Whether it takes place because of a continuous partitioning is an open question. But the discovery of cubic membranes strongly suggests that continuous intracellular membranes are much more frequently adopted than is normally suggested. -

Cell space organisation and communication is an active area of research, yet there exists no homogenous model. Specifically, there seems to be no suggestion that the intracellular space can be optimised through the use of membranes as continuous space partitioners, even though this issue has already been recognised implicitly, inasmuch as one accepted function of a membrane is to isolate and contain organelles. How can intracellular space be organised, taking the observation of cubic m e m b r a n e s into account, particularly from a global perspective? Classical ideas, recycled in most cell biology texts, remain respectable. But closer examination of the problem of organising an immense number of different molecules, and an equally baffling array of spatio-temporarily distinct chemical reactions, reveals a need for a higher level of organisation than containment alone. The concept of cell-space organisation, which follows from a multicontinuous partitioning of organelles due to hyperbolic membranes, is analogous to organisation of a city. Many different locations (homes, work, stores, etc.) are essential for efficient infrastructure. When driving to work, it is wise enough to follow the usual route, despite the fact that this may not offer the shortest path. Certainly we do not drive in random directions (at least not without pain of arrest), such as those adopted by a diffusion process, even if that could in principle lead to some organisation. Recognition is essential, both to avoid collisions, and to keep a time-schedule! Similarly, in the three dimensions within a cell, the partitioning of the intraceUular space into sub-volumes by means of membranes - leading to an interpenetrated pipeline system - could in principle be orchestrated by the genetic code, which

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ultimately controls spatio-temporal activities, and thus the biogenesis of membranes. Such a solution is obviously beneficial to the organisation of intracellular space, and in the intracellular transport and communication network. A further analogy would be to describe the cytoskeleton network as a dynamic telephone network. A conceivable conjecture would then be to interpenetrate the cytoskeleton with the membrane pipeline system, giving rise to a highly efficient communication and transport network. The telephone network could have regular nodes where it was anchored to the membrane (formed by the cytoskeleton with, e.g. the plasma membrane) which would allow for intra-spatial telephone calls. An interesting solution to the sorting that has to take place in a certain space would then be by the recognition of symmetry and curvature, as long- and short-range sorting mechanisms respectively. Fast a n d / o r important messages would be sent via the protein network, while mass cooperative transport as well as synthesis, storage, etc., would be spatially located by the membranes outlining the particular space in which the activity takes place. Clearly, the classical idea of having spatially separate domains (organelles) floating around without any topological relation to their surroundings would cause problems for the Centre of the Communications Department (the gene code). That Centre would have trouble maintaining and updating its inventory (e.g. the number of mitochondria and their locations). But also it would represent an inefficient system in that each domain is unrelated to each other. Some organelles are conceivably still isolated, such as lysozymes, mitochondria, etc. Consider, for example, ATP produced in a particular mitochondrion for use in a certain spatio-temporaUy controlled activity in, say, the Golgi. If in a classical "random" cell model we put in an order for ATP, this would not only require an immense number of questions and messages about relative orientation between the Golgi and the particular mitochondrion, but it would also take very long time. Imagine now a mitochondrion located in a certain recognised space and connected to the telephone network, and Golgi, also connected, but not necessarily on the same line, meaning that the Golgi space does not contain a mitochondrion, and that ATP must be imported. At least two scenarios can now be t h o u g h t of; a direct connection to the mitochondrion, or a connection via the centriole (or the nuclear envelope) through which the signal is sent to reach the mitochondrion. This kind of solution would not only lead to efficient communication, but also to equally efficient transport, which is now restricted in space - if necessary guided over the NE, for relocation, or directly transported via some particular transmembrane mechanism. This kind of cell model d e m a n d s sophisticated control and regulation of its topology, a need particularly easy to imagine in cells that are known to change shapes frequently (such as nerve cells), since it requires an extensive reconstruction of the intracellular networks, while maintaining its intertwined topology. If this kind of hypothetical cell membrane model was to optimise its interaction between the membranebounded spaces and the network of cytoskeleton, the most efficient solution is to organise it according to an interpenetrating lattice model, such as that in a multicontinuous cubic membrane with at least one space in which the telephone network is located, (indicated in, e.g. Fig. 7.3). We point out that

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this model of spatio-temporally controlled intraceUular space can very well accommodate specific nonequilibrium self-organisational efforts. Recognition of the existence of topologically distinct spaces through the formation of cubic membranes implies that cell space is restricted and predetermined, in the fashion outlined above. This hypothesis leads to many consequences. For example, an alternative mechanism for intracellular transport and organisation has been suggested [56] which does not depend upon the formation of vesicles (outlined in Chapter 3), though that is not excluded. Very briefly, the existence of multiple hyperbolic membranes implies the existence of discrete intracellular spaces (bearing, for convenience different "colours" depending solely upon their origin). Thus, in the case of a bicontinuous symmetric membrane (whose mean curvature, measured at the mid-surface of the membrane, vanishes), the "black" space can be viewed as a black copy of its the congruent "white" space, on the other side of the bilayer. Since it is generally agreed that the ER is continuous with the ONE we may assume under some topological conditions that ribosomes are distributed in both spaces, differing only with respect to location, i.e. colour. During translocation of proteins, ribosomes are thought to be membrane bound located in the black space, say, but the newly synthesised protein is thought to cross the membrane, and thus must necessarily be designated as white, belonging to the white space. Similarly, ribosomes designated as white transcribe proteins that are necessarily black. Thus intracellular space and its topology can in principle be physically predetermined. If this indeed is the case, targeting and secretion of proteins could very well be predetermined, depending only on the colour of the ribosomes used for transcription. Although membrane continuity, and thus space continuity, is far from proven, such concepts have appeal, since protein targeting would be confined to a specific subvolume of the cell. Similarly, one space could, as in the case of several "intracellular" cubic membranes, be continuous with the extracellular space, even though physically located in an apparently intracellular site (cf. Fig. 7.26). Secretion, receptor recruitment, etc. could thus occur simply by a topological invariant membrane out-folding or membrane flowing mechanism. This would be energetically more attractive than present putative mechanisms, which are dependent upon vesicular transport. Generally, if a multicontinuous and asymmetric membrane (curved unequally towards both sides) is imagined, the spaces differ not only in colour, but also c o n t e n t - though we must assume that certain essential molecules are common to both spaces. Similar mechanisms to those outlined above would apply to virtually any transport problem, including organelle biogenesis, at any site within the cell. The anticipated flippases, which are thought to be responsible for the formation and maintenance of bilayer asymmetry regarding lipid molecules, could thus be envisioned to have a regulatory as well as constructive capacity. We stress that transportation vesicles are not necessarily excluded by these considerations. Rather, they need not be the sole biomolecular porters, alone

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327

responsible for the immense problem of intracellular sorting and transport. The suggested mechanism is of particular interest in cells that need more efficient transport systems than those provided by vesicles. Though conceptually simple, transport systems involving vesicles are necessarily physically and chemically very complex. The use of transport vesicles requires energy input (for their formation, transportation, and fusion), as well as delicate control systems, involving several steps. These include control of the content of the vesicle (including the membrane of the vesicle itself prior to fission), control of vesiculation and pathways, control of specific and non-specific recognition along the pathway as well of the target; control of targeting and the acceptance of the cargo, control of membrane fusion - including the maintenance of membrane asymmetry at the point of fusion, control of membrane topography after fusion and lastly, control of membrane recycling. The ultimate and single control mechanism in our topological model is that setting the "colour" of the space, which most likely is genetically controlled, though it can be considered as generated as an effect of the topology of the NE. We may even have to accept a higher degeneration of intracellular space than has so far been suggested. Perhaps each nuclear pore or some number of pores, is controlled so that it is continuous with a particular space of the ER, allowing for a set of different sub-ER's. Clearly, many similar ideas can be put forward, based on the notion of topological complexity within the cell volume, induced by cubic membranes. We stress again that these concepts remain at this stage speculative.

7.13 Specific structure-function relations There are several general functionalities that can be conjectured for cubic membranes. In particular, a cubic membrane could serve as a regulator of chemical or physical potential over the particular membrane. This is indicated in several cases in which the cubic membranes have been found to be sensitive to the growth medium. Though this can be explained by other factors, it is conceivable that such effects are due either to a chemical or physical induction, or to the specific stage of differentiation of the cells, which might be changed as growth conditions alter. A pressure-regulative capacity of a PM-associated cubic membrane can be imagined, which would serve to reduce osmotic shocks, just as the hyperbolic continuous morphology of the calcite skeletal network of sea-urchins affords protection against mechanical impacts (discussed in the next Chapter). Such a function is consistent with the recognition of cubic membranes in a variety of epithelial cells. Cubic membranes can be involved in curvature-controlled activation of certain enzymes, as well as control of enzyme activity. It is tempting to suggest the latter as a general function of cubic membranes since proteins could conceivably be located at regular points in the lattice. This could enhance transport efficiency of both product and substrate. Such mechanisms are particularly well suited for mass cooperative synthesis, such as those

328

Chapter 7

taking place in, e.g. mitochondria and the SER of certain steriodogenic cells. Both of these organeUes are represented in Table 7.1 and 7.2. We have noted repeatedly that the stage of differentiation seems to influence the formation of cubic membranes. Cells that actively undergo differentiation, such as the differentiating sieve elements, the spermatogenesis, germ cells, and tumour cells are all represented frequently in Table 7.1 and 7.2. Clearly, during differentiation events, there is an extreme need for sophisticated regulative and spatio-temporal transport, as well as communication. This is particularly well illustrated in spermatogenesis. It is therefore tempting to speculate on the importance of cubic membranes as space partitioners to direct intra-cellular traffic. In the case of differentiating sieve elements, cubic membranes might have a particular role in the establishment of the mechanism of assimilated transport, which has yet to be explained [98]. Many of the cells listed in Table 7.1 and 7.2 are involved in active membrane flow and other mass-cooperative transport phenomena. Since cubic membranes offer a high surface to-volume ratio, they may also be actively involved in these processes, perhaps as membrane storage bodies, or as transport guides. It is of interest to note that aggregates of "synaptic vesicles" often resemble cubic membranes (see Chapter 5 and [136]). This can be taken as an indication of a possible on-off mechanism of membrane continuity, which might account for a regulative capacity of the release of transmitter substance. As already pointed out, there are certain cubic membranes whose particular structures are strongly linked to function, such as the D-PCS structure of the PLB in oxygen-evolving photosynthetic cells. A physical role has been conjectured for the selection of the D-surface geometry [4, 56]. This is based on theoretical considerations on how the geometry of crystals can control the emission or absorption of certain wavelengths of light through the existence of photonic band-gaps (see, e.g. [137]). Briefly, although the D-surface is isotropic in bulk, its geometry is such that it can trap photons. The associated energy could be used by certain molecules positioned along a particular lattice direction, which would then trigger the PLB to revert to the active thylakoid membrane geometry. This conjecture has recently been supported by experiments (Guo, Y., Zhao, G., Zhow, J., Ho, J.T., Mieczkowski, M., and Landh, T., work in progress). It is probable that the particular molecules responsible for this functionality are conserved through the evolutionary process and this could be the origin for the selection of the D-PCS membrane. The phase behaviour of the lipids in thylakoid membranes has been investigated in some detail [138]. The cubic phase formed in the three-component system of purified monogalactosyl-, digalactosyl-diacylglycerols, and water has been determined to be of the gyroid type (space group Ia3d), rather than the D-surface (Fm3m). It is however conceivable that the addition of proteins to the lipid matrix induces the formation of a cubic phase following the topology of the D-surface. Further the extension of the cubic phase region in the phase diagram is very small,

Structure-function relations

329

while that of the reversed hexagonal phase region of the phase diagram is much larger. However, most of the proteins in the light harvesting complexes and electron transport systems favour fiat geometries - away from the highly curved reverse hexagonal geometry and towards that of bicontinuous cubic phases, viz., the PLB. This is further supported by the observation of a significant decrease of the lipid/protein ratio in the PLB as compared to the thylakoid membrane (for a recent review on thylakoid lipids see [139]). A similar structural-functional role can be argued for in the selection of the primitive cubic membrane system in the bioluminescent scaleworms, whose structure is apparently also invariant. As seen in Table 7.1 and 7.2, there are, in addition to these two examples, several other photoactive cells in which we have identified cubic membranes. In the case of the mitochondrial cubic membrane in Tupaia glis we can speculate that the use of an isotropic structure allows efficient capture of the incoming light, which has to pass the giant mitochondria before reaching the outer segment. Similar reasons may guide the choice of a cubic membrane system in the lens of certain scaleworms. Epithelial cells are one of the cell types in which cubic membranes have been identified frequently. It seems that it is often the PM that invaginates, as in the sensory cells of the vomero-nasal organ, in intestinal epithelial cells, or in the ER of retinal pigment epithelial cells. We speculate that the formation of a cubic membrane in conjunction with the PM in these cells can increase adsorption in addition to the advantages that accrue to the general hypotheses above. This offers a neat transport mechanism for fat adsorption. In the same context, Luzzati has argued that bicontinuous cubic membranes, consisting of monoglycerides and fatty acids, may well arise in vivo to aid fat digestion [140]. There are many plausible curvature-controlled enzyme systems, such as phospholipase C, for which the activity might be highly dependent on the bicontinuous nature of a cubic membrane. The most intriguing case is the strong correlation between the second messenger role of diacylglycerols, which are thought to activate protein kinase C (PKC), and its propensity to form negatively curved structures. It has been shown that the activity of PKC is not specifically dependent on the propensity of lipids to induce reversed hexagonal phase (see, e.g. [141], and references therein) and, thus, reversed cubic phases. However, most of the experiments have been performed in vesicular, (or micellar) systems, whose topology and structure are not necessarily preserved throughout the time of the experiments, due to the induction of phase transformations caused by the lipids that favour highly curved structures. In view of the ubiquity of cubic membranes it would be of interest to study the influence of PKC on the structure of cubic phases, as well as the protein activity in different cubic phases, to eventually correlate the geometry of the e n v i r o n m e n t as one factor of protein activity to be considered.

330

Chapter 7

The hyperbolic architecture of a variety of enzymes has been pointed out in Chapter 6. It seems natural then that enzymes encapsulated in a hyperbolic (lipid) environment (of suitable dimensions) are able to adopt their favoured architecture, and thus perform their biochemical task efficiently. In this context, it is worth noting that remarkable increases in enzymatic efficiency of biochemical synthesis have been achieved by containment of enzymes in cubosomes, rather than conventional (fiat) liposomes. Many other structure-function relations for cubic membranes remain to be postulated. We close this survey of cell membranes with a remarkable observation that adds support to this novel picture of cytomembrane shape. In Chapter 4 (section 4.13), it was noted that many bacteria are shrouded in a mesh-like protein coat, which often displays a regular, crystallographic form. The most exotic examples of bacteria are the thermophilic archaebacteria, that thrive at temperatures between 70~176 in sulfur-rich hot-springs and mud holes. (So anachronistic are these single-celled organisms, that they are sometimes taxonomically classified as a distinct Kingdom.) It appears that the dimensions of the protein layers in species of these bacteria, Solfolobus solfataricus, are in "precise epitaxial coincidence" with the lattice parameters of a bicontinuous cubic phase, formed in excess water with the membrane lipids predominant in this organism (in vitro) [140]. Such a coincidence is indeed difficult to reconcile with the usual notion of a flat, neutral, cytomembrane, whose sole function is to support the real stuff of life, the proteins. Abbreviations:

AL: annulate lameUae.

ER, PER, RER, SER: endoplasmic reticulum, paracrystalline/rough/smooth endoplasmic reticulum. IPMS, PMS, PCS, PNS: All abbreviations for periodic hyperbolic surfaces: infinite periodic minimal surfaces, periodic minimal surfaces, periodic cubic surfaces and periodic nodal surfaces respectively. (The abbreviations P-, Dand G- prepended to these indicate the topology and symmetry of the periodic surface, corresponding to the relative tunnel arangements and black-white sub-group of the P-surface, the D-surface and the gyroid respectively. NE, ONE, INE: nuclear envelope, inner/outer nuclear envelope. PLB: prolamellar body. PM: plasma membrane. TRS: tubulo-reticular structure. UMS: undulating membranous structures.

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F. Gonzales-Crussi and H.J. Manz, Cancer, (1972). 29: pp. 1272-1280.

162.

C. Espana, M.A. Brayton, and B.H. Reubner, Exp. Mol. Pathol., (1971).

15: pp. 34-42.

163. U. Pfeifer, R. Thomssen, K. Legler, U. B6ttcher, W. Gerlich, E. Weinmann, and O. Klinge, Virchows Arch. B. Cell Path., (1980). 33: pp. 233243. 164. F.A. Murphy, A.K. Harrison, G.W. Gary, S.G. Whitfiled, and F.T. Forrester, Lab. Invest., (1968). 19: pp. 652-662. 165. B. Tandler, R.A. Erlandson, and C.M. Southam, Lab. Invest., (1973). 28: pp. 217-223. 166. J.S. Munroe, F. Shipkey, M.D. Erlandson, and W.F. Windle, Cancer Inst. Monogr., (1964). 17: pp. 365-390. 167. E. Horvath, K. Kovacs, S. Szabo, B.D. Garg, and B. Tuchweber, Zellforsch., (1973). 146: pp. 223-235.

Natl. Z.

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Some Miscellaneous Speculations

ur journey is nearly over now. We have come from an apparently Pesoteric branch of mathematics: differential geometry and topology of minimal surfaces, and found hyperbolic geometries over a broad sweep of materials: from inorganic and organic chemistry, zeolites, membrane self-assemblies, to protein and lipid structure and function. Knowledge of molecular forces and their subtlety combined with curvature really does seem to provide new insights. The case for intrinsic curvature in many condensed materials, we claim, is clear. Before concluding the reader might allow us to go a little further, onto more speculative ground. We discuss first the role of curvature in templating and in inorganic condensed systems. Finally we mention a class of more complicated supra-assembled aggregates, and a role for curvature in the origin of life itself.

O

8.1.1

Templating and curvature: DNA templating

Templating is ubiquitous in nature. A prominent and well-characterised example can be found in the mechanism of DNA replication. Genetic information is transferred to the next generation by means of spatially confined chemical messages, embodied in the DNA double helix. Whenever information transfer is called for, as in cell growth or protein synthesis, the double helix is split along the hydrogen bonded seam and the resulting single strands are utilised as templates. In the case of cell division, the single strands template the formation of complementary strands, reforming the original DNA, albeit in twice the quantity. In protein synthesis the single strand templates the formation of a suitable RNA strand that in turn is used as a secondary template in order to effect the actual synthesis. This particular templating process works by careful matching of hydrogen bonds in complementary nucleic acid strands. In other words, the identification of local properties is used to achieve a global end. A less specific form of templating - due to the electric field about ions in solution- can be recognised in a variety of organic and inorganic syntheses.

8.1.2

Templating by electric fields: equipotential and tangential field surfaces

Templating generally refers to the process whereby a molecular form is constructed from a pattern set by a (templating) molecule. However, a more subtle form of templating may be responsible for the atomic arrangement of many structures, from silicates to organic cryptate molecules.

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In these cases, the atomic structure of the templated species is set by the electric field* of the templating material, rather than the physical surface of the template. For certain arrangements of the templating material, the field lines traversing the volume lie on a variety of unusual hyperbolic geometries, close to those of minimal surfaces. These geometries can then be transferred to the templated species, resulting in hyperbolic molecules and arrangements in atomic crystals. This concept can be seen further as follows. Imagine a single ion placed into a solution from which a crystal is being grown. If the crystal precursor interacts electrostatically with the ion, we expect the atomic array of the resulting crystal to be perturbed in the neighbourhood of the ion, under the influence of electrostatic field due to the ion. Now, imagine that the mother liquor from which the crystal is grown contains a uniform density of these "templating ions". The resulting crystal is forced to grow around each ion, producing a potentially novel crystalline structure. Consider the situation where the templating ions form an ordered lattice in the solution. What is the relation between periodic minimal surfaces and electric fields about charged lattices? A substantial body of calculations by von Schnering and Nesper has established that the zero equipotential surfaces (POPS) of many common atomic structures are similar to periodic minimal surfaces [1]. (In fact, we know that the equipotentials are not identical to periodic minimal surfaces, although they are geometrically very similar [2].) Clearly, crystalline precursors are not located along equipotentials, since the electric field is everywhere perpendicular to the equipotential surface. We expect them to lie on surfaces for which the field is everywhere tangential to the surface; we call these surfaces Tangential field surfaces (TFS). It turns out that there is an exact correspondence between: (i)POPS for a cation/anion array (ii)TFS for the cation/cation (or anion/anion) array where the charges are in identical locations to those for the POPS, if and only if the POPS is itself a minimal surface [3].

8.1.3

Diffusion within fast-ion conductors

This correspondence between equipotential and tangential field surfaces leads to a simple construction for the likely diffusion trajectories of ions within fast-ion conductors, also called "solid electrolytes". These solids (typically binary salts) are electrically highly conducting (the electrical conductivity of The term "electric field" to describe and visualise the templating process is used loosely. In general the process is not determined solely by the electrostatic field. Rather, it is the entire set of inter-oscillations of the electromagnetic field that sets the zero stress surface, cf. 3.2.4.

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the solid is close to that of the molten salt), due to the presence of a significant proportion of ions within the solid which are in a state of constant motion. They are described best as partially melted solids: one portion of the solid is frozen into lattice sites (e.g. cations), while the rest of the material is in a liquid state, diffusing continuously within the frozen lattice [4]. Consider the simple case of a mobile ion diffusing through space under the influence of a pair of like charges. In this case, most field lines due to the pair of charges end up on one of these charges. However, some tangential field trajectories lie in the mirror plane between the pair and this plane is a TFS for the fixed ion-pair. By analogy then, the TFS for the frozen sub-lattice often similar to IPMS - is expected to define likely positions and trajectories of the mobile ions.

Figure 8.1. The diffusion along IPMS by mobile ions ill binary solid electrolytes (a): Partition of the body-centred cubic (bcc) lattice into two primitive cubic sub-lattices, separated by the Psurface. Average positions of the (mobile) silver cations, detected from X-ray measurements, in the solid electrolyte 0t-Agl are indicated by the smaller red spheres on the surface. The larger dark red spheres define the bcc lattice of the (frozen) iodine ions (only two occupied sites of the bcc array are shown). (b:) The bcc lattice can also be partitioned into two diamond sub-lattices by the D-surface. The curved net on the surface described trajectories of mobile Ag + ions on the D-surface; vertices of the net locate the average I" positions.

The earliest known solid electrolyte, a-AgI, discovered last century, called the "Nernst glower" is a typical example. In this material, the iodine anions remain frozen in a bcc lattice and the silver ions are in a liquid-like state, diffusing through the bcc lattice. One possible TFS that partitions the iodine lattice into two sub-lattices is the P-surface (close to an equipotential surface for the bcc CsCl structure, Figure 2.6); another is the D-surface, (which partitions the iodine lattice into two diamond sub-lattices). The "average" silver positions - found by X-ray diffraction measurements [5] - occupy sites on both the D- and P-surfaces (Fig. 8.1(a), (b)) [6]. In both cases, spiral-like curves linking these average positions (along asymptotes) result. These conclusions are in agreement with numerical calculations based on atomic dynamics [7].

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Another typical solid electrolyte is ~-PbF2, which exhibits solid-electrolyte behaviour typical of many salts with fluorite-type structures [8]. In this material, the fluorine anions "float" within a face-centred cubic (fcc) lattice of lead cations. Tetragonal distortions of the D-surface (the tD-surface, described in Chapter 1), partition the fcc lattice into two sub-lattices (each a tetragonal distortion of the diamond lattice), lying on opposite sides of the surface. This surface is close to the periodic equipotential surface half-way between cations and anions set up by the hypothetical neutral fcc structure that results if the charges on one of the tetragonally deformed (cationic) diamond sub-lattices are reversed (giving an anionic sub-lattice) while the other sub-lattice remains unchanged. Evidently, the electrostatic field lines due to this fcc array are approximately perpendicular to the tD-surface. If the anionic sublattice is replaced by cations, recovering the Pb 2+ fcc array, the field lines rotate by 90", so that they now lie in the tD-surface. The tD-surface is a TFS for the lead lattice and an ion traversing through the lead lattice is expected to follow this surface. If the mobile species is a cation, any path along the tD-surface is a stable orbit within a fcc cationic lattice. In the case of ~-PbF2, the mobile species is anionic, in which case paths on the tD-surface represent average trajectories of the fluoride ions under the influence of the (assumed) electrostatic field set up by the fcc lead lattice (due to the presence of shortrange repulsive barriers between unlike charges) [3]. In this case too, the simplest paths apparent within a three-dimensional euclidean perspective between average positions of the mobile ions are avoided (e.g. lines through the octahedral voids linking average positions), and spiral trajectories which are simplest within the hyperbolic two-dimensional picture afforded by the TFS construction - are preferred. Correlations between the most frequently occupied sites on the TFS and the Gaussian curvature of the TFS exist. The measured locations of maximum fluoride density in ~-PbF2 are precisely the "saddle-points" on the tD-surface, i.e. the sites where the magnitude of the Gaussian curvature is largest. Similarly, the average locations of the silver ions in oc-AgI are at the saddle points of the D-surface. This suggests a link between the speed of the mobile ions, and the curvature sampled along the surface.

8.1.4

The templating of zeolites

We can go beyond the case of fast-ion conductors and apply these principles to understand the role of templating species in inorganic and organic synthesis. A number of examples of zeolite frameworks have been shown in Chapter 2 to follow closely IPMS. The crystallisation of zeolites invariably requires the presence of templating species. A wide variety of templates have been used, from sodium ions, tetra-alkyl ammonium ions to crown ethers. A simple analogy between fast-ion conductors and zeolite synthesis can be drawn as follows. Assume the templates play the role of the frozen lattice in

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solid electrolytes, i.e. the TFS is set up by the templating array in the alumino-silica solution from which the zeolite is eventually crystallised. Further, assume that the alumino-silica species are free to diffuse within the solution, before silica polymerisation and crystaUisation. If the templates assemble under their mutual interactions (usually repulsive) to form a (locally) ordered array, the TFS are expected to resemble IPMS. (The actual IPMS formed depends on the coordination geometry of the templates, which is a function of the template concentration, ionic strength, temperature and so on.) Due to the interactions between alumino-silica and the templates, the former species is confined to regions close to the TFS, before and during crystallisation. The resulting zeolite framework is then a tessellation of the TFS, and it is thus not surprising that these frameworks resemble IPMS [9]. This model is crude in that we have assumed that the templated species do not interact with each other; surely an unreasonable assumption given the super-saturated concentrations of these species during crystallisation! In fact, there is some numerical evidence to suggest that surfaces approaching zero mean curvature are also favoured if we assume that the templated precursor species interact only amongst themselves. Calculations [10] suggest that minimal surfaces represent stable geometries for interacting species. (This is intuitively sensible: if the mean curvature vanishes, the force component normal to the surface is zero since the surface is equally concave and convex.) On these grounds, we expect templating species to be of critical importance to the formation of particular structures. As the nature and coordination of the template are varied, the resulting crystal structure changes. The geometry of TFS need not be confined to periodic hyperbolic surfaces, although these are responsible for the characteristic zeolite structures, containing arrays of tunnels. Some consequences for zeolite structures follow immediately from this templating model. If the TFS is a hyperbolic surface free of self-intersections, the maximum site symmetry of T-atoms in templated zeolite frameworks is restricted, (e.g. higher-order r o t a t i o n - roto-inversion sites are forbidden, since these would generate multiple sheets of the surface at that point, leading to intersections). The sites of the templating species can also be of higher symmetry to those of the T-atom framework (for example, compare the symmetries of iodine positions in 0~-AgI with those of the average silver cation sites (Fig. 8.1)). Precisely such a relation between site symmetries of Tatoms and templating species has been noticed in zeolite frameworks [11]. This observation is a baffling one within the conventional Euclidean threedimensional description, but an immediate consequence of a hyperbolic surface description. Indeed, by analogy with the behaviours of soap films, catastrophic changes of the TFS topology can occur as the mutual arrangement of the templating ions is changed continuously. (For example, the TFS can j u m p from a bicontinuous network to disconnected sheets.) This effect m a y well be

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responsible for the varieties of silicate structures-chain, sheet and framework silicates - formed in nature.

8.1.5

Templating organic molecules: the caesium effect

The hyperbolic nature of a number of large organic molecules such as carcerands has been demonstrated in Chapter 2. The synthesis of carcerands is enhanced by the presence of caesium salts (typically caesium carboxylate), and the efficacy of caesium in improving synthetic yields has been dubbed the "caesium effect" [12]. Since the caesium ions do not participate directly in the reaction chemistry, the effect has been ascribed to characteristics of this large ion alone. Again, this effect can be understood in terms of the templating of hyperbolic structures, due to the presence of hyperbolic TFS, set up by the caesium ions in solution. For example, the carcerand molecule resides on the P-surface (Fig. 2.21), which is a common TFS in aqueous zeolite syntheses. The improved yield of large ring compounds in the presence of cesium is also understandable on this basis, since the precursors to these compounds prefer to wrap around tunnels of the TFS [13].

8.1.6

Templating of the morphology of a calcite crystal

Echinodermata are a phylum of marine animals that are distinguished by protective mineral skeletons made exclusively of magnesium-rich calcite. In the living animal, this skeleton forms a complex convoluted network structure, which is interpenetrated by a matrix of living tissue (the "stroma"). Calcium and magnesium ions are selectively extracted from seawater by specific proteins, and ingested into the growing regions of the shell, where they precipitate out of solution as magnesium-rich calcite onto the skeleton. The skeleton of echinoderms consists of interlocking plates which (in some cases) support protective spines of the animal. In the late 1960's, striking electron micrographs of the surface morphology of magnesium-rich calcite skeletons of sea-urchin spines and shells were first published [14]. Unlike classical crystals, the "faces" of these convoluted networks bear no relation to the crystallographic planes in the calcite atomic structure; they are continuously curved. In other species of the echinoderm phylum (sea-stars, sea cucumbers, etc.) large single crystals of calcite are synthesised within the stroma of the animal, forming two-dimensional hexagonal networks, or less ordered "fenestrated" structures. With the single exception of echinoderm teeth, the skeletal units within echinoderms are composed of large single crystals. These observations are a striking case of exotic hyperbolic structures that occur in physical systems.

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The biomineralisation process in echinoderms is a superb example of a templating process in living systems and illustrates the supreme engineering skills of nature. The very survival of echinoderms depends upon the ability of the mineral skeleton to be able to withstand high impact stresses, and the smoothly curved crystal surface distributes impact loads throughout the entire spine and base-plate, so that these minerals have an enormous strength-to-weight ratio, far stronger than that of conventional calcite, indeed, superior to reinforced concrete! The most striking images have been recorded in Cidaris rugosa skeletons. In this sea-urchin, apparently single crystals of calcite - whose atomic structure has trigonal s y m m e t r y form extraordinarily complex hyperbolic morphologies, which are very similar to the cubic P-surface. Further, the unit cell edge of the atomic lattice is about 50~, while the surface of the mineral has a unit cell edge of about I ~m. Some of these images are shown in Fig. 8.2. The ultrastructure of the skeleton is certainly not set by the atomic lattice. It is impossible to cleave the mineral along its usual crystallographic cleavage planes, presumably a desirable trait for a living being! Even at high levels of magnification, electron scanning micrographs are not able to detect habit planes of the mineral. The most likely source of these curious violations of the usual phenomena of crystal growth is the biological tissue that fills the interstices within the calciferous deposit in the living creature. The stroma consist of a range of essential c o m p o n e n t s of the o r g a n i s m - i n c l u d i n g a m p h i p h i l i c macromolecules, whose self-assembly properties are expected to be similar to those of synthetic copolymers discussed in Chapter 4. The domain geometries of these synthetic macromolecules are strikingly similar to the ultrastructure of the echinoderm skeletal networks. We have seen that both mesh and sponge morphologies can occur in polymer melts, depending upon the composition of the system. The domains in biological polymers are of hydrophobic or aqueous characters, as in lipid assemblies. It is certain then that the tissue within the echinoderm may self-assemble to form water networks that are similar to constant mean curvature surfaces related to the lower topology periodic minimal surfaces, such as the P-surface, or twodimensional networks, depending upon the relative hydrophilicity and ionic content of the tissue, and the amount of water taken up within the tissue. In this fashion, the organic material in the animal actually determines the ultrastructure of the precipitated inorganic salts by a remarkably simple templating mechanism. These salts must precipitate within the aqueous

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Figure 8.2: Scanning electron micrographs of the base-plates in the sea-urchin cidaris ru~sJ.. The images (at various magnifications) reveal cubic symmet D" within the (single crystal) calcite network. The tunnels in these images are of the order of I l~m in diameter. Note the loss of meso-crystallinity towards the surface of the shell (top left image). (These pictures have been kindly provided by Hans-Udde Nissen.)

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domains of the amphiphilic matrix, so they naturally follow the morphology of the constraining tissue. Clearly, to synthesise single crystals over years in the sea, the animal must maintain an exquisite structural buffer, so that the bio-polymer assembly remains stable. But this is a pre-requisite for continued life in the animal, so there are no doubts about the ability of the creature to retain the optimal chemical environment! This concept of organic templating to induce biomineralisation is a largely unknown field at the moment. It is very difficult to reproduce the in vivo conditions of an echinoderm in the laboratory, so our hypothesis about the structure of the tissue self-assembly cannot be confirmed. Preliminary results suggest that growth habit of related aragonite and vaterite are very sensitive to additional templating species, cf. [15]. Our proposal offers a simple explanation how the mineral manages to adopt this exotic morphology. There are m a n y subtleties within this p h e n o m e n o n that remain to be investigated. The lack of crystallographic facets suggests that the organic matrix also acts at a micro-level, inducing the mineral to form smoothly curved surfaces. The nature of this interaction is not known, although it may well involve an epitaxis mechanism to promote crystallisation along the stroma surface. (For further discussions of this fascinating field see [16].) We are only just beginning to study the interplay between organic and inorganic matter in living systems - we have discussed only the most regular examples. The humble sea-urchin alone offers a wealth of structural variation. The elegant techniques which nature has devised to create optimal materials depend upon the variety of forms that can arise in self-assembled systems. The ultrastructure of echinoderms is nothing more than "a diagram of forces" (to adopt a concept of D'Arcy Thompson), and the hyperbolic interconnected geometries offer best protection against destruction of the skeleton due to these forces. The presence of these geometries in evolved living species is a good argument for the utility of hyperbolic geometries in materials. If we could mimic the action of echinoderms in the laboratory, we could rightly claim to have made a substantial advance in the engineering of materials. Work is u n d e r w a y to achieve this end; to date people have concentrated on surfactant-directed templates (e.g. synthesis of mesoporous "zeolites", discussed in Chapter 2), it may be more profitable to aim for larger templates, such as those of macromolecular assemblies. The extraordinary variety of shapes in the silicate skeleton of radiolaria micro-organisms (which are found in abundance in the oceans) represents a further example of templating by self-assembled molecules (present in the living organisms), and biomineralisation. To date, scanning EM images reveal a variety of characteristically porous hyperbolic forms - from "baskets" to "funnels" and "cages" (Figs. 8.3).

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Figure 8.3: SEM images of radiolaria collected from the PacificOcean (images courtesy of Roger Heady and Michael Ciszewski),

8.2 Supra Self-Assembly 8.2.1

Biological superstructures based on self-assembly

The description of structure in complex chemical systems necessarily involves a hierarchical approach: we first analyse microstructure (at the atomic level), then mesostructure (the molecular level) and so on. This approach is essential in many biological systems, since self-assembly in the formation of biological structures often takes place at many levels. This phenomenon is particularly pronounced in the complex structures formed by amphiphilic proteins that spontaneously associate in water. For example myosin molecules associate into thick threads in an aqueous solution. Actin can be transformed in a similar way from a monomeric molecular solution into helical double strands by adjusting the pH and ionic strength of the aqueous medium. The superstructure in muscle represents a higher level of organisation of such threads into an arrangement of infinite twodimensional periodicity. We will consider below self-assembled superstructures, like collagen, plant cell walls, starch and the contractile macromolecular complex. These super structures are also liquid crystals, evident from their birefringence and rheological properties. With increasing knowledge of the molecular mechanisms behind function, however, there has been an unfortunate tendency to neglect the role of overall structure.

Supra self-assembly

8.2.2

349

Collagen and plant cell walls

Collagen is the main component of connective tissues in animals. The first level of association involves the formation of a triple helical structure of three peptide chains into tropocollagen, with a length of about 30 nm and a pitch of 3 nm. The lateral arrangement of tropocollagen molecules into fibrils is not definitely known. According to the "quarter-stagger pentafibril model", which seems to be most favoured, the molecules in the cross-section show five-fold coordination and are successively displaced 67 nm along the fibre axis. Five such units (335 nm) represent the true periodicity along the fibril axis. As the molecular length is shorter (about 300 nm) there is a gap between tropocollagen molecules in the long axis direction. The gross structure of collagen in many tissues, for example tendons, involves wave-like curved fibrils, which provide elasticity. This sort of curvature occurs also when fibrils in two directions are linked into a planar net. A highly ordered collagen structure is found in the so-called "decemats membrane", which is the basement membrane of corneal endothelial cells. Stacks of hexagonal lattices have been observed which are parallel to the basement membrane [17]. We think that the structural principles that underlie quasicrystals (discussed in Chapter 2) are relevant in order to describe the superstructure of tropocollagen molecules, which must accommodate the wide variety of tissue structures that can be formed. There are no possibilities of packing triple helices in a space-filling arrangement. A structure with five-fold symmetry in the plane and with perfect periodicity in the perpendicular direction, however, which is consistent with the quarter stagger pentafibril model, is in fact a quasicrystal of so-called T-type (also known as the decagonal phase). The plant cell wall consists of cellulose fibrils anchored in the plasma membrane, where the cellulose biosynthesis takes place. These fibrils are arranged in fibres, which - like starch - exhibit the remarkable property of alternation of crystalline and amorphous zones. Outside the cellulose fibres there is a complex medium containing pectins and hemicellulose. With age this medium is mechanically strengthened by lignin; an aromatic polymer. The detailed microstructure of the cell wall is not known, although it is generally considered to be bicontinuous with regard to the cellulose fibres and their outside medium. Periodic hyperbolic surfaces may offer a fruitful approach towards understanding the complex interpenetrating structure of this important component of plants. Such a surface is expected to have a periodicity of a few hundred nanometres, corresponding to the periodicity along the cellulose fibre, and the variation in Gaussian curvature over the surface may account for the alternation of crystalline zone with amorphous regions.

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8.2.3

Clu~ter 8

The molecular packing in native starch

More than half of the energy intake of the world population comes from starch. Starch provides a very efficient form of energy storage in plants, with a semi-crystalline packing of glucose polymers in granules. The linear polymer amylase (with (z-l,4-glucoside links) is an amorphous space-filler whereas the branched amylopectin molecules are arranged in crystalline clusters. The amylopectin molecule is the largest known molecule in nature, and a hyperbolic surface description has proved to be most valuable to characterise the ordered structure. The short-range packing of glucose units in the semi-crystalline starch granule has been determined from X-ray diffraction data [18]. The most important conclusion was that the glucan chains form double helices, which are hexagonally arranged (cf. Fig. 8.4). A later remarkable finding from C 13NMR studies was that the branched component, amylopectin, is responsible for the crystallinity. The possibility then exists of combining chemical knowledge about the branched structure of amylopectin with knowledge from X-ray studies on the double-helical close-packing of glucan chains. This has led to the discovery that the arrangement is similar to the structure of quartz [19], and these two structures can both be described by a surface that has been christened the "Q* surface" [1]. The helical packing into concentrically arranged units is shown in Fig. 8.4. The amylopectin molecule forms a cluster of branches, as mentioned above. One long chain (called, appropriately enough the "long B-chain", B) joins the cluster with that underneath. New B-chains are linked to these long chains by 0~-1,6 glucoside links. The final branch is called an A-chain. The proportion of these different branch configurations is 35 B-chains to 39 A-chains per crystalline unit; their arrangement is shown in Fig. 8.4. The thickness of such a unit is about 50 A (determined by electron microscopy), which corresponds to 21/2 helical turns, or five c-periods. A remarkable feature of this structure is that the van der Waals packing distance between double helices is in agreement with the glucoside link. This is a direct consequence of the IPMS character of the structure. The position of the polyglucan chain in relation to this surface is illustrated in Fig. 8.5. Just as in quartz, there is a packing distance between the double helices that is related to the helical units. These links correspond to the labyrinths joining the double helices. The nature of these labyrinths of the Q* surface space group P6422 is considered further below. This extraordinary structure leads to an understanding of some important mechanisms in the biosynthesis of starch. There is evidence for formation of a so-called "coacervate", in other words a phase separation. The tendency of units like those shown in Fig. 8.4 to crystallise is obvious- the possibility of close-packing of adjacent units exists, as well as close packing within each unit. A direct consequence of lateral packing of such units is that the enzyme

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for synthesis will be squeezed out during this process, and synthesis of new units can proceed.

Figure 8.4: Concentric arrangement in the starch granule of crystalline amylopectin clusters shown to the left. In the middle the cross-sectional packing of double-helices in these clusters are shown (A, B are different branch types). The double helix arrangement of glycose units is shown to the right.

Another consequence of this model of the starch structure is a better understanding of starch gelatinisation, one of the most fundamental effects involved in food production. The gelatinisation p h e n o m e n o n is basically due to the thermally driven release of e n t r a p p e d amylose from the amorphous regions between the clusters into the continuous water phase and a hydration/dissociation of the amylopectin molecules. This latter effect may vary, from swelling of the native granules to complete solution of all amylopectin molecules. In either event, there is a tendency towards recrystallisation when the water solution (gel) is cooled. This effect has been described technically as retrogradation, and it is the factor that, for example, is responsible for bread turning stale. If amylose and amylopectin are mixed in water, an amylose-rich upper phase and an amylopectin-rich bottom phase are formed. We can regard the starch gelatinisation itself as such a phase separation, where amylopectin remains in the water-swollen granules and amylose forms the outside solution. It is interesting to see how the shape of the granules changes successively during this process, which is illustrated in the case of wheat starch in Fig. 8.6. Thus there is a clear tendency of deformation so as to obtain an outer surface with

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zero average curvature. When some kind of equilibrium situation is reached, the soft surface zone represents changes in interfacial energy. Hydrostatic equilibrium requires zero average curvature at the interface.

8.2.4

Saddles in the kitchen: bread from wheat flour

The metamorphosis from wheat flour to bread is a remarkable example of molecular self-assembly, where curvature and shape on the colloidal level determine the result. The average composition of wheat flour is 65% starch, 12% protein, 14% water, 2% lipids and the remainder consists of non-starch polysaccharides and inorganic components. The unique component of wheat responsible for its baking performance is the water-insoluble storage proteins, consisting of a very complex mixture of related proteins: gliadins and glutenins. When exposed to water, they swell and form the characteristic gluten gel. These proteins are very amphiphilic and exhibit self-assembly properties very similar to those of lipid molecules. Thus this gel contains a certain amount of water, and can coexist in equilibrium with excess water. The structure of the individual molecules is dominated by [3-spirals, and one possible structure is a three-dimensional packing of these helices according to the gyroid structure

[20]. The lipids, although present in a very small amount, are also of critical importance for the bread making process. The lipids themselves form HIL L(x and L3 liquid-crystalline and liquid phases respectively with increasing water content. In short, the kneading of dough introduces air cells, and during fermentation the carbon dioxide will increase the volume of these cells until a foam is obtained. In the oven the foam opens to give a pore system and the structure is "fixed" due to starch gelatinisation. The process involves different steps where proteins and lipids compete at the interface, which we will consider in more detail. The amount of water used in dough results in the formation of two aqueous phases. One is comprised of starch granules surrounded by water, the other is what used to be called the gluten gel, consisting of wheat proteins with dispersed lipids in the form of liposomes. Entrapped air cells during mixing localised in this gluten gel are thereby enabled to expose hydrophobic surfaces. During fermentation, the gas-water interface changes, with smaller protein molecules (gliadins) being replaced by larger ones (glutenin molecules). In the last period of fermentation (good cooks will realise that the dough must rise twice)! lipids finally take over the interface. The microstructure of bread is illustrated in Fig. 8.7.

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From a macroscopic point of view, the m o s t striking c h a n g e s are the o p e n i n g of the foam of gas cells to give pores, a n d solidification of the a q u e o u s b u l k m e d i u m . This solidification is partly d u e to the loss of cohesiveness w h e n the gluten gel is t r a n s f o r m e d into a coagel, a n d partly d u e to starch gelatinisation, giving no excess of w a t e r outside the starch granules.

Figure 8.5(a): Part of the Q* surface around the line of intersection (00z) together with a superimposed double helix of amylose. (b): Part of the quartz surface around (00z) showing the four wings of two double helices of which two are used by the amylose double helix. (c): Part of one single helix of amylose and of the quartz surface, which appears to be the average "surface" of the molecule cutting through two opposite atoms of the six-membered rings.

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There are two important functions of the pore system, beside the provision of an acceptable texture. One is the heat transfer during baking, which takes place by evaporation and condensation over the pore network. The pore curvature will directly influence this process. The second function is the accumulation of flavour compounds formed by Maillard (browning) reactions during the final period in the oven. Non-polar lipids form a surface film in the pore system, where these flavour compounds are absorbed - just as porous inorganic materials such as zeolites adsorb incoming species. (The relation between absorption and curvature was discussed in Chapter 2.) The crumb structure consists of lamellae, which are symmetric like a soap film, separating the gas pores. The average curvature is constant or nearly so (as determined by the pressure gradient during heating), and the periodicity means that the structure resembles a disordered version of the periodic structures described in Chapter 4 (the L3 "sponge" phase in surfactant-water system is a close analogue). Often the pores become smaller towards the centre of the bread, and this lattice dilatation reflects a persistence of a pressure gradient until the pore structure is "fixed" [20].

Figure 8.6(a)- Schematic drawing of successive shape changes of partly gelatinised wheat starch as seen in the electron microscope.

During heating the surface layer of glutenin molecules in a gel structure is crosslinked by disulfide bridges, and the layer breaks up into rubber-like units; a coagel. The polar lipids take over at the interface during this process as indicated in Fig. 8.7.

Supra self-assembly

355

Figure 8.7: Structure of bread illustrated by lamellae between gas cells/pores. The dotted regions denote coagel formed by gluten proteins [20].

8.2.5 Muscle contraction

Myosin and actin are remarkable molecules, responsible for mobility in animals, from amoeba to mammals. The hexagonal arrangement of the helical structures of actin, myosin and the interpenetrating actin/myosin regions have been well characterised from ultrastructural X-ray and electron microscopy studies. The hexagonal cross-section is shown in Fig. 8.8. The structure is highly hydrated, with short-range disorder combined with long-range order, which is similar to that in lipid systems. The liquid-crystalline character is obvious from the viscoelastic rheological behaviour and the optical birefringence. Despite the fact that the liquid-crystalline nature of the muscle cell was pointed out by Lehmann at the beginning of this century, it seems to have been neglected since then. As will be demonstrated below, it is especially fruitful to consider the long-range periodicity of the structure to understand the mechanisms involved in the contraction process. The molecular events underlying the contraction of muscle involves release of calcium ions from the sarcoplasmatic reticulum as a result of acetylcholine release from the corresponding nerve. Binding of ATP to the myosin heads is the first step in the cycle of relative movement of actin/myosin. Following that, ATP-myosin heads bind to actin molecules, dephosphorylation (the myosin head also functions as an ATP-ase) and dissociation of the actin-myosin complex and

356

Chapter8

of ADP from the myosin. A force, resulting in the relative movement of myosin/actin is produced during these steps, and the cycles are repeated about 50 times per second. The movement in each cycle is about 10 nm.

O @

9

9

O 9

O 9

0

9

.

9

0

@

. 0

9

~ O @

9

~ O 9

- 0

9

~ 9

~ @

-

Figure 8.8: Cross-section of the hexagonal arrangement of myosin (large circles) and actin (small filled circles) in the muscle cell.

"Free" myosin molecules associate spontaneously into bundles like those in the muscle cell. The molecule is one of the longest known in nature (155 nm) with twin-heads extending from the bundle in a helical fashion and the tail consisting of a coiled coil double helix (pitch 7.5 nm). The distal part of this tail (110 nm) forms the bundle and the heads are pointed outwards by a "hinge" (45 nm). The period between the heads along the axis of a bundle is 14.3 nm. It should be remarked, however, that calcium plays a crucial role in the actinmyosin association/dissociation process. Calcium has a binding site on troponin, located on the surface of the actin thread, and binding of calcium triggers the contraction cycle. (Without calcium there is no association whatsoever.) The most intriguing problem in this process is the mechanism behind the production of force and motion from ATP-hydrolysis. It is of interest here to cite Pollard [21]: "Those who believe the lovely textbook drawings that depict tilting

crossbridges pulling actin filaments past myosin thick filaments may even think that the problem has been solved. Much has been learned, but the secrets of cross-bridge motion have resolutely evaded the efforts of a generation of biophysicists and biochemists". Various theories have been introduced to explain the relative movement. Huxley proposed that the myosin head changed its orientation after binding to actin (the "rotating-head" model). According to the helix-coil transition, the normally 0~helical coiled-coil structure "melts" to a random coil conformation, which implies a reduction in length. Movements of the myosin tails, as well as conformation changes of the actin have also been considered. It now seems clear, however, that the force production takes place in or very near the myosin heads [22]. Furthermore, among the three different states of myosin - empty actin site, ATP bound to the actin site and ADP-Pi bound to the actin site - all of which can

Supra self-assembly

357

either be free or bound to actin, there is evidence that the motion is produced by dissociation by the products of ATP hydrolysis. Rigor cross-bridges are considered to represent the final conformation of the ATP-driven contraction. A detailed three-dimensional picture of the rigor structure of actin and myosin in insect flight muscles has been recently reported [22]. It resembles a (principally) hyperbolic surface, defining the interface between actin as well as myosin and the water medium. Furthermore, different conformations of the myosin heads seem to reflect different states of the contraction cycle. A structural dilemma lies in combining a phase concept of the molecular arrangement with the molecular description of the contraction process. It is known that the movements of different myosin heads on a particular myosin thread occur asynchronously, which is evident for example, from X-ray diffraction experiments. That means that a force is generated constantly with time. The coordination of movement between every muscle cell (down to the individual actin/myosin molecule) is controlled by the excitation via the tubuli system, which is a cubosome type of membrane system, described in the previous chapter. The interpenetrating structure (Fig. 8.9) wherein the mobility originates can be described by the Q* surface (discussed above). The long-range periodicity, which is a consequence of such a structural description, is relevant to understand the cooperativity and the requirement of synchronisation of the movement of the individual myosin/actin. The phase/curvature approach to the structure of the muscle cell outlined here not only has a didactic value, but adds a new dimension to the discussion of function mechanisms. The muscle cell contains predominantly water, and the organisation of the selfassembled actin and myosin threads, which is based on hydrophobic interaction, must follow the general principles outlined in Chapter 4. In contrast to the simple liquid-crystalline systems considered there, we do not know the curvature of the interface. A description based on surfaces with constant average curvature nonetheless appears the most reasonable line of attack, due to the analog), with other self-assembled systems. The myosin heads are helically distributed and the actin molecules form helical double strands. There are also additional helical elements attached to the actin threads, but they can be ignored in this context. The cross-sectional arrangement of actin and myosin shown in Fig. 8.8 is consistent with the Q* surface. The myosin molecules are centred on the 62 axes and actin on 31 axes, which occur in the proportion 2:1. The Q* surface partitioning of space into helical channel systems corresponds to the position of the myosin threads. There is thus no connection between adjacent channel systems, i.e. between neighbouring myosin threads. To understand the connections between these channels, we can consider the rectangular nets, which span this surface, shown in Fig. 8.9. The channels exhibit four-coordination; alternatively we can regard the units as four-armed.

358

Chapter8

Figure 8.9(a): The rectangularnets formingthe Q*surface. Two such nets seen along the c-axisare shown above (distance c/6) (broken and full lines). (b) The hexagonalnetwork, characteristic of the desmin network. Two of these directions correspond to up and down along the myosin thread, and the connections in the two lateral directions are also directed upwards and downwards. It is natural then to relate these four-coordinated channel regions of the Q* surface to the myosin heads. Thus the possible connections via the 31centred actin threads of myosin threads have only two directions, which fulfil the crystallographic symmetry of the surface. It is proposed here that these two directions correspond to the initial direction of the myosin head before movement and the end direction after movement, respectively. In this model of the contraction, the time-phase of mobility represents a transient disorder condition of adjacent structure elements, whereas the structure as a whole fulfils the required crystallographic symmetry. If the individual molecular conformational changes must be accommodated within the periodicity of the surface, the necessary perfect long-range synchronisation of mobility over the entire muscle seems a natural consequence. There is a further interesting structural feature of the muscle periodicity. The actin threads in the opposite two directions of the perpendicular cross-section are linked by the so-caUed "desmin" network, containing a rectangular cross-section. The change from a rectangular to a hexagonal arrangement can take place continuously between two surfaces produced by the same network, shown in Fig. 8.9(b). If the adjacent networks are not twisted we obtain the rectangular surface, which can go over to the Q* surface on twisting. There is an important difference between the macroscopic actin/myosin structure in muscle cells discussed above and the separate actin/myosin threads involved in the locomotion of individual cells. This form of myosin (myosin I) is also quite different, with one head instead of two as in muscle cells (myosin II), further it lacks a tail. Another interesting property of myosin I is its membrane

Supra self-assembly

359

association: it can even bind to phospholipid vesicles via anionic phospholipids [23]. Genetic engineering has recently provided a promising approach to the understanding of these functions. A myosin head fragment has been expressed [24] that was found to display actin-activated ATP activity. Future results should improve our understanding of this crucial physico-chemical process.

8.3

The origin of life- a role for cubosomes?

Since Urey reported that small organic molecules resembling amino acids can be formed electrochemically in a mixture of gases that occurred during the early history of earth, numerous theories concerning the origin of life have been forwarded. Following the discovery of hot springs, deep in the oceans, there have been different suggestions of their role in the evolution. The so-called hydrothermal-vent origin-of-life hypothesis assumes a process based on three steps: 1. Synthesis of amino acids and other essential organic compounds in submarine hot springs at > 350~ 2. Synthesis of peptides and nucleotides by thermal dehydration. 3. Synthesis of RNA-like molecules and conversion of these proto-life forms and quenching to colder temperatures of the primitive oceans. Recently, Miller and Bada have questioned this hypothesis, as they showed that these conditions induce decomposition of small organic compounds rather than synthesis [25]. Many reports propose that clay has played an important role, both as a catalyst for condensation of amino acids into peptides [26] and even for information storage. Recall that clays may well exhibit a hyperbolic architecture at a mesostructural level (Chapter 2). Just as zeolites can be templated by hyperbolic surfactant self-assemblies, lipid assemblies can be templated by clays. The molecules that formed the first most primitive form of life had to selfassemble in the "primeval soup". Besides the obvious requirement of selfreproduction, there must have been an encapsulating membrane, separating inside from outside. The cubosome (a dispersed bicontinuous cubic phase, cf. Chapter 5) provides a number of remarkable properties that make it a candidate as an organisational assembly for the earliest forms of life. Hydrocarbon chains attached to polar heads would certainly have been formed frequently in the "primeval soup". The possibility that these self-assembling molecules would form a cubic structure is only a question of molecular dimensions, evident from the discussions in Chapter 5. Further, catalytic properties of a surface with high (negative) Gaussian curvature are likely, as

360

Chapter8

discussed in Chapter 2. A cubic structure could certainly have been catalytically active in the synthesis of these lipid-like molecules producing a cubic bilayer type of structure. Furthermore, only small modifications in the molecular structure, e.g. shorter hydrocarbon chains, would result in the L0~ structure of the bilayer structure. A combination of Lot and cubic bilayers are required for cubosome formation (see Chapter 5). Thus, cubosomes could spontaneously bud off. The cubosome itself, being a catalytic surface for the synthesis, could thus be regarded as a simple template for self-reproduction. A further step in this development might be enzymes, providing the possibility of specificity in the synthesis, thus allowing reproduction of identical cubosomes. We think that lipid-like molecules based on amino acids and hydrocarbon chains could yield cubosomes with the ability to catalyse the synthesis of peptides or RNA-like molecules. The simple amphiphiles produced in the soup might have accidentally included these molecules (for example an amino acid ester of a longchain alcohol). Such compounds have been shown to be very reactive in associated states. For example, long-chain esters of glycine and alanine, spread as monolayers at the air/water interface, have been found to exhibit rapid polycondensation [27]. Lysine hexadecyl ester (dichloride) was observed to undergo fast hydrolysis in water when the concentration exceeds the cmc, whereas the monomeric solution is quite stable [K. Larsson, unpublished data]. Condensation reactions, which have a reaction direction opposite to that determined by the water activity, can be expected to be influenced in the same way by association into micelles or bilayers. Two types of lipid-like molecules are candidates for peptide synthesis in cubosomes; amino acids forming amide links to fatty acids, and amino acids that are ester bound to long-chain alcohols. It is also possible that the regions with highest (negative) Gaussian curvature not only provide a catalytic effect on the condensation reaction, but also will determine molecular selection at the reaction site, due to geometric constraints in this region. Such a curvature-based specificity in reactivity may even provide a simple genetic principle. The full development of enzymes might have taken a long time, with the cubosome providing an effective compartment for this process. The similarities between cubosome and enzyme structures have already been pointed out in Chapter 6. Recall also that enzyme activity is enhanced in cubosomes. There are many alternative ways in which molecular enzyme systems could have developed and taken over the synthesis of the membrane system and managed to self-reproduce. These molecules were not necessarily proteins. The discovery of catalytic activity of RNA by Altman and Cech has heralded the possibility that the first bio-catalytic molecules with ability to produce replicas of itself were RNA. In that case, synthesis of proteins developed later in biomolecular evolution, but once formed, would have taken over enzymatic function, being more effective. This argument still requires a lower level of catalytic function, where the cubosome could have been employed. The curved bilayer may have catalysed the synthesis of lipid-like molecules and RNA-like molecules, until they reached the sophistication required for self-replication.

Origins of life

361

The discovery of archaebacteria as a third kingdom of life (beside the eukaryotes and the prokaryotic eubacteria) has recently changed the picture of the evolution process itself [28]. There are no fossils (or other direct evidence) of preprokaryotic forms of life from the formation of the earth 45 billion years ago until the evolution of microorganisms. The oldest microfossils of prokaryotic cells found are 3.5 billion years old. The earliest fossils of eukaryotic cells are 1.3 billion years old, and multicellular organisms only about 600 million years old, according to macrofossil deposits. The archaebacteria existing today represent very primitive forms of life. The cell membrane lipids of archaebacteria are fundamentally different from those of the other two groups, the eukaryotic cells and eubacteria. They are ether lipids with methyl-branched hydrocarbon chains. Luzzati and coworkers have studied the structural properties of membrane lipids from thermoacidophiles; one group of archaebacteria [29]. It is interesting to note that under physiological conditions, these lipids form cubic phases. Another group of archaebacteria is methanogens, which produce methane from carbon dioxide and hydrogen. These bacteria are killed by oxygen and therefore occur only in anaerobic environment. In the earliest stage of evolution, when there was no oxygen atmosphere (more than 2 billion years ago), these bacteria could have existed everywhere. Their cell walls are periodic open structures ("mesh" phases) of self-assembled S-proteins, discussed in section 4.13. Hopanoids (the most common organic natural product on earth) must have been involved in the evolution of the biomembrane itself. All known membranes contain terpene derivatives, such as cholesterol or carotenoids, which belong to, or can be derived from, hopanoids. However, we still do not know their biological function. Their most commonly proposed mechanism is to regulate membrane fluidity. Another obvious effect is their influence on the lipid bilayer (or monolayer in the case of archaebacteria) curvature. The different types of hopanoids occurring will certainly favour the relative stability of either the planar or of the intrinsically curved membrane conformation. The ether lipids of archaebacteria, which are hopanoid derivatives, forming curved bilayers as discussed above, therefore provide evidence for cubosomes as the first organised form of life. An interesting recent study of a phospholipid-nucleoside conjugate [30] shows the possibility of spontaneous formation in water of super-helical strands. These structures are of interest with regard to the origin of life, as they can function as templates for polymerisation of nucleic acids. We observe that the helicoid is a zero cur~,ature surface with the different relevant and seemingly necessary properties also possessed by cubosomes; catalytic effects, etc. These helical assemblies are therefore alternatives to cubosomes as candidates of prebiotic assemblies, crucial to the first level of evolution. In any case, two-dimensional hyperbolic forms certainly offer many features essential to the most primitive forms of life.

362

8.4

Chapter8

A final word

In 1830, when Poisson took issue with Laplace on the question of liquid structure at interfaces, August Comte, a famous French philosopher of the time wrote: "If mathematics should ever take a prominent place in chemistry, an aberration which is happily almost impossible, it would occasion a widespread and rapid degeneration of that science". Again, and in contrast, Kant wrote of the chemistry and physiology of his day that it was scientia not Scientia, by which he meant and said that unless mathematics and physics figured prominently in chemistry, it was doomed to empiricism. We have come some way since those times, but the gap between, and the very" different aesthetics that divide physics from chemistry, chemistry from biochemistry, biochemistry from biology, have remained and even broadened. It was not always so. Last century the generally informed public could attend meetings of the British Association for the Advancement of Science and follow with interest debates ranging from Speke and Burton's expedition to the sources of the Nile, to Huxley on Darwinism, new theories of electricity, and even the Reverend Challis on capillary action. Indeed the good Master of Trinity College coined the term Mathematical Physics for this last - colloid science and surface chemistry, surfaces and their interactions - he euphemistically called the "highest department of science". The gap was evidently not so large. Mathematics and physics were inextricably interdependent and the early founders of the cell theory made repeated pleas that the role of forces in determining shape and form be taken into account in understanding nature. Mutual incomprehensibility, and a babel of tongues characterise our times. The reasons lie beyond the present questioning of the foundations that concern physicists on the one hand, and evolutionists on the other. They lie beyond the fact that what is axiomatic to one discipline can hardly be accepted for another. For example the "law of mass action" is a beginning for a chemist and the end for a physicist who can go no further without questioning the foundations. It is only in the last decade that some real insights have been gleaned on the real nature and subtlety of molecular forms, and importantly, how they conspire with molecular geometry to set curvature; shape and form. Our perhaps too strident thesis has it that the resulting language of shape brings a certain unity to the scheme of things, the mosaic becomes less blurred. Is that the truth? It is our truth and if the function of science is to provide a predictive dictionary and ordering of events, we thought it worth a try. Perhaps in the end mathematics remains the queen of the sciences, and with Descartes: "cure Deus calculat, J~t mundir' we can agree. But not with that French Comte.

References

363

References 1. H.-G. von Schnering and R. Nesper, Angew. Chem. Int. Ed. Engl., (1987). 26: pp. 1059-1080. 2. I.S. Barnes, S.T. Hyde, and B.W. Ninham, J. Phys. (France), Colloque., (1990). C-7: pp. i9-24. ~

0

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97-99.

S.T. Hyde, Phil. Mag. B, (1988). 57(6): pp. 691-702. M. O'Keeffe and B.G. Hyde, Phil. Mag., (1976). 33: pp. 219-224. M.J. Cooper and M. Sakela, Acta Cryst., (1979). A35: pp. 989-991. S. Andersson, S.T. Hyde, and J.-O. Bovin, Z. Kristallogr., (1985). 173: pp.

7. R.F. Cava, F. Riedinger, and B.J. Wuensch, in "Fast Ion Transport In Solids", P. Vashista, J.N. Mundy, and G.K. Shenoy, Editors. (1979), New Holland: Amsterdam. p. 217. .

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C.E. Derrington and M.O. O'Keeffe, Nature, (1973). 246: p. 44. Z. Blum and S.T. Hyde, Chem. Mat., (1990). 2: pp. 312-314.

10.

D. Greenspan, Appl. Math. Modell., (1983). 7: pp. 423-426.

11.

G.O. Brunner, Zeolites, (1990). 10: pp. 612-614.

12. G. Dijkstra, W.H. Kruizings, and R.M. Kellogg, J. Org. Chem., (1987). 52: p. 4230. 13.

Z. Blum, Acta Chem. Scand., (1989). 43: pp. 248-250.

14.

H.-U. Nissen, Science, (1969). 166: pp. 1150-1152.

15. S.D. Sims, J.M. Didymus, and S. Mann, Chem. Comm., (1995). 1995: pp. 1031-1032. 16.

S. Mann, Struct.Bond., (1993). 54: pp. 125-174.

17.

H. Sawada, H. Konomi, and K. Hirosawa, J. Cell Biol., (1990). 110: p. 219.

18.

A. Sarko and H.C.H. Wu, Starch/Sttirke, (1978). 30-p. 73.

19. A.-C. Eliasson, K. Larsson, S. Andersson, S.T. Hyde, R. Nesper, and H.-G. von Schnering, Starch/Stiirke, (1987). 39: pp. 147-152. 20. A.-C. Eliasson and K. Larsson, "Cereals and BreadmakingMolecular~Colloidal Approach'. (1993), New York: Marcel Dekker. 21.

T.D. Pollard, Cell, (1987). 48"p. 908.

A

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22. 953.

T.R. Hynes, S.M. Block, B.T. White, and J.A. Spudich, Cell, (1987). 48: p.

23.

M.S. Mooseker, Nature, (1989). 340 : p. 505.

4.

25. 6.

D.J. Manstein, K.M. Ruppel, and J.A. Spudich, Science, (1989). 246 : p. 656. S.L. Miller and J.L. Bada, Science, (1988). 334 : p. 809. N. Lahav, D. White and S. Chang, Science, (1978). 201: p. 67.

27. K. Fukuda, Y. Shibasaki, and H. Nakahari, J. Macromol. Sci. Chem., (1981). A15 (5): p. 999. 28.

C.R. Woese, Sci. Amer., (1981). 244 : p. 94.

29. A. Gulik, V. Luzzati, M. DeRosa, and A. Gambacorta, J. Mol. Biol., (1988). 201: p. 429-435. 30. H. Yanagawa, Y. Ogawa, H. Furuta, and A. Tsuno, J. Am. Chem. Soc., (1989). 111 :p. 4567.

Index

I n d e x ( A l l n a m e s are capitalised.)

a-AgI, 341 a-domain structure, 239 a-helix, 237 a/13 barrel structure, 240 Acer, 282

acetylcholine, 189, 249, 355 actin, 355 -myosin complex, 355 action potential, 218 adsorption and Gaussian curvature, 53 adsorption energies, 53 adsorption of alkanes into zeolites, 91 adsorption of gas, long-range attraction, 134 aging, 136 AL see annulate lamellae alcohols, 116 Alhambra, 44 aliphatic hydrocarbons, 73 aluminium-manganese alloys, 44 amide condensation, 237 amorphous lipoprotein matrix, 286 amylopectin, 350 amylose, 353 Anabaena, 273, 278 - variablis, infected cells oL 273 analogy between fast-ion conductors and zeolite synthesis, 342 annulate lameUae, 314 anti-microbial effect, 224 antibodies, 250 antidastic, 15 antiseptic effect, 222 antiviral agent, 247 AOT, 162 AOT-water, partial phase diagram, 162 aphroditidae, 289

apothecial ceils, 286 aragonite, 347 archaebacteria, 186, 330, 361 artefacts in electron microscopy, 259 Arthropoda , 287 ascobolus, 284

asymmetric electrolytes, 126 asymmetric surfaces of nonzero mean curvature, 207 asymmetry between monolayers in cytomembranes, 322 asymptotic directions, 30 ATP, 355 austenite, 56 Aues, 298 axon, squid, 219 ~-PbF2., 342 ~sheet, 237-240 13-sheet- and barrel proteins, 240 B-cells, 250 bacterial sheaths, 186

365

366

Index

bacteriorhodopsin, 242 Bain deformation, 57 balance surfaces, 18 barrel proteins, 247 basolateral membrane of endothelial cells, 259 bending energy, 158 bends, the, 134 BERNOULLI J., 82 bicontinuous cubic phases, 163 bicontinuous structures, 18, 26,157 bilayers, 149 - buckling, 152 - cubic symmetry, 164 - curvature, 152 -hyperbolic, 152,154 bile salts, 207 binding energies of rare gas crystals, 91 bioluminescence, 289 bioluminescent scaleworms, 329 biomineralisation process -in echinoderms, 327, 345 -in radiolaria, 347 biosynthesis of starch, 350 blue fog mesophase, 190 blue phase, 212, 190 - dispersed, 212 DNA, 253 D-surface and the gyroid, 191 blue-green algae, 272 body-centred cubic arrays of reverse spherical micelles, 156 BOLYAI J., 15 bonding in non-ionic crystals, 94 bonds, covalent, 95 Bonnet transformation, 20, 30, 65, 204 cascade oL 253 catenoid and the helicoid, 27 D- and P-surfaces, 26 - Enneper's surface, 33 IPMS, 31 handedness, 244 - hydrocarbon chains, 73 martensite transformation, 56ff. solvation shells, 252 - twinning, 58 Bonnet-Kovalevsky formulae, 9 Born electrostatic self-energy, 89 Born-Oppenheimer approximation, 93 boutons, 220 BPI, BPII, see blue phase brain astrocytes, potassium channels, 227 Brassica napus, 226 bread, 352 bristleworms, 289 bromide as counter-ion, 107 Brunauer-Emmet-Teller theory, 53 brush copolymers, 177 bubble-bubble interactions, 111 -

-

-

-

-

-

-

-

-

Index

bubbles - coalescence, 129 - stability produced by salts, 129 Burgess shales, 135 C 2D conformation associated with the action potential, 219 C 2D membrane, 214 membrane fusion, 215 CaAl2Si208, 63 caesium effect, 344 calcite skeletal network of sea-urchins, 327 calcite, 344 calcium - actin-myosin association/dissociation process, 356 concentration in the plasma membrane, 322 - membrane fusion, 220 calixarenes, 75 carcerands, 75, 78 -

-

Carcinus maenas, 253

carotenoids, 361 cascade of Bonnet transformations, 253 caseins, 207 catenoid, 19, 151, 240 Bonnet transformation, 27 ends, 34 Weierstrass parametrisation, 34 cavitands, 75 cell differentiation, 287 cells, 258 cubic membranes in tumour, 308 explosion and the critical micelle concentration, 222 germ, 275, 287 213 metastasis, 224 muscle, 356 skeletal muscle, 321 spatial organisation, 324 topology, 319 cellulose, 349 cetyl pyridinium chloride, 223 cetyl trimethyl ammonium bromide (CTAB), 225 chain entropy of copolymers, 177 CHALLIS REV., 124 Charae, 278 chemical potential of an aggregate, 119 - regulation of by cubic membranes, 327 chiral molecules in thermotropic liquid crystals, 189 chirality - molecular twist, 187 selectivity, 246 chitosan, 114 chloroform, 221 chloroplast, 226, 267 cholera toxin, 229 cholesteric mesophase, 189, 211 - DNA, 253 -

-

-

-

-

-

L ,

-

-

-

-

-

-

-

367

368

Index

cholesterol, 215, 361 deprivation, 310 esters, 211 oleate, 211 skeletons, 211 - synthesis, 310 Chordata, 294 chromatin, 251 condensation, 252 - in chicken erythrocytes, 252 chromosome, 251 Cidaris rugosa, 345 clathrasils, 61 clathrates of ice, silicon and germanium, 64 clathrin, 229 clay catalysis, 359 cloud point in polyoxyethylene surfactants, 111 CLP-surface, 24, 36, 39 cmc see critical micelle concentration coagel, 354 coated pits, 229 coated vesicles, 229 coesite, 62, 64 collagen, 349 compactin, 310 COMTE A., 362 condensation of amides, 237 of chromatin fibres, 252 conductivity of DDAB microemulsions, 171 conformal symmetry, 69, 72 conformal transformations, 68 conformation pressure on proteins, 216 constant mean curvature surfaces, 257 coordination number in bicontinuous microemulsions, 173 copolymers, 177 sponge and mesh structures, 176,180,184 - strength, 185 coronands, 77 covalent bonds, 95 crab cuticle, 253 cracking of hydrocarbons, 53 cristobalite, 45, 51, 64 critical micelle concentration, 115 and cell explosion, ??? crown ethers, 75, 342 crumbs, structure of, 354 cryptands, 75 cl~/stallisation of zeolites, 342 of a protein, 280 CsCI, 49, 341 CTAB see cetyl trimethyl ammonium bromide Cu4Cd3, 46, 51 cubic ice (I), 51 cubic membranes, 257 - asymmetric unfolding, 321 -

-

-

-

-

-

-

-

-

-

-

Index

cubic membranes, contimw.d - comparison with cubic phases, 322 - in normal cells and tissue, 272 - multicontinuous, 277, 282, 305, 321 - pathological, 285, 307 - regulation of chemical potential, 327 topololD,, 260 cubic phases, 163, 361 - block copolymers, 176, 180 - discontinuous, 156 - in monogalactosyl-, digalactosyl-diacylglycerols, and water, 328 lattice parameter and Gaussian curvature, 164 lipid shape, 205 lipid-water systems, 203, 205 monoglyceride-cytochrome c - water system, 206 - comparison with cubic membranes, 322 lattice parameter and concentration, 166 lattice parameter, 164 rod structures, 203 cubic surfaces, periodic, 257 cubosome, 128, 207, 228, 359 curvature, 3 -comparison with Gibbsian parameters, 118 - control of enzyme systems v/a, 329 -

-

-

-

-

-

-

-

- Gaussian, 5 - geodesic, 7

- integral, 6,14 - intrinsic, of solids, 49, 65 -mean, 5 normal, 4, 187 in atomic nets, 61 - of a bilayer, 152 - polymer chain entropy, 177 - principal, 4 relation to surfactant parameter, 145 ring size, 65 - microemulsions, 175 cydodextrins, 223 cyclosporin A, 223 cytoskeletal network, 258 D-surface, 24, 39, 50, 159,165 - block copolymers, 181 in glycerol monooleate-water mixtures, 166 in solid electrolytes, 341 - membrane system of the prolamellar body, 261 -

-

-

-

-

-

see also I P M S

DDAB, 165,170 "bare" surfactant parameter, 172 - microemulsion conductivity, 171 DDAB-cyclohexane-water system, partial phase diagram, 166 de-gassing, 134 Debye correlation energy, 99 Debye length, 103,109 Debye-Hiickel theory, 89 decagonal symmetry in lyotropic liquid crystals, 71 decemats membrane, 349 -

369

370

Index

defects -disclinations, 161,190 - in solids, 46 pores in membranes, 152 dense silicates, 61 density of frameworks, 59 density of pores and molecular shape, 152 Derjaguin-Landau-Verwey-Overbeek theory of colloid stability, 96,104, 123,131 DESCARTES R., 12, 362 designer-molecules, 76 desmin network, 358 detergents, 207 di-2-ethylhexyl sulfosuccinate, see AOT diacylglycerols, 200 diagram of forces, 347 diamond, 51 diamond network, 25 didodecyl phosphatidylethanolamine-water, partial phase diagram, 160 didodecyldimethyl ammonium bromide see DDAB -

Didymium, 284

differential geometry, 146 differentiating sieve elements, 280ff. digalactosyl-diacylglycerols, 328 digalactosyldiglyceride, partial phase diagram 201 dilatational symmetry, 69 Dioscorea reticulata, 280

dioxin, 247 dipole-dipole interactions, 97 Dirichlet region, 183 disclinations, 161,190 disjoining pressure, 98 disorder and Gaussian curvature of bilayer membrane, 217 disorder in solids, 72 dispersion forces, 97ff. dispersion self-energy, 90 dissolved gas and emulsion stability, 134 DNA, 98,112 and the helicoid, 17 blue phase, 253 cholesteric mesophase, 253 - folding, 251 - mitochondrial, 275 replication, 339 dodecahedrane, 78 dodecahedron, pentagonal, 44 double-layer forces, 97,107,125 drug delivery agents, 209 echinoderm biomineralisation, 327, 344 Ediacara extinctions, 135 E/sen/af0et/da, 293 elastic moduli, 159 electrolytes, 127,176 electron microscopy and artefacts, 259 elliptic points, 15 emulsion stability and dissolved gas, 134 endocytosis, 87, 227, 229 endoplasmic reticulum 267, 287, 309, 314 -

-

-

-

Index

- of tumour and infected cells, 308 endothelial cells, 318 ends in the catenoid, 34

Enneper's surface, 33 entropy of amphiphilic aggregates, 169 enzymes, 216, 221, 243 -curvature, 329 epithelial cells, 294, 305, 329 equipotential surfaces, 340 see also POPS ER, see endoplasmic reticulum erythrocyte membrane phospholipids, 214 ethyl ether, 221 ethylene, 226 eubacteria, 186, 361 Euclid's "Elements", 15 eukaryotes, 361 Euler's theorem, 60, 78 EULER L., 12,19 Euler-Poincar~ characteristic, 13,164 Ewald sum, 94 extinctions in the Ediacara, Permian eras, 135 F-RD surface, 38 F-surface, 141 see a/so D-surface fast-ion conductors, 340, 342 see also solid electrolytes fat adsorption, 329 fatty acids, 225, 246, 248 faujasite, 52, 54 hexagonal, 64 see also zeolite Y ferroelectric phases, 193 Fibonacci sequence, 69, 81 five-fold rotational symmetry, 44, 81 fixing of proteins, 217 fiat points, 5 flip-flop rate, 116, 223 flippase, 217 fluorite, 342 focal surfaces, 147, 156 focussing of interactions, 76, 245 fossils, 361 framework density, 59 frameworks, 62 - dense, 64 - interrupted, 47, 63 free radical production, 127 freezing transition, 72 fructose, 135 frustration in cholesteric mesophases, 190 fuUerenes, 78 fungus, 284 fusion of vesicles, 286 gap junctions, 186 gastro-intestinal system, 229 Gauss map, 6, 21 -

371

372

Index

G A U S S C., 15 Gauss-Bonnet theorem, 10,146 Gaussian curvature, 5 and latticeparameter, 164 and membrane disorder,217 of tangential field surfaces, 342 - link to normal curvature and the geodesic torsion, 60 general anaesthetic, 220 genus, 13,16 of IPMS, 151 geodesic, 8 - curvature, 7 torsion, 9,187 - and framework density, 62 Gergonne's surface, 24 germ cells, 275, 287 germanium, 48 giant single-walled vesicles, 210 Gibbsian parameters and curvature, 118 gismondine, 45 gliadin, 207, 352 globin fold, 239 glucose, 135, 249, 350 gluten, 352 glutenins, 352 glycerol monoolein-water system, 162, 203 - D-surface in, 166 - partialphase diagram, 165 glycolate oxidase, 255 glycolipids,201 glycoproteins, 229 G M O see glycerolmonoolein golden Fibonacci sequences, 68 golden ratio,67, 81 golden rhombuses, 67 Golgi complex, 294, 314 gramicidins, 225 graphite, 78 to diamond transition,56 gyroid, 28,159,165 cholesterolskeletons,212 h-CLP surface,35 H-surface, 24, 35, 39 H-surfaces - see constant mean curvature surfaces H-T surface,38 haem, 239 haemoglobin, 254 halothane, ~ 9 Hamaker constant,99 Hamaker function, 102 handles in surfaces,13,16 hard-sphere van der Waals liquids,105 head-group area, 146 Helfrich force,112 helicoid,20,151, 241, 361 Weierstrass parametrisation, 34 Bonnet transformation, 27, 244 -

-

-

-

-

-

-

-

-

Index

hexagonal faujasite, 64 hexagonal phases, 116, 156, 184, 202, 280 HIV virus, 316 holes in surfaces, 13, 16 homogeneity and symmetry of mesophases, 147,164 homogeneity index, 59, 151,154, 164 homogeneous IPMS, 31, 147 hopanoids, 361 hormones, 248 host-guest chemistry, 76 hydration forces, 98, 107,126 hydrocarbon chains, Bonnet transformation, 73 hydrogen-bond systems in (z-helical and ~-sheet regions, 238 hydrophilic, 115 -lipophih'c balance, 115 hydrophobic, 115 zeolites, 52 forces, effects of dissolved gases, 133 free energy of transfer, 120 interaction, 110,127 gas adsorption, 136 hydrophobicity, immunosuppression, 223 hyperbolic, bilayers, 152,154 - phases in block copolymers, 180 - points, 15 I-WP surface, 37, 39, 50, 165 ice -

-

-

-

-

-

-

clathrates, 64

-cubic (I), 35 frameworks, 64 - Ih,64

-

-

I X , 5 1

icosahedral symmetry, 44, 69 icosahedron, 44, 71 icosidodecahedron, 71

immune system, 250 immunosuppression, 136 - and hydrophobicity, 223 cationic surfactants, 222 incommensurate structures, 44 -

Infinite Periodic .Minimal Surfaces see I P M S insulin, 107, 238, 248 integral curvature, 6 - and topology, 14 inter-lamellar attachments, 226 interfacial topology, 146 intermediate phases, 163, 167 intrinsic curvature - in solids, 51 - bonding geometry, 63 intrinsic geometry, 6,146 invaginations of cell membranes, 317 ion channels, 218

ion-binding, 98, 107 - via NMR, 108 ionic and zwitterionic surfactants, 176

373

Index

374

ionic crystals, formation energy, 94 IPMS, 18, 24, 38,151,159 - CLP-surface, 24, 36, 39 - D-surface, 24, 39, 50,159, 165 F-RD surface, 38 genus, 151 - gyroid, 28, 39,159,165 H-surface, 24, 35, 39 H-T surface, 38 homogeneous, 31,147 I-WP surface, 37, 39, 50,165 - irregular, 27 - mDCLP surface, 39 - mPCLP surface, 39 - mPD surface, 39 - Neovius surface, 37 - oCLP surface, 39 oDa surface, 39 oDb surface, 39 oPa surface, 39 oPb surface, 39 - P-surface, 24, 25, 35, 39,159,165 rPD surface, 39,159,168 - T-surface, 24 tD-surface, 39, 342 tP surface, 39 - VAL surface, 39 isometry, 7, 68 KANT I., 362 keatite, 64 KEPLER J., 66 kneading of dough, 352 KOWALEWSKI J., 66 Krafft temperature, 116 Kurdjomov-Sachs plane correspondence, 58 L-cells, 230 L3 ("sponge") phase, 354 LAGRANGE J., 18 Lamb shift in hydrogen, 91 lamellar lattice, 275 lipid-water structure, 201 membranes, 201 phases, 116 LAPLACE P.-S., 124, 362 lattice bodies, 284 lattice parameter of cubic phases and concentration, 166 cubic phases, 164 - cubic phases vs. cubic membranes, 322 LDL see low density lipoproteins l e c i ~ n s , 167,188 LEHM O., 355 Lemur catta, 307 Leptomonas collosoma, 273 lidocain hydrochloride, 221 Lifshitz theory, 100,125 -

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

Index

light harvesting complex, 329

~8~n, 349 Linde-A, 52 Lindemann's rule, 55 lipids, 176,199 bilayer in membranes as a passive barrier, 213 bilayer, effect of temperature, 202 - composition of synaptic membranes, 220 temperature and pressure effects on, 216 - mobility, 217 molecular shape in cubic mesophases, 205 - theory of the anaesthetic effect, 222 - tubules, 189 poloxamer systems, 322 protein interactions, 215 protein systems, 322 protein-water system, formation of P-, D- and gyroid structures, 204 water cubic phases, 203 lipidic particles, 226 lipopolysaccharides, 322 liposomes, 121, 208 in monoglycerides, 209 liquid crystalline phases in protein aggregates, 193 liquid crystals, 142 LOBACHEVSKY N., 15 Local anaesthetics, 221 local/global phase diagram, 157 -

-

-

c

o

m

p

o

s

i

t

i

o

n

,

-

-

-

-

-

-

-

Locusta migratoria , 253

low density lipoprotein, 212, 229 lubrication of guns, 56 luciferase, 221 luminous scaleworrns, 289 lyotropic liquid crystals, 162, 201 lysophospholipids, 205 MaiUard reactions during baking, 354 mannose receptors, 229 martensite transformation and the Bonnet transformation, 56ff. MAXWELL J., 106,124 MCM-41 "zeolites", 58, 63 mDCLP surface, 39 mean curvature, 5 melanophlogite, 63 membrane cristae, 258 fluctuations, 112 -

-

nu~d~ty, 36x

-

folding, 320 - fusion, 220, 226 - C2D conformation and, 215 calcium ions, 220 lattice, 264, 273 lipids, 200, 213 proteins, 217, 254, 319 storage bodies, 328 tubules, 273, 285 -

-

-

-

-

-

-

375

376

Index

mesh s t z u ~ e s , 17, 156, 163,167,184, 229, 361 - disordered, 168 - in copolymers, 184 in lecithin-water system, 167 square, 168 - vs. IPMS stability, 168 see also T and R phases mesomorphism of ionic surfactants, 162 mesophases, 161 blue, 191, 212, 253 blue fog, 190 - C 2D, 214, 215 cholesteric phase, 189, 211 hexagonal, 156,184, 202, 280 homogeneity and symmetry, 164 - hyperbolic in block copolymers, 184 intermediate, 163 L3, sponge, 354 lamellar, 201 - liquid crystals, 142 - mesh, 17,156,163,167,184,186, 229, 361 miceUar, hexagonal and lamellar, 116 - precholesteric, 253 - reversed spherical and cylindrical micelles, 155 - twisted grain boundary, 192, 253 mesoscopic, 142 metastasis of cells, 224 -

-

-

-

-

-

-

-

-

-

-

Metazoa, 286

methanogens, 361 methyl transferase, 216, 221 MEUSNIER J., 19, 20 micellar phases, 116 miceUes, 143 microemulsion - coordination number of microtubes, 173 - curvature, 175 - definition, 170 DDAB, 170,174 scattering spectra, 172 mid-surface of bilayers, 164 minimal surface, 19 minimal surface, area compared with parallel surfaces, 32 mitochondria, 258, 275, 287 photoreceptor cells of Tupaia glis, 303, 329 respiration reactions, 226 spermatids, 287 steriodogenic cells, 328 - DNA, 275 P-membrane, 274, 286 MnAI4, 72 molecular shape and pore density, 152 molecular shape and tilt, 187 Mollusca, 293 monkey saddle, 24 monocaprin, 225 monogalactosyl-diacylglycerols, 328 -

-

s

-

-

-

-

-

-

t

r

u

c

t

u

r

e

,

Index

monoglyceride-water systems, liposomal dispersions, 209 monoglycerides, 201, 205 monoglyceride--cytochrome c - water cubic phase, 206 monolaurin, 225 monolayers, 154 - in cytomembranes, asymmetry, 322 monoolein, 205, 221 monoolein-water-lysozyme system, 206 mPCLP surface, 39 mPD surface, 39 multicontinuous cubic membranes, 277, 282, 305, 321 muscle cell, 356 - contraction, 355 myosin, 355 NaCd2, 46 NaCI, 43, 49 nautilus shell, 69 N E see nuclear envelope Neovius surface, 37 Nernst glower, 341 nets, 13, 65 networks of protein, 278 nicotinic acetylcholine receptor, 248 NMR, 108, 203, 224 nodal surfaces, periodic, 260 non-Euclidean geometries, 15 non-linear Poisson Boltzmann equation, 104 normal curvature, 4,187 in atomic nets, 61 nuclear envelope, 287, 318 Nymphoides peltata, 283 oCLP surface, 39 oDa surface, 39 oDb surface, 39 oligoglycan, 243 O N E see outer nuclear envelope one- and two-periodic surfaces, 17, 151 one-periodic surfaces, 17 ONSAGER L., 111 oocytes, 287, 289 oPa surface, 39 oPb surface, 39 orientable surfaces, 14 osmiophilic bodies, 286, 294 osmotic shock, 327 outer nuclear envelope, 275, 309 P- and gyroid membrane coexistence, 283 P-surface, 24, 25, 35, 39, 159, 165 lipid monolayer, 203 parabolic point, 15 paracrystalline endoplasmatic reticulum, 289, 291 parallel surfaces, 32,145 pathogenic plant cells, 283 pathological cells and tissues, 272 PAULING L., 60, 80, 238, 254 paulingite, 45, 52 -

-

-

377

378

Index

PC see phosphatidylcholine

PCS see periodic cubic surfaces PE see phosphatidylethanolamine pectins, 349 Pelomyxa carolinensis, 273 Penrose tiling, 67 pentagonal dodecahedron, 44 pentagonal tilings, 44 peptide, 237 perfluorinated surfactant, 168 periodic cubic surfaces, 257 periodic nodal surfaces, 260 periodic zero potential surface see POPS, equipotential surfaces Permian extinctions, 135 phase diagram for amphiphiles in solution, generic, 116,158 phosphatidylcholine (PC), 200, 205, 214, 219, 228 phosphatidylethanolamine (PE), 200, 205, 216, 219 phosphatidylinositol (PI), 215, 226 phosphatidylserine (PS), 215 phospholipase, 214, 218, 329 phospholipids, 201 photoreceptors, 291 photosynthetic centre, 186, 242 lamellae, 272 PI see phosphatidylinositol planar tilings, 44 plasma membrane - glycocalyx, role of, 332 calcium ions, 322 PLB see prolamellar body PM see plasma membrane PNS see periodic nodal surfaces POISSON S.-D., 124, 362 Poisson summation formula, 80 Poisson-Boltzmann theory, 105, 107,121 polyamide, 237 polychaetae, 289 polyelectrolyte, 114,176 polyhedra of five-fold point symmetry, 71 polymers, 176ff. polyoxyethylene surfactants, cloud point, 111 polypeptide, 237 POPS, 49, 94, 340 see also surfaces of zero electrostatic potential, zero equipotential surfaces potassium channels in brain astrocytes, 227 - current in snail neurons, 222 precholesteric phase, 253 pressure and membrane lipid composition, 216 effect on trans/gauche ratio, 216 primary structure of proteins, 237 primeval soup, 359 principal curvatures, 4 principal directions, 4, 5,188 prolameUar body, 215, 226, 261, 275, 327 -

-

-

-

-

-

Index

proteins, 176, 237 - adsorption, 135 coats of viruses, 186 conformation pressure, 216 - crystallisation, 254, 280 - cytoclxrome-c, 107, 206 - fixing, 217 - globin fold, 239 kinase C, 214, 329 - lipid systems, 322 - liquid crystal phases, 193 - membrane rigidity, 319 membrane, 254 networks, 278 phase transitions in lipid membranes, 323 - primary structure, 237 9 - quaternary structures, 237, 253 - secondary structure, 237 structural, 238 -TCDD complex, 247 - tertiary structure, 237 PS see phosphatidylserine pyridinium surfactants, 222 pyrochlore, 71 Pyrrhocoris apterus, 289 Q* surface, 350, 353, 358 quarter-stagger pentafibril model, 349 quartz, 64, 350 quasi-lattice, 68 quasicrystals, 67, 80, 349 - in liquid crystals, 71 quaternary ammonium surfactants, 222 quaternary structure of proteins, 237, 253 radii of gyration of polymer chains, 177 rad/o/ar/a, skeletons of, 347 random coil configuration, 237 rare gas crystals, binding energies of, 91 receptor-ligand binding, 229 red blood cells, 210, 318 regular arrays of tubules, 296 RER see rough endoplasmatic reticulum retinal pigment epithelial cells, 293 reverse micellar cubic phase, 156 reversed bilayer, 153 reversed spherical and cylindrical micelles, 155 Rh7Mg44, 46 rhombicosidodecahedron, 71 rhombohedron, 66, 69, 71 rhombohedral and tetragonal intermediate phases, 162, 167 ribosomes, 326 RIEMANN B, 15, 21 Riemann surface, 21 Riemannian geometry, 141 rings in nets, curvature, 65 ripening of plants, 226 RNA, 339, 360 rod networks and cubic phases, 166, 203 -

-

-

-

-

-

-

379

380

Index

rod structure of square mesh, 168 rotating-head model, 356 rough endoplasmatic reticulum, 314 rPD surface, 39, 159, 168 rutile transition, 56 S-layers, 186 Salmo fario, 298 sarcoplasmic reticulum, 321, 355 scaleworm photocytes, 288 scaleworms, bioluminescence, 329 SCHERK H., 21 Scherk surfaces, Weierstrass parametrisation, 34 Scherk's first and second surfaces, 21 SCHOEN A., 18, 28 SCHWARZ H., 23, 78 schwarzites, 78 SDS see also s o d i u m dodecyl sulfate sea urchins, skeletal network oL 327, 344 second messenger role of diacylglycerols, 329 secondary hydration force, 97,108 secondary structure of proteins, 237 self-assembly, avoidance of, 254 self-intersections, 26 separation of hydrophobic proteins, 135 SER see smooth endoplasmic reticulum shape of molecules and tilt, 187 sickle-ceU anaemia, 254 sieve elements, 275 silica frameworks, 60, 64 silicalite, 64 silicates, dense, 61 see also coesite, cristobalite, keatite, quartz, tridymite, zeolites silicon, 64 single twist chiral packing, 189 skeleton - of echinoderms, 344 - of radiolaria, 347 slime mould, 284 SM see sphingomyelin smooth endoplasmic reticulum, 295, 307 sodalite, 45, 52 sodium dodecyl sulfate-water system, 162 - partial phase diagram, 163 Solfolobus solfataricus, 330

solid electrolytes, 340 see also fast ion conductors solids, intrinsic curvature, 49, 65 solvation shells, Bonnet transformation, 252 solvation, 98 sp 3-, sp 2- and sp-hybridised carbons, 73 spatial requirements of lipid bilayers, 204 specific ion effects, 114, 126 spermatids,287, 293 spherands, 75, 78 sphingomyelin (SM), 219

hutex

spiral - equiangular, 82 - logarithmic, 69 sponge, 17 see also bicontinuous structures spontaneous cavitation, 110, 134 Spurilla nepolitana, cubic membrane, 292 squid axon, 219 star copolymers, 177 starch, 350 - biosynthesis, 350 stellar tetrangulae, 50 sterilising agents, 222 steroid-secreting cells, 307 STESSMAN B., 39 Streptococcus aureus, 222 streptomyces, 230 stroma, 344 structural proteins, 238 Shjela, 294 substrate, 243 sucrose, 135 sugars, 130, 229 sunflower, 69 super-coiling of a telephone cord, 252 surface charge regulation, 125 surfaces balanced, 18 of infinite genus, 17 of nonzero mean curvature, 140, 257 of zero electrostatic potential (POPS), 49, 94, 340 constant mean curvature, 257 parallel, 32,145 - triply periodic, 156 surfactant parameter, 118,144,155 relation to curvatures, 145 surfactants, 115 ionic and zwitter-ionic, 176 symmetry - conformal, 69, 72 decagonal, 71 dilatational, 69 eight- and ten-fold quasisymmetry, 71 - five-fold, 44, 81 - icosahedral, 44, 69 of mesophases and homogeneity, 164 template and zeolite frameworks, 343 synaptic vesicles and cubic membranes, 328 T and R phases, 163, 168 T-cells, 233, 250 T-surface, 24 T-tubule membrane - polymorphism, 302 complex, 299 tangential field surfaces, 340 TCDD, see dioxin tD surface, 39, 342 -

-

-

-

-

-

-

-

-

-

-

-

-

-

381

382

Index

temperature and membrane lipid composition, 216 temperature, effect on bilayer thickness, 202 template, 254, 318 tendons, 349 tenside, 115 tertiary structure of proteins, 237 tetra-alkyl ammonium ions, 342 TFS see tangential field surfaces thermoacidophiles, 361 thermophilic archaebacteria, 330 thermotropic liquid crystals, 161, 189, 201 0-point of polymers, 93,114 THOMPSON D'A., 1, 82,122, 347 three-periodic minimal surfaces see IPMS three-periodic surfaces, 17 thylakoid membranes, 226, 273 topology, 12 and global constraints, 146 and integral curvature, 14 cubic membranes, 260, 317 cytomembranes, 259 genus of IPMS, 151 of cells, 319 - orientability, 14 torsion, 8 9,187 torus, 12, 14, 69 tP surface, 39 trans-membrane channels, 248 trans/gauche ratio in hydrocarbons, effect of pressure, 216 transport and membrane proteins, crystallisation of, 254 triacontahedron, 66 tridymite, 64 triglycerides, 224, 227 tropocoUagen, 335 TRS see tubuloreticular structures tubular cristae, 273 network, 294, 298 tubules (lipid), 189 tubuloreticular structures, 314, 316 tumour cells, 309, 314 Tupaia glis, 319 tweed, 58 twinning and the Bonnet transformation, 58 twist, 187 - number and melting of chiral liquid crystals, 192 - of ~.-sheets, chirality, 240 twisted grain boundary phases, 192, 253 two-dimensional minimal surface with holes, 2.15 two-periodic, 17 ultrafilters, 186 ultrastructure of biological membranes, 266 umbilics, 5, 21 Urchordata, 294 UT-1 cells, 309, 313 VAL surface, 39 -

-

-

-

-

-

- g e o d e s i c ,

-

-

Index

van der Waals and double-layer forces, 104 energy, 89 forces, 97, 125 vesicles, 121, 209 coated, 229 - didodecyldimethyl a m m o n i u m hydroxide, 210 fusion, 219, 286 - giant, 210 - thermodynamic stability, 211 - transport, 258 -

-

-

-

-

Vibrio paranaemolyticus, 225 viral protein coats, 186 viruses, 71, 186, 247 Voronoi cells, 183 W3Fe3C, 50 WAll2, 71 water frameworks, 64 structure, 135 Weierstrass equations, 21, 33 the catenoid, 34 the helicoid, 34 Scherk surfaces, 34 - IPMS, 24ff. WEIERSTRASS K., 4 wheat starch, 351 X-ray diffraction patterns, 202 Xenopus laevis, 299, 315 zeolites, 45, 51, 61 - alkane adsorption, 91 catalysis, 55 - crystallisation, 342 faujasite, 52, 54 gismondine, 45 hexagonal faujasite, 64 - hydrophobic, 52 - Linde A, 52 - MCM materials, 58, 63 - N , 45,52 paulingite, 45, 52 - silicalite, 54 - sodalite, 45, 52 - template symmetry, 342 62, 64 - Z2<5, 45, 52 ZSM-5, 64 zero-point energies, 95 ZKS, 45, 52 ZSM-5, 64 Zygnema, 275 -

-

-

-

-

-

-

-

-

-

-

-

Y

,

383

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