Methods in Enzymology 487 - Computer Methods, Part 3

METHODS IN ENZYMOLOGY Editors-in-Chief

JOHN N. ABELSON AND MELVIN I. SIMON Division of Biology California Institute of Technology Pasadena, California Founding Editors

SIDNEY P. COLOWICK AND NATHAN O. KAPLAN

Academic Press is an imprint of Elsevier 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 32 Jamestown Road, London NW1 7BY, UK First edition 2011 Copyright # 2011, Elsevier Inc. 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 Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: permissions@ elsevier.com. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice 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. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made For information on all Academic Press publications visit our website at elsevierdirect.com ISBN: 978-0-12-381270-4 ISSN: 0076-6879 Printed and bound in United States of America 11 12 13 14 10 9 8 7 6 5 4 3 2 1

CONTRIBUTORS

Wassim Abou-Jaoude´ INRIA Sophia-Antipolis, Sophia-Antipolis, France Kurt S. Anderson Computational Dynamics Lab, Mechanical, Nuclear and Aerospace Engineering Department, Rensselaer Polytechnic Institute, Troy, New York, USA Jordan Ang Department of Chemical and Physical Sciences, and Institute for Optical Sciences, University of Toronto Mississauga, Mississauga, Ontario, Canada F. Angulo-Brown Departamento de Fı´sica, Escuela Superior de Fı´sica y Matema´ticas, Instituto Polite´cnico Nacional, Me´xico D.F., Me´xico David Baker Department of Biochemistry, University of Washington, Seattle, Washington, USA Yih-En Andrew Ban Arzeda Corporation, Seattle, Washington, USA John F. Beausang Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania, USA Monica Berrondo Rosetta Design Group, Fairfax, Virginia, USA Kishor D. Bhalerao Department of Mechanical Engineering, The University of Melbourne, Victoria, Australia Mustafa Burak Boz School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia, USA Philip Bradley Fred Hutchinson Cancer Research Center, Seattle, Washington, USA

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Contributors

Patrik R. Callis Department of Chemistry and Biochemistry, Montana State University, Bozeman, Montana, USA Gregory S. Chirikjian Department of Mechanical Engineering, Johns Hopkins University, Baltimore, Maryland, USA Carson C. Chow Laboratory of Biological Modeling, NIDDK/CEB, National Institutes of Health, Bethesda, Maryland, USA Seth Cooper Department of Computer Science, University of Washington, Seattle, Washington, USA Jacob E. Corn Department of Biochemistry, University of Washington, Seattle, Washington, USA Michelle N. Costa Pacific Northwest National Laboratory, Computational Biology and Bioinformatics Group, Richland, Washington, USA Rhiju Das Stanford University, Stanford, California, USA Ian W. Davis GrassRoots Biotechnology, Durham, North Carolina, USA Batsal Devkota Research Collaboratory for Structural Biology, Department of Chemistry and Chemical Biology, Rutgers University, Piscataway, New Jersey, USA Edward J. Dougherty Steroid Hormones Section, NIDDK/CEB, National Institutes of Health, Bethesda, Maryland, USA Hassan M. Fathallah-Shaykh Department of Mathematics, and Department of Neurology, and Department of Cell Biology, The University of Alabama at Birmingham; The UAB Comprehensive Neuroscience and Cancer Centers, Birmingham, Alabama, USA Sarel J. Fleishman Department of Biochemistry, University of Washington, Seattle, Washington, USA Samuel C. Flores Simbios Center, Bioengineering Department, Stanford University, Clark Center S231, Stanford, California, USA

Contributors

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Yale E. Goldman Pennsylvania Muscle Institute, and Department of Physiology, University of Pennsylvania, Philadelphia, Pennsylvania, USA Didier Gonze Laboratoire de Bioinformatique des Ge´nomes et des Re´seaux, Universite´ Libre de Bruxelles, Bruxelles, Belgium Jeffrey J. Gray Chemical & Biomolecular Engineering and the Program in Molecular Biophysics, Johns Hopkins University, Baltimore, Maryland, USA Maria Luisa Guerriero Centre for Systems Biology at Edinburgh, University of Edinburgh, Edinburgh, United Kingdom Shailendra K. Gupta Department of Systems Biology and Bioinformatics, University of Rostock, Rostock, Germany, and Indian Institute of Toxicology Research (CSIR), Lucknow, India L. Guzma´n-Vargas Unidad Profesional Interdisciplinaria en Ingenierı´a y Tecnologı´as Avanzadas, Instituto Polite´cnico Nacional, Me´xico D.F., Me´xico Jose´ Halloy Service d’Ecologie Sociale, Universite´ Libre de Bruxelles, Bruxelles, Belgium Stephen C. Harvey School of Biology, and School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia, USA James J. Havranek Washington University, St. Louis, Missouri, USA John K. Heath School of Biosciences and Centre for Systems Biology, University of Birmingham, Edgbaston, Birmingham, United Kingdom R. Herna´ndez-Pe´rez SATMEX, Av. de las Telecomunicaciones S/N CONTEL Edif. SGA-II, Me´xico D.F., Me´xico Brian Ingalls Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada Ron Jacak Department of Biochemistry, University of North Carolina, Chapel Hill, North Carolina, USA

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Contributors

John Karanicolas Center for Bioinformatics, and Department of Molecular Biosciences, University of Kansas, Lawrence, Kansas, USA Kristian Kaufman Department of Biochemistry, Vanderbuilt University, Nashville, Tennessee, USA Caner Kazanci Department of Mathematics/Faculty of Engineering, University of Georgia, Athens, Georgia, USA David E. Kim Department of Biochemistry, University of Washington, Seattle, Washington, USA Tanja Kortemme University of California, San Francisco, California, USA Brian Kuhlman Department of Biochemistry, University of North Carolina, Chapel Hill, North Carolina, USA Don Kulasiri Department of Molecular Biosciences, Centre for Advanced Computational Solutions (C-fACS), Lincoln University, Lincoln, Christchurch, New Zealand Alain Laederach Department of Biomedical Sciences, University at Albany, and Developmental Genetics and Bioinformatics, Wadsworth Center, Albany, New York, USA Oliver F. Lange Department of Biochemistry, University of Washington, Seattle, Washington, USA Andrew Leaver-Fay Department of Biochemistry, University of North Carolina, Chapel Hill, North Carolina, USA Daniel Levine Department of Biochemistry, Robert Wood Johnson Medical School, UMDNJ, and The Center for Advanced Biotechnology and Medicine, Piscataway, New Jersey, USA Steven M. Lewis Department of Biochemistry, University of North Carolina, Chapel Hill, North Carolina, USA Feng Li Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, Maryland, USA

Contributors

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Sergey Lyskov Chemical & Biomolecular Engineering and the Program in Molecular Biophysics, Johns Hopkins University, Baltimore, Maryland, USA Daniel J. Mandell University of California, San Francisco, California, USA Alberto Marin-Sanguino Department of Membrane Biochemistry, Max Planck Institute of Biochemistry, Martinsried, Germany David McMillen Department of Chemical and Physical Sciences, and Institute for Optical Sciences, University of Toronto Mississauga, Mississauga, Ontario, Canada Jens Meiler Department of Biochemistry, Vanderbuilt University, Nashville, Tennessee, USA Stuart Mentzer Objexx Engineering, Boston, Massachusetts, USA Radhakrishnan Nagarajan Division of Biomedical Informatics, University of Arkansas for Medical Sciences, Little Rock, Arkansas, USA Vikas Nanda Department of Biochemistry, Robert Wood Johnson Medical School, UMDNJ, and The Center for Advanced Biotechnology and Medicine, Piscataway, New Jersey, USA Philip C. Nelson Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania, USA Karen M. Ong Laboratory of Biological Modeling, NIDDK/CEB, National Institutes of Health, Bethesda, Maryland, USA Djomangan Adama Ouattara INERIS, Parc Technologique Alata, Vermuil-en-Halatte, France, UMR-CNRS 6600, Universite´ de Technologie de Compie´gne, France Anton S. Petrov School of Biology, Georgia Institute of Technology, Atlanta, Georgia, USA Zoran Popovic´ GrassRoots Biotechnology, Durham, North Carolina, USA

and

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Contributors

Mohammad Poursina Computational Dynamics Lab, Mechanical, Nuclear and Aerospace Engineering Department, Rensselaer Polytechnic Institute, Troy, New York, USA P. Douglas Renfrew Center for Genomics and Systems Biology, New York University, New York, USA Haluk Resat Pacific Northwest National Laboratory, Computational Biology and Bioinformatics Group, Richland, Washington, USA I. Reyes-Ramı´rez Unidad Profesional Interdisciplinaria en Ingenierı´a y Tecnologı´as Avanzadas, Instituto Polite´cnico Nacional, Me´xico D.F., Me´xico Joshua S. Richman Department of Medicine, Division of Preventive Medicine, University of Alabama School of Medicine, Birmingham, Alabama, USA Florian Richter Department of Biochemistry, University of Washington, Seattle, Washington, USA M. Santilla´n Centro de Investigacio´n y Estudios Avanzados del IPN, Unidad Monterrey, Parque de Investigacio´n e Innovacio´n Tecnolo´gica, Apodaca, NL, Me´xico, and Centre for Applied Mathematics in Bioscience and Medicine, 3655 Promenade Sir William Osler, McIntyre Medical Building, Montreal, Canada Elizabeth Y. Scribner Department of Mathematics, The University of Alabama at Birmingham, Birmingham, Alabama, USA Franc¸oise Seillier-Moiseiwitsch Infectious Disease Clinical Research Program, Department of Preventive Medicine and Biometrics, Uniformed Services University of the Health Sciences, Bethesda, Maryland, USA Harish Shankaran Pacific Northwest National Laboratory, Computational Biology and Bioinformatics Group, Richland, Washington, USA Will Sheffler Department of Biochemistry, University of Washington, Seattle, Washington, USA

Contributors

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S. Stoney Simons Jr. Steroid Hormones Section, NIDDK/CEB, National Institutes of Health, Bethesda, Maryland, USA Colin A. Smith University of California, San Francisco, California, USA James Thompson Department of Biochemistry, University of Washington, Seattle, Washington, USA Adrien Treuille Computer Science Department, Carnegie Mellon University, Pittsburgh, Pennsylvania, USA Michael Tyka Department of Biochemistry, University of Washington, Seattle, Washington, USA Meenakshi Upreti Division of Radiation Oncology, University of Arkansas for Medical Sciences, Little Rock, Arkansas, USA Julio Vera Department of Systems Biology and Bioinformatics, University of Rostock, Rostock, Germany Eberhard O. Voit Integrative BioSystems Institute, and The Wallace H. Coulter Department of Biomedical Engineering, Georgia Institute of Technology, Atlanta, Georgia, USA Fei Xu Department of Biochemistry, Robert Wood Johnson Medical School, UMDNJ, and The Center for Advanced Biotechnology and Medicine, Piscataway, New Jersey, USA Necmettin Yildirim Division of Natural Sciences, New College of Florida, Sarasota, Florida, USA Sohail Zahid Department of Biochemistry, Robert Wood Johnson Medical School, UMDNJ, and The Center for Advanced Biotechnology and Medicine, Piscataway, New Jersey, USA E. S. Zeron Centro de Investigacio´n y de Estudios Avanzados del IPN, Departamento de Matema´ticas, Av. Instituto Polite´cnico Nacional 2508, Me´xico DF, Me´xico

PREFACE

The general process of conducting scientific research involves the comparison of experimental results with hypothesis-driven predictions of the experimental results. Traditionally, these hypothesis-driven predictions were commonly based upon “back of the envelope” calculations and “hand waving” predictions. Fortunately, these traditional methods have been replaced with more rigorous mathematical and computational approaches. The use of computers and computational methods has become ubiquitous in biological and biomedical research. This has been driven by numerous factors, a few of which follow: One primary reason is the emphasis being placed on computers and computational methods within the National Institutes of Health (NIH) Roadmap; another factor is the increased level of mathematical and computational sophistication among researchers, particularly amongst junior scientists, students, journal reviewers, and Study Section members; another is the rapid advances in computer hardware and software which make these methods far more accessible to the rank and file research community. A common perception is that the only applications of computers and computer methods in biological and biomedical research are either basic statistical analysis or the searching of DNA sequence databases. While these are obviously important applications, they only scratch the surface of the current and potential applications of computers and computer methods in biomedical research. The various chapters within this volume include a wide variety of applications that extend well beyond this limited perception. The training of the majority of senior MDs and Ph.Ds in clinical or basic disciplines at academic research and medical centers commonly does not include advanced coursework in mathematics, numerical analysis, statistics, or computer science. Thus, the chapters within this volume, and our previous volumes, have been written in order to be accessible to this target audience. MICHAEL L. JOHNSON AND LUDWIG BRAND

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VOLUME I. Preparation and Assay of Enzymes Edited by SIDNEY P. COLOWICK AND NATHAN O. KAPLAN VOLUME II. Preparation and Assay of Enzymes Edited by SIDNEY P. COLOWICK AND NATHAN O. KAPLAN VOLUME III. Preparation and Assay of Substrates Edited by SIDNEY P. COLOWICK AND NATHAN O. KAPLAN VOLUME IV. Special Techniques for the Enzymologist Edited by SIDNEY P. COLOWICK AND NATHAN O. KAPLAN VOLUME V. Preparation and Assay of Enzymes Edited by SIDNEY P. COLOWICK AND NATHAN O. KAPLAN VOLUME VI. Preparation and Assay of Enzymes (Continued) Preparation and Assay of Substrates Special Techniques Edited by SIDNEY P. COLOWICK AND NATHAN O. KAPLAN VOLUME VII. Cumulative Subject Index Edited by SIDNEY P. COLOWICK AND NATHAN O. KAPLAN VOLUME VIII. Complex Carbohydrates Edited by ELIZABETH F. NEUFELD AND VICTOR GINSBURG VOLUME IX. Carbohydrate Metabolism Edited by WILLIS A. WOOD VOLUME X. Oxidation and Phosphorylation Edited by RONALD W. ESTABROOK AND MAYNARD E. PULLMAN VOLUME XI. Enzyme Structure Edited by C. H. W. HIRS VOLUME XII. Nucleic Acids (Parts A and B) Edited by LAWRENCE GROSSMAN AND KIVIE MOLDAVE VOLUME XIII. Citric Acid Cycle Edited by J. M. LOWENSTEIN VOLUME XIV. Lipids Edited by J. M. LOWENSTEIN VOLUME XV. Steroids and Terpenoids Edited by RAYMOND B. CLAYTON xxiii

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VOLUME XVI. Fast Reactions Edited by KENNETH KUSTIN VOLUME XVII. Metabolism of Amino Acids and Amines (Parts A and B) Edited by HERBERT TABOR AND CELIA WHITE TABOR VOLUME XVIII. Vitamins and Coenzymes (Parts A, B, and C) Edited by DONALD B. MCCORMICK AND LEMUEL D. WRIGHT VOLUME XIX. Proteolytic Enzymes Edited by GERTRUDE E. PERLMANN AND LASZLO LORAND VOLUME XX. Nucleic Acids and Protein Synthesis (Part C) Edited by KIVIE MOLDAVE AND LAWRENCE GROSSMAN VOLUME XXI. Nucleic Acids (Part D) Edited by LAWRENCE GROSSMAN AND KIVIE MOLDAVE VOLUME XXII. Enzyme Purification and Related Techniques Edited by WILLIAM B. JAKOBY VOLUME XXIII. Photosynthesis (Part A) Edited by ANTHONY SAN PIETRO VOLUME XXIV. Photosynthesis and Nitrogen Fixation (Part B) Edited by ANTHONY SAN PIETRO VOLUME XXV. Enzyme Structure (Part B) Edited by C. H. W. HIRS AND SERGE N. TIMASHEFF VOLUME XXVI. Enzyme Structure (Part C) Edited by C. H. W. HIRS AND SERGE N. TIMASHEFF VOLUME XXVII. Enzyme Structure (Part D) Edited by C. H. W. HIRS AND SERGE N. TIMASHEFF VOLUME XXVIII. Complex Carbohydrates (Part B) Edited by VICTOR GINSBURG VOLUME XXIX. Nucleic Acids and Protein Synthesis (Part E) Edited by LAWRENCE GROSSMAN AND KIVIE MOLDAVE VOLUME XXX. Nucleic Acids and Protein Synthesis (Part F) Edited by KIVIE MOLDAVE AND LAWRENCE GROSSMAN VOLUME XXXI. Biomembranes (Part A) Edited by SIDNEY FLEISCHER AND LESTER PACKER VOLUME XXXII. Biomembranes (Part B) Edited by SIDNEY FLEISCHER AND LESTER PACKER VOLUME XXXIII. Cumulative Subject Index Volumes I-XXX Edited by MARTHA G. DENNIS AND EDWARD A. DENNIS VOLUME XXXIV. Affinity Techniques (Enzyme Purification: Part B) Edited by WILLIAM B. JAKOBY AND MEIR WILCHEK

Methods in Enzymology

VOLUME XXXV. Lipids (Part B) Edited by JOHN M. LOWENSTEIN VOLUME XXXVI. Hormone Action (Part A: Steroid Hormones) Edited by BERT W. O’MALLEY AND JOEL G. HARDMAN VOLUME XXXVII. Hormone Action (Part B: Peptide Hormones) Edited by BERT W. O’MALLEY AND JOEL G. HARDMAN VOLUME XXXVIII. Hormone Action (Part C: Cyclic Nucleotides) Edited by JOEL G. HARDMAN AND BERT W. O’MALLEY VOLUME XXXIX. Hormone Action (Part D: Isolated Cells, Tissues, and Organ Systems) Edited by JOEL G. HARDMAN AND BERT W. O’MALLEY VOLUME XL. Hormone Action (Part E: Nuclear Structure and Function) Edited by BERT W. O’MALLEY AND JOEL G. HARDMAN VOLUME XLI. Carbohydrate Metabolism (Part B) Edited by W. A. WOOD VOLUME XLII. Carbohydrate Metabolism (Part C) Edited by W. A. WOOD VOLUME XLIII. Antibiotics Edited by JOHN H. HASH VOLUME XLIV. Immobilized Enzymes Edited by KLAUS MOSBACH VOLUME XLV. Proteolytic Enzymes (Part B) Edited by LASZLO LORAND VOLUME XLVI. Affinity Labeling Edited by WILLIAM B. JAKOBY AND MEIR WILCHEK VOLUME XLVII. Enzyme Structure (Part E) Edited by C. H. W. HIRS AND SERGE N. TIMASHEFF VOLUME XLVIII. Enzyme Structure (Part F) Edited by C. H. W. HIRS AND SERGE N. TIMASHEFF VOLUME XLIX. Enzyme Structure (Part G) Edited by C. H. W. HIRS AND SERGE N. TIMASHEFF VOLUME L. Complex Carbohydrates (Part C) Edited by VICTOR GINSBURG VOLUME LI. Purine and Pyrimidine Nucleotide Metabolism Edited by PATRICIA A. HOFFEE AND MARY ELLEN JONES VOLUME LII. Biomembranes (Part C: Biological Oxidations) Edited by SIDNEY FLEISCHER AND LESTER PACKER

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VOLUME LIII. Biomembranes (Part D: Biological Oxidations) Edited by SIDNEY FLEISCHER AND LESTER PACKER VOLUME LIV. Biomembranes (Part E: Biological Oxidations) Edited by SIDNEY FLEISCHER AND LESTER PACKER VOLUME LV. Biomembranes (Part F: Bioenergetics) Edited by SIDNEY FLEISCHER AND LESTER PACKER VOLUME LVI. Biomembranes (Part G: Bioenergetics) Edited by SIDNEY FLEISCHER AND LESTER PACKER VOLUME LVII. Bioluminescence and Chemiluminescence Edited by MARLENE A. DELUCA VOLUME LVIII. Cell Culture Edited by WILLIAM B. JAKOBY AND IRA PASTAN VOLUME LIX. Nucleic Acids and Protein Synthesis (Part G) Edited by KIVIE MOLDAVE AND LAWRENCE GROSSMAN VOLUME LX. Nucleic Acids and Protein Synthesis (Part H) Edited by KIVIE MOLDAVE AND LAWRENCE GROSSMAN VOLUME 61. Enzyme Structure (Part H) Edited by C. H. W. HIRS AND SERGE N. TIMASHEFF VOLUME 62. Vitamins and Coenzymes (Part D) Edited by DONALD B. MCCORMICK AND LEMUEL D. WRIGHT VOLUME 63. Enzyme Kinetics and Mechanism (Part A: Initial Rate and Inhibitor Methods) Edited by DANIEL L. PURICH VOLUME 64. Enzyme Kinetics and Mechanism (Part B: Isotopic Probes and Complex Enzyme Systems) Edited by DANIEL L. PURICH VOLUME 65. Nucleic Acids (Part I) Edited by LAWRENCE GROSSMAN AND KIVIE MOLDAVE VOLUME 66. Vitamins and Coenzymes (Part E) Edited by DONALD B. MCCORMICK AND LEMUEL D. WRIGHT VOLUME 67. Vitamins and Coenzymes (Part F) Edited by DONALD B. MCCORMICK AND LEMUEL D. WRIGHT VOLUME 68. Recombinant DNA Edited by RAY WU VOLUME 69. Photosynthesis and Nitrogen Fixation (Part C) Edited by ANTHONY SAN PIETRO VOLUME 70. Immunochemical Techniques (Part A) Edited by HELEN VAN VUNAKIS AND JOHN J. LANGONE

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VOLUME 71. Lipids (Part C) Edited by JOHN M. LOWENSTEIN VOLUME 72. Lipids (Part D) Edited by JOHN M. LOWENSTEIN VOLUME 73. Immunochemical Techniques (Part B) Edited by JOHN J. LANGONE AND HELEN VAN VUNAKIS VOLUME 74. Immunochemical Techniques (Part C) Edited by JOHN J. LANGONE AND HELEN VAN VUNAKIS VOLUME 75. Cumulative Subject Index Volumes XXXI, XXXII, XXXIV–LX Edited by EDWARD A. DENNIS AND MARTHA G. DENNIS VOLUME 76. Hemoglobins Edited by ERALDO ANTONINI, LUIGI ROSSI-BERNARDI, AND EMILIA CHIANCONE VOLUME 77. Detoxication and Drug Metabolism Edited by WILLIAM B. JAKOBY VOLUME 78. Interferons (Part A) Edited by SIDNEY PESTKA VOLUME 79. Interferons (Part B) Edited by SIDNEY PESTKA VOLUME 80. Proteolytic Enzymes (Part C) Edited by LASZLO LORAND VOLUME 81. Biomembranes (Part H: Visual Pigments and Purple Membranes, I) Edited by LESTER PACKER VOLUME 82. Structural and Contractile Proteins (Part A: Extracellular Matrix) Edited by LEON W. CUNNINGHAM AND DIXIE W. FREDERIKSEN VOLUME 83. Complex Carbohydrates (Part D) Edited by VICTOR GINSBURG VOLUME 84. Immunochemical Techniques (Part D: Selected Immunoassays) Edited by JOHN J. LANGONE AND HELEN VAN VUNAKIS VOLUME 85. Structural and Contractile Proteins (Part B: The Contractile Apparatus and the Cytoskeleton) Edited by DIXIE W. FREDERIKSEN AND LEON W. CUNNINGHAM VOLUME 86. Prostaglandins and Arachidonate Metabolites Edited by WILLIAM E. M. LANDS AND WILLIAM L. SMITH VOLUME 87. Enzyme Kinetics and Mechanism (Part C: Intermediates, Stereo-chemistry, and Rate Studies) Edited by DANIEL L. PURICH VOLUME 88. Biomembranes (Part I: Visual Pigments and Purple Membranes, II) Edited by LESTER PACKER

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VOLUME 89. Carbohydrate Metabolism (Part D) Edited by WILLIS A. WOOD VOLUME 90. Carbohydrate Metabolism (Part E) Edited by WILLIS A. WOOD VOLUME 91. Enzyme Structure (Part I) Edited by C. H. W. HIRS AND SERGE N. TIMASHEFF VOLUME 92. Immunochemical Techniques (Part E: Monoclonal Antibodies and General Immunoassay Methods) Edited by JOHN J. LANGONE AND HELEN VAN VUNAKIS VOLUME 93. Immunochemical Techniques (Part F: Conventional Antibodies, Fc Receptors, and Cytotoxicity) Edited by JOHN J. LANGONE AND HELEN VAN VUNAKIS VOLUME 94. Polyamines Edited by HERBERT TABOR AND CELIA WHITE TABOR VOLUME 95. Cumulative Subject Index Volumes 61–74, 76–80 Edited by EDWARD A. DENNIS AND MARTHA G. DENNIS VOLUME 96. Biomembranes [Part J: Membrane Biogenesis: Assembly and Targeting (General Methods; Eukaryotes)] Edited by SIDNEY FLEISCHER AND BECCA FLEISCHER VOLUME 97. Biomembranes [Part K: Membrane Biogenesis: Assembly and Targeting (Prokaryotes, Mitochondria, and Chloroplasts)] Edited by SIDNEY FLEISCHER AND BECCA FLEISCHER VOLUME 98. Biomembranes (Part L: Membrane Biogenesis: Processing and Recycling) Edited by SIDNEY FLEISCHER AND BECCA FLEISCHER VOLUME 99. Hormone Action (Part F: Protein Kinases) Edited by JACKIE D. CORBIN AND JOEL G. HARDMAN VOLUME 100. Recombinant DNA (Part B) Edited by RAY WU, LAWRENCE GROSSMAN, AND KIVIE MOLDAVE VOLUME 101. Recombinant DNA (Part C) Edited by RAY WU, LAWRENCE GROSSMAN, AND KIVIE MOLDAVE VOLUME 102. Hormone Action (Part G: Calmodulin and Calcium-Binding Proteins) Edited by ANTHONY R. MEANS AND BERT W. O’MALLEY VOLUME 103. Hormone Action (Part H: Neuroendocrine Peptides) Edited by P. MICHAEL CONN VOLUME 104. Enzyme Purification and Related Techniques (Part C) Edited by WILLIAM B. JAKOBY

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VOLUME 105. Oxygen Radicals in Biological Systems Edited by LESTER PACKER VOLUME 106. Posttranslational Modifications (Part A) Edited by FINN WOLD AND KIVIE MOLDAVE VOLUME 107. Posttranslational Modifications (Part B) Edited by FINN WOLD AND KIVIE MOLDAVE VOLUME 108. Immunochemical Techniques (Part G: Separation and Characterization of Lymphoid Cells) Edited by GIOVANNI DI SABATO, JOHN J. LANGONE, AND HELEN VAN VUNAKIS VOLUME 109. Hormone Action (Part I: Peptide Hormones) Edited by LUTZ BIRNBAUMER AND BERT W. O’MALLEY VOLUME 110. Steroids and Isoprenoids (Part A) Edited by JOHN H. LAW AND HANS C. RILLING VOLUME 111. Steroids and Isoprenoids (Part B) Edited by JOHN H. LAW AND HANS C. RILLING VOLUME 112. Drug and Enzyme Targeting (Part A) Edited by KENNETH J. WIDDER AND RALPH GREEN VOLUME 113. Glutamate, Glutamine, Glutathione, and Related Compounds Edited by ALTON MEISTER VOLUME 114. Diffraction Methods for Biological Macromolecules (Part A) Edited by HAROLD W. WYCKOFF, C. H. W. HIRS, AND SERGE N. TIMASHEFF VOLUME 115. Diffraction Methods for Biological Macromolecules (Part B) Edited by HAROLD W. WYCKOFF, C. H. W. HIRS, AND SERGE N. TIMASHEFF VOLUME 116. Immunochemical Techniques (Part H: Effectors and Mediators of Lymphoid Cell Functions) Edited by GIOVANNI DI SABATO, JOHN J. LANGONE, AND HELEN VAN VUNAKIS VOLUME 117. Enzyme Structure (Part J) Edited by C. H. W. HIRS AND SERGE N. TIMASHEFF VOLUME 118. Plant Molecular Biology Edited by ARTHUR WEISSBACH AND HERBERT WEISSBACH VOLUME 119. Interferons (Part C) Edited by SIDNEY PESTKA VOLUME 120. Cumulative Subject Index Volumes 81–94, 96–101 VOLUME 121. Immunochemical Techniques (Part I: Hybridoma Technology and Monoclonal Antibodies) Edited by JOHN J. LANGONE AND HELEN VAN VUNAKIS VOLUME 122. Vitamins and Coenzymes (Part G) Edited by FRANK CHYTIL AND DONALD B. MCCORMICK

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VOLUME 123. Vitamins and Coenzymes (Part H) Edited by FRANK CHYTIL AND DONALD B. MCCORMICK VOLUME 124. Hormone Action (Part J: Neuroendocrine Peptides) Edited by P. MICHAEL CONN VOLUME 125. Biomembranes (Part M: Transport in Bacteria, Mitochondria, and Chloroplasts: General Approaches and Transport Systems) Edited by SIDNEY FLEISCHER AND BECCA FLEISCHER VOLUME 126. Biomembranes (Part N: Transport in Bacteria, Mitochondria, and Chloroplasts: Protonmotive Force) Edited by SIDNEY FLEISCHER AND BECCA FLEISCHER VOLUME 127. Biomembranes (Part O: Protons and Water: Structure and Translocation) Edited by LESTER PACKER VOLUME 128. Plasma Lipoproteins (Part A: Preparation, Structure, and Molecular Biology) Edited by JERE P. SEGREST AND JOHN J. ALBERS VOLUME 129. Plasma Lipoproteins (Part B: Characterization, Cell Biology, and Metabolism) Edited by JOHN J. ALBERS AND JERE P. SEGREST VOLUME 130. Enzyme Structure (Part K) Edited by C. H. W. HIRS AND SERGE N. TIMASHEFF VOLUME 131. Enzyme Structure (Part L) Edited by C. H. W. HIRS AND SERGE N. TIMASHEFF VOLUME 132. Immunochemical Techniques (Part J: Phagocytosis and Cell-Mediated Cytotoxicity) Edited by GIOVANNI DI SABATO AND JOHANNES EVERSE VOLUME 133. Bioluminescence and Chemiluminescence (Part B) Edited by MARLENE DELUCA AND WILLIAM D. MCELROY VOLUME 134. Structural and Contractile Proteins (Part C: The Contractile Apparatus and the Cytoskeleton) Edited by RICHARD B. VALLEE VOLUME 135. Immobilized Enzymes and Cells (Part B) Edited by KLAUS MOSBACH VOLUME 136. Immobilized Enzymes and Cells (Part C) Edited by KLAUS MOSBACH VOLUME 137. Immobilized Enzymes and Cells (Part D) Edited by KLAUS MOSBACH VOLUME 138. Complex Carbohydrates (Part E) Edited by VICTOR GINSBURG

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VOLUME 139. Cellular Regulators (Part A: Calcium- and Calmodulin-Binding Proteins) Edited by ANTHONY R. MEANS AND P. MICHAEL CONN VOLUME 140. Cumulative Subject Index Volumes 102–119, 121–134 VOLUME 141. Cellular Regulators (Part B: Calcium and Lipids) Edited by P. MICHAEL CONN AND ANTHONY R. MEANS VOLUME 142. Metabolism of Aromatic Amino Acids and Amines Edited by SEYMOUR KAUFMAN VOLUME 143. Sulfur and Sulfur Amino Acids Edited by WILLIAM B. JAKOBY AND OWEN GRIFFITH VOLUME 144. Structural and Contractile Proteins (Part D: Extracellular Matrix) Edited by LEON W. CUNNINGHAM VOLUME 145. Structural and Contractile Proteins (Part E: Extracellular Matrix) Edited by LEON W. CUNNINGHAM VOLUME 146. Peptide Growth Factors (Part A) Edited by DAVID BARNES AND DAVID A. SIRBASKU VOLUME 147. Peptide Growth Factors (Part B) Edited by DAVID BARNES AND DAVID A. SIRBASKU VOLUME 148. Plant Cell Membranes Edited by LESTER PACKER AND ROLAND DOUCE VOLUME 149. Drug and Enzyme Targeting (Part B) Edited by RALPH GREEN AND KENNETH J. WIDDER VOLUME 150. Immunochemical Techniques (Part K: In Vitro Models of B and T Cell Functions and Lymphoid Cell Receptors) Edited by GIOVANNI DI SABATO VOLUME 151. Molecular Genetics of Mammalian Cells Edited by MICHAEL M. GOTTESMAN VOLUME 152. Guide to Molecular Cloning Techniques Edited by SHELBY L. BERGER AND ALAN R. KIMMEL VOLUME 153. Recombinant DNA (Part D) Edited by RAY WU AND LAWRENCE GROSSMAN VOLUME 154. Recombinant DNA (Part E) Edited by RAY WU AND LAWRENCE GROSSMAN VOLUME 155. Recombinant DNA (Part F) Edited by RAY WU VOLUME 156. Biomembranes (Part P: ATP-Driven Pumps and Related Transport: The Na, K-Pump) Edited by SIDNEY FLEISCHER AND BECCA FLEISCHER

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VOLUME 157. Biomembranes (Part Q: ATP-Driven Pumps and Related Transport: Calcium, Proton, and Potassium Pumps) Edited by SIDNEY FLEISCHER AND BECCA FLEISCHER VOLUME 158. Metalloproteins (Part A) Edited by JAMES F. RIORDAN AND BERT L. VALLEE VOLUME 159. Initiation and Termination of Cyclic Nucleotide Action Edited by JACKIE D. CORBIN AND ROGER A. JOHNSON VOLUME 160. Biomass (Part A: Cellulose and Hemicellulose) Edited by WILLIS A. WOOD AND SCOTT T. KELLOGG VOLUME 161. Biomass (Part B: Lignin, Pectin, and Chitin) Edited by WILLIS A. WOOD AND SCOTT T. KELLOGG VOLUME 162. Immunochemical Techniques (Part L: Chemotaxis and Inflammation) Edited by GIOVANNI DI SABATO VOLUME 163. Immunochemical Techniques (Part M: Chemotaxis and Inflammation) Edited by GIOVANNI DI SABATO VOLUME 164. Ribosomes Edited by HARRY F. NOLLER, JR., AND KIVIE MOLDAVE VOLUME 165. Microbial Toxins: Tools for Enzymology Edited by SIDNEY HARSHMAN VOLUME 166. Branched-Chain Amino Acids Edited by ROBERT HARRIS AND JOHN R. SOKATCH VOLUME 167. Cyanobacteria Edited by LESTER PACKER AND ALEXANDER N. GLAZER VOLUME 168. Hormone Action (Part K: Neuroendocrine Peptides) Edited by P. MICHAEL CONN VOLUME 169. Platelets: Receptors, Adhesion, Secretion (Part A) Edited by JACEK HAWIGER VOLUME 170. Nucleosomes Edited by PAUL M. WASSARMAN AND ROGER D. KORNBERG VOLUME 171. Biomembranes (Part R: Transport Theory: Cells and Model Membranes) Edited by SIDNEY FLEISCHER AND BECCA FLEISCHER VOLUME 172. Biomembranes (Part S: Transport: Membrane Isolation and Characterization) Edited by SIDNEY FLEISCHER AND BECCA FLEISCHER

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VOLUME 173. Biomembranes [Part T: Cellular and Subcellular Transport: Eukaryotic (Nonepithelial) Cells] Edited by SIDNEY FLEISCHER AND BECCA FLEISCHER VOLUME 174. Biomembranes [Part U: Cellular and Subcellular Transport: Eukaryotic (Nonepithelial) Cells] Edited by SIDNEY FLEISCHER AND BECCA FLEISCHER VOLUME 175. Cumulative Subject Index Volumes 135–139, 141–167 VOLUME 176. Nuclear Magnetic Resonance (Part A: Spectral Techniques and Dynamics) Edited by NORMAN J. OPPENHEIMER AND THOMAS L. JAMES VOLUME 177. Nuclear Magnetic Resonance (Part B: Structure and Mechanism) Edited by NORMAN J. OPPENHEIMER AND THOMAS L. JAMES VOLUME 178. Antibodies, Antigens, and Molecular Mimicry Edited by JOHN J. LANGONE VOLUME 179. Complex Carbohydrates (Part F) Edited by VICTOR GINSBURG VOLUME 180. RNA Processing (Part A: General Methods) Edited by JAMES E. DAHLBERG AND JOHN N. ABELSON VOLUME 181. RNA Processing (Part B: Specific Methods) Edited by JAMES E. DAHLBERG AND JOHN N. ABELSON VOLUME 182. Guide to Protein Purification Edited by MURRAY P. DEUTSCHER VOLUME 183. Molecular Evolution: Computer Analysis of Protein and Nucleic Acid Sequences Edited by RUSSELL F. DOOLITTLE VOLUME 184. Avidin-Biotin Technology Edited by MEIR WILCHEK AND EDWARD A. BAYER VOLUME 185. Gene Expression Technology Edited by DAVID V. GOEDDEL VOLUME 186. Oxygen Radicals in Biological Systems (Part B: Oxygen Radicals and Antioxidants) Edited by LESTER PACKER AND ALEXANDER N. GLAZER VOLUME 187. Arachidonate Related Lipid Mediators Edited by ROBERT C. MURPHY AND FRANK A. FITZPATRICK VOLUME 188. Hydrocarbons and Methylotrophy Edited by MARY E. LIDSTROM VOLUME 189. Retinoids (Part A: Molecular and Metabolic Aspects) Edited by LESTER PACKER

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VOLUME 190. Retinoids (Part B: Cell Differentiation and Clinical Applications) Edited by LESTER PACKER VOLUME 191. Biomembranes (Part V: Cellular and Subcellular Transport: Epithelial Cells) Edited by SIDNEY FLEISCHER AND BECCA FLEISCHER VOLUME 192. Biomembranes (Part W: Cellular and Subcellular Transport: Epithelial Cells) Edited by SIDNEY FLEISCHER AND BECCA FLEISCHER VOLUME 193. Mass Spectrometry Edited by JAMES A. MCCLOSKEY VOLUME 194. Guide to Yeast Genetics and Molecular Biology Edited by CHRISTINE GUTHRIE AND GERALD R. FINK VOLUME 195. Adenylyl Cyclase, G Proteins, and Guanylyl Cyclase Edited by ROGER A. JOHNSON AND JACKIE D. CORBIN VOLUME 196. Molecular Motors and the Cytoskeleton Edited by RICHARD B. VALLEE VOLUME 197. Phospholipases Edited by EDWARD A. DENNIS VOLUME 198. Peptide Growth Factors (Part C) Edited by DAVID BARNES, J. P. MATHER, AND GORDON H. SATO VOLUME 199. Cumulative Subject Index Volumes 168–174, 176–194 VOLUME 200. Protein Phosphorylation (Part A: Protein Kinases: Assays, Purification, Antibodies, Functional Analysis, Cloning, and Expression) Edited by TONY HUNTER AND BARTHOLOMEW M. SEFTON VOLUME 201. Protein Phosphorylation (Part B: Analysis of Protein Phosphorylation, Protein Kinase Inhibitors, and Protein Phosphatases) Edited by TONY HUNTER AND BARTHOLOMEW M. SEFTON VOLUME 202. Molecular Design and Modeling: Concepts and Applications (Part A: Proteins, Peptides, and Enzymes) Edited by JOHN J. LANGONE VOLUME 203. Molecular Design and Modeling: Concepts and Applications (Part B: Antibodies and Antigens, Nucleic Acids, Polysaccharides, and Drugs) Edited by JOHN J. LANGONE VOLUME 204. Bacterial Genetic Systems Edited by JEFFREY H. MILLER VOLUME 205. Metallobiochemistry (Part B: Metallothionein and Related Molecules) Edited by JAMES F. RIORDAN AND BERT L. VALLEE

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VOLUME 206. Cytochrome P450 Edited by MICHAEL R. WATERMAN AND ERIC F. JOHNSON VOLUME 207. Ion Channels Edited by BERNARDO RUDY AND LINDA E. IVERSON VOLUME 208. Protein–DNA Interactions Edited by ROBERT T. SAUER VOLUME 209. Phospholipid Biosynthesis Edited by EDWARD A. DENNIS AND DENNIS E. VANCE VOLUME 210. Numerical Computer Methods Edited by LUDWIG BRAND AND MICHAEL L. JOHNSON VOLUME 211. DNA Structures (Part A: Synthesis and Physical Analysis of DNA) Edited by DAVID M. J. LILLEY AND JAMES E. DAHLBERG VOLUME 212. DNA Structures (Part B: Chemical and Electrophoretic Analysis of DNA) Edited by DAVID M. J. LILLEY AND JAMES E. DAHLBERG VOLUME 213. Carotenoids (Part A: Chemistry, Separation, Quantitation, and Antioxidation) Edited by LESTER PACKER VOLUME 214. Carotenoids (Part B: Metabolism, Genetics, and Biosynthesis) Edited by LESTER PACKER VOLUME 215. Platelets: Receptors, Adhesion, Secretion (Part B) Edited by JACEK J. HAWIGER VOLUME 216. Recombinant DNA (Part G) Edited by RAY WU VOLUME 217. Recombinant DNA (Part H) Edited by RAY WU VOLUME 218. Recombinant DNA (Part I) Edited by RAY WU VOLUME 219. Reconstitution of Intracellular Transport Edited by JAMES E. ROTHMAN VOLUME 220. Membrane Fusion Techniques (Part A) Edited by NEJAT DU¨ZGU¨NES¸ VOLUME 221. Membrane Fusion Techniques (Part B) Edited by NEJAT DU¨ZGU¨NES¸ VOLUME 222. Proteolytic Enzymes in Coagulation, Fibrinolysis, and Complement Activation (Part A: Mammalian Blood Coagulation Factors and Inhibitors) Edited by LASZLO LORAND AND KENNETH G. MANN

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VOLUME 223. Proteolytic Enzymes in Coagulation, Fibrinolysis, and Complement Activation (Part B: Complement Activation, Fibrinolysis, and Nonmammalian Blood Coagulation Factors) Edited by LASZLO LORAND AND KENNETH G. MANN VOLUME 224. Molecular Evolution: Producing the Biochemical Data Edited by ELIZABETH ANNE ZIMMER, THOMAS J. WHITE, REBECCA L. CANN, AND ALLAN C. WILSON VOLUME 225. Guide to Techniques in Mouse Development Edited by PAUL M. WASSARMAN AND MELVIN L. DEPAMPHILIS VOLUME 226. Metallobiochemistry (Part C: Spectroscopic and Physical Methods for Probing Metal Ion Environments in Metalloenzymes and Metalloproteins) Edited by JAMES F. RIORDAN AND BERT L. VALLEE VOLUME 227. Metallobiochemistry (Part D: Physical and Spectroscopic Methods for Probing Metal Ion Environments in Metalloproteins) Edited by JAMES F. RIORDAN AND BERT L. VALLEE VOLUME 228. Aqueous Two-Phase Systems Edited by HARRY WALTER AND GO¨TE JOHANSSON VOLUME 229. Cumulative Subject Index Volumes 195–198, 200–227 VOLUME 230. Guide to Techniques in Glycobiology Edited by WILLIAM J. LENNARZ AND GERALD W. HART VOLUME 231. Hemoglobins (Part B: Biochemical and Analytical Methods) Edited by JOHANNES EVERSE, KIM D. VANDEGRIFF, AND ROBERT M. WINSLOW VOLUME 232. Hemoglobins (Part C: Biophysical Methods) Edited by JOHANNES EVERSE, KIM D. VANDEGRIFF, AND ROBERT M. WINSLOW VOLUME 233. Oxygen Radicals in Biological Systems (Part C) Edited by LESTER PACKER VOLUME 234. Oxygen Radicals in Biological Systems (Part D) Edited by LESTER PACKER VOLUME 235. Bacterial Pathogenesis (Part A: Identification and Regulation of Virulence Factors) Edited by VIRGINIA L. CLARK AND PATRIK M. BAVOIL VOLUME 236. Bacterial Pathogenesis (Part B: Integration of Pathogenic Bacteria with Host Cells) Edited by VIRGINIA L. CLARK AND PATRIK M. BAVOIL VOLUME 237. Heterotrimeric G Proteins Edited by RAVI IYENGAR VOLUME 238. Heterotrimeric G-Protein Effectors Edited by RAVI IYENGAR

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VOLUME 239. Nuclear Magnetic Resonance (Part C) Edited by THOMAS L. JAMES AND NORMAN J. OPPENHEIMER VOLUME 240. Numerical Computer Methods (Part B) Edited by MICHAEL L. JOHNSON AND LUDWIG BRAND VOLUME 241. Retroviral Proteases Edited by LAWRENCE C. KUO AND JULES A. SHAFER VOLUME 242. Neoglycoconjugates (Part A) Edited by Y. C. LEE AND REIKO T. LEE VOLUME 243. Inorganic Microbial Sulfur Metabolism Edited by HARRY D. PECK, JR., AND JEAN LEGALL VOLUME 244. Proteolytic Enzymes: Serine and Cysteine Peptidases Edited by ALAN J. BARRETT VOLUME 245. Extracellular Matrix Components Edited by E. RUOSLAHTI AND E. ENGVALL VOLUME 246. Biochemical Spectroscopy Edited by KENNETH SAUER VOLUME 247. Neoglycoconjugates (Part B: Biomedical Applications) Edited by Y. C. LEE AND REIKO T. LEE VOLUME 248. Proteolytic Enzymes: Aspartic and Metallo Peptidases Edited by ALAN J. BARRETT VOLUME 249. Enzyme Kinetics and Mechanism (Part D: Developments in Enzyme Dynamics) Edited by DANIEL L. PURICH VOLUME 250. Lipid Modifications of Proteins Edited by PATRICK J. CASEY AND JANICE E. BUSS VOLUME 251. Biothiols (Part A: Monothiols and Dithiols, Protein Thiols, and Thiyl Radicals) Edited by LESTER PACKER VOLUME 252. Biothiols (Part B: Glutathione and Thioredoxin; Thiols in Signal Transduction and Gene Regulation) Edited by LESTER PACKER VOLUME 253. Adhesion of Microbial Pathogens Edited by RON J. DOYLE AND ITZHAK OFEK VOLUME 254. Oncogene Techniques Edited by PETER K. VOGT AND INDER M. VERMA VOLUME 255. Small GTPases and Their Regulators (Part A: Ras Family) Edited by W. E. BALCH, CHANNING J. DER, AND ALAN HALL

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VOLUME 256. Small GTPases and Their Regulators (Part B: Rho Family) Edited by W. E. BALCH, CHANNING J. DER, AND ALAN HALL VOLUME 257. Small GTPases and Their Regulators (Part C: Proteins Involved in Transport) Edited by W. E. BALCH, CHANNING J. DER, AND ALAN HALL VOLUME 258. Redox-Active Amino Acids in Biology Edited by JUDITH P. KLINMAN VOLUME 259. Energetics of Biological Macromolecules Edited by MICHAEL L. JOHNSON AND GARY K. ACKERS VOLUME 260. Mitochondrial Biogenesis and Genetics (Part A) Edited by GIUSEPPE M. ATTARDI AND ANNE CHOMYN VOLUME 261. Nuclear Magnetic Resonance and Nucleic Acids Edited by THOMAS L. JAMES VOLUME 262. DNA Replication Edited by JUDITH L. CAMPBELL VOLUME 263. Plasma Lipoproteins (Part C: Quantitation) Edited by WILLIAM A. BRADLEY, SANDRA H. GIANTURCO, AND JERE P. SEGREST VOLUME 264. Mitochondrial Biogenesis and Genetics (Part B) Edited by GIUSEPPE M. ATTARDI AND ANNE CHOMYN VOLUME 265. Cumulative Subject Index Volumes 228, 230–262 VOLUME 266. Computer Methods for Macromolecular Sequence Analysis Edited by RUSSELL F. DOOLITTLE VOLUME 267. Combinatorial Chemistry Edited by JOHN N. ABELSON VOLUME 268. Nitric Oxide (Part A: Sources and Detection of NO; NO Synthase) Edited by LESTER PACKER VOLUME 269. Nitric Oxide (Part B: Physiological and Pathological Processes) Edited by LESTER PACKER VOLUME 270. High Resolution Separation and Analysis of Biological Macromolecules (Part A: Fundamentals) Edited by BARRY L. KARGER AND WILLIAM S. HANCOCK VOLUME 271. High Resolution Separation and Analysis of Biological Macromolecules (Part B: Applications) Edited by BARRY L. KARGER AND WILLIAM S. HANCOCK VOLUME 272. Cytochrome P450 (Part B) Edited by ERIC F. JOHNSON AND MICHAEL R. WATERMAN VOLUME 273. RNA Polymerase and Associated Factors (Part A) Edited by SANKAR ADHYA

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VOLUME 274. RNA Polymerase and Associated Factors (Part B) Edited by SANKAR ADHYA VOLUME 275. Viral Polymerases and Related Proteins Edited by LAWRENCE C. KUO, DAVID B. OLSEN, AND STEVEN S. CARROLL VOLUME 276. Macromolecular Crystallography (Part A) Edited by CHARLES W. CARTER, JR., AND ROBERT M. SWEET VOLUME 277. Macromolecular Crystallography (Part B) Edited by CHARLES W. CARTER, JR., AND ROBERT M. SWEET VOLUME 278. Fluorescence Spectroscopy Edited by LUDWIG BRAND AND MICHAEL L. JOHNSON VOLUME 279. Vitamins and Coenzymes (Part I) Edited by DONALD B. MCCORMICK, JOHN W. SUTTIE, AND CONRAD WAGNER VOLUME 280. Vitamins and Coenzymes (Part J) Edited by DONALD B. MCCORMICK, JOHN W. SUTTIE, AND CONRAD WAGNER VOLUME 281. Vitamins and Coenzymes (Part K) Edited by DONALD B. MCCORMICK, JOHN W. SUTTIE, AND CONRAD WAGNER VOLUME 282. Vitamins and Coenzymes (Part L) Edited by DONALD B. MCCORMICK, JOHN W. SUTTIE, AND CONRAD WAGNER VOLUME 283. Cell Cycle Control Edited by WILLIAM G. DUNPHY VOLUME 284. Lipases (Part A: Biotechnology) Edited by BYRON RUBIN AND EDWARD A. DENNIS VOLUME 285. Cumulative Subject Index Volumes 263, 264, 266–284, 286–289 VOLUME 286. Lipases (Part B: Enzyme Characterization and Utilization) Edited by BYRON RUBIN AND EDWARD A. DENNIS VOLUME 287. Chemokines Edited by RICHARD HORUK VOLUME 288. Chemokine Receptors Edited by RICHARD HORUK VOLUME 289. Solid Phase Peptide Synthesis Edited by GREGG B. FIELDS VOLUME 290. Molecular Chaperones Edited by GEORGE H. LORIMER AND THOMAS BALDWIN VOLUME 291. Caged Compounds Edited by GERARD MARRIOTT VOLUME 292. ABC Transporters: Biochemical, Cellular, and Molecular Aspects Edited by SURESH V. AMBUDKAR AND MICHAEL M. GOTTESMAN

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VOLUME 293. Ion Channels (Part B) Edited by P. MICHAEL CONN VOLUME 294. Ion Channels (Part C) Edited by P. MICHAEL CONN VOLUME 295. Energetics of Biological Macromolecules (Part B) Edited by GARY K. ACKERS AND MICHAEL L. JOHNSON VOLUME 296. Neurotransmitter Transporters Edited by SUSAN G. AMARA VOLUME 297. Photosynthesis: Molecular Biology of Energy Capture Edited by LEE MCINTOSH VOLUME 298. Molecular Motors and the Cytoskeleton (Part B) Edited by RICHARD B. VALLEE VOLUME 299. Oxidants and Antioxidants (Part A) Edited by LESTER PACKER VOLUME 300. Oxidants and Antioxidants (Part B) Edited by LESTER PACKER VOLUME 301. Nitric Oxide: Biological and Antioxidant Activities (Part C) Edited by LESTER PACKER VOLUME 302. Green Fluorescent Protein Edited by P. MICHAEL CONN VOLUME 303. cDNA Preparation and Display Edited by SHERMAN M. WEISSMAN VOLUME 304. Chromatin Edited by PAUL M. WASSARMAN AND ALAN P. WOLFFE VOLUME 305. Bioluminescence and Chemiluminescence (Part C) Edited by THOMAS O. BALDWIN AND MIRIAM M. ZIEGLER VOLUME 306. Expression of Recombinant Genes in Eukaryotic Systems Edited by JOSEPH C. GLORIOSO AND MARTIN C. SCHMIDT VOLUME 307. Confocal Microscopy Edited by P. MICHAEL CONN VOLUME 308. Enzyme Kinetics and Mechanism (Part E: Energetics of Enzyme Catalysis) Edited by DANIEL L. PURICH AND VERN L. SCHRAMM VOLUME 309. Amyloid, Prions, and Other Protein Aggregates Edited by RONALD WETZEL VOLUME 310. Biofilms Edited by RON J. DOYLE

Methods in Enzymology

VOLUME 311. Sphingolipid Metabolism and Cell Signaling (Part A) Edited by ALFRED H. MERRILL, JR., AND YUSUF A. HANNUN VOLUME 312. Sphingolipid Metabolism and Cell Signaling (Part B) Edited by ALFRED H. MERRILL, JR., AND YUSUF A. HANNUN VOLUME 313. Antisense Technology (Part A: General Methods, Methods of Delivery, and RNA Studies) Edited by M. IAN PHILLIPS VOLUME 314. Antisense Technology (Part B: Applications) Edited by M. IAN PHILLIPS VOLUME 315. Vertebrate Phototransduction and the Visual Cycle (Part A) Edited by KRZYSZTOF PALCZEWSKI VOLUME 316. Vertebrate Phototransduction and the Visual Cycle (Part B) Edited by KRZYSZTOF PALCZEWSKI VOLUME 317. RNA–Ligand Interactions (Part A: Structural Biology Methods) Edited by DANIEL W. CELANDER AND JOHN N. ABELSON VOLUME 318. RNA–Ligand Interactions (Part B: Molecular Biology Methods) Edited by DANIEL W. CELANDER AND JOHN N. ABELSON VOLUME 319. Singlet Oxygen, UV-A, and Ozone Edited by LESTER PACKER AND HELMUT SIES VOLUME 320. Cumulative Subject Index Volumes 290–319 VOLUME 321. Numerical Computer Methods (Part C) Edited by MICHAEL L. JOHNSON AND LUDWIG BRAND VOLUME 322. Apoptosis Edited by JOHN C. REED VOLUME 323. Energetics of Biological Macromolecules (Part C) Edited by MICHAEL L. JOHNSON AND GARY K. ACKERS VOLUME 324. Branched-Chain Amino Acids (Part B) Edited by ROBERT A. HARRIS AND JOHN R. SOKATCH VOLUME 325. Regulators and Effectors of Small GTPases (Part D: Rho Family) Edited by W. E. BALCH, CHANNING J. DER, AND ALAN HALL VOLUME 326. Applications of Chimeric Genes and Hybrid Proteins (Part A: Gene Expression and Protein Purification) Edited by JEREMY THORNER, SCOTT D. EMR, AND JOHN N. ABELSON VOLUME 327. Applications of Chimeric Genes and Hybrid Proteins (Part B: Cell Biology and Physiology) Edited by JEREMY THORNER, SCOTT D. EMR, AND JOHN N. ABELSON

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VOLUME 328. Applications of Chimeric Genes and Hybrid Proteins (Part C: Protein–Protein Interactions and Genomics) Edited by JEREMY THORNER, SCOTT D. EMR, AND JOHN N. ABELSON VOLUME 329. Regulators and Effectors of Small GTPases (Part E: GTPases Involved in Vesicular Traffic) Edited by W. E. BALCH, CHANNING J. DER, AND ALAN HALL VOLUME 330. Hyperthermophilic Enzymes (Part A) Edited by MICHAEL W. W. ADAMS AND ROBERT M. KELLY VOLUME 331. Hyperthermophilic Enzymes (Part B) Edited by MICHAEL W. W. ADAMS AND ROBERT M. KELLY VOLUME 332. Regulators and Effectors of Small GTPases (Part F: Ras Family I) Edited by W. E. BALCH, CHANNING J. DER, AND ALAN HALL VOLUME 333. Regulators and Effectors of Small GTPases (Part G: Ras Family II) Edited by W. E. BALCH, CHANNING J. DER, AND ALAN HALL VOLUME 334. Hyperthermophilic Enzymes (Part C) Edited by MICHAEL W. W. ADAMS AND ROBERT M. KELLY VOLUME 335. Flavonoids and Other Polyphenols Edited by LESTER PACKER VOLUME 336. Microbial Growth in Biofilms (Part A: Developmental and Molecular Biological Aspects) Edited by RON J. DOYLE VOLUME 337. Microbial Growth in Biofilms (Part B: Special Environments and Physicochemical Aspects) Edited by RON J. DOYLE VOLUME 338. Nuclear Magnetic Resonance of Biological Macromolecules (Part A) Edited by THOMAS L. JAMES, VOLKER DO¨TSCH, AND ULI SCHMITZ VOLUME 339. Nuclear Magnetic Resonance of Biological Macromolecules (Part B) Edited by THOMAS L. JAMES, VOLKER DO¨TSCH, AND ULI SCHMITZ VOLUME 340. Drug–Nucleic Acid Interactions Edited by JONATHAN B. CHAIRES AND MICHAEL J. WARING VOLUME 341. Ribonucleases (Part A) Edited by ALLEN W. NICHOLSON VOLUME 342. Ribonucleases (Part B) Edited by ALLEN W. NICHOLSON VOLUME 343. G Protein Pathways (Part A: Receptors) Edited by RAVI IYENGAR AND JOHN D. HILDEBRANDT VOLUME 344. G Protein Pathways (Part B: G Proteins and Their Regulators) Edited by RAVI IYENGAR AND JOHN D. HILDEBRANDT

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VOLUME 345. G Protein Pathways (Part C: Effector Mechanisms) Edited by RAVI IYENGAR AND JOHN D. HILDEBRANDT VOLUME 346. Gene Therapy Methods Edited by M. IAN PHILLIPS VOLUME 347. Protein Sensors and Reactive Oxygen Species (Part A: Selenoproteins and Thioredoxin) Edited by HELMUT SIES AND LESTER PACKER VOLUME 348. Protein Sensors and Reactive Oxygen Species (Part B: Thiol Enzymes and Proteins) Edited by HELMUT SIES AND LESTER PACKER VOLUME 349. Superoxide Dismutase Edited by LESTER PACKER VOLUME 350. Guide to Yeast Genetics and Molecular and Cell Biology (Part B) Edited by CHRISTINE GUTHRIE AND GERALD R. FINK VOLUME 351. Guide to Yeast Genetics and Molecular and Cell Biology (Part C) Edited by CHRISTINE GUTHRIE AND GERALD R. FINK VOLUME 352. Redox Cell Biology and Genetics (Part A) Edited by CHANDAN K. SEN AND LESTER PACKER VOLUME 353. Redox Cell Biology and Genetics (Part B) Edited by CHANDAN K. SEN AND LESTER PACKER VOLUME 354. Enzyme Kinetics and Mechanisms (Part F: Detection and Characterization of Enzyme Reaction Intermediates) Edited by DANIEL L. PURICH VOLUME 355. Cumulative Subject Index Volumes 321–354 VOLUME 356. Laser Capture Microscopy and Microdissection Edited by P. MICHAEL CONN VOLUME 357. Cytochrome P450, Part C Edited by ERIC F. JOHNSON AND MICHAEL R. WATERMAN VOLUME 358. Bacterial Pathogenesis (Part C: Identification, Regulation, and Function of Virulence Factors) Edited by VIRGINIA L. CLARK AND PATRIK M. BAVOIL VOLUME 359. Nitric Oxide (Part D) Edited by ENRIQUE CADENAS AND LESTER PACKER VOLUME 360. Biophotonics (Part A) Edited by GERARD MARRIOTT AND IAN PARKER VOLUME 361. Biophotonics (Part B) Edited by GERARD MARRIOTT AND IAN PARKER

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VOLUME 362. Recognition of Carbohydrates in Biological Systems (Part A) Edited by YUAN C. LEE AND REIKO T. LEE VOLUME 363. Recognition of Carbohydrates in Biological Systems (Part B) Edited by YUAN C. LEE AND REIKO T. LEE VOLUME 364. Nuclear Receptors Edited by DAVID W. RUSSELL AND DAVID J. MANGELSDORF VOLUME 365. Differentiation of Embryonic Stem Cells Edited by PAUL M. WASSAUMAN AND GORDON M. KELLER VOLUME 366. Protein Phosphatases Edited by SUSANNE KLUMPP AND JOSEF KRIEGLSTEIN VOLUME 367. Liposomes (Part A) Edited by NEJAT DU¨ZGU¨NES¸ VOLUME 368. Macromolecular Crystallography (Part C) Edited by CHARLES W. CARTER, JR., AND ROBERT M. SWEET VOLUME 369. Combinational Chemistry (Part B) Edited by GUILLERMO A. MORALES AND BARRY A. BUNIN VOLUME 370. RNA Polymerases and Associated Factors (Part C) Edited by SANKAR L. ADHYA AND SUSAN GARGES VOLUME 371. RNA Polymerases and Associated Factors (Part D) Edited by SANKAR L. ADHYA AND SUSAN GARGES VOLUME 372. Liposomes (Part B) Edited by NEJAT DU¨ZGU¨NES¸ VOLUME 373. Liposomes (Part C) Edited by NEJAT DU¨ZGU¨NES¸ VOLUME 374. Macromolecular Crystallography (Part D) Edited by CHARLES W. CARTER, JR., AND ROBERT W. SWEET VOLUME 375. Chromatin and Chromatin Remodeling Enzymes (Part A) Edited by C. DAVID ALLIS AND CARL WU VOLUME 376. Chromatin and Chromatin Remodeling Enzymes (Part B) Edited by C. DAVID ALLIS AND CARL WU VOLUME 377. Chromatin and Chromatin Remodeling Enzymes (Part C) Edited by C. DAVID ALLIS AND CARL WU VOLUME 378. Quinones and Quinone Enzymes (Part A) Edited by HELMUT SIES AND LESTER PACKER VOLUME 379. Energetics of Biological Macromolecules (Part D) Edited by JO M. HOLT, MICHAEL L. JOHNSON, AND GARY K. ACKERS VOLUME 380. Energetics of Biological Macromolecules (Part E) Edited by JO M. HOLT, MICHAEL L. JOHNSON, AND GARY K. ACKERS

Methods in Enzymology

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C H A P T E R

O N E

Predicting Fluorescence Lifetimes and Spectra of Biopolymers Patrik R. Callis Contents 1. Introduction 1.1. General principles for predicting fluorescence properties 2. Qualitative Concepts: Developing Intuition 2.1. Trajectories 2.2. How to recognize a quenching environment 3. Methods 3.1. Philosophy 3.2. MD simulations 3.3. Quantum mechanics 3.4. Interface between QM and MD 4. Nonexponential Fluorescence Decay 5. Final Remarks Acknowledgments References

2 3 7 7 10 14 14 14 14 15 25 28 34 34

Abstract Use of fluorescence in biology and biochemistry for imaging and characterizing equilibrium and dynamic processes is growing exponentially. Much progress has been made in the last few years on the microscopic understanding of the underlying principles of what controls the wavelength and quenching of fluorescence in biopolymers, both of which are central to the utility of fluorescent probes. This chapter is concerned with the quantitative microscopic understanding and prediction of the fluorescence wavelength and/or intensity of a fluorescent probe molecule attached to a biopolymer as revealed by hybrid quantum and classical mechanical computation procedures. The aim is not only to provide a recipe, but also even more importantly, to communicate the qualitative basic concepts of interpretation of fluorescence. These are surprisingly simple, although not broadly appreciated at this time. In addition, an effort has been made to show how these techniques have led to an emerging Department of Chemistry and Biochemistry, Montana State University, Bozeman, Montana, USA Methods in Enzymology, Volume 487 ISSN 0076-6879, DOI: 10.1016/S0076-6879(11)87001-1

#

2011 Elsevier Inc. All rights reserved.

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Patrik R. Callis

understanding of the relation between time-dependent wavelengths shifts due to solvent relaxation and population decay of conformational subensembles.

1. Introduction Use of fluorescence is enjoying rapid growth and development in its application to an array of fundamental and practical studies in biological sciences. The usefulness of fluorescence falls roughly into three broad categories: (1) imaging (2) quantifying equilibrium systems (3) following dynamic processes This chapter deals uniquely with the quantitative microscopic understanding and prediction of the fluorescence wavelength and/or intensity of a fluorescent probe molecule attached to a biopolymer while in an equilibrium state or during a dynamic process. (Of course, one is often now quantifying equilibrium and dynamic processes with the aid of imaging). Even at equilibrium, most biopolymers are increasingly perceived as an ensemble of conformations (Luo et al., 2006; Prakash and Marcus, 2007, 2008; Yang et al., 2003), which means that there are fluctuations in conformation on long-time scales and apparent heterogeneity of fluorescence properties on short-time scales. The theme of this chapter is that sensitivity to the precise local environment of the intensity, that is, quantum yield (Ff), excited state lifetime (tf), and wavelength (lmax) of a probe molecule often lies in responsiveness to the local electric field strength and direction. Or more usefully, it is the change in electric potential (volts) between the space initially occupied by the electron and the space occupied after the electronic transition that determines whether the fluorescence will be red- or blue-shifted and whether an electron acceptor will or will not quench the fluorescence. Electrons, of course, will be at lower energy if the potential is more positive (less negative), and such positive differences in potential can be provided by ions and electric dipoles. For example, the ruggedness of the electrostatic landscape in proteins is mirrored in the ruggedness of the tryptophan (Trp) fluorescence quantum yield and lifetime landscape. This is quite pertinent to the study of proteins because the diversity and strength of electric fields in proteins underlies their function as enzymes. (Kamerlin et al., 2009; Roca et al., 2008; Warshel et al., 2006). In practice, the mechanism of the detected change in wavelength or intensity of fluorescence is almost always unknown in detail, or plausibly surmised at best. While this does not limit measurement of the rate of the

Predicting Fluorescence

3

process, lack of knowledge can lead to incorrect conclusions as to the nature of the process. For example, folding of a protein is often followed by the extent of quenching of a fluorophore by a quencher residue that is thought to be spatially close in the folded state, but this alone does not assure that the intensity change precisely follows the folding process. Much progress has been made in the last few years on the microscopic understanding of the underlying principles of what controls the wavelength and quenching of fluorescence in biopolymers (Beierlein et al., 2006; Callis, 2009; Callis and Liu, 2004; Callis et al., 2007; Clark et al., 1999, 2002; Muin˜o and Callis, 2009; Rusu et al., 2008; Vivian and Callis, 2001). This chapter attempts to relate the surprisingly simple ideas that have emerged from quantum mechanics–classical molecular dynamics (QM–MD) simulations applied to proteins, including insight into correlation of wavelength and subpopulation decay, and to present details of the procedures.

1.1. General principles for predicting fluorescence properties All light may be formally classified as fluorescence inasmuch as photons are emitted from their source during downward energy transitions in matter. In this chapter, we take fluorescence to have the usual meaning common to chemistry and biology: light emitted from a molecule by virtue of being electronically excited by a substantially allowed transition (i.e., in times <1 ms), typically by absorption of UV or visible light. As with everything involving electrons in our physical world, understanding and predicting fluorescence emission wavelengths, spectral distribution, and intensity can, in principle, be reduced to a combination of stationary and time-dependent QM applied to the system of interest. The most effective probes, however, exhibit great sensitivity to their microenvironment, meaning that the property being detected will fluctuate with high amplitude and display a large variance (Loring, 1990), reflecting the dynamic nature of the polymer. Thus, any method seeking to simulate response requires simulating the microenvironment in an appropriately dynamic manner that converges to correct average values. While it is becoming feasible to apply QM to a single biopolymer including solvent using specialized GPUs (Ufimtsev and Martinez, 2009), the large number of QM calculations needed to adequately simulate the fluctuating environment in biopolymers requires, in practice, a classically propagating set of point charges while treating only the probe molecule with QM. 1.1.1. Wavelength prediction The wavelength at which the probe absorbs or emits photons comes directly from the original Planck expression relating frequency to quantum energy level difference, DE:

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Patrik R. Callis

DE ¼ hn ¼

hc l

in which h is Planck’s constant, c the speed of light, n the frequency, and l is the wavelength. This simple expression, however, begs the question as to which energy levels. At ambient temperatures, the assumption that molecules begin in the zero-point energy is reasonable, but because the geometry of a molecule changes considerably upon excitation, and electronic transitions are fast compared to the motion of nuclei (vertical excitation), there is a range of photon wavelengths that can take the molecule to the new electronic state in which some degree of vibrational excitation makes up the DE. Therefore, the final states are known as vibronic states. Generally, the vibronic state with strongest intensity is that corresponding to vertical excitation. Calculating DE at a fixed geometry, therefore, leads to the relation of direct interest: DE ¼ hn ¼

hc l max

ð1:1Þ

wherein lmax is the wavelength at which the absorptivity or emission intensity is a maximum. The bandwidth and vibronic structure are directly related to the shape and amplitude of the geometry change, with the relative amplitudes of intensity given by the Franck–Condon (FC) factors. These can be modeled quite precisely by simple ab initio quantum methods (Callis et al., 1995), but are normally not needed to make predictions of lmax in a semiempirical manner.

1.1.2. Intensities and lifetimes The ratio of photons absorbed to photons emitted is called the quantum yield, Ff, and determines the integrated fluorescence signal relative to a perfectly efficient emitter during steady illumination. Ff ¼

kr ¼ kr tf ðkr þ kn Þ

where kr is the radiative rate constant (which is proportional to the integrated extinction coefficient) and kn is the total nonradiative rate constant describing how fast the molecule becomes de-excited by any means other than emission of fluorescence from the emitting excited state. Time-resolved methods are increasingly used because of ease and higher information content (Beechem and Brand, 1985; Eftink, 1991; Lakowicz, 2006; Prendergast, 1991; Ross et al., 2007). If only a single

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Predicting Fluorescence

species and process is involved, the fluorescence lifetime is given by tf ¼ (kr þ knr)–1. The strategy for predicting Ff and tf is as follows: The rate constant describing the spontaneous emission of photons, kr, associated with an electronic transition between two states is relatively insensitive to environment. The distinguishing property of intensity-sensitive probes, therefore, is that the nonradiative rate constant, kn, must be strongly modulated by the microenvironment, that is, we must be in the business of calculating kn. A crucial—and not widely appreciated—aspect of calculating kn is to recognize that kn virtually always consists of two components: kn0, which is intrinsic to the chromophore and relatively insensitive to the environment, and what may be called a quenching constant, kq, which is strongly modulated by environmental factors and is reasonably easy to calculate. kn0 stems from strong internal coupling of the electrons and the vibrations of the molecule, leading to intramolecular transitions from the fluorescing state to a high vibrational state of a lower electronic state (e.g., triplet or ground state), that is, what is commonly called intersystem crossing and internal conversion, and is generally still not predictable in a reliable manner.1 Intensity-sensitive probes function because kq is extremely sensitive to the environment in some way. The most common examples involve quenching caused by electron transfer (ET) and resonance energy transfer, both of which are effectively treated using the Fermi golden rule form of time-dependent QM. Other types of quenching will be detailed below. At the lowest order, the above expressions for quantum yield and lifetime become Ff ¼


kr  kr þ kn0 þ kq


1 and tf ¼ kr þ kn0 þ kq :

ð1:2Þ

where typically the environmental dependence is virtually all carried in kq, which is computed as a function of the electrostatic environment and/or the distance to the quencher in the methods to be considered in this chapter. The two primary mechanisms by which the environment modulates kq are modulation of the spatial distance between the fluorophore and the 1

A footnote is to be added to this paragraph regarding the “insensitivity” of the kr and kn0 to the environment. Because both rate constants depend on the chromophore excited state wavefunction, which in turn depends somewhat on the environment, factors on the order of 2 or 3 can be observed. This is not the extreme sensitivity one normally seeks in a fluorescent probe, and often is ignored. As a tangible example, for the Trp indole ring, kr increases by a factor of 3 going from water to hydrocarbon solvent, with only a factor of 1.8 accounted for by the frequency cubed factor going with the blue shift from 360 to 310 nm (Meech et al., 1983). The remaining factor is that the transition dipole is larger as the charge transfer character of the wavefunction decreases in a nonpolar environment. kn0 also increases considerably going from water to nonpolar solvent, for Trp. As a result, the quantum yield remains relatively independent of surroundings (near 0.3 in the absence of quenching), while the lifetime becomes shorter by a factor of 2–3 for Trps buried in hydrophobic pockets (e.g., Trp48 of azurin), relative to those in nonquenching polar environments.

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quencher, and modulation of the energy gap (DE00) by the local electrostatic fields and potentials affecting the fluorescing state and the final state created either by ET or excitation transfer. The energy gap is crucial because, as for all processes in the universe, energy must be conserved, that is, transitions only happen when there is resonance between the initial and the final states. The general time-dependent quantum mechanical rate for many processes is given by the Fermi rule: kq ðt Þ ¼

2p V ðr ðt ÞÞ2 rðDE00 ðt ÞÞ ¼ 4p2 cV ðr ðtÞÞ2 rðDE00 ðtÞÞ ℏ

ð1:3Þ

where V is the Hamiltonian matrix element that connects the initial and final electronic states, and r(DE00) is the density of final states, which physically may be thought of as the number of states in resonance with the initial fluorescing state. In the convenient right-hand version, c is the speed of light in cm s–1, V is in energy units of cm–1, and r is in states per cm–1. V is shown here as a function of the fluctuating distance (r) between the chromophore and quencher, and r is given as a function of time to emphasize the fluctuating energy gap due to the fluctuating local electric field. Special cases of kq include ET and (Fo¨rster or “fluorescence”) resonance energy transfer (FRET). Ð For ET: V ¼ CS1HCCTdv, where H is the Hamiltonian operator and CS1 and CCT are the wavefunctions of Ð the fluorescing and charge transfer states; r(DE00) is given by r(DE00) ¼ FIFAdDE00 where FI and FA are FC weighted densities of states for the ionizations of the S1 state of the donor and of the acceptor anion, respectively. For FRET:
! !
! !  ð m D r m A r 1 ! ! V ¼ CD A HCDA dv ¼ 3 m D mA  3 r r2 where in the initial state, D*A, the chromophore is excited, and in the final state, DA*, excitation has been transferred to the acceptor A. ! m D and ! mA are the transition dipole vectors for the ground ! S1 excitations of donor chromophore D and acceptor A, and ! r is the intermolecular distance vector. Procedures outlined below are mostly about ET, but they map closely onto what has been done in QM–MD predictions of FRET (Beierlein et al., 2006). Computations of FRET are, in principle, somewhat more straightforward, because an accurate form of V is easy to compute, and r(DE00) is simply the well-known overlap integral of the fluorescence spectrum of the donor with the absorption spectrum of the acceptor.

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2. Qualitative Concepts: Developing Intuition In this section, we present a tour of what the methods presented in this chapter will provide. We use the properties of Trp fluorescence in proteins as the main example, but the concepts and procedures remain the same for any probe molecule. We have ported this technology to the quenching of flavins (Callis and Liu, 2006) and of fluorescein in an antibody (Hutcheson, 2009). Note that the quenching of flavins and dyes in proteins involves ET from Trp and also from tyrosine (Tyr). Figure 1.1 shows plots of calculated versus experimental quantum yields from Algorithms I and II applied to 24 Trps in 17 proteins. Table 1.1 matches numbers with PDB codes. Although the agreement is far from perfect, the method can clearly distinguish between high- and low-fluorescence quenching by the closest backbone amides.

2.1. Trajectories Figure 1.2 shows some typical QM–MD trajectories of the semiempircal QM-computed transition energies for the fluorescing state S1 (points) and the CT state (upper trace) for typical high- and low fluorescing cases of single-Trp proteins. The predictions of quantum yield come from these trajectories, with the energy gap between the S1 and CT states and the amplitude of the fluctuations being the primary controlling factors. In Fig. 1.2 are shown the trajectories for two weakly emitting Trps and two strongly emitting Trps in single-Trp proteins. Note the fluctuations of both states. In three of the four cases, the S1 state fluctuates on the order of 4000 cm 1, corresponding to the well-known broad featureless fluorescence spectra of Trp in most proteins. The mean value of the S1 energies provides a prediction of the fluorescence lmax. The CT state fluctuations are much larger: on the order of 8000 cm 1 or 1 eV. This reflects the larger amount of charge transferred and the larger distance of transfer. Both the CT–S1 energy gap and its fluctuations are key determinants of kq and therefore the fluorescence quantum yield and lifetime. A small energy gap and/or large fluctuations signify a low quantum yield, because there are many opportunities for the CT and S1 states to have the same energy. This is clearly more the case in Fig. 1.2A and B for the weak emitters when compared to the examples for high quantum yield cases shown in Fig. 1.2C and D, where the gap is large, and even large fluctuations of the CT state will not be sufficient to bring the CT state into resonance with the fluorescing state. Looking again at Fig. 1.2A for Trp126 of dsba, we focus on a quenching event. Arbitrarily at the 150 ps point, a transition to the CT state is simulated by removing the restraint that the Trp charges are those of S1. This is just to

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A

0.35 0.30 21 24

Calculated yield

0.25 17

0.20 0.15

16 18

13

0.10 8 12 9 456 10 213 7 11

0.05 0.00 0.00

15

19

14

0.05

22 23

20 0.10

0.15

0.20

0.25

0.30

0.35

Experimental yield B

0.35 23

Calculated quantum yield

0.30

24

0.25

17 19

0.20

21 12

0.15

9

18 15

0.10 6 0.05

20

4 5 11 28 7

0.00 0.00

0.05

14 0.10

22

16 0.15

0.20

0.25

0.30

0.35

Experimental quantum yield

Figure 1.1 Plots of calculated versus experimental quantum yields for 24 Trps in 17 proteins by two algorithms. (A) Algorithm I: a constant V ¼ 10 cm 1 is used and the CT–S1 gap is made 4000 cm 1 smaller than given by the vertical Zindo energies. Table 1.1 matches numbers with PDB codes. (B) Algorithm II: D95 ab initio hV2i values and a slightly modified FC density of states scheme. The error bars are the standard deviations for three 50 ps MD trajectories. Here, 4700 cm 1 is added to the Zindo CT energy computed at CT geometry, and a Gaussian with standard deviation ¼ 3000 cm 1 is used for the amide electron attachment FC density spectrum. For the buried Trp59 of rnt (#23) only, the quantum yield was computed from the small equilibrium constant (positive DG0) for the electron transfer process stemming from its small reorganization energy (the relaxed CT state lies above the 1La state). For dsb76(#20) and pfk(#22), the present accuracy of our DG0 estimates leaves open the possibility that equilibrium will favor the 1 La state, and the quantum yield predictions will be much higher.

9

Predicting Fluorescence

Table 1.1

Key to names for 24 Trps in 17 proteins for Fig. 1.1A and B

Num

Description, pdb code

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

T4 lysozyme W158-asn2 1lyd dsBa W126 1dsb Barnase W94-H18 pH5 1a2p Human cyclophilinA 2cpl TrpCage 1l2y (1 & 2) fkb506 binding protein 1d6o Subtilisin C 1sbc Phospholypase A2 2bpp NSCP W57 W4F W170F dsba W126 Q74A, N127A T4 lysozyme W138 1lyd T4 lysozyme W126 1lyd Barnase W94 pH8 1a2p Cobra toxin 1ctx Melittin 2mlt Glucagon 1gcn Barnase W71 his neutral 1a2p Parvalbumin 1b8r Barnase W35 his neutral 1a2p dsba W76 1dsb,1fvk average Staph. nuclease 1stn Phosphofructokinase 6pfk Ribonuclease T1 9rnt Apo-azurin W48 1azb

see what would happen. At the 156 ps point, a random fluctuation causes the CT state to become S1, and the atom charges of the chromophore are switched by the software to those given by the QM for the lowest CT state. Solvent reorganization about the drastically different charge distribution, in which the indole ring has become þ1 and the amide has a charge of 1, stabilizes the CT state to well below the S1 state within less than 100 fs. This is quenching. The system will return to the ground state well before being able to mount the large barrier to return to the S1 state. Figure 1.2B shows the trajectories for another weak emitter, Trp3 of phospholipaseA2, which displays similar behavior. Figure 1.2C and D shows trajectories for Trp140 of staphylococcal nuclease (SNase) and for Trp48 of azurin, both of which have quantum yields near 0.30, corresponding to kq ¼ 0. These two cases show how a high quantum yield is possible for both despite the quite different environments. For SNase, the Trp is near the protein surface and is surrounded by several charged residues. The fluctuations are therefore very large, yet the

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Patrik R. Callis

B 45 CT with S1 charges S1

40 35 30 25

CT with CT charges

20 15 0

20

40

60

80

100 120 140 160 180 200

Time, ps

C

45 40 35 30 25 20 15 0

10

20

D

45

Transition energy, cm–1/1000

Transition energy, cm–1/1000

Transition energy, cm–1/1000

Transition energy, cm–1/1000

A

40 35 30 25 20

30 Time, ps

40

50

60

45 40 35 30 25 20 15

15 0

10

20

30 Time, ps

40

50

60

0

10

30

20

40

50

60

Time, ps

Figure 1.2 Typical QM–MD trajectories of S1(points) and CT(solid lines) state energies from Zindo. (A) For Trp126 Dsba (weakly fluorescent); (B) phospholipaseA2 (weakly fluorescent); (C) Trp140 in Staph. nuclease (strongly fluorescent); and (D) Trp48 in apo-azurin (strongly fluorescent). In each panel, during the early part of the trajectory, the Trp charge distribution is that of the S1 (fluorescing state). The sharp drop at the later end is the response of the system due to arbitrarily at some point changing the charge distribution to that of the CT state, as if an electron transfer to the amide had taken place. The sharp drop is caused by the classical response by the environment (mainly water) through the MD to the new charge distribution.

charged groups conspire to create a large energy gap. In contrast, for Trp48 of azurin, the environment is extremely hydrophobic and impenetrable by water. This has the effect of greatly reducing the fluctuations of the CT state. There is a corresponding reduction in the reorganization stabilization of the CT state following simulated ET. The small fluctuations and lack of reorganization imply that no quenching can take place.

2.2. How to recognize a quenching environment One is naturally going to ask what is it that makes these dramatic differences in computed properties, which seem to correlate well with observed fluorescence yields. As the title of this section suggests, in most cases of extremely strong or weak fluorescence, it is possible to see the reason by inspection using a simple and natural criterion: that electrons are at lower energy if they are near positive charge (i.e., at positive voltage) or far from negative charge (small negative voltage).

11

Predicting Fluorescence

The large energy fluctuations of the CT state seen in Fig. 1.2 are because the fluctuations in electric potential difference (voltage difference) across the region of electron density shift for the ground ! CT transition for Trp in proteins are enormous, spanning a range of 8000 cm 1 (¼1 eV ¼ 23 kcal/ mol ¼ 96 kJ/mol), corresponding to electric fields 5  107 V/cm. To put this in perspective, only1.4 kcal/mol in DG0 (equivalent to about one-third of a typical H-bond) changes an equilibrium constant or a rate constant by a factor of 10. Figure 1.3 is a collection of cartoons illustrating major factors leading to quenching (or no quenching) of Trp fluorescence by the nearby backbone B

A

+

Lys132 Lys132

+ H2O H2O

-

-

-

Asp 123

Asp 71

C

Asp 71 D

+ Ala 1

-

Asp 123

+

H2O

+ Lys 110

+ Lys 10

+ Lys 133

Glu 129

Figure 1.3 Cartoons of major electrostatic interactions that dictate whether the amide will or will not quench Trp fluorescence in three proteins. (A) Dsba in a stabilizing environment corresponding to a low CT energy in Fig. 1.2A. (B) Dsba in a destabilizing environment corresponding to a high CT energy in Fig. 1.2A. (C) Phospholipase A2 showing stabilization of the CT state by the helix dipole, and by two nearby waters and positive charges. (D) Staph. nuclease. The CT state is destabilized primarily by two Lys residues near the indole ring. Arrows indicate the path of electron transfer.

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Patrik R. Callis

amide. The arrow shows the path of ET. Transfer toward a positive charge or away from a negative charge stabilizes the CT state, leading to quenching by the amide. For Trp126 of Dsba seen in Fig. 1.3A, the amide CT state is greatly stabilized by the positioning of Lys132, Asp71, Asp123, and a water molecule. In the CT state, the ring is positive and the amide C is negative. Therefore, the negative charges near the ring are stabilizing. Likewise, the positive charge of the Lys132 and the nearness of the positive H atom of the water to the amide C equally well stabilize the CT state. Such H-bonds— either from water or protein donors—are quite effective, because the large partial positive charge on the H (modeled as a þ0.4 point charge) can get quite close to the C when it H-bonds to the carbonyl O of the amide (Fig. 1.4). Figure 1.3B shows the same Trp and surrounding groups 20 ps earlier at a time of an extreme upward fluctuation. A major factor in the higher energy of this configuration is that the positive-charged end of the flexible ˚ ngstroms more distant, and the water has a different HLys132 is a few A bond partner and has almost no influence on the CT energy. Note that these cartoons only show the few most prominent interactions which capture the overall effect of the environment. In fact, the net effect is

0.5 volts on amide

+0.4 +0.4 –0.8 CT, vacuum: 0 volts on amide 0 volts on ring CT, +0.5 volts = 1 H-bond

ET ET

CT, +1 volts on amide S1 ET is possible

Figure 1.4 Separation of CT and S1 energy for Trp as given by Zindo calculations in vacuum and when the amide O is H-bonded by a single water. The single water makes the amide more positive by 0.5 V, thereby lowering the CT state by about 0.5 eV or 4000 cm 1. A second interaction of this magnitude would certainly cause quenching.

Predicting Fluorescence

13

always a small difference of two extremely large numbers of stabilizing and destabilizing interactions. For example, in the frame in Fig. 1.3A, the four entities in the cartoon contribute 2000 cm 1 to the CT state energy (stabilize). But all together, protein residues contribute a total 30,000 cm 1 of stabilizing interactions and þ18,000 cm 1 of destabilizing interactions, resulting in a net stabilization of 12,000. A similar pattern is seen for water. There is a total of 44,000 cm 1 stabilization and 43,000 cm 1 destabilization for a net stabilization of 1000 cm 1. In the frame of Fig. 1.3B, corresponding to a high energy fluctuation, the net stabilization by protein is only 7000 cm 1, and the water has a net destabilization of 4000 cm 1. Figure 1.3C shows the essential features of the environment of Trp3 of phospholipase A2, another highly quenched Trp. Here, the Trp amide is at the N-terminus of a section of helix. The collective dipoles from the backbone atoms reinforce to create a large positive potential at the amide in such a case. The nearby positively charged N-terminal Ala and Lys10 also contribute strongly to stabilizing the CT state, as do two waters whose negative O atoms are near the indole ring. In contrast, Fig. 1.3D shows the main contributors to the CT state destabilization for a high quantum yield case, Trp140 of SNase. Here, the positively charged Lys133 and Lys110 close to the indole ring strongly destabilize the CT state, dominating the stabilizing interaction of the water and the Glu129. No cartoon is needed to illustrate why Trp48 of azurin has a high quantum yield. As noted above in relation to Fig. 1.2D, there are simply no waters and charge groups within 10 A˚ of this Trp, which lies in an extremely hydophobic pocket and has one of the most blue-shifted lmax values known. Figure 1.4 illustrates more quantitatively that even a single H-bond from water in a strategic location can affect the quantum yield of fluorescence. The numbers presented in Fig. 1.4 are based on an actual quantum calculation of the CT–S1 energy gap for the model Trp þ backbone chromophore system used in our QM–MD simulations. Without the water (as modeled by three point charges), the energy of the CT state lies about 1 eV (8000 cm 1) above S1, and no quenching is expected. The presence of the water protons in the H-bonded complex with the amide O stabilizes the CT state by 0.5 eV. When augmented by other stabilizing groups or another such water H-bond, quenching will be efficient. The above discussion focused on electrostatic stabilization as it modulates fluorescence intensity. Entirely parallel ideas govern fluorescence wavelength (Vivian and Callis, 2001). In the case of Trp, the electron density shifts from the pyrrole ring to the benzene ring during excitation to S1. Therefore, positive charges near the benzene ring or negative charges near the pyrrole ring contribute a shift to lower energy (longer wavelengths). The reverse will contribute a blue shift. As with the CT state stabilization, the net result is the small difference of large numbers of contributions. Figure 1.2D shows how the S1 energy of Trp48 of azurin in a hydrophobic pocket exhibits a high

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Patrik R. Callis

mean value and much lower fluctuations than is the case for the more water exposed Trps in Fig. 1.2A–C. This corresponds well with the extreme blueshifted fluorescence (308 nm) and the presence of vibrational structure seen in the experimental fluorescence. Note, however, that the character of the fluorescence is 1La, as shown by recent fluorescence anisotropy experiments (Broos et al., 2007).

3. Methods 3.1. Philosophy The general scheme is to perform hybrid quantum mechanics–classical molecular dynamics (QM–MD) simulations of sufficient length that the configuration space of the surroundings of the chromophore is adequately sampled. The choice of QM is a compromise between accuracy and speed required to sample the changing environment over long simulations.

3.2. MD simulations There exist now several widely accepted and used MD packages. A useful introduction to the use of MD methods, including a recent compendium of modeling software sources, has been published in this series (Saxena et al., 2009). We have had direct experience primarily with Discover, Charmm, and Gromacs and have not seen a significant difference in performance with regard to predictive capability.

3.3. Quantum mechanics The choice of the QM method is conditioned by the criteria that it must be reasonably accurate for a variety of low-lying electronic state types, while also being simple enough to deliver energies and eigenfunction information on several excited states in a short time (ideally, in a few seconds). We have used Michael Zerner’s spectroscopically calibrated INDO/S-CIS method (Zindo) (Cory et al., 1998; Ridley and Zerner, 1973; Thompson and Zerner, 1991), incorporating a modification that allows for input of electrostatic potentials and fields generated from the partial charges of every atom in the environment, for example, protein and solvent. Zindo and similar semiempirical methods have been used in earlier QM–MM studies (Gehlen et al., 1994; Marchi et al., 1993; Warshel, 1982, 1991; Cory et al., 1998), and these studies have influenced our choices to various extents. Another essential requirement of the QM is the ability to incorporate the differing electric potentials at the chromophore atoms generated by (typically) the point charges assigned to the atoms of the surrounding biopolymer

Predicting Fluorescence

15

and solvent. The one-electron Hamiltonian is modified to include electric fields and potentials at each atom center computed from the Coulomb sum of point charges provided from the forcefield for all non-QM atoms, including all waters (Sreerama et al., 1994; Theiste et al., 1991). Alternatively, the original program has the option to read the point charges as part of the input file. An absolutely necessary addition to the parameter set for oxygen has been made for Zindo and must be used whenever oxygen plays a significant role in a chromophore (Li et al., 1999). The response of Zindo to electric fields (Callis and Burgess, 1997) is quite accurate, as indicated by a more fundamental quantum chemical study on indole (Donder-Lardeux et al., 2003). For the methods described here, it is essential that the quantum mechanical calculation be relatively accurate and faithful to the electrostatic perturbations from the environment. In the case of Trp, CASPT2 (Andersson et al., 1992, 2002) calculations for several formamide–indole pairs show that Zindo is effective at tracking the ring ! amide CT state energies as a function of electrostatic environment (Liu et al., 2005). Zindo has been incorporated into Gaussian (Frisch et al., 2002) and is also available in the VAMP (Clark et al., 2008) package, but care must be taken that the parameters of Li et al. are included. The VAMP package contains other semiempirical methods based on the PM3 and AM1 Hamiltonians that are likely quite suitable for the type of QM–MD computations described here. Regardless of the QM and MD method, quantum calculations are usually performed at 10 fs intervals on an appropriate chromophore fragment clipped from the biopolymer and capped with hydrogens. QM bond lengths are typically fixed in the MD at values representative of either the ground or of the CT state. It is useful to use the CT state geometry when the average CT energy lies well above the S1 state to avoid excessive mixing with the large number of excited states at high energy. CASPT2 calculations for the formamide–indole pairs also show that the S1 state energy increases by 4000 cm–1 relative to the CT state energy as the geometry is increased from that at the 1La energy minimum to that at the CT energy minimum (Liu et al., 2005). In addition, CASPT2 CT state energies are found to be 1000–3000 cm–1 higher than those estimated using Zindo (Liu et al., 2005). Therefore, Zindo-computed vertical excitation energies with the CT state geometry should be increased by about 6000  1000 cm–1 to make a realistic estimate of the true difference in CT and S1 zero-point energies.

3.4. Interface between QM and MD The prediction scheme we use has a number of steps, none of which is especially daunting. A block diagram gives an overview in Fig. 1.5. Whereas the MD and QM procedures in this scheme are generic and can, in principle, be used “off the shelf,” the procedures labeled with bold

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Patrik R. Callis

Raw .pdb file

1

Edit for MD

3 Unequilibrated coords

MD: min/equil

Atom charges

4

Update chromophore charges

Coord time = t

Create atom charges and group charge differences; identify S1 and lowest CT state

S1, CT energies, and group charges at time t

State energies and molecular orbitals

5

QM

S1, CT energies, and group charges as function of time

QM input file

MD: trajectory

Coords at time t + Δt

Refine and output S1, CT energies and group chg diffs at time t

2

Cut/cap the chromophore; Add point charges of env.

6

Compute electron transfer rates, lifetimes, quantum yields, and spectra

Figure 1.5 Flow chart indicating components of the general procedure. The cylinders represent files and rectangles represent programs. The MD and QM are generic. The procedures identified with the bold numbers represent components of the interface that were programmed in Fortran locally. If QM charges are fed back to the MD, the block arrow is ignored, and the dashed arrows complete the loop. If MD charges are kept fixed, the block arrow is followed and the solid arrows represent procedures that may proceed asynchronously behind the MD cycles if the option of updating charges on the chromophore is not elected.

numbers in Fig. 1.5 represent Fortran programs written by the author’s group that interface the MD and QM, and extract the necessary information to make wavelength and/or intensity predictions. A brief synopsis of the different modules follows. 3.4.1. Editing the raw structure file This is the hardest step to automate because of the nonuniform nature of X-ray structure files that are available from the RCSB Protein Data Bank, for example, (http://www.rcsb.org/pdb), combined with nonuniformity of input formats used by different MD platforms. Examples of typical special editing tasks are (1) most crystal structures have multiple protein molecules per unit cell; if the molecule is a monomer

Predicting Fluorescence

17

in solution, one must separate out one of these for the calculation. (2) Very few X-ray structures provide information about the location of hydrogens. These must be added by the program; (3) specifying one of the three common forms of histidine; the structure seldom specifies which tautomer and charge form is likely; (4) crystallographic waters are quite often of value, given the importance of water in determining wavelength and quenching; these may or may not be present, and which ones to retain is not so clear when there are multiple molecules per unit cell; (5) often there are disulfide bridges that must be connected; (6) metal ions may or may not need to be retained; (7) cocrystallizing agents of no relevance may appear in the structure and should be removed. (8) We have defined new atom types used for CT or S1 geom. These are available by request from the author. The entire protein is then solvated. We have most often used a sphere of TIP3P model waters, with radius such that all parts of the protein were solvated to a depth of at least 5 A. A quartic confining potential localized on the surface of the spherical droplet prevented “evaporation” of any of the waters during the course of the trajectory. The fully solvated protein structure is energyminimized and equilibrated before the production simulation. Use of periodic boundary conditions is preferred for simulations longer than 100 ps. 3.4.2. Program to create the chromophore QM input file with environment point charge information (1) When the chromophore is not covalently bonded to the polymer, the process is quite straightforward. The appropriate states of the chromophore are found by the QM procedure as augmented by either adding the electric potential at each atomic center due to all the point charges of the nonchromophore atoms (the environment), or by supplying as part of the input the coordinates and numerical values of the point charges. (2) When the chromophore is covalently attached to the polymer, for example, Trp in a protein, one must artificially create a realistic model of the chromophore in which atoms that have minimal influence on the spectral properties are omitted. This has been implemented with most MD packages by defining “linker atoms” that insulate the chromophore of interest from the rest of the polymer. In our procedure, the chromophore remains attached to the polymer at all times during MD propagation just as in the original structure file. The input file of coordinates for the QM procedure is changed by judiciously changing carbons defining the “point of attachment” of the chromophore to polymer into hydrogens, and shortening the bonds accordingly. The resulting coordinates, point charge information, and other details required by the QM procedure of choice are written to a file in a format appropriate to the QM method. All other atoms in the system form the environment point charge array.

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Patrik R. Callis

In either case, we commonly constrain the bond lengths of the chromophore so that fluctuations of the energy reflect more closely changes in the environment alone. In most of our applications to date, we fed the QM charge information back to MD system using the USERSB subroutine provided in the Charmm program. This is formally necessary if wavelengths are to be computed without sizable empirical adjustment. Aside from simply reading in the coordinates and charges from the generic files of the MD program, one must map chromophore atom numbers for QM to the MD system numbers and change the connecting atoms to H. There are three important subroutines that are reproduced below. One of these is called to shorten bonds to the capping hydrogen atoms. The other two make the Coulomb sums to create potentials and fields at each chromophore atom. SUBROUTINE BOND(XX,YY,ZZ,I,J,RR) c This routine scales a bond that connects the chromophore to the c polymer wherein a heavy atom j, initially located at xx,yy,zz c and is connected to the chromophore atom i becomes an H, and is c assigned new xx,yy,zz values that make the bond an appropriate c X-H bond length, where X = C, N,... dimension xx(50),yy(50),zz(50) c....Scale the i-->j bond... c create the unit vector pointing from the chromophore heavy atom, i, to the polymer c atom, j, that has already been changed to H in the main program. c x,y,z components are: xvec=xx(j)-xx(i) yvec=yy(j)-yy(i) zvec=zz(j)-zz(i) c rvec = length of vector rvec=xvec*xvec + yvec*yvec + zvec*zvec rvec=sqrt(rvec) c make the unit vector xvec=xvec/rvec yvec=yvec/rvec zvec=zvec/rvec c new xx(j),yy(j),zz(j) : position atom j along original bond from

Predicting Fluorescence

c

19

atom i so the bond length = rr xx(j)=xx(i)+xvec*rr yy(j)=yy(i)+yvec*rr zz(j)=zz(i)+zvec*rr return end

The potentials at chromophore atoms are the trivial Coulomb sum over the nonchromophore atoms, seen below as the double loop, in which the outer loop is the sum over all atoms and the inner loop is over chromophore atoms. Self-terms are avoided by the index id_chrom(k), which is equal to 1 if k corresponds to a chromophore atom: do k=1,natoms if(id_chrom(k).eq.0)then !...then include the contribution; do kk=1,nchrom r=((x(k)-xx(kk))**2 + (y(k)-yy(kk))**2+(z(k)-zz (kk))**2)**0.5 call potential(x,y,z,k,xx,yy,zz,kk,r,q,vv) vtrp(kk)=vtrp(kk) + vv call field (x,y,z,k,xx,yy,zz,kk,r,q,ex,ey,ez) extrp(kk)=extrp(kk) + ex eytrp(kk)=eytrp(kk) + ey eztrp(kk)=eztrp(kk) + ez enddo endif enddo SUBROUTINE POTENTIAL(X,Y,Z,K, XX,YY,ZZ,KK,R,Q, V) c computes the potential in volts between environment atom k c and chromophore atom kk, which are located at (x,y,z) and (xx,yy,zz) with charges in units of e q(k) and q(kk) c respectively and a distance apart r in Angstroms dimension x(30000),y(30000),z(30000), xx(50), yy (50), zz(50) dimension q(30000) v = q(k)/r*14.399644 return end SUBROUTINE FIELD(X,Y,Z,K, XX,YY,ZZ,KK, R,Q, EX, EY, EZ)

20

Patrik R. Callis

c computes the electric field vector in volts/angstrom between c environment atom k and chromophore atom kk, which are located at c (x,y,z) and (xx,yy,zz) with charges in units of e q(k) and q(kk) c respectively and a distance apart, r, in Angstroms dimension x(30000),y(30000),z(30000), xx(50), yy (50), zz(50) dimension q(30000) ex = ((xx(kk)-x(k))*q(k)/r**3)*14.399644 ey = ((yy(kk)-y(k))*q(k)/r**3)*14.399644 ez = ((zzr**3)*14.399644/r**3)*14.399644 return end

The chromophore dynamics and atomic coordinates are governed entirely by the MD (perhaps with QM-modified charges). The transition energy calculation, with the potentials and fields added, is performed on the chromophore only. The effect of the environment surrounding the chromophore atoms is incorporated directly into the QM calculation through a straightforward modification of the matrix elements of the Fock operator in vacuum involving atomic orbitals m and n: 0 Fmm ¼ Fmm  eVa ! 0 Fmn ¼ Fmn þ e Ea r!mn

in which e is the electronic charge, Va is the electrostatic potential at quantum mechanical atom a created by all non-QM protein and solvent atoms, k and Ea is the associated electrostatic field. The potential and field are evaluated at the QM atoms, a, via straightforward Coulomb sums: P qk Va ¼ k rak   ! X qk ! r ak Ea ¼ 3 rak k where the summations extend over all non-QM atoms. No dielectric constant is assumed and electronic polarizability is not included. We find that the potentials make by far the most important contribution to the state energy shifts. In practice, the fields may be considered optional. The effect of the fields is typically only to create a minor

21

Predicting Fluorescence

additional hybridization of local s and p function orbitals if a minimal basis set is used. Electronic excitation of the chromophore is simulated by instantly switching the charges (and in some studies also the geometry) on the QM system to the excited state (S1) values. Most of the internal Stark effect (Callis and Burgess, 1997; Lockhart and Kim, 1992) is expressed implicitly by the difference of the potentials at different atoms. The chromophore atoms are assigned charges given by the electron density matrix. For example, in the Lo¨wdin basis (which assumes orthogonal atomic orbitals) from the INDO/S calculation: ! X qa ¼ e Pmm  Za m

where Pmm are diagonal elements of the density matrix for all valence atomic orbitals, m, centered on atom a, Za is the atomic core charge. The charge distribution of the excited state is sufficiently sensitive to the local field (potential differences) that these charges should be updated by a QM calculation at least every 10 fs of simulated time to capture the fastest relaxation times ( Jimenez et al., 1994; Maroncelli, 1993; Muino and Callis, 1994). 3.4.3. Identifying the S1 and CT states The QM output from a CIS treatment (a good compromise) minimally would be the electronic transition energies and the canonical molecular orbitals. We have adapted the original Zerner subroutine ONEOP, which creates the electron density matrix for the ground and excited states, thereby giving the approximate charge on each atom in each state. What is of interest is the charge difference at least at the resolution of atoms accompanying each transition. The QM program may output the permanent dipole difference, but this should only be used to help identify states. The point dipole approximations are likely to be poor, since some of the stronger interactions involve distances on the order of the dipole length. A small file, for example, name_t.eden at each time step t is created which lists the first 50 excited state energies, oscillator strengths, and the charge differences experienced by the groups of interest upon excitation to the state. The groups are chosen typically as the electron donor and acceptor (s), such that charge transfer (CT) states are readily identified by inspection by eye and by software. The large number of states is necessary because the CT states fluctuate over a range of 2 eV in some cases, but must be identified to compute the ET rate. Another reason this is necessary is that the CT states repeatedly cross other states that are less sensitive to the local fields, and so

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Patrik R. Callis

are often mixed with other states to different extents. This problem is dealt with in Section 3.4.5. 3.4.4. Update MD charges The interface for informing the MD of the QM charges (if chosen) is accomplished by a user written program, either by modifying the MD code, or by a provided subroutine, for example, the USERSB subroutine in the case of CHARMM. This is most pertinent in the case of simulating transient and steady-state Stokes shifts of fluorescence spectra caused by solvent and protein response to electron density changes due to excitation. We find that the polarization of the excited state provided by the feedback of QM charges affects the shift by 20–30%. Qualitative information concerning the dynamics of the relaxation can probably be deduced without the feedback, and fluorescence maxima can be corrected to provide good estimates. 3.4.5. Program to refine CT energies The primary job of this program is to examine each of the name_t.eden files of Section 3.4.3 and create a charge difference-weighted average lowest CT energy value. If necessary, a similar weighted average for the S1 state might be necessary if the S1 state is often mixed. This is a crucial program because the energy gap between the lowest CT state and the S1 state is a key variable in determining the ET rate. At least two factors enter the process. The process is complicated typically by the existence of multiple CT states arising either from there being more than one electron donor/acceptor in the quenching process or from “excited CT states” corresponding to the “hole” residing in alternate high energy occupied MOs. The process consists of averaging over the lowest CT states whose ET does not exceed one electron by more than 20%. The output of this program is a single trajectory file, in which the charge-weighted average CT and S1 energies are tabulated as a time series along with the group charge differences. 3.4.6. Compute lifetimes, quantum yields, and wavelengths ET rate, fluorescence lifetime, and quantum yield: As in earlier work (Callis and Liu, 2004; Callis and Vivian, 2003; Callis et al., 2007), the fluorescence quantum yield (Ff) and lifetime (tf) are estimated from Eq. (1.2). For quenching by ET, kq becomes kET, and can be computed according to different assumptions. Algorithm I: Our first procedure (Callis and Liu, 2004; Callis and Vivian, 2003) was somewhat ad hoc and specialized to cases such as Trp

23

Predicting Fluorescence

in proteins wherein the donor–acceptor distance is nearly constant and the coupling large, such that the rate could be assumed to be controlled primarily by the energy gap. The operative form of the golden rule became    ð  2p  2
1 DE00  hDE00 i 2 2 1=2 jV j 2ps rFC ðDE00 Þ exp  dDE00 hkET i ¼ ℏ 2 s

ð1:4Þ where hDE00i is the mean energy gap between the zero-point energies of the CT and S1 states, s is the standard deviation of DE00, and rFC is the FC weighted density of states for the ET process, which is a function of the energy gap. The Gaussian is normalized and gives the probability of finding a particular value of DE00. rFC was estimated following the method we used in several recent studies (Callis et al., 1995; Fender et al., 1999; Short and Callis, 2000), using a direct product vibrational space established for the geometry changes for the electron ejection from indole and electron attachment to formamide with optimization using HF/3-21g and CIS/3-21g. The vibrational frequencies and modes used were those for the neutral ground-state molecules, determined at the B3LYP/6-311Gpd level. This formula gives an excellent agreement with isolated molecule vibronic spectra for isolated indole and is assumed to do the same for ET, in which the process is treated as simultaneous electron ejection and attachment. In other words, it is the overlap of the photoelectron spectrum of the indole ring with that of the acceptor anion. This is precisely analogous to the overlap of the donor emission and acceptor absorption spectra in the more familiar Fo¨rster resonance excitation energy transfer (FRET) (Fo¨rster, 1959, 1971), as emphasized by Hopfield (1974). In this procedure, the unknown Vel and DE00 values were essentially treated as empirically adjusted constants. DE00 ¼ DEvert þ D, where DEvert is the CT–S1 energy gap as read off the trajectories of Fig. 1.2, and D and Vel are the adjustable constants. A good fit is shown in Fig. 1.1A using V ¼ 10 cm 1 and D ¼ 4000 cm 1. Algorithm II: Recently, we refined the procedure to be more general by computing V from ab initio Gaussian CIS Hamiltonian elements, while continuing to use D as a fitting parameter. The typical ab initio V values are in the range 200–1000 cm 1, which would give rates 4  102 to 104 times those with 10 cm 1. The best D was on the order of þ 5000 cm 1, which is physically more reasonable, given that CT geometry was used in these calculations. Using CT geometry would lower the CT state energy while raising the S1 energy relative to the zero-point energies. The result of computing hkETi gives hkET i ¼

2p  2 V hrFC ðDEvert þ DÞi ℏ

ð1:5Þ

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Patrik R. Callis

in which hV2i comes from the average of 500 ab initio values of V taken from 15,000 points of a 150 ps trajectory, and the average density of states came from all 15,000 points. Calculation of lmax and Fluorescence Spectra. lmax values are calculated simply as the reciprocal of the average over the corresponding MD trajectory of raw Zindo vertical S1 transition energies. Often the molecular geometry is not that of the S1 state, which means that the vertical S1 transition energy is not a good approximation for lmax. We find that relative shifts for different electrostatic environments are rather insensitive to the actual geometry used. Therefore, good estimates may be found by displacing the directly calculated value by a constant. Simulated fluorescence spectra may be constructed from the S1energy distributions from the MD trajectory. To construct authentic fluorescence spectra from the distributions of vertical 1La energies, we use a vibronically resolved ab initio computed 1La spectrum that has been shown to well represent resolved 1La fluorescence of 3MI-complexed with tri-ethylamine in a cold jet (Short and Callis, 2000). This 3MI spectrum is convoluted with the instantaneous vertical 1La energies from the trajectory and weighted by frequency cubed to produce the computed fluorescence spectrum on the energy scale. Statistical noise from a limited number of points can be reduced by smoothing the resultant histograms. A wavelength to compare with experiment comes from the average over the trajectory of calculated transition energies following equilibration. Examination of transition energy time correlation functions and directresponse results do not reveal a relaxation component beyond 5 ps of the S0 ! S1 excitation that is significant compared to fluctuations during a single trajectory. Averaging several hundred trajectories of ns length are now possible (Li et al., 2007, 2008) and are able to capture slower relaxations and presumably could result in more precisely predicted wavelength maxima, and ns relaxation times (Toptygin et al., 2001, 2006). 3.4.7. Contributions to S1 and CT energies relative to vacuum As noted above, the wavelength and CT energy predictions from the QM–MM method come from approximating the electrostatic environment of each atom of the QM molecule by augmenting the Fock matrix diagonal terms in the AO representation with the electric potential coming from the Coulomb’s Law summation over the point charges of all non-QM atoms in the simulation. The environment-caused energy change of a particular state is therefore easily divided into contributions from individual amino acid residues and water molecules using the previously described analysis tool (Vivian and Callis, 2001), in which the contribution from a given peptide residue or water is given by the sum of contributions from the individual atoms of the residue. This is done effectively at a given point in a trajectory from the scalar projection of the electric potentials at the QM atoms and the

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Predicting Fluorescence

electron density changes at those atoms accompanying excitation or ET. The energy change relative to vacuum, DEj, caused by the point charges of residue j is given by X DEj ¼ Vaj Dra a

4. Nonexponential Fluorescence Decay Understanding the ubiquitous nonexponential decay exhibited by Trp fluorescence in proteins is becoming crucial for interpretation of ultrafast decay experiments, especially when the power of such experiments is used to extract dynamical information about water near protein surfaces (Halle and Nilsson, 2009; Li et al., 2007; Zhang et al., 2009). In principle, the phenomenon of nonexponential decay is expected for any useful fluorescent probe. As noted in Section 1, the high sensitivity of a useful fluorescent probe to a certain environment property in conjunction with the multiple conformers typical of biopolymers often leads to multiexponential (or nonexponential) fluorescent decay. No consensus exists as to the cause of the complex decay. For Trp, opinions divide mostly between the view that discrete ground-state subpopulations (often different rotamers) exhibit different decay times (heterogeneity model) (Dahms et al., 1995; Donzel et al., 1974; Engh et al., 1986; Gordon et al., 1992; Szabo and Rayner, 1980), and at the other extreme, the view that the excited population is homogeneous, but has a time-dependent fluorescence spectrum that shifts to longer wavelengths on a nanosecond time scale (relaxation model) (Lakowicz, 2000, 2006). Other views have also been presented (Bajzer and Prendergast, 1993; Hudson et al., 1999; Wlodarczyk and Kierdaszuk, 2003). Heterogeneity in the form of different rotamers was modeled in a QM– MD simulation of FRET for Trp43 in the TetR-tetracycline complex and found to give rise to nonexponential decay primarily due to different orientation factors of V (Beierlein et al., 2006). The observation by Xie and coworkers (Luo et al., 2006; Yang et al., 2003), of 100- to 1000-fold fluctuations in lifetime while tracking the fluorescence decay times of single molecules of FAD and of fluorescein in proteins at 100 ms intervals for time periods of seconds, emphasizes the generality of this phenomenon. For these bulky chromophores, dynamic interchange of rotamer conformations at the ms time scale is not attractive, though perhaps possible, and they have suggested that the heterogeneity is in the chromophore–quencher intermolecular distance. This is reasonable considering that energy gap heterogeneity may not be so important as in the

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Trp case. The single-molecule fit to four exponential decay constants was found to agree well with bulk measurements. Entangled in this division of views regarding Trp is a strong correlation of fluorescence lifetime and fluorescence wavelength, such that the lifetime virtually always becomes longer at longer wavelengths. This intriguing correlation is beginning to be understood, Pan et al., (2010) and plays a central role in defining interpretations. Relaxation models naturally predict such a correlation (Lakowicz, 2000; Maroncelli, 1993) because simple solvent response to an excited state dipole always lowers the excited state energy. The spectrum shifts in time to lower energies, and the loss of intensity at short wavelengths adds a fast decay component at the short-wavelength edge and adds a complementary rising component at the long-wavelength side— both in addition to an underlying excited state population decay. A rising component appears as a negative amplitude if one is globally fitting the decay profiles as a linear combination of exponentials with different decay constants. The heterogeneity model can cause apparent shifts of the fluorescence spectrum in time as the faster decaying populations disappear to leave the longer decaying components. In contrast to the relaxation mechanism, the heterogeneity model cannot produce negative amplitudes. For proteins, the heterogeneity model often seems more appropriate because there are very few cases in which negative amplitudes are found. Two studies on the protein monellin highlight the difficulty of interpretation in the 10–100 ps range (Xu et al., 2006; Peon et al., 2002). Heterogeneity has appeared in more clear-cut cases. In the weakly fluorescing Trp68 of human gamma D- and gamma S-crystallins, a major 50 ps decay component appears along with substantial amplitudes of 0.2 and 2 ns components with no measureable shift in lmax over a time range of 0–300 ps (Xu et al., 2009; Chen et al., 2009). Strong evidence for this heterogeneity is reported in that study from a 2-ns QM–MD simulation, in which a water modulates ET to the amide by forming and breaking an H-bond at intervals on the order of 100 ps. Extreme lifetime-wavelength correlation is also seen in cyclic hexapeptides on the few ns time scale (Pan et al., 2006), a time scale in which solvent relaxation for exposed Trps is not expected. In contrast to the relaxation model, no comparably attractive principle explains why heterogeneity would lead to the correlation in the absence of relaxation. The absence of plausible hypotheses for the underlying principle is used as evidence against the standard heterogeneity (rotamer) model (Lakowicz, 2000, 2006), and the absence of rising long-wavelength fluorescence (or negative amplitudes) is used as evidence against the relaxation model. Our work since finding that the short-range ET coupling in these systems is strong (Callis et al., 2007), with concomitant greatly increased sensitivity of the ET quenching rate to the energy gap between S1 and CT states (Callis, 1991, 1997), leads naturally to the missing underlying

27

Predicting Fluorescence

universal physical principle accounting for the strong correlation between lmax and tf (Pan et al., 2010). As seen in Fig. 1.2, the familiar broad fluorescence spectrum of a solvent-exposed chromophore is actually an ensemble average of single molecular lmax values, fluctuating on a femtosecond time scale typically over 3000–4000 cm–1 (fwhw) or 40 nm. This leads naturally to a picture in which those conformers having shorter wavelength emission spectra, that is, higher average energy, have an increased probability for transient fast quenching during large fluctuations in environment that bring the nonfluorescent CT state and the fluorescing state into resonance. Figure 1.6 illustrates what has just been stated. The lower panel shows the distribution of lmax values from the trajectory in the top panel. Also on the lower panel is a bar graph that shows the instantaneous kq values at each point in time, plotted versus the S1 transition energy. One sees that the bulk of the quenching probability comes from the most extreme high energy

Transition –1 energy, kcm

44 40 36 32 28 300 Rate, ns–1

250 0

20

40

80 60 Time, ps

100

120

140

20

40

60

100

120

140

200 150 100 50 0 0

Rate, ns-1

300

80 Time, ps

Distribution of instantaneous S1 transition energies

DSBa

250 200

Rates vs. S1 energy

150 100 50 0 28

29

30

31

32

33

34

1La energy, kcm-1

Figure 1.6 Explanation of the physical principle underlying the correlation of lifetime and wavelength expected in cases of heterogeneity. The lower panel shows the histogram of S1 energies (solid curve), coming from the distribution seen in the upper panel. The vertical lines on the lower panel represent a bar graph of instantaneous electron transfer rates at the S1 energy indicated on the horizontal axis. Instantaneous rates as a function of time are plotted on the middle panel.

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Patrik R. Callis

fluctuations, that is, those having the greatest probability of being in resonance with the CT state. Therefore, one concludes that subpopulations with the most blue-shifted spectra will decay fastest, providing the subpopulations persist for the excited state lifetime and have comparable fluctuation amplitudes. The principle leading to the strong correlation is actually restricted to cases, for example, Trp, for which the CT state lies well above the S1 state, and the wavelength is quite sensitive to the local electric field. In these cases, heterogeneity and relaxation give superficially similar behavior, making heterogeneity a rather elusive concept. But, two studies using nonnatural amino acids underscore the reality of heterogeneity by contrasting behavior. The fluorescence decays of 5-fluoroTrp incorporated in proteins are almost always monoexponential (Broos et al., 2004). This probe is not as easily quenched by ET as Trp because it has a higher ionization potential (Liu et al., 2005), but it retains wavelength sensitivity to solvent quite close to that of Trp. Apparently, suppressing ET quenching removes the major environmentally sensitive nonradiative decay pathway of Trp in proteins (i.e., causes kq to become quite small). Another probe has behavior opposite to Trp in regard to lifetime-wavelength correlation. Boxer and coworkers (Abbyad et al., 2007) find that time-resolved fluorescence spectra of Aladan incorporated at several sites in the protein GB1 shift to shorter wavelength on the nanosecond time scale. This striking behavior unequivocally reveals ground-state heterogeneity, because relaxation always requires a red shift in time; this is consistent with the observation that tf for this probe decreases with increasing solvent polarity by an internal mechanism.

5. Final Remarks The QM–MD studies over the past 7 years using the methods described here have produced an understanding of the phenomenally large range of fluorescence wavelengths, lifetimes, and quantum yields observed for Trp in proteins. No competing explanation has been put forward to our knowledge. The method is not just a “black box.” A visual examination of structure files of solvated and equilibrated proteins often produces excellent qualitative predictions as to the nature of wavelength and intensity, providing one has a minimal grasp of electrostatic interactions at the level of Figs. 1.3 and 1.4. The principles put forth are quite general within a broad class of fluorescent probes. Improvements in the form of including ab initio coupling elements and longer simulations out to 25 ns have not helped the ragged fit of calculation versus experiment in the intermediate region between low and very high quantum yield. Future studies will have to examine at least three

Predicting Fluorescence

29

independent questions: (1) whether the Fermi rule is adequate in all cases; (2) whether ab initio quantum methods are needed; and (3) whether the MD parameters used so far are adequate.

Appendix: Ab Initio Computation of Electron Transfer Coupling Matrix Elements Obtaining better estimates of ET coupling elements has enabled an understanding the ubiquitous correlation of lmax and tf in the absence of solvent relaxation that can arise from heterogeneity in ET rates for Trp in proteins. This Appendix details a procedure for quickly estimating ab initio ET coupling elements, V in Eq. (1.3). Using Gaussian 03, revision D.01 diabatic electronic matrix elements were taken from the singles configuration interaction (CIS) Hamiltonian matrix element coupling the two single configurations that constitute nearly pure S1 (1La) and amide CT states: the highest occupied molecular orbital (HOMO) ! ring p* lowest unoccupied molecular orbital(LUMO) and HOMO ! amide LUMO (p*). We find that a CI basis consisting of only the three excitations from HOMO to LUMO þ n (n ¼ 0, 1, 2) adequately span the desired space for the systems studied here, because mixing of ring and amide MOs is not extensive. Subroutine MrgCIS in link 914 was slightly modified using the recommended link modification procedure, so it will always write the CI < AA,BBjAA,BB> matrix, its eigenvalues, and its eigenvectors independent of other print options. The corresponding singlet and triplet adapted CI matrices were constructed from the output and were diagonalized to ensure that the results agreed with the normally printed singlet and triplet transition energies. The largest difficulty was the mixing between the amide p* and ring LUMO þ 1 MOs and the corresponding mixing of the configurations involving these MOs. This problem was solved by determining which linear combination of the mixed MOs resulted in maximum amide p* density; this linear combination of CI matrix elements then gave the desired interaction element. Three basis sets were compared: STO-3G, 3-21G, and the Dunning/Huzinga full double-z basis, D95. Below is the result of the Linux diff command: diff l914.F l914.F. modified > c modified by callis 27may06so it will always write the CI matrix and its eigenvecs 3140c3141 < If(IPrint.ge.6) then — >c If(IPrint.ge.6) then

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3144c3145 < endIf — >c endIf 5059,5060c5060,5061 < If(IPrint.ge.6) < $ Call OutMtS(IOut,’ matrix after diagonals:’,0,0, — >c If(IPrint.ge.6) > Call OutMtS(IOut,’ matrix after diagonals:’,0,0, The following .gjf file requires about 35 s with 1 cpu and about 10 s with the STO-3G basis. %nproc¼1 %Mem¼100MW %subst l914 /home/callis/l914 # hf/3-21g cis(singlet,icdiag,rw) pop¼reg iop(6/7¼0,6/8¼2,6/9¼2) v12 hf/3-21g for villin 01 1 32.81637 33.61985 32.28627 6 33.63000 33.29000 32.88000 1 34.50000 33.68000 32.36000 6 33.48000 33.99000 34.24000 1 33.65000 35.06000 34.12000 1 32.49000 33.90000 34.69000 6 34.36000 33.69000 35.40000 6 34.00000 32.96000 36.48000 1 33.00000 32.56000 36.54000 6 35.78000 33.96000 35.61000 7 35.11000 32.64000 37.24000 1 35.11000 31.94000 37.97000 6 36.24000 33.29000 36.78000 6 36.74000 34.73000 34.92000 1 36.44000 35.20000 34.00000 6 37.50000 33.47000 37.35000 1 37.81000 33.05000 38.29000 6 38.06000 34.89000 35.36000 1 38.69000 35.51000 34.75000 6 38.38000 34.19000 36.53000 1 39.38000 34.40000 36.91000 6 33.83000 31.78000 32.93000 8 32.78000 31.14000 32.99000

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31

7 35.03000 31.20000 32.87000 1 35.21010 30.21638 32.87693 1 35.90063 31.62840 32.62816 50,53 The program getv.f, reproduced below reads the .log file produced from a run using the above input file from standard input, assembles the nonspinadapted matrix elements into the correct singlet and triplet linear combinations, and outputs the desired single matrix element. This is a “debugging” version, and checking is commented out for production. c 29may06 getv.f callis: program to test algorithm for constructing c singlet CI Hamiltonian matrix elements from the CI matrix elements c from Gaussian03. This gives exactly the values for singlet and c triplet CIS energy eigenvalues and eigenvectors in the Gaussian c output. program getv IMPLICIT REAL*8(A-H,O-Z) character line*80 dimension H(6,6),HL(21), C(6,6),E(6) dimension V(3,3),VL(6), CS(3,3), ES(3) c dimension of HL ¼ n*(nþ1)/2 if H(n,n) 10 read(5,’(a80)’)line if(line(1:4).ne.’ < AA’)goto 10 read(5,*) 900 format(i7,5g13.6) 901 format(i3,2x,6f10.6) 902 format(2x,6f10.6) do i¼1,6 read(5,900)ii,(H(i,j),j¼1,5) enddo read(5,*) do i¼1,6 read(5,900)ii,H(i,6) enddo write(6,’(6i10)’)(i, i¼1,6) do i¼1,6 write(6,901)i,(H(i,j),j¼1,6)

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Patrik R. Callis

enddo write(6,*) write(6,*) c elements of the matrix being diagolized are stored in a linear c array h in order 11, 12, 22, 13, 23, 33, 14, 24, 34, 44, ..... c c c c

if i¼row and j¼col: K¼I þ J*(J-1)/2 IF(I.GT.J)K¼JþI*(I-1)/2 H(K)¼HIJ do i¼1,6 do j¼1,i k¼iþj*(j-1)/2 IF(I.GT.J)K¼JþI*(I-1)/2 HL(k)¼H(i,j) enddo enddo

c check the eigenvalues,vectors call givens(HL,C,E,6,6) write(6,902)(E(i),i¼1,6) write(6,*) do i¼1,6 write(6,901)i,(C(i,j),j¼1,6) enddo c

get the singlet eigenvalues and vectors V(1,1)¼ H(1,1)þabs(H(1,4)) V(2,2)¼ H(2,2)þabs(H(2,5)) V(3,3)¼ H(3,3)þabs(H(3,6)) V(1,2)¼ H(1,2)þH(1,5) V(1,3)¼ H(1,3)þH(1,6) V(2,3)¼ H(2,3)þH(2,6) print*,’ Singlet CI Hamiltonian’ do i¼1,3 write(6,901)i,(V(i,j),j¼1,3) enddo write(6,*) write(6,*) do i¼1,3 do j¼i,3 k¼iþj*(j-1)/2

Predicting Fluorescence

33

IF(I.GT.J) K¼JþI*(I-1)/2 VL(k)¼V(i,j) print*,k,VL(k) enddo enddo call givens(VL,CS,ES,3,3) print*, ’Singlet eigenvalues and vectors’ print*,’ Coef. divided by root 2 as in G03 output’ print*,’ ’ write(6,902)(ES(i),i¼1,3) c in eV write(6,902)(ES(i)*27.2116,i¼1,3) write(6,*) do i¼1,3 write(6,901)i,(CS(i,j)/2**.5,j¼1,3) enddo c check Triplet eigenvalues V(1,1)¼ H(1,1)-abs(H(1,4)) V(2,2)¼ H(2,2)-abs(H(2,5)) V(3,3)¼ H(3,3)-abs(H(3,6)) V(1,2)¼ H(1,2)-H(1,5) V(1,3)¼ H(1,3)-H(1,6) V(2,3)¼ H(2,3)-H(2,6) print*,’ Triplet CI Hamiltonian’ do i¼1,3 write(6,901)i,(V(i,j),j¼1,3) enddo write(6,*) write(6,*) do i¼1,3 do j¼i,3 k¼iþj*(j-1)/2 IF(I.GT.J) K¼JþI*(I-1)/2 VL(k)¼V(i,j) print*,k,VL(k) enddo enddo c givens is a commonly used routine for finding eigenvalues and c eigenvectors of symmetric matrices, as result from the linear

34

Patrik R. Callis

c variation method. call givens(VL,CS,ES,3,3) print*, ’Triplet eigenvalues and vectors’ print*,’ Coef. divided by root 2 as in G03 output’ print*,’ ’ write(6,902)(ES(i),i¼1,3) c in eV write(6,902)(ES(i)*27.2116,i¼1,3) write(6,*) do i¼1,3 write(6,901)i,(CS(i,j)/2**.5,j¼1,3) enddo end

ACKNOWLEDGMENTS This work was supported by NSF grants MCB-013306, MCB-0446542, and MCB0847047. The author thanks Pedro Muino, James Vivian, Tiqing Liu, Alexander Petrenko, Jose Tusell, Ryan Hutcheson, and Carl Fahlstrom for work contributing to this chapter; he is also indebted to Andy Albrecht, Bruce Hudson, Lenny Brand, Mary Barkley, Frank Prendergast, Jay Knutson, Jaap Broos, Dongping Zhong, Jonathan King, Yves Engelborghs, and J. Michael Schurr for inspiration and discussions. Final thanks are to the late Michael Zerner for generously providing his program so many years ago.

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C H A P T E R

T W O

Modeling of Regulatory Networks: Theory and Applications in the study of the Drosophila Circadian Clock Elizabeth Y. Scribner* and Hassan M. Fathallah-Shaykh*,†,‡,§ Contents 1. Introduction 2. Developmental History of the Drosophila Circadian Clock 3. Comparative Analysis of Three Network Regulatory Models 3.1. Michaelis–Menten enzyme kinetics: “The gold standard” 3.2. A probabilistic model for the Drosophila circadian clock 3.3. A new regulatory network model 4. The CWO Anomaly and a New Network Regulatory Rule 5. Concluding Remarks References

40 42 51 51 53 58 63 67 69

Abstract Biological networks can be very complex. Mathematical modeling and simulation of regulatory networks can assist in resolving unanswered questions about these complex systems, which are often impossible to explore experimentally. The network regulating the Drosophila circadian clock is particularly amenable to such modeling given its complexity and what we call the clockwork orange (CWO) anomaly. CWO is a protein whose function in the network as an indirect activator of genes per, tim, vri, and pdp1 is counterintuitive—in isolated experiments, CWO inhibits transcription of these genes. Although many different types of modeling frameworks have recently been applied to the Drosophila circadian network, this chapter focuses on the application of continuous deterministic dynamic modeling to this network. In particular, we present three unique systems of ordinary differential equations that have been used to successfully model different aspects of the circadian network. The last model incorporates the newly identified protein CWO, and we explain how this model’s unique mathematical equations can be used to explore and resolve the CWO * Department of Mathematics, The University of Alabama at Birmingham, Birmingham, Alabama, USA Department of Neurology, The University of Alabama at Birmingham, Birmingham, Alabama, USA Department of Cell Biology, The University of Alabama at Birmingham, Birmingham, Alabama, USA } The UAB Comprehensive Neuroscience and Cancer Centers, Birmingham, Alabama, USA { {

Methods in Enzymology, Volume 487 ISSN 0076-6879, DOI: 10.1016/S0076-6879(11)87002-3

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2011 Elsevier Inc. All rights reserved.

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Elizabeth Y. Scribner and Hassan M. Fathallah-Shaykh

anomaly. Finally, analysis of these equations gives rise to a new network regulatory rule, which clarifies the unusual role of CWO in this dynamical system.

1. Introduction The Drosophila circadian clock is a self-regulating intracellular network with a chain of biochemical reactions that generate oscillations of its key molecular components, with peak-to-peak time periods of approximately 24 h. Major advances in molecular biology and genome sequencing over the past few decades have identified the underlying architecture of the circadian clock, thus making this network an archetype for mathematical modeling and simulation. In the section that follows, we present several static diagrams of the Drosophila circadian network as it evolved with the discovery of new clock components. Red arrows denote activation of a molecule, whereas blue lines with bars denote repression, or negative feedback. Blue lines with an “x” denote degradation of one molecule by another. These molecules consist of mRNAs and proteins, which are represented with lowercase italics (per, tim, vri, etc.) and uppercase font (PER, TIM, VRI, etc.), respectively. While these static diagrams provide useful illustrations of the network’s architecture, however, a more quantitative approach is required in order to understand the system’s behaviors and complex dynamics (Kitano, 2002). In particular, we are interested in using quantitative modeling to resolve an anomaly in the clock’s architecture. Like any scientific investigation, a good mathematical model begins with a question. In the case of the Drosophila circadian network, we are trying to resolve the counterintuitive effects of a recently identified network molecule (clockwork orange, CWO) on its direct target genes (per, tim, pdp1, vri) (Kadener et al., 2007). Direct target genes in this network are defined as mRNAs, which are transcriptionally activated or repressed by regulating proteins. The per and tim transcripts encode for proteins PER and TIM, respectively, which participate in a negative feedback loop that suppresses the expression of their own mRNA (Gekakis et al., 1995). We, henceforth, refer to this loop as the PER/TIM negative feedback loop. The vri and pdp1 transcripts encode for proteins VRI and PDP1, respectively. VRI suppresses its own transcription in a negative feedback loop, while PDP1 activates its expression in a feed-forward loop (Blau and Young, 1999; Cyran et al., 2003). These three loops (PER/TIM, VRI, PDP1), modular in structure, comprise the basic framework of the clock’s architecture. Very recent studies have revealed the existence of an additional clock network gene, cwo (Kadener et al., 2007). Laboratory experiments

Modeling of Regulatory Networks

41

performed in vitro show that, in addition to suppressing its own expression in a negative feedback loop (like PER/TIM and VRI), cwo’s encoded protein CWO represses the expression of its other direct target genes per, tim, pdp1, and vri. However, detected levels of these proteins at the peak of their oscillations are higher in wild-type (wt) flies than in cwo-mutant flies, suggesting that CWO acts as an activator (rather than a repressor) of its direct target genes in the overall network (Matsumoto et al., 2007; Richier et al., 2008). This anomaly highlights the limitations of isolated in vitro experiments when trying to answer broader questions about a molecule’s overall role in a dynamic network (Endy and Brent, 2001). Like any engineering system, it is impossible to fully understand the function of an individual clock component (like cwo) without analyzing its behavior holistically within the network. In fact, the clock’s underlying control mechanisms exhibit many of the properties characteristic of robust engineering systems, including its ability to respond to environmental stimuli, the presence of negative and positive feedback loops, redundancy among these loops in the event of one component’s failure, and finally modularity (Kitano, 2002). For this reason, many scientists have attempted to model the behavior of the clock using mathematical equations and computer simulations. One of the most important characteristics of any mathematical model is its ability to replicate the observed biology of a network system (Szallasi et al., 2006). In Section 3, we present a comparative analysis of three different mathematical models, which have succeeded in reproducing 24-h oscillations of the key network components used in each model. The first model we present utilizes a system of rate equations with Michaelis–Menten and Hill-type kinetics, which refer to as the “Gold Standard,” given its prolific use among models of intracellular regulatory networks. These types of equations, however, require several variables and parameters to model a single molecular interaction, thus making the method prohibitive—or at least very difficult to understand—when the number of network molecules is large, as in the case of the Drosophila circadian clock (see Fig. 2.5, Section 2). Therefore, for comparison, we present two other models with very different types of rate equations that also simulate the Drosophila clock network. The second model utilizes first-order kinetic equations that incorporate the binding probabilities of transcription factors (both repressors and activators) to gene promoter regions, also known as E-boxes (Xie and Kulasiri, 2007). However, this particular model was developed prior to the discovery of the new clock gene cwo, and hence it does not help in resolving the aforementioned anomaly in the clock’s architecture. Finally, we present a third mathematical model with a novel set of rate equations, which are simplistic yet effective in replicating the observed biology of the circadian clock, such as its response to light stimuli as well as the phenotypes

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Elizabeth Y. Scribner and Hassan M. Fathallah-Shaykh

associated with various gene mutations (Fathallah-Shaykh et al., 2009). In addition to reproducing experimental observations of network behaviors, a good model attempts to answer a proposed question. This model incorporates the new network gene cwo and its corresponding protein CWO, with the added advantage that the proposed system of equations allows for easy analysis of the effects of CWO on its direct target genes (per, tim, vri, pdp1) at the peaks of their oscillations. We explain the unique properties of these equations and the results of their analysis in Section 4. Finally, a good model not only generates insights into the dynamic behavior of a system but also helps to make predictions about network behaviors in similar systems (Endy and Brent, 2001). Indeed, it is reasonable to assume that control mechanisms employed by the Drosophila circadian clock might also be effective in regulating other types of intracellular networks, such as the mammalian circadian clock. With this in mind, we use the insights gained from our analysis of the CWO anomaly to develop a new network regulatory rule, which can be applied generally to any network system exhibiting control mechanisms similar to that of the Drosophila clock.

2. Developmental History of the Drosophila Circadian Clock Until recently, the molecular mechanism of the Drosophila circadian clock has eluded scientists. In the 1980s, scientists used genetic screens to identify the first of many key molecular actors in this complex dynamic network (Hall and Rosbash, 1987). Aware of the clock’s endogenous nature (Moore-Ede et al., 1982), they were looking for a molecule whose biochemical structure might shed light on the nature of this self-sustaining oscillator. The discovery of the per gene offered new insights by revealing a negative feedback loop, in which the per transcript and PER protein oscillate in abundance (Baylies et al., 1987; Hardin et al., 1990). Peaks in concentration of per mRNA and PER protein are separated by a 4–6 h gap, after which activation of per expression begins to decline, suggesting that the PER protein either directly or indirectly inhibits the transcription of its own mRNA (Zerr et al., 1990). In an effort to understand the mechanism of this interaction in more detail, Goldbeter (1995) proposed a continuous, deterministic mathematical model of a single negative feedback loop in the Drosophila circadian clock. A schematic diagram of his 1995 model is shown in Fig. 2.1. The study particularly focused on the function of PER phosphorylation, which delays entry of the protein into the nucleus, and hence the repression of per transcription (Goldbeter, 1995). A more detailed analysis of this model and its accompanying system of rate equations will be presented in Section 3.

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Modeling of Regulatory Networks

VS

–

E-boxper

PN

k1

k2

per

kS Vm

V1 P0

V3 P1

V2

P2

Vd

V4

Figure 2.1 1995 Goldbeter model. per mRNA is transcribed and transported to the cytoplasm at a rate VS, where it translates unphosphorylated PER protein (P0) at a rate kS and degrades at a rate Vm. Multiple and reversible phosphorylation of PER (P0, P1, P2) delays entry of the fully phosphorylated protein (P2) into the nucleus (PN), where it represses the transcription of per mRNA by binding to E-box. P0 is phosphorylated into P1 and P2 at rates V1 and V3, respectively; while P2 is unphosphorylated into P1 and P0 at rates V4 and V2, respectively. P2 is transported into the nucleus at a rate k1, and PN is transported back into the cytoplasm at a rate k2.

In the same year, Gekakis et al. (1995) isolated a second circadian network gene—timeless (tim)—whose protein TIM binds with PER. Moreover, Saez and Young (1996) soon discovered that cytoplasmic assembly of the PER/TIM complex is a prerequisite for nuclear transport of either protein, without which oscillations cease to exist. Further studies revealed that light stimulates the degradation of TIM and thereby helps to regulate the phase of the circadian cycle (Hunter-Ensor et al., 1996; Lee et al., 1996). TIM’s response to light is mediated by a separate oscillating protein CRY, whose expression peaks during daylight hours in response to external stimuli (Emery et al., 1998). Intuitively, the addition of a second tim feedback loop in the circadian network would increase the system’s resistance to small perturbations or variability. With the help of Leloup, Golbeter proceeded to verify this hypothesis by updating his 1995 model to fit these new discoveries. The updated schematic diagram is shown in Fig. 2.2. Not included in their expanded model is the exact mechanism through which light degrades phosphorylated TIM via the photoreceptor CRY. Nevertheless, Leloup and Goldbeter (1998) discovered that the domain in parameter space was indeed increased with the addition of a second feedback loop. Meanwhile, Bae et al. (1998) reported the discovery of a new Drosophila circadian network gene dClock (Clk), homologous to the circadian clock gene found in mammals. A previously identified 69-bp promoter region (or E-box) upstream of the per gene suggested the presence of a transcriptional activator (Hao et al., 1997), and scientists soon confirmed that the Drosophila CLOCK protein (CLK) drives expression of both PER and TIM by

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Elizabeth Y. Scribner and Hassan M. Fathallah-Shaykh

Light

tim

E-boxtim

T0

–

T1

T2

P-TN

P-T

E-boxper

per

P0

P1

P2

Figure 2.2 1998 Goldbeter and Leloup model (updated from 1995). The expanded diagram includes 10 molecules. Like PER, TIM is phosphorylated multiple times (T0, T1, T2) before binding with fully phosphorylated PER (P2) to form the PER–TIM complex (P–T). After being transported to the nucleus, the complex (P–TN) represses transcription of both the per and tim genes by binding to E-box.

binding with the protein CYCLE (CYC) to form a heterodimer and transcriptional activator CLK–CYC. Accumulation of the PER–TIM dimer in the nucleus subsequently inhibits CLK–CYC’s transcriptional regulation of the per and tim genes (Darlington et al., 1998), as shown in Fig. 2.3. These findings are consistent with evidence that PER and TIM do not bind directly to their own DNA (Sasson-Corsi, 1998). Another mammalian homolog double-time (dbt) was found to encode for a protein kinase DBT, which positively regulates the accumulation and nuclear-transport of PER–TIM through cytoplasmic phosphorylation of the dimer, without which per and tim cease to oscillate (Kloss et al., 1998; Price et al., 1998). More recent inquiries into the role of DBT have revealed that it actually binds to PER, using the molecule as a vehicle for entry into the nucleus where it directly suppresses the activity of CLK–CYC (Kim et al., 2007; Yu et al., 2006). A year later, Glossop et al. (1999) observed that protein concentrations of CLK oscillate in counter phase to PER and TIM concentrations, suggesting

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Modeling of Regulatory Networks

tim

TIM

E-boxtim

– P-TN

C-CN

DBT

P-TP

E-boxper

C-C

CLK CYC per

clk PER

Figure 2.3 Expanded model of the circadian network. Protein kinase DBT positively regulates PER–TIM (P–T) accumulation and transport through phosphorylation of the dimer (P–TP). The nuclear PER–TIM dimer (P–TN) then inhibits the transcriptional activity of the nuclear CLK–CYC heterodimer (C–CN), which otherwise binds to promoter regions (E-boxes) in the per and tim genes to activate transcription.

the presence of an additional feedback loop with transcriptional activators and/or repressors of the clk gene. These predictions were supported by reports of a novel clock-controlled gene vrille (vri), whose transcripts oscillate in the same phase as per and tim (Blau and Young, 1999). In 2001, Ueda et al. (2001) revised the mathematical model of the Drosophila circadian clock to include this additional clk feedback loop, using equations similar to those introduced by Goldbeter (1995) and Leloup and Goldbeter (1998). Though this model confirmed the feasibility of interlocked feedback loops, it was not until 2003 that the exact role of the protein VRILLE as a transcriptional repressor of CLK was confirmed (Glossop et al., 2003). In the same year, Cyran et al. (2003) identified the Par-Domain-Protein (pdp1) gene as part of positive feedback loop that regulates the expression of the clk gene. In turn, CLK activates the expression of the pdp1 gene, without which clock components become arrhythmic, or cease to oscillate. Following this discovery, Smolen et al. (2004) proposed an updated mathematical model of the Drosophila clock that incorporated the vrille negative feedback loop and the newly discovered Pdp1 positive feedback loop into the network. In 2007, scientists Xie and Kulasiri modeled the same network using a novel set of continuous, deterministic mathematical equations, which will be outlined in more detail in Section 3. Figure 2.4 provides a schematic diagram of the additional feedback loops used in the model by Xie and Kulasiri (2007).

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Elizabeth Y. Scribner and Hassan M. Fathallah-Shaykh

clk

CLK

CYC PER

+ PDP1

PER-TIM

VP-BOXclk

–

–

TIM CLK-CYC

VRI

vri

pdp1

E-BOXvri

E-BOXpdp1

CLK-CYC-PER-TIM

tim

E-BOXtim

E-BOXper

per

Figure 2.4 2007 Xie and Kulasiri model. Two negative loops (PER–TIM and VRI) and one positive loop (PDP1) regulate this network of 14 molecules. PER–TIM binds to CLK–CYC to inhibit the transcriptional activity of this E-box promoter. VRI binds to the clk VP-box to inhibit transcription of the clk gene. Meanwhile, PDP1 competes with VRI to activate transcription of clk. For simplification, phosphorylation and compartmentalization are ignored in this diagram, as are the effects of both light on TIM degradation and the protein DBT on PER/TIM transport.

The latest update to the Drosophila circadian clock occurred in 2007 with the much anticipated discovery of the cwo gene, a core clock component that functions as both an activator and repressor of clock gene expression (Kadener et al., 2007; Richier et al., 2008). To date, the only mathematical model to incorporate this new discovery was published in 2009 by Fathallah-Shaykh, Bona, and Kadener. Figure 2.5 provides a schematic diagram of this new model. A more detailed analysis of the accompanying mathematical equations used to model the network in Fathallah-Shaykh et al. (2009) will be presented in Section 3. In the diagram, CWO acts as a direct transcriptional repressor of pdp1, vri, tim, per, and cwo itself, indicating that cwo-mutants should exhibit higher expression levels of each gene compared to wt Drosophila. However, several key studies indicate that the opposite is true (Matsumoto et al., 2007; Richier et al., 2008). The absence of cwo actually increases peak levels of the four transcripts (pdp1, vri, tim, and per) indicating that cwo also functions as an activator—an anomaly that will be resolved in more detail in Section 4, using the above model (Fathallah-Shaykh et al., 2009). All the aforementioned models in this section and those discussed in Sections 3 and 4 are continuous and deterministic, meaning the behavior of each molecular species is analyzed in the absence of unknown perturbations (or noise). Each model is accompanied by a set of ordinary differential equations, which measure the time-evolution of the concentration of the

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Modeling of Regulatory Networks

CYC clk

+

DBT

CLK CLK-CYC

PDP1

–

PER

PER-TIMP

– cry

CRY

TIM

cwo

VRI

CWO vri

tim pdp1

per

Figure 2.5 A schematic diagram incorporating the regulatory effects of cwo. For simplicity, transcriptional activation and repression through binding to E-box is summarized with single red arrows and blue lines, connecting the transcriptional regulator to the final mRNA transcript produced by the regulated gene. CLK–CYC activates expression of all gene products, while CWO directly inhibits expression of each gene product. The model also incorporates the protein CRY, which is known to accelerate the degradation of TIM, indicated by a blue X. DBT positively regulates phosphorylation and transport of PER–TIM into the nucleus, where is represses CLK–CYC.

mRNAs and proteins involved in the proposed network. These rate equations are functions of both kinetic parameters—which include the expression and degradation rates of each molecule—and the varying concentrations of the other molecular species. Using numerical integration, these proposed models are capable of producing robust oscillations of the mRNAs and proteins, which replicate the period and phase-peak timing observed in nature. However, given the unpredictable variation (or stochasticity) in gene expression, which is not taken into account in the deterministic model, questions have been raised regarding the effects of noise on the dynamical behavior of the system (Gonze et al., 2002; Li and Lang, 2008). To address these concerns, several stochastic models have been proposed that examine the Drosophila circadian rhythm’s resistance to molecular noise (see Table 2.1). The majority of these models utilize a birth and death chain stochastic process governed by a chemical master equation (CME), which measures the time evolution of the probability function of the system’s state. !; t Þ, computes the likelihood that the This probability function, P ð! x; tjx 0 0 quantity of molecules of each variable (or species) in the system will assume certain values ! x within the parameter space O at time t, given an initial state ! x0 at time t0. Here, ! x and ! x0 are n-dimensional vectors representing the number of molecules of the n molecular species in the network.

Table 2.1 An historical selection of mathematical models and computer simulations (both stochastic and deterministic) of the Drosophila circadian clock over the past two decades Authors, date

Det. vs. stoch.

Methods/equations

Molecules/loops

Goldbeter (1995) Leloup and Goldbeter (1998) Ueda et al. (2001)

Deterministic Deterministic

Michaelis–Menten and Hill-Type Michaelis–Menten and Hill-Type

PER negative feedback loop PER/TIM negative feedback loop

Deterministic

Michaelis–Menten and Hill-Type

Smolen et al. (2001)

Deterministic

Gonze et al. (2002)

Stochastic

Smolen et al. (2002)

Deterministic/ stochastic (comparison)

Gonze et al. (2003)

Stochastic

Smolen et al. (2004)

Deterministic

Leise and Moin (2007)

Deterministic

Interlocked PER/TIM and CLK–CYC negative feedback loops Michaelis–Menten and Hill-Type Interlocked PER feedback loop (negative) w/delay terms and CLK feedback loop (positive) Birth and death chain governed by a PER negative feedback loop CME, and solved using the Gillespie algorithm Michaelis–Menten with delay terms Interlocked PER feedback loop (negative) and CLK feedback loop (positive) (deterministic); birth and death chain governed by a CME, and solved using both the Gillespie algorithm and a fixed-time algorithm (stochastic) Birth and death chain governed by a PER/TIM negative feedback loop CME, and solved using the Gillespie algorithm Michaelis–Menten and Hill-Type Interlocked PER and VRILLE negative feedback loops and PDP1 positive feedback loop Michaelis–Menten and Hill-Type Interlocked PER/TIM and CLK–CYC negative feedback loops

Xie and Kulasiri (2007)

Deterministic

First order reactions w/binding probability equations

Li and Lang (2008)

Deterministic/ stochastic (comparison)

Fathallah-Shaykh et al. (2009)

Deterministic

Michaelis–Menten with time delays (deterministic); birth and death chain governed by a CME, and solved using the Gillespie algorithm and the Gillespie Chemical Langevin Equation (Gillespie, 2000) (stochastic) Regulatory equations w/sigmoidal and logistic functions

Interlocked PER/TIM and VRILLE negative feedback loops and PDP1 positive feedback loop Interlocked PER feedback loop (negative) and CLK feedback loop (positive)

Interlocked PER/TIM, VRILLE, and CWO negative feedback loops and the PDP1 positive feedback loop

50

Elizabeth Y. Scribner and Hassan M. Fathallah-Shaykh

!; t Þ=@t, canUnfortunately, the time evolution of this function, @P ð! x; tjx 0 0 not be solved analytically for a system as complex as the Drosophila circadian network (Szallasi et al., 2006). These models, therefore, must utilize predesigned algorithms in order to produce numerical simulations of the CME. To predict the system’s advancement from one reaction state space to the next, most of these models implement the stochastic simulation algorithm (SSA), or Gillespie algorithm, which is capable of producing numerical solutions to the CME (Gillespie, 1977; Szallasi et al., 2006). This algorithm is a discrete method of analysis that replaces the differentiable concentrations of each molecular species in the deterministic model with the probabilities that a given reaction will occur in a given time step. These “transitional probabilities” are proportional to the number of molecules of each species in the network as well as the kinetic rate parameters from the deterministic model (Gonze et al., 2002). Using transition probabilities, the algorithm determines stochastically the next reaction that will take place as well as the time interval to the next reaction (Li and Lang, 2008). Meanwhile, the transition probabilities and numbers of molecules are updated at each time step, as different species are produced and consumed in what is commonly referred to as the birth and death chain (Levin et al., 2009). A more detailed description of these methods can be found in the models listed in Table 2.1, as well as the sources referenced above. Several of the stochastic models listed in Table 2.1 introduce slight variations to the Gillespie algorithm by advancing the system by a predetermined time step t, instead o f the time to the next reaction event (Li and Lang, 2008; Smolen et al., 2002). Intended to increase computational speed, this courser-grained approximation is equally capable of producing sustained oscillations under proper choices of t (Szallasi et al., 2006). Nevertheless, numerical analysis of these models is very time-consuming with increasing numbers of mRNAs and proteins in the network (Li and Lang, 2008). As a result, most of these stochastic simulations only incorporate one or two feedbacks loops of the Drosophila circadian network into their model, thus limiting their ability to perform system-wide analysis of the entire network (Fig. 2.5). For this reason, we focus on the use of continuous and deterministic models as tools to explain the behavior of the system as a whole. Important results, however, can be drawn from the conclusions of these stochastic simulations. Namely, these models are indeed capable of producing sustained oscillations when the numbers of mRNAs and proteins are relatively low—in the tens and hundreds, respectively (Gonze et al., 2002, 2003). Moreover, the results of these simulations largely agree with the predictions of the deterministic model, thus validating its use as a tool for system-wide analysis of the network (Gonze et al., 2002). Given the aforementioned limitations of stochastic simulations, however, and our desire to resolve a system-wide anomaly in the Drosophila circadian network, we now turn to the use of continuous and deterministic models of regulatory networks.

Modeling of Regulatory Networks

51

3. Comparative Analysis of Three Network Regulatory Models Having established the advantages of continuous deterministic models of regulatory networks in the previous section, this section presents examples of three such models, which have been used to simulate the Drosophila circadian clock over the last 15 years. More specifically, we introduce these models within the framework of the discovery of the various components of the circadian network, and we offer analysis relating the methods used to the known complexity of the system. In our third example, we introduce a new and more simple network model that is capable of both addressing the complicated dynamics of the system and answering important questions about the relationships among various components of the circadian network. In an effort to optimize our presentation of the mathematics used in the second and third models, we have elected to use reduced versions of the current network model shown in Fig. 2.5, Section 2. For comparison, these reduced networks each consist of a single positive and a single negative feedback loop.

3.1. Michaelis–Menten enzyme kinetics: “The gold standard” In 1995, Goldbeter proposed a model for a single negative feedback loop exerted by the PER protein in the Drosophila circadian clock on the transcription of its own per mRNA (Goldbeter, 1995). Many of the other transcription factors, proteins, and feedback loops were unknown at the time. However, scientists had made important advances in elucidating the underlying mechanism of cellular rhythmic behavior, which was and continues to be associated with the periodic repression of the network’s various components. The model (Fig. 2.1, Section 2) correctly hypothesizes that multiple phosphorylation of the PER protein (P0, P1, P2) in the cytoplasm delays entry of the protein (PN) into the nucleus (dashed line), where it then represses the transcription of its own mRNA. Since the publication of this model, further analysis has revealed that the system is more complicated than a single negative feedback loop. Despite limiting knowledge of the Drosophila circadian clock, however, Goldbeter succeeded in using his model to produce sustained periodic oscillations of both total PER protein (PT ¼ P0 þ P1 þ P2 þ PN) and per mRNA (per) through numerical integration of a system of five differential equations. These equations estimate the rates of change of the five system molecules (P0, P1, P2, PN, MP) by employing Michaelis–Menten and Hill-type enzyme kinetics—a method that we refer to as the “Gold Standard” given its frequent use among the continuous deterministic models listed in Table 2.1.

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Elizabeth Y. Scribner and Hassan M. Fathallah-Shaykh

In 1913, chemists Michaelis and Menten used rate equations and the Law of Mass Action to conclude that the rate of any enzyme-catalyzed reaction can be expressed as: dS dP k2 E0 S k max S ¼ ¼ ¼ dt dt Km þ S K m þ S where S is the concentration of substrate that reacts with the enzyme (E0) to produce the product (P), and where Km ¼ (k 1 þ k2) / k1 is known as the Michaelis–Menten constant (Michaelis and Menten, 1913). A detailed derivation of this equation can be found in a number of sources, including Murray (2002) and Ellner and Guckenheimer (2006). The reaction diagram in Fig. 2.6 illustrates the transformation of substrate S into product P, according to reaction rates k1, k 1, and k2. Comparison of these equations with Goldbeter’s equations reveal that both degradation and phosphorylation rates of the molecules (P0, P1, P2, per) in his model adhere to Michaelis– Menten kinetics, while translation and nuclear transport of the PER protein are linear according to the Law of Mass Action, which states that the transcription (or transport) of a protein occurs at a rate proportional to the concentration of its mRNA (or protein, respectively): ½transcription  ½degradation n dðper Þ per ¼ vs K1  dt K1n þ KNn Km þ per h i h i phos: unphos: ½translation  P0  þ P P P 1 0 1 !  dP0 ¼ P0 P1 dt ks per  V1 þ V2 K1 þ P0 K2 þ P 1 phos: unphos: phos: unphos: ½P0  ! P1   ½P0  P1   ½P1 ! P2  þ ½P1  P2  dP1 ¼ V1 P0  V2 P1  V3 P1 þ V4 P2 dt K1 þ P 0 K2 þ P1 K3 þ P1 K4 þ P2

ð2:1Þ

ð2:2Þ

ð2:3Þ

phos: unphos: ½P1  ! P2   ½P1  P2   ½nuc: transport þ ½cyto: transport  ½degradation dP2 ¼ V3 P1  V4 P2  k1 P2 þ k2 PN  vd P2 dt K3 þ P1 K4 þ P 2 Kd þ P2

ð2:4Þ dPN ½nuc: transport  ½cyto: transport ¼ k1 P2  k2 PN dt k1 ES

S+E

k2

ð2:5Þ

P+E

k–1

Figure 2.6 Enzyme-catalyzed transformation of substrate S into product P.

Modeling of Regulatory Networks

53

Equation (2.2), for example, models the rate of change in concentration of the unphosphorylated protein P0. The protein is produced at a rate (ks), proportional to the concentration of cytoplasmic per mRNA. In the second term, P0 is phosphorylated into P1, according to Michaelis–Menten kinetics, where V1 ¼ vmax and K1 is the Michaelis–Menten constant. Likewise, P1 is unphosphorylated back into P0 in a similar fashion with V2 ¼ vmax and K2 as the Michaelis–Menten constant. In order to model the negative feedback by PER on the concentration of per mRNA, Goldbeter introduced a Hill-type equation with degree of binding cooperativity n ¼ 4. The equation H(n) ¼ K1n / (K1n þ PNn) produces a sigmoidal function that assumes a value close to H(n) ¼ 1, and hence the maximum rate of transcription vs, when the concentration of PN is low. As the concentration of nuclear PER increases, H(n) ! 0, and the rate of transcription becomes negligible. Hence, in Goldbeter’s Eq. (2.1), the transcription of per mRNA is regulated by a Hill sigmoidal function, and the molecule degrades at a rate vmMP / (Km þ MP), where vm is the maximum rate of degradation, and Km is the Michaelis–Menten rate constant. Goldbeter selected parameter values that would produce sustained oscillations of PER and per that were close to a period of 24 h, and his mathematical model provides an excellent illustration of the regulatory effects of an isolated negative feedback loop. In 1998, he and Leloup expanded this model to incorporate the formation of the PER/TIM complex (Fig. 2.2, Section 2), and more recent discoveries of additional negative and positive feedbacks loops in the circadian network have prompted even more detailed mathematical models that employ Michaelis–Menten and Hill-type kinetics (see Table 2.1). Some of these latest models include up to five interlocking positive and negative loops, the dynamics of which can grow quite complicated and become difficult to verify when using equations like the ones above. Other criticisms of the use of Michaelis–Menten kinetics to model protein–protein interactions in regulatory networks have been raised by Maini et al. (1991) and Sabouri-Ghomi et al. (2008). They point out that the Michaelis–Menten rate law assumes that the concentration of the regulating element, or enzyme, is much less than the concentration of the element being regulated, or substrate (see Fig. 2.6). Hence, in Eq. (2.2), for example, one must paradoxically assume that the concentration of unphosphorylated PER protein (P0) and phosphorylated PER protein (P1) are simultaneously in excess of one another. Nevertheless, as in Goldbeter’s model, these models can be quite effective in reproducing oscillations of clock molecules with 24 peak-to-peak times, as observed in nature.

3.2. A probabilistic model for the Drosophila circadian clock In 2006, with the knowledge of additional positive and negative regulatory loops in the Drosophila circadian network, Xie and Kulasiri proposed a new and robust mathematical model with a set of equations that incorporate

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Elizabeth Y. Scribner and Hassan M. Fathallah-Shaykh

first-order reactions for translation, degradation, and dissociation, as opposed to Michaelis–Menten and Hill-type equations (Xie and Kulasiri, 2007). In their model, binding probabilities of transcription factors to E-box determine the rates of transcription of the mRNAs, rather than Hill-type equations, which they argue imply an unnatural “switch-like behavior” of transcription. Xie and Kulasiri also use first-order linear expressions to model both translation and degradation, as opposed to Michaelis–Menten equations. The schematic diagram in Fig. 2.4 from Section 2 describes the feedback loops included in their model. CYC–CLK activates the transcription of four genes—pdp1, vri, tim, and per. The protein PDP1 exerts a positive feedback loop on the transcription of its regulator by binding to the V/P box for the clk gene. VRI, however, competes with PDP1 as a transcriptional repressor of clk, thereby exerting a negative feedback loop on the system. PER and TIM also introduce negative feedback loops into the network by inhibiting the transcriptional effects of CLK–CYC. Xie and Kulasiri make several assumptions in their model. For example, unlike the Goldbeter model, they disregard separate nuclear and cytoplasmic compartments, and hence the nuclear transport of proteins like PER– TIM. Also, for simplification, they do not consider phosphorylation of proteins but instead focus on the effects of transcriptional regulation. Even with this simplification, however, their entire model consists of 19 different equations—13 rate equations for the different proteins and mRNAs (CYC is assumed to be constant) and 5 binding probability equations. In an effort to explain their method, we will consider the reduced schematic diagram shown in Fig. 2.7, which only consists of one positive PDP1 loop and one negative VRI loop. Notice that even with this reduced model, there are already 28 parameters to account for in the system (Fig. 2.7). With the exception of CYC (assumed constant), each of these molecules is associated with a degradation rate (di). Other parameters include binding rates (bi) and unbinding rates (ui) to E-boxes, translation rates (ki), transcription rates (ai and si), disassociation/ association rates of CLK/CYC (vi), and finally the number of E-boxes in gene promoter regions (ngene). The number of E-boxes in the vri promoter (nvri) and the pdp1 promoter (npdp1) is 4 and 6, respectively. Xie and Kulasiri also include rate of change equations for the binding probabilities of transcription factors to promoter regions, or E-boxes. Here, the fraction of E-boxes occupied by CLK–CYC in the promoter regions of vri and pdp1 is Pr1 and Pr2, respectively. Hence, (1  Pr1) and (1  Pr2) represent the fraction of unoccupied E-boxes in the vri and pdp1 gene promoter regions, respectively. Finally, Pr3 and Pr4 denote the fraction of E-boxes occupied by vri and pdp1, respectively, in the gene promoter region for clk, which encodes for CLK. Figure 2.8 provides a pictorial cartoon of this interaction between transcriptional regulator CLK–CYC (C/C) and the promoter region of the vri gene. The solid purple balls represent bound and unbound

55

Modeling of Regulatory Networks

b1 = binding rate of CLK-CYC to E-boxvri b2 = binding rate of CLK-CYC to E-boxpdp1 b3 = binding rate of PDP1 to VP-boxclk b4 = binding rate of VRI to VP-boxclk

k3 clk

(+) b3 PDP1

CLK

CYC

VP-BOXclk b4

v1

v2

(–) CLK–CYC

VRI k2

k1 vri

pdp1

b1 E-BOXvri b2

u1 u2 u3 u4

= = = =

unbinding unbinding unbinding unbinding

rate rate rate rate

of of of of

CLK-CYC to E-boxvri CLK-CYC to E-boxpdp1 PDP1 to VP-boxclk VRI to VP-boxclk

d1 = degradation rate of vri mRNA d2 = degradation rate of pdp1 mRNA d3 = degradation rate of clk mRNA d4 = degradation rate of CLK d5 = degradation rate of VRI d6 = degradation rate of PDP1 d7 = degradation rate of CLK-CYC dimer k1 = translation rate of vri mRNA k2 = translation rate of pdp1 mRNA k3 = translation rate of clk mRNA a1 = transcription rate of CLK/CYC-activated vri gene a2 = transcription rate of CLK/CYC-activated pdp1 gene a3 = transcription rate of PDP1-activated clk gene s1 = transcription rate of VRI-repressed clk gene s2 = transcription rate deactivated vri, pdp1 s3 = basal transcription rate unbound clk gene

E-BOXpdp1

v1 = association rate of CLK/CYC dimer v2 = disassociation rate of CLK/CYC dimer npdp1 = Number of E-boxes in pdp1 promoter nvri = Number of E-boxes in vri promoter

Figure 2.7 Reduced diagram of the circadian network model. One negative loop () and one positive loop (þ) regulate this network of eight molecules.

CLK–CYC heterodimers, which bind to E-box sites (yellow boxes) to enhance vri transcription. In this figure, Pr1 ¼

CC  bound Ebox 2 ¼ Total vri Ebox 4

The probability binding equations for our truncated system are listed below. In these equations, the CYC–CLK dimer is denoted as CC. Each of these equations takes the general form: rate of change ¼ binding rate  unbinding rate. For example, the first term, or binding rate, in the rate of change Eq. (2.6) for the binding probability of CC to the promoter region of the vri gene is the fraction of unoccupied E-box, multiplied by the binding rate of CC (b1) and the concentration of CC available to bind to these unoccupied sites. The second term, or unbinding rate, in the equation measures the decrease in bound E-box sites, which is the product of the

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Elizabeth Y. Scribner and Hassan M. Fathallah-Shaykh

C/C

C/C

E-Box

E-Box

Unbound C/C vri transcription E-Box

E-Box

vri promoter region

Figure 2.8 Cartoon depicting activity in the promoter region of the vri gene. The fraction of occupied E-boxes over the total number of E-boxes, or P1, is 2:4 or 1/2 in this diagram of the vri gene and its promotor region. The rate of change in this ratio is represented by the binding rate of unbound C/C minus the unbinding rate of the bound C/C. (P1 ¼ binding rate  unbinding rate). The rate of vri transcription is the sum of its C/C activated rate and its basal transcription rate (or the transcription rate of unbound E-box).

fraction of occupied E-box and the unbinding rate of CC (u1). In Fig. 2.8, the binding rate at this particular time is 0.5  b1  CC, and the unbinding rate is 0.5  u1. A more detailed derivation of these probability rate equations can be found in the Appendix of Xie and Kulasiri (2006): dðPr1 Þ ½binding rate  ½unbinding rate ¼ ½ð1  Pr1 Þ  b1  CC  ½ðPr1 Þ  u1  dt

ð2:6Þ

dðPr2 Þ ½binding rate  ½unbinding rate ¼ ½ð1  Pr2 Þ  b2  CC  ½ðPr2 Þ  u2  dt

ð2:7Þ

dðPr3 Þ ½binding rate  ½unbinding rate ¼ ½ð1  Pr3  Pr4 Þ  b3  VRI   ½ðPr3 Þ  u3  dt

ð2:8Þ

dðPr4 Þ ½binding rate  ½unbinding rate ¼ ½ð1  Pr4  Pr3 Þ  b4  PDP1  ½ðPr4 Þ  u4  dt

ð2:9Þ

The remaining rate equations for the change in concentrations of the mRNA and protein molecules are shown below. Equations (2.10)–(2.12) represent the rates of change of the mRNA concentrations, while Eqs. (2.13)–(2.16) model the change in concentration of the four proteins and protein complexes. Recall that the concentration of CYC is assumed to be constant in these equations: ½activated transcription þ ½repressed transcription þ½basal transcription  ½degradation dðclkÞ ¼ ½Pr4  s1  þ ½Pr3  a3  þ ½ð1  Pr4  Pr3 Þ  s3   ½d3  clk dt ð2:10Þ

Modeling of Regulatory Networks

57

dðvriÞ ½activated transcription þ ½basal transcription    ½degradation ¼  4 4 1  ð1  Pr1 Þ  a1 þ ð1  Pr1 Þ  s2  ½d1  vri dt ð2:11Þ ½activated transcription þ ½basal transcription  ½degradation     dðpdp1Þ  ¼ 1  ð1  Pr2 Þ6  a2 þ ð1  Pr2 Þ6  s2  ½d2  pdp1 dt ð2:12Þ dðCLKÞ ½translation  ½association CC þ ½disassociation CC  ½degradation ¼ ½k3  clk  ½v1  CLK þ ½v2  CC  ½d4  CLK dt

ð2:13Þ dðVRIÞ ½translation  ½degradation ¼ ½k1  vri  ½d5  VRI dt

ð2:14Þ

dðPDP1Þ ½translation  ½degradation ¼ ½k2  pdp1  ½d2  PDP1 dt

ð2:15Þ

dðCCÞ ½association CC  ½disassociation CC  ½degradation ð2:16Þ ¼ ½v1  CLK  ½v2  CC  ½d7  CC dt

In Eq. (2.10), for example, the first term is the rate of transcription of VRI-bound clk VP-box. The second term is the rate of transcription of PDP1-bound clk VP-box; and the third term is the basal transcription rate of unbound VP-box. The last term in every equation is the rate of degradation of each molecule (di  molecule), which is a linear rate as opposed to the Michaelis–Menten expressions used in Goldbeter’s model. In Eq. (2.11), (1  Pr1)4 is the probability that all four vri gene E-boxes are unbound, and likewise, (1  (1  Pr1)4) is the probability that all four E-boxes are bound by CLK–CYC. Hence, the first term, or CC-activated transcription, is described by (1  (1  P1)4)a1, where a1 is the transcription rate of a CC-saturated vri promoter region, when P1 ¼ 1. Note that when P1 ¼ 1, all E-boxes are bound and hence the expression (1  (1  P1)4) ¼ 1. The expression, (1  P1)4s2, describes the basal transcription rate of vri, which is equal to s2 when all E-boxes are unoccupied, and hence P1 ¼ 0. Equations (2.13)–(2.16) follow simple first-order reaction rates. For example, in Eq. (2.13), the first term is the rate of translation of CLK, the second term is the association rate of the CLK–CYC (CC) complex, the third term is the disassociation rate of CC, and the last term is the degradation rate of CLK. Like Goldbeter’s model, which employed very different types of rate equations, this model is capable of producing sustained 24-h periodic oscillations of the interacting circadian network molecules. Also similar to other proposed models in Table 2.1, this model replicates observed biology by

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Elizabeth Y. Scribner and Hassan M. Fathallah-Shaykh

responding to changes in light cycles with phase-adjustments (or entrainment) in the periodic rhythms. The results of various simulated mutations in the per gene were also consistent with experimental observations (Xie and Kulasiri, 2007). Hence, this model validates the idea that models with very different mathematical equations can replicate the same observed biological behaviors of an intracellular regulatory network (Murray, 2002). Hence, when studying a regulatory network, it is prudent to select an appropriate model, which best serves as a tool for exploration and understanding of the question at hand. With this in mind, we now introduce a new intracellular network model whose appeal lies in its relative simplicity in comparison to the previously introduced models.

3.3. A new regulatory network model In this section, we introduce a new set of equations, originally presented by Fathallah-Shaykh et al. (2009) to model the CWO-expanded network in Fig. 2.5 of Section 2. For comparison, however, we will only use these equations to model a simplified version of the full network. The schematic diagram in Fig. 2.9, shown below, displays one positive and one negative

VRI

aPDP1-CC

aVRI-CC (–)

CLK-CYC

avri-VRI aCC-vri

PDP1

(+) apdp1-PDP1

aCC-pdp1 pdp1

vri a = strength of CC-activation of vri CC-vri avri-VRI = strength of vri-activation of VRI aVRI-CC = strength of VRI-repression of CC aCC-pdp1 = strength of CC-activation of pdp1 apdp1-PDP1 = strength of pdp1-activation of PDP1 a = strength of PDP1-activation of CC PDP1-CC

s 1 s2 s3 s 4 s

b1 = maximum rate of formation of vri mRNA b = maximum rate of formation of pdp1 mRNA 2

d = degradation rate of vri mRNA 1 d = degradation rate of pdp1 mRNA

b3 = maximum rate of formation of VRI b4 = maximum rate of formation of PDP1 b5 = maximum rate of formation of CLK-CYC dimer

5

= = = = =

saturation saturation saturation saturation saturation

level level level level level

of of of of of

vri mRNA pdp1 mRNA VRI PDP1 CLK-CYC dimer

2

d3 = degradation rate of VRI d4 = degradation rate of PDP1 d = degradation rate of CLK-CYC dimer 5

Figure 2.9 Reduced diagram of the circadian network model in Fig. 2.5, Section 2. One negative loop () and one positive loop (þ) regulate this network of five molecules.

Modeling of Regulatory Networks

59

feedback loop, with a shorter list of 11 parameters, in comparison to the 28 parameters used in the previous model. Note also that this model only includes five network molecules, as opposed to eight, thus reducing the number of rate equations and simplifying the task of numerical computation. The saturation levels of each molecule at its site of action (si) and the degradation rates of the molecules (di) are all fixed at 100% and 0.01, respectively. Hence, each of the five network molecules is associated with one individual parameter, bi, which measures the maximum rate of formation of the molecule. The remaining six parameters measure the activation and repression strengths of each molecular interaction in the diagram. This model (Fig. 2.9) preserves the fundamental structures, or modules, of the previous schematic diagram (Fig. 2.7), while simplifying intermediary tasks—like phosphorylation and binding to E-box—with single rate constants. Module identification in genetic regulatory networks can help reduce a dynamic system to a simpler network that maintains the important molecular interactions in the network, while also preserving the observed biology of the system (Szallasi et al., 2006). Contributing authors, Szallasi, Periwal, and Stelling in System Modeling in Cellular Biology, explain that modules in genetic regulatory networks can be characterized by their circular feedback structure, where each molecule in a particular module is linked to at least two other molecules in that same modular structure. Figure 2.9 is a reduced model of the more detailed circadian network shown in Fig. 2.5 (Section 2). In this modified version of the model, the positive PDP1-feedback loop and the negative VRI-feedback loop represent two separate modules. Unlike the previous model, however, this reduced diagram does not ignore compartmentalization of molecules and diffusion rates across different cellular membranes. Rather, it incorporates these rates into single activation/ repression parameters. In the VRI loop, for example, CLK–CYC-activation of vri transcription, vri mRNA translation, and VRI-repression of CLK– CYC have all been reduced to single activation/repression parameters (aCC  vri, avri  VRI, aVRI  CC, etc.). These rates (axy) are either positive (þ) or negative (), depending on whether the action of molecule x on molecule y is positive (activation) or negative (repression). Hence, we can dissect avri  VRI into a sum of several positive growth curves, which include the rate of translation of VRI and the transport of VRI to the nucleus, where it then represses CLK–CYC. An analogy can be made to a car factory that puts ready-to-sell cars onto a car dealer’s lot at a rate a. In this case, a is the sum of the rate at which the factory produces cars and the rate of transportation of these cars to the dealer lot. As a result, the five molecular symbols (vri, VRI, etc.) only denote the concentration of each molecule at the site of its action in the network system (Fathallah-Shaykh et al., 2009). However, if one desires to further investigate the influence of nuclear transport on the dynamics of the network, these parameters can be broken down into their

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Elizabeth Y. Scribner and Hassan M. Fathallah-Shaykh

components, and one can compare cytoplasmic and nuclear concentrations of each molecule. Another advantage to this simplified system is that the probability binding rates from Xie and Kulasiri (2007) are directly incorporated into the rate equations for each molecule. The rate of change for each molecule takes the general form: dxi ðt Þ ui ðtÞ ¼ bi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xi ðtÞðsi  xi ðtÞÞ dt 1 þ ui ðtÞ2

ð2:17Þ

In this general Eq. (2.17), n represents the total number of molecules in n X aji xj ðtÞ  di xi ðtÞ the system (in this case, n ¼ 5), and ui ðtÞ ¼ j¼1 1jn models the overall sum of the regulatory influences, including degradation, on a given molecule xi at time t. If molecule j does not regulate molecule i, then aji ¼ 0, a rate that is also used to simulate null mutations of specific circadian network genes in Fathallah-Shaykh et al. (2009). Equations (2.18)– (2.22) are the five rate of change equations needed to analyze the system shown in Fig. 2.9: dðvriÞ uvri ¼ b1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðvriÞðs1  vriÞ; dt 1 þ u2vri

½activation=repression  ½degradation uvri ¼ ½aCCvri  CC  ½d1  vri

ð2:18Þ updp1 dðpdp1Þ ¼ b2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðpdp1Þðs2  pdp1Þ; dt 1 þ u2 pdp1

½activation=repression   ½degradation  updp1 ¼ aCCpdp1  CC  ½d2  pdp1

ð2:19Þ dðVRIÞ uVRI ¼ b3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðVRIÞðs3  VRIÞ; dt 1 þ u2VRI

½activation=repression  ½degradation uVRI ¼ ½avriVRI  vri  ½d3  VRI

ð2:20Þ dðPDP1Þ uPDP1 ðPDP1Þðs4  PDP1Þ; ¼ b4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dt 1 þ u2PDP1

½activation=repression   ½degradation    uPDP1 ¼ apdp1PDP1  PDP1  ½d4  PDP1

ð2:21Þ dðCCÞ uCC ¼ b5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðCCÞðs5  CCÞ; dt 1 þ u2CC

½activation=repression  ½degradation uCC ¼ ½ðaPDP1CC  PDP1Þ þ ðaVRICC  VRIÞ  ½d5  CC

ð2:22Þ

61

Modeling of Regulatory Networks

u

√1 + u2

Odd sigmoidal function 1 0.8 0.6 0.4 0.2 0 – 0.2 – 0.4 – 0.6 – 0.8 –1 – 10

PRI >> NRI

Sum of PRI = Sum of NRI PRI << NRI –8

–6

0 –4 –2 2 4 u = Sum of regulatory influences

6

8

Figure 2.10 Sigmoidal graph. The range of the function g is between 1 and  1.

10

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The odd sigmoidal function, gðui ðtÞÞ ¼ ui ðt Þ= 1 þ ui ðtÞ2 , is bounded by 1 and 1, and hence the rate of formation and degradation—bi g(ui(t))—of a given product xi fluctuates between bi, its maximum rate, and bi at any given time t. Figure 2.10 displays a graphical representation of this function g. This function g exhibits similar behavior to the binding probabilities Pi in the previous model, which fluctuate between 0 and 1 depending on the fraction of occupied E-boxes in the promoter region of each gene. Note that g(ui(t)) ¼ 0 when ui(t) ¼ 0, and hence the sum of the negative regulatory influences (NRI) (degradation and repression) is equal to the sum of the positive regulatory influences (PRI) (activation). The curve approaches g(ui(t)) ¼ 1 when the sum of the PRI is much greater than that of the NRI. Likewise, when repression and degradation outweigh activation, the curve approaches g(ui(t)) ¼ 1, and hence the maximum rate of decay. The overall structure of Eqs. (2.18)–(2.22) is similar to that of a logistic equation, dx(t) /dt ¼ rx(t)(s  x(t)), where r is the growth or decline rate of population x and s is the carrying capacity, or saturation level. However, in the equations used by Fathallah-Shaykh et al. (2009), the constant rate r has been replaced by a varying rate of change, big(ui(t)), which is dependent on the summation of a molecule’s regulatory signals at a given time t. This rate of change is essential to generating oscillations in the concentrations of each molecular species: dxi(t) /dt ¼ big(ui(t))xi(t)(si  xi(t)). At any given time, the population (or concentration of a molecular species) is growing or decaying at a rate big(ui(t)) ¼ ri(t). Since g(ui(t)) fluctuates between 1 and 1, the varying rate ri(t) likewise fluctuates between a maximum rate, ri(t) ¼ bi, and a minimum rate, ri(t) ¼  bi. The closer this rate is to the maximum rate of formation, for example, the faster molecule i will reach its saturation level si. When integrating the simple logistic equation in Matlab with initial conditions x0 at time t ¼ 0, the graph of the concentration of

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A 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

B x0 = 1

r(t2) = –0.0015

r(t1) = –0.0035 0

5

10

15

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1 x0 = 0.01 0.9 0.8 0.7 0.6 r(t1) = 0.35 0.5 0.4 r(t2) = 0.15 0.3 0.2 0.1 0 0 10 20 30

30

40

50

60

Figure 2.11 Simulations to the logistic equation: dx(t) /dt ¼ rx(t)(s  x(t)). In (A), with initial condition x0 ¼ 1, the concentration of molecular species x(t) declines to zero at different negative rates r(t1) ¼  0.0035 and r(t2) ¼  0.0015. In (B), with initial condition x0 ¼ 0.01, the population grows and approaches it saturation level 1 with positive rates r(t1) ¼ 0.35 and r(t2) ¼ 0.15. The larger the absolute value of the growth rate r, the faster the concentration x(t) approaches 0 or 1.

molecule xi approaches either si ¼ 1 or 0, when the rate big(ui(t)) is either positive or negative, respectively. To illustrate this behavior, Fig. 2.11 displays the graphs of several solutions x(t) with varying parameter rates big (ui(t1)) ¼ r(t1) and big(ui(t2)) ¼ r(t2). One of the many advantages of this modeling method is that the regulatory equation (ui(t)), which models the time-dependent relationship between the regulatory molecules that act on molecule i, has a special property at the peaks and troughs of each molecule’s oscillation. Here, the rate of change equation for each molecule is equal to zero. Thus, as long as the concentration of each molecular species never reaches its saturation level or zero, this regulatory equation is also equal to zero; and hence, the concentration of each molecule can be represented by a linear equation at these peaks and troughs. This linearity affords a greater understanding of network interactions, particularly when analyzing the comparative strengths of competing regulatory signals on molecule i. When the molecule reaches its peak concentration, the rate of change (or derivative) of the function xi(t) is zero, and hence the sum of the positive and NRI must be zero as well. For example, when the concentration of CLK–CYC (CC) is at its peak: uCC ¼0 ¼ ðaPDP1CC  PDP1Þ þ ðaVRICC  VRIÞ  ðd5  CCÞ and hence, ðaPDP1CC  PDP1Þ þ ðaVRICC  VRIÞ ¼ ½CC d5 The simplicity of the model is also useful when trying to answer a big-picture question about a network, such as why the absence of a direct

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repressor CWO might actually cause a decrease in peak levels of certain molecules, like per and PER, which it directly represses. Section 4 explains the activity of CWO in more detail and illustrates how these equations were used in Fathallah-Shaykh et al. (2009) to solve this Drosophila circadian network anomaly.

4. The CWO Anomaly and a New Network Regulatory Rule As explained in the history section of this paper, CWO was not identified as a Drosophila circadian network molecule until 2007, and its counterintuitive effects on the overall network are now well documented. Kadener et al. (2007) recorded that the protein CWO directly inhibits CLK–CYC-mediated transcription of all direct target genes (per, tim, vri, pdp1), as well as cwo itself, by binding to and repressing E-boxes. Almost synonymously, Matsumoto et al. (2007) determined that CWO’s negative transcriptional effects actually contribute to sustained high-amplitude oscillations of direct target genes, suggesting that CWO behaves as an activator of these direct target genes in the overall network. In 2009, Fathallah-Shaykh and his colleagues devised the system of equations presented in Section 3 to model this CWO-expanded version of the Drosophila circadian network (Fig. 2.5, Section 2). Their goal was to develop a better understanding of CWO’s regulatory influences in the overall network and to resolve this apparent anomaly in its behavior as both an activator and a repressor of direct target genes. Parameters for the rate equations (Eq. (2.17), Section 3.3) were chosen so that numerical integration yielded timely peaks of all target mRNAs (per, tim, vri, pdp1, cwo) and proteins, with a period of 24 h—consistent with observed biology. Their model was also robust in the sense that it replicated several other behaviors and control mechanisms intrinsic to the clock, like entrainment. For example, the computer model responded to a 12 h time shift in the light–dark (LD) cycle—simulated by advancing the level of cry mRNA at midnight—with a complete phase reversal over the course of 4.5 days. This entrainment period is consistent with that observed in the mammalian clock in nature (Chen et al., 2008). Also, enhancing the activity of the gene promoter CLK–CYC on direct target genes confirms observed increases in peak mRNA levels and shortened periods in constant darkness (DD) (Kadener et al., 2008). Finally, the model duplicates the clock’s response to null mutations, simulated in silico by setting the appropriate ax  y ¼ 0. For example, the clock is arrhythmic, or ceases to oscillate, with constant high levels of PER

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A

B Wild type

CWO mutant

CLK-CYC

CLK-CYC a CC- per

per

aCC- per a CWO- per

CWO

per

Figure 2.12 Wild-type and CWO-mutant model of a simplified per feedback loop. (A) CWO directly represses per transcription by competing with the transcriptional enhancing activities of CLK–CYC. (B) In the CWO mutant, only CLK–CYC regulates per transcription. In both models, per represses the activity of CLK–CYC.

and TIM, when a mutation of the DBT-binding domain on PER is introduced (aDBT  PER/TIM ¼ 0) (Kim et al., 2007). Mutations in the cwo gene are also consistent with biological data, resulting in depressed amplitudes in the oscillations of the per, tim, vri, and pdp1 mRNAs. However, the peak level of cwo is higher in the cwo-mutant model than the wt model, which is also observed in nature (Matsumoto et al., 2007). These observed results present an interesting paradox, as CWO is known to directly repress four direct target genes (per, tim, vri, and pdp1), whose loops all intersect at CLK– CYC. In consequence, intuition might suggest that the peak levels of per, tim, vri, and pdp1 would increase in the absence of CWO. Figure 2.12A presents a simplified network diagram of CWO’s interaction with one of these feedback loops in both the wild-type model. Supported by biological observations and the results of their model simulations, Fathallah-Shaykh et al. (2009) characterized the unusual effects of CWO on the network, which we now present as a novel network regulatory rule. Namely, Box 2.1

In the network presented in Fig. 2.12A, a weak direct repressor (CWO) of per, in competition with a strong direct activator (CLK–CYC), can become an indirect activator of per in the network, if certain conditions apply (see Box 2).

To explain this anomaly, we consider a generic model of the simplified network from Fig. 2.1, where a strong activator (SA) competes with a weak repressor (WR), like CWO, to regulate a target gene (per), which in turn represses the activity of SA. The illustrative cartoon in Fig. 2.13 provides a phase portrait of this interaction across five discrete time intervals, as the activity of WR is downregulated to simulate a cwo mutation. Although the target gene (tg) initially increases with the downregulation of WR, the

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t0

t1

t2

t3

•

t4

Strong activator

Strong activator

Strong activator

Strong activator

Strong activator

tg

tg

tg

tg

tg

WR

WR

WR

WR

WR

Figure 2.13 Illustrative cartoon of new regulatory network rule. At time t0, all molecules are oscillating and peaking at normal concentration levels. At time t1, we introduce a WR mutation, and WR protein levels drop. This decrease causes a temporary increase in tg concentration at t2, as WR normally competes with SA to bind to and repress tg E-box. At t3, concentration in SA declines, since tg inhibits the activity of SA. Finally, at t4, tg levels decline because the decrease in SA (i.e., CC) outweighs the effects of the change in WR (i.e., CWO), and system levels remain low for tg, thus preventing high amplitude oscillations (see Boxes 1 and 2).

consequent increased repression of SA causes a long-term decrease in peak concentration of tg. Mathematical analysis of the model confirms this phenomenon. One of the advantages of the model proposed by Fathallah-Shaykh et al. (2009) is that their master equation (Eq. (2.17), Section 3.3) affords easy analysis of the behavior of the network at peak concentration levels of the direct target genes, like per. As explained in the previous section, since the rate equation’s derivative (or rate of change) is zero at the peaks and troughs, the concentration of per at its peak in wt flies is given by:     WT WT aCCper  CCpeak  jaCWOper j  CWOpeak ½per WT ¼ ð2:23Þ dper WT where CCWT peak and CWOpeak represent the concentrations of CLK–CYC and CWO, respectively, in wt flies at the time when per mRNA concentration peaks (Fig. 2.12A). Likewise, in cwo mutant flies, the concentration of per at its peak is given by:       cwo  ja aCCper  CCcwo j  ½ 0  a  CC CWOper CCper peak peak ½per cwo ¼ ¼ dper dper

ð2:24Þ where again, CCcwo peak represents the new concentration of CLK–CYC in cwo-mutant flies at the time when per mRNA concentration peaks, and CWOWT peak ¼0 (Fig. 2.12B). Hence, the difference in Eqs. (2.23) and (2.24),

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h ½per WT  ½per cwo ¼

 i  i h cwo WT aCCper  CCWT  CC j  CWO  0  ja CWOper peak peak peak dper

is positive (indicating a decrease in peak per levels in cwo mutants) if [per]WT > [per]cwo, or: h  i h  i cwo WT > ja aCCper  CCWT  CC j  CWO  0 CWOper peak peak peak A more detailed algebraic proof of this relationship can be found in the supporting material for (Fathallah-Shaykh et al., 2009), but the results can be summarized with the following inequality: Box 2.2

Activating signal of strong activator > repressive signal of weak repressor.

In other words, if the activating signal from the change in peak levels cwo of SA (DCC ¼ CCWT peak  CCpeak ) is greater than the repressive signal from the change in peak levels of WR (DCWO ¼ CWOWT peak  0 ¼ CWOpeakWT), then per will peak at higher levels in the wild-type model. These activating and repressive signals are described by aCC  per  DCC and aCWO  per  DCWO, respectively. Hence, if (see Box 2.2) is satisfied and the signal from the competing activator (i.e., CC) is stronger, then the network can transform a WR (i.e., CWO) into an activator (see Box 2.1). Indeed, (Fathallah-Shaykh et al., 2009) observed that peak levels of CLK–CYC are higher in the wild-type model than in the cwo-mutant cwo model, and thus: DCC ¼ (CCWT peak  CCpeak ) > 0. Moreover, the observed depressed amplitudes of per in the simulated cwo mutants imply that the above inequality is satisfied and that the positive regulatory signals from DCC are stronger than the repressive signals from DCWO. Though oscillations may still persist in the cwo-mutant model, as verified by previous models preceding the discovery of this unusual network molecule, CWO functions to mollify the repressive effects of the vri and per/tim negative loops, as evidenced by the increase in peak CLK–CYC levels in wt flies. Further, by lowering this threshold of CLK–CYC repression, the Drosophila clock is able to maintain high-amplitude oscillations of the four direct target genes that intersect at this transcriptional activator. In addition to these findings, Fathallah-Shaykh has recently completed a second analysis of these equations, which further clarifies the role of CWO in the Drosophila circadian network (Fathallah-Shaykh, 2010). Namely, theoretical results suggest that CWO regulates an antijitter control system,

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which detects and then stabilizes variability in the oscillating rhythms of network molecules. As explained in Section 2, CWO’s four target genes (per, tim, vri, and pdp1) oscillate with a 24-h peak-to-peak time. However, slight variations in peak-timing, also known as jitter, occurs naturally and is also predicted by the model. These variations are not unlike the “jitters,” or deviations in period signals, observed in digital clocks. To eliminate these discrepancies, most digital clocks employ an antijitter device that detects and then corrects minor deviations in peak-timing. Analysis of the model predicts that CWO may function in a similar manner. Fathallah-Shaykh (2010) cites evidence that, in simulations of the CWO mutant model, cycleto-cycle peak-time variability of direct target genes is driven by the variability of CLK–CYC, since CLK–CYC directly stimulates the production of these genes. In simulations of the wt model, however, CWO appears to reduce cycle-to-cycle jitter in the direct target genes because its own variability, proportional to these direct target genes, is subtracted from the variability induced by CLK–CYC. Again, the linearity of each molecule’s regulatory equation at the peak of its oscillation was used to develop the theory of this causal relationship, which can be found in the supplementary material for (Fathallah-Shaykh, 2010).

5. Concluding Remarks In this chapter, we introduced the theory and applications behind the modeling of regulatory networks. Although we focused on modeling’s particular application to the dynamics of the Drosophila circadian clock, its spectrum of use is extensive and includes the study of population dynamics, the spread of infectious diseases, tumor growth, and numerous other dynamical systems (Murray, 2002; Stamper et al., 2010). In particular, we have discussed the use of different deterministic and continuous mathematical models, whose purpose is to both simulate and better understand the network behavior of circadian clock molecules. First, we explained the theory behind the use of Michaelis–Menten and Hill-type equations in models of intracellular regulatory networks. The majority of the models listed in Table 2.1 employ Michaelis–Menten and Hill-type equations, whose primary limitation lies in the large number of parameters needed to simulate a single molecular interaction. Second, we introduced a probabilistic model, with binding probability equations and first order reactions in place of Michaelis–Menten and Hill-type equations. Finally, the novel set of equations presented in Section 3.3 eliminates many of these parameters, with the added advantage that the relationship between regulatory signals that act on a particular network molecular is linear at the peaks and troughs of the molecule’s oscillations. This linearity affords new opportunities for

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analysis of the system’s behavior, thus leading to testable hypotheses, as demonstrated in the discussion of Section 4. We explained how these equations helped resolve an anomaly in the clock’s architecture. The revelation that CWO—a direct WR of specific network genes—actually activates these direct target genes led to the formulation of a new network regulatory rule, which can now be applied to network constructs in general. Moreover, analysis of these equations in Fathallah-Shaykh (2010) suggested that CWO acts as an antijitter regulator of direct target genes. These findings demonstrate the many ways that models may serve as tools for exploration of intracellular network behavior, thus saving much of the expensive and time-consuming work that must otherwise be done in vivo or in vitro (Szallasi et al., 2006). The findings of Fathallah-Shaykh et al. (2009) also have significant implications for the broader study of intracellular molecular networks. First and foremost, because the regulatory equation g is linear at the peaks and troughs of a molecule’s oscillation, one is able to quantify the repressive and activating signals, which can lead to the development of novel network regulatory rules, like the one in Section 4. In addition, their model underscores the importance of looking at a regulator’s broader role within a network as opposed to its isolated interaction with specific genes or proteins. As we have seen, the elimination of a direct repressor in a system may lead to unintended consequences in the overall network. We observe this phenomenon on a larger scale in the study of predator–prey populations, where, for example, the removal of one predator opens the doors for another, and perhaps more lethal, predator to eliminate the prey. On a molecular level, understanding the complete behavior of a system over time and its reactions to small perturbations or mutations can aid in drug development by identifying potential systematic side effects that could counteract the intended use of the drug (Endy and Brent, 2001). For example, Robe et al. (2009) recently published a study of a clinical trial of the anticancer drug Sulfasalazine, which exhibited strong preclinical evidence of its ability to treat progressing malignant gliomas in animals. However, results of human trials revealed no beneficial effects at best, and several patients actually developed lethal side effects. The authors concluded that future trials should exercise extreme caution in testing the effects of Sulfasalazine on human patients. A more complex dynamic model of this drug’s interaction with its target molecules might have prevented a harmful human clinical trial. As Kitano et al. (2002) explains, it is probable that one day the U.S. Food and Drug Administration may require dynamic network simulation of all drugs in the same way that large-scale engineering projects undergo comprehensive environmental impact assessments before receiving their final stamp of approval. In the long run, such precautions may not only save lives but also reduce the financial impact of large-scale human trials.

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Strategies for Articulated Multibody-Based Adaptive Coarse Grain Simulation of RNA Mohammad Poursina,* Kishor D. Bhalerao,† Samuel C. Flores,‡ Kurt S. Anderson,* and Alain Laederach§,} Contents 1. Introduction 2. Need for the Development of Adaptive Coarse-Graining Machinery 2.1. Model description 2.2. Inadequacy of the static coarse graining based on simulation results 3. Metrics to Guide Transitions in Adaptive Modeling 3.1. Metrics to guide transitions from finer to coarser models 3.2. Metrics to guide transitions from coarser to finer models 4. Adaptive Modeling Framework in DCA Scheme 5. Conclusions Acknowledgments References

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Abstract Efficient modeling approaches are necessary to accurately predict large-scale structural behavior of biomolecular systems like RNA (ribonucleic acid). Coarsegrained approximations of such complex systems can significantly reduce the computational costs of the simulation while maintaining sufficient fidelity to capture the biologically significant motions. However, given the coupling and nonlinearity of RNA systems (and effectively all biopolymers), it is expected that different parameters such as geometric and dynamic boundary conditions, and applied forces will affect the system’s dynamic behavior. Consequently, static * Computational Dynamics Lab, Mechanical, Nuclear and Aerospace Engineering Department, Rensselaer Polytechnic Institute, Troy, New York, USA Department of Mechanical Engineering, The University of Melbourne, Victoria, Australia { Simbios Center, Bioengineering Department, Stanford University, Clark Center S231, Stanford, California, USA } Department of Biomedical Sciences, University at Albany, Albany, New York, USA } Developmental Genetics and Bioinformatics, Wadsworth Center, Albany, New York, USA {

Methods in Enzymology, Volume 487 ISSN 0076-6879, DOI: 10.1016/S0076-6879(11)87003-5

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2011 Elsevier Inc. All rights reserved.

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coarse-grained models (i.e., models for which the coarse graining is time invariant) are not always able to adequately sample the conformational space of the molecule. We introduce here the concept of adaptive coarse-grained molecular dynamics of RNA, which automatically adjusts the coarseness of the model, in an effort to more optimally increase simulation speed, while maintaining accuracy. Adaptivity requires two basic algorithmic developments: first, a set of integrators that seamlessly allow transitions between higher and lower fidelity models while preserving the laws of motion. Second, we propose and validate metrics for determining when and where more or less fidelity needs to be integrated into the model to allow sufficiently accurate dynamics simulation. Given the central role that multibody dynamics plays in the proposed framework, and the nominally large number of dynamic degrees of freedom being considered in these applications, a computationally efficient multibody method which lends itself well to adaptivity is essential to the success of this effort. A suite of divide-and-conquer algorithm (DCA)-based approaches is employed to this end. These algorithms have been selected and refined for this purpose because they offer a good combination of computational efficiency and modular structure.

1. Introduction Development and application of the efficient techniques to model highly complex biomolecular systems are pursued by engineers and scientists in an effort to predict and understand different structural dynamics of the systems and explain various biological processes (Chen, 2008; Dill et al., 2008; Lebrun and Lavery, 1998; Parisien and Major, 2008; Scheraga et al., 2007). Among various types of biomolecular systems, nucleic acids, in particular, RNA, play a central regulatory role in the cell (Grundy and Henkin, 2006; Schroeder et al., 2004; Tucker and Breaker, 2005; Ying and Lin, 2006). Key to RNA function is structure, in particular, its ability to fold into a functional molecule capable of gene regulation and catalysis (Guo and Cech, 2002; Woodson, 2002; Zaug et al., 1998). Structural dynamics of biomolecular systems including RNA can be modeled using a variety of different techniques. Conventional molecular dynamics (MD) (Haile, 1992; Leach, 2001), which benefits from the fully atomistic representation of the system as shown schematically in Fig. 3.1A, is conceptually the simplest approach, and results from the direct application of Newton’s laws of motion. As such, these fully atomistic models capture all the dynamics of the system. The formulation and solution of the associated equations are trivial, once the forcing terms (force field calculations) have been determined. The overwhelming majority of the computational cost in MD simulations is in these force field calculations. Additionally, due to the high frequency motion of the atoms, it is necessary to use exceedingly

B A

Fully atomistic representation

Flexible body representation of groups of atoms

Rigid body representation of groups of atoms

Figure 3.1 Different types of modeling of biomolecular systems: (A) conventional MD simulations with fully atomistic representation of the system and (B) using articulated multibody concepts with different rigid and flexible subdomains connected to each other via kinematic joints.

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small temporal integration step sizes when explicit integrators are used. High-stability implicit schemes for stiff differential equations, such as implicit-Euler (IE) (Hairer and Wanner, 1996; Peskin and Schlick, 1989; Schlick and Peskin, 1989) and implicit-midpoint (IM) (Mandziuk and Schlick, 1995), are also unsatisfactory for proteins and nucleic acids at atomic resolution at large time steps because of numerical damping (Nyberg and Schlick, 1992; Schlick and Peskin, 1995; Zhang and Schlick, 1993). Consequently, these techniques are not recommended to model the systems with large number of degrees of freedom. Alternatively, development of the coarse-grained models with the idea of introducing superatoms (beads) (Praprotnik et al., 2005) or using articulated multibody dynamics (Chun et al., 2000; Mukherjee et al., 2007) shown in Fig. 3.1B reduces the cost of the simulation while still capturing the overall conformational motion. These models can contain multiple resolutions ranging from fine scale atomistic domains, coarse-grained macromolecules, to the continuum level system descriptions. Eliminating high frequency modes of motion within certain subdomains of the system allows one to significantly increase the size of the temporal integration steps. Additionally, it is not necessary to solve the equations governing the dynamics of the system for cartesian coordinates of all existing atoms. The dynamics needs only be solved for much smaller number of degrees of freedom of the system which can be more efficient in capturing the overall conformational motion. Furthermore, the structure of these models is such that the force field calculations may be performed more efficiently. To roughly estimate and compare the computational costs associated with the force field calculations between traditional atomistic MD simulation and the coarse-grained approximation, let us consider a system with n number of atoms. MD simulation needs (n2/2) (1
(1/n)) force field calculations to find either Van der Waals or electrostatic forces. If the system is modeled by the multibody dynamics approach with s rigid substructures each containing p atoms, the force field calculations (Van der Waals or electrostatic) are limited to the interactions between the atoms not located in the same rigid body (see Fig. 3.2). As such, the force field calculations reduce to (n2/2)(1
(1/s)). This rough estimate of the force field calculations in the coarse-grained models demonstrates that significant reduction in the number of the rigid substructures in the modeling (i.e., s  n) can result in the significant decrease in the computational cost corresponding to the force field calculations. In biological systems of nucleic acids, and proteins, which tend to possess many (nominally O(103) – O(107)) degrees of freedom, important physical phenomena can occur at vastly different spatial and temporal scales. The small oscillations of individual tightly bonded atoms with high frequency content provide the sub-femto second O(10
16) temporal domain while the conformational motions of interest occur in milliseconds O(10
3) or larger time scales. Additionally, RNA molecules are not static, but in fact are

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Figure 3.2 Force field calculation using multirigid-body dynamics approach. The interactions between the atoms inside the same rigid bodies are ignored.

highly dynamic (Kent et al., 2000). The dynamic behavior of RNA is complex because of the dominant effect of base-pairing and stacking, which leads to regions of the polymer being highly rigid, connected by flexible loops (Ra´zga et al., 2004, 2006). Stacking and pairing forces are cumulative, and with time, different regions of the RNA can become rigid or flexible (Shcherbakova et al., 2008). These important properties indicate that the quasi-static approach (Redon and Lin, 2006; Rossi et al., 2007) is not appropriate to model such systems, and consequently, more challenges are introduced in the development of the adaptive multiscale methods to model the dynamics of RNA. In the adaptive multiscale framework presented herein, the underlying dynamics formulation used within each system subdomain, as well as the subdomain definitions themselves, needs to be intelligently chosen such that the system level solutions are determined with an acceptable accuracy in a timely manner. For instance, Fig. 3.1B provides an example system which comprises multiple subdomains, each containing potentially different forms of models. In such a system level model, the system is treated as multi-rigid and/or flexible bodies connected to each other via kinematic joints. In this case, the coarse-graining algorithm should as a minimum be capable of adaptively adding/removing degrees of freedom to/from the model, as well as changing the definition of the flexible and rigid regions of the model during the simulation to efficiently provide the appropriate simulation results. Physics-, mathematics-, or knowledge-based internal metrics (Rossi et al., 2007) initiate and guide the change in the number and definition of the system subdomains. Additionally, these metrics may be used to assess the

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model performance and guide the types of models (atomistic, articulated multi-rigid, and/or flexible body) used within those subdomains to obtain the optimal combination of speed and accuracy. Thus, mechanisms need to be put in place for identifying critical locations where constraints can be added or removed as needed to enhance the simulation performance. Finally, a multibody formulation needs to be used, which both lends itself well to adaptive (on the fly) changes in model and domain definition, while being efficient when applied to large complex systems. In this chapter, important features of adaptive coarse grain modeling and simulation built off of articulated multibody dynamics with the application to RNA are addressed.

2. Need for the Development of Adaptive Coarse-Graining Machinery In this section, we investigate the effects of specific parameters on the behavior of different kinds of RNAs establishing the necessity for developing general and robust adaptive machineries to model complex RNA systems.

2.1. Model description We study different sequences of nucleotides adenine (A), cytosine (C), guanine (G), and uracil (U) and construct representative single-stranded RNA segments, each 18 nucleotides long, as shown in Fig. 3.3. All the simulations are performed using RNABuilder (Flores et al., 2010). The RNABuilder package is written using the Simbody (Schmidt et al., 2008)

Figure 3.3 RNA with 18 residues shown in different colors. The prescribed motion is applied to the first residue located at the left side of the RNA. The last residue at the right side of the RNA is fixed in space.

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internal coordinate mechanics library and its molecular mechanics extension, Molmodel, both available from SimTK.org. RNABuilder provides a series of mobilizer commands which set the flexibility of the molecule (Flores et al., 2010). In this work, we use the “rigid” mobilizer to rigidify each residue in the system. As such, each nucleotide (residue) is treated as a rigid body, being attached to its neighboring nucleotides by kinematic joints producing an articulated multibody system. Defining the BondMobility as “torsion,” each nucleotide is connected to its parent residue via a revolute joint whose joint axis is coincident with the local bond. Each of the spatial 18 residue segments investigated may be viewed as a portion of a very much longer RNA, and the prescribed motions at the segments boundaries represent the dynamic effect of the rest of the system on the 18 residue RNA segment. For the performed simulations, one boundary (18th residue) is fixed in space using the constraint called “Weld.” The other boundary of the RNA (first residue) receives a prescribed motion in the form of simple harmonic functions in both x and y directions (in Newtonian frame) as xðtÞ ¼ 0:05½ cosðo1 t Þ
1;

ð3:1Þ

yðtÞ ¼ 0:05½cosðo2 t Þ
1;

ð3:2Þ

where x and y are measured in nanometers. This task has been performed using the command “prescribed motion” and modifying the class “function” to create the functions indicated in Eqs. (3.1) and (3.2). We conduct the simulations on RNAs with various sequences described previously with two different prescribed motions for 20 ps. The first prescribed motion is defined with the same frequencies in both x and y directions as o1 ¼ o2 ¼ 20 rad/ps, while the second prescribed motion is composed of the frequency content of o2 ¼ 2o1 ¼ 40 rad/ps. The values of the frequencies and the amplitudes are selected such that the range of the prescribed motion is in accordance with the real motion. The simulations are performed at 300 K, and the AMBER (Parm99) potential field (Wang et al., 2000) is used to determine the bond and nonbond forces. The overall conformation of the RNA is important to identify the structural dynamics of the system. This overall conformation is determined by the values of the generalized coordinates (internal coordinates) defined at the joints connecting the consecutive residues. Since, in our truth model, all these joints are revolute, and torsional motion is allowed about the axis of the bonds which connect the consecutive residues, the joint angles are monitored during the course of the simulation to assess the overall conformation.

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2.2. Inadequacy of the static coarse graining based on simulation results Some representative simulation results for the systems described previously are provided here to convey the nature of the associated dynamic behavior. Figure 3.4 illustrates how differently the representative joint angles at various locations of the system and at different instants may behave, when the systems are excited by the prescribed motion characterized by o2 ¼ 2o1 ¼ 40 rad/ps. In Fig. 3.4A, high amplitude motion is observed in dynamic behavior of the joint angle between residues 1 and 2. Consequently, this joint should not be locked in the coarse-graining process. Figure 3.4B shows that after the passage of a couple of picoseconds, the joint angle between residues 8 and 9 experiences a transition from one regime of motion (experiencing little relative motion) to another regime of motion where the motion across the joint is significantly greater. Therefore, the coarse-grained model in which this joint is initially locked, though potentially valid at the beginning of the simulation, does not provide a reliable conformation for the entire course of the simulation. The prescribed motion at the RNA segment boundary represents the effect of the dynamics of the rest of the system as an input to the portion of interest. To study the effects of the changes in the input parameters on the response of the system in more detail, the behavior of the selected joint angles at different locations of the poly-A RNA is shown in Fig. 3.5. Due to the subtle change in one of the input frequencies, an interesting behavior is observed which had not been predicted. It is expected that the joint angle between residues 1 and 2 that is located most closely to the prescribed motion would never be a good candidate for being locked within the course of the simulation. However, the results shown in Fig. 3.5A indicate that when the input motions contain the same frequencies as o1 ¼ o2 ¼ 20 rad/ps, this joint can be locked after the passage of time, while the same coarsening is inappropriate when the system is driven by the prescribed motions in which the frequency associated with the y component is doubled to 40 rad/ps. As another example, in Fig. 3.5B, the dynamics of the joint angle between residues 8 and 9 is compared when the system is excited by two different prescribed motions. This joint can be locked through the course of the simulation due to its small variations when both frequencies in the prescribed motions are chosen 20 rad/ps. However, the results in Fig. 3.5B indicate that if the value of o2 changes to 40 rad/ps, for the same system, this joint angle cannot be locked during the whole course of the simulation. The simulation results affirm the nonlinear nature of the systems described previously. They demonstrate that the dynamic behavior of each joint angle is highly time variant and is significantly affected by the

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Figure 3.4 Illustration of the nonlinear dynamics of RNAs with sequences: 18 As (red), 18 Us (blue), 9 GCs (green), and 9 AUs (black), when the prescribed motion is characterized by o2 ¼ 2o1 ¼ 40 rad/ps. (A) High amplitude motion of the joint angle between residues 1 and 2. (B) Significantly different regimes of motion of the joint angle between residues 8 and 9.

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–95 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 Time (ps) Time (ps) Figure 3.5 Effect of changing the input dynamics from o1 ¼ o2 ¼ 20 rad/s (red) to o2 ¼ 2o1 ¼ 40 rad/s (blue) on behavior of the poly-A RNA with sequence of 18 As. (A) Joint angle between residues 1 and 2, the closest one to the excitation point, can be locked after a passage of time when o2 ¼ 20 rad/ps; however, it cannot be locked when o2 ¼ 40 rad/ps. (B) Joint angle between residues 8 and 9 can be frozen for the whole course of the simulation when o2 ¼ 20 rad/ps; however, when o2 ¼ 40 rad/ps, the dynamic behavior of this joint angle prevents the model reduction at this location.

0

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changes in the dynamics of the rest of the system. Consequently, even for a specific structure, a suggested coarse model may not be valid when the dynamics of the boundaries changes, or the conformation of the structure changes with time. As such, the static (i.e., time invariant) coarse graining may not provide the appropriate results, and the coarse grain modeling of these systems must be implemented in an adaptive framework. These conclusions can be generalized to other biomolecular systems such as DNAs and proteins. In this process, some degrees of freedom of the system are adaptively constrained or released at different instants and different locations of the system. Additionally, based on the behavior of different subdomains of the system, the definition of the rigid and/or flexible regions may change.

3. Metrics to Guide Transitions in Adaptive Modeling Key to this effort is the need to develop metrics that guide the adaptive machinery. These metrics may be knowledge- (derived empirically), math(derived from strictly mathematical relations), and/or physics-based (derived directly from physical laws). Herein, we introduce two different types of metrics for guiding the model transitions to the coarser and finer models.

3.1. Metrics to guide transitions from finer to coarser models The behavior of the individual degrees of freedom of the model can be used to assess which of them may be removed, while the resulting reduced order model still produces essentially the same behavior, but with less computational effort. As such, the significant (active) degrees of freedom are retained in the coarsened model while the less significant degrees of freedom are identified and removed. In biomolecular systems, the high frequency modes of motion that are related to the finer fidelity models provide large instantaneous relative velocities and accelerations. However, such modes do not contribute significantly to the global conformation of the system. Thus, velocity- and acceleration-based metrics are not well suited for identifying those degrees of freedoms which are more significant in the overall conformational motions. Additionally, dynamics of biomolecular systems is highly nonlinear and chaotic, and as such, using coarse-graining metrics based on the instantaneous values of the states of the system are not expected to (and have been shown not to) yield improved results. Therefore, we propose to monitor the moving-window statistical properties of the generalized coordinates of the system for math-based metrics to assess and guide the coarse-

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graining process instead of instantaneous velocity- and acceleration-based metrics described in Redon and Lin (2006) and Rossi et al. (2007). The metric of choice for determining if an existing joint should be kept or removed is the standard deviation of the generalized coordinates defined at the joint collected within the sliding window, as given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn ðxk
x
w Þ2 Sw ¼ : k¼1 n

ð3:3Þ

In the above equation, x
w is the moving-window average of the sequence of data within the window of the size of n. Each generalized coordinate contributes differently to the overall conformation. In other words, the overall conformation of the system may be more sensitive to some specific generalized coordinates, while this sensitivity varies with time. Consequently, if the weighted (scaled) value of the moving-window standard deviation of any generalized coordinate defined at the joint of the system (i.e., internal coordinate) is less than a predefined threshold, then the associated degree of freedom is considered less significant, and thus eligible to be frozen. In this work, since the torsional motion is the only allowable motion by the kinematic joints of the system, if the moving-window standard deviation of any specific joint angle is insignificant in comparison to the predefined threshold, the entire joint will be locked. Herein, we pick the Poly-GC 18 nucleotide RNA to perform the coarse-graining process based on the results obtained during a 1-ps simulation. Both frequencies in the prescribed motions are considered to be 20 rad/ps. The size of the window to calculate the standard deviation of the joint angles, as well as the value of the threshold on the standard deviation, is both system dependent. In this case, the values of the joint angles are sampled every 0.01 ps, and the length of the moving-window is chosen to be 100 sampling times. We also ignore the weights on the standard divinations associated with different joint angles of the system. Examining different values for the threshold, and comparing the results of the associated coarse model to those of the truth model, we lock the joints of the system if the moving-window standard deviation of their angles within the first 100 sampling times is less than 1 . Based on this threshold, the joints between residues 5 and 6, 7 and 8, 9 and 10, 11 and 12, 13 and 14, 14 and 15, 16 and 17, and 17 and 18 are allowed to be frozen. The adequacy of the proposed coarse graining can be assessed by comparing the results achieved by the coarse-grained model with those obtained exclusively from the fine grained “truth model” over the extended simulation periods. The simulation of the suggested coarse model is, therefore, conducted for 30 ps with the same initial conditions and prescribed motions applied to the truth model. Figure 3.6 shows the behavior of some

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Figure 3.6 Comparison between the dynamic behavior of the fine system (red) and the suggested coarse model (blue) for poly-GC RNA. (A) Joint angle between residues 2 and 3, (B) Joint angle between residues 9 and 10, (C) Joint angle between residues 11 and 12, and (D) Joint angle between residues 12 and 13.

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representative joint angles for both models. The joint angle between residues 2 and 3 in both models experiences the same behavior as shown in Fig. 3.6A. Although the joint angle between residues 9 and 10 has been locked in the coarsening process, Fig. 3.6B indicates that the associated behavior in the proposed coarse model is very close to that obtained for the finer model. Based on the results shown in Fig. 3.6C, the joint angle between residues 11 and 12 in the coarse model deviates from that of the truth model as time passes. According to the required accuracy, it may be needed to unlock this joint during the adaptive coarse-graining process. Finally, in Fig. 3.6D, similar trends are observed in the dynamics of the relative motion between residues 12 and 13 for both models.

3.2. Metrics to guide transitions from coarser to finer models An inadequate coarse-grained model as a result of constraining certain critical degrees of freedom of the system can lead to an incorrect representation of the system behavior and conformation. An important metric to check the validity of the selected coarse-grained model is the spatial constraint loads (forces and moments) acting on all the kinematic joints of the system and intermediate locations within the rigid and flexible bodies which represent segments of the articulated molecular system. The system’s internal and constraint loads arise from the interactions between the bodies, the imposed boundary conditions, and the kinematic constraints imposed on adjacent body-to-body motions by the connecting joints as shown in Fig. 3.7. For instance, consider the coarse-grained model provided in Fc

Joint

Body k

Body k + 1

Imposed motion due to the dynamics of the rest of the system

Figure 3.7 Magnitude of constraint load gives an estimate of errors introduced due to locking the joint.

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Section 3.1. If the frequency content in the y-direction of the prescribed motion in Eq. (3.2) changes from 20 to 60 rad/ps at t ¼ 10 ps, the behavior of the constraint torques change significantly in some joints of the system locked previously in the coarse-graining process. For instance, Fig. 3.8 shows the significant increase in the value of the constraint torque in the direction of the bond connecting residues 16 and 17 which has been locked in the coarsening process. Thus, the constraint load magnitude gives an indication of the degree to which the body or joint in question is attempting to be deformed at a location. If during the course of the simulation, the value of the spatial constraint load at any location of the system exceeds the nominal load which figuratively causes the mechanical failure, the associated joint is released. This effectively releases (removes) the constraint and permits the adjacent segments to move relative to one another in a manner permitted by the joint and in response to the system forcing terms. The required mechanisms to change the definition of the joints and the derivation of the appropriate math for the model transitions remain a contemporary challenge in the field. Adding degrees of freedom into the system poses additional challenges. In real mechanical systems, since energy is not “created” by unlocking a

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Figure 3.8 Constraint torque between residues 10 and 11. The constraint torque about the bond direction (locked in the coarsening process) changes significantly when a modest change occurs in the frequency of the prescribed motion at t ¼ 10 ps.

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joint or the failure of a member, the “creation” of a new mode of motion by the removal of a constraint does not pose any problem. Therefore, there is no jump in the system velocity variables. However, in the coarse-graining process of biomolecular systems, naturally existing higher modes of motion are ignored in the modeling because the internal metric had previously indicated these modes as less relevant. Therefore, in the transition from the coarse model to the finer model, the current value of the kinetic energy of these ignored modes must be estimated and considered appropriately. Details of choosing the optimum solution and the various issues in adding joints in the model have been discussed in detail in Anderson and Poursina (2009a) and Poursina et al. (2009).

4. Adaptive Modeling Framework in DCA Scheme Within the adaptive framework, a divide-and-conquer algorithm (DCA) is used to solve the forward dynamics problem for the articulated multibody systems composed of rigid and flexible bodies with any arbitrary configurations such as kinematically open loops and multiple closed loops (Featherstone, 1999a,b; Mukherjee and Anderson, 2007b,c). These DCAbased methods are used in the context of the large-scale adaptive molecular problems because: (1) they are relatively efficient for large-scale sequential computer implementation; (2) these formulations have a highly modular structure, which makes their implementation and use within an adaptive framework relatively straightforward; and (3) these methods are highly parallelizable. The computational complexity of the algorithm is O(n) and O(log(n)) in serial and parallel implementations, respectively, where n denotes the number of degrees of freedom of the system. The basic idea of the DCA is to treat a large multibody system by recursively assembling adjacent articulated bodies/subsystems (Fig. 3.9A) into larger encompassing subsystems (Fig. 3.9B). The brief overview of the method is provided here. Consider the two consecutive articulated bodies k and k þ 1 shown in Fig. 3.9A which are connected to each other via the kinematic joint Jk. The term handle, which appears in the paper continually, is any selected point on the body used in modeling the interactions of the body with the environment. The handles on the body can correspond to the joint locations, center of mass or any desired reference points. Each body can have any number of handles on it. For the algorithm presented here, the joint locations are chosen as the handles on the body. The two-handle equations of motion for each body are those in which the spatial acceleration of each handle of the body is expressed as a linear combination of the spatial constraint forces applied to the inward and outward handles. These equations for bodies k and k þ 1 are expressed as

F2kc

A

H2k Body k

B

F1kc+1

Jk

H1k + 1

H1k

H2k + 1 F1kc

Body k

Body k + 1

F2kc+1

Body k+1 H2k+1

H1k F1kc

F2kc+1

Figure 3.9 Assembling of two consecutive bodies to form a new body; (A) consecutive bodies k and k þ 1 connected to each other via a kinematic joint and (B) fictitious body formed from two distinct bodies by eliminating the constraint force and spatial accelerations corresponding to the common joints.

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k

k

ð3:4Þ

k

k

ð3:5Þ

Ak1 ¼ fk11 ℱ 1c þ fk12 ℱ 2c þ fk13 ; Ak2 ¼ fk21 ℱ 1c þ fk22 ℱ 2c þ fk23 ; and kþ1

kþ1

ð3:6Þ

kþ1

kþ1

ð3:7Þ

kþ1 kþ1 ¼ fkþ1 Akþ1 11 ℱ 1c þ f12 ℱ 2c þ f13 ; 1 kþ1 kþ1 ¼ fkþ1 Akþ1 21 ℱ 1c þ f22 ℱ 2c þ f23 ; 2

respectively. In the above equations, all the coefficients fij ’s are known quantities at each time step. The terms fij (i, j ¼ 1, 2) are associated with the inertia of the body and, consequently, are constant for each body within the course of the simulation if expressed in body basis. The terms fi3 (i ¼ 1, 2) are associated with the known applied forces, as well as centripetal and coriolis terms which should be updated at each time step. At the joint Jk, the kinematical constraint at the acceleration level is expressed as: Jk

Akþ1 ¼ Ak2 þ P J u_ J þ P_ uJ ; 1 k

k

k

ð3:8Þ

k

joint free modes of where P J is the known matrix associated with the k motion (Roberson and Schwertassek, 1988), and u J represents the known generalized speeds defined at the jointk Jk. Using the relations provided above, and introducing the matrix D J as the orthogonal complement of the joint free-motion map at Jk, one can arrive at the two-handle equations governing the dynamics of the assembly k:k þ 1 as k

kþ1

Ak1 ¼ fk:kþ1 ℱ 1c þ fk:kþ1 ℱ 2c þ fk:kþ1 ; 11 12 13 k

kþ1

¼ fk:kþ1 ℱ 1c þ fk:kþ1 ℱ 2c þ fk:kþ1 ; Akþ1 21 22 23 2

ð3:9Þ ð3:10Þ

where   fk:kþ1 ¼ fk11
Wfk21 ; 11   fk:kþ1 ¼ Wfkþ1 12 12 ;   fk:kþ1 ¼ fk13
WY ; 13   fk:kþ1 ¼ Zfk21 ; 21   kþ1 fk:kþ1 ¼ fkþ1 22 22
Zf12 ;

ð3:11Þ ð3:12Þ ð3:13Þ ð3:14Þ ð3:15Þ

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  fk:kþ1 ¼ fkþ1 23 23 þ ZY ; h i  T  J k
1 h J k iT k kþ1 Jk Jk D f22 þ f11 D D ; X ¼D Jk

ð3:16Þ ð3:17Þ

_ J Y ¼ fk23
fkþ1 13 þ P u ;

ð3:18Þ

W ¼ fk12 X ;

ð3:19Þ

Z ¼ fkþ1 21 X :

ð3:20Þ

k

The DCA is implemented in two main passes: assembly and disassembly as shown in Fig. 3.10. The assembly process starts at the level of the individual bodies (leaf nodes) by coupling together pairs of adjacent bodies to form the assemblies. Proceeding in this manner, one can recursively eliminate both unknown constraint loads and joint accelerations at the common joints of the consecutive bodies/assemblies to form the twoLeaf level of the binary tree

A 2

1

4

3

5

8

Assembly

7−8

5−6

3−4

1−2

7

6

5−6−7−8

1−2−3−4

1−2−3−4−5−6−7−8

Root node

B

7

8

Disassembly process of these three new bodies stops at this level Disassembly

7−8

5−6

3−4

1−2

5−6−7−8

1−2−3−4

1−2−3−4−5−6−7−8

Figure 3.10 Adaptive framework in DCA scheme. Red joints are locked, and red assemblies are composed of the consecutive bodies whose connecting joint is locked: (A) assembly process by coupling together pairs of adjacent bodies/subassemblies to form the new assemblies until reaching the all-encompassing node and (B) disassembly process to find the spatial constraint forces and the generalized speeds at the common joints.

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handle equations of the resulting assemblies. This process works hierarchically exploiting the same structure as that of a binary tree. At the end of the hierarchic assembly process, the whole articulated system may be modeled in terms of the two-handle equations of motion of a single all encompassing assembly. Different types of boundary conditions may be applied to this root node of the binary tree. It is possible for each boundary of the system to be either free floating or constrained by a kinematic joint or prescribed motion. The disassembly process starts by applying the boundary conditions to the system and solve for the unknown spatial constraint forces and/or accelerations of the terminal handles of the whole system. Solving this problem is addressed in detail in Mukherjee and Anderson (2007c). Then, these known values of the constraint loads and accelerations of the terminal handles are substituted into the two-handle equations of the associated subassemblies (Eqs. 3.4–3.7) to find the values of the spatial constraint forces and the accelerations at the common joints of the subassemblies. This process is repeated in a hierarchic disassembly of the binary tree where the known boundary conditions are used to solve the two-handle equations of the subassemblies, until the constraint loads and the generalized speeds of all bodies in the system are determined. In performing the adaptive modeling of the dynamics of the system in DCA framework, if based on the values of the moving-window standard deviation of the joint angles, it is desired to lock (remove) a joint of the system, the associated orthogonal complement of the joint free-motion map becomes the identity matrix. As such, Eq. (3.17) is simplified as:
1 X ¼ ½fk22 þ fkþ1 11  :

ð3:21Þ

If based on monitoring the values of the constraint loads, it is deemed necessary to change the definition of the joint within the adaptive modeling, the only change occurs at the leaf level is changing the joint freek motion map P J of the associated joint. For instance, consider the current coarse model shown in Fig. 3.11A composed of eight bodies. All the connecting joints are revolute, except the last one shown in red which is locked. The desired model is formed by releasing the joint between bodies 7 and 8 due to the corresponding high strain. Additionally, in this model, the joints between bodies 1 and 2, 3 and 4, and 5 and 6 are to be locked as shown in Fig. 3.11B because they are determined to be making an insignificant contribution to the overall conformation of the system. In this case, the spatial joint free-motion maps of the locked joints become zero matrices. Consequently, the associated orthogonal complement of the joint free-motion map is replaced by the identity matrix if expressed in the joint coordinate system. For the revolute joints of the system, the spatial joint free-motion map expressed in joint basis becomes

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A 1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

B

Figure 3.11 Illustration of transitioning between coarse models; red joints are locked. (A) Current coarse model, in which a high strain is measured in the joint between bodies 7 and 8. (B) New coarse model, where the joint between bodies 7 and 8 is released and the joints between bodies 1 and 2, 3 and 4, and 5 and 6 are locked.

2 3 1 607 6 7 607 Jk 7 P ¼6 6 0 7: 6 7 405 0

ð3:22Þ

As such, the associated orthogonal complement of the joint free-motion map is expressed as 2 3 0 0 0 0 0 61 0 0 0 07 6 7 60 1 0 0 07 Jk 6 7 D ¼6 ð3:23Þ 7 60 0 1 0 07 40 0 0 1 05 0 0 0 0 1 The assembly and disassembly processes are performed as described before using the appropriate matrices characterizing the joints of the desired coarse model. In these processes, subassemblies shown in red in Fig. 3.10 are those composed of consecutive bodies whose connecting joint is locked. For the new coarse model, the disassembly process of the three new bodies stops at one level prior to the leaf level of the binary tree as shown in Fig. 3.10B, as all the associated spatial constraint forces and generalized speeds are available at this level. However, this process continues to the leaf level for the determination of the unknown values of the constraint loads and generalized speeds associated with common joint of bodies 7 and 8. Any violation in the conservation of the generalized momentum of the system in the transition between different models leads to nonphysical results since the instantaneous switch in the system model definition is

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incurred without the influence of any external forces. In other words, the momentum of each differential element projected onto the space of admissible motions permitted by the more restrictive model (whether pre- or post-transition) when integrated over the entire system must be conserved across the model transition (Kane and Levinson, 1985). The jumps in the system partial velocities (free modes of motion; Roberson and Schwertassek, 1988) due to the sudden change in the model resolution result in the jumps in the generalized speeds corresponding to the new set of degrees of freedom. Therefore, the adaptive framework should be equipped with the machinery to provide the momenta balance equations. The formation of the impulse–momentum equations within the transitions can also be performed in divide-and-conquer scheme (Mukherjee and Anderson, 2007a). Similar to the DCA, the DCA-based generalized momentum assembly–disassembly procedures are performed on the two-handle impulse–momentum equations of each body as follows (Mukherjee and Anderson, 2007a) DV k1 ¼ Fk11 DV k2

¼

Fk21

ð tþ t


ð tþ t


k

ℱ 1c dt þ Fk12 k ℱ 1c dt

þ

Fk22

ð tþ t


ð tþ t


k

ð3:24Þ

k

ð3:25Þ

ℱ 2c dt þ Fk13 ; ℱ 2c dt þ Fk23 ;

where Fijk is an intermediate known term. The terms DV ki (i ¼ 1, 2) indicate the jumps in the spatial velocities of the ith handle of body k. The spatial impulsive constraint forces acting on body k at its inward (parent) and outward (child) handles within theÐ instantaneous time period Ðtransition þ þ from t
to tþ are represented by tt
ℱ1ck dt and tt
ℱ2ck dt, respectively. Since the impulses associated with the external loads, as well as the coriolis k and centripetal terms can be ignored, the terms F13 and F23k do not appear in the equations governing the impulse–momentum of the individual bodies at the leaf level of the binary tree. Additionally, at this level of the binary tree, for each body, the coefficients of the spatial constraint forces, as well as the impulsive constraint forces which are essentially functions of the inertia and the geometry are the same, in other words: Fkij ¼ fkij

i; j ¼ 1; 2

ð3:26Þ

As such, setting up the generalized momentum balance equations is fast and easy. This algorithm is also capable of efficiently providing the required equations for the knowledge-, math-, or physics-based optimization problem in the transitions from the coarse models to the finer models as explained in detail in Anderson and Poursina (2009b).

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5. Conclusions We have addressed different issues associated with the adaptive coarse grain modeling of biomolecular systems using articulated multibody dynamics concepts. Based on the results of the simulations of different types of RNAs, it has been demonstrated that the static (time invariant) coarse graining of models are not likely to provide the appropriate information about the structural behavior of the system for the entire course of the simulation due to the nonlinearities embedded in the system’s dynamics. Therefore, there is a need for the development of an adaptive molecular modeling and simulation framework. The adaptive algorithm should be capable of identifying the critical locations of the system to remove and add degrees of freedom, or change model types and definitions as necessary. In this chapter, we have suggested the moving-window standard deviation of the joint angles, as well as the values of the constraint loads as metrics to guide the transitions to the coarser and finer models, respectively. Since the system is treated as the articulated multibody system, using an efficient algorithm is necessary for the modeling. DCA has been proposed as a convenient technique for the efficient modeling of the system. This method is highly modular and, as such, lends itself well to adaptivity and massive parallelization. The computational complexity of the method is O(n) and O (log(n)) in serial and parallel implementations, respectively, where n is the number of the degrees of freedom of the system.

ACKNOWLEDGMENTS This work was supported through the NSF award No. CMMI-0757936 to Kurt Anderson and in part through the US National Institutes of Health grant R00 GM079953 (NIGMS) to Alain Laederach. The authors would like to thank the funding agencies and also show their gratitude to Dr. Russ Altman and Michael Sherman from Simbios group at Stanford University for their help in this effort.

REFERENCES Anderson, K. S., and Poursina, M. (2009a). Energy concern in biomolecular simulations with transition from a coarse to a fine model. Proceedings of the Seventh International Conference on Multibody Systems, Nonlinear Dynamics and Control, ASME Design Engineering Technical Conference 2009, (IDETC09), IDETC2009/MSND-87297, San Diego, CA. Anderson, K. S., and Poursina, M. (2009b). Optimization problem in biomolecular simulations with dca-based modeling of transition from a coarse to a fine fidelity. Proceedings of the Seventh International Conference on Multibody Systems, Nonlinear Dynamics and

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Control, ASME Design Engineering Technical Conference 2009, (IDETC09), IDETC2009/MSND-87319, San Diego, CA. Chen, S. J. (2008). RNA folding: Conformational statistics, folding kinetics, and ion electrostatics. Annu. Rev. Biophys. 37, 197–214. Chun, H. M., Padilla, C. E., Chin, D. N., Watenabe, M., Karlov, V. I., Alper, H. E., Soosaar, K., Blair, K. B., Becker, O. M., Caves, L. S. D., Nagle, R., Haney, D. N., et al. (2000). MBO(N)D: A multibody method for long-time molecular dynamics simulations. J. Comput. Chem. 21, 159–184. Dill, K. A., Ozkan, S. B., Shell, M. S., and Weikl, T. R. (2008). The protein folding problem. Annu. Rev. Biophys. 37, 289–316. Featherstone, R. (1999a). A divide-and-conquer articulated body algorithm for parallel O (log(n)) calculation of rigid body dynamics. Part 1: Basic algorithm. Int. J. Rob. Res. 18, 867–875. Featherstone, R. (1999b). A divide-and-conquer articulated body algorithm for parallel O (log(n)) calculation of rigid body dynamics. Part 2: Trees, loops, and accuracy. Int. J. Rob. Res. 18, 876–892. Flores, S., Wan, Y., Russell, R., and Altman, R. B. (2010). Predicting RNA structure by multiple template homology modeling. Proceedings of the Pacific Symposium on Biocomputing. 216–227. Grundy, F., and Henkin, T. (2006). From ribosome to riboswitch: Control of gene expression in bacteria by RNA structural rearrangements. Crit. Rev. Biochem. Mol. Biol. 41, 329–338. Guo, F., and Cech, T. (2002). Evolution of tetrahymena ribozyme mutants with increased structural stability. Nat. Struct. Biol. 9, 855–861. Haile, J. (1992). Molecular Dynamics Simulation: Elementary Methods. Wiley Interscience, New York. Hairer, E., and Wanner, G. (1996). Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, 2nd ed. Springer Series in Computational Mathematics. Vol. 14. Springer-Verlag, New York. Kane, T. R., and Levinson, D. A. (1985). Dynamics: Theory and Application. Mcgraw-Hill, New York. Kent, O., Chaulk, S., and MacMillan, A. (2000). Kinetic analysis of the M1 RNA folding pathway. J. Mol. Biol. 304, 699–705. Leach, A. R. (2001). Molecular Modelling Principles and Applications, 2nd ed. Prentice Hall, Harlow, England. Lebrun, A., and Lavery, R. (1998). Modeling the mechanics of a DNA oligomer. J. Biomol. Struct. Dyn. 16, 593–604. Mandziuk, M., and Schlick, T. (1995). Resonance in the dynamics of chemical systems simulated by the implicit-midpoint scheme. Chem. Phys. Lett. 237, 525–535. Mukherjee, R. M., Crozier, P. S., Plimpton, S. J., and Anderson, K. S. (2007). Substructured molecular dynamics using multibody dynamics algorithms. Int. J. Non. Mech.: Non. Mech. Dyn. Macromol. 43(10), 1040–1055. Mukherjee, R. M., and Anderson, K. S. (2007a). Efficient methodology for multibody simulations with discontinuous changes in system definition. Multibody Syst. Dyn. 18, 145–168. Mukherjee, R. M., and Anderson, K. S. (2007b). A logarithmic complexity divide-andconquer algorithm for multi-flexible articulated body systems. J. Comput. Nonlinear Dyn. 2, 10–21. Mukherjee, R. M., and Anderson, K. S. (2007c). An orthogonal complement based divideand-conquer algorithm for constrained multibody systems. Nonlinear Dyn. 48, 199–215. Nyberg, A., and Schlick, T. (1992). Increasing the time step in molecular dynamics. Chem. Phys. Lett. 198, 538–546.

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Parisien, M., and Major, F. (2008). The MC-fold and MC-Sym pipeline infers RNA structure from sequence data. Nature 452, 51–55. Peskin, C. S., and Schlick, T. (1989). Molecular dynamics by the backward Euler’s method. Commun. Pure Appl. Math. 42, 1001–1031. Poursina, M., Bhalerao, K. D., and Anderson, K. S. (2009). Energy concern in biomolecular simulations with discontinuous changes in system definition. Proceedings of the ECCOMAS Thematic Conference Multibody Systems Dynamics, Warsaw, Poland. Praprotnik, M., Site, L., and Kremer, K. (2005). Adaptive resolution molecular-dynamics simulation: Changing the degrees of freedom on the fly. J. Chem. Phys. 123, 224106–224114. Ra´zga, F., Spackova, N., Re´blova, K., Koca, J., Leontis, N., and Sponer, J. (2004). Ribosomal RNA kink-turn motif-a flexible molecular hinge. J. Biomol. Struct. Dyn. 22, 183–194. Ra´zga, F., Zacharias, M., Re´blova, K., Koca, J., and Sponer, J. (2006). RNA kink-turns as molecular elbows: Hydration, cation binding, and large-scale dynamics. Structure 14, 825–835. Redon, S., and Lin, M. C. (2006). An efficient, error-bounded approximation algorithm for simulating quasi-statics of complex linkages. Comput. Aided Des. 38, 300–314. Roberson, R. E., and Schwertassek, R. (1988). Dynamics of Multibody Systems. SpringerVerlag, Berlin. Rossi, R., Isorce, M., Morin, S., Flocard, J., Arumugam, K., Crouzy, S., Vivaudou, M., and Redon, S. (2007). Adaptive torsion-angle quasi-statics: A general simulation method with applications to protein structure analysis and design. ISMB/ECCB (Supplement of Bioinformatics), 408–417. Scheraga, H. A., Khalili, M., and Liwo, A. (2007). Protein-folding dynamics: Overview of molecular simulation techniques. Annu. Rev. Phys. Chem. 58, 57–83. Schlick, T., and Peskin, C. S. (1989). Can classical equations simulate quantum-mechanical behavior? A molecular dynamics investigation of a diatomic molecule with a morse potential. Commun. Pure Appl. Math. 42, 1141–1163. Schlick, T., and Peskin, C. (1995). Comment on: The evaluation of LI and LIN for dynamics simulations. J. Chem. Phys. 103, 9888–9889. Schmidt, J. P., Delp, S. L., Sherman, M. A., Taylor, C. A., Pande, V. S., and Altman, R. B. (2008). The simbios national center: Systems biology in motion. Proc. IEEE 96, 1266–1280, special issue on Computational System Biology. Schroeder, R., Barta, A., and Semrad, K. (2004). Strategies for RNA folding and assembly. Nat. Rev. Mol. Cell Biol. 5, 908–919. Shcherbakova, I., Mitra, S., Laederach, A., and Brenowitz, M. (2008). Energy barriers, pathways, and dynamics during folding of large, multidomain RNAs. Curr. Opin. Chem. Biol. 12, 655–666. Tucker, B., and Breaker, R. (2005). Riboswitches as versatile gene control elements. Curr. Opin. Struct. Biol. 15, 342–348. Wang, J., Cieplak, P., and Kollman, P. A. (2000). How well does a restrained electrostatic potential (resp) model perform in calculating conformational energies of organic and biological molecules? J. Comput. Chem. 21, 1049–1074. Woodson, S. (2002). Recent insights on RNA folding mechanisms from catalytic RNA. Cell. Mol. Life Sci. 57, 796–808. Ying, S., and Lin, S. (2006). Current perspectives in intronic micro RNAs (miRNAs). J. Biomed. Sci. 13, 5–15. Zaug, A., Grosshans, C., and Cech, T. (1998). Sequence-specific endoribonuclease activity of the tetrahymena ribozyme: Enhanced cleavage of certain oligonucleotide substrates that form mismatched ribozyme-substrate complexes. Biochemistry 27, 8924–8931. Zhang, G., and Schlick, T. (1993). LIN: A new algorithm combining implicit integration and normal mode techniques for molecular dynamics. J. Comput. Chem. 14, 1212–1233.

C H A P T E R

F O U R

Modeling Loop Entropy Gregory S. Chirikjian Contents 1. Introduction 1.1. Literature review 1.2. Statistical mechanics 1.3. Mathematics review 2. Computing Bounds on the Entropy of the Unfolded Ensemble 2.1. End-to-end position and orientation distributions and the Cartesian conformational entropy of serial polymer chains 2.2. Modeling excluded volume effects 2.3. Bounding Cartesian conformational entropy 3. Approximating Entropy of the Loops in the Folded Ensemble 4. Examples 4.1. Model 1: Long loops modeled as Gaussian chains 4.2. Model 2: Short loops modeled as semiflexible polymers 4.3. From covariance matrices to entropy 5. Conclusions Acknowledgments References

100 101 103 107 113 113 116 118 119 120 120 122 124 127 128 128

Abstract Proteins fold from a highly disordered state into a highly ordered one. Traditionally, the folding problem has been stated as one of predicting “the” tertiary structure from sequential information. However, new evidence suggests that the ensemble of unfolded forms may not be as disordered as once believed, and that the native form of many proteins may not be described by a single conformation, but rather an ensemble of its own. Quantifying the relative disorder in the folded and unfolded ensembles as an entropy difference may therefore shed light on the folding process. One issue that clouds discussions of “entropy” is that many different kinds of entropy can be defined: entropy associated with overall translational and rotational Brownian motion, configurational entropy, vibrational entropy, conformational entropy computed in internal or Cartesian coordinates (which can even be different from each other), conformational entropy computed on a lattice, each of the above with different Department of Mechanical Engineering, Johns Hopkins University, Baltimore, Maryland, USA Methods in Enzymology, Volume 487 ISSN 0076-6879, DOI: 10.1016/S0076-6879(11)87004-7

#

2011 Elsevier Inc. All rights reserved.

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solvation and solvent models, thermodynamic entropy measured experimentally, etc. The focus of this work is the conformational entropy of coil/loop regions in proteins. New mathematical modeling tools for the approximation of changes in conformational entropy during transition from unfolded to folded ensembles are introduced. In particular, models for computing lower and upper bounds on entropy for polymer models of polypeptide coils both with and without end constraints are presented. The methods reviewed here include kinematics (the mathematics of rigid-body motions), classical statistical mechanics, and information theory.

1. Introduction In a classic observation, Anfinsen observed the spontaneous and repeatable folding of a protein from a highly disordered state into a highly ordered one (Anfinsen, 1973). From this result and others that followed, it has been inferred over the years that similar processes work for wide classes of proteins. But exactly how unstructured is the unfolded/denatured state? And how structured is the native state? New evidence suggests that the ensemble of unfolded forms may not be as disordered as once believed and that the native form may not be as rigid as one might expect. In this light, protein folding is a transformation of a high-conformational entropy ensemble into a lower one. But how high is high, and how low is low? To answer such questions, some new mathematical and computer models will be helpful. Therefore, new mathematical tools for the approximation of conformational entropy in the unfolded and folded ensembles are introduced here. A number of related tools already exist in other fields. These are reviewed, adapted, and developed further. In particular, lower and upper bounds on entropy are derived for polymer models of polypeptide chains, both with and without, constraints on the positions and orientations of the ends. The methods reviewed here include kinematics (the mathematics of rigid-body motions as studied in the field of Robotics), information theory, and functional analysis on Lie groups (which, in part, considers how probability density functions of group-valued argument combine and propagate). In particular, we attach reference frames to polypeptide chains, as shown in Fig. 4.1, where the origin of the ith frame is located at the ith Ca atom with a unique orientation defined by the Ca–C0 bond and the plane defined by the Ca–C0 ¼O atoms, as in Lee and Chirikjian (2005). We use the distributions of relative motion between consecutive residues to characterize backbone conformational entropy. Side-chain motions are computed relative to these reference frames and we show how to compute the associated side-chain entropy. These new and powerful methods make it

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Side chain R

R2

O H

f

C⬘ Ca

N

R1

H

H H

y Ca

H

Ca

H C⬘

Ca

H H

O

OH

N

N

R3

Amino group

C⬘ O

Carboxyl group

Figure 4.1 Reference frames attached to a polypeptide chain: (left) dihedral angle definitions; (right) attaching frames to Ca atoms in a canonical way.

possible to approximate changes in entropy between relatively ordered and disordered states without using traditional sampling techniques. To summarize, the main contributions of this work are 

A method for generating the distribution of relative positions and orientations of polymer-like polypeptide coils is presented, building on prior work in the Robotics literature.  The distribution of end-constrained loop conformations is obtained from this information by applying Bayes’ rule.  Quantitative bounds on the associated loop entropy are derived, and the change in loop entropy resulting from constraining one end relative to the other is computed. The computational complexity of this approach is low enough that it can be implemented on a single-processor personal computer running standardized software such as Matlab. The remainder of this work is structured as follows: A comprehensive review of the literature is provided in Section 1.1. This is followed by a review of the necessary concepts from Statistical Mechanics in Section 1.2, and of background mathematics in Section 1.3. Section 2 applies these techniques to compute lower and upper bounds on the entropy of the unfolded ensemble of polypeptide conformations. Section 3 develops bounds on the entropy of the folded ensemble. Section 4 demonstrates the methodology with some closed-form examples. Finally, Section 5 summarizes the results and maps out future directions.

1.1. Literature review Protein folding is often viewed graphically as a funnel from the polymerlike ensemble of unfolded states to the native state (Bryngelson et al., 1995). Changes in backbone entropy between unfolded and native states have been measured experimentally (D’Aquino et al., 1996). And NMR has been

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shown to be a useful tool for experimentally observing conformational fluctuations in proteins in general (Li et al., 1996; Palmer, 1997; Yang and Kay, 1996). A growing body of literature suggests that “the native state” of certain proteins may not be as ordered as once believed (Bracken et al., 2004; Dunker et al., 2001, 2005; Radivojac et al., 2004; Vucetic et al., 2005). On the other hand, recent studies suggest that the unfolded ensemble is not as disordered as once believed (Shortle and Ackerman, 2001), and that sequential interactions and sterics provide strong constraints on possible folding pathways (Baldwin and Rose, 1999; Fitzkee and Rose, 2004, 2005; Gong and Rose, 2005; Pappu et al., 2000). Furthermore, conformational entropy of the native ensemble is believed to play an important role in binding (Boehr et al., 2009; Frederick et al., 2007). For these reasons, the development of analytical and computational models of entropy in protein loops with and without end constraints provides a way to compare the relative amount of disorder in the folded and unfolded cases. Many statistical mechanical treatments of protein folding have been performed (e.g., Crippen, 2001, 2004; Dill et al., 1993; Wang and Crippen, 2004). In some studies, full chemical detail is used in molecular dynamics (Karplus and Weaver, 1976; Levitt, 1983), yet it appears that this level of detail may not be required for successful prediction of folding (Rhee and Pande, 2006). Furthermore, when computing statistical quantities such as entropy, sufficient data must be obtained in high-dimensional configuration or phase spaces in order to obtain robust results. This is almost impossible to do at a fully detailed level. Therefore, simplified statistical models such as those presented in this work may be useful. Furthermore, while the emphasis here is loop entropy in proteins, the methodology presented here can in principle be applied to RNA structures. Models of loop entropy in nucleic acids have been presented in Chirikjian (2010); Liu and Chen (2010); Zhang et al. (2008). The author’s original field is the kinematic geometry of snakelike (or “hyper-redundant”) robot arms with many degrees of freedom (Chirikjian and Burdick, 1992, 1994). A tool which is useful for the analysis of all positions and orientations reachable by the “gripper” at the distal end of this kind of arm is noncommutative harmonic analysis (Chirikjian and Kyatkin, 2001). This mathematical tool combines ideas from group theory and Fourier analysis (Gel’fand et al., 1963; Miller, 1968; Sugiura, 1990; Talman, 1968; Vilenkin and Klimyk, 1991), and can be used to compute convolutions and diffusions of functions on Lie groups, such as the rotation group or rigid-body motion group (Chirikjian and Kyatkin, 2001; Wang and Chirikjian, 2004). This is a particularly useful tool in the quantitative analysis of the distribution of all possible reachable gripper positions and orientations. Such quantities are quite similar to those encountered in polymer statistical mechanics. In polymer theory, distributions of relative end-to-end distance and orientation of backbone points and their tangents play central roles, as described in Birshtein and Ptitsyn (1966), Boyd and

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Phillips (1993), de Gennes (1979), des Cloizeaux and Jannink (1990), Doi and Edwards (1986), Flory (1969), Grosberg and Khokhlov (1994), Mattice and Suter (1994), and Skliros and Chirikjian (2008). With the tool of noncommutative harmonic analysis, distributions in all six dimensions of rigid-body motion (three translational and three rotational) can be obtained, and marginals of these distributions can be taken to yield those that are commonly of interest in polymer physics (such as the distribution of endto-end distances or relative orientations; Chirikjian, 2001). This approach has been taken by the author in a series of papers, particularly concerned with semiflexible polymers in which there is internal bending and torsional stiffness (Chirikjian and Kyatkin, 2000; Chirikjian and Wang, 2000). The case of statistical distributions when semiflexible polymers have internal joints and rigid bends has also been addressed using these methods (Zhou and Chirikjian, 2003, 2006). And it has been shown that this method can be applied to more general polymers, including unfolded polypeptide chains (Kim and Chirikjian, 2005). Similar tools can be used to analyze large amounts of geometric data in the protein data bank (Berman et al., 2000) such as statistics of helix–helix crossing angle (Lee and Chirikjian, 2004) and the relative pose (position and orientation) between alpha carbons in proteins (Chirikjian, 2001; Lee and Chirikjian, 2005). Of course, the author is not the only (and not even the first) member of the robotics community to attempt to transfer theoretical and computational tools from that field to study structural biology and biophysical phenomena. Lozano-Perez and coworkers have applied methods from robot motion planning and artificial intelligence to a number of problems in structural biology and rational drug design (Rienstra et al., 2002; Wang et al., 1998). Latombe and his students have applied methods from robot motion planning to explore configuration spaces and do energy minimization in the context of protein structures (Hsu et al., 1999; Lavalle et al., 2000; Lotan et al., 2004). These build on the method of probabilistic roadmaps (Kavraki et al., 1996). Amato and coworkers (Amato and Song, 2002; Amato et al., 2003; Tang et al., 2005; Thomas et al., 2005) and Kavraki (Das et al., 2006; Shehu et al., 2006; Teodoro et al., 2001; Zhang et al., 2005) have been leaders in the application of robotics techniques in computational biology. The cyclic descent algorithm for robot kinematics has been applied to protein loops (Canutescu and Dunbrack, 2003), as has other methods from kinematics (Kazerounian, 2004; Kazerounian et al., 2005; Manocha et al., 1995). Of late, it has been fashionable in engineering to consider proteins as examples of molecular machines (Kim et al., 2005; Mavroidis et al., 2004).

1.2. Statistical mechanics In classical equilibrium statistical mechanics, the Boltzmann distribution is defined as

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f ðp; qÞ ¼

1 expðbHðp; qÞÞ; Z

where the partition function is defined as ð ð expðbHðp; qÞÞdpdq: Z¼ q p

ð4:1Þ

ð4:2Þ

Here, b ¼ 1/kBT (kB is the Boltzmann constant and T is the temperature measured in degrees Kelvin), pi ¼ p  ei is the momentum conjugate to the ith generalized coordinate qi ¼ q  ei, H is the Hamiltonian for the system, and dpdq ¼ dp1dpNdq1dqN for a system with N degrees of freedom. The range of integration is over all possible states of the system. The Boltzmann distribution describes the probability density of all states of a system at equilibrium. The full set of generalized coordinates, {q}, describes the configuration of the system, which includes the overall rigid-body motion, and the intrinsic structural degrees of freedom. These intrinsic degrees of freedom can be further broken down into “hard” degrees of freedom such as bond angles and bond lengths that do not vary substantially from referential values, and “soft” degrees of freedom such as torsion angles that can vary widely. The hard degrees of freedom describe vibrational states and the soft degrees of freedom describe conformational changes, that is, motions due to rotations around covalent chemical bonds. While the words, “configuration” and “conformation,” are often used interchangeably in the literature, the distinction between them as defined above is important in this work. For any classical mechanical system, the Hamiltonian is of the form  1  Hðp; qÞ ¼ pT M 1 ðqÞ p þ V ðqÞ; 2

ð4:3Þ

where V(q) is the potential energy and M(q) is the mass matrix (also called the mass metric tensor; Patriciu et al., 2004).The Gibbs formula for entropy of an ensemble described by f(p, q) is ð ð S ¼ kB f ðp; qÞ log f ðp; qÞdpdq: ð4:4Þ p q

Mathematically, “continuous entropy” as defined above can take on negative values (and the entropy in the limiting case of a Dirac delta function goes to negative infinity). As explained in Chirikjian (2009), this is very different than discrete entropy. Physically, continuum theory and classical mechanics break down at very small scales in phase space. By definition, a discretization of phase space is chosen such that S ¼ 0 corresponds to the

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Modeling Loop Entropy

most ordered system that is physically possible, which is when all the states in an ensemble are contained in the same smallest possible element of discretized phase space. This is not the same as discretizing conformational space on a coarse lattice, as is often done in polymer simulations. The effects of discretization of continuous entropy are discussed in Chirikjian (2009). As a practical matter, there are several limitations in using Eq. (4.4) as a computational tool. First, there is some debate about what molecular potentials to use. On the one hand, the accuracy of ab initio potentials derived from first principles for small molecules and then applied to macromolecular simulations can be questioned. On the other hand, the accuracy of statistical potentials derived from structural data is limited by the richness of the databases from which they are extracted. For different perspectives on this debate, see Fang and Shortle (2005), Jernigan and Bahar (1996), Kortemme et al. (2003), Lazaridis and Karplus (1999), Moult (1997), and Vajda et al. (1997). Second, the number of degrees of freedom in macromolecules is so high (many thousands for a protein in continuum solvent, and perhaps millions when including explicit solvent degrees of freedom) that it is not possible to approximate f(p, q) with any degree of fidelity. (If the number of sample values required to accurately estimate a pdf in one degree of freedom is K, then one would expect to need K2N samples to approximate a pdf in a 2N-dimensional phase space.) If K is on the order of 10–100 and N ranges from thousands to millions, this is clearly intractable. One way to circumvent this problem is to compute only marginals of the full Boltzmann distribution, which as explained below, allows one to establish bounds on the true value of entropy. Due to the structure of the Hamiltonian (Eq. (4.3)), it is easy to see that in general, the Boltzmann distribution cannot be separated into a product of configurational and momentum distributions, so f ðp; qÞ 6¼ fp ðpÞfq ðqÞ (due to the dependence of the mass matrix on configuration), and so the thermodynamic entropy is bounded by the entropies of each marginal as1 S  Sp þ Sq where the configurational entropy is ð Sq ¼ kB fq ðqÞ log fq ðqÞdq; ð4:5Þ q

and it is often assumed that Sp is constant. In fact, when the generalized coordinates are the Cartesian coordinates of the positions of all atoms in a macromolecule so that q becomes the 3n-dimensional vector of all such T positions, denoted here as x ¼ fxT1 ; . . . ; xTn g , then f ðp; xÞ ¼ fp ðpÞfx ðxÞ and S ¼ Sp þ Sx. Furthermore, in this special case, the mass matrix is diagonal and constant, M(x) ¼ M0, and 1

Using results from information theory (Chirikjian, 2009; Shannon, 1948).

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Gregory S. Chirikjian

  1 1 T 1 fp ðpÞ ¼ exp  p M0 p=kB T Zp 2 and so Sp ¼ logfð2pekB T Þ3n=2 jM0 j1=2 g

ð4:6Þ

is in fact constant at constant temperature, without having to assume anything. It follows that DS ¼ DSx, which is not necessarily true for general choices of coordinates, including dihedral angles. Under a change of coordinates x ¼ x(q), it is generally the case that Sx 6¼ Sq because the computation of ð Sx ¼ kB fx ðxÞ log fx ðxÞdx ð4:7Þ x

in an alternative coordinate systems (such as dihedral angles) becomes ð Sx ¼ kB fx ðxðqÞÞ log fx ðxðqÞÞj det J ðqÞjdq q

which is not generally equal to Sq unless jdet J(q)j ¼ 1. Therefore, when referring to configurational entropy, it is important to distinguish between Cartesian configurational entropy and dihedral configurational entropy unless jdet J(q)j ¼ 1. In some scenarios, it is convenient to subdivide the configurational degrees of freedom into the categories: rigid body, hard, and soft, so that q ¼ (qrb, qhard, qsoft). It can be shown that the determinants of the mass and Jacobian matrices for chain structures can be written as functions proportional to the form w1(qrb)  w2(qhard). Similarly, it is a common modeling assumption that for a system not subjected to an external force field, and with sufficiently hard degrees of freedom, that V ðqrb ; qhard ; qsoft Þ ¼ V1 ðqhard Þ þ V2 ðqsoft Þ: Assumptions such as these lead to the separability of the partition function into a product, and the separability of entropy into a sum of terms: S ¼ Srb þ Shard þ S soft :

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Modeling Loop Entropy

Since the rigid-body term is the same for all ensembles of a given system in the same volume and temperature, DS ¼ DShard þ DS soft : We will focus on methods for computing Cartesian conformational entropy, DSx, using the concept of convolution on the rigid-body motion group. When the hard degrees of freedom are treated as rigid, DShard ! 0 and DSx ! DS soft. In the remainder of this work, we will examine Sx for (a) polymer-like ensembles with rotatable bonds and free ends and (b) polymer-like loop regions with end constraints.

1.3. Mathematics review When considering models of polypeptide chains, it often will be convenient to treat parts of the chain as rigid. For example, the plane of the peptide bond can be considered rigid, as can a cluster of side-chain atoms such a methyl group. At a coarser level, one might consider an alpha helix to be a rigid object. At a coarser level still, a whole domain might be approximated as a rigid body. Therefore, it is clear that at various levels of detail, when characterizing the conformational entropy of a protein, it is conceivable that attaching reference frames to the rigid elements and recording the set of all possible rigid-body motions between these elements is a way to describe the conformational part of the Boltzmann distribution, and therefore get at the conformational entropy via Gibbs’ formula. In this section, a coordinate-free review of rigid-body motions is presented. More detailed reviews and comparisons of various parametrizations such as Euler angles and Cayley parameters can be found in Chirikjian and Kyatkin (2001). 1.3.1. Mathematics of rigid-body motion The group of rigid-body motions, which is also called the special Euclidean group and is denoted SE(3), is the semidirect product of (R3, þ) (threedimensional Euclidean space endowed with the operation of vector addition) with the special orthogonal group, SO(3), which consists of 3  3 rotation matrices together with the operation of matrix multiplication. In both instances, the word “special” means that reflections are excluded and only physically allowable isometries of three-dimensional space are allowed. We denote elements of SE(3) as g ¼ ða; AÞ 2 SEð3Þ where A 2 SO(3) and a 2 R3 . For any g ¼ (a, A) and h ¼ ðr; RÞ 2 SEð3Þ, the group law is written as g∘h ¼ ða þ Ar; ARÞ, and g1 ¼ ðAT a; AT Þ. Alternately, one may represent any element of SE(3) as a 4  4 homogeneous transformation matrix of the form

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Gregory S. Chirikjian

 g¼

A 0T

 a ; 1

in which case the group law is matrix multiplication. The bottom row in these matrices, which consists of three zeros (i.e., 0T is the transposed, or row, vector corresponding to the column vector of zeros, 0) and the number one, is a placeholder which ensures that the matrix multiplication reproduces the correct group operation. In the above matrix, A 2 SO(3) denotes rotations and a 2 R3 denotes translations of a reference frame which, when attached to a rigid body, represent the motion of that body from the reference position and orientation defined by the identity element e ¼ (0, I). In Lie theory,2 the exponential mapping from the Lie algebra to a corresponding Lie group plays an important role (Chirikjian and Kyatkin, 2001). In the current context, the Lie group of interest is SE(3), and the corresponding Lie algebra is se(3), which consists of all matrices formed by linear combinations of the following basis elements: 0 1 0 1 0 0 0 0 0 0 1 0 B 0 0 1 0 C B 0 0 0 0C C B C E1 ¼ B @ 0 1 0 0 A; E2 ¼ @ 1 0 0 0 A; 00 0 0 01 0 0 0 0 01 0 1 0 0 0 0 0 1 B1 0 0 0C B0 0 0 0C C B C E3 ¼ B @ 0 0 0 0 A; E4 ¼ @ 0 0 0 0 A; 00 0 0 01 00 0 0 01 0 0 0 0 0 0 0 0 B0 0 0 1C B0 0 0 0C C C E5 ¼ B E6 ¼ B @ 0 0 0 0 A; @ 0 0 0 1 A: 0 0 0 0 0 0 0 0 For small (infinitesimal) motions around the identity (null motion), g  I þ X 2 SE(3) where X 2 se(3). However, for larger motions, this is not true. For those unfamiliar with his terminology, definitions and properties important to our formulation have been provided in the book (Chirikjian and Kyatkin, 2001). The essential thing to know is that elements of se(3) and SE(3) can both be viewed as 4  4 matrices; however, while it makes sense to add elements of se(3) (i.e., velocities add), it only makes sense to multiply elements of SE(3). Furthermore, by the matrix exponential mapping, it is

2

Named after Norwegian mathematician Marius Sophus Lie (1842–1899).

109

Modeling Loop Entropy

possible to produce elements of SE(3) from those in se(3), and vice versa using the matrix logarithm: exp : seð3Þ ! SEð3Þ and log : SEð3Þ ! seð3Þ: Figure 4.2(left) illustrates that the composition of rigid-body motions is not a commutative operation. Figure 4.2(right) shows the relationship between the Lie algebra se(3), consisting of infinitesimal motions (which form a linear vector space), and SE(3), consisting of large motions (which form a curved manifold, which is a Lie group). For small translational (rotational) displacements from the identity along (about) the ith coordinate axis, the homogeneous transforms representing infinitesimal motions look like expðeEi Þ  I þ eEi ;

ð4:8Þ

where I is the 4  4 identity matrix, jej  1, and exp(X) ¼ I þ X þ X2 /2 þ    is the matrix exponential defined by the Taylor series of the usual exponential function evaluated with a matrix rather than a scalar. For example, 0 1 0 1 cosy  siny 0 0 1 0 0 0 B siny cosy 0 0 C B C C and expðyE5 Þ ¼ B 0 1 0 y C; expðyE3 Þ ¼ B @ 0 A @ 0 1 0 0 0 1 0A 0 0 0 1 0 0 0 1

Frame 2′ SE(3) g1 g′0,2

g

Frame 1′ g2 Frame 0′ Frame 2

exp

log

g2 g0,2

X g1

Frame 0

Frame 1 se (3)

Figure 4.2 (left) Rigid-body transformations between reference frames form a noncommutative Lie group (g1 ∘ g2 6¼ g2 ∘ g1); (right) the exponential map.

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Gregory S. Chirikjian

and for small values expanding sin y  y and cos y  1, it is easy to see that Eq. (4.8) holds for the example on the left. For the example on the right, Eq. (4.8) holds even for large values of y. The “exponential parametrization” ! 6 X g ¼ gðw1 ; w2 ; . . . ; w6 Þ ¼ exp ð4:9Þ wi Ei i¼1

is a useful way to describe relatively small rigid-body motions because, unlike the Euler angles, it does not have singularities near the identity. One defines the “vee” operator, _, such that for any 6 X w i Ei ; i¼1 0 1 w1 B w2 C B C X _ ¼ B . C: @ .. A

X¼

w6 The 6



6 adjoint matrix, Adg, is defined by the expression  _ Adg ðX _ Þ ¼ gXg1 ;

and explicitly if g ¼ (a, A) then  Adg ¼

 A 0 ; aA A

where a  A denotes the matrix resulting from the cross product of a with each column of A. The vector of exponential parameters, x 2 R6 , can be obtained from g 2 G with the formula x ¼ ð log gÞ_ :

ð4:10Þ

The action of an element of the motion group, g ¼ (a, A), on a vector x in three-dimensional space is defined as gx ¼ Ax þ a. In contrast, given a function f(x), we can translate and rotate the function by g as f ðg1 xÞ ¼ f ðAT ðx  aÞÞ. The fact that the inverse of the transformation applies under the function (rather than the transformation itself) in order to implement the desired motion is directly analogous to the case of translation

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Modeling Loop Entropy

-1

f1,2 (h2

f (g-1⭈ x) →

⬚ g)

f1 (h -1 ,2 4

-1

f1,2 (h1

⬚ g)

-1

f1,2 (h3

⬚ g)

⬚ g) (h f 1,2

-1

g

g)

⬚

f0,2 = f0,1 * f 1,2

f0,1 (h)

→

f (x)

Figure 4.3 (left) Action of a motion on a function; (right) convolution of functions of rigid-body motion.

on the real line. For example, given a function on the real line, f(x), with its mode at x ¼ 0, if we want to translate the whole function in the positive x direction by amount x so that the mode is at x ¼ x, we compute f(x  x) (not f(x þ x)). This is a very important point to understand in order for the rest of this work to make sense. Figure 4.3(left) illustrates the shifting of a function under rigid-body motion geometrically. 1.3.2. Manipulations of functions of rigid-body motion Suppose that three rigid bodies labeled 0, 1, and 2 are given, with reference frames attached to each, and assume that only sequentially adjacent bodies interact. Suppose also that body 0 is fixed in space and the ensemble of all possible motions of body 1 with respect to 0 are recorded, and motions of 2 with respect to 1 are also recorded. Then, we have two functions of motion, f0,1(g) and f1,2(g) which together describe the conformational variability of this simple system. If we are interested in knowing the probability distribution describing the ensemble of all possible ways that body 2 can move relative to body 0, how is this obtained? In fact, it is computed via the convolution on SE(3) (Chirikjian and Kyatkin, 2001): ð     f0;2 ðgÞ ¼ f0;1 f1;2 ðgÞ ¼ f0;1 ðhÞf1;2 h1 ∘g dh: ð4:11Þ G

What this says is that the distribution f1,2(g) is shifted through all possible rigid-body motions, h, weighted by the frequency of occurrence of these motions, f0,1(h), and integrated over all values of h 2 G (G is just short for “Group,” which throughout this work is the group of rigid-body motions, SE(3)). Figure 4.3(right) illustrates this geometrically.

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Gregory S. Chirikjian

Explicitly, what is meant by this integral? Let us assume for the moment that rotations are parameterized using Euler angles. The range of the Euler angles is 0  a, g  2p, and 0  b  p. In this parametrization, the volume element for G is given by dg ¼

1 sinb dadbdgdr1 dr2 dr3 ; 8p2

which is the of the volume elements for R3 (dr ¼ dr1dr2dr3), and  product  for SO(3) dR ¼ 8p1 2 sinb Ð dadbdg . The normalization factor in the definition of dR is such that SOð3Þ dR ¼ 1. The volume element for SE(3) can also be expressed in the exponential coordinates described in Section 1.3.1, in which case dg ¼ jJ ðxÞjdw1   dw6 ; where jJ (x)j is a Jacobian determinant for this parametrization. The Jacobian can be computed using the formula  J ðxÞ ¼

g

1

@g @w1

_

 _ 1 @g ;...; g @w6

and it can be shown that jJ(0)j ¼ 1 and so close to the identity that the Jacobian factor in this parametrization can be ignored (which is not true for many other parametrizations, including the Euler angles). The fact that the volume element is invariant to right and left translations, that is, dg ¼ dðh∘gÞ ¼ dðg∘hÞ is well known in certain communities (see, e.g., Sugiura, 1990; Vilenkin and Klimyk, 1991). A convolution integral of the form in Eq. (4.11) can be written in the following equivalent ways: ð ð       f0;1 f1;2 ðgÞ ¼ f0;1 z1 f1;2 ðz∘gÞdz ¼ f0;1 g∘k1 f1;2 ðkÞdk; ð4:12Þ G

G

where the substitutions z ¼ h 1 and k ¼ h 1 ○ g have been made, and the invariance of integration under shifts and inversions is used. The concept of convolution on SE(3) will be central in the formulation that follows: 

One can define a Gaussian distribution on the six-dimensional Lie group SE(3) much in the same way as is done on R6 provided that (1)

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Modeling Loop Entropy

the covariances are small and (2) the mean is located at the identity. The reason for these conditions is because near the identity, SE(3) resembles R6 which means that dg  dw1    dw6 and we can define the Gaussian in the exponential parameters as   1 1 T 1 f ðgðxÞÞ ¼ exp  x S x : ð4:13Þ 2 ð2pÞ3 jSj1=2 

Given two such distributions that are shifted as fi,iþ1(gi,iþ11 ∘ g), each with 6  6 covariance Si,iþ1, then it can be shown that the mean and covariance of the convolution f0,1(g0,11 ∘ g) f1,2(g1,21 ∘ g), respectively, will be of the form g0,2 ¼ g0,1 ∘ g1,2 and (Wang and Chirikjian, 2008) T S0;2 ¼ Ad1 g1;2 S0;1 Adg1;2 þ S1;2 :

ð4:14Þ

This provides a method for computing covariances of two concatenated segments, and this formula can be iterated to compute covariances of chains without having to compute convolutions directly. This is demonstrated numerically in the context of robotic arms in Wang and Chirikjian (2008).

2. Computing Bounds on the Entropy of the Unfolded Ensemble 2.1. End-to-end position and orientation distributions and the Cartesian conformational entropy of serial polymer chains Consider a polymer consisting of a serial chain of n þ 1 essentially rigid monomer units numbered from 0 to n. Attach a frame of reference to the ith such unit. Let gi denote the rigid-body motion from the reference frame of the zeroth unit to that attached to the ith. Let gk,kþ1 denote the relative motion from body k to body kþ1. Then, gi ¼ g0,i ¼ g0,1 ∘ g1,2 ∘    ∘gi1,i will be the cumulative motion from body 0 to body i. The relationship between these reference frames is described in Fig. 4.4. In a purely pairwise energy model, only the interactions between adjacent units are important. In this simplest model, the probability of the relative pose gi,iþ1 ¼ gi 1 ∘ giþ1 taking a particular value is given by        fi;iþ1 gi;iþ1 ¼ 1=Zi;iþ1 exp bV gi;iþ1 :

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Gregory S. Chirikjian

g3,4 g2,3

Body i g4 = g0,4

®

gi

xi1

g3 = g0,3 g2 = g0,2 ®

®

xi4

g1 = g0,1

xi2 ®

xi3 g0 = e

Figure 4.4 (left) Relative and absolute reference frames attached to the chain; (right) the relative positions of mass points within body i.

Then, the conformational distribution described in terms of rigid-body poses is f ðg1 ; g2 ; . . . ; gn Þ ¼

n1 Y

  fi;iþ1 gi1 ∘giþ1 ;

ð4:15Þ

i¼0

where g0 ¼ e, the identity. This is related to the end-to-end position and orientation distribution   f0;n ðgn Þ ¼ f0;1 f1;2    fn1;n ðgn Þ; ð4:16Þ which is an n-fold convolution of the form in Section 1.3.2, by marginalization of Eq. (4.15) as ð ð f0;n ðgn Þ ¼    f ðg1 ; g2 ; . . . ; gn Þdg1   dgn1 : G

G

This is illustrated in Fig. 4.5. Equation (4.15) represents a generalization of the classical polymer models in which only pairwise interactions are considered. If the frames of reference gi and giþ 1 are attached at the Ca atoms of residues i and iþ1 in a polypeptide, then the function fi,iþ1(gi,iþ1) would be the six-dimensional generalization of a Ramachandran map that could include small bond angle bending, warping of the peptide plane, and even bond stretching. If one chooses not to model these effects, then the classical Ramachandran map

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Modeling Loop Entropy

Copies of gi + 1

f0,n (g) f0,k (g) gi

f0,1 (g)

Figure 4.5 Kinematic covariance propagation: (left) in the absence of other constraints, distributions describing the allowable rigid-body motions between consecutive residues “add” by convolution, resulting in a spreading out of probability density in position and orientation, f0,i(gi), as i increases; (right) a zoomed-in view of the probabilistic relationship between reference frames i and iþ1 embodied by the functions fi, iþ1(gi, iþ1).

(Ramachandran et al., 1963) can be reflected by appropriately defining fi,iþ1(gi,iþ1), as has been done in Kim and Chirikjian (2005). This is consistent with the Flory isolated pair model (Flory, 1969), which has been challenged in recent years (Pappu et al., 2000). However, as an upper bound on conformational entropy, it may still be useful in some contexts. Note that since gn ¼ (r, R) describes both the end-to-end position and orientation of the distal end of the chain relative to the proximal end, we can marginalize further to obtain quantities such as the end-to-end distance distribution or end-to-end orientational distribution. These quantities (or several of their moments) can be measured directly from a variety of experimental measurements. In order to convert these probabilities into a form that is directly useful for computing Cartesian conformational entropy, we must know the positions of all atoms in each of the i rigid monomer units. Given f(g1, g2, . . ., gn) in Eq. (4.15) and given the family of probability density functions fDi ðxi1 ; . . . ; xik Þg, each of which describes the distribution of motions of the ik  i1 þ 1 atoms within body i, it is possible to compute the full Cartesian conformational distribution as ð rðx1 ; . .. ;xN Þ ¼

ð  

G

f ðg1 ;g2 ; . .. ; gn Þ G

n Y   Di gi1 xi1 ; .. . ;gi1 xik dg1  dgn ; i¼1

ð4:17Þ

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Gregory S. Chirikjian

where N ¼ in is the total number of atoms in the chain and h iT T T xi ¼ xi1 ; . .. ; xik is the composite vector of Cartesian coordinates of all positions in the ith body. Di ðxi1 ; . . . ; xik Þ is a probability density on 3(ik  i1 þ 1)-dimensional Euclidean space. In other words, ð ð    Di ðxi1 ; . . . ; xik Þdxi1   dxik ¼ 1: R3

R3

As an example, when the ith body is modeled as being perfectly rigid, ik

 Y 0 d xj  xj ; Di ðxi1 ; . . . ; xik Þ ¼ j¼i1

where x0j is the fixed position of atom j as seen in the frame of reference gi affixed to rigid body i. In contrast, if body i is an articulated side chain, averaging over all of its conformational states would result in a Di which is not a sum of Dirac delta functions. In some cases, it may be useful to compute the full pose entropy of the chain: ð ð Sg ¼     f ðg1 ; g2 ; . . . ; gn Þ log f ðg1 ; g2 ; . . . ; gn Þdg1   dgn : ð4:18Þ G

G

2.2. Modeling excluded volume effects The phantom polymer chain model in which the effects of excluded volume are ignored is clearly not a realistic model, but it can be used as a baseline onto which self-avoidance can be built. In a polypeptide, residue i interacts with residues i þ 1,. . .,i þ 4 substantially as well as more sequentially distant residues. These interactions are not only responsible for the formation of secondary structures, but also substantially winnow down the available conformational space (Fitzkee and Rose, 2005). Clearly, this has implications for the entropy. More specifically, polymer models can be used to compute upper bounds on the conformational entropy in polypeptides. And these bounds can be made tighter by incorporating the effects of steric clash into modified versions of the conformational probability distributions. To begin, let us compute the density of body i. This can either be done directly by, for example, averaging body i over all possible side-chain conformations. Or, it can be done by first computing each marginal of the density function Di as

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Modeling Loop Entropy

  dij xij ¼

ð

ð xi1 2R3



ð

xij1 2R3 xijþ1 2R3

ð 

xik 2R3

Di ðxi1 ; . . . ; xik Þdxi1   xij1 xijþ1   dxik : Then, the average density of body i (normalized to be a probability density) is di0 ðxÞ ¼

ik X 1 di ðxÞ: ik  i1 þ 1 ij ¼i1 j

The overlap of bodies in the chain is illustrated in Fig. 4.6. Therefore, if body i is moved by rigid-body motion gi, and likewise for body j, we can compute an estimate of their overlap (averaged overall deformations of the bodies) as ð     

wij gi ; gj ¼ di0 gi1  x dj0 gj1  x dx: R3

A general property of integration over all of three-dimensional space is that it is invariant under rigid-body motions. Therefore, if we make the change of variables y ¼ gi1 ∘x, then we find that

     wij gi ; gj ¼ wij e; gi1 ∘gj ¼ wij gj1 ∘gi ; e : Clearly, when the two bodies do not overlap, wij ¼ 0. Otherwise, they will have some positive value. One can imagine evaluating wij as the argument of a “sigmoid function,” which sharply ramps up from zero to one, where it

Figure 4.6 Conformations to be removed from the phantom chain ensemble: (left) local overlaps; (right) nonlocal overlaps.

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Gregory S. Chirikjian

then plateaus at higher values. The resulting Wij(gi 1 ∘ gj) ¼ 1  exp ( (wij(gi, gj))2 /2s2) (for some small value of s) would effectively window-out all values of the rigid-body motions gi and gj that contribute to nonphysical overlaps. Then, the original f(g1, g2, . . ., gn) in Eq. (4.15) could be replaced with one of the form fex ðg1 ; g2 ; . . . ; gn Þ ¼ Cf ðg1 ; g2 ; . . . ; gn Þ

n  Y   1  Wij gi1 ∘gj ;

ð4:19Þ

i<j

where C is the normalization required to make fex a pdf. Note that the product in this expression is not only over sequentially local pairs of bodies, but rather all bodies, where the i < j simply avoids double counting. In this way, a phantom polymer model that generates f(g1, g2, . . ., gn) can be viewed as the starting point for a more realistic model that includes steric constraints.

2.3. Bounding Cartesian conformational entropy Practically speaking, computing such high-dimensional integrals as in Eq. (4.17) or (4.19) can impose a computational problem, except when simple closed-form expressions such as Gaussians are used. If we seek an upper bound on Cartesian conformational entropy, marginals can be computed and information-theoretic bounds can be employed. Performing such marginalization, one finds ð   ri ðxi Þ ¼ f0;i ðgi ÞDi gi1  xi1 ; . . . ; gi1 xik dgi : ð4:20Þ G

In the case, when one representative point is chosen per residue (e.g., the Ca atom, which is where the reference frame for the residue is usually attached), we have k ¼ 1. Then, i ¼ i1, and since x0i ¼ 0 due to the way the reference frame is attached, we can write ð   ri ðxi Þ ¼ f0;i ðgÞd g1  xi dg: G

If g ¼ (r, R), then dðg1 xi Þ ¼ dðRT ðxi  rÞÞ ¼ dðxi  rÞ and so we can get the positional distribution of the ith Ca atom by marginalizing the full pose distribution over orientations as ð ð ð f0;i ðr; RÞdðxi  rÞdrdR ¼ f0;i ðR; xi ÞdR: ð4:21Þ ri ðxi Þ ¼ SOð3Þ R3

SOð3Þ

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Modeling Loop Entropy

The conformational entropy of the backbone represented by Ca atoms is then bounded from below by the entropy of individual marginals (with the tightest lower bound resulting from the maximum of these). The loop entropy will be bounded from above by the sum of entropies from all of the marginals. Therefore, max Si  Sx  i

n X

ð Si ; where

Si ¼ 

i¼1

R3

ri ðxi Þ logri ðxi Þdxi

ð4:22Þ



T and x ¼ xT1 ; xT2 ; . . . ; xTn .

3. Approximating Entropy of the Loops in the Folded Ensemble The native ensemble of a protein is characterized by a relatively high degree of order. However, the native form is not completely rigid. In particular, loop/coil regions connecting secondary structures can exhibit large motions. Here, we model the ends of these loops as being fixed at specific positions and orientations, as illustrated in Fig. 4.7. Bounds on the contribution of loop motions to overall entropy are discussed here. If f(g1,. . .,gn) is the conformational distribution function describing the positions and orientations of all bodies in the system with respect to the proximal end of the chain, then if we fix the distal end at a specific pose, gend, the resulting distribution will be the conditional density

gend

fix

f0,i (gi ; gend)

Figure 4.7 Using density information to determine probabilities of conformations that obey end constraints.

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Gregory S. Chirikjian

f fix ðg1 ; g2 ; . . . ; gn1 ; gend Þ ¼ f ðg1 ; . . . ; gn jgn ¼ gend Þ ¼ f ðg1 ; . . . ; gn1 ; gend Þ=f0;n ðgend Þ:

ð4:23Þ The entropy of this distribution, in some cases, can either be computed directly, or each of the marginals can be computed as   fix f0;i ðgi ; gend Þ ¼ f0;i ðgi Þfi;n gi1 ∘gend =f0;n ðgend Þ:

ð4:24Þ

The reason for this is that in the definition of f(g1,. . .,gn), the variable gi appears in only the two multiplied terms: fi1, i(gi11 ∘ gi)  fi,iþ1(gi1 ∘ giþ1). Marginalizing over g1 through gi 1 results in f0,i(gi). If gi were the identity element, then marginalizing over giþ1 through gn1 would yield fi,n(gend). However, since in general gi ¼ 6 e, this result is shifted by gi to yield fi, n(gi1 ∘ gend). Division by f0,n(gend) is the normalization required to make the result a pdf (since integration of the numerator in Eq. (4.24) over gi is a convolution). This denominator is carried along from Eq. (4.23), which is a statement of Bayes’ rule. Intuitively, the entropy of a chain with fixed ends must be smaller than that of a chain with freely moving ends. This can be quantified when using Eqs. (4.23) and (4.24), as will be shown in the examples in the next section.

4. Examples In this section, examples are used to illustrate the formulation presented earlier in this work. In both of these examples, a piece of flexible loop/coil connects relatively rigid structures. In the first example, the loop is considered to be a long phantom chain, whereas in the second, it is considered to be a semiflexible polymer. The reduction in conformational entropy associated with constraining the ends in both cases is examined.

4.1. Model 1: Long loops modeled as Gaussian chains Perhaps, the most common model for the distribution of end-vector distribution in polymer theory is the Gaussian distribution: 

 3r 2 f ðgÞ ¼ W ðrÞ ¼ exp  2 2hr i   1 T 1 exp  r S r ; ¼ 2 ð2pÞ3=2 jSj1=2 3 2phr 2 i 1

3=2

ð4:25Þ

where g ¼ (r, R) 2 SE(3) and the chain is so flexible that the orientational part of the distribution is constant, and S ¼ (hr2i /3)I, where I is the 3  3

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Modeling Loop Entropy

identity matrix. This distribution is spherically symmetric (and hence depends only on r ¼ jrj). It is normalized so that it is a pdf, ð ð1 W ðrÞdV ¼ 4p W ðrÞr 2 dr ¼ 1; R3

0

satisfying ð R3

W ðrÞjrj2 dV ¼ 4p

ð1

  W ðrÞr 4 dr ¼ r 2 :

0

The pdf for a freely jointed chain with n links, each of length l can be approximated as a Gaussian random walk with  2 r ¼ nl 2 : If we denote Wn(r) to be the function for n links, then it is clear that in this simple model, Wn1 Wn2 ¼ Wn1 þ n2. Therefore, in this simple model, the conformational entropy is bounded as n     3 3X 2 log 2pekl 2 =3 log 2penl =3  Sr  2 2 k¼1

using Eq. (4.22) where r takes the place of x. The conformational entropy of the phantom chain that gives rise to this distribution is bounded from below by the entropy of the pdf of the location of the terminal end, and from above by the sum of probabilities for each link from base to end, since these are marginals of the total conformational distribution f(g1,. . .,gn), which for this case is a function Q W ðr1 ; . . . ; rn Þ ¼ n1 i¼0 W1 ðriþ1  ri Þ with r0 ¼ 0 where W1 is the effective one-bond Gaussian distribution with covariance matrix S1 ¼ (l2/3)I. Therefore, it is possible to compute the Cartesian conformational entropy in this model exactly in closed form as ð ð Sr ¼     W ðr1 ; . . . ; rn Þ logW ðr1 ; . . . ; rn Þdr1    drn : r1

rn

Since the chain is assumed to be uniform and the product of Gaussians is a Gaussian, we can use the fact that there is a closed-form formula for the entropy of a Gaussian in terms of its covariance matrix, together with the fact that in this example

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Gregory S. Chirikjian

0

S1

2I

B I B B 3B 0 ¼ 2B B l B0 B. @ .. 0

I

0

0



2I

I .. . .. . .. . 0

0 .. . .. .

 .. . .. .

I 0

2I I

I 0 .. .



1 0 .. C . C C 0 C C: C 0 C C I A

ð4:26Þ

I

In principle, the entropy of a Gaussian chain with end position constraints can be computed using Eqs. (4.23) and (4.24). In practice, there are some details that need to be addressed, which are addressed in Section 4.3. Whereas here a Gaussian chain in which orientations diffuse rapidly was considered, the opposite extreme of a stiff chain is considered in the following section.

4.2. Model 2: Short loops modeled as semiflexible polymers Suppose that we have a semiflexible loop, that is, one that has local resistance to bending and twisting that reflects sequentially local steric constraints. Then, each gi will deviate only a relatively small amount from a constant reference pose, hi, and so we write gi ¼ hi expxi where jjxi jj < 1. This sort of assumption is consistent with findings in the literature. For example, (Zhou, 2001) validated the use of semi-flexible polymer models to describe protein loop motions. The relative motion between adjacent reference frames will be gi1 ∘giþ1 ¼ exp  xi ∘h1 i ∘hiþ1 ∘ expxiþ1 : If the probability density fi,iþ 1 is a Gaussian with mean at hi 1 ∘ hi þ 1, then it will be of the form

 1  fi;iþ1 ðgÞ ¼ FSi;iþ1 h1 ∘g ; i ∘hiþ1 where FSi;iþ1 ð expxÞ ¼

1 ð2pÞ3 jSi;iþ1 j1=2

  1 exp  xT S1 x : i;iþ1 2

Therefore,

    1 1 ∘h ∘ exp  x ∘h ∘h ∘ expx fi;iþ1 gi1 ∘giþ1 ¼ FSi;iþ1 h1 iþ1 iþ1 i iþ1 : i i For small motions between adjacent bodies, the approximation

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Modeling Loop Entropy

h

 i_ 1 1 log h1 ∘h ∘ exp  x ∘h ∘h ∘ expx iþ1 iþ1 i iþ1 i i  xiþ1  Ad1 h1 ∘hiþ1 xi i

has been proved to be accurate (Wang and Chirikjian, 2008). If as shorthand, we define Ai,iþ1 ¼ Adhi 1∘hiþ1, then  

1 1 exp  xTi ; xTiþ1 fi;iþ1 gi1 ∘giþ1 ¼ 3 1=2 2 ð2pÞ jSi;iþ1 j ! ð4:27Þ 1 1 T 1 Ai;iþ1 Si;iþ1 Ai;iþ1 AT xi i;iþ1 Si:iþ1 : 1 xiþ1 S1 S1 i;iþ1 Ai;iþ1 i;iþ1 This, together with the product in Eq. (4.15), leads f(g1, Tg2, . . ., gn) to be a Gaussian distribution in the variable x ¼ xT1 ; . . . ; xTn with an inverse covariance of the form 2

0

1 S1 0;1 þ S1;2 6 6 6 S1 A1 1;2 1;2 6 6 6 60 6 6 . S1 ¼ 6 6 .. 6 6 60 6 6 6 60 4

0

1 AT 1;2 S1;2



0

0 ..

.

..

.

..

.

0

.

..

.

0

1 T 1 S1 1;2 þ S2;3 A2;3 S2;3 0 1 S1 2;3 A2;3

.. ..

.

..

.

..

0

0

.

0

..

..

0

0

1 S1 n2;n1 An2;n1

1 S1 n2;n1 þ Sn1;n

0



0

0

0

.

7 7 7 7 7 7 7 0 7 7 .. 7; 7 . 7 7 7 0 7 7 7 1 7 T An1;n Sn1;n 5 S1 n1;n 0

0 .. .

.

3

0

1 1 1 1 T S1 n3;n2 An3;n2 Sn2;n1 þ Sn3;n3 An2;n1 Sn2;n1 0

ð4:28Þ 0

where Si,iþ1 ¼ Ai,iþ1Si,iþ1ATi, iþ1. The entropy Sg for the case with free ends is then given by the formula (4.18), which is relatively efficient to compute due to the block-tridiagonal form of S 1. This will be discussed further in Section 4.3. Obtaining the entropy for the case when both ends are fixed in position and orientation is also possible within this model. Using the fact that the adjoint is a homomorphism, that is, Αd(g1∘g2) ¼ Ad(g1)Ad(g2) and Ad(g 1) ¼ Ad 1(g), this generalizes to the concatenation of n reference frames that vary around values hi as n X T Ad1 ð4:29Þ S0 1  n ¼ hk;n Sk Adhk;n : k¼0

To compute Sg for the case when the distal end of the chain is fixed at gn ¼ gend, we would use Eq. (4.23) with the covariance of f0,n(g) being given by Eq. (4.29). Conditioning of Gaussians by Gaussians yields Gaussians, the

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entropy of which can be computed in closed form in principle. However, there are some subtle issues that need to be addressed, as discussed below.

4.3. From covariance matrices to entropy It is one thing to have bounds such as Eq. (4.22). It is another to have a closed-form expression for the actual quantity of interest. Here, Eqs. (4.26) and (4.28) are used to compute entropy. This follows from the fact that for a d(n)-dimensional Gaussian distribution with covariance S, the entropy is given as (Chirikjian, 2009; Shannon, 1948) S ¼ logfð2peÞdðnÞ=2 jSj1=2 g:

ð4:30Þ

Here, we use the notation d(n) to denote the dimension of the covariance matrix, which is d(n) ¼ 3n for positional Gaussian, and d(n) ¼ 6n for the semiflexible case. In other words, we can write d(n) ¼ d0  n. We will consider the case when entropy change is due to fixing the ends. Consider a chain (either Gaussian or semiflexible), and let x1,. . .,xn denote the variables describing the kinematic state of the n segments (i.e., xi ¼ ri 2 R3 for the Gaussian chain and xi ¼ xi 2 R6 for the semiflexible

T chain). Let us denote x ¼ xT1 ; . . . ; xTn1 , y ¼ xn 2 Rd0 , and T z ¼ ½xT ; yT  . Then, f(x1,. . .,xn) can be written as   1 1 T 1 exp  z S z : f ðzÞ ¼ 2 ð2pÞd0 n=2 jSj1=2 The entropy of this distribution is computed simply as Eq. (4.30). However, the end-constrained case is somewhat more involved. The conditional probability density describing the ensemble of endconstrained conformations is of the form f ðxjyÞ ¼ f ðx; yÞ=f ðyÞ; where f(x, y) ¼ f(z) (with y held fixed rather than being a variable) and the marginal distribution f(y) is given by   1 1 T 1 f ðyÞ ¼ exp  y Syy y ; 2 ð2pÞd0 =2 jSyy j1=2 where 

Sxx S¼ Syx

Sxy : Syy

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Modeling Loop Entropy

The conditional distribution will then be   1 1 T 1 f ðxjyÞ ¼ exp  ½x  x0  L ½x  x0  ; 2 ð2pÞd0 ðn1Þ=2 jLj1=2

ð4:31Þ

where L ¼ Sxx  Sxy S1 yy Syx ¼

h  i1 S1 xx

and x0 ¼ Sxy S1 yy y: In principle, we now have everything we need to compute entropy differences. However, in practice, there is an implicit assumption about polymer distribution functions that must be addressed. Namely, even though the chain length is L, and hence the distal end cannot reach outside a ball or radius L centered at the proximal end, for the sake of convenience, we will accept distributions with infinitely long tails. Another way to say this is that as long as the pdf decays rapidly enough and all integrals over a ball of radius L centered at the origin can be replaced by integrals over an infinitely large ball, then things will work out fine. Such calculations include computing probabilities and entropies from probability densities. In other words, we have simplifications such as ðL

x2

L

e

dx 

ð1

ex dx: 2

1

While this is perfectly reasonable when the Gaussians are centered at the origin, it will no longer be the case when we shift them by significant amounts. In other words, even though the value of an integral over an infinite range is invariant under shifts, this is not the case for integration over finite intervals: ðL L

ðxL=2Þ2

e

dx 6¼

ðL L

ex dx: 2

This is important in the context of the current discussion because the conditional pdf in Eq. (4.31) is shifted from the origin by a vector x0. In other words, if we fix the distal end of the chain at an arbitrary y, then this distribution of interest in d0  (n  1)-dimensional space will not be

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centered at the origin, and the infinite integral used to approximate integration over a ball of radius L ¼ nl centered at the origin that resulted in the normalization constant [(2p)d0(n  1) /2jLj1/2] 1 will no longer be a valid approximation. The computation of this constant and the computation of entropy then become a problem when jjyjj (and hence, jjx0 jj) is not very small relative to total chain length, L ¼ nl. However, when it is very small, the integral can still be approximated as being over infinite-dimensional space because the overwhelming majority of the mass under the pdf will still be contained in the finite ball of radius L. 4.3.1. Entropy for the Gaussian chain For the Gaussian chain, effectively the end constraint rn ¼ 0 means that the chain forms a closed loop because the vectors {ri} in this case are absolute positions of the ith residue with respect to the proximal end. The entropy difference between two ensembles described by Gaussians with dimensions d(n) and d(n  1) in the unconstrained and end-constrained states, respectively, will be DS ¼ S2  S1 ¼ logfð2peÞdðnÞ=2 jS2 j1=2 g  logfð2peÞdðn1Þ=2 jS1 j1=2 g 2 # " # 1=2 1 jS j 1 jS j 2 d =2 d 1 0 0 : ¼ log ð2peÞ ¼ log4ð2peÞ 2 jS1 jS1 j1=2 2 j

ð4:32Þ

The last equality means that there is no need to invert the matrices in Eqs. (4.26) and (4.28) when computing entropy differences between the ensembles with free and fixed ends. This is useful, because in practice, one usually is interested only in entropy differences, and the determinants of block-tridiagonal matrices can be computed very efficiently (in O(n) computations for a chain of length n), whereas computing their inverses followed by taking the determinant can be an O(n3) operation. 4.3.2. Entropy of a semiflexible chain The entropy being considered is that defined in Eq. (4.18). For the semiflexible chain, the set {xi} describes the relative small rigid-body displacements of the ith residue with respect to a referential configuration. Therefore, in this case, the same tools developed in this work can be used to describe the entropy differences between the free and end-constrained cases for a somewhat different scenario than the Gaussian chain model. Namely, we can compute multiple reference conformations and consider small deviations around each. The reduction in entropy due to constraining both ends of the chain is then due to eliminating motions around multiple reference conformations (each with free ends) and only allowing motions

127

Modeling Loop Entropy

around the one reference conformation that satisfies the required end conditions. This discussion is quantified below. Imagine sampling the relative poses between adjacent amino acids in a loop at their K most populated isolated peaks. For example, K might be equal to 3 if we sample at the centers of the a, b, and O regions of the f–c plane. If each of these peaks are isolated, and the distributions around them are modeled as SE(3) Gaussian distributions with small covariances and essentially nonoverlapping tails, then the entropy associated with each of these conformational ensembles will be given by Eq. (4.30) where S is defined by Eq. (4.28). If the loop has n residues, then each reference conformation for i ¼ 1,. . .,Kn will have its own entropy, Si defined by these equations. If the relative weights of each of these reference conformations are given by w1, . . ., wKn, and if each conformation is disjoint, and there is minimal overlap between the associated conformational distributions around each, then the total entropy in the case of free ends can be approximated as Sfree  

Kn X i¼1

wi logwi þ

Kn X

wi Si :

i¼1

The entropy for the case when the distal end is fixed can be approximated by adding contributions from the subset of baseline conformations that approximately satisfy the end constraints.

5. Conclusions This work reviewed and built on techniques from the fields of robotics, information theory, and theoretical polymer science, and applied these to model conformational entropy in protein loops. At the core of this presentation was the mathematics of rigid-body motion and associated statistical computations, as well as the use of inequalities from information theory for developing a rigorous mathematical treatment of the entropy of unfolded, partially folded, and fully folded proteins. Models of conformational statistics in these three kinds of ensembles were reviewed and developed. These models were then applied to compute entropy differences. The various concepts of entropy in statistical mechanics, computational polymer science, and information theory were reviewed. The distinction between conformational entropy computed in internal and Cartesian coordinates was made. Inequalities to bound the value of entropy from below and above were presented in cases when exact computations were judged to be intractable.

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ACKNOWLEDGMENTS This work was performed with support from the National Institutes for Health under grant R01GM075310 and R01GM075310-04S1. The author thanks Profs G. Rose and E. Lattman for their valuable comments and Dr W. Park for proofreading and Dr. S. Lee and Ms. Y. Wang for creating some of the figures.

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Radivojac, P., Obradovic, Z., Smith, D. K., Zhu, G., Vucetic, S., Brown, C. J., Lawson, J. D., and Dunker, A. K. (2004). Protein flexibility and intrinsic disorder. Protein Sci. 13(1), 71–80. Ramachandran, G. N., Ramakrishnan, C., and Sasisekharan, V. (1963). Stereochemistry of polypeptide chain configurations. J. Mol. Biol. 7(1), 95–99. Rhee, Y. M., and Pande, V. S. (2006). On the role of chemical detail in simulating protein folding kinetics. Chem. Phys. 323, 66–77. Rienstra, C. M., Tucker-Kellogg, L., Jaroniec, C. P., Hohwy, M., Reif, B., McMahon, M. T., Tidor, B., Lozano-Perez, T., and Griffin, R. G. (2002). De novo determination of peptide structure with solid-state magic-angle spinning NMR spectroscopy. PNAS 99(16), 10260–10265. Shannon, C. E. (1948). A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, pp. 623–656. Shehu, A., Clementi, C., and Kavraki, L. E. (2006). Modeling protein conformational ensembles: From missing loops to equilibrium fluctuations. Proteins: Struct. Funct. Bioinform. 65(1), 164–179. Shortle, D., and Ackerman, M. S. (2001). Persistence of native-like topology in a denatured protein in 8 M urea. Science 293(5529), 487–489. Skliros, A., and Chirikjian, G. S. (2008). Positional and orientational distributions for locally self-avoiding random walks with obstacles. Polymer 49(6), 1701–1715. Sugiura, M. (1990). Unitary representations and harmonic analysis. 2nd edn. Elsevier Science Publisher, The Netherlands. Talman, J. (1968). Special Functions. W. A. Benjamin, Inc., Amsterdam. Tang, X. Y., Kirkpatrick, B., Thomas, S., Song, G., and Amato, N. M. (2005). Using motion planning to study RNA folding kinetics. J. Comput. Biol. 12(6), 862–881. Teodoro, M., Phillips, G. N., Jr., and Kavraki, L. E. (2001). Molecular docking: A problem with thousands of degrees of freedom. Proceedings of the 2001 IEEE International Conference on Robotics and Automation (ICRA 2001), pp. 960–966. IEEE Press, Seoul. Thomas, S., Song, G., and Amato, N. M. (2005). Protein folding by motion planning. Phys. Biol. 2(4), S148–S155. Vajda, S., Sippl, M., and Novotny, J. (1997). Empirical potentials and functions for protein folding and binding. Curr. Opin. Struct. Biol. 7(2), 222–228. Vilenkin, N. J., and Klimyk, A. U. (1991). Representation of Lie Group and Special Functions, Vols. 1–3. Kluwer Academic Publishers, The Netherlands. Vucetic, S., Obradovic, Z., Vacic, V., Radivojac, P., Peng, K., Iakoucheva, L. M., Cortese, M. S., Lawson, J. D., Brown, C. J., Sikes, J. G., Newton, C. D., and Dunker, A. K. (2005). DisProt: A database of protein disorder. Bioinformatics 21(1), 137–140. Wang, Y., and Chirikjian, G. S. (2004). Workspace generation of hyper-redundant manipulators as a diffusion process on SE(N). IEEE Trans. Robot. Autom. 20(3), 399–408. Wang, Y., and Chirikjian, G. S. (2008). Nonparametric second-order theory of error propagation on the Euclidean group. Int. J. Rob. Res. 27(1112), 1258–1273. Wang, J. Y., and Crippen, G. M. (2004). Statistical mechanics of protein folding with separable energy functions. Biopolymers 74(3), 214–220. Wang, C. S. E., Lozano-Perez, T., and Tidor, B. (1998). AmbiPack: A systematic algorithm for packing of macromolecular structures with ambiguous distance constraints. Proteins: Struct. Funct. Genet. 32(1), 26–42. Yang, D., and Kay, L. E. (1996). Contributions to conformational entropy arising from bond vector fluctuations measured from NMR-derived order parameters: Application to protein folding. J. Mol. Biol. 263, 369–382.

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Zhang, M., White, R. A., Wang, L., Goldman, R., Kavraki, L. E., and Hassett, B. (2005). Improving conformational searches by geometric screening. Bioinformatics 21(5), 624–630. Zhang, J., Lin, M., Chen, R., Wang, W., and Liang, J. (2008). Discrete state model and accurate estimation of loop entropy of RNA secondary structures. J. Chem. Phys. 128, 125107. Zhou, H.-X. (2001). Loops in proteins can be modeled as worm-like chains. J. Phys. Chem. B 105, 6763–6766. Zhou, Y., and Chirikjian, G. S. (2003). Conformational statistics of bent semiflexible polymers. J. Chem. Phys. 119(9), 4962–4970. Zhou, Y., and Chirikjian, G. S. (2006). Conformational statistics of semi-flexible macromolecular chains with internal joints. Macromolecules 39(5), 1950–1960.

C H A P T E R

F I V E

Inferring Functional Relationships and Causal Network Structure from Gene Expression Profiles Radhakrishnan Nagarajan* and Meenakshi Upreti† Contents 134 136

1. Introduction 2. Methods 2.1. Synthetic two-gene network modeled as a first-order bivariate VAR 2.2. Diagnostic tests 3. Results 3.1. Impact of noise variance on GC analysis of a synthetic two-gene network 3.2. GC analysis of cell-cycle gene expression profiles 4. Conclusions References

136 140 141 141 143 144 145

Abstract Inferring functional relationships and network structure from the observed gene expression profiles can provide a novel insight into the working of the genes as a system or network as opposed to independent entities. Such networks may also represent possible causal relationships between a given set of genes, hence can prove to be a convenient abstraction of the underlying signaling mechanism. The discovery of functional relationships from the observed gene expression profiles does not rely on prior literature, hence useful in identifying undocumented relationships between a given set of genes. Several techniques have been proposed in the literature. The present study investigates the choice Granger causality (GC) and its extensions in modeling the network structure between a given pair of genes from their expression profiles. The impact of noise variance on GC relationships is investigated. VAR parameter estimation is proposed to obtain a finer insight into the functional relationships inferred * Division of Biomedical Informatics, University of Arkansas for Medical Sciences, Little Rock, Arkansas, USA Division of Radiation Oncology, University of Arkansas for Medical Sciences, Little Rock, Arkansas, USA

{

Methods in Enzymology, Volume 487 ISSN 0076-6879, DOI: 10.1016/S0076-6879(11)87005-9

#

2011 Elsevier Inc. All rights reserved.

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using GC tests. The results are presented on synthetic networks generated from known vector-autoregressive (VAR) models and those from cell-cycle gene expression profiles that can be modeled as a first-order bivariate VAR.

1. Introduction Classical biological studies had focused on understanding the changes in the expression of specific genes (i.e., differential gene expression) across distinct biological states (e.g., control tissue vs. cancer tissue). Such studies, while useful, provide limited insights into the interaction between the genes. Capturing the interaction or functional relationships (Gardner et al., 2003) between genes is critical since they work in concert as a system (Kitano, 2002) and mediate specific phenotypic outcomes. Such a system-level understanding can also prove to be a convenient abstraction of the underlying signaling mechanism. Modeling the interaction by integrating the results across mutually exclusive gene expression studies can be challenging. Recent studies have encouraged differential gene expression studies in conjunction with text mining of published literature from large repositories such as PUBMED/ MEDLINE in order to arrive at empirical pathways and identify enriched pathways (Ashburner et al., 2000; Subramanian et al., 2005). These approaches integrate knowledge from heterogeneous data from disparate sources. However, discrepancy in findings across laboratories, protocols, tissues, cell lines, and species may impose constraints on constructing empirical pathways from a specific experimental study. Cross talk (Sachs et al., 2005), involvement of multiple pathways (Nagarajan et al., 2010), and noncanonical signaling mechanisms (Korswagen, 2002) also discourage a straightforward construction of pathways from mutually exclusive studies. The above issues are compelling enough to explore alternate approaches to construct gene networks and pathways. The recent explosion of high-throughput assays enable simultaneous measurement of transcriptional (e.g., microarrays), translational (e.g., protein arrays), and posttranslational (e.g., kinase arrays) activities from biological systems. Simultaneous measurement of genes is necessary, since the superposition of results of knowledge across single genes may not provide sufficient insight into their working as a system. Recent studies have used highthroughput assays in conjunction with suitable computational and statistical techniques (Butte et al., 2000; Eisen et al., 1998; Friedman, 2004; Gardner et al., 2003; Pe’er, 2005; Sachs et al., 2005) to infer the functional relationships from the observed gene expression profiles. It is important to note that computational approaches can prove to be useful initial screening tools, recommend new relationships, and confirm established relationships. However, they cannot substitute rigorous biological validation.

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Gene expression data generated from a high-throughput assay such as microarray fall under observational data. Such data sets are passive and do not necessarily lend themselves to active perturbation. This, in turn, renders the inference problem challenging. In spite of the above limitations, several studies have successfully used computational and statistical approaches to infer functional relationships especially from gene expression data under implicit assumptions. Gene expression data obtained from high-throughput assays can be with or without temporal information. Lack of temporal information provides only a snapshot of the biological process (i.e., static data). In contrast, temporal data or time series captures the evolution of the gene expression dynamics as a function of time (i.e., dynamic data). Techniques such as clustering are widely used to group static as well as dynamic gene expression profiles based on some similarity or correlation metric. The underlying hypothesis being genes that share the same expression profile may be functionally related (Eisen et al., 1998). Such an approach may also be useful in attributing possible function to expressed sequence tags (ESTs) whose expression profile is similar to a gene with known function. However, correlation metrics are symmetric and do not provide information about the directionality or causation between the genes. Correlation between a given pair of genes may also manifest after a finite delay as their regulation need not necessarily be synchronized. Popular clustering techniques are oblivious to such delays, hence have clear limitations. In the case of static data sets and replicate measurements, probabilistic approaches such as Bayesian structure learning techniques prove to be ideal candidates for modeling the network structure. Justification behind Bayesian approaches can be attributed to inherent uncertainty in the gene expression measurements across the replicates. Several Bayesian structure learning techniques have been used widely for gene network inference (Pe’er, 2005; Sachs et al., 2005). However, inferring the network structure from joint probability distributions may result in a family of networks or equivalent structures (Pe’er, 2005) as opposed to a unique network. Network inference from static data sets may also fail to accommodate possible feedback mechanisms between the genes, rendering the resulting network acyclic. On a related note, feedback mechanisms are ubiquitous in gene networks/signaling cascades and play a critical role in its control and stability. Network inference from temporal gene expression profiles or dynamic data sets alleviate some of these concerns faced in static data sets. More importantly, inference with temporal data is useful in modeling feedback relationships. Recent studies have used Granger causality (GC, Granger, 1969) and its extensions to model functional relationship from multivariate time series across distinct biological systems (Brovelli et al., 2004; Gregoriou et al., 2009; Seth et al., 2006; Sridharan et al., 2008) including gene expression profiles (Fujita et al., 2007; Guo et al., 2008; Mukhopadhyay and Chatterjee, 2007; Nagarajan, 2009; Nagarajan and Upreti, 2010). It is

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important to note that causality as defined by GC essentially represents the forecasting ability between a given pair of processes. The ability of GC to accommodate inherent delays between the processes and ease of implementation may be possible reasons for its widespread use across several scientific domains, including gene expression analysis. GC is ideally suited to investigate relationships in bivariate vector-autoregressive process (VAR). A zero magnitude of an off-diagonal element of a bivariate VAR indicates absence of relationship in the corresponding direction. However, in practical settings, statistical tests such as the F-test (Hamilton, 1994; Lutkepohl, 2006), which depends on the ratio of the mean-squared forecast errors, is used for identifying significant GC relationships from observed time series data.

2. Methods GC represents the forecasting ability between a given pair of processes. In order to reject the null hypothesis that the given process “xt does not granger cause yt” statistical tests such as F-test (Hamilton, 1994; Lutkepohl, 2006) that rely on the ratio of the mean-squared forecast errors gx ! z ¼ s0 / s1 are used, where s0 is the mean-squared forecast error in predicting yt from its own past and s1 is the mean-squared forecast error in predicting yt from its own past and the past of xt. If xt contributes to the forecasting ability, then we expect to observe a decrease in the mean-squared forecast error such that s1 < s0 and inflation of gx ! y. Since the expression of F-test statistic is a function of gx ! y, we restrict the subsequent discussion to the ratio of the mean-squared forecast errors. In the following discussion, we investigate the contribution of the VAR process parameters to the forecast errors.

2.1. Synthetic two-gene network modeled as a first-order bivariate VAR Consider the first-order bivariate VAR [15, 16], Fig. 5.1, given by        xt a1 b1 xt1 e ¼ þ t yt b2 a2 yt1 t

ð5:1Þ

In Eq. (5.1), (xt, yt) represent the mRNA expression of the genes (x, y), respectively, at time t. The diagonal elements represent autoregulatory feedback strengths a1, a2 2 R, whereas the off-diagonal elements b1, b2 2 R represent the transcriptional coupling strengths between the genes. The terms et, t represent normally distributed zero-mean uncorrelated noise with finite variance attributed to inherent uncertainties in the transcriptional mechanisms (Elowitz et al., 2002; Kaern et al., 2005) and artifacts

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b1 Y

X a1

a2

b2 ht

et

Figure 5.1 A synthetic two-gene network (Eq. 5.1) where (b1, b2) represent the coupling strengths between the genes, (a1, a2) represent the autoregulatory feedback strengths, and (et, t) represent their corresponding noise variances.

(Okoniewski and Miller, 2006; Tu et al., 2002) ubiquitous in microarray expression studies. In order to estimate whether xt Granger causes yt (i.e., x GC y), we follow the following steps: From Eq. (5.1), we have xt ¼ a1 xt1 þ b1 yt1 þ et xt ¼

1 ½b yt1 þ et  ð1  a1 BÞ 1

ð5:2Þ

where B is the backshift operator such that Bxt ¼ xt  1. Power series expansion (Eq. 5.2) converges provided ja1j < 1. From Eq. (5.1), we also have yt ¼ b2 xt1 þ a2 yt1 þ t Substituting for xt  1 from Eq. (5.2) in Eq. (5.3), we get   1 y t ¼ b2 ½b yt2 þ et1  þ a2 yt1 þ t ð1  a1 BÞ 1 b2 b1 b2 yt2 þ a2 yt1 þ et1 þ t ¼ ð1  a1 BÞ 1  a1 B

ð5:3Þ

ð5:4Þ

From Eq. (5.3), forecasting yt from the past of yt as well as xt results in yt f 0 ¼ b2 xt1 þ a2 yt1

ð5:5Þ

Therefore, the mean-squared forecast error on forecasting yt from the past of yt as well as xt from Eqs. (5.3) and (5.5) is s0 ¼ s2

ð5:6Þ

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From Eq. (5.4), forecasting yt from the past of yt alone results in yt f 1 ¼

b2 b1 yt2 þ a2 yt1 ð1  a1 BÞ

ð5:7Þ

Therefore, the error in forecasting yt from its past from Eqs. (5.4) and (5.7) is
b2 et1 þ t ¼ 1 þ a1 B þ a21 B2 þ    b2 et1 þ t ð1  a1 BÞ
¼ b2 1 þ a1 et2 þ a21 et3 þ    þ t

ð5:8Þ

From Eq. (5.8), the mean-squared forecast error on forecasting yt from its past is s1 ¼

b22 s2 þ s2 ð1  a21 Þ e

ð5:9Þ

From Eqs. (5.6) and (5.9), the ratio of the mean-squared forecast errors corresponding to xt Granger cause yt is given by gx!y ¼

s1 b22 s2e ¼ þ1 s0 ð1  a21 Þ s2

ð5:10Þ

The mean-squared forecast error corresponding to yt Granger causes xt; a similar approach is given by gy!x ¼

s1 b21 s2 ¼ þ 1; s0 ð1  a22 Þ s2e

where ja2 j < 1

ð5:11Þ

Several important observations can be made based on the functional forms of the mean-squared forecast error ratios (Eqs. 5.10 and 5.11). 

These ratios (Eqs. 5.10 and 5.11) have significant contributions from the autoregulatory feedback strengths (a1, a2) and the ratio of the noise variances (et, t) in addition to the coupling strengths (b1, b2).  The ratios (Eqs. 5.10 and 5.11) are also immune to the sign of the bivariate VAR process parameters.  As expected, the larger the coupling strength, the larger the corresponding ratio of the mean-squared forecast error (e.g., larger b2 implies larger gx ! y). Therefore, the gene with a larger coupling

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strength has a greater tendency to act as a cause compared to the gene with the smaller coupling strength in the bivariate VAR representation.  From Eqs. (5.10) and (5.11), it should also be noted that a larger value of jaij < 1, i ¼ 1, 2 implies a larger value of the corresponding ratio. Consider the case where the coupling strength and noise variance held constant across the genes, that is, b1 ¼ b2 and s2 ¼ se2. However, only one of the genes has an autoregulatory feedback, that is, a1 6¼ 0, a2 ¼ 0. For these choices of the parameters, we have gx!y ¼

gy!x 1  a21

Since ja1j < 1, gx ! y is always greater than gy ! x for identical coupling strengths and noise variances between the genes. In essence, the gene with an autoregulatory feedback has a greater tendency to act as the cause as opposed to the gene with no autoregulatory feedback for the same choice of parameters. 

Unlike autoregulatory feedback and coupling strength, the magnitude of the noise variance affects the ratio of the mean-squared forecast errors along either direction; that is, changes in the magnitude of the noise variance on a given gene affect not only gx ! y but also gy ! x and genes with larger noise variance have a greater tendency to act as the cause. Consider the expressions (Eqs. 5.10 and 5.11) where the coupling strength and autoregulatory feedback are held constant across the genes, that is, b1 ¼ b2 ¼ b and a1 ¼ a2 ¼ a. Let us assume that the noise variances are markedly different such that se ¼ ks, k > 1. For these choices of the parameters, we have gx!y ¼ gy!x ¼

b2 k2 þ1 ð 1  a2 Þ

b2 þ1 ð1  a2 Þk2

Therefore, gx!y s4 / k4 ¼ 4e gy!x s

ð5:12Þ

It is important to recall that noise can be an outcome of stochastic mechanisms underlying transcription or artifacts prevalent in high-throughput

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assays such as microarrays. These artifacts can be an outcome of the sensitivity of the probes, nonspecific binding, cross-hybridization, and measurement noise inherent in biological assays (Okoniewski and Miller, 2006; Tu et al., 2002).

2.2. Diagnostic tests VAR parameter estimation and statistical tests for identifying significant GC relationships are valid only under certain implicit assumptions (Hamilton, 1994; Lutkepohl, 2006) and can lead to spurious conclusions when these assumptions are violated. For instance, the F-test used widely for GC inference implicitly assumes the gene expression profiles of interest to be normally distributed. In the present study, we use a series of diagnostic tests as sanity checks prior to GC analysis of the expression profiles. Since we focus on gene expression profiles that can be modeled as first-order bivariate VAR, the optimal lag (topt) between any given pair of gene expression profiles was confirmed to be the one (topt ¼ 1) using Schwarz criterion [16]. Subsequently, the stability of the bivariate VAR was investigated and diagnostic tests were used to check the validity of normality assumptions and absence of serial correlation in the residuals (Lutkepohl, 2006). VAR process whose reverse characteristic polynomial had roots outside the unit circle were deemed as stable. Multivariate normality assumptions were investigated using Jarque-Bera test. Serial correlation in the residuals up to lag 10 was investigated using adjusted Portmanteau test (Lutkepohl, 2006). Pairs that passed the above diagnostics tests were subjected to tests for GC (F-test) as well as instantaneous GC (IGC, w2 test) (Lutkepohl, 2006). Pairs that exhibited significant IGC were not considered for further investigation. The significance level for all the GC tests (F-tests) was fixed at (a ¼ 0.01). A Monte-Carlo approach (Nagarajan, 2009; Nagarajan and Upreti, 2010) was subsequently used to estimate the statistical power of GC relationships generated from synthetic networks where the functional form is known. The statistical power is estimated as the number of successful rejections across (N ¼ 100) independent realizations of the bivariate time series generated from the synthetic network modeled as a bivariate VAR with nonzero coupling strengths. In the case of the cell-cycle gene expression profiles, a similar approach was used to determine what is termed as the proportion (C). Since the functional form of the bivariate VAR is unknown in the case of cell-cycle gene expression profiles, we first estimate the firstorder bivariate VAR process parameters from the given gene expression profiles. We first estimate the first-order bivariate VAR process parameters from the given gene expression profiles. Subsequently, the number of rejections (i.e., proportion) across (N ¼ 100) independent realizations of the bivariate time series generated from the estimated VAR parameters represents the proportion (C). Similar to statistical power, the proportion

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of rejections is bounded in [0, 1], with a large value indicating the possible presence of a significant GC relationship between the given pair of processes in that specific direction, whereas a small value implies the possible absence of relationship.

3. Results 3.1. Impact of noise variance on GC analysis of a synthetic two-gene network Consider the first-order bivariate VAR inspired by the example from Granger’s seminal article (Granger, 1969, Section 5) given by        xt 0 b1 xt1 e ¼ þ t ð5:13Þ yt b2 0 yt1 t Following the frequency-domain approach outlined under (Section 5 in Granger, 1969), the causality-coherence along either direction is given by Cx!y ðoÞ ¼ 

b22 s4e   s2e þ b21 s2 s2 þ b22 s2e

b21 s4   Cy!x ðoÞ ¼  s2e þ b21 s2 s2 þ b22 s2e For equal coupling strength (b1 ¼ b2 ¼ b), the ratio of the causalitycoherence takes the form Cx!y ðoÞ s4e / Cy!x ðoÞ s4

ð5:14Þ

Of interest is to note the similarity between the expressions (Eq. 5.14) in the frequency-domain to those in the time-domain (Eq. 5.12). To illustrate the impact of noise on the statistical conclusions, we consider the first-order bivariate VAR (Eq. 5.13) with identical coupling strengths (b1 ¼ b2 ¼ b) where b is a real number, zero-mean Gaussian noise (et, t), and absence of autoregulatory feedback (a1, a2 ¼ 0).        0 b xt1 e xt ¼ þ t ð5:15Þ b 0 zt1 zt t

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Stability of the bivariate VAR (Eq. 5.15) is guaranteed provided the roots of the reverse characteristic polynomial     1 0 0 b l ¼0 0 1 b 0 is such that (i.e., l1, 2 ¼  1/ b, jl1, 2j > 1);that is, the coupling strength should be such that jbj < 1. The number of samples were fixed to be the same as that of the cell-cycle gene expression profiles (i.e., n ¼ 48). The noise variance was fixed as s ¼ 1 across one of the processes yt, whereas it was scaled across the other xt by a noise factor k, such that se ¼ ks. The statistical power (a ¼ 0.01, F-test) along either direction estimated from 100 independent realizations of Eq. (5.14) as a function of the noise factor (k ¼ 1.0, 2.0, 3.0, 4.0) and coupling strength (b ¼  0.8, 0.8, 0.4, 0.2) is shown in Fig. 5.2. For equal noise variance (i.e., k ¼ 1), the statistical power corresponding to x GC y and y GC x is similar irrespective of the coupling strengths (b ¼  0.8, 0.8, 0.4, 0.2) (Fig. 5.1A–D). This is to be expected, as for these choice of the VAR process parameters the ratio of the mean-squared A

b = – 0.8

1

B

0.8

0.8

0.6

0.6

0.4

0.4

y GC x x GC y

0.2

0.2 0

0

Power

b = 0.8

1

1

2

3 b = 0.4

C

4

1 1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 1

2

3

3

4

b = 0.2

D

1

0

2

0 1 4 Noise factor (k)

2

3

4

Figure 5.2 Variation in statistical power as function of noise factor (k) for the synthetic two-gene network (Eq. 5.14) as function of coupling strength (b ¼  0.8, 0.8, 0.4, 0.2). The solid and dotted lines correspond to statistical power for y GC x and x GC y, respectively.

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forecast errors (gx ! y ¼ gy ! x), hence the statistical power is unchanged across x GC y and y GC x, from Eqs (5.10) and (5.11). However, as k increases, considerable discrepancy in the statistical power is observed along either direction. For larger coupling strengths, (i.e., b ¼  0.8, 0.8), irrespective of the sign of the coupling strength, the discrepancy in power is not pronounced for the various choices of k, Fig. 5.1A and B. However, a marked discrepancy in the statistical power as a function of increasing noise factor (k ¼ 3, 4) is observed at smaller coupling strengths (i.e., b ¼ 0.2, 0.4, Fig. 5.1C and D), respectively. This is to be expected since an increase in k increases the mean-squared forecast error gx ! y while decreasing gy ! x. Thus, GC relationship, where the noisier gene tends to be the cause, has a higher statistical power. From the above case study, it is clear that a bidirectional network with equal coupling strengths (b) can be rendered unidirectional provided there is sufficient discrepancy in the noise variance between the genes. Thus, understanding the contribution of the noise may be critical in avoiding spurious conclusion of functional relationship between a given pair of genes using GC analysis.

3.2. GC analysis of cell-cycle gene expression profiles In this section, we identify functional relationships from cell-cycle gene expression profiles (Whitfield et al., 2002) using GC analysis. We also investigate the impact of the noise variance on the GC conclusions. Only those gene expression profiles that pass the diagnostic tests (Section 2.2) and can be modeled as a first-order bivariate VAR are considered. The proportions (C) (Section 2.2) are estimated from the given gene expression profiles as well as those generated after constraining the noise variance to be equal (se ¼ s ¼ 1, k ¼ 1) sampled from a zero-mean Gaussian noise. UBE2C, TLOC1: UBE2C represents the ubiquitin-conjugating enzyme E2C and involved in the G2 phase of the cell-cycle. TLOC1 represents the translocation protein 1 and involved in the G1/S phase of the cell-cycle (Whitfield et al., 2002). This gene pair satisfied the diagnostic tests and could be modeled as a first-order bivariate VAR (Section 2.2). F-test (a ¼ 0.01) revealed statistically significant GC relationship along either directions UBE2C ! TLOC1 as well as TLOC1 ! UBE2C. The proportion of rejections across 100 independent realizations (Section 2.2) along either direction was cUBE2C ! TLOC1  0.95 and cTLOC1 ! UBE2C  0.89, indicative of a significant GC relationship along either direction, that is, bidirectional network. However, VAR parameter estimation revealed considerable discrepancy in the noise variance between these genes with sUBE2C  sTLOC1 and k  4.3 >> 1. On constraining the noise variance to be equal (k ¼ 1, sUBE2C ¼ sTLOC1 ¼ 1), the proportion of rejections along either directions changed to cUBE2C ! TLOC1  0.54 and cTLOC1 ! UBE2C  1.0. These results raise concern as to whether the high proportion estimate of UBE2C ! TLOC1 obtained from the given

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expression profile was solely due to discrepancy in noise variance sUBE2C  sTLOC1, with the noisier gene having a tendency to act as the cause. This case study elucidates the nontrivial impact of noise variance on functional relationships inferred using GC analysis. From the above analysis, the relationship TLOC1 ! UBE2C as opposed to UBE2C ! TLOC1 may be of interest. RGS3-RNPS1: RGS3 represents the regulator of G protein signaling 3 and involved in the G2 phase of the cell-cycle. RNPS1 represents the RNAbinding protein S1 and involved in the G2/M phase of the cell-cycle (Whitfield et al., 2002). This pair satisfied the diagnostic tests and had an optimal delay (topt ¼ 1), and hence can be modeled as a first-order bivariate VAR. GC analysis revealed significant GC relationship only in the direction RGS3 ! RNPS1 (a ¼ 0.01, F-test), rendering the relationship acyclic. The proportion of statistically significant GC relationships across 100 independent realizations estimated using the Monte-Carlo approach were cRGS3 ! RNPS1 ¼ 0.88 and cRNPS1 ! RGS3 ¼ 0.25. However, the noise factor from VAR parameter estimation was k  4.3, with one of the genes exhibiting higher noise variance than the other, that is, sRGS3  sRNPS1. Also, the gene with higher noise variance (i.e., RGS3) acted as the cause in the resulting acyclic approximation, RGS3 ! RNPS1. Constraining the noise variance to be equal (i.e., se ¼ s ¼ 1, k ¼ 1) completely reversed the conclusions with proportions sRGS3  sRNPS1 ¼ 0.19 and cRNPS1 ! RGS3 ¼ 0.94, respectively. Therefore, the functional relationship RGS3 ! RNPS1 on GC analysis of the given gene expression profiles should be regarded with caution. IKKa, NFkΒ: A recent study (Guo et al., 2008) reported functional relationships between these genes involved in NF-kB signaling (Fig. 3 in Guo et al., 2008) using pair-wise GC analysis. We reinvestigated this pair (IKKa, NFkB) that had nonzero expression values across the 48 time points, passed the VAR diagnostic tests, and could be modeled as a first-order bivariate VAR. However, our analysis failed to reveal significant GC in any direction (a ¼ 0.01) between these genes in contrast to the findings by Guo et al. (2008). The proportions estimated by the Monte Carlo approach across 100 independent realizations (Section 2.2) were low as expected cIKKa ! NFkB ¼ 0.21 and cNFkB ! IKKa ¼ 0.23. VAR estimation revealed discrepancy in the noise variance, with noise factor k  1.5. Constraining the noise variance to be equal (i.e., s ¼ s ¼ 1, k ¼ 1) did not sufficiently change the conclusions.

4. Conclusions Understanding the working of genes as a system is critical, as phenotypic outcomes are mediated by the concerted working of genes. Several techniques have been proposed in the literature that infers functional

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relationships between genes from their observed temporal gene expression profiles obtained from high-throughput assays. In this chapter, we briefly investigated the choice of GC analysis in inferring functional relationships between the genes that can be modeled as first-order bivariate VAR. Subsequently, the impact of the noise variance on GC results was elucidated. VAR parameter estimation was proposed to obtain a finer interpretation. The results were demonstrated on a synthetic gene network with known functional form and cell-cycle temporal gene expression profiles.

REFERENCES Ashburner, M., et al. (2000). Gene ontology: Tool for the unification of biology. Nat. Genet. 25(1), 25–29. Brovelli, A., et al. (2004). Beta oscillations in a large-scale sensorimotor cortical network: Directional influences revealed by Granger causality. Proc. Natl. Acad. Sci. USA 101, 9849–9854. Butte, A. J., et al. (2000). Discovering functional relationships between RNA expression and chemotherapeutic susceptibility using relevance networks. Proc. Natl. Acad. Sci. USA 97 (22), 12182–12186. Eisen, M. B., et al. (1998). Cluster analysis and display of genome-wide expression patterns. Proc. Natl. Acad. Sci. USA 95(25), 14863–14868. Elowitz, M. B., et al. (2002). Stochastic gene expression in a single cell. Science 297(5584), 1183–1186. Friedman, N. (2004). Inferring cellular networks using probabilistic graphical models. Science 303, 799–805. Fujita, A., et al. (2007). Time-varying modeling of gene expression regulatory networks using the wavelet dynamic vector autoregressive method. Bioinformatics 23(13), 1623–1630. Gardner, T. S., et al. (2003). Inferring genetic networks and identifying compound mode of action via expression profiling. Science 301, 102–105. Granger, C. W. J. (1969). Investigating causal relations by econometric models and crossspectral methods. Econometrica 37(3), 424–438. Gregoriou, G. G., et al. (2009). High-frequency, long-range coupling between prefrontal and visual cortex during attention. Science 324, 1207–1210. Guo, S., et al. (2008). Uncovering interactions in the frequency domain. PLoS Comput. Biol. 4(5), e1000087. Hamilton, J. D. (1994). Time Series Analysis. Princeton University Press, Princeton, NJ. Kaern, M., et al. (2005). Stochasticity in gene expression: From theories to phenotypes. Nat. Rev. Genet. 6, 451–464. Kitano, H. (2002). Systems biology: A brief overview. Science 295, 1662–1664. Korswagen, H. C. (2002). Canonical and non-canonical Wnt signaling pathways in Caenorhabditis elegans: Variations on a common signaling theme. Bioessays 24(9), 801–810. Lutkepohl, H. (2006). New Introduction to Multiple Time Series Analysis. Springer-Verlag, Heidelberg. Mukhopadhyay, N. D., and Chatterjee, S. (2007). Causality and pathway search in microarray time series experiment. Bioinformatics 23(4), 442–449. Nagarajan, R. (2009). A note on inferring acyclic network structures using Granger causality tests. Int. J. Biostat. 5, 1.

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Nagarajan, R., and Upreti, M. (2010). Granger causality analysis of human cell-cycle gene expression profiles. Stat. Appl. Genet. Mol. Biol. 9(1), 31. Nagarajan, R., et al. (2010). Functional relationships between genes associated with differentiation potential of aged myogenic progenitors. Front. Physiol. 1, 21. Okoniewski, M. J., and Miller, C. J. (2006). Hybridization interactions between probesets in short oligomicroarrays lead to spurious correlations. BMC Bioinformatics 7, 276. Pe’er, D. (2005). Bayesian network analysis of signaling networks: A primer. Science STKE 281, p14. Sachs, K., et al. (2005). Causal protein-signaling networks derived from multiparameter single-cell data. Science 308(5721), 523–529. Seth, A. K., et al. (2006). Theories and measures of consciousness: An extended framework. Proc. Natl. Acad. Sci. USA 103, 10799–10804. Sridharan, D., et al. (2008). A critical role for the right fronto-insular cortex in switching between central-executive and default-mode networks. Proc. Natl. Acad. Sci. USA 105 (34), 12569–12574. Subramanian, A., et al. (2005). Gene set enrichment analysis: A knowledge-based approach for interpreting genome-wide expression profiles. Proc. Natl. Acad. Sci. USA 102(43), 15545–15550. Tu, Y., et al. (2002). Quantitative noise analysis for gene expression microarray experiments. Proc Natl. Acad. Sci. USA 99(22), 14031–14036. Whitfield, M. L., et al. (2002). Identification of genes periodically expressed in the human cell cycle and their expression in tumors. Mol. Biol. Cell 13(6), 1977–2000.

C H A P T E R

S I X

Numerical Solution of the Chemical Master Equation: Uniqueness and Stability of the Stationary Distribution for Chemical Networks, and mRNA Bursting in a Gene Network with Negative Feedback Regulation E. S. Zeron* and M. Santilla´n†,‡ Contents 1. 2. 3. 4.

Introduction The Chemical Master Equation Irreducible Chemical Reaction Systems Stability of the Chemical Master Equation Stationary Probability Distribution 5. Two Different Algorithms to Calculate Stationary Probability Distributions for the Chemical Master Equation 6. Gene Expression with Negative Feedback Regulation 7. Concluding Remarks References

148 150 152 153 157 160 167 167

Abstract In this work, we introduce a couple of algorithms to compute the stationary probability distribution for the chemical master equation (CME) of arbitrary chemical networks. We further find the conditions that guaranty the algorithms’ convergence and the unicity and stability of the stationary distribution. Next, we employ these algorithms to study the mRNA and protein probability distributions in a gene regulatory network subject to negative feedback regulation. In particular, we analyze the influence of the promoter activation/deactivation * Centro de Investigacio´n y de Estudios Avanzados del IPN, Departamento de Matema´ticas, Av. Instituto Polite´cnico Nacional 2508, Me´xico DF, Me´xico Centro de Investigacio´n y Estudios Avanzados del IPN, Unidad Monterrey, Parque de Investigacio´n e Innovacio´n Tecnolo´gica, Apodaca, NL, Me´xico { Centre for Applied Mathematics in Bioscience and Medicine, 3655 Promenade Sir William Osler, McIntyre Medical Building, Montreal, Canada {

Methods in Enzymology, Volume 487 ISSN 0076-6879, DOI: 10.1016/S0076-6879(11)87006-0

#

2011 Elsevier Inc. All rights reserved.

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speed on the shape of such distributions. We find that a reduction of the promoter activation/deactivation speed modifies the shape of those distributions in a way consistent with the phenomenon known as mRNA (or transcription) bursting.

1. Introduction The emergence of disciplines like systems and synthetic biology has ushered a renewed interest in the development of mathematical models to study the processes taking place at the intracellular level (Breitling, 2010; Robeva, 2010). Numerous modeling approaches have been proposed to study the dynamic behavior of such systems. Nonetheless, all of them rely on the conceptualization of the cell as a complex reactor, where a huge number of interlinked chemical reactions take place. Another common feature to all modeling approaches is the assumption that the global chemical-reaction network in a cell can be theoretically decomposed into a set of relatively independently functioning subsystems, all of which have network structures of their own. Henceforth, one of the most frequently employed strategies consists in picking up a given subsystem and constructing the corresponding mathematical model from the underlying chemical-reaction network (Robeva, 2010). Most of the models developed to analyze cellular systems can be classified into one of two categories: (A) those that employ ordinary differential equations (ODE) to deterministically model the dynamics of the system’s chemical-species concentrations, and (B) those that utilize the chemical master or the chemical Langevin equation to model the time evolution of the system probability distribution function (Simpson et al., 2009). In general, deterministic models can be described as follows: Let X(t) be a vector denoting the system state; that is, the entries of X correspond to the concentrations of all the chemical species in the network to be modeled. The dynamics of X are then governed by the following ODE system: dX ¼ F ðX Þ; dt where the vectorial function F(X ) is determined by the rates of all the chemical reactions taking place within the system, as well as by the corresponding stoichiometric coefficients. As a matter of fact, the specific form of F(X ) for a given network can be derived by means of the law of mass action of chemical kinetics. The resulting ODE systems are often nonlinear and impossible to solve analytically, except for a few exceptional cases. However, the available numerical algorithms provide accurate and

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149

computationally cheap solutions in most situations. Moreover, nonlinear dynamics and bifurcation analysis are powerful analytical tools to study the dependence of the system behavior on the parameter values. In other words, there are several numerical and analytical techniques to analyze deterministic ODE models, and this makes them an effective technique to study the dynamics of chemical networks. On the other hand, deterministic models have one serious flaw. As we have mentioned, this modeling approach relies on the law of mass action, which is only strictly valid when the count of molecules involved in all chemical reactions is of the order of Avogadro’s number. Unfortunately, this is not the case in many intracellular processes. In particular, the number of protein molecules involved in gene regulation can be as low as a few dozens. In consequence, the predictions from deterministic models fail to reproduce the fluctuations in molecule numbers (also known as biochemical noise) originated by the stochastic nature of individual chemical reactions (Arkin et al., 1998; Elowitz et al., 2002; Paulsson, 2004; Pedraza and van Oudenaarden, 2005; Shahrezaei and Swain, 2008; Swain et al., 2002; Thattai and van Oudenaarden, 2001). In fact, the solution X(t) of a given ODE model corresponds in general to the average over an ensemble of identical chemical reactors (Simpson et al., 2009). Biochemical noise has become the object of numerous theoretical and experimental studies aimed at answering questions like how does noise affect cell functioning? How has the cell chemical circuitry evolved to minimize noise when it is detrimental? How is noise important for the functioning of some cellular networks? Most of these questions remain unanswered to this day, and their solution will necessarily involve an accurate modeling framework (Arkin et al., 1998; Elowitz et al., 2002; Paulsson, 2004; Pedraza and van Oudenaarden, 2005; Shahrezaei and Swain, 2008; Simpson et al., 2009; Swain et al., 2002; Thattai and van Oudenaarden, 2001). The most fundamental theoretical treatment for studying biochemical noise is the chemical master equation (CME). In it, the state of the system is defined by the vector x(t), whose entries correspond to the number of molecules of each chemical species at time t. A detailed description of the CME shall be presented in the next section, but let us say for now that it is a linear differential equation governing the time evolution of the probability P(x, t) that the system is in state x at time t. Despite its linearity, the CME is impossible to solve analytically, except for a few special cases. Therefore, one has to recur to numerical solutions and/or make use of simplifying assumptions (Simpson et al., 2009). For instance, while x(t) is a discrete variable, it can be treated as a continuous one when the mean values of the molecular counts in the system are large, and so the CME can be approximated by the so-called chemical Langevin equation (Gillespie, 2000, 2002). Another common approach

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consists in developing stochastic simulation algorithms (SSA) that are equivalent to Monte Carlo simulations of the CME. Most of the available algorithms are derivations of the celebrated algorithm proposed by Gillespie (1977). While the Gillespie algorithm is exact in the sense that it generates statistically correct trajectories, it happens to be too computationally expensive in most situations. Much work has been aimed at improving stochastic algorithms by means of sound approximations (Burrage et al., 2004; Cao et al., 2005a,b; Liu and Vanden-Eijnden, 2005; Zhou et al., 2008). Although approximation techniques show exceptional promise to accelerate stochastic simulations, understanding when to use these algorithms as well as the magnitude of the error they introduce is an area of active research. Still another strategy consists in numerically solving the CME. The advantage of this approach is that it directly returns the system probability distribution function that, in many cases, is more adequate than stochastic simulations to analyze the system dynamic behavior. Munsky and Khammash (2006, 2007) have proposed the so-called finite state projection algorithm to solve the CME, while Macnamara et al. (2008) have worked out improvements for this algorithm. In a previous work (Zeron and Santilla´n, 2010), we introduced a different algorithm, based on the Jacobi method, to numerically solve the CME. We tested such algorithm by studying the influence of negative feedback regulation on intrinsic noise in a simple gene regulatory network. Comparison with equivalent results from stochastic simulations validated the proposed algorithm. However, we were unable to provide at that time a formal demonstration for the algorithm convergence. In the present work, we expand the results in Zeron and Santilla´n (2010) by introducing a couple of algorithms to calculate the steady-state solution of the CME and analyzing their convergence. The chapter is organized as follows: In Section 2, we introduce the CME. In Section 3, we define what irreducible chemical systems are and analyze them. In Section 4, we study the existence, uniqueness, and (asymptotic) stability of the stationary distribution for irreducible chemical systems. In Section 5, we introduce two different algorithms to find the stationary probability distribution function of arbitrary chemical systems, and prove their convergence.

2. The Chemical Master Equation Consider a chemical system (or reactor) in which n different chemical species are involved in m different chemical reactions. As previously mentioned, the state of this system can be represented by a vector of natural numbers x(t) 2 Nn, such that the molecule count at time t of the ‘th chemical species is represented by the ‘th entry of the vector x(t) (for

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which we use the notation (x‘(t))). Furthermore, each one of the chemical reactions taking place in this reactor can be represented as usual: aðk;xÞ

x ! x þ v½k;

k ¼ 1; 2 . . . m;

ð6:1Þ

where a(k,x)  0 and v[k] 2 Zn denote the propensity and the stoichiometric vector of the kth reaction, respectively. The product a(k,x)dt is the probability that the kth reaction takes place during the time interval [t, t þ dt] (dt  0), while each of the entries of v[k] 2 Zn determines the change produced in the molecule count of the corresponding species by the occurrence of the kth chemical reaction. Here and thereafter, we assume that every propensity a(k,x) does not depend on time explicitly. In accordance with the fact that negative molecule counts are impossible, a(k,x) vanishes whenever any entry of x or x þ v[k] is negative. Similarly, we assume the existence of a natural number s  1 such that a(k,x) vanishes whenever any entry of x or x þ v[k] is larger than s. The above constrains can be restated as follows: a(k,x) vanishes whenever x and/or x þ v[k] are not in Sn, with S :¼ f0; 1; 2; . . . ; sg:

ð6:2Þ

Let P(x,t)  0 be the probability that the system state at time t is x. In agreement with the constrains discussed in the previous paragraph, we assume that P(x,t) vanishes whenever x 2 = Sn. From the previous definitions, the time evolution of P(x,t)  0 is governed by the so-called CME (or Forward Kolmogorov equation): m @P ðx; tÞ X ½aðk; x  v½kÞP ðx  v½k; t Þ  aðk; xÞP ðx; tÞ: ¼ @t k¼1

ð6:3Þ

For a deeper discussion, see Gillespie (1976, 1977, 1992) or Higham (2008). The normalization condition for the probability distribution function P(x,t) implies that X P ðx; tÞ ¼ 1: x2Sn

It further follows from this that X @P ðx; tÞ x2Sn

@t

 0:

The last result is in agreement with the fact that

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X

aðk; x  v½kP ðx  v½k; tÞ ¼

x2Sn

X

aðk; xÞP ðx; tÞ:

x2Sn

Finally, by definition, any stationary distribution P*(x) must satisfy @P*(x)/ @t ¼ 0, and so P ðxÞ

m X j¼1

aðx; jÞ ¼

m X

aðk; x  v½kÞP ðx  v½kÞ:

ð6:4Þ

k¼1

3. Irreducible Chemical Reaction Systems A chemical reaction system is said to be irreducible if and only if, given any arbitrary pair of states x and y in Sn, there exists a finite set of reactions of the form   aðkj ;xj Þ xj ! xjþ1 :¼ xj þ v kj ;

j ¼ 1; 2 . . . q;

such that a(kj,xj) > 0 for all j. Moreover x1 ¼ x, and xqþ 1 ¼ y. In other words, aðkq ;xq Þ aðk1 ;x1 Þ aðk2 ;x2 Þ aðk3 ;x3 Þ x ¼ x1 ! x2! x3! ! xqþ1 ¼ y:

ð6:5Þ

The adjective irreducible originates from the fact that the matrix representing the set of chemical reactions in Eq. (6.1) is irreducible for these systems. To see this, let us sort the elements of the finite set Sn using the lexicographic order. Then, the stationary distribution P*(x) can be seen as a vector, where each element x 2 Sn is an index of P*. The chemical reactions represented in Eq. (6.1) can be similarly codified into a square matrix whose entries A[x, y] are given by  aðk; xÞ; if y ¼ x þ v½k for some k; A½x; y :¼ ð6:6Þ 0; otherwise: Notice that a given entry A[x, y] is strictly positive if and only if there is a chemical reaction in Eq. (6.1) that takes the system from state x into state y and has a strictly positive propensity a(k,x) ¼ A[x, y]. It is straightforward to see that the system of chemical reactions in Eq. (6.1) is irreducible if and only if the matrix A in Eq. (6.6) is irreducible according to the characterization given in Bapat and Raghavan (1997, p. 4) or Berman and Plemmons (1979,

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p. 30); the reason being that the associated graphic G(A) is strongly connected via the paths (Eq. (6.5)) transforming the system from any arbitrary state x into any other arbitrary state y. We shall use later the fact that the transpose matrix A0 is irreducible if and only if A is irreducible—see for example Bapat and Raghavan (1997, p. 4) or Berman and Plemmons (1979, p. 27). The irreducibility of a matrix only depends on which entries are zero or strictly positive, and so it is invariant under the multiplication of every entry by strictly positive numbers. Consider, for instance, the invertible diagonal matrix  dðxÞ; if y ¼ x; D½x; y :¼ 0; otherwise; where d(x) > 0 is a collection of positive numbers indexed by x 2 Sn. Then, the entries of the product AD are given by ðDAÞ½x; y ¼ dðxÞA½x; y: Since any entry (DA)[x, y] vanishes if and only if A[x, y] also vanishes, it is easy to prove that the matrix A is irreducible if and only if DA is irreducible. The importance of analyzing irreducible chemical reaction systems arises from the fact that, as we shall see later, these systems have a unique stationary distribution.

4. Stability of the Chemical Master Equation Stationary Probability Distribution We begin by regarding the CME as a differential equation where the dependent variable is a matrix. Let us endow the set Sn with the lexicographic order and define a pair of square matrices D and B (D being diagonal) as follows:  aðk; y^Þ; if y^ ¼ x^  v½k for some k; B½x^; y^ :¼ ð6:7Þ 0; otherwise; and D½x^; y^ :¼

8X m < að j; x^Þ; :

if x^ ¼ y^;

j¼1

0;

otherwise:

ð6:8Þ

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Notice that the entries D[^ x, y^] and B[^ x, y^] are indexed by the vectors x^ and y^ in Sn. The diagonal matrix D is invertible because for every state x^ 2 Sn there exists at least one strictly positive propensity function a(^ x, j) > 0 for some index j. Furthermore, observe that the matrix A in Eq. (6.6) is nothing else but the transpose of B in Eq. (6.7), since  aðk; xÞ; if y ¼ x þ v½k; 0 B ½x; y ¼ B½y; x ¼ A½x; y ¼ ð6:9Þ 0; otherwise: Finally, it follows from the definitions of D and B that the CME can be rewritten as m @P ðx; tÞ X ½aðk; x  v½kÞP ðx  v½k; tÞ  aðk; xÞP ðx; t Þ ¼ @t k¼1 P ¼ y2Sn ½Bðx; yÞ  Dðx; yÞP ðy; tÞ:

ð6:10Þ

The analysis below is strongly based on the fact that D 1A is a stochastic matrix according to the definitions in Meyer (2000, p. 687) and Seber (2008, p. 212). That is, all the entries of this matrix are nonnegative and the sum of all elements in a given row equals one: X y2Sn

" #1 " # m m X X  D A ½x; y ¼ að j; xÞ aðk; xÞ ¼ 1: 1

j¼1

ð6:11Þ

k¼1

As a consequence of the above fact, each eigenvalue w of D 1A has an absolute value smaller than or equal to one—see Seber (2008, p. 212) and notice that jwj jwpj þ p 1, with p the minimum value in the main diagonal of D 1A. On the other hand, w0 ¼ 1 is a semisimple eigenvalue of D 1A—see Meyer (2000, p. 696). That is, the Jordan blocks associated with w0 ¼ 1 are all trivial, that is, there is an invertible matrix M satisfying the following equation:   1 1 Id 0 D A¼M M; ð6:12Þ 0 S where Id is the identity matrix and S is an upper-diagonal matrix such that no element in the main diagonal is equal to one—see Elaydi (1999, p. 143), Meyer (2000, p. 510), or Seber (2008, p. 92). Finally, given that P*(x) is a solution of Eq. (6.4) and so it satisfies BP* ¼ DP*, every stationary distribution P*(x)  0 lies in the kernel of B  D. In the same way, each nonzero

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eigenvector V(x)  0 in the kernel of B  D yields a stationary distribution after renormalization: V ðxÞ P ðxÞ ¼ X V ðyÞ  0;

so that

X

P ðxÞ ¼ 1:

ð6:13Þ

x2Sn y2Sn

We have seen that a stationary probability function P*(x) can be constructed from every nonzero vector V(x)  0 in the kernel of B  D. In the following theorem, we prove the existence of at least one such a vector. Theorem 6.1 The identity jwj 1 holds for every eigenvalue w of the matrix product BD 1 defined in Eqs. (6.7) and (6.8). Each nonzero eigenvalue l 6¼ 0 of the difference B  D has a negative real part: ℛ (l) < 0. There exists at least one nonzero nonnegative vector V(x)  0 in the kernel of B  D. Finally, l1 ¼ 0 is always a semisimple eigenvalue of B  D, and it is a unique eigenvalue when the system of chemical reactions (Eq. (6.1)) associated with B  D is irreducible according to the definition given in Section 5. Proof We begin by showing that every nonzero eigenvalue l 6¼ 0 of B  D has a negative real part: ℛ(l) < 0. For the sake of simplicity, we analyze the transpose B0  D0 ¼ A  D instead of B  D, and use the fact that both of them have the same eigenvalues. Notice that D0 ¼ D since this matrix is diagonal, and recall that A0 ¼ B according to Eq. (6.9). Let l be any eigenvalue of A  D and let U 6¼ 0 be its associated eigenvector. Hence, AU ¼ lU þ DU. Choose x^ 2 Sn such that jU(^ x)j  jU(x)j holds for every index x 2 Sn (observe that U(^ x) 6¼ 0 because U 6¼ 0). Therefore, m X X aðk; x^Þ jU ðx^Þj ¼ A½x^; yU ðyÞ l þ y2Nn k¼1 " # m m X X aðk; x^ÞU ðx^ þ v½kÞ

aðk; x^Þ jU ðx^Þj: ¼ k¼1 k¼1 Since a(k, x^)  0 for all k and U(^ x) 6¼ 0, the inequality in the previous equation may only hold if l ¼ 0 or ℛ(l) < 0. We showed in Eq. (6.11) that D 1A is a stochastic matrix (i.e., it is nonnegative and the sum of all the elements in a given row equals one). We also have that BD 1 and its transpose D 1A have exactly the same eigenvalues and Jordan forms. That is, w0 ¼ 1 is a semisimple eigenvalue of BD 1

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and jwj 1 for every eingenvalue w of BD 1—see Meyer (2000, p. 696) and Seber (2008, p. 212). Since BD 1 is also nonnegative, the Perron– Frobenius theorem grants the existence of a nonzero eigenvector U(x)  0 such that (see Bapat and Raghavan, 1997, p. 35 or Serre, 2002, p. 81)  1  BD U  U and ðB  DÞD1 U  0: ð6:14Þ The last equation further implies that the vector V ¼ D 1U lies in the kernel of B  D. Furthermore, V is nonzero and nonnegative given that, according to Eq. (6.8), D is a diagonal matrix with strictly positive entries in the main diagonal. Since w0 ¼ 1 is a semisimple eigenvalue of BD 1, the Jordan blocks associated with w0 ¼ 1 are all trivial according to Eq. (6.12)— see also Elaydi (1999, p. 143), and so there exists a square invertible matrix M satisfying   0 1 1 Id BD ¼ M M; 0 T where Id is the identity matrix and T is a lower diagonal matrix such that no element in its main diagonal equals one. This result further implies that   0 1 0 BD¼M MD: 0 T  Id Notice that MD is invertible and T  Id is lower diagonal with no zero element in its main diagonal. Therefore, the number of zero eigenvalues l1 ¼ 0 of the matrix B  D is equal to the dimension of its kernel, and so l1 ¼ 0 is a semisimple eigenvalue of B  D—see Elaydi (1999, p. 143), Meyer (2000, p. 510), or Seber (2008, p. 92). Finally, the matrix A in Eq. (6.6), the product D 1A, and its transpose 1 BD , are all irreducible whenever the system of chemical reactions (Eq. 6.1) is irreducible according to the definition given in Section 3. It then follows from the Perron–Frobenius theorem—see Bapat and Raghavan (1997, p. 17), Berman and Plemmons (1979, p. 27), or Serre (2002, p. 82)—that the eigenvalue w0 ¼ 1 of BD 1 and the eigenvalue l1 ¼ 0 of B  D are both unique. Recall that the number of eigenvalues l1 ¼ 0 of the matrix B  D, the dimension of its kernel, and the number of eigenvalues w0 ¼ 1 of BD 1 are all equal. □ To end this section, the foregoing theorem gives the sufficient conditions for the existence, uniqueness, and stability of the stationary distributions P*(x) of the system in Eqs. (6.3)–(6.10). We wish to emphasize that this is the main result of the present section.

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Theorem 6.2 The system (Eqs. (6.3)–(6.10)) is stable and has at least one nonzero stationary distribution P*(x)  0 satisfying Eq. (6.4). Furthermore, the stationary distribution P*(x) is unique and asymptotically stable whenever the system of chemical reactions (Eq. 6.1) is irreducible in the sense of Section 3. Proof Theorem 6.1 implies that the real part ℛ(l) of any nonzero eigenvalue l of the square matrix B  D defined in Eqs. (6.7) and (6.8) is strictly negative. Moreover l1 ¼ 0 is a semisimple eigenvalue of B  D. Then, the results in Amann (1990, p. 201) or Brugnano and Trigiante (1998, p. 4) imply that any stationary distribution of the system (Eqs. (6.3)–(6.10)) is stable. Theorem 6.1 also grants the existence of at least one nonzero nonnegative vector V(x)  0 that lies in the kernel of B  D. A stationary distribution P*(x) can then be constructed by normalizing each V(x)—see Eq. (6.13). On the other hand, the eigenvalue l1 ¼ 0 of B  D is also unique when the system of chemical reactions (Eq. (6.1)) is irreducible according to the definition given in Section 3. This finally implies that the distribution P*(x) is unique and asymptotically stable. The unicity follows from the fact that the kernel of B  D has unitary real dimension. □

5. Two Different Algorithms to Calculate Stationary Probability Distributions for the Chemical Master Equation We introduce now an algorithm, based on the Jacobi method, to calculate the steady-state probability distribution P*(x) of an irreducible system. Let p^0 ðxÞ  0 be an arbitrary initial distribution and define the sequence of nonnegative functions p^q ðxÞ  0 via the following iterative process: p^qþ1 ðxÞ :¼

X B½x; y^ pq ðyÞ y2Sn

D½x; x

¼

m X aðk; x  v½kÞ k¼1

D½x; x

p^q ðx  v½kÞ;

ð6:15Þ

where B and D are given by Eqs. (6.7) and (6.8), respectively. Then, p^qþ1

  ¼ D1 B p^q

with D½x; x ¼

m X j¼1

að j; xÞ > 0:

ð6:16Þ

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It follows from Eq. (6.16) that p^q ¼ ðD1 BÞ p^0 for every integer q  0. It is important to emphasize that the distributions p^q ðxÞ  0 do not need to be normalized because the 1-norms k D^ pqþ1 k and k D^ pq k coincide. That is, q

X x2Sn

D½x; x^ pqþ1 ðxÞ ¼

m X X

aðk; yÞ^ pq ðyÞ ¼

k¼1 y2Sn

X

D½y; y^ pq ðyÞ:

y2Sn

In the following theorem, we study the sufficient conditions for the convergence of p^q ðxÞ, as q ! 1, to a positive multiple of a stationary distribution P*(x) satisfying Eq. (6.4). Theorem 6.3 Let p^0 ðxÞ  0 be an arbitrary initial distribution. The sequence of nonnegative functions p^q ðxÞ  0 defined via the iterative process (Eqs. (6.15)–(6.16)) converges to a positive multiple of a stationary solution p^ ðxÞ  0 in Eq. (6.4) as p ! 1 and the following two conditions hold A. The vector D^ p0  0 has nontrivial projection onto the eigenspace associated with the eigenvalue w0 ¼ 1 of BD 1. B. w0 ¼ 1 is the only eigenvalue w of BD 1 (counting multiplicity) with absolute value jwj ¼ 1. The fulfillment of condition A is immediate because, given any initial distribution p^0 ðxÞ, the projection of D^ p0 onto the eigenspace associated with the eigenvalue w1 ¼ 1 is nontrivial with probability one. Moreover, Theorem 6.1 and the existence of a nonzero eigenvector U  0 satisfying Eq. (6.14) together imply that w1 ¼ 1 is an eigenvalue of BD 1. Proof Notice that D^ pq ¼ ðBD1 Þ D^ p0 for every integer q  1. Theorem 6.1 and condition B automatically imply that the eigenvalues wj of BD 1 can be organized in a decreasing order as follows: q

w0 ¼ 1 > jw1 j  jw2 j  jw3 j  The power method (see Groetsch and King, 1988, p. 274 or Golub and Van Loan, 1996, pp. 330–332) implies that p^q ðxÞ  0 converges to a nonzero vector V(x)  0 as q ! 1. In consequence, DV is an eigenvector associated with the maximal eigenvalue w0 ¼ 1 of BD 1. To obtain the stationary solution P*(x), we only need to normalize V(x)  0 using Eq. (6.13), the reason being that  1  BD DV  DV ; if and only if ðB  DÞV  0: h

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In order to assure that the sequence p^q converges as q ! 1, it is essential that the matrix BD 1 has a unique eigenvalue l with absolute value jlj ¼ 1. To see this necessity, consider the following example. Let r be an arbitrary real number in the interval [0,1], take the following matrices B and D, and use the following initial distribution p^0 :    

 0 1 1 0 1r B :¼ ; D :¼ ; and p^0 :¼ : 1 0 0 1 r It is straightforward to prove that the eigenvalues of BD 1 are 1, and that the iteratively calculated functions p^q oscillate between



 1r r and : r 1r To finish this section, we introduce a second algorithm (also based on the Jacobi method) to calculate the stationary distribution P*(x). The advantage of this second algorithm is that it does not need condition B in Theorem 6.3. Let p^0 ðxÞ  0 be an arbitrary initial distribution and let p0(x)  0. Define the function sequences p^q ðxÞ  0 and pq(x)  0 via the iterative processes pqþ1 ðxÞ :¼ pq ðxÞ þ p^q ðxÞ;

p^qþ1 ðxÞ :¼

m X aðk; x  v½kÞ k¼1

D½x; x

p^q ðx  v½kÞ; ð6:17Þ

where D[x, x] ¼ Sjm¼ 1 a( j, x) > 0 is as given by Eq. (6.8) or Eq. (6.16). In the next theorem, we prove that pq(x)/q always converges, as q ! 1, to a positive multiple of a stationary distribution P*(x) satisfying Eq. (6.4). Theorem 6.4 Let p^0 ðxÞ  0 be an arbitrary distribution and let p0(x)  0. The sequence of nonnegative functions pq(x)/q  0 defined via the iterative process (Eq. 6.17) converges to a positive multiple of a stationary solution P*(x)  0 satisfying Eq. (6.4) as q ! 1, provided the following condition holds: A. The vector D^ p0  0 has nontrivial projection onto the eigenspace associated with the eigenvalue w0 ¼ 1 of BD 1. The fulfillment of condition A is immediate because, given any initial distribution p^0 ðxÞ  0, the projection of D^ p0 onto the eigenspace associated with the eigenvalue w0 ¼ 1 is nontrivial with probability one.

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Proof Consider the matrices A, B, and D defined in Eqs. (6.6)–(6.8). We showed in Eq. (6.11) that the product D 1A is a stochastic matrix because every entry of D 1A is nonnegative and the sum of all elements in a given row is equal to one (see Bapat and Raghavan, 1997, p. 45 or Meyer, 2000, p. 687). Furthermore, the theorem 1.9.7 in Bapat and Raghavan (1997, p. 50) (see also Meyer, 2000, p. 697) implies the existence of a square matrix Q such that Q ¼ lim

q!1

q1 k X ðD1 AÞ

q

k¼0

and

  Q D1 A ¼ Q:

ð6:18Þ

Since D is diagonal and A0 ¼ B—according to Eq. (6.9)—the transpose Q0 is equal to BD 1Q0 . Let us define now the square matrix Y :¼ D 1Q0 D. Then, Y ¼ lim D1 q!1

q1 k X ðBD1 Þ

q

k¼0

D ¼ lim

q!1

q1 k X ðD1 BÞ k¼0

q

:

On the other hand, it is straightforward to prove that pq ¼

q1 X 

k D1 B p^0  0:

k¼1

Finally, the two previous equations imply that pq/q converges to a nonnegative vector V :¼ Y^ p0 as q ! 1. To obtain the stationary distribution P*(x), we only need to normalize V(x)  0 using Eq. (6.13). The rationale behind this assertion is that DV ¼ DY^ p0 ¼ Q0 D^ p0 ¼ BD1 Q0 D^ p0 ¼ BY^ p0 ¼ BV : Recall that Y ¼ D 1Q0 D and Q0 ¼ BD 1Q0 .

□

6. Gene Expression with Negative Feedback Regulation In this section, we introduce a simple gene regulatory network with negative feedback regulation at the transcriptional level, and analyze its stochastic dynamic behavior by means of the previously introduced

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techniques. A schematic representation of this system is given in Fig. 6.1. We assume that the promoter can be either active (Da) or inactive (Di), that transcription can only be started in—and so that mRNA molecules (M) can only be synthesized from—an active promoter, that proteins P—translated from mRNA molecules—cause an increase in the number of metabolite molecules (R) either by catalyzing their production or by uptaking them from the environment, and that this metabolite interacts with the transcription initiation complex turning the promoter inactive. Riboswitch B12 is a good example for the type of regulation illustrated in Fig. 6.1. In this case, one of the genes under regulation of the corresponding promoter codes for a protein is involved in the uptake of vitamin B12. This vitamin further interacts with the transcription initiation complex, prematurely terminating transcription and avoiding the production of mature mRNA. (Santilla´n and Mackey, 2005).

gr

kr

Inactive promoter (Di) Metabolites (R) kd–

kd+

Proteins (P)

mRNA (M) kp

km Active promoter (Da) gm

gp

Figure 6.1 Schematic representation of a simple gene network with negative feedback regulation. See the main text for more details.

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As for the parameters in Fig. 6.1, km is the probability per unit time of transcription, kp the probability per unit time of translation, gm the probability per unit time of mRNA degradation, gp is the probability per unit time of protein degradation, kr denotes the probability per unit time that a protein P catalyzes the production of a molecule R or uptakes it from the environment, gr is the probability per unit time of metabolite degradation, kdþ the probability per unit time that a molecule R binds the active promoter, and kd is the probability per unit time of the R:Da complex dissociation. Of all the reactions depicted in Fig. 6.1, the synthesis and degradation of metabolites are much faster processes than transcription, translation, mRNA degradation, protein degradation, and the promoter flipping between states Da and Di. If we take into account that the metabolite molecule count is of the order of tens of thousands, then its number does not change noticeably when one of them binds Da. From all these considerations, we can perform a separation of time scales to make an adiabatic approximation—emulating (Zeron and Santilla´n, 2010)—as follows. Notice that only the metabolite count nR is affected by the fast reactions. For the sake of convenience, we introduce the variable y ¼ nR. Let us denote the number of active promoters as nD, the mRNA count as nM, and the protein count as nP. Then, the vector x ¼ (nD, nM, nP) represents the state of all the chemical species affected by the slow reactions. According to Zeron and Santilla´n (2010), the dynamics of this system are governed by the following reduced CME m @P ðx; tÞ X ½aðk; x  v½kÞP ðx  v½k; tÞ  aðk; xÞP ðx; t Þ; ¼ @t k¼1 0

ð6:19Þ

where m0 ¼ 6 is the number of slow reactions, while a(k, x) is the effective propensity of the kth reaction, given the state x of the slow subsystem. The effective propensities are given on their own by X aðk; xÞ ¼ bðk; x; yÞP ðyjx; 1Þ; ð6:20Þ y with b(k, x, y) the propensity of the kth reaction and P(yjx, 1) the steadystate probability distribution for the fast variable y, given the state x of the slow subsystem. The propensity for the metabolite-synthesis reaction is krnP, while that for the metabolite degradation reaction is grnR. Therefore, these processes are equivalent to those of mRNA synthesis and degradation in a constitutive gene, and so the stationary distribution for nR (given the protein count nP) is given by the following Poisson distribution (Shahrezaei and Swain, 2008; Thattai and van Oudenaarden, 2001):

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Numerical Solution of the Chemical Master Equation

P ðnR jnP ; 1Þ ¼

lnR el ; nR !

where l ¼ krnP/gr. With this, the effective-propensity vector for the slow reaction subsystem comes out to be 0

1 k d ð1  nD Þ B þ kr nP C B kd nD C B gr C B C C aðxÞ ¼ B B km nD C: B g nM C m B C @ kp nM A gp nP

ð6:21Þ

The rows in this propensity vector respectively correspond to the following reactions: promoter activation, promoter inactivation, mRNA synthesis, mRNA degradation, protein synthesis, and protein degradation. In the derivation of the above propensities, we have used the fact that the mean of the variable nR is given by 1 X nR ¼0

nR

lnR el kr nP ¼ l¼ : nR ! gr

On the other hand, the stoichiometric matrix corresponding to the slow reaction subsystem is 0 1 1 0 0 B 1 0 0 C B C B 0 1 0 C B C: v¼B ð6:22Þ C 0 1 0 B C @ 0 0 1 A 0 0 1 The rows in this matrix correspond to the different chemical reactions in the system, as in the propensity vector (Eq. (6.21)), while each column determines the change produced by the system chemical reactions on the molecule count of the following chemical species: active promoter, mRNA, and protein. To perform the forthcoming analysis, we consider the parameter values estimated in Zeron and Santilla´n (2010) (where a very similar system is studied) and tabulated in Table 6.1.

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Table 6.1 Parameter values for the gene regulatory network here analyzed

km  1.0 min 1 kp  10.0 min 1 kr/gr  100 molecules

gm  1.0 min 1 gp  0.1 min 1

Table 6.2 Combinations of kd and kdþ values employed in this work

kd(min 1) 500 50 5 0.5 0.05 0.005

kdþ(min 1molecules 1) 1 0.1 0.01 0.001 0.0001 0.00001

As for parameters, kdþ and kd, Zeron and Santilla´n (2010) estimated that KD ¼

  k 6 d þ 2 1; 10 molecules; kd

where KD ¼ 1 molecule corresponds to a very strong feedback loop, while KD ¼ 106 molecules corresponds to a very week feedback loop. Here, we consider KD ¼ 5000 molecules, which represents an intermediately strong negative feedback regulation. Zeron and Santilla´n (2010) implicitly assumed that kd, kdþ  1, and so regarded the promoter activation and deactivation reactions as fast ones while reducing the system by means of an adiabatic approximation. In the present work, we dismiss such assumption and consider the combinations of kd and kdþ tabulated in Table 6.2, all of which satisfy that kd / kdþ ¼ 5000 molecules. Since KD remains constant, the probability that the promoter is active at a given time is the same in all cases. However, the activation and deactivation processes become slower as kd decreases. Therefore, by considering all the kd values in Table 6.2, we can investigate the influence of slowing down the promoter flipping between Da and Di on the mRNA and protein probability distributions. We implemented both of the algorithms introduced in Section 5 in Python to find the stationary probability distribution functions for the system given by Eqs. (6.19), (6.21), and (6.22), with the parameter values tabulated in Tables 6.1 and 6.2. Since both algorithms rendered exactly the same results, we only plot one curve per each set of parameters in Fig. 6.2.

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A

0.8

Probability

0.6

0.4

0.2

0 0 B

1

2 3 mRNA count

4

5

0.025

Probability

0.02 0.015 0.01 0.05 0

0

50

100 Protein count

150

200

Figure 6.2 Stationary probability distribution functions for the mRNA (A) and protein (B) counts, corresponding to the system represented in Fig. 6.1, with the parameter values given in Tables 6.1 and 6.2. The plot colors correspond to the different kd values as follows: kd ¼ 500 min  1, black; kd ¼ 50 min  1, blue; kd ¼ 5 min  1, red; kd ¼ 0.5 min  1, green; kd ¼ 0.05 min  1, magenta; and kd ¼ 0.005 min  1, brown. Finally, the distributions plotted in yellow were calculated with the model proposed by Zeron and Santilla´n (2010), in which the promoter activation and deactivation processes are regarded as fast ones while making an adiabatic approximation to simplify the model—see the main text for details.

Note that the distributions corresponding to kd ¼ 500 min  1 and ¼ 50 min  1 overlap, and both of them agree with the simulation from the model analyzed in Zeron and Santilla´n (2010), in which the promoter activation and deactivation are regarded as fast processes. Nonetheless, as kd and kdþ decrease (keeping constant the ratio KD ¼ kd / kdþ), the promoter activation and deactivation processes turn slower, and both

kd

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the mRNA and the protein distribution functions change. The probability of finding three or more mRNA molecules increases, as well as that of finding no molecules at all. In contrast, the probabilities for one or two mRNA molecules decrease as the promoter activation and deactivation processes slow down. Regarding the protein distribution functions, they widen as kd and kdþ decrease. This happens because the probabilities for very low and very high protein counts increase while kd and kdþ decrease. In particular, we see that the protein probability distribution function becomes bimodal for very low kd and kdþ values. This phenomenon can be explained by considering that the promoter flips between the active and inactive states, but stays for long periods of time in each one of them. Thus, when the promoter is active, many transcriptions are initiated, and so the mRNA and protein counts highly increase. Nonetheless, when the promoter is inactive, no transcription is initiated for a long period of time, and so the mRNA and protein counts decrease because of degradation. The above-described behavior is congruent with the phenomenon known as mRNA (or transcription) bursting (Golding et al., 2005). New experimental techniques allowing the investigation of gene expression at the single molecule level have demonstrated that mRNA synthesis takes place in short periods of high activity, followed by long inactive intervals. Statistically, mRNA bursting induces a widening of the expected mRNA probability distribution function. That is, the observed standard deviation is larger than expected, given the average mRNA count. mRNA bursting has been observed both in eukaryotic and prokaryotic cells. In eukaryotic cells, chromatin remodeling has been invoked as the cause of transcription bursting. However, no satisfactory explanation exists for this phenomenon in prokaryotic cells. In agreement with the possible causes speculated by Golding et al. (2005), our results suggest that mRNA bursting can be explained by the promoter activation and deactivation processes being very slow. The possible causes for the slowing down of the promoter flipping between the active and inactive states is an issue that deserves further investigation. For instance, Choi et al. (2008) argue that, in the lac operon, it can be due to the simultaneous interaction of a repressor molecule with two separate operator sites along the DNA chain. From this consideration, it could be possible that other instances of cooperativity, like that observed between two different repressors in the phage l switch (Dodd et al., 2005) or in the tryptophan operon (Grillo et al., 1999), can also lead to mRNA bursting. Finally, one interesting prediction from our results is the fact that, when promoter activation and deactivation become too slow, the stationary probability distribution function for proteins becomes bimodal. Would it be possible to find in nature or to engineer promoters showing this bimodal behavior, their biological implications would be worth studying.

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167

7. Concluding Remarks Summarizing, we have introduced in this work a couple of algorithms to find the stationary probability distribution for the CME of arbitrary chemical networks. Furthermore, we found the conditions that guaranty the algorithms’ convergence and the unicity and (asymptotic) stability of the stationary distribution. The most popular approach employed in the study of the so-called biochemical noise in chemical-reaction networks involves the obtention of individual trajectories by means of the Montecarlo method. However, this procedure becomes impractical to calculate stationary probability distributions, since plenty individual simulations are required for that purpose. In that sense, the approach here introduced provides an efficient and more economical way to directly find the system stationary probability distribution. We further employed the proposed algorithms to study the mRNA and protein probability distributions in a gene regulatory network subject to negative feedback regulation. In particular, we analyzed the influence of the promoter activation/deactivation speed on the shape of such distributions. We found that a reduction of the promoter activation/deactivation speed modifies the shape of those distributions in a way consistent with the phenomenon known as mRNA (or transcription) bursting. This phenomenon has only been experimentally observed in prokaryotic genes expressing constitutively, but our results predict that it should also be observed in gene networks with negative feedback regulation at the transcriptional level. One final prediction from our results is the fact that, if the promoter switching between the active and the inactive states becomes too slow, the protein probability distribution may become bimodal.

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C H A P T E R

S E V E N

How Molecular Should Your Molecular Model Be? On the Level of Molecular Detail Required to Simulate Biological Networks in Systems and Synthetic Biology Didier Gonze,* Wassim Abou-Jaoude´,† Djomangan Adama Ouattara,‡,§ and Jose´ Halloyk Contents 1. Introduction 2. Michaelis–Menten Kinetics Revisited 2.1. Michaelis–Menten equation for an isolated reaction 2.2. Embedded Michaelis–Menten kinetics 2.3. Stochastic simulation of the Michaelis–Menten system 3. Use of the Hill Kinetics for Transcription Rate 4. Repressilator 4.1. Original version: Hill-based model 4.2. Developed version: Cooperative binding sites 4.3. Deterministic simulation of the Repressilator 4.4. Stochastic simulation of the Repressilator 5. Toggle Switch 5.1. Original version: Hill-based model 5.2. Developed version: Cooperative binding sites 5.3. Deterministic simulation of the Toggle Switch model 5.4. Stochastic simulation of the Toggle Switch model 6. Discussion 6.1. Qualitative models are useful to understand dynamical properties of gene regulatory networks 6.2. Michaelis–Menten and Hill assumptions are often reasonable

172 175 176 178 182 185 190 190 191 193 196 198 198 199 200 201 204 204 205

* Laboratoire de Bioinformatique des Ge´nomes et des Re´seaux, Universite´ Libre de Bruxelles, Bruxelles, Belgium INRIA Sophia-Antipolis, Sophia-Antipolis, France { INERIS, Parc Technologique Alata, Verneuil-en-Halatte, France } UMR-CNRS 6600, Universite´ de Technologie de Compie`gne, France k Service d’Ecologie Sociale, Universite´ Libre de Bruxelles, Bruxelles, Belgium {

Methods in Enzymology, Volume 487 ISSN 0076-6879, DOI: 10.1016/S0076-6879(11)87007-2

#

2011 Elsevier Inc. All rights reserved.

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6.3. The level of details depends on the question asked 6.4. The notion of elementary reaction steps in biology is elusive 6.5. Stochastic simulations can be performed using compact kinetics 6.6. Spatial effects are important and may imply complex kinetics 7. Conclusion Acknowledgments References

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Abstract The recent advance of genetic studies and the rapid accumulation of molecular data, together with the increasing performance of computers, led researchers to design more and more detailed mathematical models of biological systems. Many modeling approaches rely on ordinary differential equations (ODE) which are based on standard enzyme kinetics. Michaelis–Menten and Hill functions are indeed commonly used in dynamical models in systems and synthetic biology because they provide the necessary nonlinearity to make the dynamics nontrivial (i.e., limit-cycle oscillations or multistability). For most of the systems modeled, the actual molecular mechanism is unknown, and the enzyme equations should be regarded as phenomenological. In this chapter, we discuss the validity and accuracy of these approximations. In particular, we focus on the validity of the Michaelis–Menten function for open systems and on the use of Hill kinetics to describe transcription rates of regulated genes. Our discussion is illustrated by numerical simulations of prototype systems, including the Repressilator (a genetic oscillator) and the Toggle Switch model (a bistable system). We systematically compare the results obtained with the compact version (based on Michaelis–Menten and Hill functions) with its corresponding developed versions (based on “elementary” reaction steps and mass action laws). We also discuss the use of compact approaches to perform stochastic simulations (Gillespie algorithm). On the basis of these results, we argue that using compact models is suitable to model qualitatively biological systems.

1. Introduction The recent advance of genetic studies and the rapid accumulation of molecular data led the biologists to draw detailed pictures of complex regulatory networks, including the cell cycle, the circadian clock, signaling pathways, and developmental processes. These biological networks often involve many components (genes, proteins, and small molecules) which are coupled through multiple regulatory interactions. A recurrent motif displayed by such gene regulatory networks are positive and negative feedback loops, which are necessary to provide systems with rich dynamical properties, like self-sustained oscillations and multistability (Thomas and D’Ari, 1990).

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Understanding the dynamics of such large networks by sheer intuition is often difficult. To fully grasp the role of the multiple feedback loops in such complex networks, it is necessary to resort to mathematical modeling. These models tend to be molecular: they describe the time evolution of the concentrations of identified species and they capture as many molecular details as possible. Dynamical models for metabolic pathways and genetic regulatory networks proposed in systems and in synthetic biology are generally based on classical enzyme kinetics, including Michaelis–Menten and Hill rate functions (Alon, 2007; Cornish-Bowden, 1995; Kaern et al., 2003; Segel, 1976). These equations have been shown to be good approximations of more detailed reaction schemes under the assumption that some reactions are fast compared to others. A further motivation to use these functions in mathematical models is that they provide the necessary nonlinearity to make the dynamics nontrivial. Nonlinearity is indeed required to observe behaviors like limit-cycle oscillations or multistability (Goldbeter, 1996; Murray, 2003). The use of Michaelis–Menten and Hill rate equations raises several related questions. First, they have been derived for isolated enzyme reactions and describe accurately the asymptotic behavior of the system (Cornish-Bowden, 1995). We may wonder whether these kinetic equations are still valid if the enzymatic reaction is embedded in a larger reaction network in which the substrate and the product of the reaction are themselves involved in other biochemical reactions. Furthermore, in natural conditions, the systems do not remain constantly at steady state. They must respond adequately to transient perturbations or to oscillatory inputs. It is thus necessary to check whether the Michaelis–Menten and Hill descriptions, established for an isolated system at steady state, still hold for dynamical systems undergoing temporary or periodic changes in time. Second, Michaelis–Menten and Hill rate equations have been extensively studied in the context of enzymatic reactions, but they are also often used to describe processes like mRNA synthesis (gene transcription). Simple models of gene regulation show that Michaelis–Menten-like kinetics can indeed describe transcription rates (Keller, 1995). In reality, however, gene transcription involves elaborate molecular machineries, allowing the proper processing of transcription and translation (Lewin, 2010; Ptashne, 2004). Since the kinetic details of the multiple steps involved in these processes are most of the time not known, the question of the validity of Michaelis– Menten and Hill approximations remains open. Classical enzyme kinetic functions can only be considered as phenomenological equations, hopefully capturing the essential dynamical properties of such processes, at least in a qualitative manner. The question of how to use this phenomenological approach is of interest for the modeling of any biomolecular network. Can the enzyme kinetic functions be used as such or should they be developed

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into a detailed reaction scheme in which each reaction is described by the mass action law? Clearly, the developed version of these functions would more explicitly show the assumptions underlying the compact version (i.e., constraints on the parameter values). Moreover, developed models would also allow working in a region of parameter space where the underlying assumptions are lifted. Nevertheless, given that molecular details are often not available, we may question the need for such a decomposition. Third, due to the low number of macromolecules in cellular processes, the dynamics of biological systems is likely to be affected by molecular noise (Barkai and Leibler, 2000; Elowitz et al., 2002; McAdams and Arkin, 1997, 1999; Raj and van Oudenaarden, 2008; Raser and O’Shea, 2005). Stochastic equations are then necessary to assess the potentially important effect of such molecular noise. A common approach to simulate stochastic systems is the Gillespie algorithm (Gillespie, 1977). In this approach, a propensity is associated with each reaction step, and the reactions occur probabilistically according to their propensities (see Appendix). This approach is considered to be valid for elementary reaction steps (Gardiner, 2004; van Kampen, 2007). However, several stochastic models compute the reaction propensities with the Michaelis–Menten or Hill functions (Gonze et al., 2003; Kraus et al., 1992; Ouattara et al., 2010; Song et al., 2007). The question remains whether the compact form of kinetic equations or the developed version is the more appropriate choice for the system at hand. Can we use compact, nonlinear functions as propensities of the reactions? Do such compact stochastic models give the same results as their developed version? In the present chapter, we address these questions by systematically comparing the compact and developed versions of various models, and analyzing their deterministic and stochastic behaviors. We first recall the derivation of the Michaelis–Menten equation for an isolated system, highlighting the underlying hypotheses. We then slightly modify this system to examine the conditions of validity of the Michaelis–Menten equation when the system is open (i.e., when the substrate and product are themselves involved in other reactions). We will also examine the cases where the synthesis rate of the substrate undergoes temporary or periodic changes in time. In the second part, we briefly discuss one possible derivation of the Hill function for the transcription rate. We then compare the compact and developed versions of the Repressilator (Elowitz and Leibler, 2000) and the Toggle Switch (Gardner et al., 2000), two prototypical genetic models, displaying oscillations and bistability, respectively. For all these models, we compare the compact and developed versions using both deterministic and stochastic simulations. Note that these models should not be regarded as models for precise molecular systems, but rather as generic and representative models.

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Many papers already discuss in details some aspects mentioned above. In the present paper, our goal is not to present a detailed theoretical study on enzyme and gene kinetics, but rather to recall and highlight some hypotheses implicitly done when using these standard kinetics in modeling biological systems. In the discussion, we also give some reasons why we believe that models based on Michaelis–Menten and Hill functions are a good approach to analyze deterministic and stochastic dynamics of biological systems.

2. Michaelis–Menten Kinetics Revisited We start with the analysis of the Michaelis–Menten function commonly used to describe the kinetics of enzymatic reactions (CornishBowden, 1995; Segel, 1976, 1988). Enzymatic reactions characterize not only most metabolic reactions but also posttranslational modifications such as phosphorylation–dephosphorylation of proteins by kinases and phosphatases. By regulating the activity of the proteins, these transformations can play crucial roles in the dynamics of genetic networks. In the mammalian circadian clock, for example, the Clock protein needs to be phosphorylated to enter the nucleus, where it acts as a transcriptional inhibitor (Leloup, 2009; Tamanini et al., 2005). Phosphorylation (among other posttranslational processes) affects both activity and stability of the p53 protein, a key regulator of the cell cycle and apoptosis (Brooks and Gu, 2003). When modeling metabolic pathways (Brooks and Storey, 1992; Pettersson, 1993) or genetic networks such as circadian clocks (Goldbeter, 1995; Leloup and Goldbeter, 2003) or the p52–Mdm2 network (Abou-Jaoude´ et al., 2009), Michaelis–Menten functions are often taken by default as standard kinetic equations. It should be stressed, however, that in these systems, the precise molecular mechanism of the reactions is often not known, and these equations should be regarded as a phenomenological description of saturation kinetics. The use of such equations implies hypotheses on the molecular mechanism. These hypotheses are rarely discussed, neither shown to be satisfied in the systems under study. We briefly recall here the derivation of the Michaelis–Menten equation for an isolated enzymatic reaction. We then discuss its validity in open systems, that is, systems in which the level of substrate and product are dynamically regulated by “external” reactions. To this end, for each model, we compare the compact model (based on Michaelis–Menten equations) and its corresponding developed model (i.e., detailed molecular mechanism). We also compare the stochastic time series generated by the stochastic simulation of the two versions of the model.

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2.1. Michaelis–Menten equation for an isolated reaction We assume that an enzyme E catalyzes the transformation of a substrate S into a product P (where S and P represent the dephosphorylated and phosphorylated forms of a protein and E a kinase): E

S!P

ð7:1Þ

For an isolated system, the rate of production of P is equal to the rate of consumption of S, and the time evolutions of the concentration S and P are thus often written as v¼

dP dS S ; ¼  ¼ vm dt dt KM þ S

ð7:2Þ

where vm is the maximum reaction rate (reached when S is very large) and KM is the Michaelian constant. Note that in this compact formulation, the concentration of the enzyme, E, does not appear explicitly. The classical reaction scheme proposed to describe the molecular mechanism by which the enzyme E catalyzes the conversion of S into P is the following: k1

k3

E þ SÐC!E þ P k2

ð7:3Þ

The evolution equations for the concentration of the different species follow the mass action law: dS ¼ k1 ES þ k2 C; dt dE dC ¼ ¼ k1 ES þ k2 C þ k3 C; dt dt dP ¼ v ¼ k3 C: dt

ð7:4Þ

The quasi-steady-state assumption (QSSA) stipulates that dC/dt ¼ dE/ dt ¼ 0 (Briggs and Haldane, 1925; Cornish-Bowden, 1995). This assumption derives from the singular perturbation theory (Bowen et al., 1962; Segel and Slemrod, 1989), which shows how the different time scales that characterize different processes can be exploited to reduce the dimensionality (i.e., the number of variables) in a model. Under the QSSA, assuming that the total enzyme concentration is constant, Etot ¼ E þ C, and much smaller than the substrate concentration S (allowing to consider S ¼ S0 þ

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C  S0 where S0 is the concentration of the free substrate), it can be shown that the rate of synthesis of the product is well described by Eq. (7.2) provided that vm ¼ k3 Etot ; k2 þ k3 ; KM ¼ k1 Etot  S:

ð7:5Þ

In fact, the derivation of the Michaelian function from the QSSA was due to Briggs and Haldane (Briggs and Haldane, 1925; Cornish-Bowden, 1995) and slightly differs from the original derivation by Michaelis and Menten, which was based on the rapid equilibrium between E þ S and C (i.e., k1, k2  k3). The latter leads to the same general equation, where KM is replaced by KS ¼ k2 / k1  KM (Michaelis and Menten, 1913). Since the precise molecular mechanism of the biochemical reactions modeled in systems biology is rarely known, this distinction is almost never done. Another point to notice is that when Eq. (7.2) is used in a large model, S implicitly represents the total concentration of the substrate. However, when Eq. (7.2) is derived from Eq. (7.4), the quantity S appearing in the denominator is the free form of the enzyme, the total substrate concentration being S þ C. However, if the condition S  Etot holds, then the fraction of substrate bound to the enzyme is small and the approximation is valid. Different extensions of the Michaelis–Menten equation to the case of high enzyme concentration have been proposed by other authors (Cha, 1970; Schnell and Maini, 2000). Variants of the QSSA, namely, based on the so-called total quasi-steady-state assumption (tQSSA), were also shown to more accurately describe systems in which the levels of substrate and enzyme are within the same range (Borghans et al., 1996; Ciliberto et al., 2007; Macnamara et al., 2008). In Fig. 7.1, we compare the time evolution of S and P for the compact model (Eq. (7.2), dashed curves) and the developed model (Eq. (7.4), solid curves) for the case where the conditions (Eq. (7.5)) are satisfied (Etot low, panels A and B) or not (E tot high, panels C and D). In both cases, parameter values have been chosen such that vm ¼ k3Etot and KM ¼ (k2 þ k3)/k1. Clearly, in the latter case, the compact model is not a good approximation of the developed reactional scheme. In the developed model, when Etot is high, a large quantity of the substrate rapidly binds the enzyme to form the complex C, and the catalytic constant k3 is small so that the product is released in a slower fashion. The initial reaction rate is thus small because the complex C rather tends to dissociate into S and E than to produce the product P (k2 high, k3 small) (panel D, solid curve) Conversely, in the compact model, as soon as the substrate S is present, it is converted into the product P at a high rate (panel D, dashed curve).

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B

A 1

0.5

P Reaction rate, v

Concentration

0.8 0.6 0.4

0.4 0.3 0.2 0.1

0.2

v (compact) = v (dev)

S 0

0

5

10

0

15

0

5

D

C 1

Concentration

Reaction rate, v

P

0.8 0.6 0.4 0.2 0

5

15

10

0.5 0.4 0.3 v (compact)

0.2 0.1

S 0

10 Time

Time

15

0

v (dev) 0

Time

5

10

15

Time

Figure 7.1 Michaelis–Menten kinetics for an isolated reaction: comparison of the kinetics obtained by the compact model (Eq. (7.2), dashed curves) and by the corresponding developed model (Eq. (7.4), solid curves). Parameter values: (A, B) vm ¼ 1 nM min 1, KM ¼ 2 nM, k1 ¼ 55 nM 1 min 1, k2 ¼ 100 min 1, k3 ¼ 10 min 1, and Etot ¼ 0.1 nM. (C, D) vm ¼ 1 nM min 1, KM ¼ 2 nM, k1 ¼ 0.55 nM 1 min 1, k2 ¼ 1 min 1, k3 ¼ 0.1 min 1, and Etot ¼ 10 nM. In (A) and (C) are shown the time evolution of the concentration of the substrate, S, and of the product, P. In (B) and (D) is plotted the product production rate, v ¼ dP/dt. Note that in (A) and (B), a perfect agreement between the two descriptions is obtained: the dashed and the solid curves are superimposed. Note that the units of the parameters here and in the subsequent figures are arbitrary.

2.2. Embedded Michaelis–Menten kinetics Unless studied in vitro, a biochemical reaction as Eq. (7.1) is not expected to take place in a cell without being connected to other reactions. In vivo enzymatic reaction (7.1) is likely embedded in larger biochemical networks (metabolic pathway, signaling cascade, circadian clock, etc.). In such systems, the substrate S is generally the product of an upstream reaction, and the product P serves as a substrate for downstream reactions. In this condition, since the substrate is continuously supplied, the system can evolve

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towards a non-trivial steady state. In the present section, we modified the previous model by incorporating an inflow of the substrate S and an outflow of the product P. We first check whether, in such an open system, the compact version (based on Michaelis–Menten function) is still in agreement with the corresponding developed reaction scheme. In particular, since the substrate S varies with time (and can reach low levels, possibly smaller than the enzyme), it is important to check the validity of the Michaelis–Menten equation when the concentration of the enzyme is not smaller than the concentration of the substrate. We also perform stochastic simulations using the compact and the developed model and compare both descriptions. We modified model (7.1) by considering that the substrate S is produced at some rate vs and the product P is consumed in another reaction or degraded, and thus disappears at a rate vd (which will here assumed to be constant): vs ! S S ! vd P!

½E; vm ; KM

P

ð7:6Þ

Assuming that the QSSA holds, the time evolution of S and P is governed by the following evolution equations: dS S ; ¼ vs  vm dt KM þ S dP S  vd P: ¼ vm dt KM þ S

ð7:7Þ

Decomposing the enzymatic reaction into elementary steps according to Eq. (7.3), we obtain the following reactional scheme: vs ! S k3 E þ SÐC!E þ P k1 k2

ð7:8Þ

P! vd

For this developed version of the model, the time evolution of the species follows the mass action law: dS ¼ vs  k1 ES þ k2 C dt dE dC ¼ ¼ k1 ES þ k2 C þ k3 C dt dt dP ¼ k3 C  vd P dt

ð7:9Þ

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In Fig. 7.2, we compare the time evolution of S obtained with the compact model (Eq. (7.7), dashed curves) and the developed model (Eq. (7.9), solid curves). Panels A and D display the time evolution of S and P. In panel A, the total enzyme concentration, Etot, is five times larger than the concentration of the substrate at steady state, while in panel D, the concentration of the enzyme is about 50 times larger than the concentration of the substrate. In these simulations, we also imposed the constraints vm ¼ k3Etot and KM ¼ (k2 þ k3)/k1 (cf. Eqs. 7.5). In both cases, the steady state reached in the developed version is the same as the steady state reached in the compact model, but the transients are a bit longer when the enzyme concentration is high (panel D). In the developed version of the model, the additional steps create a delay in the response of the system. This delay is smaller and the match between the two descriptions is better if k1, k2, and k3 are increased, provided that vm ¼ k3Etot and KM ¼ (k2 þ k3)/k1 (cf. Eqs. 7.5) (panel A). The condition that the enzyme concentration is smaller than the substrate concentration is thus not required. If we are interested in the asymptotical properties of the system (i.e., at the steady state), this delay effect has no impact. In natural conditions, however, most biological systems do not operate at the steady state. They must respond adequately to external stimuli (e.g., the biosynthesis of amino acids must be regulated depending on the requirement and the availability of amino acids) or to periodic forcing (biological rhythms regulate numerous output pathways and signaling cascades). In panels B and E, we study the effect of a transient synthesis of the substrate (i.e., a pulse of vs). As expected from the above observations, the detailed model does not respond as fast as the compact model. Consequently, the peaks reached by S and P in the developed version of the model are slightly delayed and of smaller amplitude than in the compact version. This delay is very small if the enzyme concentration is low (panel B), but becomes important when the enzyme concentration is high (panel E). In panels C and F, we study the effect of a periodic forcing of the synthesis rate of the substrate S:

  A 2pt vs ð t Þ ¼ sin þ1 ; ð7:10Þ 2 t where A and t are the amplitude and period of the oscillations, respectively. The amplitude of the oscillations observed in the concentration of the substrate, S, and of the product, P, are slightly reduced in the developed model compared to the compact model. This reduction is more significant when the enzyme concentration is high (panel F). We also noticed that increasing A leads to a better agreement between the two versions, while changing t has not much effect (not shown).

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A

D 1

0.8 0.6 0.4 0.2 0

Concentration

B

Concentration

P

0

20

40 Time

60

P

0.8 0.6 0.4 0.2

S

0

80 E

0.4 0.3

Concentration

Concentration

1

0.2 P 0.1

S 0

20

C

0

10

20 30 Time

0.3 0.2 P 0.1

40

0

50 F

Concentration

Concentration

5 P S

3 2 1 0

80

S

6

4

60

0.4

S 0

40 Time

0

10

20 30 Time

40

50

6 5 P

4

S

3 2 1

0

20

40 Time

60

80

0

0

20

40 Time

60

80

Figure 7.2 Deterministic simulation of the simple, embedded, one-reaction Michaelis–Menten-based model. (A, D) Evolution of S and P toward the steady state obtained with the compact model (Eq. (7.7), dashed curves) and with the developed model (Eq. (7.9), solid curves), starting with S(0) ¼ P(0) ¼ 0 nM. (B, E) Response of the system to a pulse of vs:vs increases from 0 to 0.1 nM min 1 during t ¼ 5 and t ¼ 10 min. (C, F) Entrainment of the system by a periodic forcing of vs:vs undergoes sine oscillations, as defined by Eq. (7.10) with A ¼ 1 and t ¼ 15. Parameter values: (A–C) vm ¼ 1 nM min 1, KM ¼ 2 nM, k1 ¼ 5.5 nM 1 min 1, k2 ¼ 10 min 1, k3 ¼ 1 min 1, Etot ¼ 1 nM, vs ¼ 0.1 nM min 1, and vd ¼ 0.1 min 1. (D–F) Idem except k1 ¼ 0.55 nM 1 min 1, k2 ¼ 1 min 1, k3 ¼ 0.1 min 1, and Etot ¼ 10 nM.

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In all the above cases, a better agreement between the two descriptions can easily be achieved by increasing the constant rates k1, k2, and k3 (and decreasing Etot accordingly to ensure that vm ¼ k3Etot). We also noted that the condition that the enzyme concentration is smaller than the substrate concentration is not absolutely required to obtain a qualitative agreement between the developed versions and the Michaelis-Menten-based models. A quantitative agreement is obtained when the kinetic constants k1, k2, and k3 are sufficiently large (which also implies that Etot is small). A more detailed and quantitative analysis of the validity of the Michaelis–Menten approximation for open systems was recently published by Stoleriu et al. (2004, 2005), and their conclusion is in agreement with the present observations. More elaborate Michaelis–Menten-based models, including linear chains of reactions and models incorporating feedbacks, are currently under investigation.

2.3. Stochastic simulation of the Michaelis–Menten system The second question was to determine whether compact models can be used to perform stochastic simulations. More precisely, it is still a matter of debate whether we can compute the propensities of the reactions in the Gillespie algorithm (see Appendix) with nonlinear rate equations (e.g., Michaelis–Menten function; Barik et al., 2008; Goutsias, 2005; Grima, 2009; Macnamara et al., 2008; Rao and Arkin, 2003). First, note that resorting to such a description would reduce the number of variables as well as the number of reactions (and thus the number of propensities to be computed). In addition, if the constant rates k1, k2, and k3 are high, those reactions will occur very often, and this will be CPU-consuming. For large biomolecular networks, this intensive computation may be prohibitive. It is therefore tempting to use compact models to perform stochastic simulations. Stochastic versions of the compact and of the developed models are given in Tables 7.1 and 7.2. In Fig. 7.3, we present the results of the simulations of the stochastic versions of the models. Parameter values and initial conditions correspond to the deterministic situation shown in Fig. 7.2D. The system size was set to O ¼ 100, which yields a number of molecules of enzyme Etot ¼ 1000, that is, 50 times larger than the number of molecules of the substrate at steady state. Panels A and C show plots of the time series of S and P generated by the stochastic simulation of the compact and developed models, respectively. Panels B and D display the corresponding histograms of S and P at the steady state. Mean and standard deviation of both variables for each version are similar. The amplitude of the fluctuations is also similar in both models. The frequency of the fluctuations at the level of S is higher in the developed model because of the fast S–E binding/unbinding.

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Table 7.1 Stochastic version of the compact version of the simple Michaelis–Mentenbased model No.

Reaction

Propensity

Parameter values

1 2 3

!S S!P P!

w1 ¼ vsO S w2 ¼ vm O KM OþS w3 ¼ vdP

vs ¼ 0.1 nM min 1 vm ¼ 1 nM min 1, KM ¼ 2 nM vd ¼ 0.1 min 1

Default parameter values are given in the right column (parameter values are the same as in Fig. 7.2).

Table 7.2 Stochastic version of the developed version of the simple Michaelis– Menten-based model No.

Reaction

Propensity

1 2 3 4 5

!S SþE!C C!SþE C!P P!

w1 w2 w3 w4 w5

¼ ¼ ¼ ¼ ¼

vsO k1SE/O k2C k3C v dP

Parameter values

vs ¼ 0.1 nM min 1 k1 ¼ 0.55 nM 1 min 1 k2 ¼ 1 min 1 k3 ¼ 0.1 min 1 vd ¼ 0.1 min 1

The total enzyme concentration, Etot ¼ E þ C ¼ 0.1O.

One goal of stochastic simulations is to assess the robustness of the behavior when the number of molecules is reduced. In the compact version of the model, the enzyme concentration is implicit and does not contribute to the stochasticity. In the developed version of the model, the number of enzyme molecules appears explicitly, and one may wonder whether a low number of enzyme molecules affect the results of the stochastic simulations. In order to check whether the above conclusion still holds when E is small, we performed stochastic simulations for small Etot and thus large k1, k2, and k3 to satisfy the constraints vm ¼ k3Etot and KM ¼ (k2 þ k3)/k1 (Fig. 7.4). For Etot ¼ 10 (panels A–C) and Etot ¼ 1 (panels D–F), for a system size O ¼ 100, the results are highly similar to the ones shown in Fig. 7.3. The very low level of enzyme does not lead to higher stochasticity because it is compensated by a very high turnover rate, leading to a barcode-like pattern of the enzyme level (the enzyme present in a single molecule frequently switches between the free and the complex form; panel E). A similar behavior was also reported in a stochastic study of a model for circadian rhythms (Gonze et al., 2004). If the system size O is further decreased, as in Fig. 7.4G–I, the number of enzyme molecules that theoretically corresponds to the deterministic value becomes smaller than 1. Since the number of enzyme molecules must be an integer, we have two choices. Either, the number of enzymes is set to 0 and there is no reaction, or the number of enzyme is set to 1, which leads to a product k3Etot which is larger than vm.

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120

Number of molecules

Mean(P) = 101.12 Std(P) = 7.98

0.3 Frequency

80 60 40

S

0.25 P

0.2 0.15

S

0.1

20

0.05 0

20

40 60 Time

80

0

100

0

50 100 Number of molecules

150

D

140

Mean(M) = 21.6 Std(M) = 4.58

0.35

120

P

80 60 40

Mean(P) = 96.96 Std(P) = 8.25

0.3

100

S

0.25 P

0.2 0.15

S

0.1

20 0

Mean(M) = 23.7 Std(M) = 5.45

0.35

P

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0

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B

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Didier Gonze et al.

0.05 0

20

40 60 Time

80

100

0

0

50 100 Number of molecules

150

Figure 7.3 Stochastic simulation of the “embedded” Michaelis–Menten-based model. (A) Evolution of S and P for the compact model (obtained with the Gillespie algorithm for the model given in Table 7.1). (C) Evolution of S and P for the developed model (Table 7.2). (B, D) Histograms of S and P at the steady state for the corresponding time evolution. For each model, the mean and standard deviation is indicated above the histograms. Parameter values and initial conditions correspond to the deterministic values given in Fig. 7.2D (cf. Tables 7.1 and 7.2, with system size O ¼ 100).

In the latter case (panels G–I), the discrepancy between the developed and the compact model is not due to the noise but to the fact that the hypotheses underlying the Michaelis–Menten equations (vm ¼ k3Etot) are no longer satisfied. These simulations show that the compact and developed version of the model give very similar results, provided that the conditions underlying the Michaelian kinetics are satisfied. A good agreement is observed even for very low enzyme numbers as long as this low level is compensated by high kinetic rates. These observations thus justify the use of Michaelian functions as propensities of enzymatic biochemical reactions in stochastic simulations.

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A

B

100 80 60 S

20

40

60

80

Number of molecules

Number of molecules

120

P

100 80 60 40

S

20 0

0

20

40

60

80

P

12 8 S 4 0

0

20

40

60

Time

80

100

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30

40

P

S 0.1 0

50

0

50

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F 0.3

1

0

10

20

30

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0.2

P S

0.1 0

50

Time

H Number of molecules

Number of molecules

16

10

Time

0

100

Time

G

0

E

140

0.2

2 0

100

Time

D

4

Frequency

0

8 6

0

50

100

150

Number of molecules

I 0.3

1

0

Frequency

40 20

0.3

10 Frequency

Number of molecules

Number of molecules

P

120

0

C 12

140

0

10

20

30

Time

40

50

0.2

P

S 0.1 0

0

5

10

15

20

Number of molecules

Figure 7.4 Stochastic simulation of the “embedded” Michaelis–Menten-based model. (A–C) O ¼ 100, k1 ¼ 55, k2 ¼ 100, k3 ¼ 10, Etot ¼ 0.1  O ¼ 10; (D–F) O ¼ 100, k1 ¼ 550, k2 ¼ 1000, k3 ¼ 100, Etot ¼ 0.01  O ¼ 1; and (G–I) O ¼ 10, k1 ¼ 550, k2 ¼ 1000, k3 ¼ 100, Etot ¼ 0.1  O ¼ 1. Other parameter values are as in Table 7.2. Panels A, D, and G show the time evolution of S and P. Panels B, E, and H show the time evolution of the free form of the enzyme, E. Panels C, F, and I show the histograms of S and P at steady state.

3. Use of the Hill Kinetics for Transcription Rate Hill functions are commonly used to describe the kinetics of enzymatic reactions in which an enzyme has several, cooperative binding sites (Cornish-Bowden, 1995; Segel, 1976). Sigmoidal functions have also been applied to describe transcriptional regulation (Alon, 2007; Keller, 1995), and to model mRNA synthesis in circadian genetic clocks (Goldbeter, 1995; Leloup and Goldbeter, 2003), in the p53–Mdm2 network (Abou-

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Jaoude´ et al., 2009), and in various synthetic genetic networks (Elowitz and Leibler, 2000; Gardner et al., 2000). Various molecular mechanisms have been invoked as possible explanations for the occurrence of such sigmoidal gene transcriptional activity. They include cooperativity of binding sites (Keller, 1995), multimer formation (Alon, 2007; Yang et al., 2007), competition between a repressor and an activator (Rossi et al., 2000), or DNA looping (Narang, 2007). The purpose of the present section is to present one possible molecular mechanism that leads to Hill kinetics for gene expression and to compare the compact and the corresponding developed version of this Hill kinetics. In the next sections, we will decompose the Repressilator (a minimal genetic oscillator; Elowitz and Leibler, 2000) and the Toggle Switch (a bistable genetic system; Gardner et al., 2000) according to this mechanism. We consider here the case of transcriptional inhibition. Let us assume that a gene G can be active (i.e., can be transcribed) or inactive. Transition from the active to the inactive form is induced by a repressor I. Only the gene in its active form can be transcribed and can produce mRNA (M ) at a rate vs. I

GðactiveÞ Ð GðinactiveÞ vs ½GðactiveÞ ! M

ð7:11Þ

The square brackets denote the fact that the gene G is not consumed in the reaction. In most models of biological systems, the gene is not considered explicitly. The rate of synthesis of mRNA is often modeled by the following sigmoidal function: v¼

dM Kn ¼ vs n I n ; KI þ I dt

ð7:12Þ

where I is the concentration of the inhibitor. Parameter vs is the maximum transcription rate (reached in the absence of inhibitor), KI is referred to as the “inhibitory constant” and n is the Hill coefficient which quantifies the steepness of the function. The term KIn / (KIn þ In) thus represents the fraction of active G. Here, we focus on one of the many possible mechanisms, the one based on cooperative binding of the repressor to several binding sites of the gene promoter. More specifically, we will assume that n molecules of inhibitor I cooperatively bind to the promoter of the gene and that only the free form of G is active. For the case of two cooperative binding sites, the molecular mechanism can be decomposed as follows:

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ka1

G þ I ÐGI kd1 ka2

GI þ I ÐGI2

ð7:13Þ

kd2 vs

½G !M The kinetic equations for this reaction scheme are dG ¼ ka1 GI þ kd1 GI; dt dGI ¼ ka1 GI  kd1 GI  ka2 GII þ kd2 GI2 ; dt ð7:14Þ dGI2 ¼ ka2 GII  kd2 GI2 ; dt dI ¼ ka1 GI þ kd1 GI  ka2 GII þ kd2 GI2 : dt In these equations, I represents the concentration of the free inhibitor. G represents the concentration of free gene, while GI and GI2 are the concentrations of gene bound to one repressor and gene bound to two repressors, respectively (averaged over a large cell population). Variable G (expressed in concentration units) can be interpreted as the gene activity, or the probability of the gene to be transcribed (Alon, 2007). We assume that the total gene “concentration” is constant: G þ GI þ GI2 ¼ Gtot ¼ constant:

ð7:15Þ

Here, we consider the “extreme” case where only the free form, G, of the gene is transcribed (at a rate vs), whereas GI and/or GI2 are inactive (not transcribed). Of course, we could consider the more general case where GI has some transcriptional activities, but as we will see later, in the case of high cooperativity, this form is present in small amounts, so that its contribution to the transcription rate can be neglected. Thus, the transcription rate is given by v¼

dM ¼ vs G: dt

ð7:16Þ

Applying the quasi-steady-state approximation for G and I (implying that dG/dt ¼ dGI/dt ¼ dGI2/dt ¼ dI/dt ¼ 0), we can express G as a function of Gtot and find v¼

dM K1 K2 ; ¼ vs Gtot 2 I þ 2K2 I þ K1 K2 dt

ð7:17Þ

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where K2 ¼ kd2/ka2 and K1 ¼ kd1/ka1 are the dissociation constants. Assuming that Gtot ¼ 1, the correspondence between the Eqs. (7.17) and (7.12) is obtained if the term K2I can be neglected, that is, if K2  K1 (a rigorous development will pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi be presented in a subsequent paper), by setting n ¼ 2 and KI ¼ ðK1 K2 Þ. The relation K2  K1 defines cooperativity: the binding of j molecules to the promoter of the gene favors the binding of the ( j þ 1)th molecule. It can be shown that if there is no cooperativity (K2 K1), the term K2I is equal to or is greater than I2 and this leads to a kinetic of the type described by Eqs. (7.12) with n ¼ 1. In practice, the cooperativity is never maximal, so that 1 n number of binding sites. Saying that the Hill coefficient n is equal to the number of binding sites thus implicitly assumes a very high cooperativity. It should be stressed here that variable I in the developed model (7.14) represents the concentration of the free inhibitor, whereas variable I in the compact model (7.12) is often implicitly considered as the total concentration of inhibitor. Thus, a good agreement between the two descriptions is obtained when the concentration of inhibitor is large enough (compared to Gtot) so that the fraction of inhibitor bound to the gene promoter is negligible. In Fig. 7.5, we compare the Hill function defined by Eq. (7.12) for the case n ¼ 2, with the developed model described by Eq. (7.14). In panel A, we plotted the Hill kinetics as a function of the inhibitor I for the compact model (7.12) (thin dashed line) as well as the steady regime of v ¼ vsG obtained by numerical integration of Eq. (7.14) (black dots). Parameters have been chosen to account for a high cooperativity (K1 ¼ 10 and K2 ¼ 0.1), so that a good agreement between the two versions is obtained. The main difference between the two situations pertains to the transients which appears in the developed model (Fig. 7.5B, solid lines). A consequence of these transients is that the system will not adapt instantaneously to a pulse of inhibitor I (Fig. 7.5C). This delay effect also affects the response of the system to an external periodic forcing. Let us assume now that the synthesis of I is periodic. The time evolution of I is described by: dI ¼ ks  kr I dt

ð7:18Þ

with A ks ¼ 2





  2pt sin þ1 : t

ð7:19Þ

In the developed model, equation for dI /dt in Eq. (7.14) is then replaced by dI ¼ ks  kr I  ka1 GI þ kd1 GI  ka2 GII þ kd2 GI2 : dt

ð7:20Þ

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A

B 1 Transcription rate, v

Transcription rate, v

1 0.8 0.6 0.4 0.2 0

0

1

2 3 Inhibitor

4

C

0.4 0.2 0

1

0

10

2

3 Time

I=3 4

5

6

40

50

D I=0

1

I=0

Transcription rate, v

1 Transcription rate, v

0.6

0

5

I = 0.5

0.8

0.8 0.6 0.4 I=3

0.2 0

0

5

10 Time

15

20

0.8 0.6 0.4 0.2 0

20 30 Time

Figure 7.5 Comparison of the Hill kinetics and the corresponding developed molecular mechanism based on cooperative binding sites. (A) Transcription rate v for the compact model (Eq. (7.12), thin dashed curve) and for the developed model (numerical integration of Eq. (7.14), dots) as a function of the concentration of the inhibitor I. (B) Transcription rate v (thin dashed lines, Eq. (7.12)) or time evolution of v (solid curves, Eq. (7.14)) for different concentrations of the inhibitor I (from top to bottom: I ¼ 0.5, 1, 1.5, 2, 2.5, and 3 nM). (C) Variation of rate v for the compact model (thin dashed curve) and for the developed model (solid curve) observed when the concentration of inhibitor I transiently increases from 0 to 3 nM during t ¼ 5 and t ¼ 12 min. (D) Oscillations of v when the synthesis of I undergoes oscillations. Parameter values: vs ¼ 1 nM min 1, KI ¼ 1 nM, n ¼ 2, ka1 ¼ 1 min 1 nM 1, kd1 ¼ 10 min 1, ka2 ¼ 10 min 1 nM 1, and kd2 ¼ 1 min 1. In panel D, the evolution equation for I is given by Eqs. (7.19) and (7.20) with A ¼ 3, t ¼ 10, and kr ¼ 1.

The results presented in Fig. 7.5D show that the oscillations of v obtained with the developed model have a slightly reduced amplitude compared to the oscillations obtained with the compact model. As for Michaelis-Menten kinetics (see previous section), increasing the binding and unbinding constant rates ka1, kd1, ka2, kd2 (while respecting the pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi constraints KI ¼ ðK1 K2 Þ) reduces the transients, hence leading to a better

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agreement between the compact and the developed version of the model (not shown). A good correspondence between the developed and the compact versions requires several assumptions and constraints on the parameter values. Namely, we need cooperativity, a sufficiently high level of inhibitor, and high binding/unbinding constants. The latter condition is necessary to reduce the transients (not shown). The molecular details being often unknown, it is usually impossible to check whether these requirements are satisfied. The Hill function should thus be regarded as a phenomenological equation, which is probably realistic in many situations. Our conclusions can be extended to systems with a higher Hill coefficient (n > 2), if we consider multiple, cooperative binding sites. Alternative molecular schemes, for example, based on inhibitor dimer (or multimer) formation, can also lead to the occurrence of Hill kinetics at the level of transcription (Alon, 2007).

4. Repressilator The Repressilator consists of three genes, cyclically regulated in such a way that each gene codes for a repressor protein which inhibits the transcription of the following gene in the cycle (Elowitz and Leibler, 2000) (Fig. 7.6). A theoretical model was used to design and to guide the experimental construction of a synthetic gene circuit exhibiting self-sustained oscillations in Escherichia coli (Elowitz and Leibler, 2000). In the present section, we develop the Repressilator model into “elementary” reaction steps, according to the molecular mechanism described in the previous section. We then compare the deterministic and stochastic dynamics of the original and developed versions of the model.

4.1. Original version: Hill-based model The original model of the Repressilator comprises six evolution equations governing the time evolution of each mRNA and protein level (Elowitz and Leibler, 2000). In the original formulation of the model, variables and

P2

P1 M1

G1

P3

M2

G2

Figure 7.6 Scheme of the Repressilator.

M3 G3

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time were rendered dimensionless. For our present purpose, we have rewritten the equations in terms of concentrations: dM1 Kn  dM1 ; ¼a n dt K þ P3n dM2 Kn  dM2 ; ¼a n dt K þ P1n dM3 Kn ¼a n  dM3 ; dt K þ P2n dP1 ¼ bM1  gP1 ; dt dP2 ¼ bM2  gP2 ; dt dP3 ¼ bM3  gP3 : dt

ð7:21Þ

Variables Mi and Pi (with i ¼ 1, 2, 3) represent the concentration of mRNA and protein of the three components of the Repressilator. Parameters a, d, b, and g are the maximum transcription rate, mRNA degradation rate, protein synthesis rate, and protein degradation rate, respectively. The repression of the transcription is described by a Hill kinetics with an inhibition threshold K and a cooperativity degree n. For simplicity, each kinetic parameter was set to an identical value for each gene and protein. (Elowitz and Leibler, 2000) showed that such a symmetry favors the occurrence of oscillations. The stochastic version of this model (given in Table 7.3) consists of 12 reaction steps. Here, we computed the propensities of mRNA synthesis with the Hill kinetics.

4.2. Developed version: Cooperative binding sites As shown in the previous section, Hill functions used in Eq. (7.21) can be obtained, for example, if we assume that two binding sites are present in the promoter of each gene and that the binding of a repressor molecule is cooperative. We further assume that only the free gene can be transcribed at a maximum rate a, while the gene bound to one or two repressor proteins is not transcribed. A similar assumption was considered when developing a model for circadian rhythms (Gonze et al., 2002a). We thus obtain the following developed model:

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Table 7.3 Stochastic version of the original version of the Repressilator model No.

Reaction

1

[G1] ! M1

2 3

[G2] ! M2 [G3] ! M3

4 5 6 7 8 9 10 11 12

M1! M2! M3! [M1] ! P1 [M2] ! P2 [M3] ! P3 P1 ! P2 ! P3 !

Propensity w1

¼

Parameter values

ðKOÞn aO ðKO Þn þP3n n

ðKOÞ w2 ¼ aO ðKO Þn þP1n ðKOÞn w3 ¼ aO ðKOÞn þP n 2 w4 ¼ dM1 w5 ¼ dM2 w6 ¼ dM3 w7 ¼ bM1 w8 ¼ bM2 w9 ¼ bM3 w10 ¼ gP1 w11 ¼ gP2 w12 ¼ gP3

a ¼ 40 nM min 1, K ¼ 1 nM, n ¼ 2

d ¼ 5 min 1 b ¼ 5 min 1 g ¼ 5 min 1

Variables in brackets indicate species which are necessary for the reaction but which are not consumed in the reaction.

dG1 ¼ ka1 P3 G1 þ kd1 G1P ; dt dG1P ¼ ka1 P3 G1  kd1 G1P  ka2 P3 G1P þ kd2 G1PP ; dt dG2 ¼ ka1 P1 G2 þ kd1 G2P ; dt dG2P ¼ ka1 P1 G2  kd1 G2P  ka2 P1 G2P þ kd2 G2PP ; dt dG3 ¼ ka1 P2 G3 þ kd1 G3P ; dt dG3P ¼ ka1 P2 G3  kd1 G3P  ka2 P2 G3P þ kd2 G3PP ; dt dM1 ¼ aG1  dM1 ; dt dM2 ¼ aG2  dM2 ; dt dM3 ¼ aG3  dM3 ; dt

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How Molecular Should Your Molecular Model Be?

dP1 ¼ bM1  gP1  ka1 P1 G2 þ kd1 G2P  ka2 P1 G2P þ kd2 G2PP ; dt dP2 ¼ bM2  gP2  ka1 P2 G3 þ kd1 G3P  ka2 P2 G3P þ kd2 G3PP ; dt dP3 ¼ bM3  gP3  ka1 P3 G1 þ kd1 G1P  ka2 P3 G1P þ kd2 G1PP : ð7:22Þ dt In these equations, Gi, GiP, and GiPP represent the concentration of the free gene, the gene bound to one repressor protein, and the gene bound to two repressor proteins (average over a large cell population). We assume that the total gene amount is constant: G1T ¼ G1 þ G1P þ G1PP ; G2T ¼ G2 þ G2P þ G2PP ; G3T ¼ G3 þ G3P þ G3PP :

ð7:23Þ

Parameters kai and kdi are the rates of binding and unbinding of the repressor to the gene promoter. It is reasonable to assume that the values of these kinetic rates are large compared to the other kinetic rates, because binding/ unbinding are fast compared to transcription/translation p and degradation pro-ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cesses. In addition, we also impose the constraint K ¼ ðkd1 =ka1 Þðkd2 =ka2 Þ as well as the high cooperativity, i.e., kd2/ka2  kd1/ka1 to make the developed model consistent with the original Hill-based model with n ¼ 2 (cf. Eq. (7.21)). The stochastic version of this developed model is given in Table 7.4. Note that the number of reaction steps is doubled compared to the compact model. We should stress that this model is taken here as a toy example. We do not aim at developing a precise model for the biological system itself. In the Repressilator constructed experimentally, there is no indication that the repressors cooperatively bind several sites in the promoter of their target genes. The sigmoidal shape of the transcription function may also arise from the dimerization of the protein prior to the binding. Other developments of the Repressilator into “elementary” reaction steps, for example, based on dimerization of the repressors prior to their binding, could also have been considered (as done for example by Chen et al., 2004 or by Loinger and Biham, 2007).

4.3. Deterministic simulation of the Repressilator The results of deterministic simulations of the original, compact model are given in Fig. 7.8A and B. Figure 7.8A shows the deterministic limit-cycle oscillations obtained by numerical integration of Eq. (7.21). Because of the symmetry in the model and in the parameters values, each mRNA

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Table 7.4 Stochastic version of the developed version of the Repressilator model No.

Reaction

Propensity

Parameter values

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

G1 þ P3 ! G1P G1P ! G1 þ P3 G1P þ P3 ! G1PP G1PP ! G1P þ P3 G2 þ P1 ! G2P G2P ! G2 þ P1 G2P þ P1 ! G2PP G2PP ! G2P þ P1 G3 þ P2 ! G3P G3P ! G3 þ P2 G3P þ P2 ! G3PP G3PP ! G3P þ P2 [G1] ! M1 [G2] ! M2 [G3] ! M3 M1! M2! M3! [M1] ! P1 [M2] ! P2 [M3] ! P3 P1! P2! P3!

w1 ¼ ka1P3G1/O w2 ¼ kd1G1P w3 ¼ ka2P3G1P/O w4 ¼ kd2G1PP w5 ¼ ka1P1G2/O w6 ¼ kd1G2P w7 ¼ ka2P1G2P/O w8 ¼ kd2G2PP w9 ¼ ka1P2G3/O w10 ¼ kd1G3P w11 ¼ ka2P2G3P/O w12 ¼ kd2G3PP w13 ¼ aG1 w14 ¼ aG2 w15 ¼ aG3 w16 ¼ dM1 w17 ¼ dM2 w18 ¼ dM3 w19 ¼ bM1 w20 ¼ bM2 w21 ¼ bM3 w22 ¼ gP1 w23 ¼ gP2 w24 ¼ gP3

ka1 ¼ 10 nM 1 min 1 kd1 ¼ 100 min 1 ka2 ¼ 100 nM 1 min 1 kd2 ¼ 10 min 1

a ¼ 40 min 1 d ¼ 1 min 1 b ¼ 5 min 1 g ¼ 5 min 1

Variables in brackets indicate the species which are necessary for the reaction but which are not consumed in the reaction. The total gene concentration is G1T ¼ G2T ¼ G3T ¼ 1O.

concentration oscillates with the same amplitude, but the oscillations are 120 out-of-phase. For each gene, the protein level directly follows the mRNA level (not shown). For the parameter values chosen, the period of the oscillations is about 7.3 min. Figure 7.7B presents the bifurcation diagram as a function of the maximum transcription rate, a (taken identical for each gene). A Hopf bifurcation occurs at a ¼ 7.5 nM/min. Limit-cycle oscillations occur above this threshold value. Their amplitude increases somewhat linearly with a. The results of deterministic simulations of the developed model are given in Fig. 7.7C and D. Parameter values are the same as for the original model. To reduce delay effects, the value of the parameters for the binding and unbinding rates is 10 times higher than the ones used in Fig. 7.5. The oscillations produced by this developed model are very similar to the ones produced by the original model. The Hopf bifurcation is located at a ¼ 5.4 nM/min

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A

B 25

20 M1 = M2 = M3

20 Concentration

25

M1 M2 M3

15 10 5 0

15 HB a = 7.5

10 5

0

10

20

30

40

0

50

0

10

30

40

50

40

50

D 25

M1 M2 M3

20 15 10 5 0

25 20 M1 = M2 = M3

C

Concentration

20

Transcription rate, a

Time

15 HB a = 5.4

10 5

0

10

20

30 Time

40

50

0

0

10

20

30

Transcription rate, a

Figure 7.7 Deterministic simulation of the Repressilator. (A, B) Original model. (C, D) Developed model. (A) Deterministic oscillations obtained by numerical integration of Eq. (7.21). Parameter values: a ¼ 40 nM min 1, d ¼ 1 min 1, b ¼ 5 min 1, g ¼ 5 min 1, n ¼ 2, K ¼ 1 nM, and G1T ¼ G2T ¼ G3T ¼ 1 nM. The period is 7.3. (B) Bifurcation diagram as a function of the transcription rate, a (kept identical for all mRNA). HB denotes a Hopf bifurcation. On the left of this point, the solid curve gives the stable steady state; on the right on this point, the dashed curve corresponds to the unstable steady state, and the two solid curves are the maximum and the minimum of the mRNA oscillations. (C) Deterministic oscillations obtained by numerical integration of Eq. (7.21). Parameter values: ka1 ¼ 10 min 1 nM 1, kd1 ¼ 100 min 1, ka2 ¼ 100 min 1 nM 1, and kd2 ¼ 10 min 1; other parameters are as in panel A. The period is 7.97. (D) Bifurcation diagram as a function of the transcription rate, a.

(Fig. 7.7D), a value slightly smaller than for the compact model. Nevertheless, the shape of the envelope of the oscillations is very similar in both versions. If binding and unbinding rates are attributed smaller values (i.e., ka1 ¼ 1, kd1 ¼ 10, ka2 ¼ 10, kd2 ¼ 1, as in Fig. 7.5), then the oscillations present a slightly larger amplitude and period (period ¼ 10.4 min) than in the original model, but the structure of the bifurcation diagram is preserved. The comparison between Fig. 7.7A and B and Fig. 7.7C and D shows that the original and the developed model share the same qualitative features (limit-cycle oscillations, bifurcation structure). They only slightly differ in

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some quantitative aspects. Of course, the best agreement between the two versions of the model is obtained when the parameter values of the developed model are consistent with the compact model (fast binding/unbinding rates, high cooperativity). The developed model offers the possibility to explore the behavior of the system when these conditions are not fulfilled (slow binding/unbinding rates, absence of cooperativity). If, for example, we choose ka1 ¼ kd1 ¼ ka2 ¼ kd2 ¼ 1, the oscillations are preserved and even the structure of the bifurcation diagram is kept (HB at a ¼ 7.2 nM/min) (not shown). If we choose ka1 ¼ 10, kd1 ¼ 1, ka2 ¼ 1, kd2 ¼ 10, then, for the default parameter values, the oscillations are damped (no limit-cycle) (not shown). They can, however, be restored at very high values of a (a > 1700 nM/min). Thus, cooperativity and fast binding/unbinding rates are not required to generate limit-cycle oscillations.

4.4. Stochastic simulation of the Repressilator The results of stochastic simulations for the original model are given in Fig. 7.8A and B. Figure 7.8A shows the stochastic oscillations obtained with the Gillespie algorithm for the stochastic version of the model presented in Table 7.3. Note that in this version, the propensities of the transcriptional processes are computed using the Hill functions. In the case illustrated on the figure, the system size has been fixed to O ¼ 100, which yields a number of molecules of a few thousands mRNA and proteins. For this number of molecules, the oscillations are clearly observable, but their amplitudes and periods undergo some variability. The effect of noise may be quantified by computing the distribution of the periods and by measuring its standard deviation (Fig. 7.8B). For the compact version, the mean period is 7.33 and the standard deviation is 0.25 (the coefficient of variation CV ¼ standard deviation/mean ¼ 3.41%). The results of stochastic simulations for the developed model are given in Fig. 7.8C and D. As for the original model, the systems size was set to O ¼ 100. The number of mRNA and protein molecules is thus comparable to those of the original model and the number of gene copies for each species is equal to G1T ¼ G2T ¼ G3T ¼ O. Comparing the histogram of the period distribution (Fig. 7.8D) with the histogram obtained for the original model (Fig. 7.8B), we notice that the effect of noise is roughly similar in both cases (mean period ¼ 7.98, standard deviation ¼ 0.33, CV ¼ 4.13%). The fact that the period is slightly increased is consistent with the prediction from the deterministic model (see legend of Fig 7.7A and C). In panels C and D, the total gene concentration was scaled by the system size O. However, we may want to set the number of gene copies equal to 1. To rescale variables Mi and Pi while keeping G1T ¼ G2T ¼ G3T ¼ 1, we do the following change: we multiply all the binding and unbinding rates by

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B

2500

M1 M2 M3

2000 1500 1000 500 0

120 100

Occurence

Number of molecules

A

Mean = 7.33 Std = 0.26

80 60 40 20

0

10

20

30

40

0

50

4

5

6

Time

Number of molecules

Occurence

1500 1000 500

9

10

7 Period

8

9

10

7 Period

8

9

10

Mean = 7.99 Std = 0.33

80 60 40 20

0

10

20

30 Time

40

0

50

F

2500

4

1500 1000 500

5

6

120 100

2000

0

8

120 100

2000

0

E

D

2500

Occurence

Number of molecules

C

7 Period

Mean = 7.38 Std = 0.6

80 60 40 20

0

10

20

30 Time

40

50

0

4

5

6

Figure 7.8 Stochastic simulation of the Repressilator. (A, B) Stochastic oscillations and period distribution obtained for the original model (Table 7.3). (C, D) Stochastic oscillations and period distribution obtained for the developed model (Table 7.4) for the parameter values given in Table 7.4. (E, F) Stochastic oscillations and period distribution obtained for the developed model (Table 7.4) when Gtot ¼ 1 and the binding/unbinding rates (ka1, ka2, kd1, and kd2) as well as the transcription rate a are multiplied by O. For all these simulations, the system size has been set to O ¼ 100 and the period distribution is computed from a time series of 5000 min.

O and we multiply the maximum transcription rate, a, by O (see Table 7.4, parameter set 2). Stochastic simulations of this model yield the time series shown in Fig. 7.8E. The corresponding period distribution is given in Fig. 7.8F. This version of the model leads to a lower robustness to molecular noise but does not destroy the oscillations.

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In summary, developing the Repressilator in “elementary reaction steps” does not significantly affect the robustness of the oscillations to molecular noise. However, if we want to set the number of genes equal to 1, some tricks are needed to preserve the consistency of the model with its deterministic counterpart. These adaptations may have some influence on the effect of noise.

5. Toggle Switch The Toggle Switch model consists of two genes which inhibit each other (Gardner et al., 2000) (Fig. 7.9). For appropriate parameter values and sufficient nonlinearity, this model produces bistability (coexistence between two stable steady states). This model was used to design and to guide the experimental construction of a synthetic gene circuit exhibiting bistability in E. coli (Gardner et al., 2000).

5.1. Original version: Hill-based model In the original version of the Toggle Switch model, mRNA and protein were not distinguished. The model thus contained only two variables. We consider here a variant in which we distinguish the mRNA (M1 and M2) from the protein (P1 and P2): dM1 Kn  dM1 ; ¼ a1 n dt K þ P2n dM2 Kn  dM2 ; ¼ a2 n dt K þ P1n dP1 ¼ bM1  gP1 ; dt dP2 ¼ bM2  gP2 : dt

ð7:24Þ

P2

P1 M1

G1

M2

G2

Figure 7.9 Scheme of the Toggle Switch model.

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These equations are built in a similar way than the ones of the Repressilator. Variables Mi and Pi (with i ¼ 1, 2) represent the concentration of mRNA and protein coded by the two genes of the Toggle Switch system. The definition of the parameters is the same as for the Repressilator. In the experiments, the switch between the two steady states is triggered by an inducer (e.g., IPTG) which binds to one of the repressor proteins and thereby prevents the protein to bind the gene promoter. Equations can be modified to account explicitly for the inducer (Gardner et al., 2000). For the sake of simplicity, we consider here that the switch is induced by a transient increase (pulse) of the transcription rate of one of the gene (a1). Again, our goal is not to model the experimental Toggle Switch system, but rather to study a minimal model exhibiting bistability. The stochastic version of this model, which comprises eight reaction steps, is given in Table 7.5. Here again, we computed the propensities of mRNA synthesis with the Hill kinetics.

5.2. Developed version: Cooperative binding sites We developed the Toggle Switch model in a similar way as the Repressilator. We assume that two binding sites are present in the promoter of each gene and that the binding of the repressor proteins is cooperative. We thus obtain the following eight-variable ODE model: dG1 ¼ ka1 P2 G1 þ kd1 G1P ; dt dG1P ¼ ka1 P2 G1  kd1 G1P  ka2 P2 G1P þ kd2 G1PP ; dt dG2 ¼ ka1 P1 G2 þ kd1 G2P ; dt dG2P ¼ ka1 P1 G2  kd1 G2P  ka2 P1 G2P þ kd2 G2PP ; dt ð7:25Þ dM1 ¼ a1 G1  dM1 ; dt dM2 ¼ a2 G2  dM2 ; dt dP1 ¼ bM1  gP1  ka1 P1 G2 þ kd1 G2P  ka2 P1 G2P þ kd2 G2PP ; dt dP2 ¼ bM2  gP2  ka1 P2 G1 þ kd1 G1P  ka2 P2 G1P þ kd2 G1PP ; dt

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Table 7.5 Stochastic version of the original version of the Toggle Switch model No.

Reaction

Propensity

1

[G1] ! M1 w1 ¼

2 3 4 5 6 7 8

[G2] ! M2 M1! M2! [M1] ! P1 [M2] ! P2 P1! P2!

Parameter values

ðKOÞn a1 O ðKO Þn þP2n n

a1 ¼ 4 nM 1 min 1, K ¼ 1 nM, n¼2 a2 ¼ 4 nM 1 min 1 d ¼ 1 min 1

ðKOÞ w2 ¼ a2 O ðKO Þn þP1n w4 ¼ dM1 w5 ¼ dM2 w7 ¼ bM1 b ¼ 2 min 1 w8 ¼ bM2 w10 ¼ gP1 g ¼ 2 min 1 w11 ¼ gP2

Variables in brackets indicate species that are necessary for the reaction but not consumed in the reaction.

with the conservation relations: G1T ¼ G1 þ G1P þ G1PP ; G2T ¼ G2 þ G2P þ G2PP :

ð7:26Þ

The definition of the variables and parameters is the same as for the Repressilator. The stochastic version of this developed Toggle Switch model is given in Table 7.6. As for the Repressilator, the number of reaction steps is doubled compared to its compact version. As for the Repressilator, the parameter values have been chosen for consistency between the two versions of the model (high cooperativity, fast binding/unbinding constants ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and K ¼ ðkd1 =ka1 Þðkd2 =ka2 Þ).

5.3. Deterministic simulation of the Toggle Switch model We first simulate the deterministic Toggle Switch model (Fig. 7.10). For appropriate parameter values, the compact as well as the developed version of the model show a bistable behavior. In panel A, we show how a pulse of a1 can induce a switch from one steady state to the other in the original model. Panel B shows the bifurcation diagram as a function of a1 for the original model. Two saddle node bifurcations (denoted by LP on the figure), located at a1 ¼ 3.2 nM/min and a1 ¼ 5.8 nM/min, delimit the region of bistability. Panels C and D give the corresponding time series and bifurcation diagram for the developed model. The bifurcation picture is very similar to the one of the compact model with a bistable region located between a1 ¼ 3.3 nM/min and a1 ¼ 5.5 nM/min. Thus, the bistability region is slightly reduced compared to the compact model.

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Table 7.6 Stochastic version of the developed version of the Toggle Switch model No.

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G1 þ P2 ! G1P G1P ! G1 þ P2 G1P þ P2 ! G1PP G1PP ! G1P þ P2 G2 þ P1 ! G2P G2P ! G2 þ P1 G2P þ P1 ! G2PP G2PP ! G2P þ P1 [G1] ! M1 [G2] ! M2 M1 ! M2 ! [M1] ! P1 [M2] ! P2 P1 ! P2 !

w1 ¼ ka1P2G1/O w2 ¼ kd1G1P w3 ¼ ka2P2G1P/O w4 ¼ kd2G1PP w5 ¼ ka1P1G2/O w6 ¼ kd1G2P w7 ¼ ka2P1G2P/O w8 ¼ kd2G2PP w9 ¼ a1G1 w10 ¼ a2G2 w11 ¼ dM1 w12 ¼ dM2 w13 ¼ bM1 w14 ¼ bM2 w15 ¼ gP1 w16 ¼ gP2

ka1 ¼ 10 nM 1 min 1 kd1 ¼ 100 min 1 ka2 ¼ 100 nM 1 min 1 kd2 ¼ 10 min 1

a1 ¼ 4 min 1 a2 ¼ 4 min 1 d ¼ 1 min 1 b ¼ 2 min 1 g ¼ 2 min 1

Variables in brackets indicate the species that are necessary for the reaction but not consumed in the reaction.

Figure 7.10E shows the stability diagram as a function of the transcription rates a1 and a2. Clearly, the domain of bistability defined by the developed model for n ¼ 2 is very close to the bistability domain of the compact model (for n ¼ 2). For n ¼ 3, this domain is significantly enlarged. Thus, replacing the developed model with its compact counterpart affects the results much less than some parametric changes. To induce the switch from one steady state to the other, the pulse of a1 must have a sufficient amplitude and duration. In Fig. 7.10F, we plot the minimum amplitude needed to induce the switch as a function of the pulse duration for both versions of the model. Here again, the compact and developed versions lead to similar results. In summary, we obtain a good qualitative agreement between the compact and the developed versions of the Toggle Switch model. The difference between these two versions mainly concerns some minor quantitative changes, which are smaller than the effect induced by a variation of some parameters such as n. However, when modeling biological systems, the values of these parameters are often unknown.

5.4. Stochastic simulation of the Toggle Switch model We now discuss the stochastic analysis of both versions of the Toggle Switch model. Figure 7.11A displays the stochastic time series for 10 independent runs for the original model. In all cases, a pulse of a1 induces a switch from

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Figure 7.10 Deterministic simulation of the Toggle Switch model. (A, B) Original model. (C, D) Developed model. (A) Switches induced by a pulse of a1 when a1 is increased from 4 to 6.8 nM min 1 during t = 20 and t = 40 min. These time series have been obtained by numerical integration of Eq. (7.24). Parameter values: a2 = 4 nM min 1, d = 2 min 1, b = 2 min 1, g = 1 min 1, K = 1 nM, n = 2, and G1T = G2T = 1 nM. (B) Bifurcation diagram as a function of a1 (a2 = 4). LP1 and LP2 denote the saddle node bifurcations. Between these two points, the system displays bistability. (C) Same as in panel A for the developed model. Parameter values: ka1 = 10 min 1 nM 1, kd1 = 100 min 1, ka2 = 100 min 1 nM 1, and kd2 = 10 min 1; other parameter values are as in panel A. (D) Same as in panel B for the developed model. (E) Stability diagram as a function of parameters a1 and a2 for the compact model (for n = 2 and n = 3, solid curves) and for the developed model (n = 2, dashed curve). Bistability is observed in the upper right part of the diagram. (F) Conditions on the amplitude and duration of the a1 pulse to induce the switch, for the compact model (solid curve) and the developed model (dashed curve).

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Figure 7.11 Stochastic simulation of the Toggle Switch model. (A, B) Time series of 10 independent runs and distribution of the switch times (500 runs) obtained for the original model (Table 7.5). (C, D) Time series of 10 independent runs and distribution of the switch times (500 runs) obtained for the developed model (Table 7.6). (E, F) Time series of 10 independent runs and distribution of the switch times (500 runs) obtained for the developed model (Table 7.6) when Gtot ¼ 1 and the binding/unbinding rates (ka1, ka2, kd1, and kd2) as well as the transcription rate a are multiplied by O. For all these simulations, the system size has been set to O ¼ 100. The switch time is defined as the time at which M1 becomes larger than M2.

one steady state to the other one, but, due to the stochastic nature of the model, the system does not switch exactly at the same time in each run. As a measure of the effect of noise on the system, we computed the distribution

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of the switch times, defined as the time at which M1 becomes larger than M2 (Fig. 7.11B). In Fig. 7.11C and D, we show the corresponding results for the developed version of the Toggle Switch model. The main difference is that the distribution is shifted towards smaller switch times and its standard deviation is reduced compared to the one obtained for original model (Fig. 7.11B). This quantitative difference may be attributed to the fact that the Hill function and its development are not strictly equivalent (see above) and that the phase space properties (vector flows and steady states attractivities) are not identical. As described for the Repressilator, we also consider a variant of the developed model in which the number of gene copies is maintained at G1T ¼ G2T ¼ 1. The results obtained for this variant are given in Fig. 7.11E and F. The distribution of the switch times is very similar to the developed version in which the total gene concentration is scaled by O. These stochastic simulations show that the distribution of the switch times obtained for the compact model is slightly shifted toward larger values and somewhat broader than for the developed version of the model. Developing the toggle switch in detailed reaction steps does not significantly affect the robustness of the bistable behavior to molecular noise.

6. Discussion 6.1. Qualitative models are useful to understand dynamical properties of gene regulatory networks Models in biology are useful to understand the relation between the structure of a regulatory network and its dynamical properties. Qualitative models capturing the core regulatory architecture of the system are suitable to unravel these relations. Earlier models in biology allowed to associate negative circuits to oscillations and positive circuits to multistability (Thomas and D’Ari, 1990). Models for circadian clocks can be used to explore the advantage of the interlocked feedback loops (Becker-Weimann et al., 2004). Models for the p53–Mdm2 system can be used to understand the frequency tuning of the oscillations (Abou-Jaoude´ et al., 2009). Qualitative models are usually relatively simple, rely on phenomenological kinetic equations, and reveal generic properties of biological systems. In synthetic biology, models for the Repressilator and for the Toggle Switch are essential to design the genetic circuits, but they do not tell quantitatively what should be the synthesis or the degradation rate of the mRNA and proteins. They rather show how these different rates should be balanced to obtain the desired behavior and to predict the behavior of these systems when the balance is impaired. In most biological systems studied, the value of kinetic parameters is unknown. Many reactions are known to be enzymatic, but the binding/

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unbinding constants and turnover rates remains usually unknown. In most cases, even the molecular mechanism is not elucidated. For these systems, it is impossible to develop detailed quantitative models that reproduce precisely experimental time series or predict quantitatively the behavior of the system for various well-defined sets of parameters. Detailed models are not absolutely required to fit experimental data. Hill functions for transcription have been measured experimentally (Rosenfeld et al., 2005), without the molecular mechanism responsible for such threshold being elucidated. The mechanism based on cooperative binding sites discussed here is only one possible way to generate a sigmoidal function, but other mechanisms, based on the multimerization of the inhibitor, on competition of transcription factors for a given binding site (Rossi et al., 2000), or positive-feedback-loop-based models (Ferrell and Machleder, 1998), may also explain the occurrence of Hilllike kinetics. Phenomenological models based on measured kinetics are thus suitable for quantitative modeling. For instance, in a study of bistability in the lactose system in bacteria, Ozbudak et al. (2004) resort to a compact three-variable model (built upon Hill function) to fit experimental data to the nonlinear functions.

6.2. Michaelis–Menten and Hill assumptions are often reasonable When modeling a biological system, Michaelis–Menten and Hill functions are often taken as standard equations. Michaelis–Menten functions are used to describe not only rates of enzyme reactions in metabolism but also posttranslational processes (such as phosphorylation of proteins) that play crucial roles in the dynamics of genetic networks. Hill functions are employed to describe the kinetic of the multisite enzyme as well as gene transcription. The rationale behind this choice is that biochemical reactions are often enzymatic and that saturation kinetics and threshold responses are often observed experimentally. This choice is also motivated by the fact that these functions provide appropriate nonlinearity necessary for the system to generate nontrivial behaviors, like self-sustained oscillations and multistability. The assumptions that make the Michaelis–Menten equations valid are, however, rarely discussed. Many textbooks present the derivation of the Michaelis–Menten kinetics for isolated reactions (S ! P) and highlight the underlying QSSA hypothesis. Only a few works discuss the validity of these assumptions in open systems (Flach and Schnell, 2006; Stoleriu et al., 2004, 2005). We have shown here that Michaelis–Menten kinetics are still valid when the reaction of interest is embedded in a larger reaction scheme (! S ! P !), but have reported here a few observations to keep in mind. First, the enzyme does not need to be in small amount compared to the substrate. Second, the E–S binding rates must be fast compared to the

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reactions of production of the substrate and degradation of the product. If it is not the case, the relaxation time to the steady states induces quantitative changes in the system and this may affect the response of the system to both transient perturbations and periodic forcing. Upon appropriate tuning of some parameter values, it is usually possible to make the developed model consistent with its compact version. The same question arises for the Hill function. We have presented one possible molecular mechanism that can explain the occurrence of the Hill functions at the level of gene transcription. When the transcription module is embedded in a larger network, then a good agreement between the compact and the developed versions of the model is obtained, if the binding/unbinding rates are fast compared to the other processes (such as transcription or translation). This assumption is generally valid because processes like transcription/translation, which require the recruitment of many transcription factors and the polymerization of RNA and protein molecules, are much slower than the binding of one molecule of a transcription factor to the gene promoter. When these conditions are fulfilled, it is possible to reproduce the behavior predicted by the compact model with a developed model consisting of elementary reaction steps. This was illustrated here with two simple gene models, the Toggle Switch (Gardner et al., 2000) and Repressilator (Elowitz and Leibler, 2000). An agreement of two different levels of detail of a model was also obtained for the circadian clock (Gonze et al., 2002a,b). In all these systems, the deterministic behavior and the bifurcation properties are largely conserved. Compact models are thus qualitatively equivalent regarding the global dynamical properties and bifurcations structures to their corresponding developed models, provided they incorporate the same regulatory network in terms of feedbacks and delays and fulfill the necessary conditions.

6.3. The level of details depends on the question asked The main advantage of using compact models is to reduce the number of variables and parameters, making the core structure of the network more visible and facilitating the numerical analysis of the equations. Exploring the properties and robustness of the system in parameter space is much easier if the number of key parameters is reduced. Such models, although approximative, have been successfully applied to study dynamical properties of genetic networks both in systems and in synthetic biology. It is important to stress here that by “developing a model,” we mean to decompose the Michaelis–Menten and Hill function into a detailed reactional scheme. This consists of adding intermediary reaction steps and compounds. Apart from quantitative changes, these additional steps would not change the global dynamics of the system. Developing a model is justified when we want to access a particular parameter or process which

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is not explicitely taken into account in the compact model and to examine its effect on the dynamics. Of course, incorporating new regulations in a network may change the behaviour of the system. For the case of the cell cycle modeling, authors developed models mainly to add more regulations as they are discovered (Chen et al., 2004; Ge´rard and Goldbeter, 2009). Such detailed models allow to assess the relative importance of the regulations. The level of details thus depends on the question asked. Note that it might not be straightforward to explain nonstandard phenomenological functions by a detailed molecular scheme (Sabouri-Ghomi et al., 2008). In this chapter, the question of how to treat the “gene” in the developed version of the Repressilator and the Toggle Switch illustrates the difficulties that can arise when developing a model. Furthermore, biomolecular mechanisms are never “fully” developed. It is always possible to further decompose each molecular step. Indeed, a fully detailed scheme of the transcription would require taking into account the recruitment of each protein that make and activate the RNA polymerase, as well as each polymerization step. Such a detailed model was designed to study the lysis/lysogeny switch in the l phage (Arkin et al., 1998). Where to fix the limit? In theory, each reactional step can always be further detailed till we reach the level of “elementary reaction step.” But what does it mean?

6.4. The notion of elementary reaction steps in biology is elusive In physical chemistry, the Van’t Hoff law of mass action is a phenomenological model empirically valid also for nonelementary step chemical reactions (Berry et al., 2001; McQuarrie and Simon, 1999). Even in more simple inorganic reaction schemes, there is an ongoing discussion on the level of detail required to model these systems. Rigorous mathematical demonstration of the validity of such phenomenological kinetic laws is based on many assumptions such as ideal solution, homogeneity of the system, Maxwell velocity distribution, thermal equilibrium, and constant temperature in time and space. From a mathematical point of view, one may rapidly conclude that none of these assumptions are respected in real, experimental chemical systems, a fortiori, in biochemical and cellular systems. However, we do know that these phenomenological kinetic equations are good approximations in terms of the dynamics of experimental systems. One should not be confused by the mathematical proof of phenomenological laws based on stringent hypotheses with the empirical validity of such laws. These are two related but nevertheless separated issues. Elementary steps in experimental kinetics are very seldom known even in inorganic chemical reactions. Physical chemistry shows that phenomenological kinetic laws are very robust and valid in many cases that go beyond

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the assumptions made to perform rigorous mathematical demonstrations. From a pragmatic, experimental point of view, chemical kinetic laws are valid in a much larger range than their corresponding rigorous mathematical theorems based on, for example, ideal solution or perfect gases models (Berry et al., 2001; McQuarrie and Simon, 1999). This general consideration is also valid for biological models (Cornish-Bowden, 1995). We thus believe that the emphasis on the need for writing a biological model in terms of elementary steps is, for the majority of the systems studied, not justified. The need for using elementary steps is some kind of inaccessible idealization for the vast majority of chemical and a fortiori biochemical processes.

6.5. Stochastic simulations can be performed using compact kinetics To assess the impact of molecular noise resulting from the low number of molecules, it is necessary to resort to stochastic simulations. A commonly used method to simulate stochastic models is the Gillespie algorithm (Gillespie, 1977). To run the Gillespie algorithm (or any of its variants), a propensity of each reaction must be defined (see Appendix). These propensities are directly related to the rate constants. The Gillespie algorithm was shown to be mathematically correct for “elementary reaction steps.” However, the master equation describing stochastic processes may include propensities not based on mass action law. Because running this Monte Carlo-type simulation on large systems is CPU-consuming, especially for systems with fast-rate reactions, it is highly tempting to use Michaelis–Menten or Hill functions to compute these propensities, rather than developing models into multiple reaction steps. Several studies apply compact models for stochastic simulations (Gonze et al., 2003; Kraus et al., 1992; Longo et al., 2009; Ouattara et al., 2010; Song et al., 2007), but the accuracy of these models is sometimes questioned. For this reason, many authors prefer to run stochastic simulations on developed models (Cai and Yuan, 2009; Forger and Peskin, 2005; Gonze et al., 2002a; Kar et al., 2009). Here, stochastic simulations of the compact version and developed version of the Repressilator and of the Toggle Switch models show a good qualitative correspondence. In agreement with the observations in the present study, a developed model for circadian clock gives similar results as its compact version (Gonze et al., 2002b). Interestingly, the results of a detailed stochastic model for p53–Mdm2 (Cai and Yuan, 2009) were comparable to those of a compact model of the same system (Ouattara et al., 2010), although they have been established independently by two different groups. We believe that the accuracy of these approaches should be discussed in a broader context. First, as explained in the previous section, the need for “elementary reaction steps” is overemphasized. Second, the Gillespie

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algorithm (as well as their ODE counterpart) implicitly assumes that the molecules are homogeneously distributed and well stirred. It neglects spatial organization or specific diffusion processes. The validity of such simplifications should be much more questionable than the use of a compact instead of a developed model. Finally, the purpose of these simulations is often to highlight the link between the network structure and the impact of noise, to identify robustness factors, or to determine how the noise may affect qualitatively the dynamical behaviors. Being developed or compact, this kind of models should be fitted to make quantitative predictions regarding the exact level of noise. In addition, for most biological systems, the absolute number of molecules is unknown, and therefore, quantitative predictions should be based on available experimental data and may not require a lower-level description of the system (Elowitz et al., 2002; Ozbudak et al., 2002). Nevertheless, these compact models are good at estimating the effect of relative noise on the system, that is, although the quantitative level of noise may not be correct, its relative impact on different variables of the system can be analyzed. For example, the role of network regulation in damping or amplifying the relative amplitude of molecular noise can be analyzed by comparing different versions of compact models incorporating different regulatory feedbacks (Cagatay et al., 2009).

6.6. Spatial effects are important and may imply complex kinetics Biomolecular systems work with complex spatial organization. Diffusion and transport processes play an important role both in the regulation and in the timing of biological processes. Components do not move randomly in the cell (Agutter and Wheatley, 2000; Agutter et al., 1995, 2000; Florescu and Joyeux, 2010). The molecular mechanisms of such processes are complex (compartmentalization, facilitated diffusion, DNA sliding, DNA looping, actin-guided protein transport, etc.; Klenin et al., 2006; Lomholt et al., 2009; Minton, 2006; von Hippel and Berg, 1989). The mathematical foundation of chemical kinetics assumes perfect stirring, ideal solution, etc. Although some adaptations of enzyme kinetics have been suggested (through the concept of fractal kinetics; Kopelman, 1988; Savageau, 1995) and models based on partial differential equations (PDE), which include the spatial dimension, have been used, there is currently no satisfactory approach to model accurately spatial effects. Nevertheless, pragmatic kinetic approaches make use of purely phenomenological kinetic equations. Those phenomenological equations can be the result of complicated biochemical mechanisms involving large molecular assemblies and elaborate transport mechanisms. When a Hill kinetics is measured in vivo, it includes spatial aspects (Rosenfeld et al., 2005).

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We can thus assume that Michaelis–Menten or Hill-like functions in phenomenological models encapsulate this high molecular and spatial complexity, provided that hyperbolic or threshold kinetics are experimentally observed. Developing such compact high-level description into a detailed lower-level description can become a daunting task. This difficult task has to be tackled on experimental grounds, that is, measured data should be accessible.

7. Conclusion One of the major goals of modeling gene regulatory networks is to relate the network architectures (generally in terms of feedback and feedforward loops) to dynamical properties (e.g., multistability, oscillations). The question of how much molecular details must be taken into account is often debated. Our results and the considerations discussed above provide arguments that compact models constitute a suitable approach both for deterministic and stochastic modeling. Using developed models may be appropriate for specific questions pertaining to the effect of some particular variables or to explore specific parameter effects. Provided that a compact model captures all important regulations of a system, it is suitable for analyzing the global dynamical properties. Adding intermediary steps to such a model raises the issue of having experimental data about the new variables and parameters, although this development does not affect qualitatively its dynamical properties. Moreover, decomposing a compact model into lower-level processes is neither straightforward nor univocal, as many different mechanisms can correspond to the same higher-level description. To make appropriate choices of model development, it is thus important to keep in mind the biological question at stake as well as the available experimental information. Here, we have illustrated this point by showing that adding molecular details to standard compact models does not lead to important qualitative changes in the global dynamical properties such as multistability or oscillations. However, qualitative models can give quantitative insights in terms of relative values of parameters or variables, as shown by the Toggle Switch or the Repressilator, and this is why they are useful for designing synthetic systems (Elowitz and Leibler, 2000; Gardner et al., 2000). From a quantitative perspective, experimental data can be fitted by compact models (Ozbudak et al., 2004). The latter automatically encapsulate the effects of underlying molecular mechanisms and spatial aspects. In conclusion, from a phenomenological point of view, the main qualitative behaviors can be obtained by compact versions of deterministic or stochastic models. In modeling, there is high value in simplicity (May, 2004).

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ACKNOWLEDGMENTS We thank M. Kaufman for fruitful discussions as well as K. Faust and L. De Mot for helpful comments on the chapter. This work was supported by grant #3.4636.04 from the Fonds de la Recherche Scientifique Me´dicale (F.R.S.M., Belgium), by the European Union through the Network of Excellence BioSim (Contract No. LSHB-CT-2004-005137), and by the Belgian Program on Interuniversity Attraction Poles, initiated by the Belgian Federal Science Policy Office, project P6/25 (BioMaGNet).

Appendix: The Gillespie Algorithm The Gillespie method is a standard, rigorous algorithm to simulate a stochastic system (Gillespie, 1977). It associates a certain propensity with each reaction (from which a probability can be easily obtained); at each time step, the algorithm stochastically determines the reaction that takes place according to its probability as well as the time interval to the next reaction. At each time step, the numbers of molecules of the different reacting species as well as the reaction propensities are updated. In this approach, a parameter, referred to as the system size and denoted O, is used to convert the concentration into the number of molecules, and hence allows us to control the number of molecules in the system. This parameter, which has the unit of a volume, thus permits us to control the level of noise in the system. Although the original Gillespie algorithm is rigorous, running such stochastic simulations might be CPU-consuming (even for relatively small systems, millions of reactions may occur). This is why alternative, approximate, or hybrid approaches have been proposed (for a review of these methods, see Pahle, 2009). As discussed in the main text, another way to limit memory and CPU requirements is to use compact, nonlinear equations (i.e., Michaelis–Menten or Hill functions) to compute the reaction propensities.

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C H A P T E R

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Computational Modeling of Biological Pathways by Executable Biology Maria Luisa Guerriero* and John K. Heath† Contents 1. Introduction 2. Executable Modeling Languages for Biology 2.1. Petri nets 2.2. Rewriting systems 2.3. Process algebra 3. Intuitive Representation of Formal Models 3.1. The Narrative Language: A high-level executable textual language for biology 4. Case Studies 4.1. The JAK/STAT pathway 4.2. Circadian clocks 5. Conclusions and Perspectives Acknowledgments References

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Abstract “In silico” experiments (i.e., computer simulation) constitute an aid to traditional biological research, by allowing biologists to execute efficient simulations taking into consideration the data obtained in wet experiments and to generate new hypotheses, which can be later verified in additional wet experiments. In addition to being much cheaper and faster than wet experiments, computer simulation has other advantages: it allows us to run experiments in which several species can be monitored at the same time, to explore quickly various conditions by varying species and parameters in different runs, and in some cases to observe the behavior of the system at a greater level of detail than the one permitted by experimental techniques. * Centre for Systems Biology at Edinburgh, University of Edinburgh, Edinburgh, United Kingdom School of Biosciences and Centre for Systems Biology, University of Birmingham, Edgbaston, Birmingham, United Kingdom

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In the past few years there has been a considerable effort in the computer science community to develop computational languages and software tools for modeling and analysing biochemical systems. Among the challenges which must be addressed in this context, there are: the definition of languages powerful enough to express all the relevant features of biochemical systems, the development of efficient algorithms to analyze models and interpret the results, and the implementation of modeling platforms which are usable by nonprogrammers. In this chapter, we focus on the use of computational modeling to the analysis of biochemical systems. Computational modeling, in conjunction with the use of formal intuitive modeling languages, enables biologists to define models using a notation very similar to the informal descriptions they commonly use, but formal and, hence, automatically executable. We describe the main features of the existing textual computational languages and the tool support available for model development and analysis.

1. Introduction There are multiple incentives for the generation and analysis of robust models of biological processes (Kwiatkowska and Heath, 2009). At the simplest level, models provide a convenient way of storing and sharing knowledge of a particular system. Models potentially become more powerful when they offer the means to analyze processes without having to conduct laboratory experiments. If the model is an accurate representation of the natural process, dynamic analysis of model behavior allows the investigator to reason about the process in question with the objective of identifying logical flaws or areas of ignorance, test-specific hypotheses or scenarios, and reveal behaviors that might not be predicted by biological intuition. Models can also be used to simulate experimental outcomes with the purpose of prioritizing experiments when resource is limited or identify optimal experimental conditions to study particular outcomes in the laboratory. If models are a sufficiently faithful emulation of the natural process, they can be used to make confident predictions of future behavior under specific constraints which would have great practical value in, for example, predicting optimal therapeutic interventions in a disease state. The most common approach to modeling biological processes is the use of mathematical formalisms in which the relationship between quantities of reactants which change overtime are articulated in the form of equations which, for example, relate the values of an input value and an output value. Mathematical models can be studied either by analytical methods or by numerical simulation depending on their complexity. Mathematical models can be constructed from a priori descriptions of the biological pathway (Aldridge et al., 2006) or by fitting models to observed biological data ( Janes and Yaffe, 2006).

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In recent times, stimulated by the suggestions of Regev and Shapiro (2002), biological processes have been modeled using the formalisms of computer science. In this approach, the behavior of the system is described by algorithms—rules which describe the behavior of the system in the form of transitions between states. This approach has been termed “Executable” or “Algorithmic” biology (Fisher and Henzinger, 2007; Priami, 2009) as, in effect, an algorithm represents an executable computer program which can be analyzed by the large body of analytical and simulation tools developed by computer scientists. The algorithmic approach is especially useful for analyzing biological processes which are not deterministic (i.e., execution of a process may lead to different outcomes), concurrent (i.e., multiple processes execute at the same time and may exhibit interdependencies), and/or stochastic (i.e., system evolution depends on the likelihood for different reactions to occur). The emphasis in this approach to modeling is on the logic, or the temporal and causal relationship, between events in the biological process and, as discussed below, is particularly suited to in silico interrogation of biological models by changing parameter values or by formally verifying the satisfaction of specific system properties. At present, executable methods are based on a priori models based upon prevailing biological knowledge although approaches to the automated construction of executable models from biological data have been suggested (Heath, 2009; Vyshemirsky and Girolami, 2007). As a consequence of these developments in the past few years, there has been a considerable effort in the computer science community to develop computational languages and software tools for modeling and analysis biochemical systems. Several challenges need to be addressed in this context, such as the definition of languages powerful enough to express all the relevant features of biochemical systems, the development of efficient algorithms to analyze models and interpret the results, and the implementation of modeling platforms which are usable by nonprogrammers. A compromise between language expressiveness and ease of use is needed. Tools traditionally used for biochemical modeling allow users to develop models as complex and detailed as desired, but they require users to write mathematical formulas, a process which can be complex and errorprone. A number of tools for modeling and simulation have been now developed to hide as much as possible of the mathematical details from users. Some of these tools provide easy-to-use graphical user interfaces which frees the end-user from knowing the details of the underlying modeling language and of the simulation methods: systems can be modeled using the standard notation of biochemical reactions reactant1 þ . . . þ reactantn ! product1 þ . . . þ productm and reaction rates can be selected from a set of predefined rate laws. Examples of these tools are the SymBiology toolbox for MATLAB and COPASI.

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An alternative approach, on which we focus in this chapter, is the use of intuitive textual notations, that is “narrative-style” modeling languages which aim to be as close as possible to the informal descriptions biologists often use to describe models (such as Protein A binds to B to form a complex which can then phosphorylate. The phosphorylated complex becomes active.). Examples of these languages are the Narrative Language (NL; Guerriero et al., 2009) and the high-level textual interface of the SPiM tool (Kahramanogu˘lları et al., 2009). In this chapter, we focus on the existing high-level languages and in particular on the textual approach; with the help of two examples, we describe the main language features and the tool support available for model development and execution.

2. Executable Modeling Languages for Biology The issue of constructing models to represent complex biological systems involves the definition of techniques to develop and simulate models, and analyze the obtained results. In addition to simulation, formal models enable modelers to analyze systems: for example, formal properties of models can be verified to uncover causal relations between events, reachability of specific states, equivalences between different systems. Therefore, techniques for verifying such properties should also be developed to fully benefit from the formal modeling of biological systems. Several formal methods have been proposed to model, analyze, and simulate biological systems, such as Petri nets (Reisig, 1985), rewriting systems (e.g., membrane systems, Pa˘un, 2002; and Kappa, Danos and Laneve, 2004), and process algebra (e.g., the biochemical stochastic p-calculus, Priami et al., 2001; BioAmbients, Regev et al., 2004; Beta-binders, Priami and Quaglia, 2005; Bio-PEPA, Ciocchetta and Hillston, 2009). Since each of these approaches has different attributes, they can be more suitable for solving a particular class of problems or according to the desired level of abstraction. The model analysis techniques that are available differ for the various formalisms. Most of them are equipped with a discrete stochastic semantics, and some also allow for a continuous deterministic interpretation. For some of these languages, it is also possible to employ formal verification techniques such as model-checking to formally prove model correctness or identify existing errors: see Clarke et al. (1999) for an introduction on model-checking and Kwiatkowska et al. (2002) for a description of the well-known stochastic model-checker PRISM (available from PRISM). These kinds of computational models can either be analyzed statically via techniques which work at the level of model structure, or can be

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dynamically executed via stochastic simulation techniques such as, for instance, the Gillespie’s stochastic simulation algorithm (Gillespie, 1977), which produce time-course observations of amounts of the participating species. For languages which have a deterministic interpretation, numerical solution of the associated set of ordinary differential equations (ODEs) can be also performed, together with the various mathematical methods available for the analysis of ODE systems (e.g., bistability, bifurcation, and continuation analysis). Existing simulation tools, generally, also allow modelers to perform in a simple way model experimentation (parameter sensitivity analysis, components knockdown, dose response experiments, etc.). Stochasticity has been shown to be very relevant in modeling biological systems such as genetic networks, which involve a small number of molecules and consequently can exhibit significant variability due to noise. In addition, analysis via stochastic simulation can provide modelers with an interesting parallel between simulation results and experimental data: the results of a single run of stochastic simulation can be considered as the modeling counterpart of single cell data, while results obtained averaging over multiple runs is equivalent to population of cells data. These considerations are particularly interesting for systems exhibiting periodic/oscillatory behavior, where the robustness of the systems over a population of cells might be affected by phase asynchrony. In this section, we give a brief overview of the mentioned formalisms, presenting their main features, advantages, and disadvantages. In the rest of the chapter, we focus on process algebras, and in particular on Bio-PEPA and Beta-binders, but the concepts we describe can be similarly applied to the other formal languages mentioned here.

2.1. Petri nets A Petri net model of a biochemical system is an automaton whose states represent molecular species and whose transitions represent reactions transforming reactants into products. Graphically, the automaton is a graph with nodes representing states, which are connected, by arcs, to nodes representing transitions (Reisig, 1985). Petri nets are able to represent generic biochemical systems, and several variants of them (e.g., timed, stochastic, continuous, hybrid) have been defined in order to better address specific types of systems or perform specific types of analysis. Petri nets rely on well-assessed mathematical foundations which enable to obtain both transient and steady-state solutions of models. Moreover, static analysis of Petri nets models can also be used to uncover modeling errors: for instance, reachability analysis can be used to identify parts of the model which are not connected, boundedness analysis can be used to ensure uncontrolled growth of molecules is not possible, and invariant analysis can identify violations of the law of conservation of mass. See Peleg et al. (2002)

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and Heiner et al. (2008) for more details and examples. Petri nets are among the best known formalisms for biological modeling, and owe their popularity to their graphical intuitive representation, which makes models easily understandable also for nonmodelers. This strength, however, comes to a limit for big or highly interconnected models, which become too messy to represent graphically. Several tools for development and analysis of Petri nets models are available from Petri Nets World.

2.2. Rewriting systems Rewriting systems consist of a set of objects and a set of relations over them which specify how to transform a set of objects into another. In the case of biochemical systems, the objects represent molecular species and the relations represent the biochemical reactions. The dynamic evolution of a system is obtained by applying the relations starting from a set of objects initially present. Rewriting relations can be either deterministic, not deterministic, or stochastic. Though there are a number of notably different formalisms which can be grouped into the category of rewriting systems, the syntax of most of them is similar, and it is also similar to the biochemical reaction equations, traditionally used to describe biochemical systems. We mention here two of the best known rewriting system formalisms used for biochemical modeling. 2.2.1. Membrane systems Membrane systems, or P systems (Pa˘un, 2002), are computational models inspired by the notion of cellular membranes and the observation that complex biological systems are composed of independent computing processes separated by and communicating through membranes. The evolution of a membrane system starting from an initial configuration of membranes and objects is obtained by repeated application of evolution rules, representing biochemical reactions and movement of objects across membranes. Membrane systems are a powerful method for describing and simulating multicompartment models involving mainly intracompartment reactions and limited intercompartment exchange of information. Several simulators for membrane systems are available from PSystems. 2.2.2. Kappa Kappa (Danos and Laneve, 2004) is a rule-based language for molecular biology, which focuses on the description of protein interaction and complexation. Agents representing biochemical species can have explicit interfaces (i.e., interaction and modification sites), and a set of rewriting rules describes the evolution of molecules in terms of activation and complexation of their interfaces. Each rule, representing a biochemical reaction,

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specifies how agents’ interfaces change following its occurrence, and the constraints that agents need to satisfy in order to be involved in it. An interesting feature of Kappa is that rules can include agents with partially defined interfaces (called patterns), which allows modelers to define generic rules by omitting the parts of agents’ interfaces which are not relevant to their occurrence. This partial form of rules makes Kappa particularly well suited to represent systems exhibiting a high degree of symmetry: in this case, the number of pattern rules is potentially much smaller than the number of reactions, leading to a huge gain in the compactness of the system description. A set of modeling and analysis tools for Kappa, available from RuleBase, can be used to perform stochastic simulation and additional analysis, such as the identification of stories in model evolution, which represent causality chains.

2.3. Process algebra Process algebras were originally developed to give semantics to concurrent programs, and were used to model computer networks and mobile communication systems, to specifying communication and security protocols and to verifying their correctness (Milner, 1989). Recently, following Regev and Shapiro’s (2002) landmark paper, process algebras have also proved themselves as powerful tools for quantitative dynamical modeling of biological systems. The basic components of a process algebra model are concurrent processes which can interact via communication and exchange of information. Processes have internal states and interaction capabilities. When a process receives an input, its behavior is based on its internal state and on the content of the input. A direct consequence of interaction can be the modification of the internal state and of the interaction capabilities of the process. Complex entities can be described hierarchically as the composition of smaller entities. The abstraction of biochemical systems proposed in Regev et al. (2001) consists in representing the molecules composing biological systems (e.g., proteins, genes, etc.) by computational processes which can interact by means of the standard primitives of process algebras. The occurrence of a reaction is represented by an interaction between processes representing its reactants and products, whose effect is a change in the state of the involved processes to represent the transformation of reactants into products (Table 8.1). Table 8.1 The “molecules as processes” abstraction of process algebra Molecule

Process

Cell

Interaction capability Interaction Modification

Channel Communication State and/or channel change

Interaction capability Signal State change

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Process algebras are inherently discrete (i.e., the amounts of biochemical species are given in terms of molecule counts rather than in concentrations as in differential equations), and are generally equipped with a stochastic semantics (i.e., reactions are associated with a probability of occurrence rather than a rate). The behavior of models can be obtained by stochastic simulation or by other kinds of formal analysis: for instance, reachability and causality analysis enable modelers to identify interesting behaviors in a model. Moreover, process algebra come equipped with well-assessed equivalence relations which could be powerful tools for biology: for example, the equivalence of the same functional unit in different organisms could be used as a measure of behavioral and structural similarity. Several process algebras have been developed and applied to the modeling of biological systems. Though the underlying concepts are the same for all of them, there are some noteworthy differences in the considered level of abstraction and in the language operators. We mention here the process algebras which have been used more for biochemical modeling. 2.3.1. Biochemical p-calculus The biochemical stochastic p-calculus (Priami et al., 2001) is an extension of the p-calculus (Milner, 1999) specifically designed for biochemical modeling, in which interactions are represented as input/output communications between pairs of processes and the “molecules as processes” abstraction can be applied straightforwardly. The biochemical p-calculus can rely on the strong formal foundations of the p-calculus, for which causality, locality, and equivalence relations have been defined. The p-calculus style of modeling biochemical interactions and complex formations is elegant, but it can sometimes cause models to be too abstract and hard to understand. For instance, the use of binary communications does not allow modelers to represent in a straightforward way interactions involving three or more processes or reactions with arbitrary stoichiometric coefficients. Successive extensions of the biochemical p-calculus focused on the definition of new operators to make the description of specific features of systems easier (e.g., compartments and membranes, or protein binding, and allosteric sites). The two most common simulation tools for the biochemical p-calculus are SPiM and BioSPI. 2.3.2. BioAmbients BioAmbients (Regev et al., 2004) extends the biochemical p-calculus by allowing processes to nest in order to represent hierarchies of cellular and subcellular compartments and membranes. Interactions are divided into local (i.e., within one compartment) and nonlocal (i.e., between one compartment and the one enclosing it), hierarchies of compartments are dynamical, and processes can move in and out of compartments by synchronizing with membranes.

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Thanks to its notions of localization of processes and hierarchy of compartments, BioAmbients is ideal to describe systems whose focus is on movement of objects across membranes and on the modeling of dynamic rearrangement of compartments. BioAmbients models can be analyzed via stochastic simulation using the BAM tool (BAM). In addition, static analysis techniques such as control flow analysis (Nielson et al., 2007) and pathway analysis (Pilegaard et al., 2008) have been defined for BioAmbients and can be used to investigate causal properties of models. 2.3.3. Beta-binders Beta-binders (Priami and Quaglia, 2005) is another extension of p-calculus, whose focus is on the description of interaction and complexation of molecules via communication between compatible protein sites. Processes representing proteins are enclosed in boxes with interaction capabilities representing their allosteric/binding sites, and interactions between two processes are allowed only if their sites have compatible types (according to a user-defined notion of compatibility). A basic notion of localization and movement of processes within a (static) compartment hierarchy is also defined. The notion of compatibility of interaction sites and the definition of primitives for making interaction sites available or not and for describing their modification make the representation of binding reactions intuitive and the description of reactions definitely more informative than with only input/output communication as in the p-calculus. Interactions, however, are still binary, and hence generic reactions involving more than two molecular species cannot be defined explicitly, thus limiting the ability of modelers to define abstract reactions. The BetaWB is a collection of tools for modeling, simulating, and analyzing models described in the BlenX programming language, which is based on Beta-binders. 2.3.4. Bio-PEPA In the “molecules as processes” abstraction used for the process algebras mentioned so far, a process represents a molecule, and the occurrence of a reaction represents the change of state in the reactant and product molecules involved. The abstraction underlying Bio-PEPA (Ciocchetta and Hillston, 2009), called “species as processes,” is slightly different: a process represents a molecular species, and the occurrence of a reaction represents the change in available amount of the involved molecules. Processes interact by means of shared action names representing reactions and specifying their role in the reaction (reactant, product, inhibitor, etc.) and their stoichiometric coefficient for that reaction; the effect of a reaction occurrence is to decrease the amount of reactants and increase the amount of products. Generic reaction rate law and multiway synchronization can be used, thus allowing the modeling of generic reactions with any number of interacting processes. A basic notion of localization and movement similar to Beta-binders is defined.

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The syntax of Bio-PEPA is very similar to that of a system of ODEs (indeed, the translation from ODEs to Bio-PEPA is mechanical) and, therefore, it is quite easy for modelers familiar with ODE models to understand the meaning of Bio-PEPA models. On the other hand, as for ODEs, models must satisfy a precise structure: this makes it hard to exploit symmetries and patterns for making models more compact, which can be an issue for large size models. Species amounts in Bio-PEPA can either be molecule counts (discrete semantics) or concentrations (continuous semantics), hence allowing both stochastic analysis methods and numerical methods for differential equations. An intermediate representation in terms of discrete levels of concentration is also possible, whose purpose is to make the system smaller and bounded (in terms of the range of values the species can assume) so that model-checking techniques can be applied to BioPEPA models. The Bio-PEPA Eclipse Plug-in, available from Bio-PEPA, is a framework for model development and analysis, which enables modelers to perform static analysis (e.g., invariant analysis and identification of sources and syncs), dynamic time-series analysis (stochastic simulation and solution of ODEs), and to export models for analysis via other simulation, modelchecking, and analysis tools).

3. Intuitive Representation of Formal Models Central to the process algebra approach to biological modeling is the absolute requirement to articulate the biological process in a form that conforms to the rules of formal logic (thus allowing computational execution) and has an unambiguous syntax (to allow translation into a executable programme) but, at the same time, faithfully and accurately represents the biological process under consideration. It would also be advantageous if the method of model formulation was readily understood by both biologists and computer scientists. A further advantage should be that the biological formulation should be “portable,” that is formatted in a fashion which can be employed by a variety of different computational tools. Formal computer languages are, unfortunately, hard for the nonspecialist to understand and use. Moreover, in general, the more formal the language is, the more complex it is for noncomputer scientists to write and understand models. Developing a model using process algebras, for example, requires modelers to learn the meaning of a specific set of language operators (usually not straightforward, being quite different to traditional biochemical reaction description). In addition, in order to be unambiguous, formal models must comply with a precise structure and generally cannot include any sort of uncertainty in either model structure or quantitative aspects. However, formal foundations of descriptions are mandatory requirements

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in order to enhance the understanding of complex biological systems and to perform automatic simulation and analysis of models. Computer science modeling is specifically designed to meet the above requirements, but it should hide as many technical details as possible from users in order to be usable by nonexpert users. The definition of intuitive notations for modeling biological systems is an area of active investigation and several—both graphical and textual— informal notations are been proposed. The drawback of many notations classically employed in the biological literature is that they can be ambiguous in construction or syntax and conceal important features of the pathway—such as spatial or temporal features—from the naı¨ve viewer. Hence it is not possible to have a direct translation of a model developed using these notations into a computational executable model that can be simulated on a computer. Two broad classes of biological notation have been developed to address this challenge: graphical and textual notations. In graphical notations, systems are described by arrows (representing reactions) connecting boxes (representing the involved molecules), and the localization of molecules can also be visually represented. Several graphical notations have been developed, among which Kohn Molecular Interaction Maps (Kohn, 1999) and SBGN (SBGN Home Page), and several tools (e.g., CellDesigner, Funahashi et al., 2003, available from CellDesigner, VCell, KEGG, and EPE; Sorokin et al., 2006, available from EPE) allow some sort of graphical representation. The clear advantage of graphical notations is that they are (Harel, 1987) intuitive for the biologist and hence they make it easy to build and work with models. However, graphical notations tend to be ambiguous and the interpretation of the meaning of their components is often left to the simulator rather then being standardized and temporal and/or spatial aspects are not frequently portrayed. In addition, viewing large or highly interconnected models and searching through them is often cumbersome. A second graphical approach draws directly on computer science formalisms for representing processes as state machines. Statecharts (Harel, 1987) is a formal graphical method which extends state diagrams to express hierarchical, concurrent, and communicating processes and is thus particularly suited to the description of biological processes. The Statechart approach has recently been modified to explicitly encompass biological processes in the form of Biocharts (Kugler et al., 2010). They can be nested which allows representation of a process at different hierarchical levels, for example, in the biological context, the behavior of pathways within and between cells. They have proved an effective tool for representing biological processes for computational execution for both pathways (Fisher et al., 2005) and multiscale modeling of tissues (Setty et al., 2008). The alternative approach is to employ textual representations. These are intuitively closer to a set of program instructions, can be easily modularized

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and equipped with a formal syntax and grammatical structure which can be presented in a way which is biologically intuitive but retains the ability to describe concurrent, hierarchical, or temporal features. Despite not being as compact and intuitive as graphical representations, text-based descriptions of biological processes generally become more readable than graphical descriptions when employed to represent big highly interconnected systems. System Biology Markup Language (SBML Home Page; Hucka et al., 2003) is a machine readable language based on XML which has become the de facto standard for sharing biological pathway models and has the benefit of a large community of users and developers. SBML has been designed as a storage/exchange language and, hence, as a machine-readable rather than human-readable language, and its syntax is too heavy for users to read and write SBML models by hand. On the other hand, most modeling frameworks nowadays allow import/export of SBML models. SBML models are also available via an online repository (BioModels Database) for community sharing. Tools for translating SBML into process algebras have been described (Ciocchetta et al., 2008; Eccher and Priami, 2006). SPiM, one of the simulation tools for the biochemical p-calculus mentioned earlier, has been equipped with a high-level interface for model development and visualization. Models can be described either graphically or textually. In the graphical notation (Phillips et al., 2006), a system is represented as a directed graph with nodes for processes and arcs for interactions; in the high-level textual representation (Kahramanogu˘lları et al., 2009), a system is described by a set of constrained English sentences, in the style of the NL described in the following section. In both cases, these notations are automatically translated into an SPiM program which can be simulated. In the rest of this section, we describe in detail the NL, a high-level textual language which has been recently defined for biochemical modeling and from which process algebra models can be automatically generated.

3.1. The Narrative Language: A high-level executable textual language for biology A high-level textual modeling language called NL has been presented in Guerriero et al. (2009). The motivation which leads to the definition of the NL language is to allow modelers to formulate models in an intuitive way— as close as possible to the sort of informal description commonly employed by biologists to describe systems—but in such a way that this formulation could be easily translated into executable models. The NL is a narrativestyle language, in which the sequence of events occurring in a biochemical system is described by means of a sequence of semiformal textual rules. The main focus of the language is the representation of the molecules

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(e.g., proteins, genes, metabolites, enzymes) involved in cellular processes such as signaling pathways and of the dynamic changes which they undergo, in terms of state changes and of modifications of their binding/allosteric sites. Some basic geometrical constraints can be also specified by including topological information relative to the compartments in which the molecules are located. Finally, it is possible to specify different sorts of temporal/ causal relationships between events (sequential, concurrent, and competing events). Each numerical value (i.e., initial amounts, kinetic parameters, and compartment sizes) can be assigned a “reliability value,” which is a number describing its reliability, ranging from 100% for precise values obtained from high-quality experiments, to 0% for values with no experimental evidence. Reliability values are optional and do not influence the behavior of the program: they are annotations to inform use of the model. The basic entities of the language are components (proteins, genes, etc.) and compartments (cellular and subcellular locations). Molecules can interact (e.g., bind/unbind), undergo biochemical modification (e.g., phosphorylation/dephosphorylation) and move between compartments. The dynamic behavior of a model is described in the form of a narrative of events involving the basic entities, which imposes a temporal sequence and defines interdependencies between events. A compartment can represent a cellular or subcellular compartment (e.g., nucleus, cytoplasm, cell membrane) or an abstract location; it is described by an identifier, a name, the size, and the number of spatial dimensions (to distinguish between 2D compartments, i.e., membranes, and 3D ones). As an example, Table 8.2 shows the compartments involved in the gp130/ JAK/STAT model presented in Guerriero et al. (2009). A component represents a molecular species and it is identified by a name, an informal description, a list of interaction sites, a list of states, a list of locations, and the initial quantity of molecules present. Table 8.3 shows an example of model components which are part of the components of the gp130/JAK/STAT model presented in Guerriero et al. (2009). Since the effect of biochemical reactions is essentially the modification of protein sites, components are essentially seen as a list of interaction sites defined by a name and a state (e.g., active, bound, phosphorylated). States can also be associated with components to represent modifications occurring at Table 8.2 Example of model compartments Id

1 2 3 4

Name

Exosol Cell_membrane Cytoplasm Nucleus

Size

9.91 1.26 2.09 0.25

   

 12

10 10 7 10 12 10 12

Unit of measure

Dimensions

l dm2 l l

3 2 3 3

Table 8.3 Example of model components

Name

Descr

OSM gp130

Ligand Receptor

OSMR

Receptor

STAT3

Effector

SOCS3

Inhibitor

Site

Site_state

Site_act

OSM SOCS3 OSM SOCS3 Y705 gp130 OSMR

Bound Bound Bound Bound Phosphorylated Bound Bound

False False False False False False False

State

State_act

Comp

Comp_act

Init_ amount

Rel (%)

Bound Bound

False False

Exosol Cell_membrane

True True

3000 1000

100 50

Bound

False

Cell_membrane

True

1000

50

Dimer

False

Cytoplasm Nucleus

True False

3000

30

Bound

False

Cytoplasm

True

0

100

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unknown interaction sites. A label associated with each state and site specifies the status of the component at system initialization. For instance, in Table 8.3, STAT3 has a phosphorylation site, Y705, which is initially not phosphorylated, and an unknown homodimerization site represented by a state called dimer. If the location of the protein is relevant to the model, the compartments in which it can be located during the evolution of the system are specified, as references to their identifiers in the table of compartments (or their names, if unique). A label associated with each compartment defines where the component is located initially. For instance, STAT3 can be either in the nucleus or in the cytoplasm, and is initially assumed to be in the cytoplasm. Finally, the initial quantity of the component is defined in terms of molecule counts. A reaction specifies the details of a biochemical modification or binding reaction; it is described by an identifier, an informal description, the reaction type (e.g., binding, unbinding, dimerization, phosphorylation, relocation, etc.), the reaction rate (i.e., the kinetic constant), and the reaction volume (i.e., the volume in which the reaction occurs, which can be the name of a compartment, or a different value in case the reaction is known to be localized). Table 8.4 contains an example of the definition of reaction details. A narrative of events describes the evolution of the system as a sequence of events, which can be optionally grouped into processes. An event is a constrained textual description of a biochemical reaction involving at most two components: it is described by an identifier, an informal description, a semiformal description, the identifier of the reaction associated with the event (in the table of reactions), and optionally an identifier of alternative events. The semiformal description, which can be prefixed by conditions on the state of components/sites or on their position, specifies the occurring reaction (e.g., phosphorylates, relocates to, binds, unbinds) and the involved component(s). The underlying assumption is that each event involves an interaction/modification of one site of the involved components: if no site is specified, the interaction/modification is assumed to involve one of the Table 8.4 Example of model reactions

Id Type

Rate

15 16 17 18 19 20 21 25

4.8  108 M 1 min 1 0.06 min 1 0.2 min 1 0 min 1 inf min 1 inf min 1 1 min (t1/2) 0.01 min 1

Binding Unbinding Phosphorylation Dephosphorylation Unbinding Homodimerization Relocation Synthesis

Unit

Rel (%) React_vol

Rel Unit (%)

20 30 80 0 10 50 10 50

l l l l l l l l

Cytoplasm Cytoplasm Cytoplasm Cytoplasm Cytoplasm Cytoplasm Cytoplasm Nucleus

50 50 50 50 50 50 50 50

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Table 8.5 Example of model events Id Description

1

5

39 40 45

49

50

if gp130.LIF is not bound and LIF is not bound and gp130.typeI is not dimer and gp130.typeII is not dimer then LIF binds gp130 on LIF if gp130.OSM is not bound and OSM is not bound and gp130. typeI is not dimer and gp130.typeII is not dimer then OSM binds gp130 on OSM if STAT3.OSMR is bound then STAT3 phosphorylates on Y705 if STAT3 is in cytoplasm then STAT3 dephosphorylates on Y705 if STAT3 is in cytoplasm and STAT3 is dimer and STAT3. gp130 is not bound and STAT3.LIFR is not bound and STAT3.OSMR is not bound then STAT3 relocates to nucleus if STAT3 is in nucleus and STAT3 is dimer and STAT3.Y705 is phosphorylated and STAT3.PIAS3 is not bound then STAT3 synthesises SOCS3 SOCS3 degrades

React Alt

1

5

1

17 18 21

25

30

component’s generic states; if a list of sites is specified, the event represents simultaneous reactions involving different sites. Conditions on events are used to enforce the ordering of sequential events (e.g., a phosphorylation of a site of a protein is allowed only after it is bound to another protein); mutually exclusive events (e.g., competitive binding) are handled specifying which events are alternatives to each other; events that are not explicitly declared either alternative or sequential, are considered independent and are treated as concurrent events (e.g., independent events involving different proteins). Table 8.5 reports some examples of definition of events. 3.1.1. Tool support for model execution Since each entity and each description of event in the NL must satisfy rigorous language constraints, models written in the NL are formal and unambiguous. Consequently, NL models can be translated in a mechanical way into executable models (in process algebras or other formal languages), enabling modelers to easily simulate models developed in the NL. Translation of NL models into formal languages obviously requires knowledge of the target language, which means that the ability of users to perform analysis of NL models using a particular computational tool depends on their knowledge of the tool and of its underlying modeling language. Hence, to exploit the full power of the NL, automatic translation into formal languages is needed, so that users could potentially execute models without any specific knowledge of the particular target language used by different computational tools.

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The N2BB tool, available from N2BB, implements an automatic translation (Guerriero et al., 2007) of models specified in the NL into executable models compatible with the BetaWB tool. BlenX4Bio, described in Priami et al. (2009) is a tabular interface to the BetaWB, allowing modelers to define the entities of the NL via a user-friendly graphical interface. Using these tools, modelers who are not familiar with the BlenX language can build models simply filling the tables with the components, reactions, parameters, etc., and automatically obtain a model ready for execution via the BetaWB simulator, to which modelers only need to add the information on the time interval for which they want the simulation to run; once the simulation is completed, the time-course results can be either directly visualized in the BetaWB plotter or exported into other tools for further analysis.

4. Case Studies In this section, we present two case studies in order to illustrate the process algebra modeling approach and the kinds of analyses which can be performed using it. In Section 4.1, we illustrate in great detail a computational model of the JAK/STAT signaling pathway, focusing mainly on the model description in both the NL and process algebras; in Section 4.2, instead, we show some results obtained by analyzing a process algebra model of a circadian clock, which clearly show the difference between stochastic and deterministic models.

4.1. The JAK/STAT pathway As an example of modeling of signaling pathways, we describe the NL model of the gp130/JAK/STAT pathway presented in Guerriero et al. (2009). In that work, starting from the existing knowledge biologists had of the system, the NL model was developed and automatically translated into a BlenX model using the N2BB tool and simulated using the BetaWB simulator. Starting from the same NL model, in a later work (Guerriero, 2009), the model was translated into Bio-PEPA, which enabled us to obtain additional simulation and model-checking results. In this section, we briefly describe the system and report some extracts of the models and some analysis results. 4.1.1. The pathway The gp130/JAK/STAT signaling pathway is an interesting case study for computational modeling as it illustrates the role of both component compartmentalization of components and inhibitory mechanisms of different types. The pathway is also of significant biological interest due to its functions in human fertility, neuronal repair cancer, and hematological development (Mahdavi et al., 2007; Singh et al., 2006; Swameye et al., 2003) and accordingly

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it has become a focus for the development of targeted therapeutic interventions. In this respect, modeling the gp130/JAK/STAT pathway and exploring its parameter sensitivities should provide valuable insights into identifying entry points for the development of new forms of intervention. Signaling through the gp130/JAK/STAT pathway is initiated by ligands of the interleukin 6/leukemia inhibitory factor family binding to the gp130 transmembrane receptor and a second receptor which, depending on the identity of the ligand, could be a second copy of gp130 or the structurally related receptors LIFR and OSMR. The creation of transmembrane receptor dimers elicits activation of the receptor-associated JAK kinase creating a phosphorylated residue which acts as a binding site for STAT3. Phosphorylation of STAT3 results in release from the receptor complex and transport across the nuclear membrane where it is able to bind to DNA target sites. Dephosphorylation by the phosphatase TC-PTP of STAT3 results in nuclear export where the phosphorylated STAT3 is free to reengage with phosphorylated receptor. The pathway is thus a regulated nuclear-cytoplasmic shuttle. We also consider two forms of pathway inhibition: the action of SOCS3 proteins which are induced in response to the DNA bound form of STAT3, and the action of PIAS3. This informal description of the pathway is shown in the form of a diagram in Fig. 8.1 and encoded formally as described later in this section. LIF

Extracellular space

r5 LIF LIF

r3

LIF LIF

r6

LIF r8

P

SOCS3 SOCS3

P

LLI FIFRR

gpp 1133 00

LLI IFF RR

P

P

P

r17

STAT3

LIF LIF

r9 ggpp 1133 00

P

13

LLII FFR

Cell membrane

r7

gp

00

ggpp 1133 00

0 13

LIL FIR FR

r4

gp

13

LI

gp

gp

13

0

0

FR

LIF LIF r2

STAT3

LLI IFF R

LIF LIF r1

P P

P P SOCS3

STAT3

STAT3

STAT3 STAT3

SOCS3

r10

Cytoplasm P

P

STAT3

STAT3

r11

r15

STAT3

P P STAT3

r12

STAT3

STAT3

STAT3

STAT3

P P STAT3

r16 SOCS3

R

0

Nucleus

STAT3

O SM

13

r13

r14

OSM

gp

0 13 gp

LI FR

0 13 gp

LI FR

OSM

LIF

STAT3

PIAS3

P P

r18

PIAS3

STAT3

Figure 8.1 Graphical representation of the gp130/JAK/STAT pathway.

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The molecular species we consider in the model are: two ligands (LIF and OSM), three membrane-bound receptors (gp130, LIFR, and OSMR), one effector (STAT3), and two inhibitors (SOCS3 and PIAS3). The receptor-associated JAK kinase and TC-PTP phosphatase are implicitly modeled. Receptors are activated by ligand bindings, and active receptors dimerize to form receptor complexes (gp130:LIFR or gp130:OSMR; reaction r1 in Fig. 8.1). Once the receptor dimeric complex is formed, each receptor subunit (gp130, LIFR, and OSMR) can undergo JAK-mediated phosphorylation (r2). STAT3 can bind on receptors’ phosphorylated sites (r3), and the binding of STAT3 leads to its activation (phosphorylation; r4). Once phosphorylated, STAT3 dissociates from the receptor complex, and its phosphorylated site allows STAT3 to homodimerize (r5). When STAT3 is in dimeric form, it can translocate into the nucleus (r6) where it can carry out its specific functions (not modeled here): STAT3 binds to the DNA, thus activating the transcription of downstream gene targets. Nuclear STAT3 dimers are inactivated through TC-PTP-mediated dephosphorylation, which leads to the dimers’ dissociation (r7) and to STAT3 export to the cytoplasm (r8), where STAT3 can undergo additional cycles of activation. The two inhibition mechanisms considered are due to SOCS3 and PIAS3. SOCS3 is synthesized by STAT3 (r9) and it acts by competing with STAT3 in binding to receptors (r10). PIAS3 acts by binding to active nuclear STAT3 (r11). 4.1.2. A Narrative Language model We overview here the main features of the NL model presented in Guerriero et al. (2009). The full model is included in the supplementary material of that work. We describe here in more details how STAT3 is modeled and the reactions in which it is involved. Additional extracts of the NL model have been already shown in Section 3.1. Four compartments are involved in the system: the exosol (the extracellular space, where the two ligands are located), the cell membrane (location of the receptors), the cytoplasm (initial location of STAT3), and the nucleus (in which STAT3 can translocate). The information given about the compartments is reported in Table 8.2 (note in particular the field called id which can used in the rest of the model to reference the compartments, and the compartment size which is needed for stochastic simulation). Table 8.6 reports the definition of the model component representing STAT3. For our purpose, STAT3 has one phosphorylation site (Y705, initially not phosphorylated) and four binding sites for binding of the three receptors and of PIAS3 inhibitor (all initially unbound). STAT3 can be present either as a monomer or as a homodimer, so a dimer state (initially set to false) is also present. In addition, it can be either in the cytoplasm (initially set to true) or in the nucleus (initially set to false). The initial amount of STAT3 is set to 3000 and a reliability of 30% is associated with this numerical value.

Table 8.6 Component representing STAT3 Name

Descr

Site

Site_state

Site_act

State

State_act

Comp

Comp_act

Init_amount

Reliab (%)

STAT3

Effector

Y705 gp130 LIFR OSMR PIAS3

Phosphoryated Bound Bound Bound Bound

False False False False False

Dimer

False

Cytoplasm Nucleus

True False

3000

30

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237

Receptors contain one or two ligand binding sites (OSMR has only one site for OSM, while LIFR and gp130 also have one site for LIF), one binding site for SOCS3, and a number of phosphorylation sites. Moreover, receptors can be in dimeric state (an additional site in gp130 allows us to distinguish between the two types of OSM receptors). Ligands, located in the exosol, only have a bound state to represent their binding to receptors, similarly to the two inhibitors SOCS3 and PIAS3. Table 8.7 reports some of the events involving STAT3: its activation (binding to active receptors, phosphorylation, and homodimerization), its shuttling between cytoplasm and nucleus, and the reactions related to the inhibitors (synthesis of SOCS3, and binding of PIAS3). As there are a lot of replications of events involving the three different receptors, we report here only the ones for gp130, but analogous events are defined for LIFR and OSMR. Each event is defined by a reaction involving one or two components, prefixed by a sequence of conditions involving the reactants, and by a reference to the identifier of the reaction (referencing to Table 8.8). For instance, event 37, which represents the phosphorylation of site Y705 of STAT3 following the binding of STAT3 to one of the receptors, is defined as “STAT3 phosphorylates on Y705” and it can occur only “if STAT3.gp130 is bound.” The condition for the occurrence of the binding event 31 (“gp130 binds STAT3 on gp130”) is more complex as it is composed by a conjunction of conditions (“if gp130.Y767 is phosphorylated and STAT3 is in cytoplasm and STAT3 is not dimer,” etc.) that STAT3 and gp130 need to satisfy for the binding to occur. The type, kinetic rate parameter, and reaction volume of the reactions involving STAT3 are reported in Table 8.8. Rate parameters are either a number or the keyword “inf” (for instantaneous reactions), and their units of measures are also specified. Reaction volumes in this case are simply the size of the enclosing compartments. In general, many rate parameters in biological pathways are either unknown or subject to considerable uncertainty. In initially setting up the model, therefore, it is common to use generic values—or data drawn from the literature—where the relative rates of reactions are employed (see, e.g., Fisher and Henzinger, 2007) and the task of identifying accurate parameters to capture the quantitative aspects of model behavior is carried out during the model analysis phase via parameter exploration. 4.1.3. The Bio-PEPA model The described NL model was translated into a Bio-PEPA model, presented in Guerriero (2009). The translation, though done manually, is quite mechanical, due to the similarities in the structure of the two modeling languages and hence it could potentially be implemented in a tool. In Bio-PEPA, each form of the involved molecular species is modeled as a distinct component: different species components are defined to represent

Table 8.7 Events involving STAT3 Id Description

STAT3 activation 31 if gp130.Y767 is phosphorylated and STAT3 is in cytoplasm and STAT3 is not dimer and gp130 is not bound and gp130.SOCS3 is not bound and STAT3.gp130 is not bound and STAT3.LIFR is not bound and STAT3.OSMR is not bound and STAT3.Y705 is not phosphorylated then gp130 binds STAT3 on gp130 37 if STAT3.gp130 is bound then STAT3 phosphorylates on Y705 STAT3 unbinding and homodimerization 41 if gp130.Y767 is phosphorylation and STAT3 is in 3 and STAT3 is not dimer and gp130 is bound and gp130.SOCS3 is not bound and STAT3.gp130 is bound and STAT3.LIFR is not bound and STAT3.OSMR is not bound and STAT3. Y705 is phosphorylated then gp130 unbinds STAT3 on gp130 44 if STAT3.Y705 is phosphorylation and STAT3 is not dimer and STAT3.gp130 is not bound and STAT3.LIFR is not bound and STAT3.OSMR is not bound then STAT3 homodimerizes STAT3 shuttling 45 if STAT3 is in cytoplasm and STAT3 is dimer and STAT3.gp130 is not bound and STAT3.LIFR is not bound and STAT3.OSMR is not bound then STAT3 relocates to nucleus 46 if STAT3 is in nucleus and STAT3 is dimer and STAT3.PIAS3 is not bound then STAT3 dephosphorylated on Y705 47 if STAT3 is in nucleus and STAT3 is dimer and STAT3.Y705 is not phosphorylated then STAT3 dehomodimerizes 48 if STAT3 is in nucleus and STAT3 is not dimer and STAT3.Y705 is not phosphorylated then STAT3 relocates to cytoplasm SOCS3 synthesis 49 if STAT3 is in nucleus and STAT3 is dimer and STAT3.Y705 is phosphorylated and STAT3.PIAS3 is not bound then STAT3 synthesises SOCS3 PIAS3 inhibition 57 if STAT3 is in nucleus and STAT3 is dimer and STAT3.Y705 is phosphorylated and STAT3.PIAS3 is not bound and PIAS3 is not bound then PIAS3 binds STAT3 on PIAS3 58 if STAT3 is in nucleus and STAT3 is dimer and STAT3.Y705 is phosphorylated and STAT3.PIAS3 is bound and PIAS3 is bound then PIAS3 unbinds STAT3 on PIAS3

React Alt

15

17 19

20

21 22 23 24

25

28 29

Table 8.8 Reactions involving STAT3 Id

15 17 19 20 21 22 23 24 25 28 29

Type

Rate

Unit

Binding Phosphorylation Unbinding Homodimerization Relocation Dephosphorylation Dehomodimerization Relocation Synthesis Binding Unbinding

4.8  10 0.2 inf inf 1 0.04 inf 15 0.01 1.0  108 0.06 8

1

1

M min min 1 min 1 min 1 min (t1/2) min 1 min 1 min(t1/2) min 1 M 1 min 1 min 1

Reliability (%)

Reaction_volume

Unit

Reliability (%)

20 80 10 50 10 20 20 10 50 20 30

Cytoplasm Cytoplasm Cytoplasm Cytoplasm Cytoplasm Nucleus Nucleus Nucleus Nucleus Nucleus Nucleus

l l l l l l l l l l l

50 50 50 50 50 50 50 50 50 50 50

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the different states, locations, etc., of each component in the NL model. As an example, STAT3 is modeled by four distinct components representing, respectively, the cytoplasmic dephosphorylated monomeric form (STAT3_c), the cytoplasmic phosphorylated dimeric form (STAT3PD_c), the nuclear phosphorylated dimeric form (STAT3-PD_n), and the nuclear dephosphorylated monomeric form (STAT3_n); further species components are defined for each state of each complex containing STAT3. The definitions of the species representing the different forms of STAT3 are reported in Table 8.9 (again, we omit some of the duplicate reactions for the different receptors). Reactions and biochemical modifications are represented by actions executed synchronously by the involved species components, in the form “reaction_name species_role,” where reaction_name is a name identifying uniquely the reaction, and species_role describe the role of a species in the reaction: a down-arrow # indicates a reactant (i.e., its amount is decreased by the reaction), an up-arrow " indicates a product (i.e., its amount is increased by the reaction), and a plus  indicates an positive regulator (i.e., it acts as a catalyst and its amount is unaffected by the reaction). Stoichiometric coefficients are assumed to be 1 by default, and they can otherwise be specified with reactions of the form (reaction_name, stoichiometry_coefficient) species_role. For instance, the reaction representing r7 in Fig. 8.1 is modeled as the reaction dephosphorylation_dedimerization_STAT3, which is performed by STAT3-PD_n and STAT3_n, and whose effect is to decrease the amount of STAT3-PD_n and increase (with stoichiometric coefficient 2) the amount of STAT3_n. For each reaction, a functional rate specifying its kinetic rate law is defined. The ones used in the species definitions for STAT3-PD_n and STAT3_n are shown in Table 8.10 (lines starting with // are comments and are ignored by the simulator). Finally, the initial amounts for all the species must be specified in the form showed in Table 8.11. Note that, analogously to the NL, the initial amounts are given in terms of molecule counts rather than concentrations in order to allow models to be analyzed via stochastic simulation. As mentioned above, Bio-PEPA allows us to interpret the system either as continuous-deterministic or as discrete-stochastic, and the Bio-PEPA Eclipse plug-in (Duguid et al., 2009) enables us to obtain time-series results via both stochastic simulation and ODE solvers. In addition to this, BioPEPA models can be exported for analysis via different tools, in particular, PRISM for model-checking. In Guerriero (2009), some stochastic simulation results have been presented, and the satisfaction of a number of expected/desired properties of the system is verified using PRISM. Some of the results described in that work were qualitative (or semiquantitative) analysis: though this kind of analysis does not provide exact numerical values, it can often be interesting, due to its greater efficiency compared to exact quantitative methods, and to the fact that the experimental data

Table 8.9 Bio-PEPA model extract: Species definitions for STAT3

STAT3_c

STAT3-PD_c STAT3-PD_n STAT3_n

¼ binding_gp130-P-OSM-OSMR_STAT3# þ unbinding_gp130-P-OSM-OSMR_STAT3" þ binding_gp130-OSM-OSMR-P_STAT3# þ unbinding_gp130-OSM-OSMR-P_STAT3" þ binding_gp130-P-OSM-OSMR-P_STAT3# þ unbinding_gp130-P-OSM-OSMR-P_STAT3" þ binding_gp130-P-OSM-OSMR-P_STAT3# þ unbinding_gp130-P-OSM-OSMR-P_STAT3" þ binding_STAT3-gp130-P-OSM-OSMR-P_STAT3# þ unbinding_STAT3-gp130-P-OSMOSMR-P_STAT3" þ binding_gp130-P-OSM-OSMR-P-STAT3_STAT3# þ unbinding_gp130-POSM-OSMR-P-STAT3_STAT3" þ . . . // analogous set of reactions for bindings to other receptor complexes þ relocation_STAT3_n_c" ¼ unbinding_STAT3-PD_c_gp130-LIF-LIFR" þ unbinding_STAT3-PD_c_gp130-OSM-LIFR" þ unbinding_STAT3-PD_c_gp130-OSM-LIFR" þ relocation_STAT3_c_n# ¼ relocation_STAT3_c_n" þ dephospho_dedimer_STAT3# þ synthesis_SOCS3 þ binding_PIAS3_STAT3# þ unbinding_PIAS3_STAT3" ¼ (dephospho_dedimer_STAT3, 2) " þ relocation_STAT3_n_c#

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Table 8.10 Bio-PEPA model extract: Kinetic rates

relocation_STAT3_c_n ¼ [0.693/k58  STAT3-PD_c] dephospho_dedimer_STAT3 ¼ [k59  STAT3-PD_n] relocation_STAT3_n_c ¼ [0.693/k60  STAT3_n] synthesis_SOCS3 ¼ [k61  STAT3-PD_n] binding_PIAS3_STAT3 ¼ [k80/ (nucleusNA)PIAS3STAT3-PD_n] unbinding_PIAS3_STAT3 ¼ [k-80  PIAS3:STAT3-PD_n]

//STAT3-PD_c relocation cytoplasm -> nucleus //STAT3-PD_n dephosphorylation & dedimerization //STAT3-PD_n relocation nucleus > cytoplasm //SOCS3 synthesis by STAT3-PD_n //PIAS3/STAT3-PD_n binding //PIAS3/STAT3-PD_n unbinding

Table 8.11 Bio-PEPA model extract: Initial amounts

LIF[3000] OSMR [1000] STAT3_PD_c [0] SOCS3 [0] gp130:OSM:OSMR [0]

OSM [3000] gp130 [1000] STAT3_PD_n [0] gp130:LIF:LIFR [0] gp130:OSM:OSMR_P: STAT3 [0]

LIFR [1000] STAT3_c [3000] STAT3_n [0] gp130:OSM:LIFR [0] 

used for validation of models, are often qualitative themselves. In Clark et al. (2010), additional verification techniques have been applied to this system, with the main aim of identifying modeling errors (both syntactical and conceptual). Syntactic analysis, which can be performed statically (i.e., without performing any kind of simulation), can identify errors which invalidate some essential implicit hypotheses (e.g., conservation of mass, no unbounded growth of cells, etc.). Dynamic analysis (i.e., performed on simulation results) can provide additional knowledge about the system (the range in which the amount of each molecular species is expected to be, the number of times each reaction is expected to occur on average, etc.). Figure 8.2 shows the localization of STAT3 as a ratio between cytoplasmic and nuclear STAT3. Figure 8.2A is obtained from experimental data (Guerriero et al., 2009), and Fig. 8.2B is obtained from the average behavior of 10 stochastic simulation runs of the Bio-PEPA model. Figure 8.3 is a graphical representation which is automatically generated from the Bio-PEPA model, and it clearly shows the limit of graphical notations for big systems: this model is quite big, but not huge (it is

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A 100 Nuclear STAT (%)

80

Cytoplasmic STAT (%)

60 40 20 0 0

100

200

300 400 Time (min)

500

600

B 100 Cytoplasmic STAT (%) Nuclear STAT (%)

80 60 40 20 0 0

100

200

300

400

500

Figure 8.2 Localization of STAT3 molecules: comparison of (A) experimental and (B) simulation results. Panel (A) is published in the supplementary material of Guerriero et al. (2009).

composed of 63 molecular species and 118 reactions), but this size is enough for reaction arrows to be intersecting so much that they become very hard to follow. 4.1.4. The BlenX model In Guerriero et al. (2009), we have presented several results obtained by simulating the BlenX model automatically generated from the NL model described above, using the BetaWB simulation tool. Figure 8.4 is a screenshot of BetaWB showing an extract of the BlenX model (at the bottom) and its visual representation (at the top): from this short extract it is clear that the syntax of the BlenX model is not particularly intuitive and the model is not as readable as the original NL model. The modeler, however, does not generally need to care about the generated model and can instead consider it

Figure 8.3 Petri-net style graphical representation automatically generated from the Bio-PEPA model.

Figure 8.4 Automatically generated BlenX model imported in BetaWB.

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as a black-box: the model is executed and the results are reported automatically by the simulation tool. A particular emphasis in Guerriero et al. (2009) was on in silico experimentation for parameter exploration: the sensitivity of the model behavior to component removal and to changes in quantitative parameters was investigated. These results showed that the rates of dephosphorylation of nuclear STAT3 and of STAT3 nuclear export were the most influential on the system dynamics, while most of the other parameters did not have a great impact. As an example, in Fig. 8.5, we report a comparison of the

A

TC-PTP-mediated dephosphorylation 2000 Rate dephospho = 0 min–1 Rate dephospho = 0.004 min–1 Rate dephospho = 0.04 min–1 Rate dephospho = 0.4 min–1 Rate dephospho = 4 min–1

Species amount

1500

1000

500

0 0 B 900

100

200 300 Time (min)

400

500

JAK-mediated phosphorylation Rate JAK phospho = 12 min–1 Rate JAK phospho = 1.2 min–1 Rate JAK phospho = 0.2 min–1 Rate JAK phospho = 0.06 min–1 Rate JAK phospho = 0.02 min–1

800 700 600 500 400 300 200 100 0

0

100

200

300

400

500

Time (min)

Figure 8.5 Parameter sensitivity analysis: rate of (A) cytoplasmic phosphorylation and (B) nuclear dephosphorylation of STAT3. These results are published in Guerriero et al. (2009).

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effect of changing the rate of phosphorylation of cytoplasmic STAT3 and that of dephosphorylation of nuclear STAT3.

4.2. Circadian clocks Circadian clocks are a classic example of systems for which deterministic and stochastic models exhibit significantly different behavior. They are genetic networks which involve a very small number of proteins and genes, and which are present in small copy numbers. Consequently, they can exhibit a very noisy behavior, which means that the behavior of each cell in a population of cells can be very different from that of others. As mentioned earlier, the solution of a system of ODEs simulates the average population behavior, whereas a simulation of a stochastic system simulates the behavior of an individual cell. Because of the small noise, for big systems such as the signaling pathway described in the previous section, the deterministic population behavior is generally equivalent to the mean behavior of a large number of individual stochastic behaviors. For small systems such as circadian clocks and other oscillatory networks, instead, the noise can have such a big effect that the mean stochastic and the mean deterministic behaviors can be very different. Figure 8.6 shows a comparison of the results obtained from the model of the Neurospora crassa circadian clock presented in Akman et al. (2009, 2010b). The two panels refer to the same model settings with the exception of one altered parameter. The deterministic behavior (solid black line) is totally different for the two settings (in one case dampened oscillations, in the other persistent regular oscillations). The stochastic model, instead, is more robust and the oscillations are preserved in both settings: due to the stochastic noise, however, the oscillations observed in a single cell (red and A

B 700 600 500 400 300 200

ODE SSA (population) SSA (single cell 1) SSA (single cell 2)

600 FRQ protein

FRQ protein

700

ODE SSA (population) SSA (single cell 1) SSA (single cell 2)

500 400 300 200

100 0 24 72 48 96 120 144 168 192 216 240 264 288 312 336 360 Time (h)

100

0

24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 Time (h)

Figure 8.6 Individual versus population behavior: (A) and (B) refer to different parameter settings. The solid black line shows the deterministic behavior, the dashed blue line is the mean stochastic behavior, and the red and green dots are two individual stochastic behaviors.

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green dots show two independent stochastic simulation runs) are irregular, both in amplitude and in phase, and consequently the observed mean behavior (dashed blue line is the average over 1000 stochastic runs) is of dampening oscillations for both settings. Additional details of the stochastic model and additional analysis results can be found in Akman et al. (2009). Other works on models of oscillating systems which exhibit differences between stochastic and deterministic behaviors are Akman et al. (2010a), Ballarini and Guerriero (2010), and Tymchyshyn and Kwiatkowska (2008).

5. Conclusions and Perspectives Here we have reviewed some of the tools and techniques employed in the analysis of biological pathways as executable computer programmes and shown case studies of how this approach can be applied in practice to reasoning about the dynamic behavior and structure of biological pathways. The major requirement in this approach is an ability to formulate the pathway of interest in a very precise and logical fashion to enable computational execution. Although this involves constraints for the biologist, which at first sight might be discouraging, the process of developing a rigorous pathway description acts to uncover areas of ignorance or ambiguity in biological knowledge. Although the diversity of computational tools is rapidly expanding, the development of common frameworks for pathway formulation, such as NL and related approaches, should enable a much closer dialogue between biologists and computer scientists.

ACKNOWLEDGMENTS The authors thank Jane Hillston for the helpful discussions. M. L. G. is funded by the Centre for Systems Biology at Edinburgh. “The Centre for Systems Biology at Edinburgh is a Centre for Integrative Systems Biology (CISB) funded by BBSRC and EPSRC, reference BB/D019621/1.” J. K. H. is funded by the Cancer Research, UK.

REFERENCES Akman, O. E., Ciocchetta, F., Degasperi, A., and Guerriero, M. L. (2009). Modelling Biological Clocks with Bio-PEPA: Stochasticity and Robustness for the Neurospora Crassa Circadian Network, Proc. of CMSB’09. Springer, Berlin, 52–67. Akman, O. E., Guerriero, M. L., Loewe, L., and Troein, C. (2010a). Complementary approaches to understanding the plant circadian clock, Proc. of FBTC’10. Cyprus, 1–19. Akman, O. E., Rand, D. A., Brown, P. E., and Millar, A. J. (2010b). Robustness from flexibility in the fungal circadian clock. BMC Syst. Biol. 4, 88.

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SBGN Home Page http://www.sbgn.org. SBML Home Page http://www.sbml.org. Setty, Y., Cohen, I. R., Dor, Y., and Harel, D. (2008). Four-dimensional realistic modeling of pancreatic organogenesis. Proc. Natl. Acad. Sci. USA 105, 20374–20379. Singh, A., Jayaraman, A., and Hahn, J. (2006). Modeling regulatory mechanisms in IL-6 transduction in hepatocytes. Biotechnol. Bioeng. 95, 850–862. Sorokin, A., Paliy, K., Selkov, A., Demin, O. V., Dronov, S., Ghazal, P., and Goryanin, I. (2006). The Pathway editor: A tool for managing complex biological networks. IBM J. Res. Dev. 50, 561–576. Stochastic Pi Machine Home Page. http://research.microsoft.com/en-us/projects/spim/. Swameye, I., Mu¨ller, T. G., Timmer, J., Sandra, O., and Klingmu¨ller, U. (2003). Identification of nucleocytoplasmic cycling as a remote sensor in cellular signaling by databased modeling. PNAS 100, 1028–1033. The BioAmbients Machine Home Page. http://aesop.doc.ic.ac.uk/tools/bam/. The P Systems Webpage. http://ppage.psystems.eu/. Tymchyshyn, O., and Kwiatkowska, M. (2008). Combining Intra- and Inter-cellular Dynamics to Investigate Intestinal Homeostasis, Formal Methods in Systems Biology. Springer, Berlin, pp. 63–76. VirtualCell Home Page. http://vcell.org/. Vyshemirsky, V., and Girolami, M. A. (2007). Bayesian ranking of biochemical system models. Bioinformatics 24, 833–839.

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Computing Molecular Fluctuations in Biochemical Reaction Systems Based on a Mechanistic, Statistical Theory of Irreversible Processes Don Kulasiri Contents 254 256 260 262 266 269 273 277 277 277

1. Introduction 2. Theoretical Developments 3. Elementary Chemical Reactions 4. An Example of Chemical Reaction 5. Activation of Transcriptional Factors 6. Binding and Unbinding TF to E-boxes 7. Binding and Unbinding of Activated TF to E-Boxes 8. Conclusions Acknowledgments References

Abstract We discuss the quantification of molecular fluctuations in the biochemical reaction systems within the context of intracellular processes associated with gene expression. We take the molecular reactions pertaining to circadian rhythms to develop models of molecular fluctuations in this chapter. There are a significant number of studies on stochastic fluctuations in intracellular genetic regulatory networks based on single cell-level experiments. In order to understand the fluctuations associated with the gene expression in circadian rhythm networks, it is important to model the interactions of transcriptional factors with the E-boxes in the promoter regions of some of the genes. The pertinent aspects of a near-equilibrium theory that would integrate the thermodynamical and particle dynamic characteristics of intracellular molecular fluctuations would be discussed, and the theory is extended by using the theory of stochastic differential equations. We then model the Department of Molecular Biosciences, Centre for Advanced Computational Solutions (C-fACS), Lincoln University, Lincoln, Christchurch, New Zealand Methods in Enzymology, Volume 487 ISSN 0076-6879, DOI: 10.1016/S0076-6879(11)87009-6

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2011 Elsevier Inc. All rights reserved.

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fluctuations associated with the promoter regions using general mathematical settings. We implemented ubiquitous Gillespie’s algorithms, which are used to simulate stochasticity in biochemical networks, for each of the motifs. Both the theory and the Gillespie’s algorithms gave the same results in terms of the time evolution of means and variances of molecular numbers. As biochemical reactions occur far away from equilibrium—hence the use of the Gillespie algorithm—these results suggest that the near-equilibrium theory should be a good approximation for some of the biochemical reactions.

1. Introduction Molecular fluctuations in biochemical reaction systems are important within many fields, but we focus on the subsystems associated with gene expresion in this chapter. Variations in phenotypic characteristics have been observed in the genetically identical cells of organisms ranging in complexity from bacteria to mammals, and are hypothesized to be an important factor in evolution as well as in the physiological development (Kirschner and Gehart, 2005). Only in recent years, new experimental techniques in molecular biology, such as fluorescent reporters, allow stochastic gene expression to be quantified in vivo (e.g., see Austin et al., 2006; Blake et al., 2003; Elowitz et al., 2002; Kaern et al., 2005; Ozbudak et al., 2002; Pedraza and van Oudenaarden, 2005; Raser and O’Shea, 2004). These elegant experiments, along with the associated mathematical studies, have greatly facilitated our understanding of the sources and consequences of such stochasticity in genetic regulatory networks. Such stochasticity are divided into intrinsic noise and extrinsic noise; the former is associated with the gene and is dependent on the gene, whereas extrinsic noise is due to the surrounding biochemical reactions and diffusion processes within the cell, and therefore, is independent of the gene. While it is important to understand the “noise” associated with the above-mentioned molecular processes in terms of extrinsic and intrinsic components, we consider these noises to be the molecular fluctuations having mechanistic and thermodynamic characteristics. Some of these processes are thermodynamically irreversible (e.g., mRNAs are translated into proteins, but we have not seen proteins becoming mRNAs), and the fluctuations are intimately connected to the well-established theory of fluctuations and dissipation (Keizer, 1987). Mathematical models and the associated methods are essential parts of gene expression research, and some of the studies discussed above incorporate probabilistic mathematical models based on masters equations approach (Kampen, 2001), the formulation of which is based on the transition probabilities of molecules from one form to another form. Some other studies take purely a numerical simulation approach based on the most

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popular Gillespie’s algorithm (Gillespie, 1977) for reaction kinetics or many variants of it because it works well when the molecule numbers are low, and is based on sound gas-kinetics laws. The Gillespie’s algorithms are valid for the biochemical reaction systems far away from the thermodynamic equilibrium as well as near the equilibrium. We have chosen molecular reaction subsystems associated with the circadian rhythms of Drosophila to compute molecular fluctuations associated with gene expression. The biochemical dynamics of the motifs of these subsystems can be represented by chemical reactions which can be used in other applications. We hope that the elucidations within such a data-rich mechanism would enliven the applications of theories. Our discussion here is based on a deterministic mathematical model (Xie and Kulasiri, 2007) developed to represent the transcriptional regulatory networks essential for circadian rhythmicity in Drosophila. The model incorporates the transcriptional feedback loops revealed so far in the networks of the circadian clock (PER/TIM and VRI/PDP1 loops where PER, TIM, VRI, and PDP1 are key proteins in the system). The model simulates sustained circadian oscillations in mRNA and protein concentrations in constant darkness in agreement with experimental observations. The model is robust given a wide range of parameter variations. The model simulates entrainment by light–dark cycles and phase response curves resembling the experimental results. The simulated per01, tim01, and clkJrk and E-box mutations are similar to those observed in the experiments (Xie and Kulasiri, 2007). (E-boxes are CACGTG enhancers in the promoter regions of genes, and transcriptional factors (TF) bind to them to activate or repress the transcription of a gene.) One of the main differences in this model is that conventional Hill functions are not assumed to describe the regulation of genes; instead, the explicit reactions of binding and unbinding processes of transcription factors to E-boxes in the promoters were modeled. As the activity around any promoter region strongly influences the entire gene network, it is often important to investigate possible molecular fluctuations around this region. By modeling the bindings and unbindings of the TFs to E-boxes as elementary reactions would allow us to investigate the molecular fluctuations associated with E-box-mediated gene expression. The purpose of this chapter is to explore the molecular fluctuations in these promoter motifs in general mathematical settings using a nearequilibrium theory of molecular fluctuations based on irreversible thermodynamics and compare the results with those from the well-established Gillespie algorithm. We make use of the theory of stochastic differential equations (SDEs) to extend the near-equilibrium theoretical solutions, and the comparisons with the results from the Gillespie’s algorithms would indicate whether we could use the near-equilibrium theory as an approximation for these reactions, which are normally considered as happening far away from the equilibrium. The advantage of a positive comparison, that is,

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the time evolutions of means and variances are similar in both cases, would allow us to use the theory as an approximation for the investigations in molecular fluctuations in the network, which would reduce the computational cost significantly.

2. Theoretical Developments When two types of molecules react and/or combine to produce another type of molecules, two physical processes must happen: they should physically move through the solution and they should collide. The driving forces acting on molecules translate into kinetic energy, and the medium acts as the dissipater of the kinetic energy; any such energy dissipation associated with small molecules generates fluctuations. A physical ensemble of these molecules would depict behaviors that can be captured by using appropriate extensive variables. (Extensive variables depend on the extent of the system of molecules, i.e., the number of molecules, concentrations, kinetic energy, etc., whereas intensive variables do not change with the size of the system, i.e., pressure, temperature, entropy, etc.) These measurable quantities at the macroscopic level have origins at the microscopic level. Therefore, we can anticipate that molecular-level description would justify the operational models that we develop at an ensemble level. Naturally, one could expect that the statistical moments of the variables of an ensemble would lead to meaningful models of the process we would like to observe. The total differential of entropy of an idealized system of molecules can be written as
 m dU P dS ¼ þ dVT  dN; ð9:1Þ T T T where U is the internal energy; VT the volume of the system; N the number of molecules; P the pressure; m the chemical potential; and T is the absolute temperature. Equation (9.1) is a statement for a system of molecules and the system has the well-defined physical boundaries through which mass and heat transfer could occur. The momentum of the particles is included in the internal energy term, and by including the momentum (M), the total energy EΤ ¼ U þ (M2 /2m) where m is the mass of a particle. We can write Eq. (9.1) in the following form after including the momentum as a thermodynamic variable:
 m dE Τ  v  P dS ¼  dM þ dVT  dN : ð9:2Þ T T
T T

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In Eq. (9.2), v is the row vector of particle velocities and M is the column
(dS) can be
vector of particle moments. The total differential of entropy expressed in terms of partial derivatives:

 @S @S @S @S dS ¼ dM þ dVT þ dE Τ þ dN ; ð9:3Þ
@VT @E @M @N where @S=@ M indicates the row vector of @ S / @ Mi; @S=@ M, @ S / @ VT, and @
S / @ N are thermodynamically conjugate to the
respective variables in Eq. (9.3), namely, E Τ ; M; VT ; and N .
laws for irreversible processes states The Onsager principle for the linear that the rate of change of an extensive variable is linearly related to the difference of the corresponding thermodynamically conjugate intensive variable from its value at the thermodynamic equilibrium (Keizer, 1987). To simplify the notation in the linear laws, we introduce ai(t) ¼ E[xi(t)]0  E[xi(te)]0 to denote the average of the difference between an extensive variable of our choice and its value at the thermodynamic equilibrium, conditional on the initial values. Then, the Onsager linear laws can be written in terms of the relaxation matrix H governing the return of mean values of the extensive variables to equilibrium;
 da ¼ Ha; ð9:4Þ

dt which has the solution, aðt Þ ¼ eðHtÞ a0 ;



ð9:5Þ

where a0 is the initial value of the selected process. The matrix H must be
semidefinite as the entropy increases on the average during the negative relaxation process. Therefore, from a thermodynamic point of view, nearequilibrium fluctuations of an extensive variable is an exponentially decaying process from the initial value, and this fact should also be reflected in any other type of model of fluctuations. Using Eq. (9.5), we can deduce the two-time covariance function, Cðt1 ; t2 Þ,
h i T Cðt1 ; t2 Þ ¼ E aðt1 ÞaT ðt2 Þ ¼ E½ðeðHt1 Þ a0 ÞðeðHt2 Þ a0 ÞT  ¼ E½a0 a0T eðHt1 þH t2 Þ :






ð9:6Þ If t2 ¼ t1 þ t and closer to the equilibrium, the process is stationary,

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h i T h i CðtÞ ¼ E a0 aT ðtÞ ¼ E a0 a0T eðH tÞ :






ð9:7Þ

As we can see, from a thermodynamic point of view, Eqs. (9.6) and (9.7) state that the covariances have an exponentially decaying character to them. As discussed before, any dissipative process would have fluctuations in extensive variables. Let us define the fluctuations with reference to the expected value conditioned upon the initial value as dxi ðt Þ ¼ xi ðt Þ  E½xi ðtÞ0 :

ð9:8Þ

Then, it can be shown (Keizer, 1987) that, near equilibrium, the Onsager’s regression hypothesis for fluctuations can be written as ddX ¼ HdX þ ef;


dt

ð9:9Þ

where ef is a random vector. This equation is the Onsager’s regression
for fluctuations. This hypothesis is based on the thermodynamic hypothesis arguments not on the particles behavior in a physical ensemble. To complete the Onsager picture of random fluctuations in Eq. (9.9), we need to consider it as an SDE. Then, ef term can be defined in terms of the Wiener
form of an SDE. process to develop the simplest While the Onsager regression hypothesis is based on the entropy and the coefficients that form the relaxation matrix, the Boltzmann equation is, on the other hand, dependent entirely on the molecular dynamics of collisions and the resulting fluctuations. It can be shown that the linearized Boltzmann equation is a special case of Onsager theory (Keizer, 1987). In the derivation of the Boltzmann equation, we have a six-dimensional space in which the position of r and velocity, v, of the center of mass of a
single molecule are defined. We
call this six-dimensional space the m-space or molecule phase space. We can divide the six-dimensional space into small cellular volumes and each volume elements is assigned an index i ¼ 1, 2, 3, . . . as a unique number for identification purposes. The number of molecules, Ni(t) would be the macroscopic Boltzmann variable associated in the volume element i, and we choose the volume element i to be sufficiently large that Ni(t) is a large number. It is assumed that binary collisions between the molecules of only two volume elements located at r; v and r; v1 occur in the m-space. Each of these
and

volume elements located at r; v0 and volumes loses one molecule
each
r; v0 1 gain one molecule each at the end of each collision. (The
primes
denote the velocities at the center of mass velocities after the collisions.) We can define the extensive property of the number density as m-space,

259

Computing Molecular Fluctuations

rðr; v; tÞ, so that rðr; v; tÞdr dv is the number of molecules with center of

position and velocity


in
the ranges ½r; r þ dr and ½v; v þ dv. mass


Then, the Boltzmann equation gives


ð @r ^T g½r0 r01  rr1 dv1 ð9:10Þ ¼ vrr r  Frv r þ s

;
@t where rðr; v0 ; tÞ ¼ r0; rðr; v1 ; tÞ ¼ r01 ; F is an external force field acting


in the m-space, rr and
r
v are the derivatives with respect to r and v,
is the
respectively. The third term on the right hand side of Eq. (9.10) ^T is a linear operator and g is the constant dissipative effect of collisions; s relative velocity magnitude (Keizer, 1987). In the absence of an external force, Eq. (9.10) can be written as @r ¼ Vrr r þ ds ;
@t

ð9:11Þ

with ds lumps the dissipation due to collisions. Unlike the Onsager’s linear laws, which are true only near the thermodynamic equilibrium, the Boltzmann equation is true not only in the vicinity of the equilibrium but also away from the equilibrium. However, near the equilibrium these two pictures are similar, and while the Boltzmann equation is valid, strictly speaking, only for diluted gases, the Onsager linear laws are valid for any ensemble. In the vicinity of the equilibrium, we can write       r r; v; t ¼ re v þ Dr r; v; t ; ð9:12Þ




  where Dr r; v; t is a small change in the m-space density and superscript
equilibrium values. Equation (9.12) is a crucial assumption “e” denotes
the which makes the entire theory valid only near the thermodynamic equilibrium. By substituting Eq. (9.12) for Eq. (9.11) and ignoring the higher order terms of Dr, we obtain @Dr ¼ Vrr Dr þ C ½Dr;
@t

ð9:13Þ

with C[Dr] replacing the dissipation integral as a linear functional. It can be shown that, by adopting the Onsager hypothesis (Keizer, 1987),   @Dr ¼ L ½X  þ fe r; v; t ;

@t where, X ¼ kB ln

  r re

ð9:14Þ

is the local thermodynamic force in m-space;

260

Don Kulasiri


L ½X  

 ð   vre rr X þ L S v1 X1 dv1 ;

kB

ð9:15Þ

where LS is a linear operator (Keizer, 1987); and fe is a random term. The random term now can be defined with E½ feðr; V; tÞ ¼ 0 and

    E½feðr; v; tÞðfeðr0 ; v0 ; t 0 ÞÞ ¼ 2kB L S v; v1 d r  r0 dðt  t 0 Þ: ð9:16Þ







In Eq. (9.14), the m-space density increments are expressed in terms of thermodynamic forces (X). By deriving the random term fe as in Eq. (9.16), we see that it is a zero-mean stochastic process in the m-space, d-correlated in r and t, but it is influenced by the velocity of the center of mean through a
operator derived from the dissipation term, d , in the Boltzmann linear S equation. This analysis shows that the Boltzmann and Onsager pictures are united near the equilibrium. Equally importantly, Eq. (9.16) justifies the d-correlated stochastic process to model the fluctuations. Moving away from the m-space, we describe the fluctuations and dissipation using the theory of stochastic processes in an effort to develop operational models of molecular fluctuations.

3. Elementary Chemical Reactions As mentioned before, a direct collision between two molecules with velocities in the range v to v þ dv and v1 to v1 þ dv1 in the spatial volume element dr located at r is a conditional event which changes the number of molecules in the volume element dr dv, N(r, v, t), and N(r, v, t) ¼ r(r, v, t)dr dv. (The symbols in bold are vectors or matrices to distinguish the chemical reaction from the general theory.) If a collision occurs, both N(r, v, t) and N(r, v1, t) decrease by one, and both N ðr; v0 ; tÞ and N ðr; v0 1 ; tÞ increase by one; therefore, each collision produces a deterministic effect on N(r, v, t). We define any molecular process which causes a deterministic change in an extensive variable as an elementary process. A direct collision and a reverse of the same collision can be denoted by v; v1 Ð v0 ; v0 1 , and this can be considered as a single elementary process. However, to determine how elementary processes change the extensive variable(s), we need to evaluate the probability associated with each elementary process that occurs within dt. The Boltzmann equation provides the average affect of collision on the occupancy number, and the rate of change of N(r, v, t) due to the elementary process, v; v1 Ð v0 ; v0 1 , is given by

261

Computing Molecular Fluctuations

Rk ¼ ^ sT g½r0 r0 1  rr1 dv1 dv dr; which can be written as   Rk ¼  Vkþ  Vk ; where Vkþ and Vk are the transition rates. Let us define n(t) as the column vector consisting of the occupancy number of volume elements. Labeling each cell by an index i, we assume that each cell is large enough for Ni(t) to be a macroscopic variable. n(t) is a stochastic process governed by the probabilities of the elementary processes. Let v denote the vector of changes: vkþ for the forward part of the elementary process k and vk for the reverse part of the elementary process. Note that  vk ¼ vkþ  vk. As for an elementary process, the reverse process restores the effect of the direct collision. In addition, n(t) can be changed by the stream motion in m-space. Thus, the probability of the total change of n by dn1 þ dns can be defined according to ansatz by Keizer (1987)  P2 n0 ¼ n þ dns þdn; t þ dtjn; t ¼

8 > > < > > :

if dn ¼ ok ; V  dt þ Oðdt Þ;
X k þ  1 Vk þ Vk þ Oðdt Þ; if dn ¼ 0; k

0;

otherwise:

ð9:17Þ It can be shown that this ansatz gives rise to a stochastic process with characteristics of a stochastic diffusion process after nontrivial scaling arguments. Keizer (1987) developed a Fokker-Planck type equation for the fluctuation of n(t) around its mean n
ðtÞ, dnðtÞ  nðtÞ  n
ðtÞ. The resulting Fokker-Planck equation for P2 ðn
þ dn; tjn1 ; t1 Þ is, using the summation convention on repeated indices, @P2 @Hij ðn
Þdnj P2 1 @ 2 gij ðn
Þ þ ¼ @t @dni 2 @dni @dnj

ð9:18Þ

P  þ 
oki V
k  V
 k þ Si ; Hij ðn
Þ ¼ @
nj

ð9:19Þ

with @

and

262

Don Kulasiri

gij ðn
Þ 

X


 okj V
þ k  V k okj ;

ð9:20Þ

k

when n
i ðni ; t Þ solves the Boltzmann equation d
ni X 
þ
oki V k  V
 ¼ k þ Si ; dt k

ð9:21Þ

where S
i is stream effects. (Notation: the bar indicates mean values.) The Fokker-Planck equation (Eq. (9.18)) is very similar to that of the Fokker-Planck equation for an Orenstein-Uhlenbeck process, but in Eq. (9.18), Hij and gij are time-dependent; therefore, Eq. (9.18) can be considered as a generalization of an Orenstein-Uhlenbeck process. A diffusion process can always be expressed as an Ito SDE (Klebaner, 1998); therefore, the equivalent Ito SDE for Eq. (9.18) is ddn ¼ Hðn
Þdndt þ g1=2 ðn
Þdw;

ð9:22Þ

where w is a vector-valued Wiener process and g1/2 is the square root of the matrix g, defined as (g1/2)2 ¼ g. As H and g are independent of dn, Eq. (9.22) is a linear SDE. Therefore, we can write ddn ¼ Hðn
ðn1 ; tÞÞdn þ feðt Þ; dt

h i with E feðtÞ ¼ 0, and h i E fei ðtÞfej ðt0 Þ ¼ gij ðn
ðn1 ; tÞÞdðt  t 0 Þ:

ð9:23Þ

ð9:24Þ

We see that Eq. (9.23) has the same form as Onsager’s regression hypothesis, with H driven by elementary processes, g being the two-time correlation 0 function depending on both t and t  t , and feðt Þ being a colored noise. The point of difference, however, is that H matrix is not quite a relaxation matrix as we see in the example given below.

4. An Example of Chemical Reaction The following example given by Keizer (1987) is summarized here to illustrate the application of the theory. Let us consider the general bimolecular chemical reaction, which occupies the entire system: A þ B Ð C þ D:

ð9:25Þ

263

Computing Molecular Fluctuations

The thermodynamic extensive variables are the internal energy, the volume, and the molecular numbers nA, nB, nC, and nD. In this reaction, one molecule, each of A and B, is involved in the forward reaction, whereas none of the molecules of A and B is involved in the reverse reaction. The number of C and D molecules involved in the forward reaction is 0, whereas in the reverse reaction, one of each C and D molecules is involved. We can denote these numbers as follows: þ   nþ A ¼ nB ¼ 1; nA ¼ nB ¼ 0; and

þ   nþ C ¼ nD ¼ 0; nC ¼ nD ¼ 1:

In this notation, the superscript “þ” denotes the forward reaction and the superscript “” denotes the reverse reaction. Using these symbols, we can write reaction (9.25) as  þ þ þ þ     nA ; nB ; nC ; nD Ð n A ; nB ; nC ; nD ; that is, ð1; 1; 0; 0Þ Ð ð0; 0; 1; 1Þ: If we define the change of any of the number of molecules, ni, as oi  ni  niþ, oA ¼ oB ¼ 1, oC ¼ oD ¼ þ1. We can apply the mass action law to define the rates of elementary reactions using the activities for each species V þ ¼ Vkþ aA aB and V  ¼ Vk aC aD ; where ai is the activity for the species i and V. The chemical potential of a chemical species, mi, is a very useful intensive property which is related to the entropy as discussed previously (Eq. (9.2)); that is, if Fi is the intensive variable thermodynamically conjugate to the molecular number, ni, thus
 @S m Fi  ¼  i: ð9:26Þ T @ni E;V ;n The chemical potential is related to the thermodynamic equilibrium through the change in Gibbs free energy (DGe) for reaction (9.25) DGe ¼ mec þ meD  meA  meB ¼ 0: Therefore, by incorporating the chemical potential through the activity, ai, we can give a thermodynamic interpretation to the matrix, Hi and g in

264

Don Kulasiri

Eq. (9.23). The relationship between the activity, ai, and the chemical potential, mi, is given by 

ai ¼ exp mi 

m0i



  Fi m0i exp  =kB T ¼ exp  ; kB kB T

where mi0 is the standard-state chemical potential, kB is the Boltzmann constant, and T is the temperature. (“exp” denotes the exponential function.) Thus, the transition rates can be expressed as    V þ ¼ Vkþ exp  m0A þ m0B =kB T exp½ðFA þ FB Þ=kB ; and   V  ¼ Vk exp m0C þ m0D =kB T exp½ðFC þ FD Þ=kB : We can define    Oþ ¼ Vkþ exp  m0A þ m0B =kB T ; and   O ¼ Vk exp m0C þ m0D =kB T ; because both the rate constants and the standard-state chemical potentials are independent of composition. It can also be shown that Oþ ¼ O  O, which is called microscopic reversibility (Keizer, 1987). Then, the transition rates for this reaction can be written as V þ ¼ Oþ exp½ðFA þ FB Þ=kB ;

ð9:27Þ

V  ¼ O exp½FC þ FD =kB :

ð9:28Þ

and

However, we can see FA þ FB ¼

X

nþ j Fj ;

j

and FC þ FD ¼

X j

n j Fj :

265

Computing Molecular Fluctuations

Therefore, we can write Eqs. (9.27) and (9.28) as " # X V  ¼ O exp  n j Fj =kB :

ð9:29Þ

j

By using this general form of the transition rates, we can write the basic postulates of the theory as ( " # " #) X X d
ni X þ  oki Ok exp nkj mj ðn
Þ=kB T  exp nkj mj ðn
Þ=kB T ¼ dt k j j
; t Þ;  Ri ðn
; tÞ þ Si ðn ð9:30Þ ddni
; tÞdnj þ fei ; ¼ Hij ðn dt

ð9:31Þ

where, dni ðtÞ  ni ðtÞ  n
i ðtÞ, and Hij ðn
; tÞ  @Ri ðn
; t Þ=@
nj : h i E feðtÞ ¼ 0;

ð9:32Þ ð9:33Þ

h i
Þdðt  t0 Þ; E feiðtÞfej ðt 0 Þ ¼ gij ðn

ð9:34Þ

and

with gij ðn
Þ ¼

X k

( oki Ok okj

" # " #) X X þ  exp nkj mj ðn
Þ=kB T þ exp nkj mj ðn
Þ=kB T : j

j

ð9:35Þ To summarize, Eqs. (9.30)–(9.35) provide a mechanistic statistical theoretical framework to investigate the molecular fluctuations around mean. In this theory, H matrix plays an important role, and it is the gradient of the rate of change of mean molecular numbers with respect to the mean number of molecules. The theory outlined above can now be used to investigate the molecular fluctuations.

266

Don Kulasiri

5. Activation of Transcriptional Factors Before binding to E-boxes, TFs are activated by a number of ways, including phosphorylation and forming dimmers. This process can be generalized as an elementary unimolecular reaction, A Ð B, where B represents the activated form of A. We assume that the total number of the activated and nonactivated TFs, n, is a constant (n ¼ nA þ nB). This reaction can be written as (nAþ, nBþ) Ð (nA, nB), that is, (1, 0) Ð (0, 1). Therefore, oA ¼  oB ¼  1. Eq. (9.30) of the theory now takes the form



 d
nA d
nB mA ðn
A Þ mE ðn
B Þ  ¼ ¼ O exp  exp : ð9:36Þ dt dt kB T kB T The chemical potential now can be written as
i Þ; mi ðn
Þ ¼ m0i ðT Þ þ kB T lnðr
i is the number density of the ith species, and therefore, r
i ¼ n
i =V : where r By defining the following rate constants  0 O exp m =k T B A kþ ¼ ; V and  O exp m0B =kB T ; k ¼ V 

we can rewrite Eq. (9.36) as 

d
nA d
nB ¼ ¼ kþ n
A  k n
B : dt dt

ð9:37Þ

nB =dt ¼ 0; therefore nAe / nBe ¼ k / kþ, At the equilibrium, d
nA =dt ¼ d
where the superscript “e” denotes the values at the equilibrium. Given that n ¼ nAe þ nBe, nBe ¼ kþn / (kþ þ k). Thus, Eq. (9.37) can be solved:  n
B ðt Þ ¼ neB þ exp½lt  n0B  neB ;

267

Computing Molecular Fluctuations

where nB0 is the initial value of nB and l ¼ kþ þ k. Thus, the SDE governing the molecular fluctuation dnB is given by ddnB ¼ ldnB þ feB ; dt where random term is defined by h i E fei ðt Þfej ðt0 Þ ¼ gðn
B ðtÞÞdðt  t 0 Þ: g can be derived from Eq. (9.35): nB ðt Þ þ kþ n: g ¼ ðk  kþ Þ
We solve this model for dnB and the variance of dnB, s(nB0, t), conditioned on the initial conditions, using the Ito SDE theory (Klebaner, 1998). Hence, we briefly illustrate that Ito integration also leads to the same results. As mentioned before, the SDE can be written as ddnB ¼ ldnB dt þ g1=2 ðn
B ðtÞÞdw: g can now be expressed as a function of t    gðtÞ ¼ kþ n þ ðk  kþ Þ neB þ n0B  neB exp½lt  :

ð9:38Þ

ð9:39Þ

Therefore, dnB ¼ 

ðt

ldnB dt þ

0

ðt

g1=2 ðtÞdw;

ð9:40Þ

0

which is a linear SDE having a unique solution for the well-defined function of g(t) (see Klebaner, 1998). The solution of the linear SDE based on Ito integration is (see Klebaner, 1998)
 ðt 1=2 dnB ¼ expðlt Þ dnB ð0Þ þ g ðsÞ expðlsÞdw ðsÞ : ð9:41Þ 0

The integral in equation should be interpreted as an Ito integral. However, dnB ð0Þ ¼ nB ð0Þ  n
B ð0Þ ¼ 0; therefore, dnB ¼ expðlt Þ

ðt 0

g1=2 ðsÞ expðlsÞdw ðsÞ:

ð9:42Þ

268

Don Kulasiri

Further, from the mean-zero property of Ito integrals, E[dnB] ¼ 0, and let sðt Þ ¼ Þ;  E½dnB ðt ÞdnÐ Bt ðt1=2   Ðt ¼ E expðlt Þ 0 g ðsÞ expðlsÞdwðsÞ expðlt Þ 0 g1=2 ðsÞ expðlsÞdw ðsÞ ;  Ð t  2 ¼ expð2lt Þ E 0 g1=2 ðsÞ expðlsÞdwðsÞ :

From isometry property of Ito integrals, Ðt sðt Þ ¼ expð2ltÞ 0 E ½gðsÞ expð2lsÞdt; Ðt ¼ expð2ltÞ 0 gðsÞ expð2lsÞdt: Using Eq. (9.39) for g(t), after algebraic manipulation, we can show that, the variance,   ðk  kþ Þ n0B  neB k neB ð expðlt Þ  expð2ltÞÞ : ð1  expð2ltÞÞ þ sðt Þ ¼ l l

As we can see from the derivations mentioned above, the molecular fluctuation of B can be characterized fully using the initial conditions and the rate constants, k and kþ. These results are valid away from the initial conditions, and one can observe that when t ! 1, s(t) ! (knBe / l), that is, s(t) ¼ (knBe / l) ¼ (nAenBe / n). Near the equilibrium, we can obtain the variance s(t) ¼ se(1  exp ( 2lt)). The fluctuations and the variance of fluctuations decay exponentially, and the rate of decay is dependent on the parameters of the system. If there are temperature changes in the system, these are reflected through the rate constants. We have implemented the Gillespie’s Monte Carlo Simulation algorithm (Gillespie, 1977) for reaction A Ð B (for brevity, we do not provide the implementation) and compared with the theoretical results. Figure 9.1 shows the comparisons between the two methods for B; the system was simulated for 20 days, and the means and variances were computed using 10,000 realizations. The initial values are nA0 ¼ 150, nB0 ¼ 0; and the rate coefficients are kþ ¼ 0.4, k ¼ 0.2. The agreement between the two methods for the mean and variances is excellent, and this agreement holds for other initial and rate constants.

269

Computing Molecular Fluctuations

A 120

Particle number

100 80 60 40 20

Theoretical model Gillespie method

0

0

5

10

15

20

Time B 40

Variance

30

20

10 Theoretical model Gillespie method 0

0

5

10 Time

15

20

Figure 9.1 (A) Mean number of particles for B using the theory and the Gillespie’s method; (B) Comparisons for variances.

6. Binding and Unbinding TF to E-boxes As we discussed before with reference to circadian rhythms (Xie and Kulasiri, 2007), the region upstream to a promoter (promoter region) in a structural gene is inundated with a number of E-boxes. These E-boxes are

270

Don Kulasiri

where the proteins (TFs) bind either to activate or repress the transcription of the gene. We can model this situation, assuming independent binding of each E-box as a reaction, as follows: TF þ EB Ð TEB;

ð9:43Þ

where TF is the protein involved, EB is the E-box segment, and TEB is the protein-bound E-box. As this situation would occur in a large enough number of cells, we would treat the dynamics of the interaction macroscopically using the theory. Even within a cell, there may be large enough number of E-boxes in a gene to justify this approach. As the total number of E-boxes are fixed in a promoter region, we can assume nEB þ nTEB ¼ n ¼ const., where nEB is the number of free E-boxes and nTEB is the number of E-boxes bound to the transcription factor, TF. The number of TF (nTF) is assumed to be unknown as it is often the case in a cell. We can write the chemical reaction as  þ þ þ    nEB ; nTF ; nTEB Ð n EB ; nTF ; nTEB ; where nþ EB ¼ 1; n EB ¼ 0;

nþ TF ¼ 1;

n TF ¼ 0;

nþ TEB ¼ 0;

and

n TEB ¼ 1:

Therefore, reaction (9.43) can be written as (1, 1, 0) Ð (0, 0, 1), and oEB ¼ 1 ¼ oTF ; oTEB ¼ 1. The forward and backward rates of reactions with usual notation can be derived as follows:

0 

0  O mEB m þ k ¼ 2 exp exp TF ; kB T kB T V and k ¼

0  O m exp TEB : kB T V

With the restriction, nEB þ nTEB ¼ n, the following equations are obtained: (the bar indicates the mean values.) d
nEB ¼ kþ n
EB n
TF þ k n
EBT ; dt d
nTF d
nEB d
nTEB ¼ ¼ ; dt dt dt

ð9:44Þ ð9:45Þ

Computing Molecular Fluctuations

and

 n
TF ðt Þ ¼ n
EB ðtÞ þ n0TF  n0EB ;

where superscript “0” indicates the initial values of variables. We solve Eq. (9.45) analytically; the only valid solution is h pffiffiffiffi i þ fD0 exp  D=k t þ neEB h pffiffiffiffi i ; n
EB ðtÞ ¼ 1  D0 exp  D=kþ t where

271

ð9:46Þ

ð9:47Þ

 pffiffiffiffi kþ n0TF  n0EB þ D ; f¼ 2kþ   2 D ¼ kþ n0TF  n0EB þ 4kþ k n; pffiffiffiffi kþ n0EB þ kþ n0TF þ k  D pffiffiffiffi ; D0 ¼ kþ n0EB þ kþ n0TF þ k þ D

and 1 pffiffiffiffi 1  0 0 D  :  n n TF EB 2kþ 2 pffiffiffiffi  pffiffiffiffi  We see that f þ neEB ¼ D=kþ  neEB . D=kþ ; therefore, f ¼ We can see that n
EB is an exponential function reaching toward the limiting values as one would expect. The rate of movement toward the equilibrium is a function of the initial values of TF and EB, the rate constants, and the total number of E-boxes. As in the previous example, the molecule numbers are influenced by the equilibrium molecule number of the species, that is, the thermodynamics of the reactions. However, the equilibrium molecule number is a function of initial values. Even if nTF0  nEB0, it can be shown that nEBe is always positive. We can define the fluctuation from mean of EB as dnEB ðtÞ ¼ nEB ðt Þ  n
EB ðtÞ. The SDE governing the behavior of dnEB is given by neEB ¼ equilibrium value ¼

dðdnEB Þ ¼ jdnEB dt þ g1=2 ðn
EB Þdw;

ð9:48Þ

where j is obtained by differentiating the  right hand side of Eq. (9.48) with respect to n
EB : j ¼ 2kþ n
EB  kþ n0TF  n0EB  k . g is given by, according to Eq. (9.35),

272

Don Kulasiri

   g ¼ kþ n
2EB þ kþ n0TF  n0EB  k n
EB þ k n: Equation (9.48) can be solved by first transforming g and j into the functions of t using Eq. (9.47) and then solving the linear SDE. Some realizations of the solution for EB are depicted in Fig. 9.2 for a specific set of initial conditions and the reaction rates. The time evolution of the variance of dnEB can be obtained by   sðdnEB Þ ¼ E ðdnEB Þ2 ; Ð t  2 ¼ E 0 jðtÞdnEB dt þ g1=2 ðtÞdw : This can also be evaluated both numerically and analytically using the isometry property of Ito integrals. As before, we used the Gillespie’s algorithm to simulate the reaction (9.43) and the comparative results are given in Fig. 9.3. We show an extreme situation (nEB þ nTEB ¼ n ¼ 50; nTF0 ¼ 150; nEB0 ¼ 40; nTEB0 ¼ 40; and kþ ¼ 0.1; k ¼ 4) to highlight the short transient period. The system was simulated for 20 days, the mean and variance of TEB computed from 10,000 realizations are given in Fig. 9.3A and B, respectively. Even in this case, the theory and the Gillespie’s algorithms produce essentially the same results. 4

Fluctuation of EB

3 2 1 0 –1 –2 –3

0

0.1

0.2

0.3

0.4

0.5 Time

0.6

0.7

0.8

0.9

1

Figure 9.2 Three realizations of EB fluctuation for both forward and backward reaction rates are 0.4, the total number of E-boxes is 10, and the initial values of TFA and TEB are 0 and 10, respectively. Negative values indicate that current values are less than mean values.

273

Computing Molecular Fluctuations

A 40

Particle number

35 30 25 20 15 10

Theoretical model Gillespie method 0

5

10

15

20

Time B

10

Variance

8 6 4 2

Theoretical model Gillespie method

0

0

5

10 Time

15

20

Figure 9.3 (A) Mean number of particles for TEB using the theory and the Gillespie’s method; (B) Comparisons for variances.

7. Binding and Unbinding of Activated TF to E-Boxes In many situations, the TF is activated prior to binding onto the E-boxes. This situation can be modeled as two elementary reactions. TFA is activated TF, and TAB is TF bound E-boxes. TF Ð TFA EB þ TFA Ð TAB:

ð9:49Þ

274

Don Kulasiri

By following the similar procedure, we can obtain the following vectors for the theoretical equations: 2 3 n
TF N ¼ 4 n
TFA 5; n
TAB and

2

kþ 1 H ¼ 4 kþ 1 0

k 1  k1  kþ
EB 2n þ k2 n
EB

3 0 þ k
TFA 5: 2 þ k2 n þ  k2  k2 n
TFA

We also have the restriction: nEB(t) þ nTAB(t) ¼ n ¼ const. In Eq. (9.49),
0  O1 m þ exp TF ; k1 ¼ V kB T
0  O1 m exp TFA ; k 1 ¼ V kB T
0 
0  O2 mEB m þ exp exp TF ; k2 ¼ V kB T kB T and k 2 ¼


0  O2 m exp TAB : V kB T

Note that O values are different for the two different reactions. Now, the fluctuations of N relative to the mean can be obtained from the following SDE: ddN ¼ HdN dt þ g1=2 dw;

ð9:50Þ

where w is a column vector of three independent Wiener processes, g is given by X  þ gij ¼ oki V
k þ V
 k okj ; k

where for the two elementary reactions, 2 3 1 v1 ¼ 4 1 5 0

275

Computing Molecular Fluctuations

and 2

3 0 v2 ¼ 4 1 5 1 We know that þ V
þ
TF ; 1 ¼ k1 n

 V

TFA ; 1 ¼ k1 n

þ
EB n
TFA ; V
þ 2 ¼ k2 n

 V

TAB : 2 ¼ k2 n

Therefore, g11 ¼ kþ
TF þ k
TFA ; 1n 1n

g33 ¼ kþ
EB n
TFA þ k
TAB ; 2n 2n

g22 ¼ g11 þ g33 ¼ kþ
TF þ k
TFA þ kþ
EB n
TFA þ k
TAB ; 1n 1n 2n 2n g12 ¼ g21 ¼ g11 ¼ kþ
TF  k
TFA ; 1n 1n g23 ¼ g32 ¼ g33 ¼ kþ
EB n
TFA  k
TAB ; 2n 2n and 45 Mean of TF TF

Number of molecule TF

40 35 30 25 20 15 10

0

1

2

3

4

5 Time

6

7

8

9

10

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g13 ¼ g31 ¼ 0: As g is a positive semidefinite and symmetric, it has a well-defined square root. Figure 9.4 shows the time evolution of the mean of TF, and three realizations of TF are superimposed. We implemented the Gillespie’s algorithm for this reaction system, and the comparisons of the means and variances are given in Fig. 9.5 for nEB þ nTAB ¼ n ¼ 50;nTF0 ¼ 150; nTFA0 ¼ 0; nEB0 ¼ 40; nTAB0 ¼ 40; and k1þ ¼ 0.4; k1 ¼ 0.2; k2þ ¼ 0.1; k2 ¼ 4. As before, the comparisons A

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are based on 10,000 realizations. We chose to show only TAF because it couples the reactions, and the results show excellent agreement.

8. Conclusions We have reviewed the pertinent aspects of a mechanistic, statistical theory which integrates the thermodynamical and particle dynamic aspects of intracellular molecular fluctuations, and extended the theory using the theory of SDEs. The theory was used to model the fluctuations in binding and unbinding of TFs to E-boxes, which is a common motif in transcriptional regulation in circadian rhythms. The stochastic variations in biochemical reactions are ubiquitously simulated by the Gillespie’s algorithm. We compared the theoretical results with those from the Gillespie’s algorithms for the reaction systems and found them to be almost the same. Even though the theory is for near-equilibrium reactions, once extended with the SDEs, it can be used for biochemicals reactions in the motifs described here, which are usually considered as happening away from the equilibrium.

ACKNOWLEDGMENTS The funding is provided by Lincoln University LUREST grants. Yao He’s assistance is appreciated.

REFERENCES Austin, D. W., Allen, M. S., McCollum, J. M., Dar, R. D., Wilgus, R. J. R., Sayler, G. S., Samatova, N. F., Cox, C. D., and Simpson, M. L. (2006). Gene network shaping of inherent noise spectra. Nature 439(7076), 608–611. Blake, W. J., Kaern, M., Cantor, C. R., and Collins, J. J. (2003). Noise in eukaryotic gene expression. Nature 422(6932), 633–637. Elowitz, M. B., Levine, A. J., Siggia, E. D., and Swain, P. S. (2002). Stochastic gene expression in a single cell. Science 297(5584), 1183–1186. Gillespie, D. T. (1977). Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 61, 2340–2361. Kaern, M., Elston, T. C., Blake, W. J., and Collins, J. J. (2005). Stochasticity in gene expression: From theories to phenotypes. Nat. Rev. Genet. 6(6), 451–464. Kampen, V. (2001). Stochastic Processes in Physics and Chemistry. Elsevier Sci. Pub., Amsterdam, Netherlands. Keizer, J. (1987). Statistical Thermodynamics of Nonequilibrium Processes. Springer-Verlag, New York. Kirschner, M. W., and Gehart, J. C. (2005). The Plausibility of Life: Resolving Darwin’s Dilemma. Yale University Press, New Haven, CT, USA. Klebaner, F. C. (1998). Introduction to Stochastic Differential Equations. Academic Press, London.

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Ozbudak, E. M., Thattai, M., Kurtser, I., Grossman, A. D., and van Oudenaarden, A. (2002). Regulation of noise in the expression of a single gene. Nat. Genet. 31(1), 69–73. Pedraza, J. M., and van Oudenaarden, A. (2005). Noise propagation in gene networks. Science 307(5717), 1965–1969. Raser, J. M., and O’Shea, E. K. (2004). Control of stochasticity in eukaryotic gene expression. Science 304(5678), 1811–1814. Xie, Z., and Kulasiri, D. (2007). Modelling of circadian rhythms in Drosophila incorporating the interlocked PER/TIM and VRI/PDP1 feedback loops. J. Theor. Biol. 245, 290–304.

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Probing the Input–Output Behavior of Biochemical and Genetic Systems: System Identification Methods from Control Theory Jordan Ang,*,† Brian Ingalls,‡ and David McMillen*,† Contents 280 282 283 285 291 291 293 305 311 316

1. Introduction 2. System Identification Applied to a G-Protein Pathway 2.1. Construction of the frequency response 2.2. Interpretation of the frequency response 3. System Identification 3.1. Transfer function models 3.2. Applying the procedure to the G-protein pathway 3.3. Examples of experimental implementation 4. Conclusion References

Abstract A key aspect of the behavior of any system is the timescale on which it operates: when inputs change, do responses take milliseconds, seconds, minutes, hours, days, months? Does the system respond preferentially to inputs at certain timescales? These questions are well addressed by the methods of frequency response analysis. In this review, we introduce these methods and outline a procedure for applying this analysis directly to experimental data. This procedure, known as system identification, is a well-established tool in engineering systems and control theory and allows the construction of a predictive dynamic model of a biological system in the absence of any mechanistic details. When studying biochemical and genetic systems, the required experiments are not standard laboratory practice, but with advances in both our ability to measure system outputs (e.g., using fluorescent reporters) and our ability to generate * Department of Chemical and Physical Sciences, University of Toronto Mississauga, Mississauga, Ontario, Canada Institute for Optical Sciences, University of Toronto Mississauga, Mississauga, Ontario, Canada { Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada {

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precise inputs (with microfluidic chambers capable of changing cells’ environments rapidly and under fine control), these frequency response methods are now experimentally practical for a wide range of biological systems, as evidenced by a number of successful recent applications of these techniques. We use a yeast G-protein signaling cascade as a running example, illustrating both theoretical concepts and practical considerations while keeping mathematical details to a minimum. The review aims to provide the reader with the tools required to design frequency response experiments for their own biological system and the background required to analyze and interpret the resulting data.

1. Introduction Recent advances in experimental techniques have allowed ever-increasing scrutiny of the dynamic behavior of cellular mechanisms. In many cases, such as development and response to environmental changes, the dynamic nature of a biological process is the key to its function. Standard tools of biological data analysis are well equipped to address steady-state behavior (e.g., dose–response curves), but to unravel the nature of dynamic processes (e.g., transient and oscillatory responses), new analytic methods are required. A fundamental aspect of a dynamic process is the timescale on which it acts. Just as biological systems span a wide range of spatial scales (from nanometer-wide proteins to kilometer-wide ecosystems), biological processes span a wide range of temporal scales, from ligand association (seconds) to protein expression (minutes to hours) to organismal development (months to years) to ecosystem alteration (years to millennia). Investigation of any given process often reveals a network of simultaneous events occurring over a range of timescales. The resulting dynamic network behavior is often not readily apparent from the nature of the individual components. The engineering community has long been dealing with (and exploit˚ stro¨m and ing) the dynamic behavior of mechanical and electrical systems (A Murray, 2008; Haykin and Van Veen, 2005). One of the basic notions the engineers have derived for addressing time-varying processes is the frequency response, which provides a concise characterization of the manner in which a system responds to perturbations at various timescales. The frequency response relies on frequency domain analysis, which applies directly only to linear systems. A system is linear if it acts additively: the response to two simultaneous perturbations is equivalent to the sum of the responses to the individual perturbations. Linearity is a helpful conceit; no real systems are perfectly linear. (In particular, biological systems typically involve saturation, rather than additivity, of response.) Nevertheless, systems often exhibit approximately linear behavior, and in fact all systems behave approximately linearly when exposed to perturbations that are

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sufficiently small. In practice, many systems exhibit behavior that is close to linear. This is particularly true for self-regulating systems that tend to operate around a specific nominal condition; this applies both to engineered automatic feedback systems and to homeostatic biological mechanisms. The frequency response of a system can be assessed by observing the response of the system to specific inputs. This analysis is simplest when those inputs are sinusoidal oscillations at various frequencies; the corresponding responses indicate the behavior of the system over a range of timescales. In some contexts (e.g., electrical circuits), the generation of oscillatory signals is straightforward. In contrast, such signals can be difficult to produce in a biological setting, especially in the case of chemical signals (though Block et al., 1983 is an early example of work applying oscillatory signals to analyze a biochemical system). Recent advances in microfluidic technologies have placed the production of such inputs, and of other oscillatory inputs such as square waves, within broader reach (Beebe et al., 2002; Bennett and Hasty, 2009; Bennett et al., 2008; Hersen et al., 2008; Mettetal et al., 2008; Shimizu et al., 2010). In engineering applications, the frequency response can be efficiently assessed in a single experiment using an input that “excites” the system at multiple frequencies simultaneously (Ljung, 1999). (Standard examples are white noise, steps, or approximations of the “impulse function”—an infinitely tall, infinitely short pulse). The signal-to-noise ratios (SNRs) inherent in molecular biology make these experiments less useful in this setting (although see Block et al., 1982 for a successful implementation of a chemical impulse). The frequency response provides valuable insight into a system’s dynamic behavior. This includes a characterization of the bandwidth of the system: the fastest timescale on which the system can act. Moreover, the frequency response can be used to generate a transfer function model of the system’s input–response behavior. A transfer function model allows prediction of the response of the system to arbitrary inputs (provided linearity of behavior is adequately maintained). A transfer function model does not address the specific mechanisms (e.g., biochemical or genetic) underlying the input–response behavior; it is a “black box” model. The term system identification refers to the process of constructing such a model from observation of dynamic responses. In this review, we will introduce and illustrate the system identification process and highlight cases in which the method has been successfully applied experimentally. In Section 2, we present a running application—a yeast G-protein signaling cascade—and use a mathematical model of this system to illustrate how the frequency response can be assessed from idealized experimental observation of the system’s response to sinusoidal inputs, as well as how the frequency response provides insight into system behavior. In Section 3, we describe how a frequency response could be generated from real experimental observations, and then extend that analysis to the construction of a transfer function model. We close Section 3 with a brief

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discussion of examples of experimental applications of frequency response methods to biochemical systems. Finally, in Section 4, we review successful biological applications of systems identification to experimental data, and conclude with a discussion of the role that the methods we describe here may play in future efforts to understand biochemical and genetic systems.

2. System Identification Applied to a G-Protein Pathway Heterotrimeric G-protein signaling systems are a common component of eukaryotic signal transduction pathways, and are of acute clinical interest as common drug targets (McCudden et al., 2005; Oldham and Hamm, 2008; Yi et al., 2003). The G-protein component of the pheromone response pathway in the budding yeast Saccharomyces cerevisiae is a well-characterized example of this family of pathways; both the kinetic details and the dynamic behavior of the pathway have been studied (Yi et al., 2003). We will use this pathway, shown schematically in Fig. 10.1, to illustrate system identification techniques. In this section, we will examine the behavior of the mathematical model constructed by Yi et al. (2003) as an idealization of an experimental analysis. This will allow a clean presentation of the concepts underlying the frequency response.

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2.1. Construction of the frequency response As described in Section 1, the frequency response can be assessed from timeseries observation of the system’s response to oscillatory input signals. In the case of the pheromone response G-protein pathway, the input (extracellular ligand) is the level of alpha factor pheromone (assuming the target cells are mating type a). Figure 10.2 shows a model-generated dose–response curve indicating the abundance of the pathway output, active Ga-GTP as a function of extracellular alpha factor concentration (L in Figure 10.1). This curve, which matches the experimental results of Yi et al. (2003), indicates that over its active range, the signal transduction pathway’s active range is characterized by a near-linear response centered at an alpha factor level of about 1 nM. Consequently, we choose 1 nM as a nominal input level, and address the behavior of the system as the ligand input varies around this nominal value. From the curve, an alpha factor concentration of 1 nM corresponds to a Ga-GTP abundance of 510 molecules per cell. Linear systems and sinusoidal inputs share a special relationship that underlies all frequency domain analysis: the steady-state response of a linear system to a sinusoidal input is a sinusoidal output of the same frequency. This does not hold for nonlinear systems, or for other classes of inputs.

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Figure 10.3 shows a model simulation representing the time-series measurements of an experiment in which cells are exposed to a sinusoidal oscillation in ligand level. The system response follows a short transient before settling to a steady oscillatory behavior. The frequency of oscillation of the response matches that of the input, but the output oscillations do not have the same amplitude as the input, and the two signals are out-of-phase (i.e., the peaks and troughs are not aligned). The ratio of the amplitude of response to the amplitude of input is called the system gain; the difference in the phase is called the system phase shift (or just system phase). These measures of system behavior are dependent on the frequency of the input, but (as a consequence of linearity) do not depend on its amplitude. By probing the response of the system to sinusoidal inputs at various frequencies, we reveal how these two measures (gain and phase) depend on the timescale of the input. In the subsequent dynamic analysis, we will address the response of the system in terms of the deviation of the Ga-GTP level from 510 molecules per cell, to an input described by the deviation of the alpha factor level from 1 nM. Since we are simulating model behavior, we will address the response to pure sinusoids. In many cases, sinusoidal inputs cannot be experimentally achieved, and so other oscillatory inputs (e.g., square waves) must be used.

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We will defer the treatment of these more general inputs (which requires a discussion of Fourier analysis) to Section 3. Figure 10.4 shows three simulated steady-state input–response pairs. In each case, the gain and phase can be assessed by comparing the two signals. Comparing these three responses (note the change in scale on the time axis), we see that the amplitude of the response (the deviation from nominal GaGTP levels) changes with input frequency, although the amplitude of the input remains the same. We see too that the phase behavior changes: at low frequency the response lies (nearly) in phase with the input; at higher frequencies the response falls significantly out-of-phase. When this experiment is repeated over a range of frequencies, an overall picture of the system’s frequency - dependence emerges. This information is typically displayed in a pair of plots: graphs of the system gain and system phase as functions of frequency. Figure 10.5 shows the corresponding curves for this system, with the data from Fig. 10.4 labeled. Following standard practice, frequency is plotted on a logarithmic scale, as is gain, while phase shift is plotted in degrees. These are known as gain and phase Bode plots, named after the American engineer Hendrik Bode (1905–1982) who was the first to apply this analysis. (Engineers typically scale the gain in a Bode plot by a prefactor of 20, and report the result in decibels; also they generally report frequency in radians per second rather than Hertz.) In this case, the Bode plot was generated directly from the mathematical model. In application, these curves would be fit to experimental observations. We illustrate this fitting process in Section 3.

2.2. Interpretation of the frequency response The Bode plots provide direct insight into the dynamic behavior of the system. The phase plot contains information about the way in which the system acts when connected to other systems (in cascade or in feedback). We will not address such issues here, but in section 3 we will find the phase plot useful as an additional constraint on the fitting of a transfer function to frequency response data. The gain plot indicates the response of the system to perturbations at various frequencies (i.e., timescales). In particular, it reveals the (frequency) filtering properties of the system. The notion of frequency filtering follows from the fact that any signal can be expressed as a combination of pure sinusoids, via Fourier decomposition.1 As illustrated above, the gain plot shows how a system amplifies or 1

A periodic signal can be written as a sum of sinusoids over a discrete range of frequencies, via a Fourier series. A nonperiodic signal can be expressed as an integral of sinusoids over a continuum of frequencies, via the Fourier transform.

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attenuates a sinusoidal input at a given frequency. When the linear system is exposed to a general input, each of its frequency components is passed to the output according to the corresponding gain, and thus this graph gives a powerful map of the system’s input–output behavior. As a first example, consider Fig. 10.6, which shows the gain plot for a system with a resonant frequency of 0.5 Hz. This plot suggests that this system will respond strongly to sinusoids at this resonant frequency, and will

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attenuate oscillatory signals at all other frequencies. Referring to Fig. 10.7, we see an example of that behavior. The input is a Gaussian white noise signal, which is composed of oscillations over a wide range of frequencies. The output, in contrast, shows only the influence of the input at the resonant frequency: the remaining components of the signal have been so attenuated that they are no longer visible on the plot. This sort of resonant system is useful in technology2 and may appear in some biological oscillators as well (see, e.g., Westermark et al., 2009), but is not representative of the typical behavior of biochemical and genetic networks; we include it here as a direct illustration of filtering. Figure 10.8 illustrates some of the filtering behaviors more commonly exhibited by cellular processes. A single input function—a low-frequency (0.16 Hz) sinusoid corrupted by higher frequency noise (Fig. 10.8A)—has been passed through three separate systems. In each case, the nature of the resulting output signal can be predicted from the system’s frequency response, as follows. In the first case (Fig. 10.8B and C), the system acts as a low-pass filter. The gain plot is approximately piece-wise linear: a constant value at low frequencies, then a corner, followed by a linear descent with a slope of 1 on the log–log plot. The frequency at which the corner occurs is known as the system’s corner (or cut-off or break) frequency, and indicates the system’s bandwidth. Below this frequency, the system allows all components of the 2

For instance, Fig. 10.7 illustrates the phenomenon that allows a distinct radio signal to be captured from the mixture of signals that we hear as radio “static.” The tuning dial allows the listener to change the resonant frequency, thus selecting which signal will be “lifted” out of the noise.

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Figure 10.8 Filtering behavior. (A) The input signal is a sinusoid with frequency 0.16 Hz, corrupted with high-frequency noise. This input has been passed through each of the systems in the second row, resulting in the output signals in the third row. (B and C) The gain plot for B shows that this system is a low-pass filter with a corner frequency of 1 Hz and a roll-off of  1. The output C shows that the high-frequency noise has been reduced, leaving the underlying sinusoidal signal more visible. (D and E) The gain Bode plot D is again a low-pass filter, this time with a lower corner frequency (10 2 Hz). The output signal has its high-frequency components attenuated more than in panel C, resulting in a smoother sinusoid. (F and G) This system’s gain Bode plot F shows that it is a band-pass filter, with a peak at 102 Hz. The frequency of the sinusoidal oscillation in the input signal (panel A) lies well below this peak, so the filter obscures the sinusoidal nature of the original signal, leaving only the higher frequency noise components. The resulting output G is relatively flat, and somewhat less noisy than the original because the highest frequency components of the noise have also been attenuated by the band-pass filter.

input to “pass” through to the output signal. High-frequency components of the input, in contrast, are significantly attenuated. Comparing the input and output signals in this case, we see that the low-frequency oscillations are retained in the output, as is the noise around that oscillation, but the noise has been smoothed: the high-frequency components have been stripped away, leaving a signal with less “wild” changes in direction.

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More specifically, the gain plot for this low-pass filter is characterized by three parameters: the gain at low frequency, the frequency at which the “corner” occurs, and the slope after the corner (the so-called “roll-off ”); all three are informative about the system’s behavior: 

Low-frequency (DC) gain: The gain at low frequency is the degree to which constant and slowly varying inputs are amplified or attenuated. (Electrical engineers refer to this as the “DC gain” since direct current (DC) provides a constant input.) This low-frequency gain describes, for instance, the steady-state response of the system to a step input, and is often referred to as the (local, steady-state) sensitivity of the system to the input (Ingalls, 2004).  Corner frequency: As mentioned, the corner frequency indicates the system’s bandwidth: the frequency above which inputs are attenuated. For cellular processes, the bandwidth is closely related to the timescales on which component processes act (e.g., association/dissociation rates, halflives of proteins, metabolic consumption rates). When exposed to disturbances that act on timescales considerably faster than their inherent timescales, these processes are unable to “keep up,” and instead react only to the temporal average of the perturbation. In the case of sinusoidal inputs centered around a nominal level, the signal averages to the nominal input, and so the system exhibits no dynamic response to such highfrequency signals.  Roll-off: The drop beyond the corner frequency eventually reaches a straight line with a downward slope of N decades of gain per decade of frequency on this log–log plot. The slope N is called the relative degree of the system, and is a measure of how quickly the output responds to changes in the input (the larger the relative degree, the more sluggish the response). The filters considered here have a slope of one decade per decade, and so have relative degree 1. The effect of a change in the bandwidth is illustrated by the second system shown in Fig. 10.8D and E. This system is also a low-pass filter, but has a shorter bandwidth, and so allows fewer high-frequency components of an input signal to pass to the output. The response in the Fig. 10.8E illustrates this behavior: the low-frequency sinusoid is clearly seen in the output, but this filter has removed even more of the high-frequency noise. The gain plot of our third example (Fig. 10.8F and G) differs significantly from the low-pass filters: it shows attenuation at both low and high frequencies. This is referred to as a band-pass filter, and is characterized by the range (or band) of frequencies that it allows to pass from input to output. (The resonant system in Fig. 10.6 is also a band-pass filter, with a very tight band.) Referring to the corresponding output signal (Fig. 10.8G), we see a behavior that is complementary to the low-pass filters: the low-frequency sinusoid has been filtered out since its frequency does not match the

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frequencies “passed” by the band-pass filter; only the higher frequency noise has been retained (and has been somewhat smoothed by blocking of its highest frequencies). In technology, band-pass filters are implemented to allow downstream processes to access specific frequency ranges within input signals. Biological processes that exhibit band-pass behavior may have been selected to perform similar roles. In any case, the range of frequencies over which a biological band-pass filter acts is likely indicative of the kinds of signals to which it is attuned, and so speaks directly to its function and environment. A third category, the high-pass filter, has essentially the opposite form and effect as a low-pass filter: the frequency response is small for low frequencies, and rises to some maximum for high frequencies. Such filters can, much like low-pass filters, be characterized by the slope on their logscale Bode plot and the corner frequency (here representing the frequency below which the response is attenuated). Returning to our G-protein pathway model (Fig. 10.5), we see that it displays primarily low-pass behavior, with a mild band-pass around 10 3 Hz. The low-frequency (DC) gain is 200 (a ratio of molecules per cell to nM), the bandwidth is 5
10 3 Hz, and the roll-off indicates a relative degree of 2. These values reveal the types of signals to which the pathway responds most strongly, and may provide insight into its evolution or biological function. As we have seen, the gain plot provides a useful nonparametric model for the system, and could be fit using standard nonlinear regression techniques (Seber and Wild, 2003). However, the phase shift data sheds light on the construction of a more useful representation of system behavior: an input– output map known as the system transfer function, that will allow prediction of the system response to arbitrary signals

3. System Identification In this section, we present the method of system identification from data, using simulated data from a stochastic simulation of the G-protein pathway model to illustrate the method. Before discussing the details of implementation, we introduce the notion of the transfer function.

3.1. Transfer function models Most dynamic mathematical models in systems biology are based on knowledge of the kinetics of underlying chemical and genetic mechanisms (Hasty et al., 2001; Ideker et al., 2001; Kitano, 2002; Tyson et al., 2001). Here we take a complementary approach: using only our collection of

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input–response data (and assuming no knowledge of mechanism), we fit a “black box” model that accurately describes the observed input–output behavior. Such a model does not directly serve to provide insight into the mechanisms of the system, but rather provides a concise, predictive representation of its input–output behavior. The model we seek to fit is called a transfer function and takes the form of a ˚ stro¨m and rational function: a ratio of polynomials of a complex variable (A Murray, 2008). The transfer function formalism allows for a simple representation of dynamic input–output relationships. Such dynamic relationships are often captured by differential equations. For example, suppose that for a given system, the response y(t) is related to the input u(t) according to the differential equation d2 d d yðt Þ þ a1 yðtÞ þ a0 yðtÞ ¼ b1 uðtÞ þ b0 uðt Þ: 2 dt dt dt

ð10:1Þ

To avoid working with dynamic equations of this sort, we make use of the Laplace Transform L, which is an operator that maps functions of time to functions of a complex variable s, so that L[y(t)] ¼ Y(s) (Boyce and DiPrima, 2008). The Laplace transform has two key properties that we will use. The first is that it converts derivatives to products: L[(dn /dtn)y(t)] ¼ snY(s) (provided the signal, y, is zeroed and resting at its nominal value at t ¼ 0). This means that the differential equation relating y(t) to u(t) can be converted to a simpler, nondynamic, equation relating their Laplace transforms. With L[u(t)] ¼ U(s), we have s2 Y ðsÞ þ a1 sY ðsÞ þ a0 Y ðsÞ ¼ b1 sU ðsÞ þ b0 U ðsÞ;

ð10:2Þ

which can be simplified to Y ðsÞ ¼

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ð10:3Þ

Letting G(s) ¼ (b1s þ b0) / (s2 þ a1s þ a0), we call G(s) the transfer function for the system. The transfer function provides a simple relationship between the Laplace transforms, but will not be useful unless we can recover information about the original signals. Here the second key property of Laplace transforms can be employed: when evaluated at s ¼ io (where i is the square root of negative one),3 the Laplace transform yields the Fourier transform, which, as

3

Note: most control engineers follow the electrical engineering notation j ¼

pffiffiffiffiffiffiffi 1:

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we have discussed above, is a measure of the frequency content of a signal at frequency o. Thus the equation Y ðioÞ ¼ GðioÞU ðioÞ

ð10:4Þ

relates the content of the input, u, and the output, y, at frequency o. This is precisely the role of the frequency response, and in fact G(io), the value of the transfer function at s ¼ io, is precisely the frequency response of the system (at frequency o). To compare with a Bode plot, we recognize that G(io) is a complex-valued function of o: the gain at o corresponds to the modulus of this complex number, while the phase shift corresponds to its argument.4

3.2. Applying the procedure to the G-protein pathway For this illustrative example, we have generated simulated “data” from a stochastic version of the G-protein pathway model from Yi et al. (2003) (details in Appendix). Our system identification procedure seeks to recover aspects of the model behavior directly from the simulated data. In the following sections, we will present our implementation of the method shown schematically in Fig. 10.9. 3.2.1. Preliminary investigations As a first step, it is often worth performing preliminary investigations using input profiles that are standard in the lab, and therefore relatively easy to produce. These investigations represent a simple but informative "first pass" allowing us to focus on the appropriate amplitude and frequency ranges of the input signal before beginning frequency response analysis, where the necessary inputs are more challenging to produce experimentally. We begin our analysis of the system by determining the dose–response behavior for our chosen input and output. Figure 10.10 shows the results of exposing the system to a range of input values and determining the corresponding steady-state output value. As in Section 2, the input is the concentration of the alpha factor ligand and the output is the cellular abundance of the species Ga-GTP. From this graph, we can identify a roughly linear range of behavior between saturation at low and high input values. We chose to center our linear analysis around the input level 1 nM, as indicated in the figure. The corresponding nominal output value is a concentration of Ga-GTP of about 450 molecules per cell. Outside of the surrounding linear range, nonlinear effects become appreciable; these effects are also exposed in the frequency response analysis detailed below. We will 4

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi For a complex number x ¼ a þ bi, the modulus is jxj ¼ a2 þ b2 while the argument is arg(x) ¼ arctan (b / a). The argument of a complex number is sometimes referred to as its phase.

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Figure 10.9 Schematic flow chart of the sequence of steps for system identification via frequency response. The dashed box indicates steps that should be repeated over a range of frequencies in order to capture the principle features of the Bode plot. The feedback loops on the left side illustrate the iterative nature of this method. If the estimated model is not satisfactory, one should first look to different (higher order) transfer model structures. If a satisfactory model still cannot be found, then the overall experimental design may need to be modified in order to capture more detailed or more accurate information.

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Figure 10.10 Simulated G-protein pathway dose–response curve. To avoid regimes of saturation, we use 1 nM as the nominal dosage level about which to estimate a linear transfer function model.

discuss the linearity of the outputs after we have examined the system’s varying-frequency behavior. We then begin an exploration of the dynamic behavior of the system by exposing it to an input rectangular “pulse” (i.e., a step up followed by a step down). Using the knowledge gained from the dose–response analysis, we center our input values about the chosen nominal input. We observe an exponential rise when the input increases, and then exponential relaxation when the input is removed (Fig. 10.11). This exponential behavior is typical of biochemical and genetic networks. Some networks might display significant delay in the output response (indicating a high relative degree, see Section 2.2), or even damped oscillatory “ringing” rather than a monotonic rise and fall. (Such ringing is indicative of a strongly resonant frequency, as in Fig. 10.6, and would suggest that careful attention be paid to the frequency response near the frequency of the ringing.) In the case of exponential rise and fall, we are able to extract information about the timescale on which the system acts from the rise time (how long the system takes to reach its steady-state value after the step input is applied) and the decay time (how long the system takes to return to its initial level once the step input jumps back down to zero). In Fig. 10.11, the rise time shows that the pathway can respond to changes in the input signal on timescales on the order of tens of seconds, while the decay time is on the order of hundreds of seconds. Thus, to probe the system effectively, we must use oscillatory signals with periods that span this range, and ideally

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extend to significantly shorter and longer timescales as well. In the analysis to follow, we will use input signals with periods between 15 and 32,000 s (frequencies of 3.125
10 5 to 0.066 Hz). Of course in a real implementation, experimental limitations will likely provide constraints on the range of timescales that can be effectively probed. 3.2.2. Experimental design, data preprocessing, and the experimental oscillatory response We exposed the G-protein pathway model to sinusoidal oscillatory input signals over a range of frequencies (3.125
10 5 to 0.066 Hz), and observed the output response function at each frequency. In experimental implementations, square waves produced by rapid switching of computercontrolled values are often used as inputs (Hersen et al., 2008; McClean

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et al., 2009; Mettetal et al., 2008), though sinusoids have also been implemented (Bennett et al., 2008; Block et al., 1983; Shimizu et al., 2010). Using sinusoidal inputs allows us to mirror the analysis in Section 2. We discuss the case of general oscillatory inputs in Section 3.2.8. The choice of input amplitude can depend on a number of factors. The signal to noise ratio (SNR) of observations improves with increased amplitude, but increased amplitudes will also more strongly excite nonlinear effects; the Gprotein model system will exhibit linear behavior only over the range of input values indicated in Fig. 10.10. The experimental implementation may also constrain on the range of input amplitudes that can be achieved. Although the frequency response technique discussed here is used to estimate a linear model, it may not be possible to avoid exciting nonlinearities at experimentally realizable input amplitudes; such nonlinearities may be addressed by incorporating a nonlinear rectifier into the model, as presented in Section 3.2.6. We chose to probe our model system with sinusoids of amplitude half the nominal input value (i.e., alpha factor concentrations oscillating between 0.5 to 1.5 nM) When generating a time-series from the output response, note that the sampling frequency should be larger than twice the probing frequency, thus satisfying the Nyquist limit (Haykin and Van Veen, 2005) in order to avoid aliasing effects.5 Noise and nonlinearities will introduce content in the response at frequencies other than the driving frequency, but as long as the SNR is adequate, any aliasing resulting from higher frequency components should be negligible. We sampled the output response of our model system with 100 time points over the period of the input signal for the lowest frequency inputs (periods of 500 to 32,000 s), 50 time points over the input period for midrange frequencies (62.5 to 250 s), and 10 time points over the input period for high frequencies (15.625 to 31.25 s). A few more comments on data preprocessing are called for. As discussed in Section 2, for frequency response analysis, it is important to consider only the steady-state portion of the response, and not its initial transient. Once steady-state behavior is attained, it is beneficial to sample from multiple periods, to improve the SNR (Lipan and Wong, 2005); in essence, each period acts as a replicate. In our example, we waited for the system to reach the desired steady-state behavior (where the nature of the output oscillations is consistent and well established) and then sampled time-series whose length ranged from a single complete period for the lowest frequency responses (reflecting the difficulties of running long time-series experiments) to 10 complete periods for the highest frequencies. For convenience, the start time of our sampled steady-state data corresponds to the beginning of a sinusoidal cycle in the input signal. 5

Aliasing occurs when there are not enough data points during a period of a signal to faithfully represent that signal’s shape. For example, consider sin(t) sampled at t ¼ 0, p, 2p, . . ., leading to the mistaken observation of a constant signal.

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Finally, if necessary, one should also remove any obvious trends from the output data (e.g., linear trends caused by experimental drift), as well as obvious outliers in the data that are likely to arise from momentary instrument noise rather than the actual dynamics of the system. Figure 10.12 shows the results of sampling the output from our model system. 3.2.3. Fourier filtering and the experimental frequency response Unlike the idealized model in Section 2, the observed output responses here, like real experimental data, are not pure sinusoids, since they have been corrupted by nonlinearities and noise. We can isolate the contribution

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that corresponds solely to the input (driving) frequency by determining the corresponding coefficient in the Fourier series expansion of the ^ of the observed output response time-series. The Fourier coefficient , R, observed output response, R(t), at the angular input frequency o, is described by ^ðoÞ ¼ 2 R nT

ð nT

eiot RðtÞdt;

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0

where T ¼ 2p / o is the period of the input signal. Here, t ¼ 0 represents the beginning of our steady-state response data (recall that we had synced this to the beginning of a sinusoidal cycle in the input signal), and n denotes the number of full steady-state periods sampled. ^ðoÞ, whose The result of this integration is a complex number, R ^ modulus jRðoÞj is the output amplitude at frequency o, and whose argu^ðoÞ , is 90 degrees below the output phase at frequency o. ment, arg R Since Ro(t) is represented as a discrete time-series, Eq. (10.5) is evaluated by means of numerical integration. Numerical computing packages (such as MATLAB, Mathematica, and Maple) are equipped with the algorithms for handling such a problem, and a simple strategy such as the midpoint method can be easily implemented in, for example, Python. For our G-protein example, we have used the MATLAB function trapz, which implements a trapezoidal numerical integration algorithm. The system gain and phase shift is then calculated as in Section 2, as the ratio of the output to input amplitude and the difference between the output and input phase, respectively. The individual data points in Fig. 10.13 show the result of applying this analysis to the G-protein pathway model. Note that this is in close agreement with the idealized results in Fig. 10.5. 3.2.4. Fitting the transfer function model We now seek to go beyond the graphical representation provided by the Bode plots to construct a predictive model of the system’s input–output behavior. As discussed in Section 3.1, given the form of a rational function as in Eq. (10.2), we search for values for the coefficients ai and bi so that the modulus of G(io) and the argument of G(io) match the gain and phase shift observations at each frequency o. Before any fitting can be attempted, we must specify the form of the transfer function, that is, the degree of the numerator and denominator polynomials in Eq. (10.2). A transfer function model of the linear behavior of an autonomous biochemical or genetic network will take the form of a rational function in which the order of the numerator polynomial is less than the order of the

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Figure 10.13 Bode plot for the G-protein pathway determined by Fourier filtering the stochastically simulated sinusoidal response data (discrete data points) plotted alongside the fitted curve generated by the estimated linear model (solid curve).

denominator polynomial. The difference in the orders of these two polynomials is called the relative degree of the model, and corresponds, as noted earlier, to the slope of the roll-off on the gain plot. The order of the denominator polynomial is referred to as the order of the system; it represents, in some sense, a measure of the complexity of the system, and corresponds roughly to the number of corners or “kinks” in the Bode plots. (Referring back to the filters in Fig. 10.8, the two low-pass filters are first order systems—one corner—while the band-pass filter is a second order system—two corners). It is often the case that a higher order system is well approximated by a lower order model, so while the number of visible corners may not correspond directly to the order of the system, it gives a good indication of the minimal order required to provide a decent fit to the frequency response. The gain Bode plot for our G-protein example (Fig. 10.13) has a roll-off slope of approximately 2, indicating that we should choose a transfer function of relative degree 2. A simple transfer function of this sort has numerator of degree 1 and a denominator of degree 3:

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GðsÞ ¼

s3

b2 s þ b0 : þ a2 s2 þ a1 s þ a0

ð10:6Þ

The solid curve in Fig. 10.13 shows a fit of this model to the frequency response data, with parameter values of: a2 ¼ 2.0870
10 1, a1 ¼ 2.7659
10 3, a0 ¼ 3.0898
10 6, b1 ¼ 1.0914, b0 ¼ 7.6371
10 4. (We used the MATLAB function fminsearch to perform the fit, with a least squares error. Other nonlinear optimization routines, surveyed, e.g., by Moles et al. (2003) could also be used). We also fitted a fourth order model, but found little improvement in the fit. (Note that the model used to generate our “data” is in fact fourth order. Our analysis shows that a third order model provides a good approximation to the system.) 3.2.5. Simulating model trajectories The system response (beginning at the nominal steady state) can be simulated by evaluating an integral involving the system’s unit impulse response and the input function. The unit impulse response, g(t), is the system’s transient output in response to a unit impulse input at t ¼ 0, and can be found by applying the inverse Fourier transform to the frequency response, G(io). Specifically, the response y(t) can be calculated as the convolution of g(t) with an input function u(t), defined by yðtÞ ¼

ðt

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ð10:7Þ

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In practice, g(t) can be evaluated numerically from G(io) via the Inverse Fast Fourier Transform (IFFT). The IFFT routine, along with the Fast Fourier Transform (FFT) are standard components of numerical packages like MATLAB (as are routines evaluating convolution integrals). Alternatively, one can apply an inverse Laplace transform to the transfer function (by replacing the algebraic s variable with the time-derivative operator d/dt (compare Eqs. (10.2) and (10.3) to Eq. (10.1)). Output trajectories can then be determined by numerically solving the resulting differential equation. The resulting trajectories correspond to the linear model and so describe displacements from the nominal steady state. The nominal input and output values must be accounted for when comparing output model trajectories to measured data. Figure 10.14 shows the sampled oscillatory outputs alongside the oscillatory behavior predicted by our estimated model. The plots are in agreement, implying that our linear transfer function adequately represents the original oscillatory input–output data set.

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3.2.6. Applying a nonlinear rectifier In many circumstances, stronger nonlinear effects will cause discrepancies between the linearly modeled and measured data sets. If such is the case, it may be possible to address such effects after the fact, by appending a nonlinear element to the linear transfer function model. The method we present here attempts to correct for nonlinear effects by passing the linear model output through a static nonlinear rectifier. This linear–nonlinear cascade structure is referred to as a Wiener model, and is shown in block diagram form in Fig. 10.15, where the rectifier is represented mathematically by the static function fNL.6 In order to determine a form for fNL, we construct a scatter diagram that plots the linear model outputs against observations. A best fit trendline will define the function fNL. If the scatter plot indicates perfect agreement between model prediction and data, then there is no need for a nonlinear element. 6

An alternative structure that passes the original input signal through a static nonlinear element before it reaches the linear block is referred to as a Hammerstein model.

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Wiener Input u(t)

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Figure 10.15 Block diagram representing the structure of the Wiener model which places a static nonlinear block in series following the dynamic linear block (the transfer function model in our case).

The reader may be concerned that the addition of this nonlinear element will alter the model’s frequency response, painstakingly determined from the data. This is not the case. While this nonlinear element will cause lower frequencies to mix into higher frequencies, the driving (lowest) frequency component of an oscillatory input signal will propagate through unchanged. Therefore, measurements at the driving frequency will not be affected (Westwick and Kearney, 2003). The scatter plot for our G-protein example is shown in Fig. 10.16. This plot is populated with data from the oscillatory response simulations. The linear trendline—with a slope nearly equal 1 and a very small offset value of 18 molecules per cell—indicates that, over the range of inputs that were used, the estimated linear model sufficiently reproduces the true oscillatory output. Therefore, no nonlinear correction appears to be necessary in this case. In general, the nonlinearity fNL may be more complex than a linear, or even higher order polynomial correction. For example, in the work of Mettetal et al. (2008), a piece-wise defined function was chosen to compensate for both an offset and the fact that the linear model was providing predictions of negative responses within the input range of interest. 3.2.7. Model validation The final step in the system identification procedure is to compare outputs predicted by the estimated model to experimentally measured outputs for an independent set of data (that is, data that was not used for the purposes of fitting the model). Separating these data will ensure that the actual system has been modeled, rather than the specific output represented in the estimation data set. Comparisons can be done in both the time domain and the frequency domain. In Fig. 10.17, we compare the model and stochastically simulated responses in the time domain of the G-protein pathway to a rectangular pulse input. That they are in good agreement supports the validation that our model accurately captures the general dynamics of the G-protein system. Note that it is also a common engineering practice to perform a variety of statistical tests (particularly residual analysis) on the prediction errors of the estimated model as part of the model validation procedure (Ljung, 1999).

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3.2.8. Nonsinusoidal inputs In some cases, the generation of purely sinusoidal inputs will not be feasible, and so another oscillatory waveform must be used to excite the system. The system identification procedure can still be followed in this case, with some minor modifications. To illustrate how nonsinusoidal inputs may be used, we have repeated the frequency response analysis of the G-protein pathway model using square wave inputs. The driving frequency of the square wave represents the input frequency. Sampled points from the resulting output are shown in Fig. 10.18. When using nonsinusoidal oscillatory inputs, the amplitude of the input contribution at the driving frequency is not equivalent to the amplitude of

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the waveform itself. Since the input corresponds to a combination of sinusoids at different frequencies, Eq. (10.5) can be used to determine the amplitude of the contribution at the driving frequency. In the case of square waves, this calculation indicates that the amplitude of the overall wave must be scaled by a factor of 4/p in order to recover the contribution of the component at the driving frequency. Using the stochastically simulated square wave data, we performed a new fit to a transfer function model. The result is essentially the same as that obtained from the sinusoidal inputs; Fig. 10.18 shows the behavior predicted by the estimated transfer function model.

3.3. Examples of experimental implementation There is a long tradition of studying biological systems using the techniques of control theory (Bayliss, 1966; Iglesias and Ingalls, 2009; Ingalls et al., 2006; Khoo, 2000; Wiener, 1965). Techniques from system identification,

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including frequency response methods, have long been experimentally applied in physiology (Westwick and Kearney, 2003) and in bacterial chemotaxis (Block et al., 1982, 1983; Shimizu et al., 2010), and have

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more recently been applied to signaling cascades in yeast (Bennett et al., 2008; Hersen et al., 2008; Mettetal et al., 2008). 3.3.1. Bacterial chemotaxis Bacteria such as Escherichia coli respond to gradients of attractant or repellent in their environment by altering the proportion of time they spend randomly tumbling versus swimming in a straight line; this shift is implemented by changing the direction of rotation of their flagella, and has the effect of implementing a random walk that is biased in the direction of increasing concentrations of nutrients. Elegant work from Howard Berg’s group has examined the frequency response characteristics of bacterial chemotaxis for decades, initially at the whole-cell level (Block et al., 1982, 1983; Segall et al., 1986) and later targeting specific elements of the intracellular response system (Shimizu et al., 2010; Sourjik et al., 2007); we will touch here on only a small portion of the large body of work on this subject. In whole-cell investigations (Block et al., 1982, 1983; Segall et al., 1986), the experimental protocol consisted of affixing cells to a cover slip by a single flagellum, using a procedure described by Berg and Tedesco (1975). When pinned in this manner, the cells’ flagellar motors caused the entire cell to spin either clockwise (CW; corresponding to the random “tumble” mode of motion in a free-moving cell) or counterclockwise (CCW; corresponding to the linear “run” motion seen in free-moving cells) in response to the ambient chemical environment. This motion could be observed and classified (with particular attention to the timing of transitions between CW and CCW rotation) through phase-contrast microscopy. The intracellular measurement experiments (Shimizu et al., 2010) used Forster resonance energy transfer, (FRET; also known as fluorescence resonance energy transfer) to generate a fluorescence signal proportional to the physical proximity of the labeled proteins; by attaching fluorescent proteins to key players in the chemotaxis signaling system, FRET studies provided an output measurement corresponding to the level of activity of this system (Sourjik et al., 2007). Impulse inputs. In the work of Block et al. (1982), the environment was changed by introducing very brief changes in the concentration of either an attractant or a repellent, through a micropipette: the pulse of chemical stimulation passed across the population of tethered cells as a diffusive wave, providing each cell with a very brief burst of stimulation. This input approximates a mathematical “impulse”: a sharp “spike” of input (the mathematical ideal is infinitely tall but infinitely short; real impulses, such as the chemical pulses used in the work described here, serve as an approximation to this mathematical abstraction). Characterizing the response to an impulse input illustrates an alternative approach to the frequency response method we have discussed above. In ˚ stro¨m and Murray, 2008; Cluett engineering, impulse and step responses (A and Wang, 1991) are used as an alternative to explicitly sweeping the

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frequency of a series of input signals. The basis for this approach is that these signals, by virtue of their sharpness, excite a wide range of frequencies simultaneously since a sharp signal, when broken down into a sum of signals of different frequencies, requires many frequencies in order to represent it accurately. This approach succeeded in (Block et al., 1982) yielding frequency response plots without the experimental complications of generating inputs of varying frequencies; the plots showed a band-pass filtering behavior in the bacterial response to the chemical attractants and repellents, with a peak response near 0.25 Hz. More generally, it is not clear that impulse or step responses will always be sufficient to characterize other biochemical systems, especially in terms of gene expression and signal transduction. Using specific input frequencies improves the experimental SNR, allowing one to distinguish responses from noise, and to use the persistent, periodic input to average the response over multiple periods, gathering more data and thus improving SNR. Lipan and Wong (2005) argued that periodic inputs offer significant advantages over step inputs in particular: in addition to noting the SNR improvements, they carried out a theoretical analysis indicating that to obtain a given Fourier component with a desired level of confidence, significantly more experimental replicates of step inputs would be required than of periodic inputs. Further, the theoretical analysis presented by Saeki and Saito (2002) suggested that step inputs are limited in their ability to generate frequency response information, even in systems less complex than biological networks. The successful application of impulse responses by Block et al. (1982) to derive a full frequency response plot is impressive, and demonstrates that this approach is at least sometimes feasible in a biochemical response context. The “digital” nature of the output signal employed by Block et al. (1982) (only one of two outcomes is observed for any given cell, CW or CCW rotation, rather than a continuum of possible values) may have helped reduce potential problems with the SNR, since noise inside the system would be seen in the output only if it was large enough to flip the response from one rotational state to another. Exponentiated sine inputs. In the work of Block et al. (1983), computercontrolled pumps were used to generate varying-frequency input signals, similar to the sinusoidal inputs discussed in Sections 2 and 3.2, but in this case implemented as exponential functions of the form (esin ot), superimposing a periodic input onto an exponential ramp. The same work (Block et al., 1983) showed that an exponential ramp input caused the switching rate of the flagellar motor to reach and remain at a steady state over time; this motivated the choice of an exponentiated sine wave as the frequency-varying input for this system. Using such signals, they obtained a Bode plot spanning frequencies from 10 3 to 0.05 Hz, showing high-pass filtering behavior with a slope on the log–log plot of þ2.

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Another study (Shimizu et al., 2010) again employed input signals of the exponentiated sine form, and used FRET (Sourjik et al., 2007) to monitor the intracellular interactions of proteins CheY-YFP and CheZ-CFP. Phosphorylated CheY binds to the flagellar motor to affect its rotational behavior, while CheZ dephosphorylates CheY; the FRET signal, by reporting how many CheY–CheZ pairs were close to one another, thus provided a measurement of the output of the intracellular signaling system underlying the bacterial chemotactic response. The Bode plot constructed by sweeping the frequencies of the exponentiated sine inputs showed high-pass filtering behavior, qualitatively similar to that seen by Block et al. (1983) and well predicted by a mathematical model (Shimizu et al., 2010; Tu et al., 2008). A Bode plot relating the time derivative of the input to the output showed low-pass filtering behavior; the bacterial chemotaxis system appears to function as a high-pass filter of the signal itself, and a low-pass filter of its derivative, as previously predicted theoretically (Tu et al., 2008). 3.3.2. Signaling cascades in yeast Three papers using frequency response methods in yeast appeared in 2008. Two effectively simultaneous publications dealt with the high-osmolarity glycerol (HOG) osmo-adaptation pathway (Hersen et al., 2008; Mettetal et al., 2008), while the third (Bennett et al., 2008) addressed the galactose utilization response. Experimental setup. All three papers made use of microfluidic chambers to contain the cells and expose them to varying inputs: changing concentrations of extracellular sorbitol (Hersen et al., 2008), sodium chloride (Mettetal et al., 2008), or glucose (Bennett et al., 2008). In the studies of Hersen et al. (2008) and Mettetal et al. (2008), the flow was driven by pumps, and varying concentrations were obtained by computer-controlled switching of valves connecting the microfluidic chamber to different fluid reservoirs; the input signals generated were approximately square waves. In the work of Bennett et al. (2008), the flow was driven by gravity-induced pressure differences and the input concentration was regulated by a computer-controlled actuators that varied the height of two fluid reservoirs (Bennett and Hasty, 2009; Bennett et al., 2008); the input signals used were sinusoidal in shape. In all three cases, the output signals were observed through fluorescence microscopy: fluorescence of Hog1-YFP, using colocalization with the strictly nuclear protein Nrd1-RFP to isolate and measure only the Hog1-YFP present in the nucleus (Mettetal et al., 2008); fluorescence of Hog1-GFP, using colocalization with Htb2-mCherry to isolate the nuclear Hog1-GFP (Hersen et al., 2008); and whole-cell fluorescence of Gal1-yECFP fusion proteins (Bennett et al., 2008). Time-lapse fluorescence images, with image intensities quantified through various image analysis routines, generated time-series for the output signals in each case; combining these with the known input signal time-series yielded a

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complete input–output response profile for each frequency, and this procedure was repeated across a range of input frequencies.

A (w)

Wild-type 10–1

Low Pbs2 –2

10

10–2

10–1

100

w (rad/min)

101

100

w = 0.0046 Hz

B A

Nuclear hog 1-GFP fluctuations (a.u.)

3.3.2.1. Osmo-adaptation Yeast sense and respond to osmotic pressure through a multistage signaling cascade that terminates in glycerol production, activated when the protein Hog1 is phosphorylated and enters the nucleus. Mettetal et al. (2008) and Hersen et al. (2008) studied the frequency response of this system. The Bode gain plots for the HOG pathway’s response to changing osmotic pressure in the two papers are shown in Fig. 10.19A and B, respectively. At first sight, the plots appear to present different conclusions about the same pathway, showing band-pass filtering behavior in one case, and low-pass filtering in the other. Note, however, the different frequency ranges considered in the studies: Mettetal et al. (2008) considered periods ranging from 2 to 128 min, corresponding to frequencies of 1.3
10 4–8.3
10 3 Hz (shown in radians per minute in Fig. 10.19A: 0.05–3.14 rad/min), while Hersen et al. (2008) considered higher frequencies, ranging from 10 3 to 1 Hz; the two ranges overlap only slightly. The band-pass response seen in the work of Mettetal et al. (2008) (panel A) does not appear in the Hersen et al. (2008) data (panel B) because

80 60 40 20 0 0.001

0.01

0.1

1

Frequency (Hz)

Figure 10.19 The gain portions of two experimental Bode plots for the HOG osmoadaptation system in yeast. (A) Experimental results from Mettetal et al. (2008). The input signals were square waves of extracellular sodium chloride concentrations at varying frequencies, and the output was the fluorescence intensity of Hog-YFP in the nucleus. “Low Pbs2” refers to a yeast strain that expresses lower (than wild type) levels of the Pbs2 protein (an element in the signaling cascade). [From Mettetal et al. (2008). Reprinted with permission from AAAS.] (B) Experimental results from Hersen et al. (2008). The input signals were square waves of extracellular sorbitol at varying frequencies, and the output was the fluorescence intensity of Hog1-GFP in the nucleus. [Copyright 2008, National Academy of Sciences (USA), reprinted with permission.] Note that the results in panel (B) are shown on a linear vertical scale, rather than the log scale used in panel (A) and in the Bode gain plots elsewhere in this review (Figs. 10.5, 10.8, and 10.13). The asymptotic approach to zero is a curve on the linear scale, rather than the straight line seen on a log scale.

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the latter considers frequencies higher than those at which the initial increase in the amplitude response were observed; at the same time, the significantly higher frequency regime examined by Hersen et al. (2008) provides valuable information, showing very clearly that the amplitude response does continue to decline with frequency in a classic low-pass filter pattern. Combined, these two studies provide a complete and consistent picture of the frequency behavior of the HOG pathway: an initial rise in the amplitude response, to a frequency of approximately 10 3 Hz, followed by a steady decline for higher frequencies. The slope of the decline of the amplitude in the log–log plot shown in Fig. 10.19A suggests a relative degree of 1. The authors in Mettetal et al. (2008) chose a transfer function model with this relative degree. The parameters of this model were fit using experimental data, then the same parameters were used to predict the system’s response to a step input, a form of input not used in the fitting. The result was a good match between the model predictions and the experimentally observed response (Mettetal et al., 2008), validating the model and providing a nice illustration of the predictive power of transfer function models. 3.3.2.2. Carbon source utilization When its preferred carbon source, glucose, is not available, yeast can activate genes to utilize galactose instead. Bennett et al. (2008) examined the frequency response of this system was examined. The resulting frequency response showed low-pass behavior. The authors went on to formulate a detailed biochemical kinetic model of the system, and attempted to use it to reproduce the experimentally observed frequency response. They found, however, that their initial model did not provide a good match to the experimental frequency response behavior. Modifying the model to allow the degradation rates of two key components (the mRNA of genes GAL1 and GAL3) to be a function of glucose concentration yielded a much more accurate model, and subsequent experiments confirmed that these mRNA half-lives were in fact glucose-dependent, revealing a level of posttranscriptional regulation that had not previously been known. The ability to compare with a full set of frequency response experiments, rather than, for example, steady-state experiments at a series of fixed glucose concentrations, provided the important ability to isolate which portion of the model needed to be adjusted, and in this case yielded an important new insight into a well-studied biological system.

4. Conclusion The engineering concepts described in this review can serve the systems and synthetic biology communities in a number of ways.

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Some experimental projects may have a frequency response input– output characterization as their end goal. Using the transfer function models described above, such projects can result in the ability to predict a system’s response to arbitrary inputs. That capability is often valuable in itself, and is of particular interest in the engineering-oriented field of synthetic biology (Andrianantoandro et al., 2006; Hasty et al., 2002; Kærn and Weiss, 2006; Kærn et al., 2003; Khalil and Collins, 2010; Purnick and Weiss, 2009), where the design and construction of novel cellular regulatory systems can be guided and informed by the use of frequency response methods. For systems biology purposes, obtaining a frequency response profile using the methods described here will typically be part of a larger effort to understand biochemical and genetic systems at a level beyond their input– output profile. These methods have several things to offer this endeavor. Most basically, the frequency response profile provides insight into the timescales that are most relevant to the system being studied: high or low gains at particular frequency ranges will indicate which timescales the system responds to most strongly, and whether such responses are tuned to any particular frequencies. This information places constraints on the biochemical or genetic mechanisms that underlie the observed response, and thus can guide experimental investigations. When comparing computational or theoretical models to experimental results, a full frequency response profile provides a richer basis for comparison than does steady-state or step-response data. We noted above, for example, that Bennett et al. (2008) used their frequency response data to pin down the source of an experimental mismatch in their modeling work. This is not simply a question of there being more data available to compare: the data is organized in a qualitatively different manner when experimental frequency response information has been obtained, and it allows access to a set of analytical tools that would not apply in the absence of such information. In the long term, researchers may aim to use “black box” models (where inputs elicit outputs by an unknown internal process) provided by frequency response methods as a step toward “white box” models (where the internal mechanism of the system are known; biochemical kinetic models (Conrad and Tyson, 2006; Hasty et al., 2001) are an example), possibly through intermediate “gray box” models (where the black boxes are used to partially populate a model with limited internal details) (Lipan, 2008). Such an approach was adopted by Mettetal et al. (2008), using the structure of the frequency response to guide the construction of a partially filled-in kinetic model of their system. Experimentally, the generation of stable, repeatable, rapidly varying oscillatory inputs remains a challenge, but the barrier for entry is dropping quickly as microfluidic technologies become less expensive and more widely available. To date, identification of frequency response from

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experimental data in biochemical and genetic systems has been carried out using extracellular signals that are conveyed to the cells by changing the concentration of some chemical in the surrounding medium (Bennett et al., 2008; Hersen et al., 2008; Mettetal et al., 2008; Shimizu et al., 2010). Inducing oscillations in internal states that are not coupled to an extracellular signal will be a further challenge, though as Lipan (2008) notes, one path might be to use intracellular regulatory elements that can be induced by the application of photons (Shimizu-Sato et al., 2002), perhaps coupled to other synthetic tunable systems (Grilly et al., 2007). Our ability to apply oscillatory inputs to cells will only grow more sophisticated over time, increasing the number of systems to which frequency response methods can be profitably applied. Finally, we address again an apparently severe limitation of this approach: its applicability only to linear systems (ones for which the response to a sum of two inputs is the sum of the responses to each individual input). Every real physical system is nonlinear, displaying at least saturation behavior, and biological systems are particularly subject to nonlinearities. Engineers must often deal with systems that are highly nonlinear, but this has not prevented them from designing much of the modern world using techniques founded on linear approximations. They make use of the fact that systems are always approximately linear in the vicinity of their steady states (where self-regulating systems spend the bulk of their time); they use nonlinear correction blocks of the type described in Section 3.2.5; and they use the information derived from linear approaches as a starting point for understanding fully nonlinear behavior. In fact, linear approximations are at the core of the study of nonlinear systems (Glendinning, 1994): the first thing generally done when analyzing a nonlinear system is to consider its locally linear behavior. Biologists are faced with the study of the most complex nonlinear dynamic systems ever examined, and they too can benefit from the insight and foundation for future study provided by linear analysis. There can be no doubt that the methods of dynamic analysis we review here will become a valuable part of the biological toolbox.

Appendix We use a model of the G-protein pathway as developed in Yi et al. (2003) as a running example throughout the above discussion. The model has four independent states: free receptor, R; ligand-bound receptor, RL; inactive heterotrimeric G-protein, G; and active Ga-GTP, Ga. In addition, there are three dependent states: free ligand, L; free Gbg, Gbg; and inactive

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Ga-GDP, Gd. The species concentrations are measured in molecules per cell, while ligand concentrations are measured in nanomoles per liter (nM).

A.1. Deterministic model The dynamics are d ½Rðt Þ ¼ kRL ½L ðtÞ½RðtÞ þ kRLm ½RL ðtÞ  kRd0 ½RðtÞ þ kRs dt d ½RL ðtÞ ¼ kRL ½L ðtÞ½RðtÞ  kRLm ½RL ðtÞ  kRd1 ½RL ðt Þ dt d ½GðtÞ ¼ kGa ½RL ðt Þ½GðtÞ þ kG1 ½Gd ðtÞ½Gbgðt Þ dt d ½Gaðt Þ ¼ kGa ½RL ðtÞ½GðtÞ  kGd ½Gaðt Þ dt where the G-protein components are constrained by ½GbgðtÞ ¼ ½Gtotal   ½GðtÞ ½GdðtÞ ¼ ½Gtotal   ½GðtÞ  ½Gaðt Þ: The free ligand concentration L(t) is the input to the system. It is assumed that ligand uptake by the cells is fast, and has a negligible effect on the overall concentration.The parameter values used are kRL ¼ 2
106 M1 s1 ; kRLm ¼ 1
102 s1 ; kRd0 ¼ 4
104 s1 kRs ¼ 4ðmolecules per cellÞs1 ; kRd1 ¼ 4
103 s1 kGa ¼ 1
105 ðmolecules per cellÞ1 s1 ; kG1 ¼ 1ðmolecules per cellÞ1 s1 kGd ¼ 4
103 s1 L ðt Þ : variable input; nominal value 1 nM Gtotal ¼ 10; 000ðmolecules per cellÞ

A.2. Stochastic model Since all processes in the deterministic model are individual chemical events, it can be easily converted to a stochastic model. The states are the same, with abundance in molecules (and the system size being a single cell). Again, it is assumed that ligand uptake by the cells is fast, and has a negligible effect on the overall concentration. Thus we retain a molar measure of ligand concentration, and treat its effect as a variation of the effectively first order parameter kRL[L(t)], which has units of s 1.

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The set of reactions representing the model from Yi et al. (2003) is kRL

L þ R Ð RL kRLm

kRs

ðÞ Ð R kRd0

kGa

RL þ G ! Ga þ Gbg þ RL kRd1

RL ! ðÞ kGd

Ga ! Gd kG1

Gd þ Gbg ! G Numbering the reactions r1 through r8 (including reverse reactions), we have corresponding propensities a1 ¼ kRL L ðt ÞRðt Þ; a2 ¼ kRLm RL ðt Þ; a5 ¼ kGa RL ðtÞGðtÞ; a6 ¼ kRd1 RL ðtÞ; a8 ¼ kG1 Gd ðtÞGbgðtÞ Numbering the state (s1, s2, s3, s4) matrix is given as 2 1 1 1 6 1 1 0 S¼6 4 0 0 0 0 0 0

a3 ¼ kRs ; a4 ¼ kRd0 Rðt Þ a7 ¼ kGd Gaðt Þ;

¼ (R, RL, G, Ga), the stoichiometry 1 0 0 0

0 1 0 0

0 0 0 0 1 1 1 0

3 0 07 7 05 1

The species Gbg and Gd are updated according to the conservations GbgðtÞ ¼ Gtotal  GðtÞ Gd ðtÞ ¼ Gtotal  Gðt Þ  GaðtÞ: The model was simulated using Gillespie’s direct method (Gibson and Bruck, 2000; Gillespie, 1977). The resulting numerical output was sampled at uniform time intervals. To mimic nonsystematic measurement noise, we added to each value a white noise term with a coefficient of variation (CV, the standard deviation divided by the mean) of 0.2.

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Biochemical Pathway Modeling Tools for Drug Target Detection in Cancer and Other Complex Diseases Alberto Marin-Sanguino,* Shailendra K. Gupta,†,‡ Eberhard O. Voit,§,} and Julio Vera† Contents 1. Introduction and Overview 2. Biomedical Knowledge and Data Retrieval: Constructing a Conceptual Map of a Biochemical Network 3. Mathematical Modeling of Biochemical Networks: Translating Knowledge into Mathematical Equations 3.1. Network reconstruction 3.2. Flux balance analysis 3.3. Dynamic models 4. Model Calibration: Matching the Mathematical Model to Quantitative Experimental Data 5. Predictive Model Simulations as a Tool for Drug Discovery 6. Model Sensitivity Analysis as a Tool for Detecting Critical Processes in Biochemical Networks 7. Drug Target Detection Through Model Optimization 8. One Step Further: Combining Mathematical Modeling with Drug Screening via Protein Docking-Based Techniques 9. Final Remarks Acknowledgments References

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Abstract In the near future, computational tools and methods based on the mathematical modeling of biomedically relevant networks and pathways will be necessary for the design of therapeutic strategies that fight complex multifactorial diseases. * Department of Membrane Biochemistry, Max Planck Institute of Biochemistry, Martinsried, Germany Department of Systems Biology and Bioinformatics, University of Rostock, Rostock, Germany { Indian Institute of Toxicology Research (CSIR), Lucknow, India } Integrative BioSystems Institute, Georgia Institute of Technology, Atlanta, Georgia, USA } The Wallace H. Coulter Department of Biomedical Engineering, Georgia Institute of Technology, Atlanta, Georgia, USA # 2011 Elsevier Inc. Methods in Enzymology, Volume 487 ISSN 0076-6879, DOI: 10.1016/S0076-6879(11)87011-4 All rights reserved. {

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Beyond the use of pharmacokinetic and pharmacodynamic approaches, we propose here the use of dynamic modeling as a tool for describing and analyzing the structure and responses of signaling, genetic and metabolic networks involved in such diseases. Specifically, we discuss the design and construction of meaningful models of biochemical networks, as well as tools, concepts, and strategies for using these models in the search of potential drug targets. We describe three different families of computational tools: predictive model simulations as tools for designing optimal drug profiles and doses; sensitivity analysis as a method to detect key interactions that affect critical outcomes and other characteristics of the network; and other tools integrating mathematical modeling with advanced computation and optimization for detecting potential drug targets. Furthermore, we show how potential drug targets detected with these approaches can be used in a computer-aided context to design or select new drug molecules. All concepts are illustrated with simplified examples and with actual case studies extracted from the recent literature.

1. Introduction and Overview Today’s drug discovery process starts with a chemical substance and aims to identify its potential therapeutic effects. This task very often requires the screening of thousands if not millions of molecules in “blind,” undirected biological tests. Once a positive therapeutic effect is detected and the side effects are assessed to be tolerable, the compound is allowed to enter clinical trials and, if successful, is subsequently launched for treatment in the general population (Paul et al., 2010). The specific mechanisms responsible for the therapeutic effects are sometimes only identified a posteriori, possibly after a long time, as was the case of aspirin (Miner and Hoffhines, 2007). This traditional paradigm for drug discovery has been challenged in the last decade because of three significant problems. Firstly, the demanding criteria for proofs of efficacy and safety with which new drugs must comply, together with the complexity of the targeted diseases, make the drug discovery process extremely long, expensive, and, in the vast majority of cases, poorly cost-effective. To launch a new drug very often requires decades of research and roughly a billion dollars in up-front investments. The second element to be considered is the incredible progress made in quantitative techniques for molecular biology in recent years. A decade ago experimentalists struggled to produce enough data for testing a hypothesis. Now, the emergence of “omics” technologies has almost inverted the situation: Data retrieval, processing, and analysis have become the critical steps in every modern biomedical research lab. The good news is that a plethora of computational techniques, including mathematical modeling, are becoming powerful enough to help with the management and analysis

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of these enormous amounts of biological data. This computational support will continue to become stronger and increase our ability to analyze biochemical systems relevant for many diseases, thereby boosting the discovery of drugs and therapies. In addition to this, recent developments in genetic engineering and synthetic chemistry and biology promise the design of drugs a la carte in the foreseeable future. These new methods of modern allosteric synthesis and synthetic biology will enable the creation of specifically designed biomolecules that have the desired biological effect, while minimizing side effects as well as cost. The ensemble of biological and computational elements will make it possible to formulate a completely new paradigm for drug discovery in the near future, in which the traditional sequence of steps will almost be reversed. In this emerging era of computer-aided pharmacology, drug discovery will be initiated by stating the desired effect to be achieved and by subsequently modulating biochemical systems related to the target disease, first with computational and then with experimental methods. The experimental methods will primarily search for a feasible and implementable mechanism, which may restore the proper functioning of the biochemical system. The alterations of the system could possibly involve a range of actions, from diet design to finding protein drug targets, but the overriding rationale expected is the modulation of physiological processes that alleviate the pathological condition associated with the disease. This last step will involve the design of a therapy, including the chemical design of drugs that effectively interact with the intended targets. This vision of the future purports that mathematical modeling is a mandatory component of drug development, and it is necessary to back up this claim. For instance, one could think that sufficiently many good data, expert knowledge, and intuition should eventually be sufficient for solving biomedical problems with traditional and future experimental methods. The reality, however, points in a different direction. First, the investigation of complex and highly interconnected biochemical networks and the integration of high-throughput data are no longer feasible without the support of computational and mathematical modeling techniques. The enormous power of biological intuition will always be required but becomes insufficient in the era of large biological interaction networks of proteins, genes, and metabolites. Maybe even more important, the dynamic behavior of biological networks with regulatory structures like feedback loops can be counter-intuitive and change qualitatively in response to sometimes rather small quantitative perturbations of its components or possibly in the environment. Figure 11.1 illustrates this issue with a simple but relevant example, in which distinctly different behaviors in p53 signaling arise, depending on the strength of a single feedback loop inhibition, exerted by the protein MDM2. In cases of this nature, mathematical

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MDM2

p53

p53

Stress signal (DNA damage)

Time (h)

Figure 11.1 A negative feedback loop is a regulatory structure in which the activation of a signaling event regulates a process upstream in the system. The feedback often ensures homeostasis in the system, that is, the ability to keep an internal state that is robust toward small fluctuations in the input signal. Feedbacks also facilitate fast signal termination and permit sustained oscillations. These concepts are illustrated here with the case of a feedback loop involved with the tumor suppressor p53 and its inhibitor MDM2. Low levels of DNA damage trigger the synthesis and activation of p53, which is terminated, with delay, upon synthesis of its inhibitor MDM2, due to the negative feedback loop structure of the system (solid black line). By contrast, extreme levels of DNA damage can cause the same feedback loop to trigger sustained oscillations in p53 values (solid red line), which have been experimentally verified (Lahav et al., 2004).

modeling is a “must” because nonlinear properties evade our intuition and can be understood only in mathematical terms (for introductions to nonlinear biochemical systems, see Voit, 2000 and Tyson et al., 2003, Vera and Wolkenhauer, 2008). Therefore, a systemic perspective of structurally and dynamically complex biochemical pathways is needed. The mathematical framework that provides this kind of perspective is dynamical systems theory, and the language that implements this idea and permits actual exploration and analysis is mathematical modeling. Applied to biology, this framework is the core of systems biology. Systems biology addresses biomedical questions through integrating experiments in iterative cycles with mathematical modeling, computational simulation, and model analysis (www.erasysbio.net). In the future, applications of systems biology to drug development, mathematical modeling techniques, and other computational approaches are positioned to play an essential role, especially in the integration and analysis of quantitative and often heterogeneous experimental data. But modeling will also aid in the detection of key interactions in biochemical networks, which may be corrupted by disease, and in the detailed interpretation of information regarding potential biomarkers and the development of new drugs.

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Although a definitive standard workflow for drug target detection using a systems biology approach has not yet emerged, we can distinguish a number of steps, which are visualized in Fig. 11.2. The procedure can be divided into three

S

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p dXi gjk = Σ cij • gj • Π Xk k = 1 j dt

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?

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Computational methodes for drug target detection

Review Original paper Letter

Figure 11.2 Conceptual scheme of a drug target discovery work flow that includes data-based mathematical modeling as a core component.

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main conceptual blocks, namely: (1) retrieval and process of biomedical knowledge; (2) mathematical modeling; and (3) application of computational methods for model analysis and prediction. The latter uses the constructed model as a predictive tool to identify potential drug targets and may include procedures to select, design, and test small molecules with a desired therapeutic effect. In this anticipated workflow, the first step is to retrieve, select, and organize the relevant biomedical information describing the investigated biochemical processes and its physiological context. In some circumstances, this state of the art of biomedical knowledge about the system might be complemented with biological hypotheses about the structure and dynamics of the investigated biological network. Once processed, this biomedical knowledge is converted into a conceptual map of the investigated biochemical network. The map provides a first level of abstraction and specifies important information that is to be transferred into the mathematical model. Depending on the available information and data, as well as the characteristics of the investigated biochemical system, different modeling frameworks can be used (Voit et al., 2008). But no matter what model is chosen, the information in the map must be complemented with quantitative data that are to be generated in specific wet lab experiments to design and characterize the model, in a process sometimes called model calibration. The desired output of this process, which may require several iterations between modeling and experimentation, is the proper characterization of the mathematical model in terms of precise, reliable values for the parameters in the model equations. Since the aim of drug target detection is the ability to make reliable, and possibly quantitative predictions, the model must be subjected to an external validation, which may be accomplished in different ways, such as comparisons with additional data sets or tests of model predictions regarding so far untested scenarios. These validation techniques can be used iteratively in a process of model refinement that leads to increasingly more accurate models. While validation is very important, the ultimate goal of modeling based drug discovery is not to create a predictive mathematical model, but to use it in the detection of potential drug targets. Toward this end, the following sections define, explain, and illustrate the use of a number of computational and mathematical modeling tools that allow: (a) the detection of critical biochemical processes, whose modulation may overcome the pathological conditions under investigation; (b) the design of clinically feasible strategies to modulate them (Fitzgerald et al., 2006); and (c) the identification of potential new biomarkers (Wang et al., 2010). The main emphasis will be on biochemical networks, which are pertinent to several aspects of drug target detection. In order to make the text accessible to a wide audience, we will proceed as follows. Every technique is illustrated with a simple mathematical “toy” model, which allows us to make concepts easy to understand and visualize.

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At the end of each section we include and discuss one or more published cases, where these techniques have been used. This discussion will allow the reader to experience how the techniques are used in real life. For the sake of simplicity, we have removed most of the mathematical equations and technical concepts from the text, but even the few remaining technical details are not truly required to understand the applications. Readers who are not interested in mathematical detail can therefore skip Sections 3.1-3.3 and 4, where the construction of mathematical models is explained in some detail. The equations of the mathematical model that is used throughout as an example can be found in the Appendix.

2. Biomedical Knowledge and Data Retrieval: Constructing a Conceptual Map of a Biochemical Network The first step in a procedure for model-based drug target detection is to retrieve, select, and organize the relevant biomedical information describing the investigated biochemical processes and its physiological context. The aim here is to list the critical molecules, usually genes, proteins, metabolites, small molecules, and various complexes, and the biochemical interactions involved in the investigated system. Good resources for this initial step are: (a) relevant biomedical and biological publications selected manually, with the help of PubMed, or automatically retrieved using text mining tools; (b) databases containing relevant information about biochemical interactions or protein–protein interactions, such as KEGG, MetaCyc, and Brenda; and (c) expert advice of biomedical researchers, clinicians, or biologists, who have acquired deep knowledge of the investigated biological phenomenon. The Appendix presents a list of useful tools and databases for data and information retrieval. In some circumstances, the retrieved and processed biomedical knowledge of the system can be complemented with a set of biological hypotheses about the structure and dynamics of the investigated biological network that are relevant for drug target detection. Once the necessary biomedical knowledge is acquired and processed, it is converted into an annotated map of the investigated processes using standard protocols (Le Nove`re et al., 2005, 2009; Voit, 2000). Information concerning the critical ontologies associated with each interacting partner of the network can be retrieved from the Gene Ontology (GO) and used to complement the constructed annotated map of the biochemical network under analysis. A visual version of the map can be produced using software tools for pathway representation and annotation. The Appendix lists useful software tools for construction and annotation of conceptual maps.

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The map thus constructed is itself a model in the most general sense. It formalizes the knowledge of the system, enumerates the relevant variables, and identifies interactions among them. By providing a certain degree of abstraction and using a language that is accessible to everyone, these concept maps are an excellent tool for discussion between the members of interdisciplinary teams (experimentalists and modelers). However, this map contains only a “static view” of the system, and the fine details of regulation and the dynamics of the biochemical system under different experimental conditions can only be understood through developing and using a mathematical model (Goel et al., 2006, Vera and Wolkenhauer, 2008) (Fig. 11.3). Figure 11.4 demonstrates the process with a toy example, where two metabolites M1 and M2 are combined to synthesize a product called M3. A later transformation of M3 is catalyzed by an allosteric enzyme P, whose activation (P ! P*) is regulated by a kinase I, whose action is related to a High-throughput expression data

Protein–protein interaction databases

Text mining of biomedical literature

Quantitative time series data

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I P

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Figure 11.3 Model construction as a two-step process. The structure of the network (the set of metabolites, proteins, and small molecules together with the interaction between them) is first established based on the biomedical knowledge that is documented in relevant publications and databases. We call this representation the “annotated map” of the biochemical system. To characterize the network dynamics, which consists of changes in the concentrations of the compounds over time, quantitative data are required and must be integrated into a dynamic model.

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I V7 P V8

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P*

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V5

Figure 11.4 Conceptual map of a fictitious biochemical network used here as a simple case study. Solid black arrows account for transformation processes while dashed arrows account for regulatory processes (dashed black arrow: positive regulation; dashed red line: negative regulation). An example of a pathway with such a structure is the synthesis of amino acids, where M2 represents ammonia and M1 is either a common precursor for the family of amino acids through V4 or serves as substrate for the synthesis of compounds through V2.

signaling event. The system contains a feedback loop in which M3 regulates the input flux of M2. In healthy conditions, this feedback loop maintains homeostasis in M3 for a wide range of values in the input fluxes of M1 and M2. By contrast, an alleged pathological condition causes an overexpression of enzyme P. The heightened activity level leads to an imbalance in the network fluxes, which causes an accumulation of M2 to toxic levels. The toy example could be a simplified model of a biosynthetic pathway in which two precursors M1 and M2 are combined to produce a final product, as it is the case with the synthesis of amino acids. In this latter case, M2 would play the role of ammonia, entering the cell through a reversible exchange V5  V6 and M1 would be the common precursor of the family that can be converted into an amino acid through V4 or continue to feed the synthesis of the remaining amino acids in the family through V2.

3. Mathematical Modeling of Biochemical Networks: Translating Knowledge into Mathematical Equations A map of the network such as in Fig. 11.4 contains quite a bit of information. It displays the variables that are considered relevant, the processes that transform the components into one another, as well as regulatory

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M1

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Figure 11.5 Dynamics of a simple biochemical system, represented in a so-called phase space. The pathway has only two metabolites and all its states can be described as points in the plane, where the coordinates are the concentrations of M1 and M2. Given a hypothetical initial condition (red dot; initial concentrations for M1 and M2), the system evolves through different states (red line) until it reaches a steady state configuration, in which concentrations of compounds do not change anymore. In this case, the process begins with M2 decreasing very quickly, while M1 is essentially unchanged. Subsequently, both M1 and M2 increase, oscillate, and then come to rest at the steady state.

signals. The next step is to move from this qualitative description to one that is quantitative and dynamic. In other words, the map needs to be translated into a mathematical model that is able to predict responses to changes in the biochemical pathway in terms of metabolite and protein concentrations, signals, or metabolic fluxes. Within this framework of a mathematical model, every configuration of the biochemical network is defined as a state of the system. Thus, the state of a biochemical network is characterized by the magnitudes of all metabolites, interacting proteins, external signals, and fluxes. If a stimulus is applied to the biochemical network, it “moves” from one state to another, which typically means that the concentrations of the compounds will change over time. Given the state of the biochemical system at a certain instant, the model is able to predict how it will evolve over time and predict its configurations in the future (Fig. 11.5) .

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The most frequently used modeling tools for dynamic systems are sets of ordinary differential equations (ODEs). In these models, the change in the concentration of a metabolite or interacting protein Xi in the biochemical pathway or network over time can be written with an equation like Eq. (11.1) X dXi ¼ si;1 V1 þ si;2 V2 þ    þ si;p Vp ¼ si;j Vj dt j¼1 p

ð11:1Þ

The left-hand side of the equation can be read as variation of Xi per unit time, while the right-hand side is the sum of all the process leading to an increase (uptake, synthesis, etc.) or a decrease in Xi (export, degradation, etc.). The contribution of each process to the overall change in Xi has two factors: each Vi is the reaction or transport rate with which the process is taking place and the si,j are the stoichiometric coefficients of the reaction. These coefficients are positive if they contribute to processes that increase Xi or negative for those processes that decrease Xi. This kind of mathematical model can be used to simulate the dynamics of a biochemical pathway over time. Using appropriate software and quantifying the initial conditions of the system (concentrations of the metabolites and proteins), as well as any stimulus that is applied, the integration of these equations predicts the fate of the biochemical pathway after stimulation or any kinds of perturbation in a sense that the model exhibits how much every metabolite and protein concentration changes over time. Sections 3.1–3.3 describe the details of different kinds of mathematical models used for investigating biochemical networks in drug targeting.

3.1. Network reconstruction The coefficients si,j in equation 11.1 can be deduced from the map. They contain all information about the connectivity of the network and prescribe how some metabolites and protein states are transformed into others. It is helpful to collect all coefficients in a table as shown in Fig. 11.6. The rows of the table correspond to the different variables of the system and the columns to the various processes. Every entry in the table relates a variable with a process. If a process does not involve a certain variable, its entry in the table is zero. In mathematical terminology, the table is called the stoichiometric matrix (S). Setting up the stoichiometric matrix is one of the milestones in the construction of a model and integrates much biological information. The process is also referred to as network reconstruction because it translates the connectivity of the map into mathematical terms, thereby enabling its computational analysis.

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Stoichiometry M1 M2 M3 P P* I

v1 1 0 0 0 0 0

v2 –1 0 0 0 0 0

v3 –1 –1 1 0 0 0

v4 0 0 –1 0 0 0

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Kinetics, qualitative I V7 P V1

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P 0 0 0 1 0 0 1 0

P* 0 0 0 0 0 0 0 1

I 0 0 0 0 0 0 [0,1] 0

Figure 11.6 Information regarding the structure of a biochemical pathway can be condensed into a set of tables or matrices, namely the stoichiometric matrix and a table specifying the effect of every compound in any of the biochemical processes. This table can also be used to assign qualitative values to the kinetic orders (see next section).

The information on which the network reconstruction is based can come from different sources, ranging from common biochemical knowledge to annotated genomes or databases. There are, however, some ambiguities that often arise in the process and must be resolved in order to obtain a reliable and unambiguous network representation. For instance, the functions of most genes in a genome are identified by sequence similarity and have therefore limited specificity. Some examples of these limitations are the specificity of transporters and cofactors. For this reason, it is often necessary to complement the information found in the genome with additional information such as whether a certain reaction uses NADH or NADPH as an electron donor. The overall stoichiometry of aggregated processes also poses a difficulty in network reconstruction. The electron transport chain, complex biosynthetic processes and even biomass production are often lumped into a single reaction for modeling purposes and the overall stoichiometry of such a reaction cannot normally be determined by

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genomic information alone. In such cases, experimental information and educated guesses must be used to assign realistic values to the unknown coefficients. The analysis of the stoichiometric matrix provides some insight into important characteristics of the system. Mathematical properties of the matrix can be associated with conserved pools. For instance, the sum of the concentrations of all the states of a single protein in a signaling pathway is constant, as it is the case for P and P* in the illustration example. However, the stoichiometric matrix becomes most useful for analyzing possible steady states. Since all the concentrations remain constant in the steady state, the rates of the processes must balance one another in a way that is determined by the configuration of the network and, therefore, the stoichiometric matrix. An algebraic property of the stoichiometric matrix, called its rank, determines which reaction rates or fluxes must be measured in order to completely characterize a certain steady state. In the example, for instance, no steady state can be reached unless V3 ¼ V4. It is therefore enough to measure only one of the two. In most cases, the number of reaction rates or fluxes to be measured is too large for such an approach to be practical, which has prompted the development of techniques for interpreting incomplete information or even for making predictions regarding the distribution of all fluxes without measuring most of them. The most popular of such techniques is flux balance analysis (FBA) (Palsson, 2006), whose basic concepts and applications to drug discovery we discuss in the coming paragraphs.

3.2. Flux balance analysis FBA is concerned mainly with the stoichiometry of the system and the information that can be gained from it concerning the steady states of the biochemical network. At the steady state, both concentrations and reaction rates remain constant and Eq. (11.1) becomes Eq. (11.2): dX
¼ 0 ) S
v¼0 dt

ð11:2Þ

This equation must be fulfilled in every steady state. It is of note that the left part of the equation is expressed in terms of concentration variables, while the right side is formulated in terms of fluxes. Thus, the same system permits alternative (dual) representations (Marin-Sanguino et al., 2010). Therefore, of all the possible states in a phase space like Fig. 11.5, there is a region defined by Eq. (11.2) where steady states may be found. This feasible region can be further limited by complementing the steady state condition

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with additional constraints. These constraints can in principle be of any nature. For mathematical convenience, they are usually represented as linear functions of the fluxes of the system. Thus, the feasible region is normally defined by (a) Steady state condition. Since all the rates must be balanced for every node of the network, there is an equation for each node that must add up to zero. In the illustration example, for instance, the balancing at M3 implies V3  V4 ¼ 0 and that at M1 requires V1  V2  V3 ¼ 0. (b) Physiological bounds. Maximum and minimum values constrain the variables and rates in the system. Upper limits can be due to hard physical and chemical constraints (e.g., diffusion limit for oxygen) or normalizations (e.g., an arbitrary limit of 100 for glucose uptake to normalize all the fluxes with respect to it). (c) Reaction reversibility. Rates of irreversible reactions can be defined as strictly positive. These constraints are extremely important since failing to identify irreversible reactions can lead to unrealistic scenarios like a futile cycle running backward to generate ATP “for free.” (d) Experimental data. When rates can be measured, their values can be used as constraints in order to accept only those scenarios that are consistent with the observed behavior of the system. Ideally, the feasible region for the parameters of a system would reduce to a single point if enough experimental information were available. In practice this is only the case for very simple systems, and additional techniques must be used to explore relevant steady states within the feasible region. One way of moving toward physiological solutions is to define biologically meaningful goals that can be attained by the biochemical network. Such goals depend on the nature of the network, but examples include the production of energy or redox equivalents, or the synthesis of specific biomolecules. The definition of such goals enables the computation of how each goal can be optimally achieved within constraints that are physiologically meaningful. To compute optimal states, an optimization problem must be solved. For instance, the task could be the following: “Find a state in the feasible region such that ATP production is maximal.” Optimization problems generally pose hard mathematical challenges, unless they belong to one of several subclasses that can be solved efficiently. Thanks to the linear nature of the constraints described above, any objective that can be formulated as a linear combination of the fluxes will lead to a class of optimization problems called linear programming, which can be solved for very large number of variables. The solution of the optimization problem provides a steady state value for each of the fluxes of the system. This technique enables the exploration of the capabilities of the network in terms of energy production, maximal conversion yields, or other features.

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We can think of our particular example as a biosynthetic pathway that has V4 as its main output. The inputs of M1 and M2 are limited by some maximum value and V2 routes flux to other metabolic processes and should therefore be kept above a certain minimum. Due to the architecture of the network, it is necessary for any steady state that V1 ¼ V2 þ V3, V4 ¼ V3, and V5  V6 ¼ V3. These constraints enable us to represent every possible steady state in terms of only three fluxes: V1, V2, and V4. Our optimization problem can thus be stated as: maximize V4 subject to: V1 ¼ V2 þ V4 V1 < M1 maxinput V2 > min V2 V4 < M2 maxinput All other fluxes can be calculated from these three as shown above. The feasible region for this problem is shown in Fig. 11.7. In this case, all steady states are contained in a plane; in general, the solution space is multidimensional. The red polygon encloses the feasible area, where all constraints are satisfied. The solution to this problem is not unique: all states lying on the dashed line in Fig. 11.7 share the same value for the objective (V4) and differ only in the value for the alternate branch (V2). This is called a degenerate solution and often occurs in FBA, especially for large networks. Degenerate solutions can be the result of under-constrained models, which implies that additional biological constraints should be formulated. However, they may also occur when there are redundant pathways in a network that provide robustness or reroute overflows. They furthermore arise in cases such as our example, when the network can meet two different goals at once. In our case, the biosynthetic pathway can work at full capacity (max V4) while still providing flux through the alternate branch (V2) up to a certain limit. This situation would completely disappear if the maximum input of M1 were less than that for M2. In this latter case, our FBA problem would have a unique solution. FBA focuses on steady states and does not include regulatory information, such as the effects of feedback regulation. For that reason, not every state predicted by FBA is achievable by the cell. However, FBA flux distributions can be interpreted as theoretical maxima since states that are outside the feasible region are impossible to attain. For this reason, FBA can be used to check genome annotations and has been especially successful in predicting the impact of knocked out genes in the overall performance of biochemical and metabolic systems. An example is the use of FBA to predict

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180

120 V1 60

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Figure 11.7 Feasible region for the example problem. The blue plane represents constraints among the fluxes, while the red area shows the domain that satisfies all constraints.

lethal mutations in the protozoan pathogen Leishmania major (Chavali et al., 2008). In this work, the authors reconstructed the metabolic network of this parasite of mammalian organisms and, using FBA, simulated a comprehensive set of lethal single and double gene deletions in search for therapeutically relevant knockouts in the microbe. Solving the optimization problem with maximum growth as the objective, and repeating the optimization without one or two certain genes at a time, the comparison of the results presented an estimation of the importance of the gene(s) for growth. Comparisons of FBA results with experimental data have shown that most microorganisms are extremely robust to single mutations and can often reroute their metabolism to circumvent the missing step without a significant loss of performance on growth. This result seems to agree with the above-mentioned existence of redundancies in biological networks that lead to equivalent solutions. What the analysis of Chavali et al., (2008) demonstrates is that FBA offers a useful tool for identifying critical steps for which no redundancy exists. These steps are interesting candidate targets for antimicrobial drugs. Furthermore, comparing FBA results between different types of metabolic networks seems to be a promising approach to improve the selectivity of drug action. Powerful as it is, FBA has the disadvantage that it does not use information about regulation. Therefore, it is unable to predict which steady state is finally reached by the cell in a given situation or how and when it switches from one state to another. To obtain this information, a dynamic model is needed.

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3.3. Dynamic models In order to obtain a complete description of the model dynamics, the dependence of the reactions on the variables of the system must be made explicit in Eq. (11.1). This means that the equation must be expanded to take into account how every reaction rate depends on the metabolites, proteins, and small molecules involved in it. A general formulation of this task is given in Eq. (11.3): dXi X ¼ si;j Vj ðx
; I Þ dt j¼1 p

ð11:3Þ

Here x
is the vector of time-dependent variables, such as metabolites and proteins, and I accounts for the external stimulus. Vj is now a mathematical function, whose arguments are the variables x
and I. Obtaining kinetic information to characterize this dependence is much more difficult than characterizing the stoichiometry of the biochemical network. First, the kinetics of each reaction depends on many different parameters. And second, the parameter values obtained in vitro will not necessarily reproduce the behavior of the reaction in vivo. A critical decision that must be made at this point is which kind of function or “rate law” to use for mathematically representing the processes. Traditional rate laws used in enzymology were defined to characterize the mechanism and molecular properties of an enzyme and not necessarily to be used for modeling. As a result, most rate laws have many more parameters than required to reproduce the behavior of the enzyme with reasonable accuracy. The larger number of parameters is a significant issue, because it increases the amount of data required to fit the model. It also introduces too many degrees of freedom, which leads to overfitting, which means that the model is so complex that it becomes unreliable. For this reason, the use of simplified rate laws is steadily gaining popularity. Examples are power-laws (Savageau, 1969; Vera et al., 2007a, 2010a, Voit, 2000), linlog models (Nikerel et al., 2006), and the so-called Saturable-Cooperative formalism (Sorribas et al., 2007). For our discussion in this chapter, we have chosen power-law models (Savageau, 1969; Torres and Voit, 2002; Voit, 2000), which are by far the most prevalent models for this purpose. Power laws have the general form given in Eq. (11.4). Vi ¼ ki X1f1 X2f2 Xnfn ¼ ki 

Y

f

Xj i;j

ð11:4Þ

j¼1;n

Due to their similarity with mass action kinetics, the parameters ki are called rate constants and the fi,j are called kinetic orders, which here however may be any real numbers. The most remarkable property of power-law

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models is that the same power-law equation can represent different biochemically relevant dynamics by keeping its mathematical structure intact and changing only the numerical value for the kinetic orders. Specifically, (a) a negative value for a kinetic order represents inhibition; (b) a value of zero for a kinetic order indicates that the variable does not affect the modeled process; (c) a kinetic order equaling one means that the reaction behaves as in linear kinetics; (d) values between zero and one account for saturation-like behaviors in certain concentration ranges and are default settings for approximating Michaelis–Menten kinetics; (e) values higher than one often model cooperative processes or processes occurring on surfaces or in channels (Savageau, 1998; Vera et al., 2007a; Voit, 2000). The homogeneous structure of power-law representations makes it easy to construct complex mathematical models even if knowledge about the system is sparse, and also to test hypotheses about the structure and dynamics of the biochemical network (Akutsu et al., 2000; Veflingstad et al., 2004; Vera et al., 2010a; Vilela et al., 2009). It is convenient to collect all kinetic orders in a matrix that accompanies the matrix of stoichiometric coefficients (Fig. 11.6, right-bottom). Once the power-law rate has been chosen as the mathematical representation, the task of formulating a model is reduced to that of finding numerical values for the parameters k and f. This task can be accomplished in different ways. One option, which is especially useful for purely metabolic networks, consists of interpreting each rate constant as the parameter that relates the steady state values of metabolites with rates in the stoichiometric system, as shown in Eq. (11.5). ki ¼ Q

Vi0

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Therefore, if the steady state is known, ki is fully determined by the value of the kinetic orders. This is one of the reasons why performing FBA is a helpful step toward a dynamic model. When good estimates for the metabolite concentrations are available, they can be fed into Eq. 11.5) together with a flux distribution obtained from FBA. In such a case, only the kinetic orders must be estimated. Fortunately, not every metabolite has a direct effect in every reaction rate. As a result, most of the entries in the matrix of kinetic orders will be zero, which significantly reduces the complexity of the parameter estimation. Furthermore, not all kinetic orders have the same impact on the dynamics of the model and educated guesses for some of them are often sufficient to obtain a realistic model, at least as a first default that subsequently undergoes further refinement. Educated guesses are often used for the kinetic orders of substrates and products of each reaction (Voit, 2000, Chapter 5). For example, the classical biochemical assumption that the Michaelis–Menten constant of an enzyme is often close to the concentration of its substrate is analogous to setting the kinetic order to 0.5.

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In the general case, the remaining unassigned values of rate constants and kinetic orders may be estimated by integrating quantitative experimental data in the way described in the following section.

4. Model Calibration: Matching the Mathematical Model to Quantitative Experimental Data Up to this point in our discussion, the model has been an abstract construct that encodes in mathematical terms the structure and some other qualitative features of the investigated biochemical network. In most cases, the structure alone is not sufficient and the mathematical model must be matched against quantitative data that describe the dynamics of the system and its steady state configuration. Only if this matching can be accomplished in a satisfactory fashion has the model the potential of becoming a reliable predictor of the outcomes of untested scenarios. In this process of model calibration, we follow a computational approach that ultimately assigns representative values to the parameters that characterize the model equations (rate constants, kinetic orders, etc.) in such a manner that the model reproduces the behavior of the system, as it is characterized by available experimental data. The underlying strategy is to determine those values for the model parameters that make the differences between the actual measurements and the values produced by the corresponding model simulation as small as possible (Fig. 11.8). This strategy can be represented by the mathematical statement in Eq. (11.6), which calls for minimizing the residual error between model and data: exp X tp  nvar X 2 X 1 exp Xk;j ðti Þ  Xk;j ðti Þ nexp  nvar  ntp k¼1 j¼1 i¼1

n

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n

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In this equation, nexp is the number of experiments, nvar is the number of measured quantities (observables), and ntp is the number of time points where each observable was measured. Xk, j(ti) is the value of the j th observable at the ith time point obtained after numerical simulation of the model for the kth experiment and Xk, jexp(ti) is the corresponding value of the j th observable at the i th time point measured in the kth experiment. Experimental data used for model calibration can be of distinctly different types. Traditionally most dominant, especially in metabolic systems, used to be kinetic properties and steady state configurations of the pathway for a given number of biological conditions. Newer data consist of time series measurements of some

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or all variables considered in the mathematical model for a set of biological scenarios. Figures 11.9 and 11.10 illustrate both types of experimental data. Most actual parameter estimation efforts today combine a bibliographic search, where information is extracted from publications where similar experimental conditions were investigated, manual training of the parameter set, including the fine-tuning of values until the model is able to simulate the experimental data with sufficient accuracy, and algorithmic data fitting. In the latter case, which can be more or less automated, the experimental measurements are fitted by using computational techniques based on regression, the mathematical principle of maximum likelihood, or a number of other techniques. These techniques very often assign values to the model parameters in an iterative procedure that continues until the difference between the experimental data and the model simulations for the investigated biological scenarios are minimized. A variety of optimization algorithms have been specifically designed or adapted for this purpose, and they use local search methods, evolutionary algorithms, global search methods, or a number of specialized techniques. Overviews of this topic include the recent papers by Ashyraliyev et al. (2009), Chou and Voit (2009), and Srinath and Gunawan (2010). In addition, a variety of software tools are available for calibrating ODE models (see Appendix). In order to obtain reliable results, the model calibration requires a sufficient amount of quantitative, high-quality data from specifically designed experiments. The parameter estimation procedure can and should be complemented with an identifiability analysis, which assesses whether

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the parameter values have been estimated with sufficient certainty, or if other sets of values may be equally able to reproduce the available data, a situation that reduces the predictive abilities of the model. The literature contains some papers with excellent discussions on how to integrate both analyses (e.g., Balsa-Canto et al., 2010; Srinath and Gunawan, 2010).

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For our case study, we designed two synthetic experiments to illustrate these concepts. The first computational experiment (Fig. 11.9) resulted in an artificial time series, where the concentrations of the metabolites M1, M2, M3 were “measured” at different time points after a transient stimulation of the system with an input signal I. In fact, these “measurements” are simulation results, which are artificially corrupted with noise. For the model calibration, we assume not to know the parameter values used in the simulation. Optimization methods like the ones described before are used to tune the value of the model parameters until model simulations achieve a satisfactory reproduction of the data. The left-bottom panel of Fig. 11.9 shows the comparison between experimental data and the ultimate model simulation for the chosen sets of parameters, while the bottom-right panel

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of Fig. 11.9 compares data with a simulation obtained with an unsatisfactory set of parameter values. In the second experiment (Fig. 11.10) the steady state values of M1, M2, and M3 were “measured” for different values of the input flux V1 to the system. These types of data can be used as well to tune the model parameters. The bottom panel of Fig. 11.10 shows the comparison of experimental data (squares) and model simulations (line with solid dots) for the different values of V1 considered in the interval [0.25, 4]. A sufficiently validated and refined mathematical model can be used for model-based analysis of the biochemical network under investigation. Ideally, computer simulations of different biological scenarios have predictive abilities, thereby shortcutting the way to a better understanding of the system and also to detection of new drug targets. Different approaches for using mathematical models in drug target identification and drug discovery are discussed in the coming sections.

5. Predictive Model Simulations as a Tool for Drug Discovery A mathematical model that was sufficiently calibrated and validated has predictive abilities regarding not yet explored biological scenarios. Expressed differently, model simulations can be used for investigating the features of biological scenarios that have not yet been experimentally analyzed, therefore saving at least a portion of the costs of expensive and time consuming experiments. In the context of basic research, this property of mathematical models has been employed for testing hypotheses concerning features associated with the dynamics and structure of the network, and also with the subcellular compartmentalization of interacting biomolecules (for good examples, see Alvarez-Vasquez et al. 2005; Rehm et al. 2006; or Vera et al., 2010c). In this sense, predictive simulations are already useful in drug discovery by pointing to critical properties of the network under investigation. Outside dynamical modeling, the use of mathematical model simulations is not new in the context of drug discovery. For decades, pharmacokinetics has been devoted to the model-based analysis and simulation of the mechanisms by which a drug enters the body, distributes among the organs and tissues and is finally metabolized and eliminated (Rowland and Tozer, 2007). All these aspects are essential features for determining effective and secure drug doses. In more recent times the use of models in pharmacology has been extended toward pharmacodynamics, which uses simulation techniques to study the primary physiological effects of the drug and to fine-tune the drug dose with respect to the desired physiological effect (Rowland and Tozer, 2007).

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With the emergence of systems biology, the use of mathematical modeling in drug discovery can advance one step further. Namely, it is becoming feasible to expand the modeling approach of pharmacokinetics and pharmacodynamics by including detailed dynamic models that describe the intracellular pathways that are targeted or affected by the drug of interest and characterize issues related to their regulation in the cell. For this expansion, an integrative approach should include: (a) modeling of drug administration and degradation dynamics; (b) modeling of the dynamics of drug modulation of the targeted biomolecules, which are usually proteins or membrane-bound protein receptors; and (c) modeling of the dynamics of the intracellular networks (signaling, genomic, or metabolic networks) that are affected by the targeted biomolecule (Fig. 11.11). Simulations with these types of extended models would allow the finetuning of drug doses by tracing not only the metabolic fate of the drug itself, but also the effects of drug administration on critical intracellular processes that are altered by the drug administration. For our case study, we assume that a drug X is able to inhibit the activation of P through the input signal I. In this way, the administration of an adequate dose (adequate in timing and amount) of drug X can counteract the effect of the overexpression of P in a hypothesized

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Figure 11.11 Integration of different predictive simulation approaches in drug discovery and drug dose determination.

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pathological condition of the system. It is easy to perform model simulations to assess the effects of different types of drug administration in the biochemical network. For instance, the center-left panel of Fig. 11.12 shows the simulated drug dose profile of drug X, where we assume two doses per day (every 12 h) and a half-life of the drug of approximately 6 h. The center-right panel shows how model simulations can be used to investigate the effects of drug administration on the concentrations of the different components of the system (M1, M2, M3, and P*). The black solid horizontal line represents the alleged physiologically tolerable threshold for the concentration of M2 (here set to 150% of the concentration in healthy subjects). In the simulated drug dosing experiment, the values of M2 always remain below the threshold. The leftbottom panel of Fig. 11.12 exhibits predicted model values for the network compounds in healthy subjects (HS), the pathological state (PS), as well as average values after the administration of the drug as described before. Finally, the right-bottom panel of Fig. 11.12 shows the predicted average values of M2 for different doses (twice a day) of the drug compared with HS and PS. These simple results indicate how model simulations can be used to establish adequate or minimally sufficient drug doses, and they also show the effects of the drug administration in the other network components. In recent years, this promising approach for testing drug efficacy has already been applied in realistic situations. For instance, Lai et al. (2009) developed a mathematical model combining experimental data for different aspects of murine erythropoiesis. This process of red blood cell generation and differentiation is closely regulated by the blood levels of the hormone erythropoietin, Epo. The model included three kinds of biological events: (a) subcellular events associated with the Epo-stimulated JAK2-STAT5 signaling activation (Vera et al., 2008); (b) Epo-regulated dynamics of erythrocyte differentiation; and (c) hypoxia-mediated regulation of Epo blood levels. In addition, the model included equations accounting for the dose-dependent injection of exogenous Epo and the later dynamics of the hormone, which was used in the study as a drug to fight a certain kind of anemia. With the help of their quantitative model, the authors performed several predictive simulations to establish the appropriate doses of exogenous Epo injection that were required to compensate for the pathological downregulation of the crucial signaling proteins. Another example is a series of articles by Qi and collaborators that were geared toward a better understanding of neurodegenerative diseases like Parkinson’s (Qi et al., 2008a) and schizophrenia (Qi et al., 2008b, 2010a). The most prominent common link between the two diseases is the neurotransmitter dopamine, which becomes depleted in the former and is found to be excessive in the latter. Thus, the authors began by setting up a complex dynamic model of dopamine metabolism in the brain, which accounted for synthesis, compartmentalization, degradation, and the known regulatory mechanisms. While the structure of the dopamine pathway system was fairly well known, the identification of parameter values was a significant challenge,

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because metabolite levels were needed in certain cells within relatively small, defined areas of the brain. Interestingly, the authors were able to estimate parameters based on semiqualitative expert opinion and generic experience with models of this type, and the resulting model turned out to be quite predictive, as judged by comparisons with a variety of biological and clinical findings (Qi et al., 2008b). The model was subsequently used to identify the key determinants of the dopamine system, which were seen as prime candidates for drug targets. This identification consisted essentially of a global sensitivity analysis (see next section) and was performed through large scale Monte Carlo simulations that demonstrated which steps, and combinations of steps, had the greatest influence on the main metabolite, dopamine, and also on toxic by-products (Qi et al., 2009). The results reliably revealed enzymatic steps that are targeted by current drug treatments and suggested combination treatments, which were predicted to incur less severe side effects. Because dopamine metabolism only controls the neurotransmitter, but not the signal transduction process itself, the authors also studied how normal and perturbed dopamine signals are interpreted by receptor neurons (Qi et al., 2010b). Another promising application of predictive systems models in drug discovery is cronotherapy. The idea here is that mathematical modeling combined with quantitative data can be used to establish the optimal timing of drug delivery in such a way that the treatment occurs in positive synergy with the internal rhythms of the organism, thereby reducing toxicity and increasing the desired therapeutic effect. In line with this goal, Altinok et al. (2009) recently explored the idea of applied cronobiology in cancer therapy, where the efficacy of cytotoxic agents is critically affected by the regulation of cell cycle related processes. The authors developed a mathematical model to simulate the distribution of cell cycle phases and their interactions with the circadian clock molecules and used this model to investigate the optimal circadian patterns of administration of two anticancer drugs, 5-fluorouracil (5-FU) and oxaliplatin (l-OHP). In their analysis they compared various patterns of drug administration that differed in the timing of maximum drug delivery and looked for those settings displaying minimum cytotoxicity for the population of normal cells but high cytotoxicity toward a second cell population of tumor cells.

6. Model Sensitivity Analysis as a Tool for Detecting Critical Processes in Biochemical Networks Several computational and analytical tools can be used to analyze a biochemical network in search for potential drug targets. The underlying idea is to employ the predictive abilities of the mathematical model to find the set of biochemical processes within the network whose modulation is

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expected to affect the dynamics of the network most significantly. These biochemical processes later become priority candidates for drug targeting or other therapeutic approaches. Toward this end, the computation and analysis of model sensitivities can be an extremely useful tool. Sensitivity analysis is the study of how the variation in the critical outcomes of a given biochemical system can be categorized and assigned, qualitatively or quantitatively, to different sources of variation in the system (Saltelli et al., 2000). In the context of a mathematical model, we typically associate this feature to changes in the values of model parameters and look for those model parameters for which a numerical variation significantly affects critical responses of the system. These critical (i.e., especially sensitive) parameters are associated with processes in the network that are therefore considered particularly promising as potential candidates for drug-mediated modulation (Nikolov et al., 2010). We can distinguish two approaches to sensitivity analysis. In local sensitivity analysis we search for the parameters for which small variations around their nominal value have a significant influence on the critical responses of the biochemical pathway. The analysis is “local” because it is performed in the mathematical vicinity of a preferential or nominal configuration of the system, and also because only one parameter is perturbed at a time. In the conventional methods for estimating local sensitivities, the value of a given parameter is slightly modified (e.g., by 1–5%; up or down) and the model is used to compute the resultant change in the critical response of interest. The sensitivity is calculated with the equation SkX ¼

DX X ðk þ DkÞ  X ðkÞ ¼ Dk Dk

where X represents a response variable and k the modified parameter; D represents a small change. In the case of power-law models, sensitivities of the model that are related to steady state configurations of the system can be calculated using simple algebraic equations, which renders sensitivity analysis quick and computationally simple for these models (Voit, 2000). In global sensitivity analysis the search for the critical parameters considers larger regions of feasible values for the parameters rather than preferential values. In addition, several, many, or all parameters may be varied simultaneously (for a comprehensive discussion, see Saltelli et al., 2000). This approach provides an initial impression of those biochemical processes whose modulation critically affects the system (see, e.g., Qi et al., 2009). However, highly nonlinear biochemical systems may have such a structure that the parameter influence, or the criticality of the process, differs substantially among large regions of parameter values. Thus, one may find distinctive regions with rather different response features. In this case, a unique global sensitivity value for each parameter may not reflect

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biologically significant criticality and additional computational techniques must be used to dissect the relevant region of the parameter space into subspaces with similar meaningful sensitivities. Sensitivity analysis can be used for the detection of potential drug targets, because insensitive parameters would not have much effect, even if they could be altered therapeutically. This feature of sensitivities suggests a ranking of biochemical processes, along with the model parameters representing them, in terms of how effective their modulation might be for altering the input–output relationships of the system. The precise effect of any such modulation must be later investigated with additional simulations. Expressed differently, sensitivity analysis alone can point out where the most sensitive parameters are found in the model, but not necessarily what the precise effect of their modulation is; this latter aspect must be analyzed with additional simulations. As an illustration, we performed a sensitivity analysis of the mathematical model for our case study with the goal of identifying how variations in model parameters would affect the values predicted for the metabolites and P* (Fig. 11.13, top panel). The sensitivities were calculated for the pathological condition (PS) and the primary aim was to analyze the effect of parameter modulation on the value of M2, which is the metabolite critically accumulating in the pathological configuration of the system. The sign of the sensitivities is of great relevance, because it indicates whether an increase or a decrease in a parameter value has a positive or negative effect. A positive sensitivity value predicts that an increase in the parameter values raises the targeted response variable, while a negative sensitivity predicts a reduction. By inspecting the calculated results we see that critical parameters for affecting the value of M2 are k1, k4, k7, and k8. The latter two correspond to the “trivial solution” of the problem, because they suggest that, by inhibiting P activation or enhancing P* deactivation, we will affect the level of M2, which is quite obvious. However, our analysis also suggests two other means of modulating the concentration of M2, which are associated with the modification of the input flux of M1, which is governed by k1, and/or the catalytic transformation of M3, which is governed by k4. The sensitivity for k1 has a negative sign, which suggests that if we want to reduce the value of M2 we have to increase the value of the input flux of M1. This could be achieved by overexpressing the enzyme regulating the flux V1, which however is not realistically feasible, or by increasing the uptake of this metabolite, which could be accomplished with dietary restrictions. The sensitivity with respect to k4 is positive, which indicates that reduction of the flux V4 is expected to induce the downregulation of M2. This reduction of V4 could be implemented, for example, through competitive inhibition of P* or inhibition of any concurrent cofactor. The local sensitivity analysis alone does not allow us to judge whether the modulation of the sensitive biochemical interaction suffices to restore

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acceptable values of M2, and further computational simulations are required (Fig. 11.13, bottom panel). Such simulations suggest that duplication of V1 in the incoming flux of M1 can restore acceptable values of M2 (Fig. 11.13, bottom-left panel), but also that a 50% inhibition of the flux V4 can reduce M2 values to suitable levels. von Kriegsheim et al. (2009) used this technique to identify critical interactions regulating the timing and peaking of ERK signaling, a network fundamental to the regulation of important cellular events like cell proliferation, which appears to be corrupted in a number of cancers. In a nutshell, these authors set up a rather large computational model using quantitative

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proteomic data and other experimental techniques. With this model, they functionally characterized the set of ERK interacting partners and their regulation of the fine tuning of ERK signaling features, such as signal duration, intensity, and localization. The model was later analyzed using sensitivity analysis, where the aim was to detect the critical model parameters and processes, whose modulation may significantly change the features of ERK signaling. Rather than finding a unique critical interaction controlling the network performance, their analysis revealed a large set of important parameters. The authors proposed that these interactions may exert synergistic regulation of the system and suggested that coordinated modulation of several processes (e.g., via multifactorial drug treatments) may be an effective way to control ERK signaling in pathological conditions. In a recent research study very close to drug discovery, Schoeberl et al. (2009) used mathematical modeling and sensitivity analysis to therapeutically target the signaling system of ErbB receptors and the PI3K signaling pathway, which is crucial for the development of anticancer drugs. Their team, which consisted of modelers and experimentalists from a pharmaceutical company, designed a mathematical model and calibrated it with quantitative experimental data describing the ErbB/PI3K signaling network. Sensitivity analysis was used to identify the critical receptors in the family related to the activation of AKT, an important protein downstream of the signaling system. They found that ErbB3, a member of the ErbB receptor family, has a dominant role in this activation process and is a promising target for drug inhibition. They later successfully tested a new potential drug that targets the receptor by inhibiting its phosphorylation (Fig. 11.14). For complex biochemical networks, sensitivity analysis may or may not be a sufficient tool due to the emergence or disappearance of nonlinear phenomena, such as sustained oscillations or multistability. Such phenomena are associated with the passage of a parameter across a critical value, which is called a bifurcation point. These points may be treacherous for sensitivity analysis, because not only does the sensitivity value depend on the particular value of the parameter, but it is also possible that the overall dynamics of the network is altered. For example, it may happen that an oscillation totally disappears. Nikolov et al. (2010) proposed to combine sensitivity and bifurcation analysis in order to detect the critical processes regulating the main outcomes of the biochemical network, as well as changes that qualitatively alter the dynamics of the system. They used this concept to detect critical interactions in a model describing the differentiation of red blood cells controlled by the Epo-JAK2-STAT5 signaling pathway and to analyze if modulation of their representative model parameters could induce or eliminate sustained oscillations.

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7. Drug Target Detection Through Model Optimization The notion of using mathematical modeling and analysis of biochemical networks as a tool for identifying potential drug targets can be further expanded with more sophisticated approaches. These approaches often require more information about the biochemical origins of the investigated disease. For instance, it is useful to know which specific biochemical process appears to be altered and what metabolites or proteins are affected by this alteration. As an example for how these more advanced approaches can be derived, we discuss here a method proposed by Vera et al. (2007b, 2010b), in which mathematical modeling of biochemical networks was combined

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with optimization techniques to detect potential drug targets in metabolic diseases. The underlying idea was simple: in a complex metabolic network that is misregulated by a disease, it may be possible to use drugs specific for a pertinent enzyme to redirect some critical fluxes in a manner that restores in the network all critical fluxes and metabolites to values similar to the ones found in health subjects (HS). As a simplified example, Fig. 11.15 shows M1 and M2 as critical metabolites whose concentrations are mis-regulated in the investigated disease. A clinical intervention with drugs and other therapeutic strategies should try to affect the network such that the actual pathological values for M1 and M2 (PS) return to values similar to those in HS. While conceptually simple, the actual implementation of such a treatment strategy is difficult for highly complex, interconnected and regulated biochemical networks, and mathematical models and computational techniques become obligatory. The strategy proposed by Vera et al. (2003, 2007b, 2010b) works in the following manner: Step 1. A mathematical model describing the dynamics of the metabolic network under analysis is set up and calibrated with experimental data. Step 2. Pertinent biomedical knowledge about the origin and effects of the disease is retrieved and processed. Critical fluxes and metabolites that are unbalanced in the disease condition are identified, and their values in HS are assessed. Step 3. Model simulations are used to investigate which biochemical processes in the network must be modulated and how this might be accomplished, for

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instance, through inhibition or activation, in order to move the system from the current values of critical metabolites and fluxes toward health. Step 4. Simulations may consider therapeutic modulations of one reaction at a time or multiple modulations simultaneously. The simulation results offer insights guiding possible strategies for developing potential multifactorial treatments. This method requires considerable computational effort if the disease system is complex. An elegant manner to deal with the challenge is to convert the task into an optimization problem of the critical parameters and processes. In the Optimisation Drug Discovery Program suggested in Vera et al. (2007b, 2010b), the search for these critical (drug-targeted) reactions is represented with an optimization program containing the following elements: 1. A mathematical description of the functional origin of the disease. 2. A mathematical formulation for the notion of moving the current pathological values of the critical metabolites and fluxes toward the desired health state. For example, one may execute the following minimization    # p l Xj  X HS  X X Ji  JiHS   j   ; Min lj  l þ  XjHS  i¼1 i  JiHS  j¼1 "

where Xj and XjHS denote the current and the health values of the metabolites and Ji and JiHS are the corresponding values for the fluxes, respectively. The values of the positive weights lj and li are proportional to the relative importance of each key metabolite and flux. Finally, the objective function itself formalizes the minimization of the differences between the values of all critical metabolites and fluxes computed by the model in the pathological (X and J) and the healthy scenarios (XHS and JHS). Every term in the equation is scaled by its value in the healthy condition such that all contributions have comparable weights. 3. A number of conditions ensuring that the network configuration obtained in the optimization indeed represents a physiologically feasible and stable steady state. Additional conditions guarantee that the proposed interventions themselves are physiologically feasible. A more detailed mathematical description of the Optimisation Drug Discovery Program can be found in the Appendix. This optimization program can be computed using standard methodologies for nonlinear optimization (Banga, 2008) or via linear optimization techniques that benefit from the computational advantages of power-law models (MarinSanguino anf Torres, 2003; Marin-Sanguino et al., 2007; Torres and Voit, 2002; Voit, 1992). The method can be applied in an iterative manner: at

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each step we allow for the modulation of a unique process (single enzyme target) in the network and compute the solution. This solution consists of a set of computationally predicted values for metabolites, initial substrates and the modulation level of the targeted enzyme. One selects and stores only those solutions for which the computed values of critical metabolites and fluxes are sufficiently close to the ones in the healthy condition (HS). The method may also be used to analyze combined treatments that consider parallel targets consisting of two or more processes in the network, although one should keep in mind that multifactorial treatments may increase or decrease the potential for adverse side effects. In our case study, we try to identify which biochemical interactions in the network may be targeted by drug inhibitors in such a way that the network is returned to its healthy condition. Specifically, we look for means of inhibiting network interactions that primarily restore the value of M2, which accumulates to toxic levels in the pathological scenario. A secondary goal is to restore the values of the other metabolites (M1 and M3) as far as possible. These goals may be translated into mathematical terms, for instance, with the following objective:        M2  MHS  M1  MHS  M3  MHS  2  1  3     Min 1   þ 0:05  þ 0:05    MHS   MHS  MHS 2 1 3 This minimization task forces the network to move to a configuration that restores the healthy values of M1, M2, and M3 as much as possible. We have assigned a much higher weighting factor to M2 (1 vs. 0.05) because it is the critical metabolite showing toxic concentrations in the PS. We first consider the eight possible single target strategies and proceed as explained before by computing solutions per simulation. The results (Fig. 11.16) clearly demonstrate that only some of the network interactions are promising targets for drug inhibition. Strategies inhibiting V7 (trivial solution), V4, and V5 can entirely restore the value of M2 and the other two metabolites. In addition, inhibition of the interaction V2 produces a partial and possibly acceptable recovery of the values for M2, which however doubles the target value of M1. We also considered multifactorial strategies that combine the drugmediated inhibition of two interactions. The results of the set of 8  8 combinations (half of them are redundant) appear in Fig. 11.17. The results in terms of M2 do not vary in comparison with the single inhibition solutions because in this particular case only combinations of the single-inhibitor solutions (V2, V4, V5, V7) with other potential targets restore healthy M2 levels. Different solutions show different patterns regarding the concentrations of M1 and M3, which suggests potential combination therapies with lower negative effects on M1 and M3 levels. One must note that these results

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15 M1 M2 M3 Concentration (a.u.)

P* 10

5

0 V2

V1

V3

V4

V5

V6

V7

V8

Targeted interaction

Figure 11.16 Solutions of the drug target detection optimization program when single inhibition strategies are considered. Each set of bars accounts for the drug-mediated inhibition of one biochemical step (e.g., V1). M1

M2 14

V8 V7

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V1 V2 V3 V4 V5 V6 V7 V8

V1 V2 V3 V4 V5 V6 V7 V8

V1 V2 V3 V4 V5 V6 V7 V8

Targeted interaction

Targeted interaction

Targeted interaction

Figure 11.17 Solutions of the drug target detection optimization program where double inhibition strategies are explored. Each square reflects the combined inhibition of the network interactions indicated by the row and column. Values in the solution for M1 (left), M2 (center), and M3 (right).

only consider drug-mediated inhibition. If we would consider positive regulation for some or all of the processes as well, the set of promising solutions would increase. For instance, one could consider solutions that increase the value of V1, as suggested by the earlier sensitivity analysis. Vera et al. (2007b) used this methodology to identify potential enzyme drug targets for diseases associated with human purine metabolism and, in particular, hyperuricemia (Fig. 11.18). Hyperuricemia is a usually nonlethal

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1.4

Vade

Vaptr

PRPP

Ade Vhprt

Vpolyam

Vtrans

Vden

SAM Vtrans

Vimpd XMP GMP Vgmps Vgmpr GDP GTP Vrnag

Vasuc IMP

Vgdrnr

dGMP dGDP dGTP

Vgrna Vdnag

Vadna DNA Vdnaa

Vgdna

Vdgnuc Gua

Vhprt

Vgua Xa

Vx

Vdada Hx

1.1

0.9 HS

PS

Diet Vxd+diet Vampd

1

0.5

Vhx

Vxd UA

Vhxd

dAdo dAMP dADP dATP

1.2

1

S-AMP Vasli Ado AMP Vampd ADP ATP Varna Vrnaa Vada Vinuc Vadrnr

RNA Vgnuc

Vmat

Vxd (norm.)

Vgprt

1.3 Uric acid (norm.)

R5P Vprpps

Vpyr

Vua

0.5

1

Vampd (norm.)

Figure 11.18 Combining mathematical modeling of biochemical networks and optimization to detect drug targets. Vera et al. (2007b) combined mathematical modeling and optimization to identify drug targets in hyperuricemia, a disease related to human purine metabolism. Toward this end, a mathematical model describing purine metabolism (left panel) was analyzed with the methodology described in the text. The authors looked for drug targets whose inhibition could reduce uric acid (UA) to healthy levels. A number of potential targets were detected and are indicated in the biochemical diagram with a red “injection” symbol. The panel at the top-right compares the levels of uric acid in the healthy state (HS) and the pathological condition associated to hyperuricemia (PS). Some of the solutions suggest a change in “diet”; “Vxdþdiet” represents the combination of diet and inhibition of the enzyme xanthine oxidase, which is similar to the conventional treatment based on the drug allopurinol; and “Vampd” suggests the inhibition of AMP deaminase. The analysis also allowed the investigation of combinations of drug targets whose simultaneous inhibition reduces levels of UA to the HS condition. For instance, inhibition of xanthine oxidase and AMP deaminase can be combined to reduce UA levels with intermediate levels of inhibition in both enzymes (bottom-right panel; black area displays combinations of inhibition that reduce levels of UA to the healthy state).

disease that is in some cases associated with a functional defect in the enzymatic activity of phosphoribosyl pyrophosphate synthetase, an enzyme that regulates the de novo synthesis of purines. As a result of this defect, the purines are unbalanced and the level of uric acid, which is here considered the critical metabolite, increases. This increase, in turn, triggers acute episodes of arthritic pain and nephropathy. An earlier mathematical model of purine metabolism (Curto et al., 1997, 1998a,b) was used to detect critical enzymes whose inhibition with specific drugs could restore the healthy level

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of uric acid without compromising the physiological levels of the other metabolites in the network. The method detected six potential single enzyme targets, one of them coinciding with the conventional clinical treatment using the drug allopurinol, but two of them totally unexpected. When considering potential multifactorial treatments, numerous possible solutions involving the parallel inhibition of two enzymes were detected.

8. One Step Further: Combining Mathematical Modeling with Drug Screening via Protein Docking-Based Techniques As we mentioned at the beginning of this chapter, the traditional paradigm of drug discovery is becoming increasingly problematic due to the demanding criteria that new drugs must satisfy and the complexity of many diseases. We are suggesting here that mathematical modeling combined with virtual screening and followed by in vitro and in vivo experiments is a promising approach for drug discovery. In this approach, the potential drug targets that were identified by the mathematical modeling techniques are chosen preferentially as the most promising candidates in a computeraided strategy that relies on the analysis of the three-dimensional (3D) structure of the targeted protein for the design or selection of new drug molecules. The procedure can be summarized as follows: Structural analysis of the target and target site identification. The 3D structure of the targeted protein is generated based on X-ray crystallography and NMR data or homology modeling. Later, an analysis of ligand binding sites on the surface of the target protein is performed to find critical docking sites. Ideally, the target site should be a pocket or protuberance on the protein surface surrounded by amino acid residues with hydrogen bond donors, hydrogen bond acceptors, and/or the capability of hydrophobic interactions. Complex structures stored in the protein data bank (PDB) can be invaluable for this target site prediction, if initial small molecular inhibitors are available. The output of this analysis is a number of protein sites where docking of a molecule may inhibit the activity of the targeted protein. Drug design. After detailed structural analysis and binding site prediction for the target protein, several computational methods may be used to select active inhibitory molecules. Focused screening inspects known molecules that bind to the site of interest. These small molecules are modified to become inhibitors based on maximizing complementary interactions with the target site (Chan et al., 2001). In virtual screening, large libraries of small compounds are computationally docked onto the site of interest on the target molecule in order to identify the molecules with highest

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binding affinities. In a process called de novo generation, small molecule fragments are initially searched from fragment libraries and positioned in the binding site, and the linking of fragments is performed to partially build a ligand molecule. Most of the docking analysis protocols have two major components: (1) docking methods that generate optimal configurations of the ligand in the binding cavity of the protein; and (2) scoring functions, generally used to estimate the binding affinity of a ligand, based on the candidate ligand pose geometry and its docking with the target protein structure. The output of this process is a number of potential drug molecules with predictive inhibitory activity for the investigated protein. Lead optimization and prediction of pharmacokinetics properties. Those leads discovered by drug design methods need to be optimized in order to produce a list of candidates with low toxicity and improved bioavailability. Various in silico prediction tools are used to evaluate drug-likeness by analyzing the ADMET (absorption, distribution, metabolism, excretion, and toxicity) properties of the molecule. These prediction methods usually try to build a functional relationship between a set of molecular features and given ADMET properties using Quantitative Structure Activity Relationships (QSARs). Recently, substructure pattern fingerprint reorganization techniques along with support vector machines (SVMs) have been combined for the estimation of ADMET properties (Shen et al., 2010). ADMET Properties that are often evaluated include: human intestinal absorption, intestinal permeability; aqueous solubility; oral bioavailability, blood–brain barrier penetration, cytochrome P450 inhibition, hepatotoxicity, plasma protein binding, metabolic stability, active transport and toxicity. The final output of the process is a selection of compounds for experimental testing and the prediction of potential toxicity and side effects (Fig. 11.19). Laboratory testing with in vitro and in vivo experiments of the compounds completes the procedure, in which quantitative data-based mathematical modeling is used to detect potential drug targets, and docking-based techniques are subsequently used to select a number of lead molecules with inhibitory activity in the targeted proteins. Akhoon et al. (2010) combined these techniques to perform a dockingbased drug design and optimization of an anti-breast cancer oligopeptide targeting HER-2 (also known as Erb2). HER-2 overexpression is associated with increased tumor growth and metastasis, and therapeutic approaches are being developed to block these effects in breast cancer. In line with this goal, trastuzumab, a monoclonal antibody (mAb) against HER-2, was recently approved for treatment of breast cancer. Another mAb, pertuzumab, mediates the antibody dependent cytotoxic effects just like trastuzumab, but in addition to cytotoxic effects, pertuzumab binding directly

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Disease specific biochemical pathways I P′

P M1 f

M2 M3

f

Target identification

Experimental procedures for target structure

Literature mining/disease specific pathway/ microarray data analysis

Pure preparation of target in solution

Target expression

Identification of differentially expressed genes/proteins

Assay development Identification of enzymes at critical steps of disease pathways

High throughput screening

Therapeutic targets identification

Structure determination of target using X-ray crystallography/NMR Combinatorial chemistry

New drug target

Structure based drug design

In silico procedure for target structure

3D structure analysis of target

3D structure of target Identification of key amino acid residues

Homology modeling Search for suitable templates

No

Modeling successful

Ligand libraries

Structural validation

Model refinement

No

Prediction of binding sites

Yes

Build 3D model

Data mining

Docking

De novo generation

Fragment libraries

Yes Validated target structure

Lead identification

Lead optimization Biological assay for ADME and toxicity

Biological assay passed?

No

Yes Chemical synthesis of lead

Clinical trials If successfully passed Commercial drugs

Figure 11.19 Schematic diagram for computer-aided drug design, integrating system biology, and structural biology. The combined approach identifies a disease specific biochemical pathway and predicts implementable therapeutic targets through a structure based drug design strategy.

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inhibits ErbB2 association with its partner receptors, thereby blocking the signaling cascade in its initial steps. Akhoon et al. (2010) used the structure of the HER-2/pertuzumab complex as a starting point to design other oligopeptides with analogous activity. An energy minimization model was used for the identification of the active site and the contact surface between the extracellular domain of HER-2 and pertuzumab; this information was used as a target spot of the computational peptide design system. Docking analysis indicated that amino acids VAL286, SER288, LEU295, and HIS296 of HER-2 were strongly involved in binding with pertuzumab. These residues interacting with the active site with high affinity were selected for the design of new sequential oligopeptide LPRAEDTVS. The structure of potential 9mer oligopeptides was modeled and docking simulation analysis was performed in the binding cavity of HER-2 extracellular domain. In order to increase binding affinity of the designed oligopeptide, several derivatives were randomly generated by altering the positions of amino acid residues positions in the 9mer oligopeptide followed by control docking in the target binding cavity. The result was the creation of a new oligopeptide (RASPADREV), which possesses high binding affinity with a highly populated cluster of docked poses of reasonable energy; the structure is shown in Fig. 11.20. The authors later computationally analyzed various ADMET properties of the oligopeptide, such as oral bioavailability in humans, solubility in pure water, pH dependent aqueous solubility, distribution, physicochemical properties, the octanol–water partition coefficient, pKa and ion fraction values, as well as the pH dependent distribution coefficient. The “designed” oligopeptide with anti-cancer activity against HER-2 was furthermore predicted to show other desirable properties like good solubility and stability.

9. Final Remarks The landscape and perspectives of drug discovery have changed drastically over the past decade. On one hand, the traditional approach to drug discovery, consisting primarily of a mass drug screening of millions of potential biomolecules, is becoming more and more problematic as we must deal with complex, multifactorial diseases that are often chronic and almost epidemic. On the other hand, pharmacology and molecular biology have moved from a lack of sufficiently many and sufficiently good experimental data to a situation where vast amounts of quantitative experimental data face the challenging problem of information retrieval, integration, interpretation, and understanding. Our working hypothesis in this chapter was that tools and methods based on the mathematical modeling of

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A

RASPADREV peptide

Active site HER2 extracellular domain

B

RASPADREV

Hydrogen bonds

Active site

Figure 11.20 The oligopeptide RASPADREV binds to the extracellular domain of HER-2. HER-2 receptor overexpression is associated with increased tumor growth and metastasis, and several drugs are being developed to block the effects of this overexpression. Akhoon et al. (2010) used computational methods for virtual screening of drug molecules, as they were described here, to design a new oligopeptide with docking in the target binding cavity of HER-2. The result was the design of new oligopeptides, including RASPADREV, with high binding affinity and sufficient bioavailability. Figure obtained from Akhoon et al. (2010) with the agreement of the publisher.

biochemical networks and pathways are becoming essential to gain new insights into the origin and root causes of complex diseases that are needed to design efficacious therapeutic strategies. The methodology described here complements the approaches of pharmacokinetics and pharmacodynamics by expanding the level of detail in models to the full dynamics of the

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signaling, genetic, proteomic, and metabolic networks that are often involved in disease. These dynamic models, along with a number of computational techniques and strategies that we discussed here, can be powerful tools in the search for potential drug targets. While we exemplified them with specific cases studies, the concepts behind these techniques are disease independent, and the same basic methods can be applied with small variations to biochemical networks related to different kinds of pathologies, ranging from cancer to neurodegenerative diseases and metabolic pathologies. The implementation of these techniques and the application to actual disease are not trivial and require an interdisciplinary approach, with teams composed of experimental biologists and clinicians, mathematical modelers, computational biologists and bioinformaticians who work in close collaboration. Ultimately, this will lead to the development of systems biology based medicine, which will involve the integration of quantitative high-throughput data, mathematical modeling, computing, and analysis (Voit and Brigham, 2008; Wolkenhauer et al., 2010). This new paradigm is expected to boost the development of new drugs and to form the basis of the medicine of the future proposed by E. Zerhouni, the former director of the U.S. National Institute of Health, which will be predictive, personalized, preventive, and participatory.

Appendix 

Complete mathematical model description

ODE equations: dM1 ¼ V1  V2  V3 dt dM2 ¼ V5  V6  V3 dt dM3 ¼ V3  V4 dt dP ¼ V7  V8 dt dP ¼ V8  V7 dt

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Rate equations: V1 V3 V5 V7

¼ k1 ; V2 ¼ k2 M1 a ¼ k2 M1 b M2 c ; V4 ¼ k4 M3 e P d ¼ k5 M3 f ; V6 ¼ k6 M2 g ¼ k7 Ph Ii ; V8 ¼ k8 Pj

Parameter values: Kinetic orders: a ¼ 1; b ¼ 0.5; c ¼ 0.9; d ¼ 1; e ¼ 0.5; f ¼ 0.9; g ¼ 0.5; h ¼ 1; i ¼ 2; j ¼ 1. Rate constants: k1 ¼ 10; k2 ¼ 1.2; k3 ¼ 0.665; k4 ¼ 16; k5 ¼ 9; k6 ¼ 2.887; k7 ¼ 133.3; k8 ¼ 400. 

Equations for the Optimisation Drug Discovery Program in the case study    # p Xj  X HS  X l X Ji  JiHS   j  Min lj  li  HS  þ HS   X Ji j j¼1 i¼1 "

Subject to: Mathematical description of the functional origin of the disease Enzk ¼ EnzPS k Stable steady state of the disease related metabolic network dXi ¼0 dt Constraints guaranteeing physiological feasibility of any interventions: XiLB  Xi  XiUB ;

i ¼ 1; . . . ; nD ;

JkLB  Jk  JkUB ;

k ¼ 1; . . . ; nCF

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Table A.1 Databases and software tools used for biomedical knowledge retrieval and conceptual map construction

HPRD iHOP

KEGG pathway

CellDesigner

Cytoscape

GO

MetaCyc

Online database containing information on protein–protein interactions for biochemical networks in humans (www.hprd.org). Webtool providing up-to-date information on biological molecules and interactions retrieved by automatically extracting key sentences from PubMed documents (www.ihop-net.org). Online database containing a collection of pathways and network maps representing updated knowledge on biochemical reaction networks (www.genome.jp/kegg/pathway). Software tool editor for drawing gene-regulatory and biochemical networks. Networks are drawn based on the graphical notation system proposed by Kitano, and are stored using a computational standard format (www.celldesigner.org). Software platform for visualizing biological pathways and networks, integrating data concerning annotations, gene expression profiles and other state data (www.cytoscape.org). Online database that provides a controlled vocabulary of terms for describing gene product characteristics, gene product annotation data, cellular components, and molecular functions (www.geneontology.org). Online database of nonredundant, experimentally elucidated metabolic pathways curated from the scientific experimental literature. MetaCyc contains pathways involved in both primary and secondary metabolism, as well as associated compounds, enzymes, and genes (www. metacyc.org).

Table A.2 Software for mathematical modeling and simulation of biochemical systems

Sbtoolbox Potter0 s Wheel COPASI AMIGO MADONNA

www.sbtoolbox2.org www.potterswheel.de www.copasi.org Available upon request in [email protected] www.berkeleymadonna.com

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Table A.3 Important databases used for the screening of potential drug targets

Database Drug-target databases PDTD: Potential Drug Target Database

DrugBank

PharmGKB: Pharmacogenomics Knowledge Base

ddTargets: Drug targets/ Disease target Database

TTD: Therapeutic Target Database

TRMPD: Therapeutically Relevant Multiple Pathways Database

Description

PDTD is a comprehensive, webaccessible database of drug targets that focuses on drug targets with known 3D-structures. PDTD currently contains 1207 entries covering 841 known and potential drug targets with structures from the Protein Data Bank (PDB) (www.dddc.ac.cn/pdtd). This database contains detail chemical, pharmacological, pharmaceutical data, with comprehensive drug target information. The database contains nearly 4800 drug and more than 2500 non-redundant protein drug targets (www.drugbank.ca). Central repository for genetic, genomic, molecular and cellular phenotype data and clinical information from various pharmacogenomics research studies (www.pharmgkb.org). Comprehensive database on therapeutic targets. Database contains around 4100 targets classified according to drug/disease type (www.sciclips. com/sciclips/drug-targets-main.do). Information about known and exploratory therapeutic protein and nucleic acid targets, diseases, pathways and corresponding drugs directed at each of these targets. Database currently contains information on 1894 targets and 5126 drugs (http:// bidd.nus.edu.sg/group/cjttd/ TTD_HOME.asp). This database currently contains 11 entries of multiple pathways, 97 entries of individual pathways, 120 targets covering 72 disease conditions

Biochemical Pathway Modeling Tools for Drug Target Detection

STITCH: Search tool for interaction of chemicals

MutationView/ KMcancerDB OMIM: Online Mendelian Inheritance in Man

METAGENE: Metabolic and Genetic Information Centre

HGMD: Human Gene Mutation Database DEPD: Differentially Expressed Protein Database

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along with 120 sets of drugs directed at each of these targets (http://xin.cz3. nus.edu.sg/group/trmp/trmp.asp). Resource to explore known and predicted interactions of chemicals and proteins. Chemicals are linked to other chemicals and proteins by evidence derived from experiments, databases, and the literature. STITCH contains interactions for over 74,000 small molecules and over 2.5 million proteins in 630 organisms (http:// stitch.embl.de). A database of disease-causing genetic mutations and index of cancercausing mutations (http://mutview. dmb.med.keio.ac.jp). OMIM is a comprehensive compendium of human genes and genetic phenotypes. OMIM contains information on all known Mendelian disorders and over 12,000 genes. OMIM focuses on relationships between phenotype and genotype (www.ncbi.nlm.nih.gov/omim). A knowledgebase for inborn errors of metabolism providing information about the disease, genetic cause, treatment and the characteristic metabolite concentrations or clinical tests that may be used to diagnose or monitor the condition. It has data on 432 genetic diseases (www.metagene. de). A database of mutations in human genes searchable by gene and associated disease (www.hgmd.cf.ac.uk). DEPD systematically compares global protein expression profiles in disease and normal cellular states and focuses on quantitative changes that occur as a function of disease, treatment, or environment (http://protchem. hunnu.edu.cn/depd).

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Table A.4 List of some prominent computer assisted drug discovery software

Program

Description

Protein–Protein docking software 3D Dock FTDock, RPScore and MultiDock Perform rigidSuite body docking of biomolecules (www.bmm.icnet. uk/docking) ClusPro Fully automated protein-protein docking. Performs rigid docking by DOT, ZDOCK, PIPER. Clustering of complexes (http://structure.bu.edu/ Projects/PPDocking/cluspro.html) DOT Rigid docking of macromolecules including Protein, DNA, RNA (www.sdsc.edu/CCMS/DOT) FireDock Refinement and re-scoring of rigid-body protein– protein docking solutions (http://bioinfo3d.cs.tau. ac.il/FireDock) HEX Protein docking and molecular superimposition program (www.loria.fr/ritchied/hex) HADDOCK High Ambiguity Driven biomolecular DOCKing (www.nmr.chem.uu.nl/haddock) ZDock Fast Fourier Transform based protein docking program (http://zdock.bu.edu) Autodock Rigid and flexible docking. Docking of the ligand to a set of grids describing the target protein (http:// autodock.scripps.edu) DOCK Docks small molecules or fragments, allowing flexible ligand docking, and using a shape based algorithm (http://dock.compbio.ucsf.edu) FlexX Incremental construction approach of ligands; allows flexible protein–ligand docking (www.biosolveit. de/FlexX) GLIDE Complete suite for protein–ligand interactions based on systematic search techniques (www.schrodinger. com) GOLD Uses genetic algorithms for docking (www.ccdc.cam. ac.uk/products/life_sciences/gold) LigPlot Automatically plots protein–ligand interactions (www. biochem.ucl.ac.uk/bsm/ligplot/ligplot.html) VEGA Ranks ligands on the basis of receptor–ligand interaction energy (http://users.unimi.it/ddl) Discovery Flexible docking, virtual high-throughput screening, Studio de novo ligand generation (http://www.accelrys.com)

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ACKNOWLEDGMENTS J. V. is funded by the German Federal Ministry of Education and Research (BMBF) as part of the project CALSYS-FORSYS under contract 0315264 (www.sbi.uni-rostock.de/calsys). The work was partially supported by NIH project P01-ES016731 and NSF project MCB0946595.

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C H A P T E R

T W E LV E

Deterministic and Stochastic Simulation and Analysis of Biochemical Reaction Networks: The Lactose Operon Example Necmettin Yildirim* and Caner Kazanci† Contents 1. Introduction 2. Mathematical Modeling of Biochemical Reaction Networks and Law of Mass Action 2.1. Simple enzymatic reactions and Michaelis–Menten equation 2.2. Higher order kinetics and Hill equations 2.3. Steady state and linear stability analysis in one-dimensional models 2.4. Modeling coupled reactions and bistability 3. Stochastic Simulations 3.1. Cases where stochasticity matters 3.2. Stochastic simulation algorithms 4. An Example: Lactose Operon in E. coli 5. Conclusions and Discussion Acknowledgment References

372 372 374 375 377 378 381 382 384 386 393 395 395

Abstract A brief introduction to mathematical modeling of biochemical regulatory reaction networks is presented. Both deterministic and stochastic modeling techniques are covered with examples from enzyme kinetics, coupled reaction networks with oscillatory dynamics and bistability. The Yildirim–Mackey model for lactose operon is used as an example to discuss and show how deterministic and stochastic methods can be used to investigate various aspects of this bacterial circuit. * Division of Natural Sciences, New College of Florida, Sarasota, Florida, USA Department of Mathematics/Faculty of Engineering, University of Georgia, Athens, Georgia, USA

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Methods in Enzymology, Volume 487 ISSN 0076-6879, DOI: 10.1016/S0076-6879(11)87012-6

#

2011 Elsevier Inc. All rights reserved.

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1. Introduction Recent advancements in experimental techniques in biology and medicine enable high-throughput experiments acquire data that are easier, cheaper, and more accurate. As more data became available, the need for the right mathematical and computational tools for analysis and interpretation became clear. Until recently, reductionist methods and statistics were the main tools used to study biological systems. “Simpler diseases” were understood and treated this way, but the remaining complicated ones, such as cancer, AIDS, sepsis, and heart diseases require new sophisticated techniques that can cope with their complexity. Structure of biological organisms enables researchers to use various mathematical and computational approaches for modeling, simulation, and analysis. At cellular level, there are distinct biochemical mechanisms that are responsible for specific jobs, such as energy production, protein synthesis, motor functions, defense, and signaling. We present a brief introduction to deterministic and stochastic modeling of biochemical networks. Outline of this chapter is as follows: In Section 2, we describe the basics of mathematical modeling of biochemical reactions, their steady state and stability analysis using the law of mass action. Section 3 summarizes stochastic modeling techniques and describes a basic stochastic algorithm. We focus on the Yildirim–Mackey model for lac operon in Section 4 and discuss how bistability arises in this network. We then simulate the lac operon model using both the deterministic and stochastic methods with experimentally estimated parameter values to show that this system indeed displays bistable behavior for physiologically reasonable parameters values. The chapter ends with Section 5 which includes conclusions and discussion.

2. Mathematical Modeling of Biochemical Reaction Networks and Law of Mass Action There are different approaches and methodologies to studying biochemical reactions. Mass-action kinetics which results in system of differential equations are commonly used to describe the dynamics of biochemical reaction networks. This approach is a fully deterministic and it is appropriate when a system under consideration has large number of molecules and these molecules are spatially homogeneous. In this section, we briefly describe how to construct differential equation models that describe dynamics of a reaction network under these two assumptions. Suppose that A, B, and C are three proteins and when molecules of A collide with molecules of B, they may react and form C. Assume that this reaction is associated with a positive rate constant k1, quantifying how likely

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it is for such a collusion to result in a reaction. We also assume that C can break into A and B. Let us assume that the backward positive rate constant for reaction is k2. We use the following notation Eq. (12.1) to denote this chemical reaction system: k1

A þ B Ð C: k2

ð12:1Þ

Now suppose that there exist various amounts of these proteins in a constantly well-stirred pot, so that its contents remain spatially homogeneous. Here, we are interested in the temporal evolution of the molecular concentrations of these protein molecules. Let us denote the concentration of a species X with [X]. We would like to construct a system of differential equations that governs the temporal evolution of [A], [B], and [C]. Naturally, we can think of the reaction given in Eq. (12.1) as two separate k1 k2 reactions: A þ B ! C and C ! A þ C . According to mass-action kinetics, the time derivative of the concentration of protein A is equated to the difference between the sum of the gain terms (input chemical fluxes) that cause the concentration to increase and the sum of the loss terms (output chemical fluxes) that act to decrease the concentration as: X d½A X ¼ Input fluxes  Output fluxes: dt

ð12:2Þ

For the reaction given in Eq. (12.1), the mathematical model is d d ½A ¼ ½B ¼ k1 ½A½B þ k2 ½C ; dt dt d ½C  ¼ k1 ½A½B  k2 ½C : dt

ð12:3Þ

This system of differential equations can be solved to simulate temporal evolution of [A], [B], and [C] after assigning values for initial concentrations of A, B, and C for t ¼ 0. Although mass-action kinetics is extremely useful for modeling chemical reactions, biological systems benefit greatly from enzymatic kinetics. Most chemical reactions in biological organisms rely on enzymes, special molecules that enable certain reactions to occur. In general, enzymes are fast acting molecules existing in low concentrations. These properties enables derivation of simpler equations for enzymatic reactions using certain approximations. In the following two sections, we give two examples from enzyme kinetics. In the first example, the rate for the product formation is a hyperbolic function of the substrate

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concentration. In the second example, the rate of product is a sigmoidal function of the substrate concentration. Then we discuss importance of these types of functional relationships.

2.1. Simple enzymatic reactions and Michaelis–Menten equation Consider the following enzyme catalyzed reaction given in Eq. (12.4). An enzyme E binds to a substrate S and forms an enzyme–substrate complex ES with a rate constant k1. We assume this reaction is fully reversible. That is to say ES can break down into E and S, and suppose that the associated rate constant for this backward reaction is k2. In this reaction network, we assume that E can also release from ES and produce P and E. The rate constant for this final step of the reaction is k3. In this simple system, there are four time dependent variables: [E], [S], [ES], and [P]. k1

k3

E þ S Ð ES ! P þ E: k2

ð12:4Þ

We assume that the total concentrations of the enzyme and the substrate stay constant over time for this system. That gives us the following two equations: E0 ¼ ½E  þ ½ES;

ð12:5Þ

S0 ¼ ½S þ ½ES þ ½P ;

ð12:6Þ

where E0 and S0 are the initial concentrations of the enzyme and the substrate, respectively. These two equations reduce the number of free variables from four to two. Now, we can write two differential equations that describe the dynamics of the concentration of ES and P. d½ES ¼ k1 ½E ½S  ðk2 þ k3 Þ½ES: dt d½P ¼ k3 ½ES: dt

ð12:7Þ ð12:8Þ

These equations describe the dynamics of the single enzyme–substrate reaction in Eq. (12.4). However, we can further simplify this model using additional assumptions. Not all variables in a dynamic system change at same time scale. It is often the case that some variables change significantly faster than others. If we assume [ES] is a fast variable and reaches a steady state much earlier than [P], then we get d[ES] /dt  0 and hence

Deterministic and Stochastic Simulation of Biochemical Reaction Networks

½ES ¼

k1 ½E½S: k2 þ k3

375

ð12:9Þ

This is called the quasi steady state assumption on [ES]. If we substitute [ES] given in Eq. (12.9) into Eq. (12.5) and solve the resultant equation for [E], we get ½E  ¼

E0 : 1 þ ðk1 =k2 þ k3 Þ½S

ð12:10Þ

Plugging Eq. (12.9) into Eq. (12.8) after replacing [E] in Eq. (12.9) by Eq. (12.10) gives us d½P Vmax ½S ; ¼ dt Km þ ½S

ð12:11Þ

where Vmax ¼ k3E0 ¼ k2 þ k3 / k1. This equation is well known as Michaelis–Menten equation in enzyme kinetics. In Eq. (12.11), the parameter Vmax is the maximum rate that this reaction can occur and the parameter Km is defined as the values of [S] that gives half of Vmax. In other words, when [S] ¼ Km, the product formation (d[P] /dt   d[S] /dt) occurs at half of its maximum rate (Vmax). A graphical representation of Michaelis–Menten equation in Eq. (12.11) for various values of Km is depicted in Fig. 12.1. As seen in this graph, all the curves approach to a maximum value of Vmax as [S] increases. For larger Km values, the curves shift toward the right. All the curves are concave downward and the concavities of these curves do not change as [S] increases.

2.2. Higher order kinetics and Hill equations Consider the following reaction system given in Eq. (12.12). In this reaction system, n-molecules of a substrate S bind to an enzyme E and form a complex ESn with a forward rate constant k1 and a reverse rate constant k2. Then the enzyme is released and a product P is formed with a rate constant k3 k1

E þ nS Ð ESn ; k2

k3

ESn ! P þ E:

ð12:12Þ ð12:13Þ

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Km increases

d[P] dt

Vmax

[S ]

Figure 12.1 A graphical representation of Michaelis–Menten equation in Eq. (12.11) and hyperbolic kinetics for various values of Km when the maximum rate Vmax kept fixed. As Km increases, the curves move to the right and all curves are looking downward.

Let us also assume that at any time throughout the course of these reactions, the first reaction (Eq. (12.12)) is much faster compared to the second one (Eq. (12.13)). Therefore the first reaction reaches equilibrium (forward and backward reaction rates become equal) earlier than P starts to get produced. This allows us to write d½ESn  ¼ k1 ½E½Sn  k2 ½ESn  ¼ 0: dt This is called the equilibrium assumption. Then, we can write ½E  ¼

k2 ½ESn  : k1 ½Sn

ð12:14Þ

Assuming that the total amount of enzyme is conserved, we can write Etot ¼ ½E þ ½ESn :

ð12:15Þ

Substituting Eq. (12.14) into Eq. (12.15) and then solving it for [ESn], we obtain ½ESn  ¼

Etot ½Sn ; Keq þ ½Sn

Keq ¼

k2 : k1

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Keq increases

d[P] dt

Vmax

[S]

Figure 12.2 A graphical representation of Hill Equation in Eq. (12.16) and sigmoidal kinetics for various values of Keq when the maximum rate Vmax kept fixed for n ¼ 4. Similar to the Michaelis–Menten curves, as Keq values increase, the curves move to the right. Unlike Michaelis–Menten curves, the Hill function curves are concave upward for smaller values of [S] and then they become concave downward for larger values of [S].

According to mass-action law, the dynamics of the concentration of P is proportional to the concentration of [ESn] with proportionality constant k3. Hence, d[P] /dt takes the following form d½P Vmax ½Sn ; ¼ Keq þ ½Sn dt

ð12:16Þ

where, Vmax ¼ k3Etot and Keq ¼ k2 / k1. A graphical representation of Hill equation in Eq. (12.16) for various values of Keq when n ¼ 4 is shown in Fig. 12.2. As seen in this figure, all the curves approach to a maximum value of Vmax as [S] increases. For larger Keq values, the curves shift toward the right. One of the important characteristics of Hill equation curves is that they are all concave upward and then become concave downward after a threshold value of [S] for no matter what Keq values are. This important feature leads to bistability, ability of a system to rest in two steady states, as will be discussed in the following sections.

2.3. Steady state and linear stability analysis in one-dimensional models One-dimensional mathematical model has the following general form d½A ¼ f ð½AÞ: dt

ð12:17Þ

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d [A] dt

[A]

Figure 12.3 Graphical approach to the steady state and stability analysis of onedimensional model given in Eq. (12.17).

We say a point [A*] is a steady state if time derivative at that point is zero. The steady states can be computed by solving the equation f([A*]) ¼ 0, since d[A] /dt ¼ 0 at [A] ¼ [A*]. When Eq. (12.17) is plotted as a function taking [A] on x-axis and d[A] /dt on y-axis, the x-intercepts gives the steady state values (see Fig. 12.3). Now, we can think of the equation given by Eq. (12.17) a model that describes movement of an imaginary particle moving along [A]-axis and d[A] /dt is the velocity of that particle. Since d[A] /dt ¼ 0 at the steady state value, there is no change in [A] when d[A] /dt ¼ 0. If d[A] /dt < 0 for a value of [A], the arrows point to the left, otherwise they point to the right. As can be seen in Fig. 12.3, there are two types of steady states. The filled dot represents the stable steady state since the flow is toward this steady state. The open circle represent the unstable steady state since the flow is toward away from this steady state. We can conclude from Fig. 12.3 that a steady state is stable if d[A] / dt < 0 at that steady state value. It is an unstable steady state if d[A] /dt > 0 holds at that steady state value.

2.4. Modeling coupled reactions and bistability In this section, we give an example with a positive feedback loop, one of the important regulatory mechanisms in biological systems. It is capable of producing two stable steady states separated by an unstable steady state, so called “bistable system.” Bistability provides a true discontinuous switching between stable steady states. A bistable system often involves a positive feedback loop. Positive feedback loops are ubiquitous control mechanisms in gene networks. The lactose operon and the arabinose operon of Escherichia coli are two examples of this type of regulatory control networks (Lewin, 2008; Schleif, 2000). Consider the hypothetical system with positive feedback loop in Fig. 12.4. This reaction network has two proteins A and B. We use the Eqs. (12.18) and (12.19) to model the dynamics of this

379

Deterministic and Stochastic Simulation of Biochemical Reaction Networks

A

B

Figure 12.4 A cartoon for a reaction network with a positive feedback loop.

toy reaction network with positive feedback. The dynamics of the concentration of A is given by Eq. (12.18). In this network, we assume A is produced with a constant rate a1 and degraded at a rate proportional to its concentration with a proportionality constant b1. The second term in this equation is for the increase in the production rate of A due to the positive feedback and we assume that this relationship has a Hill function form with n ¼ 2. Equation (12.19) models the dynamics of [B]. It is assumed that A is required for the production of B and the production rate of A is proportional to the concentration of A with a proportionality a2. We assume that B has a decay constant b2. d½A Vm ½B2  b1 ½A: ¼ a1 þ dt Km þ ½B2

ð12:18Þ

d½B ¼ a2 ½A  b2 ½B: dt

ð12:19Þ

This system of differential equations has two time dependent variables [A] and [B] and six positive parameters a1, a2, b1, b2, Vm, and Km. 2.4.1. Steady state and stability analysis Suppose that the system given in Eqs. (12.18) and (12.19) has a steady state ([A*], [B*]). At this steady state ([A*], [B*]), d[A]/dt ¼ d[B]/dt ¼ 0 has to hold simultaneously. Therefore, we can write a1 þ

Vm ½B 2  b1 ½A  ¼ 0; Km þ ½B 2

ð12:20Þ

a2 ½A   b2 ½B  ¼ 0:

ð12:21Þ

After solving Eq. (12.21) for [A*] and plugging it back in Eq. (12.20), we get a nonlinear equation in [B*] as a1 þ

Vm ½B 2 b2  ½B : 2 ¼ b1  a2 Km þ ½B 

ð12:22Þ

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When we look at the right-hand side and left-hand side of Eq. (12.22) as two different functions of [B], we see that the right-hand side is a linear function of [B] with a positive slope m ¼ b1(b2 / a2). The left-hand side is a Hill function which equals a1 when [B] ¼ 0, and approaches to a maximum value of a1 þ Vm as [B] ! 1 (Fig. 12.5). When both of these functions are plotted in the same plane choosing x-axis as [B], where these two functions intersect each other give us the steady state value or values. It is not hard to see that these two functions can intersect each other at only one point for small and large values of m and but at three points for intermediate values of m. Since the left-hand side of Eq. (12.22) is a Hill function and it is concave upward for small values of [B] that means increase in this function is relatively small for smaller concentration of B. Then there is a sharp increase for intermediate concentration of B and after that the curve changes its concavity and becomes concave downward and finally it levels off. This feature of the curve allows possibility of having multiple steady states for intermediate values of m. Figure 12.5 shows how one, two, or three steady states can arise in this model for different values of m. The local stability analysis of the model given in Eqs. (12.18) and (12.19) can be studied mathematically by linearizing this system of differential equations around a given steady state and looking at eigenvalues of the jacobian matrix. For the sake of simplicity, let us assume [B] is a fast variable in this system and d[B] /dt ¼ 0 in Eq. (12.19). After solving d[B] /dt ¼ 0 in Eq. (12.19) for [B] and putting it into Eq. (12.18), the two-dimensional model reduces to a one-dimensional model as:

RHS LHS

[B]

Figure 12.5 A diagrammatic representation showing how one, two, or three steady states may arise in the model given by Eqs. (12.18) and (12.19). The solid line represents the left-hand side of Eq. (12.22). The dash-dotted lines are for the righthand side of Eq. (12.22) for three different values of m ¼ b1b2 / a2. As seen in this plot, there is only one steady state for either small or large values of m. However, there is a range for m in which it is possible to have three coexisting steady states.

Deterministic and Stochastic Simulation of Biochemical Reaction Networks

A

B

d [A]

d [A]

d [A]

dt

dt

dt

381

C

[A]

[A]

[A]

Figure 12.6 A diagrammatic representation of stability analysis of the bistable system modeled by Eq. (12.23). In this plot we see that when there is only one steady state (A and C), this steady state is stable (the flow is toward this steady state). When there are three steady states (B), the middle one is unstable (the flow is away from this steady state) and the other two are stable.

d½A Vm ½A2  b1 ½A: ¼ a1 þ dt Km ðb2 =a2 Þ2 þ ½A