Multidimensional Liquid Chromatography: Theory and Applications in Industrial Chemistry and the Life Sciences

MULTIDIMENSIONAL LIQUID CHROMATOGRAPHY Theory and Applications in Industrial Chemistry and the Life Sciences Edited by ...

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MULTIDIMENSIONAL LIQUID CHROMATOGRAPHY Theory and Applications in Industrial Chemistry and the Life Sciences Edited by

STEVEN A. COHEN Life Sciences R&D, Waters Corporation Milford, MA 01757, USA

MARK R. SCHURE Theoretical Separation Science Laboratory Rohm and Haas Company Springhouse, PA 19477-0904, USA

A JOHN WILEY & SONS, INC., PUBLICATION

MULTIDIMENSIONAL LIQUID CHROMATOGRAPHY

MULTIDIMENSIONAL LIQUID CHROMATOGRAPHY Theory and Applications in Industrial Chemistry and the Life Sciences Edited by

STEVEN A. COHEN Life Sciences R&D, Waters Corporation Milford, MA 01757, USA

MARK R. SCHURE Theoretical Separation Science Laboratory Rohm and Haas Company Springhouse, PA 19477-0904, USA

A JOHN WILEY & SONS, INC., PUBLICATION

Copyright Ó 2008 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada 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, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, 201-748-6011, fax 201-748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at 800-762-2974, outside the United States at 317-572-3993 or fax 317-572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www. wiley.com. Library of Congress Cataloging-in-Publication Data: Multidimensional liquid chromatography: theory and applications in industrial chemistry and the life sciences / edited By Steven A. Cohen, Mark R. Schure. p. cm. Includes index. ISBN 978-0-471-73847-3 (cloth) 1. Liquid chromatography. 2. Chemical engineering. 3. Chemistry, Technical. 4. Biochemistry. I. Cohen, Steven A., 1953- II. Schure, Mark R.; 1952QD79.C454M85 2007 543’.84–dc22 2007041576

Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

CONTENTS

Foreword Preface

xiii xv

Contributors

xvii

1 Introduction

1

1.1 Previous Literature Which Covers MDLC 1.2 How this Book is Organized References

PART I

THEORY

2 Elements of the Theory of Multidimensional Liquid Chromatography 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

Introduction Peak Capacity Resolution Orthogonality Two-Dimensional Theory of Peak Overlap Dimensionality, Peak Ordering, and Clustering Theory of Zone Sampling Dilution and Limit of Detection Chemometric Analysis

4 5 6

9

11 11 13 17 19 21 23 24 26 27 v

vi

CONTENTS

2.10 Future Directions References 3 Peak Capacity in Two-Dimensional Liquid Chromatography 3.1 Introduction 3.2 Theory 3.3 Procedures 3.4 Results and Discussion 3.5 Conclusions Appendix 3A Generation of Random Correlated Coordinates Appendix 3B Derivation of Limiting Correlation Coefficient r References 4 Decoding Complex 2D Separations 4.1 4.2

Introduction Fundamentals: The Statistical Description of Complex Multicomponent Separations 4.3 Decoding 1D and 2D Multicomponent Separations by Using the SMO Poisson Statistics 4.4 Decoding Multicomponent Separations by the Autocovariance Function 4.5 Application to 2D Separations 4.5.1 Results from SMO Method 4.5.2 Results from 2D Autocovariance Function Method 4.6 Concluding Remarks Acknowledgments References

PART II COLUMNS, INSTRUMENTATION AND METHODS DEVELOPMENT 5 Instrumentation for Comprehensive Multidimensional Liquid Chromatography 5.1 5.2 5.3 5.4

Introduction Heart-Cutting Versus Comprehensive Mode Chromatographic Hardware 5.3.1 Valves CE Interfaces 5.4.1 Gated Interface for HPLC–CE 5.4.2 Microfluidic Valves for On-Chip Multidimensional Analysis

28 30 35 35 37 41 42 49 50 54 56 59 59 62 68 74 78 81 84 88 88 88

91

93 93 95 97 97 104 104 105

CONTENTS

5.5

Columns and Combinations 5.5.1 Column Systems, Dilution, and Splitting 5.6 Detection 5.7 Computer Hardware and Software 5.7.1 Software Development 5.7.2 Valve Sequencing 5.7.3 Data Format and Storage 5.8 Zone Visualization 5.8.1 Contour Visualization 5.8.2 2D Peak Presentation 5.8.3 Zone Visualization in Specific Chemical (pI) Regions 5.8.4 External Plotting Programs 5.8.5 Difference Plots 5.8.6 Multi-channel Data 5.9 Data Analysis and Signal Processing 5.10 Future Prospects References 6 Method Development in Comprehensive Multidimensional Liquid Chromatography 6.1 6.2 6.3 6.4

Introduction Previous Work Column Variables Method Development 6.4.1 The Cardinal Rules of 2DLC Method Development 6.5 Planning the Experiment 6.6 General Comments on Optimizing the 2DLC Experiment: Speed–Resolution Trade-off Acknowledgment References 7 Monolithic Columns and Their 2D-HPLC Applications 7.1 7.2

7.3

Introduction Monolithic Polymer Columns 7.2.1 Structural Properties of Polymer Monoliths 7.2.2 Chromatographic Properties of Polymer Monolithic Columns 7.2.3 Two-Dimensional HPLC Using Polymer Monoliths Monolithic Silica Columns 7.3.1 Preparation 7.3.2 Structural Properties of Monolithic Silica Columns 7.3.3 Chromatographic Properties of Monolithic Silica Columns

vii

106 108 109 109 110 111 113 115 115 117 117 117 118 118 119 120 121

127 127 128 130 130 132 143 143 144 144 147 147 148 148 150 152 153 154 154 156

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CONTENTS

7.4

Peak Capacity Increase by Using Monolithic Silica Columns in Gradient Elution 7.5 2D HPLC Using Monolithic Silica Columns 7.5.1 RP-RP 2D HPLC Using Two Different Columns 7.5.2 RP–RP 2D HPLC Using Two Similar Columns 7.5.3 Ion Exchange–Reversed-Phase 2D HPLC Using a Monolithic Column for the 2nd-D 7.5.4 IEX-RP 2D HPLC Using a Monolithic RP Capillary Column for the 2nd-D 7.6 Summary and Future Improvement of 2D HPLC References 8 Ultrahigh Pressure Multidimensional Liquid Chromatography 8.1

Background: MDLC in the Jorgenson Lab 8.1.1 Cation Exchange–Size Exclusion 8.1.2 Anion Exchange–Reversed Phase 8.1.3 Cation Exchange–Reversed Phase 8.1.4 Size Exclusion–Reversed Phase 8.2 Online Versus Off-Line MDLC 8.3 MDLC Using Ultrahigh Pressure Liquid Chromatography: Benefits and Challenges 8.3.1 An Introduction to UHPLC 8.3.2 UHPLC for LC LC: High Speed Versus High Peak Capacity 8.3.3 LC UHPLC for Separations of Intact Proteins 8.4 Experimental Details 8.4.1 Instrumentation 8.4.2 Data Analysis 8.4.3 Chromatographic Conditions 8.4.4 Samples 8.5 Results and Discussion 8.6 Future Directions for UHP-MDLC References

PART III

LIFE SCIENCE APPLICATIONS

9 Peptidomics 9.1 State of the Art—Why Peptidomics? 9.2 Strategies and Solutions 9.3 Summary and Conclusions References

158 159 161 164 166 168 171 171 177 177 178 180 181 183 188 189 190 191 191 193 193 194 195 196 196 202 203

205 207 207 208 218 218

CONTENTS

10 A Two-Dimensional Liquid Mass Mapping Technique for Biomarker Discovery 10.1 10.2

Introduction Methods for Separating and Identifying Proteins 10.2.1 pI-Based Methods of Separation 10.2.2 Chromatofocusing-A Column Based pH Separation 10.2.3 Nonporous Separation of Proteins 10.2.4 Electrospray-Time of Flight-Mass Spectrometry 10.2.5 MALDI Peptide Mass Fingerprinting 10.2.6 Data Analysis and Recombination 10.3 Applications 10.3.1 Proteomic Mapping and Clustering of Multiple Samples—Application to Ovarian Cancer Cell Lines 10.3.2 2D Liquid Mass Mapping of Tumor Cell Line Secreted Samples, Application to Metastasis-Associated Protein Profiles 10.3.3 Identification Annotation and Data Correlation in MCF10 Human Breast Cancer Cell Lines 10.4 Summary and Conclusions Acknowledgments References 11 Coupled Multidimensional Chromatography and Tandem Mass Spectrometry Systems for Complex Peptide Mixture Analysis 11.1 SCX-RP/MS/MS 11.2 SCX/RP/MS/MS 11.3 MudPIT 11.4 Alternative First Dimension Approaches 11.5 Conclusion References 12 Development of Orthogonal 2DLC Methods for Separation of Peptides 12.1 12.2 12.3

Introduction Previous Work Developing Orthogonal 2DLC Methods 12.3.1 LC Selectivity for Peptides: Experimental Design 12.3.2 Investigation of 2DLC Orthogonality for Separation of Peptides 12.3.3 Geometric Approach to Orthogonality in 2DLC 12.3.4 Practical 2DLC Considerations in Proteome Research 12.3.5 Evaluation of Selected 2DLC MS/MS Systems

ix

221 221 223 223 225 226 228 229 230 230

230

233 235 237 238 238

243 245 248 251 254 255 255

261 261 263 264 264 266 271 275 276

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CONTENTS

12.3.6 Peak Capacity in 2DLC-MS/MS 12.3.7 Considerations of Concentration Dynamic Range 12.4 Conclusions Acknowledgment References 13 Multidimensional Separation of Proteins with Online Electrospray Time-of-Flight Mass Spectrometric Detection 13.1 13.2 13.3 13.4

Introduction Chromatographic Parameters Analyte Detection and Subsequent Analysis Building a Multidimensional Protein Separation 13.4.1 Selection of an Ion-Exchange–Reversed-Phase Separation System for Protein-Level Separations 13.4.2 Chromatographic Sorbent Considerations 13.4.3 Chromatographic Behavior of Proteins 13.5 Comprehensive Multidimensional Chromatographic Systems 13.6 Coupling 2DLC with Online ESI–MS Detection 13.6.1 Interactions between the Two Dimensions of Chromatography (Step Vs. Linear) 13.6.2 Recognizing Increased Selectivity in 2DLC Separations 13.7 Expanding Multidimensional Separations into a “Middle-Out” Approach to Proteomic Analysis 13.8 Future Directions in Protein MDLC 13.8.1 Protein Chromatography 13.8.2 MS Analysis of Proteins 13.8.3 Data Interpretation 13.9 Conclusion References 14 Analysis of Enantiomeric Compounds Using Multidimensional Liquid Chromatography 14.1 14.2

Online Achiral-Chiral LC-LC Applications 14.2.1 Analysis of Enantiomers in Plasma and Urine 14.3 Amino Acids 14.3.1 Physiological Fluids or Tissues 14.3.2 In Food, Beverages, and Other Products 14.4 Other Applications 14.4.1 Analysis of Enantiomers from Plant and Environmental Sources 14.5 Miscellaneous Applications 14.6 Conclusion References

280 282 284 284 284

291 291 293 293 294 295 295 296 296 299 304 306 308 311 312 313 314 314 315

319 320 323 323 328 328 333 334 334 336 338 339

CONTENTS

xi

PART IV MULTIDIMENSIONAL SEPARATION USING CAPILLARY ELECTROPHORESIS

345

15 Two-Dimensional Capillary Electrophoresis for the Comprehensive Analysis of Complex Protein Mixtures

347

15.1 15.2

Introduction Previous Work 15.2.1 Miniaturized IEF/SDS-PAGE 15.2.2 One-Dimensional Capillary Electrophoresis for Protein Analysis 15.3 Two-Dimensional Capillary Separations for Analysis of Peptides and Proteins 15.3.1 Capillary Liquid Chromatography Coupled with Capillary Electrophoresis for Analysis of Unlabeled Peptides and Proteins 15.3.2 Two-Dimensional Capillary Electrophoresis for Analysis of Proteins 15.3.3 High-Speed Two-Dimensional Capillary Electrophoresis 15.3.4 The Analysis of a Single Fixed Cell 15.4 Conclusions 15.5 Abbreviations References

16 Two-Dimensional HPLC–CE Methods for Protein/Peptide Separation 16.1 Introduction 16.2 Off-line Versus Online 16.3 HPLC Fractionation 16.4 2D HPLC–CE 16.5 CE–MS Detection 16.6 Applications 16.7 Concluding Remarks Acknowledgment References

PART V

INDUSTRIAL APPLICATIONS

17 Multidimensional Liquid Chromatography in Industrial Applications 17.1 17.2

Introduction Principles of Multidimensional Liquid Chromatography as Applied to Polymer Analysis

347 348 348 349 352

352 352 356 358 360 360 360

365 365 366 366 367 368 370 380 381 381

385

387 387 390

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CONTENTS

17.3 17.4

Experimental Analysis of Alkylene Oxide-Based Polymers 17.4.1 Amphiphilic Polyalkylene Oxides 17.5 Excipients 17.6 Polyether Polyols 17.7 Analysis of Condensation Polymers 17.8 Polyamides 17.9 Aromatic Polyesters 17.10 Aliphatic Polyesters References

393 395 395 399 403 406 407 414 417 420

18 The Analysis of Surfactants by Multidimensional Liquid Chromatography

425

18.1 18.2

Introduction Analytical Characterization Methods 18.2.1 CE and CGE 18.2.2 SEC 18.2.3 NPLC 18.2.4 RPLC 18.3 Detection Methods 18.4 2DLC 18.4.1 RPLC Coupled to SEC 18.4.2 NPLC Coupled to RPLC 18.5 Conclusions References

425 428 429 430 431 433 434 434 435 435 442 443

Index

447

FOREWORD

The principal rationale for multidimensional separations is that they offer a more effective as well as efficient way to generate high peak capacity and thus permit more complete resolution of complex mixtures. I suspect, however, that there is another motivation that attracts people to multidimensional separations: the resulting twodimensional chromatograms make fascinating pictures. Two-dimensional separation patterns are somehow more satisfying than a series of peaks in a one-dimensional chromatogram. The human mind is highly adept at dealing with complex information presented in the form of images and, despite the complexity, is able to quickly spot differences among such patterns. My own inspiration for pursuing multidimensional separations came from J. Calvin Giddings. I was invited to present a seminar at the University of Utah in May 1987, where I spoke on our current research project concerning liquid chromatography in open-tubular columns. That night over dinner Cal told me that he liked the work I presented and also our work on capillary electrophoresis, and suggested that I consider multidimensional chromatography as a more practical approach for the resolution of complex mixtures. On the trip back to Chapel Hill, I thought of nothing other than Cal’s recommendation and how I might set about to implement it. I was interested in analyzing samples in their entirety by two-dimensional separations, so I did not want to settle for the well-established “heart-cutting” approach, where only a single portion of the effluent from the first separation dimension is subjected to a second dimension of separation. Instead, I wanted to subject the entire sample to the full two dimensions of resolution. Also, I did not want to do two-dimensional separations in space, as in two-dimensional thin-layer chromatography, but to use coupled columns instead. This latter consideration was driven at least in part by the ready coupling to mass

xiii

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FOREWORD

spectrometry that columns (but not slabs) provide. Upon arriving home, I made plans with my graduate student, Michelle Bushey, to initiate a project on protein separation by two-dimensional liquid chromatography. Michelle also went on to develop comprehensive two-dimensional liquid chromatography-capillary electrophoresis in my lab. We searched for a suitable term to differentiate our approach from the “heartcutting’’ style of two-dimensional separations and settled for “comprehensive’’ in order to emphasize that all of the sample components were subjected to the full two dimensions of separation. Multidimensional separations have proven to be quite successful, as evidenced by the wealth of examples of hardware and applications described in this book. It is hoped that increased awareness and use of multidimensional separations will open up possibilities for meaningful analyses of truly complex samples and permit the routine analysis of thousands of components from a single sample in a single run. James W. Jorgenson Chapel Hill, North Carolina, USA September 9, 2007

PREFACE

At least two driving forces have contributed to the recent increased use and development of multidimensional liquid chromatography (MDLC). These include the high resolution and peak capacity needed for proteomics studies and the independent size and chemical structure selectivity for resolving industrial polymers. In this regard, separation science focuses on a system approach to separation as individual columns can contribute only part of the separation task and must be incorporated into a larger separation system for a more in-depth analytical scheme. Separation techniques are increasingly used to resolve molecular structure at a finer and finer scale and in chemical environments that are fundamentally complex. This applies not only to small and medium-sized (<1000 Da) molecules but also to polymers, nanoparticles, and colloidal-sized chemical structures. Of particular importance is when a small change in molecular structure modifies a response or behavior of another molecule. This is often the case in the life sciences where resolution of biomarker molecules can be a key to early disease detection. The complexity of the biomarker problem is high as the biomarker molecule may be present as a minor- or trace-level component (very large dynamic range), and its structure may be a slight modification of something present in far larger concentration (large structural diversity). One often finds that when high resolution separation schemes are utilized, other techniques and disciplines must participate in the scheme of understanding and effectively utilizing the separation with subsequent identification of the resulting zones. A rigorous and often multidimensional detection scheme such as mass spectrometry and/or fluorescence is found both for the life science and industrial polymer applications. Other disciplines including informatics and statistics are often

xv

xvi

PREFACE

found to be integral to the separation/detection scheme study when a full analysis of the information present is to be performed. In these themes we have assembled a number of recent contributions that help define the present state of the art and science of MDLC. The coverage of these topics includes instrumentation, theory, methods development, applications of MDLC in the life sciences, and applications of MDLC in industrial polymer chemistry. We have purposely narrowed the scope of all multidimensional chromatography to those techniques that incorporate separations in the liquid phase and to those in which the use of the comprehensive mode prevails but is not exclusive. This text neither incorporates elements of multidimensional thin-layer chromatography, multidimensional separations in gel media such as those commonly employed for the separation of complex mixtures of proteins, nor the techniques that utilize multidimensional gas chromatography. Some of the same principles apply, particularly in the theory section, but our emphasis is strictly on separations carried out in the liquid phase and by columns, rather than in the gas phase or in planar configurations. We would like to thank the contributors to this book for their chapters and for the ease with which we could work with these authors. We thank Heather Bergman of John Wiley Interscience for the opportunity to contribute this text in the active field of MDLC. Additionally we thank the production team of Brendan Sullivan and Robert Esposito from John Wiley and Sons, and Ekta Handa from Thomson Digital, and our respective wives, Nancy Schure and Donna Cohen, for their infinite patience. We hope that this book serves as a useful guide to the field and that it functions as both a reference and as an enticement to others to enter this field. Mark Schure, Blue Bell, PA, USA Steve Cohen, Hopkinton, MA, USA

CONTRIBUTORS

Hiroshi Aoki Department of Polymer Science and Engineering, Kyoto Institute of Technology, Matsugaski, Sakyo-Ku, Kyoto 606-8585, Japan Daniel W. Armstrong Department of Chemistry, Iowa State University, Ames, IA, USA Timothy J. Barder Eprogen, Inc., Darien, IL 60561, USA Scott J. Berger Life Sciences R&D, Waters Corporation, Milford, MA 01757, USA Nathan S. Buchanan Department of Chemistry, The University of Michigan, Ann Arbor, MI 48109, USA Kathleen Cho Department of Pathology, The University of Michigan, Ann Arbor, MI 48109, USA Steven A. Cohen Life Sciences R&D, Waters Corporation, Milford, MA 01757, USA Amy E. Daly Waters Corporation, Milford, MA 01757, USA Joe M. Davis Department of Chemistry and Biochemistry, Southern Illinois University at Carbondale, Carbondale, IL 62901-4409, USA

xvii

xviii

CONTRIBUTORS

Francesco Dondi Department of Chemistry, University of Ferrara, 1-44100 Ferrara, Italy Norman J. Dovichi Department of Chemistry, University of Washington, Seattle, WA 98195, USA Charles R. Evans Department of Chemistry, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA John C. Gebler Waters Corporation, Milford, MA 01757, USA Martin Gilar Waters Corporation, Milford, MA 01757, USA Steven Goodison Department of Pathology, University of Florida, Jacksonville, FL 32209, USA Attila Felinger Department of Analytical Chemistry, University of Pecs, Pecs, Hungary Melissa M. Harwood Department of Chemistry, University of Washington, Seattle, WA 98195, USA Ken Hosoya Department of Polymer Science and Engineering, Kyoto Institute of Technology, Matsugaski, Sakyo-Ku, Kyoto 606-8585, Japan Tohru Ikegami Department of Polymer Science and Engineering, Kyoto Institute of Technology, Matsugaski, Sakyo-Ku, Kyoto 606-8585, Japan Haleem J. Issaq Laboratory of Proteomics and Analytical Technologies, SAIC-Frederick Inc., National Cancer Institute at Frederick, Frederick, MD 21702, USA Megan Jones Department of Chemistry, University of Washington, Seattle, WA 98195, USA James W. Jorgenson Department of Chemistry, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA Hiroshi Kimura Department of Polymer Science and Engineering, Kyoto Institute of Technology, Matsugaski, Sakyo-Ku, Kyoto 606-8585, Japan James R. Kraly Department of Chemistry, University of Washington, Seattle, WA 98195, USA Paweena Kreunin Department of Chemistry, The University of Michigan, Ann Arbor, MI 48109, USA

CONTRIBUTORS

xix

David M. Lubman Department of Surgery, Comprehensive Cancer Center, The University of Michigan, Ann Arbor, MI 48109, USA Egidijus Machtejevas Institute of Inorganic Chemistry and Analytical Chemistry, Johannes Gutenberg University, Duesbergweg 10-14, 55099 Mainz, Germany Nicola Marchetti Department of Chemistry, University of Ferrara, 1-44100 Ferrara, Italy Fred R. Miller Karmanos Cancer Institute, Wayne State University, Detroit, MI 48201, USA Robert E. Murphy Kroungold Analytical, Inc., Encinitas, CA 92024, USA Petra Olivova Waters Corporation, Milford, MA 01757, USA Harald Pasch Deutsches Kunststoff-Institut (German Institute for Polymers), 64289 Darmstadt, Germany Maria C. Pietrogrande Department of Chemistry, University of Ferrara, 1-44100, Ferrara, Italy Frank Rittig BASF Aktiengesellschaft, Polymer Research, 67056 Ludwigshafen, Germany Mark R. Schure Theoretical Separation Science Laboratory, Rohm and Haas Company, Springhouse, PA 19477-0904, USA Renee J. Soukup Department of Chemistry, Iowa State University, Ames, IA, USA Nobuo Tanaka Department of Polymer Science and Engineering, Kyoto Institute of Technology, Matsugaski, Sakyo-Ku, Kyoto 606-8585, Japan Klaus K. Unger Am Alten Berg 40, 64342 Seeheim, Germany Timothy D. Veenstra Laboratory of Proteomics and Analytical Technologies, SAIC-Frederick Inc., National Cancer Institute at Frederick, Frederick, MD 21702, USA Yanfei Wang Department of Chemistry, The University of Michigan, Ann Arbor, MI 48109, USA

xx

CONTRIBUTORS

Michael P. Washburn Stowers Institute for Medical Research, 1000 E. 50th St., Kansas City, MO 64110, USA Rong Wu Department of Pathology, The University of Michigan, Ann Arbor, Ml 48109, USA

1 INTRODUCTION Mark R. Schure Theoretical Separation Science Laboratory, Rohm and Haas Company, Springhouse, PA 19477-0904, USA

Steven A. Cohen Life Sciences R&D, Waters Corporation, Milford, MA 01757, USA

Techniques commonly referred to as “Multidimensional Chromatography’’ have had a long and interesting history. One of the first examples of using two dimensions to develop higher resolution separations utilized paper chromatography with mobile phases applied at right angles in two separate development cycles. This was introduced by Martin and coworkers (Consden et al., 1944). The novelty of this technique was that the separation space was increased because a component zone was eluting through an area, not just through a one-dimensional line-like separation axis. The concept that a separation could be conducted dimensionally, that is, separations in a line (one dimension) and in an area (two dimensions), led to a number of innovations in developing separation techniques with an ever-increasing ability at resolving component zones. A great deal of thought regarding both the theoretical and experimental work in separation science has taken place since this first experiment in planar chromatography. If extractions are included in an analytical scheme, for example, extractions of cations, anions, and neutrals, higher dimensional separations are possible where the separation dimensions are greater than or equal to three. The dimensional count can be further increased through detectors that detect along a wavelength axis (e.g., a diode array detector for use in ultraviolet spectrometry) or a mass-to-charge ratio axis (as in mass spectrometry). These detectors can be coupled to

Multidimensional Liquid Chromatography: Theory and Applications in Industrial Chemistry and the Life Sciences, Edited by Steven A. Cohen and Mark R. Schure. Copyright Ó 2008 John Wiley & Sons, Inc.

1

2

INTRODUCTION

other detectors, for example, when mass spectrometers are coupled together (the socalled MS/MS detector), which increases the dimensionality even higher. Innovations in separation science continued on this theme and provided one of the most powerful separation techniques used in biochemistry, where proteins are separated with isoelectric focusing (IEF) applied in one direction, and gel electrophoresis (GE) applied at a right angle to the first separation direction (O’Farrell, 1975; Celis and Bravo, 1984). In this case, proteins are first separated according to their isoelectric point, measured in pI units, and then according to their molecular weight by gel electrophoresis. The size separation step is usually aided by addition of a surfactant, most typically sodium dodecyl sulfate (SDS), and the gel material is a polyacrylamide formulation. A typical two-dimensional (2D) gel electropherogram of proteins is shown in Figure 1.1. The resolution of this technique is quite impressive; this result was stored in one of many databases containing thousands of protein electropherograms, which have been invaluable in molecular biology and medical research. One reason for the continued use of planar 2D gels is that it gives high resolution and permits a great deal of information to be extracted about proteins. Protein migration provides information on the molecular weight and isoelectric point of the individual components. In addition, proteins can be isolated from the gel, identified, and cataloged. However,

FIGURE 1.1 A 2D-gel electrophoresis map of colorectal epithelia cells proteins from the SWISS- 2DPage database (entry CATD_HUMAN, primary access number P07339) accessible from http://www.expasy.org/swiss-2dpage.

INTRODUCTION

3

this process is time consuming, often taking two or more days for separation and processing. The isolation of the component zones is not simple, and identification of the components can take significantly large amounts of time when the components are digested and further analyzed by liquid chromatography/mass spectrometry to obtain their peptide signatures used for protein identification. This time burden does not permit the planar techniques to be used as fast, sensitive biomarker discovery systems for routine investigations. Identification of trace-level zones is particularly problematic in planar systems. Much effort has been made over the past 15 years to replace planar techniques with modern column-based techniques. The advantages of columns are many including reproducibility, speed, selectivity, and ease of use among others. Another advantage of columns is that they are much easier, in almost all cases, to interface to detectors such as mass spectrometers. The column methods are much faster and are automated so that a much larger number of samples can be processed per unit time. An example of this technology, described in more detail in Chapter 10 by Lubman and coworkers, is shown in Figure 1.2, where the first dimension is from a chromatofocusing column, which gives separations in pI much like isoelectric focusing, only here the pI axis is in bands instead of continuous pI increments. The second dimension is by reversed-phase liquid chromatography (RPLC). This technology is most important in demonstrating the huge capabilities that column chromatography can bring to the separation arsenal. Although predictions regarding the huge peak capacity that would occur when coupling multiple dimensions were in place by 1984, as discussed in the theory chapter (Chapter 2),

FIGURE 1.2 2D liquid protein expression map of the HCT-116 human colon adenocarcinoma cell line. The x-axis is in pI units from 4.0 to 7.0 in 0.2 increments. The y-axis is percent B of the RP-HPLC gradient. The gray scale of the bands represents the relative intensity of each band by UV detection at 214 nm. From Yan et al. (2003) with permission of the American Chemical Society. (See color plate.)

4

INTRODUCTION

FIGURE 1.3 Three-dimensional representation of a tryptic digest of ovalbumin. The threedimensional separation consists of size-exclusion chromatography (first dimension), reversedphase LC (second dimension), and capillary electrophoresis (third dimension). From Moore and Jorgenson, (1995) with permission of the American Chemical Society.

many investigations in polymer separations had shown the power of off-line multidimensional separations. It was the comprehensive coupling of column techniques by Erni and Frei (1978) and Bushey and Jorgenson (1990) that lead the way for the continued development of automated column-based comprehensive multidimensional methods. We use the term comprehensive to denote the repeated application of sampling the kth dimensional column effluent by the (k þ 1)th column in narrow volume elements. In this manner, the procedure resembles a planar separation but run exclusively with column methods. The versatility of columns and the understanding of how to interface columns in a multidimensional instrument were evident in obtaining a three-dimensional separation implemented approximately 10 years ago, as shown in Figure 1.3. Both life science and industrial applications benefit from the combination of multidimensional chromatography and advanced detector technology. Both of these areas of multidimensional liquid chromatography (MDLC) are covered in this book.

1.1 PREVIOUS LITERATURE WHICH COVERS MDLC Two books of interest have covered some of the ground for multidimensional liquid chromatography in various forms. These include the book Multidimensional Chromatography, edited by Cortes (1990) and Multidimensional Chromatography by Mondello et al. (2002). Cortes’ book contains a collection of chapters that discuss many of the aspects of the modern MDLC system. Specifically, Giddings’ chapter covers many of the theoretical underpinnings for the multidimensional techniques including multicolumn

HOW THIS BOOK IS ORGANIZED

5

and planar chromatography modes of operation. The role of peak capacity is specifically discussed and this has been one of the efficacy metrics to judge many of the multidimensional techniques. These developments will be elaborated on in the theory section of this book as new results have been obtained since Giddings’ chapter in 1990. In addition, the chapter on multidimensional high performance liquid chromatography by Cortes and Rothman has been a useful reference for early applications where coupled-column systems were utilized. These were not exclusively comprehensive but employed heart-cutting techniques that solved chemical problems. In the second book, Multidimensional Chromatography by Modello, Lewis, and Bartle, a number of contributions are made that cover MDLC. These include the chapter by Kazakevich and LoBrutto on industrial and polymer applications that use liquid chromatography (LC), gas chromatography (GC), and supercritical fluid chromatography (SFC) techniques. The introduction by Bartle is general to MDC and covers the topics of peak capacity, statistical overlap, resolution, and column compatibility. Multidimensional and electrodriven separations are described by Degen and Remcho. However, the emphasis in Mondello et al.’s book is undoubtedly on multidimensional gas chromatography. In the present book, we exclusively deal with liquid-phase techniques, mostly liquid chromatography with a brief focus on capillary electrophoresis (CE). Other reviews of multidimensional separations have been published. These include a book on polymer characterization by hyphenated and multidimensional techniques (Provder et al., 1995), a review on polymer analysis by 2DLC (van der Horst and Schoenmakers, 2003), and two reviews on two-dimensional techniques in peptide and protein separations (Issaq et al., 2005; Stroink et al., 2005). Reviews on multidimensional separations in biomedical and pharmaceutical analysis (Dixon et al. 2006) and multidimensional column selectivity (Jandera, 2006) were recently published. Suggested nomenclature and conventions for comprehensive multidimensional chromatography were published in 2003 (Schoenmakers et al., 2003), and a book chapter in the Advances in Chromatography series on MDLC was published in 2006 (Shalliker and Gray 2006).

1.2 HOW THIS BOOK IS ORGANIZED This book is organized into five sections: (1) Theory, (2) Columns, Instrumentation, and Methods, (3) Life Science Applications, (4) Multidimensional Separations Using Capillary Electrophoresis, and (5) Industrial Applications. The first section covers theoretical topics including a theory overview chapter (Chapter 2), which deals with peak capacity, resolution, sampling, peak overlap, and other issues that have evolved the present level of understanding of multidimensional separation science. Two issues, however, are presented in more detail, and these are the effects of correlation on peak capacity (Chapter 3) and the use of sophisticated Fourier analysis methods for component estimation (Chapter 4). Chapter 11 also discusses a new approach to evaluating correlation and peak capacity.

6

INTRODUCTION

The columns, instrumentation, and methods chapters, Chapters 5–8, include presenting the necessary background information for the reader to be brought to the modern literature on the instrumentation used in 2D methods. Specifically, Chapter 5 gives a general overview, Chapter 6 discusses the method development needed besides the usual 1D optimizations for chromatographic operation, Chapter 7 discusses monolithic columns used in 2DLC, and Chapter 8 discusses ultrahigh pressure multidimensional LC. Perhaps the biggest increase in the application and development of the MDLC technique since Cortes’s book is in life sciences, which accounts for approximately half of this book. One reason for this may be due to the high level of interest in studying the human proteome (proteomics). Proteomics is such a demanding application that the separating power needed to resolve even the normal proteins in the body is so demanding that maximum separation power is needed to provide this capability. Many aspects of separations in proteomics are discussed in Chapters 9–13, 15 and 16. Chapter 14 discusses enantiomeric compound separations by MDLC. Industrial applications of MDLC are discussed in Chapters 17 and 18. These chapters embody the types of applications used in polymer and surfactant analysis that have become mainstream in the MDLC literature. Most life science applications are currently centered on proteomics. However, other systems biology approaches are also likely to need the resolving power of MDLC. These approaches include metabolomics, the small-molecule metabolite profiles of cellular processes and glycomics, the study of oligosaccharides or chains of sugars. The developments that are given in the life sciences applications in this book set the stage for studying these other areas. In most cases, the mixtures of molecules in metabolomics and glycomics are too complex to study by any single chromatographic column methods. MDLC is well suited for these types of studies, especially when the detection is by mass spectrometry or some multidimensional version of mass spectrometry such as MS–MS. The concepts in multidimensional separations are also well suited for other biological system separations such as cell separations. Cell separations are generally done by techniques such as flow cytometry (cell sorting) magnetic encoding, sedimentation, and isopycnic (density gradient) separation. However, techniques such as field-flow fractionation (FFF) and dielectrophoresis (separation by nonuniform AC electric fields) may augment the main techniques in the context of the second dimension. Having learned a number of experimental and theoretical aspects of multidimensional techniques from 2DLC, forming new, more advanced systems that separate cells may be expedited through the application of present MDLC research.

REFERENCES Bushey, M.M., Jorgenson, J.W. (1990). Automated instrumentation for comprehensive two-dimensional high-performance liquid chromatography of proteins. Anal. Chem. 62, 161–167.

REFERENCES

7

Celis, J.E., Bravo, R. (1984). Two-Dimensional Gel Electrophoresis of Proteins. Academic Press, New York. Consden, R., Gordon, A.H., Martin, A.J.P. (1944). Qualitative analysis of proteins: a partition chromatographic method using paper. Biochemical J 38, 224–232. Cortes, H.J. (1990). Multidimensional Chromatography, Techniques and Applications, Chromatographic Series Vol. 50. Marcel Dekker Publishing, New York. Dixon, S.P., Pitfield, I.D., Perrett, D. (2006). Comprehensive multi-dimensional liquid chromatographic separation in biomedical and pharmaceutical analysis: a review. Biomed. Chromatogr. 20, 508–529. Erni, F., Frei, R.W. (1978). Two-dimensional column liquid chromatographic technique for resolution of complex mixtures. J. Chromatogr. 149, 561–569. Issaq, H.J., Chan, K.C., Janini, G.M., Conrads, T.P., Veenstra, T.D. (2005). Multidimensional separations of peptides for effective proteomics analysis. J. Chromatogr. B 817, 35–47. Jandera, P. (2006). Review: column selectivity or two-dimensional liquid chromatography. J. Sep. Sci. 29, 1763–1783. Mondello, L., Lewis, A.C., Bartle, K.D. (2002). Multidimensional Chromatography. John Wiley & Sons, Inc., New York. Moore, A.W., Jorgenson, J.W. (1995). Comprehensive three-dimensional separation of peptides using size exclusion chromatography/reversed-phase liquid chromatography/optically gated capillary zone electrophoresis. Anal. Chem. 67, 3456–3463. O’Farrell, P.H. (1975). High resolution two-dimensional electrophoresis of proteins. J. Biol. Chem. 250, 4007–4021. Provder, T., Barth, H.G., Urban, M.W., editors (1995). Chromatographic Characterization of Polymers: Hyphenated and Multidimensional Techniques, Advance in Chemistry Series 247. American Chemical Society,Washington, DC. Shalliker, R.A., Gray, M.J. (2006). In: Grushka, E., Grinberg, N., editors. Concepts and Practice of Multidimensional High-Performance Liquid Chromatography Advances in Chromatography, Vol. 44. Taylor and Francis Group, New York. Schoenmakers, P.J., Marriott, P., Beens, J. (2003). Nomenclature and Conventions in Comprehensive Multidimensional Chromatography. LCGC Europe, June 2003, 1–4. Stroink, T., Ortiz, N.C., Bult, A., Lingeman, H., de Jong, G., Underberg, W.J.M. (2005). On-line multidimensional liquid chromatography and capillary electrophoresis for peptides and proteins. J. Chromatogr. B 817, 49–66. Van der Horst, A., Schoenmakers, P.J. (2003). Comprehensive two-dimensional liquid chromatography of polymers. J. Chromatogr. A 1000, 693–709. Yan, F., Subramanian, B., Nakeff, A., Barder, T.J., Parus, S.J., Lubman, D.M. (2003). A comparison of drug-treated and untreated HCT-116 human colon adenocarcinoma cells using a 2-D liquid separation mapping method based on chromatofocusing pI fractionation. Anal. Chem. 75, 2299–2308.

PART I THEORY

2 ELEMENTS OF THE THEORY OF MULTIDIMENSIONAL LIQUID CHROMATOGRAPHY Mark R. Schure Theoretical Separation Science Laboratory, Rohm and Haas Company, Springhouse, PA 19477-0904, USA

2.1 INTRODUCTION As discussed in Chapter 1 of this book, multidimensional separations have their origin in some of the earliest works in analytical bioseparations where the separations were carried out in the planar mode of operation using paper-like media (Consden et al., 1944). One of the main innovations in these studies was the realization that using more than one simultaneous separation mechanism had enormous potential for complex separations, especially those of biological origin. Planar techniques evolved to become the mainstay of protein analysis where proteins are separated in the first dimension by isoelectric focusing according to pI and in the second dimension by gel electrophoresis, which separates on the basis of molecular weight (O’Farrell, 1975). The theoretical work that exploited the advantages of the multidimensional separation format appears to have been developed much later than the original experimental work. One of the earliest studies was conducted by Connors (1974), who assumed that the distribution of spots on a two-dimensional thin-layer chromatography (2DTLC) plate could be modeled using a Poisson distribution of data on each retention axis. He then constructed equations that related the number of chromatographic systems needed to resolve a specific number of compounds. One Multidimensional Liquid Chromatography: Theory and Applications in Industrial Chemistry and the Life Sciences, Edited by Steven A. Cohen and Mark R. Schure. Copyright 2008 John Wiley & Sons, Inc.

11

12

ELEMENTS OF THE THEORY OF MULTIDIMENSIONAL LIQUID CHROMATOGRAPHY

5

4

3 m 2

1

0

50

100

150

200

T

FIGURE 2.1 Plot of the number of chromatographic systems, m, required to resolve T compounds under a number of specific chromatographic conditions as discussed in Connors (1974). Figure reprinted with permission from the American Chemical Society.

of the results of these equations is shown in Fig. 2.1 under a number of assumptions explained in the original paper. This research implied that the challenge was to find separation systems that are uncorrelated. The correlation problem haunts multidimensional separation systems and ultimately determines their success or failure. Connors’ research was limited to different solvent systems used in each dimension of 2DTLC; however, the concepts are similar to finding uncorrelated chromatographic column systems that can be used in multidimensional chromatography (MDC), so that the separation mechanisms are different (essentially uncorrelated) among the various columns. When columns are uncorrelated we will refer to this phenomenon as an orthogonal separation. The quantification of orthogonality and correlation among the dimensions are discussed in detail here and also in the Chapters 3 and 12. The use of the Poisson distribution for this purpose predates the statistical overlap theory of Davis and Giddings (1983), which also utilized this approach, by 9 years. Connors’ work seems to be largely forgotten because it is based on 2DTLC that doesn’t have the resolving power (i.e., efficiency or the number of theoretical plates) needed for complex bioseparations. However, Martin et al. (1986) offered a more modern and rigorous theoretical approach to this problem that was further clarified recently (Davis and Blumberg, 2005) with computer simulation techniques. Clearly, the concept and mathematical approach used by Connors were established ahead of its time. Many examples of planar systems used for two-dimensional liquid chromatography (2DLC) abound in the literature. In the quest for higher performance

PEAK CAPACITY

13

separation systems, a number of investigations, specifically addressing the performance of each dimension in the context of MDC were conducted. In this regard, theoretical studies by Guiochon et al. (1983a,b) and Giddings (1984, 1987, 1990) accelerated the need to examine column-based systems to increase the performance and attempt to resolve zones at the single component level. Following these studies, the experimental realization of a comprehensive MDC column system (Bushey and Jorgenson, 1990) showed results for protein separations that demonstrated the importance of MDC, specifically 2DLC. Although relatively unknown, the instrumentation for 2DLC was conceived and implemented by Erni and Frei (1978). They reported the valve configuration presently used in most comprehensive 2DLC systems. However, they automated neither the valve nor the data conversion process to obtain a contour map or 2D peak display. They used a gel permeation chromatography (GPC) column in the first dimension and a reversedphase liquid chromatography (RPLC) column in the second dimension and studied complex plant extracts. The papers by Guiochon and Giddings emphasized the study of multidimensional resolution and peak capacity. It is these insights that have largely fueled serious attempts to exploit the potential rewards of coupling columns together. In addition, these papers provided the basis for further theory-based investigations into the mechanics of MDC that have largely superseded the original papers. In this chapter, many of the results of these older studies will be discussed along with research that provided more detailed insights into factors that control the potential of MDC. 2.2 PEAK CAPACITY One of the most common reasons for using MDC techniques is to increase the peak capacity. Peak capacity, as defined by Giddings (1967, 1991), is the number of peaks that could be put side by side at some stated resolution between two retention times or at a start and finish length, in the case of a planar separation system. The peak capacity concept was originally derived as a metric for studying the possible number of peaks in a separation system when peaks are ordered; that is, when peak retention times are some multiple so that peak spacing is a constant. In fact, peak capacity was never advocated by Giddings as something that was measurable; rather, it was a concept based on how many peaks would fit into a separation space on the basis of the width of a peak. The peak width was related to the number of plates and a fixed resolution. It is this ordering that gave the concept a theoretical bent as real separations are not ordered; the retention times in most separation techniques appear almost random across a range of separation time. The mathematical definition of peak capacity, nc, for an isocratic separation is given as (Grushka, 1970) pﬃﬃﬃﬃ N tf ln ð2:1Þ nc ¼ 1þ t1 4 where N is the number of plates and tf and t1 are the times of the final peak, and the void peak, respectively. The peak width is not constant in isocratic separations, the same is

14

ELEMENTS OF THE THEORY OF MULTIDIMENSIONAL LIQUID CHROMATOGRAPHY

true of isothermal separations in gas chromatography. Under gradient elution conditions (and temperature programmed gas chromatography), the peak width is assumed to be constant (Horvath and Lipsky, 1967), and the peak capacity can be approximated for a planar system as nc ¼

L 4sRs

ð2:2Þ

where s is the standard deviation of a Gaussian zone (in length units) and Rs is the stated resolution, that is, the resolution level that a peak will be recognized as being resolved from its neighbor. For a time-based column system undergoing gradient elution where peak width is constant (Horvath and Lipsky, 1967; Grushka, 1970), the definition of peak capacity is pﬃﬃﬃﬃ N tf 1 ð2:3Þ nc ¼ 4 t1 Constant peak width is recognized to give more peak capacity in LC and the same is true in 2DLC. The use of this equation is interesting in that the number of theoretical plates is not constant across a chromatogram of constant peak width. This is easily shown by recalling the definition of the number of plates as N¼

t2 s2

ð2:4Þ

where the denominator is a constant, that is, constant peak variance or width. A better and simpler equation for peak capacity can be derived by dividing the retention time range by some peak width, which is defined independently so that nc ¼ ðtf t1 Þ=W

ð2:5Þ

where W is the peak width. This definition is also useful for discussing the so-called “sample peak capacity’’ (Dolan et al., 1999) when one crosses over to the peak capacity of real samples with regard to the number of peaks that actually appear on the chromatogram, irrespective of whether single- or multidimensional techniques are utilized. The point cannot be overemphasized, however, that the calculated peak capacity will always overestimate the number of peaks that appear on an experimental chromatogram due to the apparent random placement of peaks. Guiochon et al. (1983a,b) gave detailed calculations of 2D and 3D peak capacities for zones in a unique chromatographic column-like device. They clearly recognized the enormous peak capacity advantage in using the multidimensional approach and how this could help resolve peak fusion. Giddings (1984) emphasized that the peak capacity for a multidimensional chromatographic system could be approximated by the simple product of individual peak capacities in each dimension: nT ¼ n1 n2 n3

ð2:6Þ

where nT is the total peak capacity of the separation system. Interestingly, this point was emphasized in a review by Freeman (1981) where he attributes the origin of this

PEAK CAPACITY

z

15

nz adjacent Gaussian profiles

y

ny adjacent Gaussian profiles

FIGURE 2.2 The peak capacity of a 2D system (Giddings, 1987) represented by the number of boxes is approximately equal to the product of the peak capacities nx and ny generated along two individual axes, as represented by the number of adjacent Gaussian profiles. Reprinted with permission from J. High Resolut. Chromatogr., Alfred Huethig Publishers.

“product rule’’ to a popular separation science textbook (Karger et al., 1973). However, the “product rule’’ seems to be largely missed in the MDC literature prior to Giddings’ work in 1984. Many researchers in the field recognize Fig. 2.2 as a radical but effective depiction of this concept. The numbers that resulted from multiplying individual column peak capacities showed the tremendous promise for this style of separation. It also fueled speculation in bringing column chromatography methods to bear on the multidimensional system instead of planar separation systems and was a large driving force in the construction of comprehensive 2DLC instrumentation that fully utilized column chromatography (Bushey and Jorgenson, 1990). As discussed below, if the two media used for separation in 2DLC are correlated with respect to the retention mechanism, the peak capacity will be lower than expected from the product approximation. The dependence of the peak capacity product on the correlation between retention mechanisms is covered in Chapter 3. Furthermore, as pointed out by Carr and coworkers (Stoll et al., 2006), if the second dimension sampling of the first dimension is undersampled, the potential peak capacity will not be able to be utilized. This is discussed in the theory of zone sampling section below. Further implications of peak capacity limitations due to sampling have been recently given by Tanaka and coworkers (Horie et al., 2007). There has been an attempt to measure the peak capacity in 1DLC and 2DLC by assigning a range of useful retention time between the unretained marker that elutes at t1 and some stated value of the retention factor k0 leading to a zone at tf and plugging in a value for the peak width W. This number is useful but will never be equal to the number

16

ELEMENTS OF THE THEORY OF MULTIDIMENSIONAL LIQUID CHROMATOGRAPHY

of experimentally accessible peaks, because the peaks are not ordered. This was the whole basis for the statistical peak overlap theory of Davis and Giddings (1983); peaks were found to be randomly placed in a complex chromatogram. However, including experimental parameters such as the gradient time, as demonstrated by Carr and coworkers (Stoll et al., 2006), in maximizing the peak capacity in 2DLC with high temperature gradient elution chromatography, allows the functional dependencies to be understood and is very useful for technique optimization. The peak capacity value does not imply the time it takes to produce these peaks. As is often the case in the chromatographic theory, one must also discuss the rate of peak capacity production as a function of time for understanding the technique efficacy, and this line of development was recognized early on by Grushka (1984). It has been noted that peak capacities of 900 in 25 min are achievable by 2DLC (Stoll et al., 2006); that is roughly one peak every 2 s. A recent detailed look at the peak capacity production rate was undertaken by Carr and coworkers (Wang et al., 2006) in the context of gradient elution chromatography for use in 2DLC. The use of the so-called “Poppe plot’’ was extended for gradient elution by these authors and this may lead to better optimization schemes, especially when gradient elution is utilized. An example of a Poppe plot is shown in Chapter 6. Since gradient elution brings such higher performance in LC, it is safe to say that this approach will become a standard operating procedure in future 2DLC experiments, especially when fast second dimension elution is needed. The concept of peak capacity is rather universal in instrumental analytical chemistry. For example, one can resolve components in time as in column chromatography or space, similar to the planar separation systems; however, the concept transcends chromatography. Mass spectrometry, for example, a powerful detection method, which is often the detector of choice for complex samples after separation by chromatography, is a separation system itself. Mass spectrometry can separate samples in time when the mass filter is scanned, for example, when the mass-to-charge ratio is scanned in a quadrupole detector. The sample can also be separated in time with a time-of-flight (TOF) mass detector so that the arrival time is related to the mass-to-charge ratio. It is not so surprising that we can talk about the peak capacity of the detector itself, as if the detector were a column or planar separator. Russell and coworkers (2002) have examinedthepeakcapacityofpeptideseparationsusingionmobility–massspectrometry (IM–MS). They have estimated that the peak capacity of IM–MS is approximately 2600 for peptides. If this were orthogonal and peak capacities multiplicative between dimensions, this would suggest that the combination of 2DLC (with n in the hundreds) and IM–MS would result in a total peak capacity somewhat below one million for peptide separations. Merenbloom et al. (2007) examined the coupling of IM–MS in two dimensions, which they call two-dimensional IMS (2D-IMS). Peak capacities ranging from480–1360are quoted for this technique that are farless than those quoted byIM–MS in a single dimension, as given above. Frahm et al. (2006) estimated the peak capacity of Fourier Transform–Ion Cyclotron Resonance (FT–ICR) and TOF mass spectrometry, specifically for proteomics applications. The concept of peak capacity is applicable to all mass spectrometry applications independent of chromatography. However, mass spectrometry may be orthogonal to the chromatographic dimensions and this offers a chance to get the multiplicative advantage of MDC peak capacity and mass spectrometry peak

RESOLUTION

17

capacity. Other works that have mentioned the peak capacity of mass spectrometry in the context of LC–MS and 2DLC–MS but have not rigorously measured it include those of Lewis et al. (1997), Shen et al. (2001), and Valentine et al. (2001). Peak capacity can also be determined for the diode array detector (DAD) often used as a multiwavelength UV detector for liquid chromatography; these detectors allow fused component zones to be resolved through chemometric techniques and do not have the large peak capacity that mass spectrometry brings. However, even doubling the peak capacity through the use of a DAD detector can provide an advantage when coupled with chemometric techniques, discussed below.

2.3 RESOLUTION Giddings (1990) presented a derivation applicable to both the planar format such as TLC that is distance-based and the comprehensive multidimensional separations that are time-based. The resolution was shown to be equal to the Euclidean norm of zone resolution components. This can be summarized as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð2:7Þ Rs ¼ Rs21 þ Rs22 þ where the subscripted resolution is that of the corresponding resolution of zone pairs in each dimension. This one-dimensional resolution for time-based systems is Rsx ¼ Dtx =4sx

ð2:8Þ

where Rsx is the resolution in dimension x, Dtx is the time difference between two peaks in dimension x, and s is the zone standard deviation in dimension x. It is assumed here that all peaks are Gaussian and the standard deviations of two adjacent peaks in a single dimension are approximately equal. In this formulation, the resolution is assumed to be independent of angle between zones. This assumption was checked by Schure (1997) as the equation for resolution as a function of angle between the two zones was previously available (Shi and Davis, 1993). This analysis shows that for a number of cases, the error is approximately less than 10% when using the Euclidean norm of resolutions, as shown in Fig. 2.3. Another way to measure resolution from experimental 2DLC data is to use a computer method to calculate the first and second moments of the zones. For highly fused zones this must be done with a parameter estimation algorithm based on some minimization criteria; usually, some form of least-squares method can be utilized to fit the zone shapes with a zone model. However, another method used to measure resolution, which is less accurate but less demanding, is to take the peak and valley measurements f and g along the line of zone intersection, shown schematically in Fig. 2.4, so as to calculate P ¼ f/g. Then Equation 2.9 is used to get the resolution rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 1P ð2:9Þ Rs ¼ ln 2 2

18

ELEMENTS OF THE THEORY OF MULTIDIMENSIONAL LIQUID CHROMATOGRAPHY

FIGURE 2.3 The error inherent in using the Euclidean norm to calculate resolution as a function of the angle made between the two zones. The conditions used for curves A, B, C, and D are described in the original paper (Schure, 1997). Reprinted by permission from J. Micro. Sep.

f g hy

3

t2,y

y

2 t1,y 1

t2, y

t1, x

0 0.00

0.50

1.20

x

1.80

2.40

3.00

FIGURE 2.4 Determining resolution based on a peak-valley measurement for twodimensional chromatography. The f and g values are measured and used to calculate P ¼ f/g giving the resolution through Equation 2.9. Reprinted with permission from Murphy et al. (1998) by courtesy of the American Chemical Society.

ORTHOGONALITY

19

This approach is reviewed in full detail for 1D chromatography by Schoenmakers (1986) and extended by Schure (1999) for dimensions of two and larger. The peak amplitudes f and g are determined either through programmed search or through inspection in a spreadsheet program. Both quantities are corrected for a finite baseline amplitude. It is interesting that this approach works for 1D, 2D, 3D and so on. — it is dimensionally invariant. Since resolution is rarely needed with great accuracy, this gives a convenient method for determining resolution in 2DLC. This approach was extended for non-Gaussian peaks (Peters et al., 2007).

2.4 ORTHOGONALITY The challenge in effectively utilizing the multidimensional peak capacity is to find different types of columns that can uniformly spread the component peaks across the separation space. This challenge means that the separation mechanism of the two columns should be as dissimilar as possible or uncorrelated. A number of experimental studies have been undertaken to examine this effect (Liu et al., 1995; Slonecker et al., 1996; Gray et al., 2002). Chapter 3 examines the effect of correlation on peak capacity in detail using simulation techniques. The mathematical development of estimating correlation is taken from Liu’s work and is a simple treatment. In 2DLC, we can form a retention matrix composed of the scaled k0 values for each zone so that each dimension (i.e., each unique column) has a k0 row vector for the compounds that were separated on it. In two dimensions, the retention representation becomes a matrix, k0 , so that k0 takes the form for a fourcomponent mixture: 0

k 11 0 k 21

0

k 12 0 k 22

0

k 13 0 k 23

0

k 14 0 k 24

ð2:10Þ

where the first index i is the column dimension number and the second index j is the component number. This matrix is scaled so that 0

k ij ¼ ðkij mi Þ=si

ð2:11Þ

where kij is the original (unscaled) retention measurement, which is equal to (tr/t0) 1. Note that the retention time tr and the void time t0 have their usual meanings and mi is the mean of the ki row vector and si is the standard deviation of the ki row vector values. Using standard statistical notation (Malinowski, 2003) the correlation matrix C is given by 1 0T 0 k k ð2:12Þ C¼ N1 where N is the number of components in each dimension and the T refers to the matrix transpose operator. The correlation matrix C is an N by N matrix with unity value diagonal elements and is symmetrical (Cij ¼ Cji), so only the upper or lower triangular

20

ELEMENTS OF THE THEORY OF MULTIDIMENSIONAL LIQUID CHROMATOGRAPHY

A Np

N2

Effective area

g b a¢

C

a Nt

FIGURE 2.5 Effective nonorthogonal 2D retention space from Liu (1995). Reprinted with permission from the American Chemical Society.

terms need to be discussed. The diagonal elements of this matrix are equal to 1 because these values are completely correlated with themselves. Experimental values of the correlation matrix (Liu et al., 1995), which were derived from gas chromatographic data, demonstrated that, at least for gas chromatography, quite dissimilar columns must be used to minimize correlation. Since retention in gas chromatography is strongly affected by the solute vapor pressure, this makes finding dissimilar columns a very important part of the method development. Similar concerns exist in MDLC, but the retention mechanisms tend to vary more among different stationary phases than in MDGC. Liu also discusses how to geometrically determine the peak spreading angle for the two-dimensional chromatogram shown in Fig. 2.5. Using the derivation in Liu’s paper, and referring to Fig. 2.5, the effective or “practical’’ peak capacity, Np, is shown to be equal to the product of peak capacities per each dimension (which gives an area) minus the areas A and C. This treatment is practical for many separation systems where there is correlation in the retention mechanisms. However, as pointed out by Gilar et al. (2005), many 2DLC systems don’t have clear demarcated regions where all of the zones occur. For these cases and as a general method, Gilar proposed to use a binning method to identify the effective area utilized over a nonuniform separation space for estimating the orthogonality and consequently the potential peak capacity for peptides in a complex proteomics characterization. This approach has many interesting applications and may see increased use in the future as a separation diagnostics. Another approach to examine orthogonality is to use information theory (IT). Slonecker et al. (1996) used a specific set of target analytes and single column separations on different types of columns to see which combinations lead to minimum correlation and maximal orthogonality. These methods were employed in a farreaching study where techniques like RPLC, supercritical fluid chromatography (SFC), gas–liquid chromatography (GLC), and micellar electrokinetic chromatography (MEKC) with a host of different column materials per technique were examined.

TWO-DIMENSIONAL THEORY OF PEAK OVERLAP

21

The idea here was to examine which pair of techniques and individual columns could lead to the best separations in 2DLC. This is achievable by using 1D separations and then comparing how the retention of each component varies across the separation space. Another innovation here was the use of IT-derived metrics such as information entropy, informational similarity, and the synentropy. As stated in this paper, “The informational similarity of 2D chromatographic systems, H(k,l), is a measure of global solute crowding or dimensional saturation.’’ In addition, this study quantitates the synentropy whereby “a synentropy of 0% describes a 2D chromatographic system whose dimensional retention mechanisms are very different. A synentropy of 100% describes a 2D chromatographic system whose dimensional retention mechanisms are identical.’’ This study showed that similar mechanisms of separation, such as those derived primarily from hydrophobicity, resulted in high values of synentropy. One of the most interesting findings of this study is that the combination of MEKC using an SDS retentive phase and an SFC system with a C1 phase showed an informational similarity of 0 and the 2D chromatogram showed a distinct lack of grouping for the solutes selected. Another study of this type was recently published by Gray et al. (2002) where the same IT techniques were used in a study of RPLC in the first dimension and a carbonaceous phase in the second dimension. This study also incorporated the measures of correlation and spreading angle from Liu’s study. No matter how sophisticated the individual columns are, the columns will be useless for separation in MDLC if their retention mechanisms have a high degree of correlation. Note that some aspects of using IT to describe multidimensional chromatographic separations were shown many years ago by Huber et al. (1979).

2.5 TWO-DIMENSIONAL THEORY OF PEAK OVERLAP The statistical theory of peak overlap (Davis and Giddings, 1983) is a great advance in understanding the true resolving power of chromatography. The theory is simple yet powerful and predicts that the probability of finding a truly resolved single component peak was approximately 13–37% for a complex chromatogram at various levels of saturation. It was well recognized that improvements in this probability of finding single peaks with single component purity would require huge increases in peak capacity. Davis and coworkers (1991, 1993) examined the ramifications of random zones placed in two-dimensional separation spaces. This work discovered that the peak capacity was not as efficiently utilized in two dimensions as opposed to onedimensional separations. Davis (1991) said ‘‘Although more than 10 times the needed peak capacity is available, approximately one-third of the zones are lost in overlap.’’

Davis goes on to say that ‘‘The loss is comparable to that found in 1D gas chromatography of complex mixtures, in which only 2 times or so peak capacity (as defined for maxima) is available than is

22

ELEMENTS OF THE THEORY OF MULTIDIMENSIONAL LIQUID CHROMATOGRAPHY

needed. One really does lose resolving power, per unit peak capacity, when one moves from one dimension to two dimensions.’’

Certainly two-dimensional techniques have far greater peak capacity than onedimensional techniques. However, the two-dimensional techniques don’t utilize the separation space as efficiently as one-dimensional techniques do. These theories and simulations utilized circles as the basis function for a two-dimensional zone. This was later relaxed to an elliptical zone shape for a more realistic zone shape (Davis, 2005) with better understanding of the surrounding boundary effects. In addition, Oros and Davis (1992) showed how to use the two-dimensional statistical theory of spot overlap to estimate the number of component zones in a complex two-dimensional chromatogram. Davis (1993) extended the statistical overlap theory to generalized n-dimensional separations with the consistent result that the separations get much better, but as dimensionality increases the efficiency of using that separation space decreases. For n-dimensional separations, Davis says ‘‘The theory shows that the maximum number of spots per unit capacity and the maximum number of any kind of multiplet per unit capacity both decrease geometrically with increasing n.’’

These are most important realizations that will guide the evolution of multiple dimension chromatographic systems and detectors for years to come. The exact quantitative nature of specific predictions is difficult because the implementation details of dimensions higher than 2DLC are largely unknown and may introduce chemical and physical constraints. Liu and Davis (2006) have recently extended the statistical overlap theory in two dimensions to highly saturated separations where more severe overlap is found. This paper also lists most of the papers that have been written on the statistical theory of multidimensional separations. Other methods based on signal processing algorithms can be used to estimate the number of component zones in 2D complex chromatograms. These are typically based on autocovariance functions and other signal processing algorithms associated with Fourier methods that are applied to experimental data. A good primer on these types of methods in chromatography are discussed by Felinger (1998). Chapter 4 by Dondi, Pietrogrande, Marchetti, and Felinger elaborates on this approach, which has been previously shown to be quite valuable in proteomics applications (Pietrogrande, 2005, 2006). The mathematical and separation mechanics of this method have been previously discussed in detail (Marchetti et al., 2004). These approaches, which originated in the Fourier-space description of complex chromatograms, were used to study the statistical overlap of peaks in one dimension. These techniques can locate order within peaks (in n-dimensional chromatography) and are additionally useful because one can also study the signal and noise properties of the chromatogram as these techniques are firmly based on signal processing. It is expected that these methods may benefit in the future from additional signal processing approaches that utilize nonlinear operators, such as wavelet transforms (Percival and Walden, 2006).

DIMENSIONALITY, PEAK ORDERING, AND CLUSTERING

23

These techniques offer even more interesting noise-filtering properties as they are capable of localizing time- and frequency-varying noise and signal components.

2.6 DIMENSIONALITY, PEAK ORDERING, AND CLUSTERING Giddings (1995) suggested that there was another way to look at what was needed for complex molecule separations and suggested the idea of sample dimensionality. Essentially this metric defines “the number of independent variables that must be specified to identify the components of the sample.’’ The concept is very useful for industrial polymers and some biopolymers. If we take a molecule that has distributions in two separate parts, for example, an alcohol ethoxylate surfactant that has alkyl chain length and polyethylene oxide length distributions, then there are two characteristics that may be resolved chromatographically. In this case, the sample dimensionality, s, is equal to two. This example is well-illustrated in Chapter 18, where the two groups comprising alcohol ethoxylates are in fact well resolved, each in its own chromatographic dimension. Giddings further discussed cases in which order in the chromatographic dimension can allow resolution where disorder clearly creates more stringent requirements for higher peak capacity and higher efficiency. Although polymeric samples may be advantageously separated this way within the confines of a small number of dimensions, extremely complex samples, such as protein tryptic digests that contain peptides, may have too large an s number or may not clearly express higher s in a chemical sense. Hence, the ordering will not occur because the chromatographic response cannot be isolated at the amino acid level. For peptides this might be possible if amino acid sequences were well defined such as 10 cysteines followed by 20 glycines, followed by 6 prolines, and so on. However, the amino acid sequences found in peptides use an almost non-repeating sequence of the amino acids so that the sample dimensionality here is 20 (for the 20 standard amino acids found in peptides), but the resolution of each type may be impossible because of the nonrepeating sequence. Standard chromatographic systems such as RPLC or ion-exchange chromatography would not see the individual amino acid functionality, but would rather discriminate over an average conformational free energy. Even affinity systems of higher dimensionality would be overwhelmed for this style of separation. When retention ordering can be established, the theoretical peak capacity could be effectively utilized in a multidimensional separation system in a far more efficient manner. However, one is reminded that with the exception of synthetic polymers and a few other special cases of small molecules, real samples have almost random retention time distributions. It is rare when the free energy, enthalpy, and entropy of interaction are determined in LC for molecules utilized in retention mechanism studies. However, the retention energetics have been determined in GC studies by Davis et al. (2000) who found that many complex samples will exhibit Poisson distributions of retention times due to a Poisson distribution in enthalpy and a compensating distribution in entropy.

24

ELEMENTS OF THE THEORY OF MULTIDIMENSIONAL LIQUID CHROMATOGRAPHY

2.7 THEORY OF ZONE SAMPLING Unlike the continuous zone development mechanism utilized in a planar separation experiment, comprehensive MDLC is a sequential operation in which finite volumes of eluant are injected into the next dimension column. Because of this finite volume aspect, the mechanism and consequences of sampling eluant from one column with subsequent injection into the next column must be understood. Undersampling would lead to a loss in two-dimensional resolution and oversampling would lead to excessively long run times as the second dimension column would be used in a very inefficient way. The first attempt at understanding the sampling relationship was made by Murphy et al. (1998). Their study examined how segmenting the continuous elution from the first dimension column would affect the resolution of the resulting two-dimensional zone shape. The main assumption made in their study was that there would be complete mixing in the sample loop as each first dimension eluent volume was collected in the sample loop. The continuous nature of the first dimension is truncated into finite volume elements as shown in Fig. 2.6. This truncation operation can be viewed as a mixing or “deseparation’’ process. The simple way to deal with this truncation mathematically is to calculate the width or standard deviation of the truncated zone as a function of the number of segments or “samples’’ across the peak. The clearest way to do this is to use the definition of the standard deviation as the square root of the second moment of the distribution. The explicit definition of the second moment is given in Equation 2.13 under the square root sign and is the integral across the peak of the truncated concentration times the difference squared between

1.00

Concentration

0.75

1 3

0.50

10 0.25 30 0.00 –0.50

–0.25

0.00 0.25 Nondimensional time

0.50

FIGURE 2.6 Segmenting the concentration signal over a Gaussian zone with 1, 3, 10, and 20 samples taken in phase to the main peak. The area under these segmented curves is equal. This figure is taken from Murphy (1998). Reprinted with permission from the American Chemical Society.

THEORY OF ZONE SAMPLING

25

time and the mean time.

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u þ¥ ð pﬃﬃﬃﬃﬃﬃ u u t sS ¼ M2 ¼ ðttÞ2 CS ðtÞdt

ð2:13Þ

¥

where CS is the normalized (unity area) sampled concentration across the first dimension column that lies in the sample loop. In Fig. 2.6, the sampling is shown to be “in-phase.’’ This refers to the zone sampling exactly coinciding with the start and end of the zone. In real applications, this will almost never be the case. Hence, we need to investigate the effect of phase in addition to how the segmentation of the zone leads to different values of sS as a function of the sampling phase. Toward this goal we write the average concentration of solute in the sample loop, hCi, accumulated over a time interval ts which is between ti and ti1 as 1 hCðti1 ; ti Þi ¼ ts

ðti

0

Cðt Þdt

0

ð2:14Þ

ti1

where t0 is the dummy variable of integration. For Gaussian zones the functional form of the Gaussian component is " # 1 ðttÞ2 ð2:15Þ CðtÞ ¼ pﬃﬃﬃﬃﬃﬃ exp 2s2 s 2p Finally, we obtain CS as the sample loop concentration by summing up the individual loop solute concentrations across all time segments as CS ðtÞ ¼

N0 X

fuðtti1 Þuðtti ÞghCðti1 ; ti Þi

ð2:16Þ

i¼1

In Equation 2.16, u(t) is the unit function such that u(t) ¼ 0 when t < 0 and u(t) ¼ 1 when t 0 and N0 is the number of samples across a peak. Equation 2.16 is then substituted into Equation 2.13 to get sS in Equation 2.13, given s in Equation 2.15. It is possible to write Equation 2.13 explicitly but the expression is very complex and reveals nothing in analytical form when written as a function of phase. Therefore, it is very easy to program the integral evaluation of Equation 2.13, as a function of phase, in a programming language such as FORTRAN using numerical techniques to integrate the functions. This has been the approach taken in the sampling study (Murphy et al., 1998). Furthermore, integrals of functions that contain Gaussian functions invariably lead to terms with the error function, erf(x). This function has known approximations (Abamowitz and Stegun, 1965) yet has no analytical form; it is usually evaluated with computer numerical methods so this approach is justified. It is well known that the second moment of a uniform density function (a function of width x and constant height) is simply x2/12 (Abamowitz, 1965). The standard deviation of this function is then the square root of the second moment that is x/3.464. If the number of samples across the peak is 1, as shown in one of the curves

26

ELEMENTS OF THE THEORY OF MULTIDIMENSIONAL LIQUID CHROMATOGRAPHY

Standard deviation relative to gaussian

5 A

4

B

3

C D 2

1 0

4 8 12 Number of samples across peak width

16

FIGURE 2.7 The ratio of ss/s, as a function of the number of samples and as a function of sampling phase, from Murphy (1998). The four phases are explained in the original paper. Reprinted with permission from the American Chemical Society.

given in Fig. 2.6, the width x ¼ 0.8. The standard deviation is then equal to 0.2309. The original (i.e., unsegmented) Gaussian zone in Fig. 2.6 has a standard deviation of 0.1 that gives the ratio of ss /s ¼ 2.309. Hence for a sample rate of 1 with sampling being done completely “in-phase,’’ there will be over two times the zone width, as measured by the standard deviation. In the limit of a large number of samples, the ratio ss /s tends toward 1. This is shown in Fig. 2.7, curve D (in-phase sampling) for one sample across the peak. Figure 2.7 shows the ratio ss /s for four different sampling phases using the numerical procedure described above. It is predicted that the loss in multidimensional resolution would be small as long as approximately four or more collection “samples’’ occur over the peak. Furthermore, the effect of phase between the four phase evaluations in Fig. 2.7 tends to diminish greatly when the number of samples is four or more. The effect of phase on the ratio ss /s for the number of samples across the peak equal to 1, 2, 3, 4, and 5 is given in the original paper where it was shown that variation with phase is greatly diminished once one samples four or more times over a peak width. A number of experimental aspects of sampling are discussed in Chapter 6. Others have looked at the zone sampling problem. The theory of incomplete sampling of the first dimension was studied by Seeley (2002). Carr has also commented about the sampling (Stoll et al., 2006) that few users of 2DLC have sampled the first dimension adequately to minimize the 2D broadening effect from incomplete sampling.

2.8 DILUTION AND LIMIT OF DETECTION One of the disadvantages of the column format for MDC is that the zone is diluted in the first column, the flow is optionally split and then saved in the sample loop with subsequent dilution in the second column separation. The extent of dilution found with columns does

CHEMOMETRIC ANALYSIS

27

not happen in planar format because the whole zone is utilized in both dimensions. This dilution can cause severe, and in a number of cases has caused, insurmountable problems with detection in techniques such as chromatography-field flow fractionation (FFF). Schure (1999) has studied the effect of multidimensional dilution for column-based separations that incorporate chromatography, capillary electrophoresis (CE), and FFF. In all of these cases, the dilution factors are multiplicative; this gives the direct result that the limit of detection for MDC is T m0;inj ¼ 5FVdil

ð2:17Þ

where m*0,inj is the mass limit of detection, F is the ratio of noise amplitude to detector sensitivity, and VTdil is the total dilution volume of the sample. This last term is equal to the original volume of injection times the product of the split ratio per each dimension, the dilution factor in each dimension, and the reciprocal of a function of the number of samples taken across a peak (Schure, 1999). The dilution volume will vary with efficiency (i.e., the number of plates N) and the retention volume, VR, for a single column as pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃ Vdil ¼ VR 2p= N ð2:18Þ For a 25-cm length column with 4.6 mm internal diameter, and other parameters taken to be typical of a modern column, the dilution factor was calculated (Schure, 1999) to be 25. This compares with the dilution factor for 2DLC of 2500 when the split ratio and finite peak sampling is factored in. Other techniques like LC/FFF give dilution factors of nearly half a million. While researchers are enthusiastic about the gain in peak capacity given by 2DLC, this comes at a price for detection limits. This is a critical aspect in applications like biomarker research in proteomics, where many biomarker molecules are thought to be present at trace concentration levels. For these cases extremely sensitive detectors are required.

2.9 CHEMOMETRIC ANALYSIS A number of aspects of chemometric approaches have been discussed by Synovec et al. (2003). This work also covers the topic of run-to-run retention time alignment where it was claimed that variations in temperature, flow rate, column degradation, and other factors cause a lack of reproducibility in 2D chromatographic data (Pierce et al., 2005). This may have been an issue in the past and in MDGC. The present situation, however, shows that tighter instrumental controls and better columns have effectively alleviated this problem as exemplified by Carr and coworkers (2006) who have measured peak retention times with reproducibility to 0.2 s. This number is even more impressive considering that a high-temperature column was utilized within the context of gradient elution. However, it has been noted that to effectively utilize single dimension LC data with MS and MS–MS strategies for proteomics database searching, it was necessary to align both LC and MS data (Jaitly et al., 2006). It is expected the same will be true when 2DLC is used as the separations front-end for database searching when the retention

28

ELEMENTS OF THE THEORY OF MULTIDIMENSIONAL LIQUID CHROMATOGRAPHY

time component is part of the search even with very exacting chromatographic reproducibility. Nonetheless, chemometric methods may be useful for the analysis of highly fused peaks. When multichannel detectors are utilized, this approach can be successful. However, with so many potential peaks occurring within a chromatographic space, the chemometric methods must be automated so as to defuse regions of a chromatogram without intensive operator intervention. This approach is difficult with 1D data and may be even more difficult to automate for higher dimensional data. Nonetheless, chemometric methods may become indispensable when 2DLC is more routinely applied in certain specific turn-key applications like proteomics research where pattern searching may be more important than a comprehensive quantitative analysis and identification of all peaks. A number of chemometric studies have used data obtained from 2DGC (Van Mispelaar et al., 2003; Sinha et al., 2004a, 2004b) but very few studies have utilized chemometric analysis of 2DLC data. One study (Fraga and Corley, 2005) utilized the Generalized Rank Annihilation method (GRAM) and parallel factor analysis (PARAFAC) methods to analyze 2DLC data. 2DLC data has been analyzed using chemometric methods by Rutan and coworkers (Porter et al., 2006) with application to metabolomics. These authors realized that the information content in 2DLC with a multichannel detector is extremely rich. Even though the second dimension separation can remove fusion from a first dimension peak, the multichannel detector can also remove peak fusion in the second dimension. The autocovariance methods discussed above in the “Two-Dimensional Theory of Peak Overlap’’section qualify as chemometric methods and these may become standard tools of MDC. If chemometric tools become routinely available, they may be utilized to classify the performance metrics of MDC systems. This is discussed in Chapter 4. Pattern recognition has not been actively applied to MDLC data but has been effectively utilized for classification of jet fuel mixtures with MDGC (Johnson and Synovec, 2002). For the analysis of zones not resolved well in 2DLC, for example, with higher molecular weight polymers that have wide distributions of both molecular weight and structural and compositional variation, analyzing the zone structure is no longer a matter of finding patterns of peaks. Here the individual zones are fused and their analysis is more directed at composite zone shapes. Many examples of this are shown in Chapter 17. For these types of zones, the “chemical variance’’ method (Vivo-Truyols and Schoenmakers, 2006) has been proposed. This transformation method, which reduces the data matrix rank by an axis transformation method, may be useful for visualizing and characterizing heavily fused zones. This method may also be useful for visualizing crowded 2D chromatograms where zones are resolved but the order is not apparent from direct visualization. Further discussion of chemometric methods used in 2DLC is given in a recent review of fast high-temperature 2DLC methods (Stoll et al., 2007).

2.10 FUTURE DIRECTIONS Many theoretical aspects of MDLC have been studied in the past 20 years and there is no question that future discussions and research along these lines will take place and

FUTURE DIRECTIONS

29

focus on optimizing the speed and resolution of the technique to serve specific applications. Microfluidics-based chromatography columns have been evolving for a number of years in chip format and the whole multidimensional system may be implemented in chip format in the future. There are advantages and disadvantages of chip-based MDLC systems. However, one clear advantage of chip-based systems is the ease with which the valves (i.e., fluidic switches), columns, sample loops, detector ports, flow system, and other components (perhaps even the detector) can be integrated at the chip level. Implementing multidimensional systems with dimension greater than two on chips may be feasible and one may find topology different from the “standard’’ comprehensive multidimensional liquid chromatograph. The term topology here refers to the connection map or graph of the columns. Presently, 2DLC just follows one column with another and turns two 1D linear spaces into a 2D area. There is no reason this connection map must be this simple and networks of separator systems with complex topology should offer unique separation abilities. This logic is somewhat analogous to what the “application specific integrated circuit’’ or ASIC has become in the electronics industry; one customizes the connection of the chip system to serve a specific purpose while the basic elements, be it transistors or in our case columns such as MEKC and CE systems, are present. Ramsey et al. (2003) have described a microfluidic device that incorporates two-dimensional separations of protein digests that incorporates MEKC as the first dimension separator and CE as the second dimension. The valve action is induced electrically, which frees one of the bulky mechanical valves used in larger scale column-based systems. The electrically-driven valve action will enable the additional complexity to be implemented with reliability. A chip-based multidimensional separation of proteins was described (Shadpour and Soper, 2006) where gel electrophoresis was utilized in the first dimension and MEKC was carried out in the second dimension. Additional examples of 2D chip systems are discussed in Chapter 5. From a theoretical point of view there is a great deal of work necessary to understand what this type of system could bring. For example, if a detector is fast and can be multiplexed, many different types of columns can be integrated into the chip system and utilized simultaneously. In addition, separations need not be completely comprehensive, for example, eluent from the second dimension column might be passed to a third column and fourth column at different times and single dimension chromatography be run on these — somewhat like heart cutting zones from the end of a 2DLC system. But the complexity that would be possible on the chip could be high and theory and simulation will be needed to evaluate the possible relationships required to run this sort of separation. It is also quite possible at that stage of separator complexity that even extremely complex samples of biological origin may be sufficiently separated to give single component separations. This would greatly help in the vast dynamic range problem that often makes the search for biomarkers extremely difficult. One example of a tree-based separator system is shown below in Fig. 2.8 where the Bethe lattice or Cayley tree is shown (Wilson, 1996). This graph can be expanded to any number of levels and can function with different types of columns and electrophoretic elements. This is not the only graph that can function as a complex multidimensional separator system. But it is an example of something with multiple

30

ELEMENTS OF THE THEORY OF MULTIDIMENSIONAL LIQUID CHROMATOGRAPHY

2

3

4

1

0

FIGURE 2.8 Separator systems cascaded to form a Bethe lattice or Cayley tree where the point of introduction is the graph vertex 0 and solute can be sampled from any of the outward nodes at position 1, 2, 3, 4, and so on. The sample loops and valves are not shown.

columns and different types of stationary phases that could function for very complex separations. These graphs, in the case of separation science, are directed graphs or digraphs (Bang-Jensen and Gutin, 2002) as the flow is unidirectional and does not change direction. Graph theory provides a number of interesting mathematical properties that may be useful for complex separation models. Trees, graphs, and networks of separator systems can be envisioned to be highly tuned for specific applications. However, a system of this type may also be ideal for methods development, as a general system with different stationary phases could be utilized adaptively for iterative method development. Besides offering the complexity needed for complex separations, chip-based systems may also be conducted with an energy input lower than the traditional externally integrated column system. Some of the aspects to this type of chip system, which incorporate discrete chromatographic media, were recently discussed (Wirth, 2007). The practicality of this approach will become apparent in the next decade as applications will drive the demand for this type of system. REFERENCES Abamowitz, M., Stegun, I.A., editors. (1965). Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Dover Publications, New York. Bang-Jensen, J., Gutin, G. (2002). Digraphs: Theory, Algorithms and Applications. Springer, New York.

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Giddings, J.C. (1995). Sample dimensionality: a predictor of order–disorder in component peak distribution in multidimensional separation. J. Chromatogr. A 703, 3–15. Gilar, M., Olivova, P., Daly, A.E., Gebler, J.C. (2005). Orthogonality of separation in twodimensional liquid chromatography. Anal. Chem. 77, 6426–6434. Gray, M., Dennis, G.R., Wormell, P., Shalliker, R.A., Slonecker, P. (2002). Two-dimensional reversed phase separations: isomeric separations incorporating C18 and carbon clad zirconia stationary phases. J. Chromatogr. A 975, 285–297. Grushka, E. (1970). Chromatographic peak capacity and the factors influencing it. Anal. Chem. 42, 1142–1147. Grushka, E. (1984). Effect of high speed on peak capacity in liquid chromatography. J. Chromatogr. 316, 81–93. Guiochon, G., Beaver, L.A., Gonnord, M.F., Siouffi, A.M., Zakaria, M. (1983a). Theoretical investigation of the potentialities of the use of a multidimensional column in chromatography. J. Chromatogr. 255, 415–437. Guiochon, G., Gonnord, M.F., Zakaria, M., Beaver, L., Siouffi, A.M. (1983b). Chromatography with a two-dimensional column. Chromatographia 17, 121–124. Horie, K., Kimura, H., Ikegami, T., Iwatsuka, A., Saad, N., Fiehn, O., Tanaka, N. (2007). Calculating the optimal modulation periods to maximize the peak capacity in twodimensional HPLC. Anal. Chem. 79, 3764–3770. Horvath, C.G., Lipsky, S.R. (1967). Peak capacity in chromatography. Anal. Chem. 39, 1893– 1893. Huber, J.F.K., Kenndler, E., Reich, G. (1979). Quantification of the information content of multi-dimensional gas chromatography and low resolution mass spectrometry in the identification of doping drugs. J. Chromatogr. 172, 15–30. Karger, B.L., Snyder, L.R., Horvath, C. (1973). An Introduction to Separation Science. WileyInterscience, New York, Chapter 19, pp. 560–564. Jaitly, N., Monroe, M.E., Petyuk, V.A., Clauss, T.R.W., Adkins, J.N., Smith, R.D. (2006). Robust algorithm for alignment of liquid chromatography–mass spectrometry analyses in an accurate mass and time tag data analysis pipeline. Anal. Chem. 78, 7397–7409. Johnson, K.J., Synovec, R.E. (2002). Pattern recognition of jet fuels: comprehensive GC GC with ANOVA-based feature selection and principal component analysis Chemom. Intell. Lab. Syst. 60, 225–237. Lewis, K.C., Opiteck, G.J., Jorgenson, J.W., Sheeley, D.M. (1997). Comprehensive on-line RPLC-CZE-MS of peptides. J. Am. Soc. Mass Spectrom. 8, 495–500. Liu, S., Davis, J.M. (2006). Dependence on saturation of average minimum resolution in twodimensional statistical–overlap theory: peak overlap in saturated two-dimensional separations. J. Chromatogr. A 1126, 244–256. Liu, Z., Patterson, D.G., Lee, M.L. (1995). Geometric approach to factor analysis for the estimation of orthogonality and practical peak capacity in comprehensive two-dimensional separations. Anal. Chem. 67, 3840–3845. Malinowski, E.R. (2003). Factor Analysis in Chemistry. 3rd edition. John Wiley & Sons, Inc., New York. Marchetti, N., Felinger, A., Pasti, L., Pitrogrande, M.C., Dondi, F. (2004). Decoding twodimensional complex multicomponent separations by autocovariance function. Anal. Chem. 76, 3055–3068.

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Q1

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Martin, M., Herman, D.P., Guiochon, G. (1986). Probability theory for number of mixture components resolved by n independent columns. Anal. Chem. 58, 2200–2207. Merenbloom, S.I., Bohrer, B.C., Koeninger, S.L., Clemmer, D.E. (2007). Accessing the peak capacity of IMS–IMS separations of tryptic peptide ions in He at 300 K. Anal. Chem. 79, 515–522. Murphy, R.E., Schure, M.R., Foley, J.P. (1998). Effect of sampling rate on resolution in comprehensive two-dimensional liquid chromatography. Anal. Chem. 70, 1585–1594. O’Farrell, P.H. (1975). High resolution two-dimensional electrophoresis of proteins. J. Biol. Chem. 250 (10), 4007–4021. Oros, F.J., Davis, J.M. (1992). Comparison of statistical theories of spot overlap in twodimensional separations and verification of means for estimating the number of zones. J. Chromatogr. 591, 1–18. Percival, D.B., Walden, A.T. (2006). Wavelet Methods for Time Series Analysis, Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, New York. Peters, S., Vivo-Truyols, G., Marriott, P.J., Schoenmakers, P.J. (2007). Development of a resolution metric for comprehensive two-dimensional chromatography. J. Chromatogr. A 1146, 232–241. Pierce, K.M., Wood, L.F., Wright, B.M., Synovec, R.E. (2005). A comprehensive twodimensional retention time alignment algorithm to enhance chemometric analysis of comprehensive two-dimensional separation data. Anal. Chem. 77, 7735–7743. Pietrogrande, M.C., Marchetti, N., Dondi, F., Righetti, P.G. (2006). Decoding 2D-PAGE complex maps: relevance to proteomics. J. Chromatogr. B 833, 51–62. Pietrogrande, M.C., Marchetti, N., Tosi, A., Dondi, F., Righetti, P.G. (2005). Decoding twodimensional polyacrylamide gel electrophoresis complex maps by autocovariance function: a simplified approach for proteomics. Electrophoresis 26, 2739–2748. Porter, S.E.G., Stoll, D.R., Rutan, S.C., Carr, P.W., Cohen, J.D. (2006). Analysis of four-way two-dimensional liquid chromatography-diode array data: application to metabolomics. Anal. Chem. 78, 5559–5569. Ramsey, J.D., Jacobson, S.C., Culbertson, C.T., Ramsey, J.M. (2003). High efficiency, twodimensional separations of protein digests on microfluidic devices. Anal. Chem. 75, 3758– 3764. Ruotolo B.T., Gillig, K.J., Stone, E.G., Russell, D.H. (2002). Peak capacity of ion mobility spectrometry: separation of peptides in helium buffer gas. J. Chromatogr. B 782, 385–392. Schoenmakers, P.J. (1986). Optimization of chromatographic selectivity. J. Chromatogr. Lib. 35, Elsevier Science Publishers, Amsterdam. Schure, M.R. (1997). Quantification of resolution for two-dimensional separations. J. Micro. Sep. 9, 169–176. Schure, M.R. (1999). Limit of detection, dilution factors, and technique compatibility in multidimensional chromatography, capillary electrophoresis, and field-flow fractionation. Anal. Chem. 71, 1645–1657. Seeley, J.V. (2002). Theoretical study of incomplete sampling of the first dimension in comprehensive two-dimensional chromatography. J. Chromgr. A 962, 21–27. Shadpour, H., Soper, S.A. (2006). Two-dimensional electrophoretic separation of proteins using poly(methyl methacrylate) microchips. Anal. Chem. 78, 3519–3527.

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Shen, Y., Tolic, N., Zhao, R., Pasa-Tolic, L., Li, L., Berger, S.J., Harkewicz, R., Anderson, G.A., Belov, M.E., Smith, R.D. (2001). High-throughput proteomics using high efficiency multiple-capillary liquid chromatography with on-line high performance ESI FTICR mass spectrometry. Anal. Chem. 73, 3011–3021. Shi, W., Davis, J.M. (1993). Test of theory of overlap for two-dimensional separations by computer simulations of three-dimensional concentration profiles. Anal. Chem. 65, 482–492. Sinha, A.E., Hope, J.L., Prazen, B.J., Fraga, C.G., Nilsson, E.J., Synovec, R.E. (2004a). Multivariate selectivity as a metric for evaluating comprehensive two-dimensional gas chromatography-time-of-flight mass spectrometry subjected to chemometric peak deconvolution. J. Chromatogr. A 1056, 145–154. Sinha, A.E., Fraga, C.G., Prazen, B.J., Synovec, R.E. (2004b). Trilinear chemometric analysis of two-dimensional comprehensive gas chromatography–time-of-flight mass spectrometry data. J. Chromatogr. A 1027, 269–277. Slonecker, P.J., Li, X., Ridgway, T.H., Dorsey, J.G. (1996). Informational orthogonality of two-dimensional chromatographic separations. Anal. Chem. 68, 682–689. Stoll, D.R., Cohen, J.D., Carr, P.W. (2006). Fast, comprehensive online two-dimensional high performance liquid chromatography through the use of high temperature ultra-fast gradient elution reversed-phase liquid chromatography. J. Chromatogr. A 1122 (1–2), 123–137. Stoll, D. R., Li, X., Wang, X., Carr, P. W., Porter, S.E.G., Rutan, S.C. (2007). Fast comprehensive two-dimensional liquid chromatography. J. Chromatogr. A 1168 (1–2), 3–43. Synovec, R.E., Prazen, B.J., Johnson, K.J., Fraga, C.G., Bruckner, C.A. (2003). Chemometric analysis of comprehensive two-dimensional separations. In: Advances in Chromatography, Vol. 42. Grushka, E., Brown, P.R., editors. Marcel Dekker, New York, pp. 1–42. Valentine, S.J., Kulchania, M., Srebalus Barnes, C.A., Clemmer, D.E. (2001). Multidimensional separations of complex peptide mixtures: a combined high-performance liquid chromatography/ion mobility/time-of-flight mass spectrometry approach. Int. J. Mass Spectrom. 212, 97–109. Van Mispelaar, V.G., Tas, A.C., Milde, A.K., Schoenmakers, P.J., van Asten, A.C. (2003). Quantitative analysis of target components by comprehensive two-dimensional gas chromatography. J. Chromatogr. A 1019, 15–29. Vivo-Truyols, G., Schoenmakers, P.J. (2006). Chemical variance, a useful tool for the interpretation and analysis of two-dimensional chromatograms. J. Chromatogr. A 1120, 273–281. Wang, X., Stoll, D.R., Carr, P.W., Schoenmakers, P.J. (2006). A graphical method for understanding the kinetics of peak capacity production in gradient liquid chromatography. J. Chromatogr. A 1125, 177–181. Wilson, R.J. (1996). Introduction to Graph Theory. Addison Wesley, Essex, England. Wirth, M.J. (2007). Separation media for microchips. Anal. Chem. 79, 800–808.

3 PEAK CAPACITY IN TWO-DIMENSIONAL LIQUID CHROMATOGRAPHY Joe M. Davis Department of Chemistry and Biochemistry, Southern Illinois University at Carbondale, Carbondale, IL 62901–4409, USA

3.1 INTRODUCTION The peak capacity of a one-dimensional separation was quantified first by Giddings (1967), who defined it as the maximum number of peaks separable in a specified region of space. The concept was generalized to two-dimensional (2D) separations by Guiochon et al. (1982, 1983) and also by Giddings (1984, 1987). In the generalizations, the peak capacity of a 2D separation is equal to, or slightly less than (Guiochon et al., 1982, 1983), the product of the one-dimensional peak capacities of the two dimensions. Peak capacities are reported routinely as figures of merit in 2D separations, including two-dimensional liquid chromatography or 2DLC although these concepts are general and apply to all two-dimensional separations including column mode and planar mode. The concept was extended to include mass spectrometry as a third dimension (Lewis et al., 1997; Shen et al., 2001; Valentine et al., 2001); see additional references in the previous theory chapter. A minor problem with 2D peak capacity is that it is based on a square or rectangular tessellation (Guiochon et al., 1982) instead of a cubic closest packed

Multidimensional Liquid Chromatography: Theory and Applications in Industrial Chemistry and the Life Sciences, Edited by Steven A. Cohen and Mark R. Schure. Copyright 2008 John Wiley & Sons, Inc.

35

36

PEAK CAPACITY IN TWO-DIMENSIONAL LIQUID CHROMATOGRAPHY

arrangement, which is the most condensed packing possible (Cotton and Wilkinson, 1972). The major problem with 2D peak capacity as a figure of merit, however, is the correlation of separation mechanisms, which causes only a fraction of the 2D space to be used. The full use of the peak capacity is possible only if the mechanisms are orthogonal, that is, if the separative displacements in the two dimensions are independent. The means to attain orthogonality have been reported in two-dimensional gas chromatography (2DGC) (Venkatramani et al., 1996) but not in 2DLC, where current success is largely empirical. Correlation was attributed by Giddings to a sample dimensionality (i.e., the number of independent variables needed to identify sample components) that is smaller than the separation dimensionality (Giddings, 1995). A few means have been proposed to determine the amount of space actually used in a 2D separation and the degree of orthogonality. Giddings qualitatively addressed the effects of correlation on the used space (Giddings, 1984). The large number of orthogonal separative displacements suggested by him was reduced markedly by Slonecker et al. (1996), who examined 105 types of 2D separations with concepts from information theory (e.g., entropy, mutual information, synentropy, and informational similarity). They showed that orthogonality among mechanisms is rare in 2D separations and correlation is common. Liu et al. (1995) proposed the means based on factor analysis to calculate the effective angle spanned by retention-time coordinates in 2D separations. The angle, which can vary from 0 to 90 , measures the correlation and was used to define a practical peak capacity. Jandera et al. (2004) suggested that the peak capacity of a 2D separation should be calculated as a weighted sum of a purely orthogonal capacity, based on the multiplication of onedimensional capacities, and a purely serial capacity, based on the square root of the sum of squares of one-dimensional capacities. The weighting factor was the ratio of free-energy differences of repeat structural units in the two dimensions. Gilar et al. (2005a) qualitatively related orthogonality to the linear correlation coefficient of 2D retention-time coordinates and defined the practical peak capacity as the product of the 2D peak capacity and the fraction of peak-capacity units occupied by coordinates (Gilar et al., 2005b). They also introduced a measure of orthogonality. Only a few researchers have used these concepts, however (Gray et al., 2002, 2003, 2004, 2005), and most 2D separations that have been described as orthogonal have not been evaluated as such. This chapter examines another measure of the space used by 2D separations subject to correlation. Some researchers use the words, peak capacity, to express the maximum number of zones separable under specific experimental conditions, regardless of what fraction of the space is used. By definition, however, the peak capacity is the maximum number of separable zones in the entire space. No substantive reason exists to change this definition. The ability to use the space, however, depends on correlation. In deference to previous researchers (Liu et al., 1995; Gilar et al., 2005b), the author adopts the term, practical peak capacity, to describe the used space. The practical peak capacity is the peak capacity, when the separation mechanisms are orthogonal, but is less than the peak capacity when they are not. The subsequent discussion is based on practical peak capacity.

THEORY

37

3.2 THEORY The question posed here is: What is an appropriate measure of practical peak capacity in 2D separations, particularly in 2DLC? Figure 3.1a, c, and e shows graphs of retention-time coordinates in three model systems of comprehensive 2DLC, which were mimicked to answer this question. The coordinates are uniformly random but also correlated, that is, the second-dimension retention time tr,2 depends on its first-dimension counterpart, tr,1. In Figure 3.1a, the coordinates form a wedge, with tr,2 confined between two lines of nonzero slope that intersect the space’s right-hand side. In Figure 3.1c, the coordinates form a fan, with tr,2 confined between two lines of nonzero slope for small tr,1 and between the line of smaller slope and an upper limiting value for large tr,1. In Fig. 3.1e, the coordinates lie within a parallelogram, with the maximum tr,1 and tr,2 values associated with a single retention-time coordinate. While no graph exactly mimics coordinates in 2DLC, all exhibit the correlated increase of tr,2 with tr,1 that often is observed. Figure 3.1b, d, and f shows graphs similar to Figure 3.1a, c, and e, but the coordinates lie in a reduced square having unit dimensions. The reduced coordinates x and y are x ¼ ðtr;1 t0;1 Þ=ðtrmax ;1 t0;1 Þ

ð3:1aÞ

y ¼ ðtr;2 t0;2 Þ=ðtrmax ;2 t0;2 Þ

ð3:1bÞ

where t0,1 and t0,2 are the void times in the first and second dimensions, as shown in Fig. 3.1a, c, and e. Times trmax ;1 and trmax ;2 are the maximum retention times in the first and second dimensions, such that 0 x 1 and 0 y 1. These reduced graphs are similar to those of other researchers (Slonecker et al., 1996; Gilar et al., 2005b). The geometrical restrictions on the reduced times are imposed by various lines having slopes b1 and b2 (b1 5 b2) and by the square’s boundaries. The slopes themselves have restricted ranges, which are reported in the figures. The spaces defined by the coordinate patterns in Fig. 3.1b, d, and f are called WEG (wedge), FAN (fan), and PAR (parallelogram). Any measure of the coordinate correlation is arbitrary. Here, the linear correlation coefficient r is used, largely because it is familiar. It measures the quality of a leastsquares fit of a line to coordinates, with a magnitude that varies from 0 (for uncorrelated random coordinates) to 1 (for fully correlated coordinates lying on the line). For the correlated coordinates in Fig. 3.1b, d, and f, theoretical relationships for r, that is, r ¼ f (b1, b2) can be derived as shown in Appendix 3B. They show that r lies between the inclusive bounds of 1/2 and 1 for WEG, and between the inclusive bounds, 0 and 1, for FAN and PAR. The practical peak capacity of these spaces can be characterized by 2D statistical overlap theory. Consider a relatively simple problem of probability. If, on average, m circles of diameter d0 are distributed randomly in a large area A, then the average number p of clusters of isolated and overlapping circles approaches (Roach, 1968)

38

PEAK CAPACITY IN TWO-DIMENSIONAL LIQUID CHROMATOGRAPHY

FIGURE 3.1 (a, c, and e) Mimics of correlated retention times tr,1 and tr,2 in comprehensive 2DLC. (b, d, and f) Reduced retention-time coordinates x and y. Equations of lines defining spatial boundaries are reported.

THEORY

p ¼ 4amexpð4aÞ=½1expð4aÞ

39

ð3:2Þ

where saturation a is P 02 =ð4AÞ ¼ m=n a ¼ mpd

ð3:3aÞ

and nP is the number of circles of area pd20 =4 that can fit in A without consideration of packing constraints. In other words, it is a geometric measure of the number of spatial nP decreases, with a resultant increase of units available. As a increases for constant m, clustering and reduction of p. In statistical-overlap theory, d0 represents the contour of single-component peaks (i.e., peaks produced by pure compounds), p represents the average number of observable peaks (with each circle cluster representing a group of overlapping single-component peaks and isolated circles representing isolated singlecomponent peaks), and nP represents one measure of peak capacity, when A is both rectangular and fully used. Equation 3.2 was proposed by Roach (1968) almost 40 years ago to model the overlap of coal particulates sampled from air onto a flat surface. The equation was verified by studying the clustering of randomly distributed circles in a square representing the reduced space of a 2D separation (Oros and Davis, 1992). It then was modified (Rowe and Davis, 1995) to study the clustering of inhomogeneous random distributions of circles (Rowe et al., 1995; Davis, 2004), in which more circles are found in parts of the reduced square than in others, and to address the clustering of ellipses and reduction of clustering that occurs near the reduced-square boundaries (Davis, 2005). For simplicity, only Equation 3.2 is used in this chapter. Obviously, zones in 2DLC are not circles; they are three-dimensional profiles (e.g., bi-Gaussians) having variable breadths and heights. With the identity, d0 ¼ 4sR*s , where R*s is the average minimum resolution between single-component peaks required for separation, Equation 3.2 can be used to model the average number p randomly of peak maxima in a rectangular area A containing (on average) m distributed bi-Gaussians having equal standard deviations s in both dimensions (Shi and Davis, 1993). The substitution of this identity into Equation 3.3a produces * 2 a ¼ 4mpðsR s Þ =A

ð3:3bÞ

Studies of both homogeneous (Shi and Davis, 1993) and inhomogeneous (Rowe and Davis, 1995) random distributions of bi-Gaussians were made to determine values of R*s at low a. These studies, however, are superceded by a recent development. Figure 3.2 is a graph of R*s versus a for 2D separations, as predicted by theory (Liu and Davis, 2006). The graph's signiﬁcance is discussed brieﬂy. Consider two neighboring peaks, each of which may contain more than one compound. Let B and C represent the two compounds closest to each other, with B in one peak and C in the other. Clearly, B and C are resolved. However, if efﬁciency is degraded and the two peaks overlap to form a single peak, then B and C are not resolved. Theory shows the minimum resolution of B and C decreases with increasing peak multiplicity (i.e., the number of compounds per peak). Since Rs is a weighted combination of all possible minimum resolutions of B and C, it also decreases with increasing multiplicity, that is, increasing a. At large a, it even is less than 0.5 (the smallest resolution separating two

40

PEAK CAPACITY IN TWO-DIMENSIONAL LIQUID CHROMATOGRAPHY

0.8 Rs* 0.7

0.6

0.5

0.4

0

0.2

0.4

a

0.6

0.8

1

FIGURE 3.2 Graph of average minimum resolution Rs versus saturation a for 2D separations of randomly distributed bi-Gaussian single-component peaks. Reprinted from Liu and Davis, copyright (2006), with permission from Elsevier.

Gaussians or bi-Gaussians of equal standard deviation and height), since the likelihood of multiplicity is high. The value of Rs at a ¼ 0 is 0.726, the average minimum resolution required to separate two Gaussian or bi-Gaussian single-component peaks having equal standard deviations and exponentially distributed heights (Creten and Nagels, 1987; El Fallah and Martin, 1987; Schure, 1991; Felinger, 1997). For complex mixtures, this height distribution has a basis that is both theoretical (Pietrogrande et al., 2000) and experimental (Nagels et al., 1983; Nagels and Creten, 1985; Dondi et al., of bi-Gaussians in area A, 1986; Pietrogrande et al., 2000). For a speciﬁed number m with standard deviation s in both dimensions and exponentially distributed heights, Equation 3.3b and Fig. 3.2 uniquely determine a (i.e., for any m , A, and s, only one value of Rs determines a from Equation 3.3b, with the coordinate (a, Rs ) also lying on the graph in Fig. 3.2). A similar theory exists for Rs in one dimension (Davis, 1997a, b). It is important to realize that statistical-overlap theory is not constrained by the contour of area A, which does not have to be rectangular as in earlier studies (in addition to previous references, see Davis, 1991; Martin, 1991, 1992). In other words, Equations 3.2 and 3.3 should apply to the spaces WEG, FAN, and PAR. In this chapter, the number of clusters of randomly distributed circles in such areas is compared to the predictions of Equations 3.2 and 3.3a to assess the relationship between nP and practical peak capacity. Similarly, the number of peak maxima formed by randomly distributed bi-Gaussians in such areas is compared to the predictions of Equations 3.2 and 3.3b, and to Fig. 3.2, to make another assessment. In general, statistical-overlap theory has been developed and applied to several systems, mostly one-dimensional ones. Two reviews summarize many developments (Pietrogrande et al., 2000; Felinger and Pietrogrande, 2001). With regards to other multidimensional separations, theories have been reported for n-dimensional

PROCEDURES

41

separations (Martin, 1991, 1995; Davis, 1993) and column switching (Peters and Davis, 1998; Samuel and Davis, 2002). Dondi and coworkers have proposed means to interpret peak overlap in 2D electrophoretic gels by partitioning them into narrow strips, to which one-dimensional theory is applied (Pietrogrande et al., 2002, 2003; Campostrini et al., 2005). In addition, they derived a theory based on autocorrelation methods (Marchetti et al., 2004), which addresses the correlation of zone centers due to similarities of chemical structure. Such correlations are not considered here.

3.3 PROCEDURES The details of computing correlated coordinates (x, y) in spaces WEG, FAN, and PAR are of marginal interest to most of the readers. They are reported in Appendix 3A. Limiting expressions of r for large numbers of coordinates in WEG, FAN, and PAR were calculated from Equation 3.B4 and expressions described in Appendix 3B for a family of b1 and b2 values and were tested by simulation. Specifically, 500 (x, y) random correlated coordinates were distributed over WEG, FAN, and PAR for specific b1 and b2, and r was calculated from the usual statistics equation (Eq. 3.B1 in Appendix 3B). The simulation was repeated 1000 times, and the average r was compared to the limiting expressions. For specific b1, b2 and r chosen as described below, two types of simulations were carried out in WEG, FAN, and PAR. In the first, the number of clusters formed from ¼ 500 circles centered at random correlated coordinates was counted in 250 m simulations for a ¼ 0.05, 0.10, 0.15, . . . , and 1.0. The areas A are reported in Appendix 3A; these areas and Equation 3.3a determined circle diameter d0. Circles having centers lying within d0 were clustered. The average number of clusters then was compared to p, Equation 3.2. In the second type of simulation, the number of peak ¼ 500 bi-Gaussians, having equal standard deviations s in both maxima formed by m dimensions and exponentially distributed heights, was counted in 100 simulations having the same saturations and coordinates as before. At any a, s was chosen to satisfy both Equation 3.3b and Fig. 3.2. Single-component peaks were computed on square grids extending from the maximum height by 6 s and containing nodes spaced by s/5. The heights of single-component peaks were exponentially random. After summing all proﬁles, peak maxima were identiﬁed by nodes having greater intensities than their eight surrounding neighbors. The average number of maxima was compared to p, Equation 3.2. In both simulations, the actual number m of coordinates in a and standard simulationpwas ﬃﬃﬃﬃ chosen in accordance with Poisson statistics (i.e., mean m ), using the Box–Muller transform (Press et al., 1992) as an approximadeviation m tion to the Poisson distribution. For simulations in WEG, b2 was set to 1 and Equation 3.B4 in Appendix 3B was solved for b1 at r ¼ 0.55, 0.60, 0.65, . . . , and 0.95. For simulations in FAN, equations described in Appendix 3B were solved with b1 ¼ 1/b2 at r ¼ 0.1, 0.2, 0.3, . . . , and 0.9. For simulations in PAR, arbitrary b2 values were chosen and equations described in Appendix 3B were solved for b1 at r ¼ 0.1, 0.2, 0.3, . . . , and 0.9.

42

PEAK CAPACITY IN TWO-DIMENSIONAL LIQUID CHROMATOGRAPHY

The random number generator in all simulations was RAN3 (Press et al., 1992). Computations were made in FORTRAN 90. To reduce cancellation errors, theoretical values of r for PAR were computed in double precision.

3.4 RESULTS AND DISCUSSION Figure 3.3a–c shows graphs of correlation coefficient r versus b2 for different b1 in WEG, FAN, and PAR. The curves are limiting values of r evaluated as described in Appendix 3B. The symbols are the average r’s from 1000 simulations of 500 random correlated coordinates for different b1 and b2. The agreement is excellent, except for r 5 0.1. The small deviation probably is caused by minor imperfections in the random number generator. Some trends merit comment. For all three spaces, r increases with increasing b1 for appropriate b2, because coordinates are distributed over increasingly confined spaces. Also, as b1 approaches b2, r approaches 1 because coordinates lie along a line. For WEG, r equals 1/2 for all positive b2, when b1 ¼ 0. For FAN, r approaches constant values as b2 approaches infinity because coordinates fill the entirety of the upper lefthand side of the space. This trend is also found in PAR. Also for PAR, r has a very shallow minimum for large b1 (e.g., 0.7 and 0.9), prior to increasing slightly with increasing b2 and then approaching an asymptote. The simulation results also show this trend, but the precisions do not establish significance. It is evident from Fig. 3.3 that the same r can be found in all three spaces for appropriate b1 and b2. Some judgment was required in choosing b1 and b2 for further study. In all cases, b2 was chosen and b1 was evaluated for periodically spaced r’s using equations reported or described in Appendix 3B. Specifically, for WEG, b2 was assigned the value 1, such that the line of greater slope intersected the upper right-hand corner of the space. For FAN, b2 was assigned the value 1/b1 to produce a symmetrical space (i.e., one having equal extents at the right vertical and upper horizontal boundaries). For PAR, b2 was selected arbitrarily to produce spaces somewhat resembling real 2DLC chromatograms. The b1 and b2 values for different r’s are reported in Tables 3.1, 3.2, and 3.3 for WEG, FAN, and PAR, respectively. Figures 3.4a-d, 3.5a-d and 3.6a-d show graphs of coordinates in WEG, FAN, and PAR, respectively, for different r, showing the effects of correlation on the coordinate distributions. Figures 3.4e and f, 3.5e and f, and 3.6e and f show contour images of biGaussians in these spaces for specific r and a. Although all bi-Gaussians have equal standard deviations in both dimensions, overlap causes observed spots to be oriented in differentdirections.Furthermore,thebi-Gaussianshavinglargeheightsformlargerspots. Figure 3.7a–f shows graphs of p versus a determined from simulations in WEG, FAN, and PAR, with p equal to the average numbers of circle clusters in Fig. 3.7a, c, and e, and equal to the average numbers of peak maxima in Fig. 3.7b, d, and f. The curve is the Roach equation, Equation 3.2. The symbols represent simulation results obtained for different r’s. In all cases, a good agreement exists between simulation and theory, with p values for simulations at different r almost superposing at the same a. The agreement is best at small r and a, simply because the spaces are larger and

(a)

WEG

b1 = 0.9

1 r

b1 = 0.7

b1 = 0.5

0.9 b1 = 0.3 0.8 b1 = 0.1

0.7 0.6 0.5

b1 = 0 0.4 0

0.2

0.4

1

0.6

b2

0.8

1

(b) FAN

r 0.8

b1 = 0.7

0.6

0.4

b1 = 0.9 b1 = 0.5 b1 = 0.3

0.2

b1 = 0.1

b1 = 0 0 0

5 (c)

10

b2

15

20

25

20

25

b1 = 0.9

PAR

1 r

b1 = 0.7 b1 = 0.5

0.8

0.6 b1 = 0.3 0.4 b1 = 0.1

0.2 b1 = 0

0 0

5

10

b2

15

FIGURE 3.3 Graph of linear correlation coefficient r versus b2 for various b1 in (a) WEG, (b) FAN, and (c) PAR. Curves are predictions; points are simulation averages. 43

44

PEAK CAPACITY IN TWO-DIMENSIONAL LIQUID CHROMATOGRAPHY

FIGURE 3.4 (a–d) Graphs of 500 coordinates in WEG for different r’s. (e and f) Contour images of 500 bi-Gaussians in WEG for specified r and a.

RESULTS AND DISCUSSION

FIGURE 3.5

As in Fig. 3.4, but for FAN.

45

46

PEAK CAPACITY IN TWO-DIMENSIONAL LIQUID CHROMATOGRAPHY

FIGURE 3.6 As in Fig. 3.4, but for PAR.

RESULTS AND DISCUSSION

47

FIGURE 3.7 Graphs of p versus a. Curves are Roach equation, Equation 3.2. Symbols are simulation averages for various r’s, with p equalling average numbers of circle clusters in (a), (c) and (e), and average numbers of maxima in (b), (d), and (f).

“boundary effects’’ are less pronounced (Davis, 2005). Also, the agreement is slightly better for circle clusters than for maxima at large a but the difference is small. The need to account for the variation of Rs with a is apparent in the insert to Fig. 3.7b , which shows the large deviation between simulation and theory that results when Rs is assigned its limiting value at a ¼ 0, 0.726. The agreement provides the answer to the question posed earlier in the chapter: a measure of practical peak capacity, as assessed by statistical-overlap theory, is nP

48

PEAK CAPACITY IN TWO-DIMENSIONAL LIQUID CHROMATOGRAPHY

TABLE 3.1 Values of b1 for Different r’s in WEG Calculated from Equation 3.B4 r

b1

0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95

0.06570 0.1301 0.1940 0.2586 0.3252 0.3957 0.4730 0.5630 0.6810

b2 ¼ 1.

TABLE 3.2 r’s in FAN r 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Values of b1 and b2 for Different b1

b2

0.09420 0.1792 0.2581 0.3333 0.4070 0.4811 0.5587 0.6444 0.7500

10.62 5.580 3.874 2.999 2.457 2.079 1.790 1.552 1.333

b2 ¼ 1/b2.

TABLE 3.3 r’s in PAR r 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Values of b1 and b2 for Different b1

b2

0.0324 0.1021 0.0929 0.2171 0.0206 0.2955 0.4419 0.5602 0.6831

15.0 10.0 5.0 5.0 2.75 2.75 2.75 2.75 2.75

b2's were chosen arbitrarily.

CONCLUSIONS

49

itself, which is the geometrical fraction of the peak capacity actually used for separation. It is not the product of the two independent one-dimensional peak capacities, unless the coordinates are uncorrelated. This should be true, even if coordinates are not random and the Roach equation does not apply. It also should be true, even though real 2DLC chromatograms are more complicated than the model systems studied here. It is instructive to compare nP to other geometrical definitions of practical peak capacity. The practical peak capacity of Liu et al. (1995) is related to the area between two lines passing through the origin and intersecting the upper- and righthand sides of the 2D space, with an angle d ¼ cos1r between the lines. Therefore, the space of Liu et al. resembles FAN. For FAN at r ¼ 0.7, d is 45.6 , which is larger than the actual angle, tan1 b2 tan1 b1 ¼ 31.6 . Therefore, the practical peak capacity of Liu et al. is larger than nP, and this trend is also found for other r’s. The practical peak capacity of Gilar et al. (2005b) is the number of peak-capacity units that are occupied by retention times. Unlike here, only the occupied units in the relevant geometry are counted, and their number depends on saturation. From their work, one can deduce that the fraction of area A containing homogeneous random retention times in area units of pd20 =4 (circular or otherwise) is 1 exp(a). Therefore, the practical peak capacity of Gilar et al. for homogeneous random retention times is nP[1 exp(a)]. These observations are not made as criticisms but to show the inconsistency among different geometrical approaches to defining practical peak capacity. Even the suggestion that it should be defined by nP is simplistic except in ideal cases. Suppose a few 2D coordinates lie outside an otherwise well-defined space containing random coordinates. What is the practical peak capacity here; that is, how does one define the area associated with the outlying, isolated coordinates? Similarly, suppose the distribution of 2D coordinates is inhomogeneously random, with many coordinates in part of the 2D space and only a few coordinates in another part of equal area. Are these two areas “used’’ equally by coordinates in defining practical peak capacity? Statistical-overlap theory suggests the answer is “yes,’’ but the author thinks this answer is not the best one.

3.5 CONCLUSIONS The author anticipates that many readers will find the results reported here to be commonplace. If so, then why do we so often report the individual peak capacities of the two dimensions and their product as the 2D peak capacity? One answer—the conservative one—is that the latter is indeed the maximum number of peaks that can be separated, in agreement with the definition. A more realistic answer is that it is easy to do and appears more impressive than it really is—especially to those who fund our work. In fact, as a practical metric it is often nonsense. Because orthogonality is so difficult to achieve, especially in 2DLC, the peak capacity is a measure of only instrumental potential, not of separation potential, and consideration of

50

PEAK CAPACITY IN TWO-DIMENSIONAL LIQUID CHROMATOGRAPHY

the instrumentation alone is deceptive. In such a situation, peak capacity has no significance; only practical peak capacity does. We should distance ourselves from peak capacity in 2DLC, except as a gauge of instrumental performance, and seek meaningful assessments of practical peak capacity. Indeed, this chapter should be read as a “call to arms’’ of appropriately minded researchers to propose a good metric for practical peak capacity. On the basis of the results reported here, the metric should reflect the area used by the separation, in agreement with others (Liu et al., 1995; Gilar et al., 2005b). The inconsistency discussed above among the different ways to express this area, however, suggests a new approach. What is the best way to measure area, especially for isolated zones? One approach simply is to adapt the definition of Gilar et al. and count the number of units of peak capacity that are occupied by retention-time coordinates. Alternatively, these coordinates could be “fuzzied’’ by probability distributions allowing for the finite breadths of peaks, since peaks that are not centered within a capacity unit extend into other units, especially if they are multiplets. Another approach might be to generalize the automated measurements of Lan and Jorgenson from a one-dimensional space to a 2D one and measure the utilized space directly (Lan and Jorgenson, 1999). If this chapter causes the reader to reflect on the appropriate definition, it will have served its purpose.

APPENDIX 3A GENERATION OF RANDOM CORRELATED COORDINATES An internet search by the author found several algorithms for generating correlated coordinates having a specified linear correlation coefficient. However, none of the sites addressed the spaces considered here or discussed the theory of coordinate generation. A constant density of correlated random coordinates (x, y) was distributed over nonrectangular areas A by equating the cumulative distribution function of area A, expressed relative to x, to a random number R (0 R 5 1) and then solving the resultant equation for x (Press et al., 1992). Ordinate y was then uniformly distributed between its bounds at that x using another random number. The cumulative distribution functions F(x) are derived simply and in the same way for all spaces. For example, study of Fig. 3.1b shows the differential area dA of WEG lying between x and x þ dx is Db x dx, where Db ¼ b2 b1. The division of dA by the total area Aw of WEG Z1 AW ¼ Db

x dx ¼ Db=2

ð3:A1Þ

0

gives 2x dx ¼ f(x)dx, where f(x) is the probability density function (pdf) of the area. F(x) then is obtained by integrating f(z) dz from the lower area bound to x, where z is a dummy variable.

FIGURE 3.A1 Partitionings of (a) FAN and (b), and (c) PAR for calculation of cumulative distributions. 51

52

PEAK CAPACITY IN TWO-DIMENSIONAL LIQUID CHROMATOGRAPHY

Consequently, for WEG, the coordinate equations are Zx z dz ¼ x2 ¼ R1

2

ð3:A2aÞ

0

where the domain of x is p 0ﬃﬃﬃﬃﬃ ﬃ x 5 1 and R1 is a random number. The solution to Equation 3.A2a (i.e., x ¼ R1 ) then is paired with ordinate y ¼ ðb1 þ DbR2 Þx

ð3:A2bÞ

where R2 is another random number, with the range of y equal to b1x y 5 b2x. Equation 3.A2b simply represents a uniform distribution of all ordinates between the lower bound, y ¼ b1x (for R2 ¼ 0), and the upper bound, y ¼ b2x (for R2 ¼ 1). In other words y ¼ lower bound þ ðupper boundlower boundÞ R2

ð3:A2cÞ

which also applies to all expressions for y below. Each evaluation of Equations 3.A2a and 3.A2b with new R1 and R2 produces a new coordinate pair (x, y) in WEG. The space FAN was partitioned into regions 1 and 2, as shown in Fig. 3.A1a, with the abscissa of the demarcation equal to b1 2 (the partition was not necessary but was made simply for convenience). Cumulative distribution functions were evaluated for both regions, and one was selected for coordinate generation by computing another random number R0. For R0 f, where f is the fractional area of region 1 (i.e., the area of region 1, divided by the summed areas of regions 1 and 2), the coordinate was computed from equations for region 1; otherwise, from equations for region 2. In region 1, the area AF,1 is Db=ð2b22 Þ and the pdf is 2b22 x, with x having the domain, 0 x 5 1/b2. Abscissa x is evaluated from Zx 2b22

z dz ¼ ðb2 xÞ2 ¼ R1

ð3:A3Þ

0

(The solutions to this and subsequent equations in x are not given here, although all are analytical.) Ordinate y again is computed from Equation 3.A2b. In region 2, the area AF,2 is ! b AF;2 ¼ 1b1 =2ðb2 Þ1 þ 12 ð3:A4aÞ 2b2 and the pdf is (AF,2)1(1 b1x), with x having the domain, 1/b2 x 5 1. Coordinates x and y are calculated from

APPENDIX 3A GENERATION OF RANDOM CORRELATED COORDINATES

ðAF;2 Þ

1

Zx ð1b1 zÞdz ¼ ðAF;2 Þ

1

xb1 x =2ðb2 Þ 2

1

þ

1=b2

b1 2b22

53

! ¼ R1 ð3:A4bÞ

y ¼ b1 x þ ð1b1 xÞR2

ð3:A4cÞ

where the range of y equals b1x y 5 1. The space PAR was partitioned among regions 1, 2, and 3, as shown in Fig. 3.A1b and c. The abscissas demarking regions 1 and 2, and regions 2 and 3, are xL and xH, respectively. They have different values, depending on b1 and b2. Specifically, for b1 þ b2 > 2, xL and xH are the abscissas of vertices at the upper left and lower right, respectively, of PAR (Fig. 3.A1b); for b1 þ b2 5 2, they are the abscissas of vertices at the lower right and upper left (Fig. 3.A1c). Their values are determined by the intersection of the appropriate lines in Figs. 3.A1b and 3.A1c, For

b1 þ b2 > 2 :

xL ¼ ð1b1 Þ=Db; xH ¼ ðb2 1Þ=Db

ð3:A5aÞ

For

b1 þ b2 5 2 :

xL ¼ ðb2 1Þ=Db; xH ¼ ð1b1 Þ=Db

ð3:A5bÞ

One notes that the xL and xH values simply are reversed for the two cases. For b1 þ b2 ¼ 2, xL equals xH and region 2 vanishes. Different regions for coordinate generation were selected as for FAN, based on comparisons of fractional areas to a random number. For region 1, the area AP,1 is Dbx 2L =2 and the pdf is 2x=x2L , with x having the domain, 0 x 5 xL. The abscissa x is calculated from 2 x2L

Zx z dz ¼ ðx=xL Þ2 ¼ R1

ð3:A6Þ

0

and ordinate y again is calculated from Equation 3.A2b. In region 2, the area AP,2 is (1 b1)(xH xL) for b1 þ b2 > 2, and is (b2 1)(xH xL) for b1 þ b2 5 2. In both cases, the pdf is (xH xL)1, with the domain of x equal to xL x 5 xH. Abscissa x is evaluated from 1

Zx

ðxH xL Þ

dz ¼ xL

For b1 þ b2 > 2, ordinate y is calculated from

xxL ¼ R1 xH xL

ð3:A7aÞ

54

PEAK CAPACITY IN TWO-DIMENSIONAL LIQUID CHROMATOGRAPHY

y ¼ b1 x þ ð1b1 ÞR2

ð3:A7bÞ

with the range, b1x y 5 b1(x 1) þ 1. For b1 þ b2 5 2, ordinate y is calculated from y ¼ b2 ðx1Þ þ 1 þ ðb2 1ÞR2

ð3:A7cÞ

with the range, b2(x 1) þ 1 y 5 b2x. For region 3, the area Ap,3 is AP;3 ¼ Dbð1=2 þ x2H =2xH Þ

ð3:A8aÞ

the pdf is (AP,3)1 Db(1 x), and the domain of x is xH x 5 1. Abscissa x is calculated from

1

Zx

ðAP;3 Þ Db xH

x2 x2 ¼ R1 ð1zÞ dz ¼ ðAP;3 Þ1 Db x xH þ H 2

ð3:A8bÞ

whereas y is y ¼ b2 ðx1Þþ1Dbðx1ÞR2

ð3:A8cÞ

with the range, b2(x 1) þ 1 y 5 b1(x 1) þ 1. Area A defining a in Equation 3.3 was calculated from Aw for WEG, the sum of AF,1 and AF,2 for FAN, and the sum of AP,1, AP,2, and AP,3 for PAR. Similar coordinate distributions can be made for any simple geometrical space of interest.

APPENDIX 3B DERIVATION OF LIMITING CORRELATION COEFFICIENT r The linear correlation coefficient r of N coordinates (x, y) is N N N P P P N xi yi xi yi i i si ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ N 2 2 N N N P P P P N x2i N y2i xi yi i

i

i

ð3:B1Þ

i

where i is an index. The limiting value of r as N approaches infinity is calculated for homogeneous random coordinates by approximating each sum in Equation 3.B1 by its equivalent integral

APPENDIX 3B DERIVATION OF LIMITING CORRELATION COEFFICIENT r

Zx2

N X

lim N! ¥

55

qi ¼ N

i

f ðxÞqðxÞdx

ð3:B2Þ

x1

where q is the continuous equivalent of coordinate or coordinate combination qi (e.g., xi becomes x, xiyi becomes xy, etc.), x1 and x2 are the minimum and maximum values of x in the region to which the summation applies, and f(x) and y are the relevant pdf and function, respectively, in Appendix 3A. In effect, Equation 3.B2 equates the discrete and continuous first moments of the distribution of random variable q; the pdf f(x) is needed to weight different q in accordance with their likelihood of occurrence. Comparison of Equations 3.B1 and 3.B2 shows that the limiting value of r is independent of N. For example, the pdf f(x) for WEG is 2x and for large N N X

Z1 xi ¼ 2N

i N X

x2 ðb1 þDbR2 Þdx ¼ Nðb1 þb2 Þ=3

ð3:B3bÞ

0

Z1 xi yi ¼ 2N

x3 ðb1 þDbR2 Þdx ¼ Nðb1 þb2 Þ=4

ð3:B3cÞ

0 N X i

i

0

yi ¼ 2N

i

N X

ð3:B3aÞ

Z1

i N X

x2 dx ¼ 2N=3

Z1 x2i

¼ 2N

x3 dx ¼ N=2

ð3:B3dÞ

0

Z1 x3 ðb1 þDbR2 Þ2 dx ¼ Nðb21 þb22 þb1 b2 Þ=6

y2i ¼ 2N

ð3:B3eÞ

0

where y is expressed by Equation 3.A2b, R2 is assigned its average value 1/2, and R22 is assigned its average value 1/3 (as determined by simulation). The final expression for r simplifies to 1 sþ1 r ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; 2 1þs2 s

s ¼ b1 =b2 ;

1=2 r 1

ð3:B4Þ

and depends only on the ratio, b1/b2. Spaces FAN and PAR were addressed similarly, with Pthe integrals evaluated for different regions. The evaluations for any sum type (e.g., N i x i ) then were added, with weights equal to the fractional areas. Consequently, numerous algebraic equations were produced, which were not simplified in evaluating r, Equation 3.B1. FORTRAN programs for their evaluation are available on request. Unlike Equation 3.B4, r for FAN and PAR does not depend on b1/b2.

56

PEAK CAPACITY IN TWO-DIMENSIONAL LIQUID CHROMATOGRAPHY

These equations and Equation 3.B4 were used to evaluate r for specified b1 and b2 (see Fig. 3.3) and to evaluate b1 for specified b2 and r (see Tables 3.1–3.3).

REFERENCES Campostrini, N., Areces, L.B., Rappsilber, J., Pietrogrande, M.C., Dondi, F., Pastorino, F., Ponzoni, M., Righetti, P. (2005). Spot overlapping in two-dimensional maps: a serious problem ignored for much too long. Proteomics 5, 2385. Cotton, F.A., Wilkinson, G. (1972). Advanced Inorganic Chemistry, 3rd edition. Interscience Publishers, New York. Creten, W.L., Nagels, L.J. (1987). Computation of determination limits for multicomponent chromatograms. Anal. Chem. 59, 822. Davis, J.M. (1991). Statistical theory of spot overlap in two-dimensional separations. Anal. Chem. 63, 2141. Davis, J.M. (1993). Statistical theory of spot overlap for n-dimensional separations. Anal. Chem. 65, 2014. Davis, J.M. (1997a). Justification of probability density function for resolution in statistical models of overlap. Chromatographia 44, 81. Davis, J.M. (1997b). Extension of statistical overlap theory to poorly resolved separations. Anal. Chem. 69, 3796. Davis, J.M. (2004). Assessment by Monte Carlo simulation of thermodynamic correlation of retention times in dual-column temperature programmed comprehensive two-dimensional gas chromatography. J. Sep. Sci. 27, 417. Davis, J.M. (2005). Statistical-overlap theory for elliptical zones of high aspect ratio in comprehensive two-dimensional separations. J. Sep. Sci. 28, 347. Dondi, F., Kahie, Y.D., Lodi, G., Remelli, M., Reschiglian, P., Bighi, C. (1986). Evaluation of the number of components in multicomponent liquid chromatograms of plant extracts. Anal. Chim. Acta 191, 261. El Fallah, M.Z., Martin, M. (1987). Influence of the peak height distribution on separation performances: discrimination factor and effective peak capacity. Chromatographia 24, 115. Felinger, A. (1997). Critical peak resolution in multicomponent chromatograms. Anal. Chem. 69, 2976. Felinger, A., Pietrogrande, M.C. (2001). Decoding complex multicomponent chromatograms. Anal. Chem. 73, 619A. Giddings, J.C. (1967). Maximum number of components resolvable by gel filtration and other elution chromatographic methods. Anal. Chem. 39, 1027. Giddings, J.C. (1984). Two-dimensional separations: concept and promise. Anal. Chem. 56, 1258A. Giddings, J.C. (1987). Concepts and comparisons in multidimensional separation. J. High Resolut. Chromatogr. Chromatogr. Commun. 10, 319. Giddings, J.C. (1995). Sample dimensionality: a predictor of order-disorder in component peak distribution in multidimensional separation. J. Chromatogr. A 703, 3.

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Gilar, M., Olivova, P., Daly, A.E., Gebler, J.C. (2005a). Two-dimensional separation of peptides using RP-RP-HPLC system with different pH in first and second separation dimensions. J. Sep. Sci. 28, 1694. Gilar, M., Olivova, P., Daly, A.E., Gebler, J.C. (2005b). Orthogonality of separation in twodimensional liquid chromatography. Anal. Chem. 77, 6426. Gray, MJ., Dennis, G.R., Slonecker, P.J., Shalliker, R.A. (2003). Evaluation of the twodimensional reversed-phase–reversed-phase separations of low-molecular mass polystyrenes. J. Chromatogr. A 1015, 89. Gray, M.J., Dennis, G.R., Slonecker, P.J., Shalliker, R.A. (2004). Comprehensive two-dimensional separations of complex mixtures using reversed-phase liquid chromatography. J. Chromatogr. A 1041, 101. Gray, M.J., Dennis, G.R., Slonecker, P.J., Shalliker, R.A. (2005). Utilising retention correlation for the separation of oligostyrenes by coupled-column liquid chromatography. J. Chromatogr. A 1073, 3. Gray, M., Dennis, G.R., Wormell, P., Shalliker, R.A., Slonecker, P. (2002). Two dimensional reversed-phase–reversed-phased separations. Isomeric separations incorporating C18 and carbon clad zirconia stationary phases. J. Chromatogr. A 975, 285. Guiochon, G., Beaver, L.A., Gonnord, M.F., Siouffi, A.M., Zakaria, M. (1983). Theoretical investigation of the potentialities of the use of a multidimensional column in chromatography. J. Chromatogr. 255, 415. Guiochon, G., Gonnord, M.F., Siouffi, A., Zakaria, M. (1982). Study of the performances of thin-layer chromatography. VII. Spot capacity in two-dimensional thin-layer chromatography. J. Chromatogr. 250, 1. Jandera, P., Novotna, K., Kolarova, L., Fischer, J. (2004). Phase system selectivity and peak capacity in liquid column chromatography—the impact on two-dimensional separations. Chromatographia 60, S27. Lan, K., Jorgenson, J.W. (1999). Automated measurement of peak widths for the determination of peak capacity in complex chromatograms. Anal. Chem. 71, 709. Lewis, K.C., Opiteck, G.R., Jorgenson, J.W., Sheeley, D.M. (1997). Comprehensive on-line RPLC-CZE-MS of peptides. J. Am. Soc. Mass. Spectrom. 8, 495. Liu, S., Davis, J.M. (2006). Dependence on saturation of average minimum resolution in twodimensional statistical-overlap theory: peak overlap in saturated two-dimensional separations. J. Chromatogr. A 1126, 244. Liu, Z., Patterson, D.G., Jr., Lee, M.L. (1995). Geometric approach to factor analysis for the estimation of orthogonality and practical peak capacity in comprehensive two-dimensional separations. Anal. Chem. 67, 3840. Marchetti, N., Felinger, A., Pasti, L., Pietrogrande, M.C., Dondi, F. (2004). Decoding twodimensional complex multicomponent separations by autocovariance function. Anal. Chem. 76, 3055. Martin, M. (1991). Que Peut-on Esperer du Couplage Entre Methods de Separation? Approache Statistique du Probleme. Proc. Congre Mesucora’91 1,3. Martin, M. (1992). Quelles Performances Attendre des Couplages en Chromatographie? Spectra 2000, 169 (Suppl.), 5. Martin, M. (1995). On the potential of two- and multi-dimensional separation systems. Fresenius J. Anal. Chem. 352, 625.

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Nagels, L.J., Creten, W.L. (1985). Evaluation of the glassy carbon electrochemical detector selectivity in high-performance liquid chromatographic analysis of plant material. Anal. Chem. 57, 2706. Nagels, L.J., Creten, W.L., Vanpeperstraete, P.M. (1983). Determination limits and distribution function of ultraviolet absorbing substances in liquid chromatographic analysis of plant extracts. Anal. Chem. 55, 216. Oros, F.J., Davis, J.M. (1992). Comparison of statistical theories of spot overlap in twodimensional separations and verification of means for estimating the number of zones. J. Chromatogr. 591, 1. Peters, M., Davis, J.M. (1998). Statistical-overlap theory for column switching. Am. Lab. 30 (18), 32C. Pietrogrande, M.C., Cavazzini, A., Dondi, F. (2000). Quantitative theory of the statistical degree of peak overlapping in chromatography. Rev. Anal. Chem. 19, 123. Pietrogrande, M.C., Marchetti, N., Dondi, F., Righetti, P.G. (2002). Spot overlapping in twodimensional polyacrylamide gel electrophoresis separations: a statistical study of complex protein maps. Electrophoresis 23, 283. Pietrogrande, M.C., Marchetti, N., Dondi, F., Righetti, P.G. (2003). Spot overlapping in twodimensional polyacrylamide gel electrophoresis maps: relevance to proteomics. Electrophoresis 24, 217. Press, W.H., Teukolsky, S.A., Vetterling, W., Flannery, B.P. (1992). Numerical Recipes in FORTRAN, 2nd edition. Cambridge University Press, Cambridge, UK. Roach, S.A. (1968). The Theory of Random Clumping. Methuen, London. Rowe, K., Davis, J.M. (1995). Relaxation of randomness in two-dimensional statistical model of overlap: theory and verification. Anal. Chem. 67, 2981. Rowe, K., Bowlin, D., Zou, M., Davis, J.M. (1995). Application of 2-D statistical theory of overlap to three separation types: 2-D thin-layer chromatography, 2-D gas chromatography, and liquid chromatography/capillary electrophoresis. Anal. Chem. 67, 2994. Samuel, C., Davis, J.M. (2002). Statistical-overlap theory of column switching in gas chromatography: application to flavor and fragrance compounds. Anal. Chem. 74, 2293. Schure, M.R. (1991). Resolution enhancement of chromatographic data. Considerations in achieving super-resolution with the constrained iterative relaxation method. J. Chromatogr. 550, 51. Shen, Y., Tolic, N., Zhao, R., Pasa-Tolic, L., Li, L., Berger, S.J., Harkewicz, R., Anderson, G.A., Belov, M.E., Smith, R.D. (2001). High-throughput proteomics using high-efficiency multiple-capillary liquid chromatography with on-line high-performance ESI FTICR mass spectrometry. Anal. Chem. 73, 3011. Shi, W., Davis, J.M. (1993). Test of theory of overlap for two-dimensional separations by computer simulations of three-dimensional concentration profiles. Anal. Chem. 65, 482. Slonecker, P.J., Li, X., Ridgway, T.H., Dorsey, J.G. (1996). Informational orthogonality of twodimensional chromatographic separations. Anal. Chem. 68, 682. Valentine, S.J., Kulchania, M., Srebalus Barnes, C.A., Clemmer, D.E. (2001). Multidimensional separations of complex peptide mixtures: a combined high-performance liquid chromatography/ion mobility/time-of-flight mass spectrometry approach. Int. J. Mass. Spectrom. 212, 97. Venkatramani, C.J., Xu, J., Phillips, J.B. (1996). Separation orthogonality in temperatureprogrammed comprehensive two-dimensional gas chromatography. Anal. Chem. 68, 1486.

4 DECODING COMPLEX 2D SEPARATIONS Francesco Dondi, Maria C. Pietrogrande, and Nicola Marchetti Department of Chemistry, University of Ferrara, 1-44100 Ferrara, Italy

Attila Felinger Department of Analytical Chemistry, University of P ecs, P ecs, Hungary

4.1 INTRODUCTION Separation and identification of complex multicomponent mixtures is a challenging task for frontier research fields, such as proteomics or metabolomics, as well as for traditional research fields (polymer, natural product, food, and environmental chemistry). Mixtures involved in these kind of analysis, can easily contain up to 5000 single components (SCs) or even more, but their exact number (m) is usually unknown. The number of SCs that can be highlighted strongly depends on two closely related factors belonging to the sample complexity: the power of the separation method (complete separation of each SC from all of the others is quite difficult to achieve) and the selected detection threshold level. Additionally, mixtures are defined as complex not only in terms of the number of SCs, but also by looking at the similarity of their SCs, the number and abundance of SC classes, for example, homologous series. These points were discussed in one of the last papers by Giddings (1995) where the concept of sample dimensionality, as a quantitative index of sample complexity, was introduced for the first time.

Multidimensional Liquid Chromatography: Theory and Applications in Industrial Chemistry and the Life Sciences, Edited by Steven A. Cohen and Mark R. Schure. Copyright 2008 John Wiley & Sons, Inc.

59

60

DECODING COMPLEX 2D SEPARATIONS

High separation capability is an increasingly requested feature in analyzing complex multicomponent mixtures: two-dimensional (2D) separation techniques are much preferred over the traditional one-dimensional separation methods such as gas chromatography (GC), liquid chromatography (LC), or electrophoresis because of the tremendous increase in peak capacity and selectivity (Schoenmakers et al., 2003). As a consequence, these separation methods produce bidimensional maps containing an elevated number of overcrowded spots. A clear example of that is displayed in Figure 4.1a, where a 2D polyacrylamide gel electrophoresis (2D PAGE) analysis1 is reported. Among the 2D separation techniques, LC LC — which constitutes the subject of this book—is the most suitable because it can combine quite different selectivity features belonging to different LC elution modes. The LC LC map of complex multicomponent mixtures contains an elevated number of SC spots: some of them are randomly spread over the 2D space, while others are organized into the so-called spot trains, as it can be seen in the several examples reported in the present book. Mastering the complexity of multicomponent 2D maps in terms of SCs and SC classes requires a specific theoretical support. Mixture identity is, in fact, exhausted not only by SC identity but also by the relationships—qualitative and quantitative—among the different SCs: such a richness of features is just revealed by the separation process and must be evaluated and managed in order to reach optimum separation results. Because of the tremendous amount of analytical– chemical information produced, an appropriate data handling method is mandatory to fully exploit multidimensional techniques. For example, in the map reported in Fig. 4.1a, 2315 spots are detected while the number of identified proteins is 865, but the reality can be much more complex than that shown in the figure. This also indicates the seriousness of the degree of overlapping and its dependence on the concentration. On the contrary, Figure 4.1b (Demalte-Annessi et al., 1999) enlightens the existence of ordered structures of compounds (see red boxes in the ordered region) as examples of compound classes embedded in a random constellation of spots. The goal of this chapter is to review the recent significant advances achieved in the study of 2D maps (Marchetti et al., 2004; Pietrogrande et al., 2002, 2003, 2005, 2006 a; Campostrini et al., 2005). Fundamental aspects concerning the intimate structure of multicomponent mixtures and separations will be discussed in Section 4.2 . Description of the methods recently developed by the present authors for characterizing the separation pattern complexity of a 2D multicomponent map will be presented in Sections 4.3 and 4.4 . These methods allow one to describe complex 2D separations in terms of SC number (m), the detection of hidden homologous series, spot shape features and separation performance. For these reasons they are named as “decoding methods.’’ In Section 4.5, the most recent achievements derived from the application

1 Human 2D PAGE database of Danish Centre for Human Genome Research, http://www.biobase.dk/cgibin/celis.

FIGURE 4.1 2D polyacrylamide gel electrophoresis (2D-PAGE) maps of protein mixtures. (a) Experimental 2D-PAGE map of keratinocytes. (Reproduced from Human 2D-PAGE database of Danish Center for Human Genome Research, www.biobase.dk/cgi-bin/celis, with kind permission.) The windows report two enlarged zones of the map showing different overcrowding conditions. (b) Example of digitized map of identified spots from colorectal adenocarcinoma cell line 2D PAGE map. Two regions of the map are enlightened showing different pI interdistance distribution: ordered and disordered. Enlarged detail: Histogram of the pI interdistance values and the estimated distribution using Equation 4.33 (red line). Reproduced from Pietrogrande et al. (2002) with permission from Wiley-VCH Verlag GmbH. (See color plate.)

62

DECODING COMPLEX 2D SEPARATIONS

of these procedures will be reviewed as application examples of the present models and methods.

4.2 FUNDAMENTALS: THE STATISTICAL DESCRIPTION OF COMPLEX MULTICOMPONENT SEPARATIONS The multicomponent mixtures considered here (indicated as “complex’’) are made by more than approximately 50 SCs: the reasons of this borderline value will be clear in the following text. These mixtures, when subjected to separation processes of medium-to-high efficiency, often produce “random’’ separation patterns. For example, their 1D chromatograms are typically made up by a random sequence of both Gaussian-shaped peaks—likely corresponding to SC peaks (singlets)—and of cluster peaks—corresponding to more than one SC (multiplet peaks). This finding is also common to 2D separations (see e.g., Fig. 4.1a and b) where “randomness’’ is spread over the two dimensions. The appearance of “randomness’’ is however a “subjective’’ impression and it does not mean that regularities are absent in the retention pattern. On the contrary, regularities are often present; in particular, they are of “statistical’’ character and thus “objective.’’ However, they are somewhat difficult to catch at first sight, since they do not exhibit themselves as “deterministic’’. In fact, statistical regularities are not located in a specific point or restricted region of the separation space, but they become evident only when the whole separation space is systematically investigated and the statistics of local observed features is made. For these purposes, a specific theoretical background and handling process are requested. One can also understand that statistics will lead the separation process itself, and great benefit in the separation strategy will result once these features will be mastered. A separation scientist has nonetheless diffidence to accept the statistical behavior in multicomponent separations. Honestly, we shared such uncertainty with some colleagues, too (Guiochon, 2005): it seems that this way of thinking is deeply rooted in our minds, since it comes rightly from our chemical way of thinking. In fact, when we are handling complex mixtures, we are very often interested only in some specific and “relevant-for-us’’ compounds, having precise, that is, deterministic— and thus not random—retention time positions. Moreover, a given group of selected SCs have often structures related to each other: the separation process, obeying well-established retention–structure relationships, will produce “deterministic’’ and not random retention patterns. This point of view can be extended to the whole separation pattern, with the result that its character should be deterministic. Randomness and its statistical description seem thus to conflict with both the practice and the “first principles’’ of separation science. These apparent paradoxes will be discussed below. All these uncertainties about the “randomness’’ in multicomponent retention patterns are overcome once its features are precisely stated. This will allow the subjective nature of the randomness to be made objective through proper analysis. We start with discussing the fact that the “pure random’’ pattern does not exist, but full

FUNDAMENTALS: THE STATISTICAL DESCRIPTION

(a)

63

S1 S2

SN–1 SN

Stot 0

6

5

m

10

3

5

m

(b)

3. f (µ) 2.

1.

0.

5.

m x 10

FIGURE 4.2 (a) Example of Poissonian sequence generation, Stot, by superimposition of elementary subsequences Si having uncorrelated phases. (b) Interdistance frequency function of the Poissonian sequence. Reproduced from Felinger et al. (1990) with permission from the American Chemical Society.

randomness is a limit condition when the multicomponent mixture is complex enough: with the help of the graphical scheme reported in Fig. 4.2 (Felinger et al., 1990) we will show how the “randomness’’ degree onsets and increases. Let us consider one homologous series—generally indicated as i-series—and characterized by a given structural increment, for example, a CH2 group. At the same time, let us separate this series through a 1D system where the separation coordinate is labeled as x. The partition free energy of the ni component of the i-series is Dmi;x ðni Þ ¼ ai;x þ bi;x ni ;

ni ¼ 1; ; nmax;i : i ¼ 1; ; imax

ð4:1Þ

64

DECODING COMPLEX 2D SEPARATIONS

where Dmi,x represents the partition free energy of the ni term of this i-series, while ai,x and bi,x are the intercepts and the slopes, respectively, here called the “phases’’ and the “frequencies’’ according to the nomenclature introduced by (Giddings 1995). i is ranging from 1 to imax (see Eq. 4.1), where imax is the number of homologous series (compound classes). In a 1D separation under good elution programming conditions, any i-series produces a retention sequence Si related to the same Equation 4.1 through a constant factor since, under these conditions, retention is proportional to partition free energy. Partition free energy in Equation 4.1 can be equally replaced by any other property proportional to the retention (e.g., electrophoretic mobility, isoelectric point, etc.). Fig. 4.2a (Felinger et al., 1990) reports the effect of superimposing several ordered sequences Si of retention points: the individual ordered patterns are lost and the overall SC sequence is seen as “random.’’ This property appears evident when three windows of equal length are separately considered (see Fig. 4.2a): they contain six, three or five points, respectively, which are clearly randomly varying numbers. We also understand that when the number of sequences, imax, increases, the randomness degree will be greater and greater, but our “subjective’’ feeling proves unable to precisely define the nature and the degree of such an increasing randomness. Randomness can be made “objective’’ by measuring some random variables over it and then performing an appropriate statistics. Let us consider the interdistances between subsequent time positions (Dx) in a superimposed pattern: they are clearly not regular and can be employed to characterize randomness (see Fig. 4.2a). In fact, when their frequency function is computed (Fig. 4.2b) a distinct and familiar regularity singles out the exponential function. f ðDxÞ ¼ l0 expðDxl0 Þ

ð4:2Þ

where l0 ¼

1 Dx

ð4:3Þ

is the frequency and Dx is the average interdistance. The distribution function F(m ¼ r) of the number of points m falling into a given interval DX, see the boxes marked in Fig. 4.2a—once computed over several intervals, will exhibit a distinct feature, which in this case is close to Poisson law (Feller, 1971), defined as Fðm ¼ rÞ ¼ m ¼ l0 DX

expðmÞmr r!

ð4:4Þ ð4:5Þ

where m is the average—or the expected—number of points in this windows. It is known that the Poisson pattern described by Equation 4.3 has also the unique property that interdistances between subsequent SC positions, Dx, are exponentially distributed (Feller, 1971) according to Equations 4.2 and 4.3. Accordingly, the feature enlightened

FUNDAMENTALS: THE STATISTICAL DESCRIPTION

65

in Fig. 4.2b proves that the randomness determined by the overlapping process of different sequences has a Poissonian character. It can be theoretically proven that the overall sequence approaches the Poisson law when imax increases (Feller, 1971). Moreover l0 is the inverse of the harmonic average value b given by l0 ¼

imax X 1 1 b bx i¼1 i;x

ð4:6Þ

Therefore, the actual randomness degree of a finite number of overlapping sequences is not yet perfectly Poissonian, as shown by the scattered point around the exponential decay in Fig. 4.2b. In this sense, both the Poisson law and the Exponential law (Eqs. 4.4 and 4.2) are thus “approximating’’ functions. In general, provided that the number of sequences (imax) are high enough, the approximation degree is good and one can replace the sample frequencies by those computed either by the Poisson law (Eq. 4.4) or the exponential law (Eq. 4.2) without having significant errors. This approximating condition means, in practice, that if one uses a statistical test (e.g., the c2 statistics) to verify the H0 hypothesis there is no statistical difference between the “actual’’ sequence of retention time positions and that one given by the Poisson law with the same parameters (Eqs. 4.2 and 4.3). Furthermore, the H0 hypothesis is not rejected (Dondi et al., 1986) and the SC retention time distribution is uniformly random (Davis and Giddings, 1985). Note that this way of proceeding does not mean that the actual sequence has features mathematically equal to those of the Poisson law, but that the Poisson law serves as an approximation, that is, a substitute most useful as will be shown below. Consequently, we can also affirm that a given multicomponent retention pattern, produced by a specific complex mixture, is “deterministic’’ in itself, but “random’’ when it is analyzed by the instrument of statistics. This clarifies the kind of information one can get by statistical investigation: only collective properties are obtained, and this is the character of information obtained by the present approach when applied to multicomponent separations. The Poissonian character of a multicomponent chromatogram assumed by Davis and Giddings (1983) as the fundamental hypothesis of their so-called Statistical Model of Overlapping (SMO) has thus well-founded grounds and it cannot be simply considered as one among the many possible models. Moreover, the SMO allowed deriving a method of estimating an important collective property of the mixture, the number of SCs, as well as other collective properties, such as the number of singlets present in the retention pattern. The SMO approach was also applied to describe 2D and 3D multicomponent separations (Davis, 1991; Oros and Davis, 1992). Mixing SCs in a mixture and overlapping many ordered sequences over a retention axis are two faces of the same coin: they are both entropy-creating processes and can equally be interpreted by using the entropy function concept (Dondi et al., 1998). Moreover, it is also worthwhile to compare similarities and differences between two processes producing randomness, the above-described Poisson type (Fig. 4.2a and Eq. 4.4), and the Gaussian one that is so often evocated in many fundamental branches of natural sciences (Feller, 1971). The latter refers to the addition of independent

66

DECODING COMPLEX 2D SEPARATIONS

random variables and is related to several important physical phenomena, for example, the diffusion process and the law of error distribution. Both these processes have the ultimate effect of destroying the order: the Poissonian one occurs by a superimpositionmixing mechanism, while the Gaussian one by an addition mechanism. Moreover both these distributions are maximum entropy type distributions (Davis, 1991; Dondi et al., 1998). Thus, both Gauss and Poisson laws play a fundamental role in separation science: they should constitute a natural way of thinking for scientists. One can consider the effect of varying either ai,x or bi,x when many ordered retention sequences are superimposed in 1D retention patterns. The four sequences considered in Fig. 4.2a differ, in fact, both in ai,x and bi,x. In Fig. 4.3a and b these two effects are

FIGURE 4.3 Example of 1D separations as the result of five superimposed homologous series having (a) constant phase (red arrow in inset) and (b) constant frequency (blue arrow in inset). (See color plate.)

FUNDAMENTALS: THE STATISTICAL DESCRIPTION

67

considered separately. Moreover, the band-broadening effect is also introduced. In particular, Fig. 4.3a represents a 1D chromatogram of a mixture having five homologous series with the same constant functional group (ai,x constant) but different bi;x increments (e.g. due to different branching, saturation degree, etc.). The second case (Fig. 4.3b) is representative of five linear homologous series having the same , OH, NO, etc.). It increment (e.g., CH2) but different functional groups (e.g., C can be seen that the result seems apparently the same, since both of them exhibit random character. Randomness degree with ai,x or bi,x in 2D retention patterns is instead quite different when one goes from 1D to 2D separations and this is clearly seen in Fig. 4.4a and b. In particular the randomness, and hence the spot overlapping degree,

FIGURE 4.4 Example of 2D separations of five homologous series having (a) constant phase (red vector in inset) and uncorrelated frequencies and (b) constant frequency (blue vector in inset) and uncorrelated phases. (See color plate.)

68

DECODING COMPLEX 2D SEPARATIONS

is greater in the case of randomness of the frequency values (bi,x; bi,y) than in the case of the only randomness of the phases values (ai;x ; ai;y ) as one can see by comparing Fig. 4.4a and b, respectively. By increasing the dimensionality of the separation system, the observed general effect is that the degree of separation increases, as one can see by comparing Figs. 4.3 and 4.4. We can say that this acts as a “demixing’’ or “decoding’’ process allowing singling out the hidden features of the complex multicomponent mixture. However this “decoding’’ effect acts quite differently according to the specific randomness in either ai,x or bi,x parameters: in the case when only the phase values of the homologous series are randomly distributed, almost complete resolution can be achieved when going from 1D to 2D (cf. Figs. 4.3b and 4.4b). The conclusion is thus that one should have alternative statistical methods to estimate the randomness degree. For example, the interdistance statistics presented in Fig. 4.2b, does not exhaust the richness of information hidden in complex retention patterns. In fact, when the Dx statistics are computed (see histograms in Fig. 4.1b), only interdistances between subsequent positions are accounted for and their order is lost. We will see that the autocovariance function discussed in Section 4.4 is able to single out the features lost by the simple interdistance statistics. Obviously, when going from 2D to 1D, we observe an inverse behavior, that is, a randomness is created. However, such a feature does not have only virtual interest but is of surprisingly practical value, as will be discussed in the subsequent paragraph. In fact the SMO approach, based on a 1D random pattern, was successfully applied to multicomponent 2D separations (Pietrogrande et al., 2002, 2003; Campostrini et al., 2005).

4.3 DECODING 1D AND 2D MULTICOMPONENT SEPARATIONS BY USING THE SMO POISSON STATISTICS The SMO developed by Davis and Giddings thoroughly defines the statistical regularities of Poissonian complex multicomponent chromatograms. Let us first consider the existence of statistical regularities in a heuristic way. This is illustrated in Fig. 4.5 by reporting three different hypothetical windows (Fig. 4.5b–d) of a multicomponent chromatogram obtained by computer simulation: the retention times obey the Poisson law and the SC peak height was assumed constant for the sake of simplicity. All these windows contain 10 SCs and each chromatographic pattern appears quite different from the others, but some statistical properties are similar. In order to explain this, we need to recall some basic separation parameters. The resolution between two SC peaks of the same height is defined as RS ¼

Dx0 4s

ð4:7Þ

where Dx0 is the interdistance between subsequent retention times and s is the peak standard deviation. In Fig. 4.5a, the well-known case of RS ¼ 1 is reported. When the well-known number of separated bands at RS ¼ 1 is estimated for Fig. 4.5b–d, by

DECODING 1D AND 2D MULTICOMPONENT SEPARATIONS

69

roughly assuming the resolution criterion shown in Fig. 4.5a, we always observe a value of 4. The number of separated bands in a given retention pattern window is obviously deterministic, but its value exhibits nonetheless regularity of a statistical type if the retention pattern is of a Poisson type. The point regarding a precise method of peak counting will be discussed below.

FIGURE 4.5 Clustering in 1D separations. (a) Resolution between adjacent SCs defines the critical distance Dx 0 . (b) interdistances between adjacent SCs (fourth through seventh SC) are considered. I > Dx0, II < Dx0, and III > Dx0. Thus fifth and sixth SC are merged in the same band (doublet) that is completely separated from the previous and subsequent bands.

70

DECODING COMPLEX 2D SEPARATIONS

The separation extent is defined as p g¼ m

ð4:8Þ

where p is the number of separated zones at a given RS value. Therefore, constant values (0.4 at RS ¼ 1) are obtained for the cases in Fig. 4.5b–d. Since g and p are related to the ensemble of SCs or the whole chromatographic pattern, they are examples of statistical parameters. Several other statistical parameters can be defined. If only one SC originates the peak, it will be defined as a singlet and its number will be called the number of singlets (s); when two SCs are overlapped together within the same separated zone, one has a doublet and its number will be d; likewise one has triplet peak number (t), and so on. Thus, the parameter p is given by p ¼ sþdþtþ

ð4:9Þ

and m ¼ s þ 2d þ 3t þ

ð4:10Þ

Not all of the above-described statistical quantities are chromatographically “observable.’’ For example, s, d, t, and m are not directly observable unless selective detectors as a mass spectrometer is employed (Campostrini et al., 2005) and thus they are “hidden’’ quantities. The point will be discussed in the third section of this chapter. A precise criterion should be established concerning the SC clustering mode that produces a peak. The criterion put forward (Davis and Giddings, 1983) is that only when interdistances between a SC peak and the other two nearby ones are greater than the critical distance Dx0 ¼ 4sRS (see Fig. 4.5a), the SC is completely separated and can be regarded as a singlet. Otherwise, SC peak clusters including two or more SCs having subsequent interdistances lower than Dx0 will generate a multiplet. In Fig. 4.5b, a doublet peak is described. Extension to the 2D case was done by Marchetti et al. (2004) and Marengo et al. (2005). Such a definition is however only for theoretical purposes and is not applicable since it implies that one knows the location of every SC inside the bands. Peaks and their number are instead currently detected and measured by an integrator or scanning devices and thus are instead “observable’’ quantities. The main drawback is that they are evaluated on the basis of established threshold and resolution criteria that very often are not precise. Nonetheless, the response values, even if poorly defined, can be used to estimate m—a “hidden’’ quantity, see above—in a random retention pattern, as described below. Fig. 4.6a reports a “line’’ chromatogram as a series of vertical lines whose location and heights correspond to the retention times and areas, respectively, of all of the peaks detected by the integrator according to a given experimental setup. There are 19 detected peaks, p0 , represented as Yi, i ¼ 1, . . ., p0 . The prime symbol (0 ) signifies the particular way of counting peaks as performed by the integrator.

DECODING 1D AND 2D MULTICOMPONENT SEPARATIONS

71

y

∆x0 = 0

(a) 2

4

6

8

10

12

14

Time

y 3

∆ x 0 = 0.14 2

2 4

6

2 8

10

12

(b)

14

Time

5 y

∆ x 0 = 0.34 2

4

6

3

2

2

(c) 8

10

12

14

Time

FIGURE 4.6 Scheme of the SMO procedure to estimate the number of SC. (a) Original line 0 chromatogram. (b, c) Multiplets are formed by applying increasing critical distance values, Dx 0 .

The following quantities can be calculated: 0

Ytot ¼

p X

Yi

ð4:11Þ

i

yobs ¼

Ytot p0

ð4:12Þ

Again we remember that each pulse of the “line chromatogram’’ (Fig. 4.6a), may be either a singlet or a muItiplet peak, and the recorded peak height is proportional to the area value of the detected cluster. can be obtained by using a procedure originally developed The number of SCs, m, by Davis and Giddings (1983, 1985) and further extended by Dondi et al. (1998) and Pietrogrande et al. (1995). It consists of generating other chromatograms, as shown by the procedure described in Fig. 4.6b and c, by increasing Dx0 0 and by merging together those peaks whose interdistance falls within Dx0 0. In Fig. 4.6b the number of detected peaks decreases from 19 to 16, since clusters were merged in three cases. The procedure is repeated with increasing Dx0 0 values: by this way, a set of values 0 ðy obs ; Dx 0 Þj is obtained, which is then employed to fit the following equation: ln yobs ¼ ln y þ m

Dx 0 DX

ð4:13Þ

where Ytot ð4:14Þ m is the estimate of the true SC number m. The method based on In Equation 4.13 m Equation 4.13 is an example of “statistical inference,’’ which allows one to obtain an yobs ¼

72

DECODING COMPLEX 2D SEPARATIONS

estimate of a “population’’ quantity. The bar over m just shows the character of average value estimate of the random variable m described by the Poisson law (see Eq. 4.2 ). In fact, the real sequence of SCs hidden in the line chromatogram should be considered as only one of the infinitely possible Poisson sequences of points pﬃﬃﬃﬃdescribed by Equation 4.2. The m value estimate has a standard deviation equal to m, which is the standard pﬃﬃﬃﬃ deviation of the Poisson variable m: for m > 50, the relative percent error (¼ 100= m) is lower than about 15%, and this justifies the above criterion put forward for defining “complex’’ multicomponent mixtures. Once m has been calculated, the so-called saturation factor in 1D separation can be obtained from a1D ¼

m n1D

ð4:15Þ

where n1D is the peak capacity of 1D separations. It is defined as n1D ¼

DX Dx0

ð4:16Þ

The SMO theories (Davis and Giddings, 1983; Pietrogrande et al., 1995) give estimate of s, d, t, . . . , and p in a given multicomponent retention pattern: s ¼ m expð2a1D Þ

ð4:17Þ

d ¼ m expð2a1D Þ½1expða1D Þ

ð4:18Þ

t ¼ m expð2a1D Þ½1expða1D Þ2

ð4:19Þ

p ¼ m expða1D Þ

ð4:20Þ

Consequently, several “hidden’’ quantities can be estimated on the basis of the SMO approach. The procedure based on Equation 4.13 can be simply extended even to 2D separations as described in Fig. 4.7. In practice, the 2D pattern, in terms of spot positions and abundances, is divided into several strips. Each strip is transformed into a 1D “line chromatogram’’ and the procedure described in Fig. 4.7 is then applied. Equation 4.13 is employed to calculate the m value of each strip from which the total m value is obtained. Applications to this procedure will be reported in Section 4.5. At this point, the reader’s attention is drawn to the fact that the procedure of transforming 2D strips into 1D chromatograms (see Fig. 4.7) once more corresponds to the overlapping mechanisms described in Fig. 4.2 and has been evocated in comparing Fig. 4.4 with Fig. 4.3. In this way, if random structures (e.g., such as those marked in Fig. 4.1b) are present, their “memory’’ is lost and the 2D pattern is reduced to a Poissonian 1D one. Therefore, the number of SCs can be correctly estimated, even if the 2D pattern was not Poissonian.

DECODING 1D AND 2D MULTICOMPONENT SEPARATIONS

0

2

4

Second separation axis, a.u.

0

6

2

73

8

4

8

6

8 6 4 2 0 0.0

0.2

0.4

0.6

0.8

1.0

First separation axis, a.u.

FIGURE 4.7 Procedure developed to reduce a 2D separation into several 1D chromatograms suitable for applying SMO method on 2D maps. As an example, only the first two strips are reported.

The SMO can be extended to the overlap probabilities of 2D spots. Davis and coworkers (Davis, 1991; Oros and Davis, 1992) determined the number of singlets, doublets, triplets, and total number of spots as s ¼ m expð4a2D Þ

ð4:21Þ

d¼

m expð4a2D Þ½1expð4a2D Þ 2

ð4:22Þ

t¼

m expð4a2D Þ½1expð4a2D Þ2 3

ð4:23Þ

p¼m

4a2D expð4a2D Þ 1expð4a2D Þ

ð4:24Þ

74

DECODING COMPLEX 2D SEPARATIONS

Equations 4.22–4.24 are the 2D equivalents of Equations 4.17–4.20. The comparison of the two sets of equations shows a surprising consequence. If the peak capacities of the 1D and 2D separation systems were identical, the 2D separation would lead to more severe overlap. In order to have the same number of components isolated as singlets with a 1D and a 2D separation system, the peak capacity of the 2D system (n2D) should be double of that of the 1D system (n1D). Ideally, in an orthogonal system n2D ¼ n21D , but part of the gain in peak capacity is lost due to the increased probability of peak overlap provided the 2D chromatogram is disordered.

4.4 DECODING MULTICOMPONENT SEPARATIONS BY THE AUTOCOVARIANCE FUNCTION Multicomponent signals often contain hidden structure or periodicities that are difficult to spot. Due to structural relationships that exist among the different sample components, the retention pattern—the distribution of peaks or spots—is characteristic, as discussed in Section 4.2. The ordered or structured distribution of the spots in a 2D separation space is often hidden to the eye but can be recognized by a proper mathematical tool. Fourier analysis has been long used to identify periodicity and order deeply buried in signals. Therefore, by calculating the power spectrum of the chromatogram, one can obtain information not directly available from the chromatogram. The statistical model of peak overlap clearly explains that the number of observed peaks is much smaller than the number of components present in the sample. The Fourier analysis of multicomponent chromatograms can not only identify the ordered or disordered retention pattern but also estimate the average spot size, the number of detectable components present in the sample, the spot capacity, and the saturation factor (Felinger et al., 1990). Fourier analysis has been applied to estimate the number of detectable components in several complex mixtures. An adaptation of Fourier analysis to 2D separations can be established by calculating the autocovariance function (Marchetti et al., 2004). The theoretical background of that approach is that the power spectrum and the autocovariance function of a signal constitute a Fourier pair, that is, the power spectrum is obtained as the Fourier transform of the autocovariance function. A multicomponent 2D chromatogram is considered as a series of 2D peaks with random position and height. For the sake of simplicity, here we assume that the peaks are modeled with bidimensional Gaussian peaks, thus the signal is expressed as ¥ X

"

ðxx0;i Þ2 ðyy0;i Þ2 hi exp f ðx; yÞ ¼ 2s2x 2s2y i¼¥

# ð4:25Þ

where hi is the random height and (x0,i, y0,i) the random position of peak i. The x and y widths of the elliptic 2D peaks are sx and sy, respectively. The autocovariance function of a 2D signal is calculated as

DECODING MULTICOMPONENT SEPARATIONS

s2 þ a2h cðx; yÞ ¼ h Tx Ty

75

Z¥ Z¥ uðv; wÞuðv þ x; v þ yÞdvdw

ð4:26Þ

¥ ¥

where ah is the mean value and s2h is the variance of the peak amplitudes; Tx and Ty are the mean x and y distances between adjacent peaks. Therefore, for f(x,y), we get the following autocovariance function: " # VT2 ðs2h =a2h þ 1Þ x2 y2 exp 2 2 cðx; yÞ ¼ 4sx 4sy 4psx sy mXY

ð4:27Þ

where m is the number of detectable components and X and Y are the lengths of the rectangular separation space. In the derivation of the above equation, we introduced the volume under a bidimensional Gaussian peak as Vi ¼ 2psxsyhi and the total volume under the entire chromatogram as VT ¼ 2psxsymah, where sx and sy are the SC spot standard deviation along the two separation axes. The above equation gives the autocovariance function of a disordered 2D chromatogram and serves as a basis to estimate the number of detectable components and the average widths of the peaks (i.e., spot sizes) along the x and y axes. In Fig. 4.8a a section of a disordered 2D separation map and its autocovariance function is plotted. The main peak is a bidimensional Gaussian peak centered at the origin (Fig. 4.9). Besides this peak in the center, some minor fluctuations of the autocovariance function are observed (Fig. 4.8b). The fluctuations can be considered as noise due to the fact that in a finite 2D chromatogram, we have a small number of spots and a finite section of the stochastic ensemble. The 2D autocovariance function can also be calculated from the 2D chromatogram acquired in digitized form, Ck;l ¼

y l x k N X k ¼ Mx ; ; 1; 0; 1; ; Mx 1 NX fi;j f Þ fiþk; jþl f l ¼ My ; ; 1; 0; 1; ; My Nx Ny i¼1 j¼1

ð4:28Þ

where Mx and My are the maximum number of lags over which the 2D autocovariance function is calculated. A nonlinear curve fitting procedure of the experimental (Eq. 4.28) to the theoretical (Eq. 4.27) 2D autocovariance function can serve to perform some fundamental characterization of the 2D separation. The total volume (VT) and the peak height dispersion (s2h =a2h ) can be readily measured in the chromatogram, thus the number of components (m) and the peak widths (sx and sy) can be estimated (Marchetti et al., 2004). The 2D autocovariance function can be utilized in a rather simple manner to get a quick estimation of the number of detectable components and the average peak widths. For that purpose only the maximum amplitude of the autocovariance function and the width at half-height should be measured (Pietrogrande et al., 2005, 2006a). The

76

DECODING COMPLEX 2D SEPARATIONS

FIGURE 4.8 Autocovariance function method computed on computer-generated 2D map. (a) Simulated disordered 2D maps containing 100 components. (b) Autocovariance function of the 2D map. Reproduced from Marchetti et al., (2004) with permission from the American Chemical Society.

autocovariance function at the origin equals cð0; 0Þ ¼

VT2 ðs2h =a2h þ 1Þ 4psx sy mXY

ð4:29Þ

Furthermore, since the xﬃ and y half-widths of the bidimensional Gaussian at halfpﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃ height are hx;1=2 ¼ 2 ln2sx ¼ 1:665sx and hy;1=2 ¼ 2 ln2sy ¼ 1:665sy , the number of components is estimated as m¼

VT2 ðs2h =a2h þ 1Þln 2 0:22 VT2 ðs2h =a2h þ 1Þ ¼ phx;1=2 hy;1=2 C0;0 XY hx;1=2 hy;1=2 C0;0 XY

ð4:30Þ

Only the central section of the autocovariance function has to be calculated (Fig. 4.9) and simple computations are required to estimate the sample complexity, that is, the

DECODING MULTICOMPONENT SEPARATIONS

77

FIGURE 4.9 The autocovariance function close to the origin of a 2D separation. The maximum amplitude and the characteristic half-widths at half-heights are used in the simplified analysis. Reproduced from Marchetti et al., (2004) with permission from the American Chemical Society.

number of SCs, m, and the separation performance represented by the average spot dimensions sx and sy. These computations, based on Equations 4.27–4.30, require the detection of the spots and the evaluation of their intensities (i.e., volumes) so that the peak height dispersion s2h =a2h is obtained from the 2D map. This problem is discussed in detail by Marchetti et al. (2004). The authors analyzed several 2D maps with the number of components up to m ¼ 1000 and saturation factors between a ¼ 0.05 and 0.3. They could estimate with accuracy the number of components and the average spot dimensions by fitting Equation 4.27 to the 2D autocovariance function calculated from the 2D separation map. Also, the simplified approach based on Equation 4.30 was successfully applied to computergenerated maps (Pietrogrande et al., 2005). The estimation of the number of components is essential for the calculation of the separation extent (see Eq. 4.8), which characterizes the completeness of the separation process. The estimation of the average spot size may be an important means to optimize the efficiency of the 2D separation and to detect overloading effects. When the positions of the spots reveal an ordered pattern on the separation map, the long-term correlations in the autocovariance function can be used to decode the ordered structure of the retention pattern. We can use a simple linear relationship to estimate the position of the nth spot (see Eq. 4.1) ðxn ; yn Þ ¼ ðDmx ; Dmy Þ ¼ ðax þ bx n; ay þ by nÞ

ð4:31Þ

The above relationship can be related to the change of partition free energies of members of homologous series due to the addition of a given structural increment or to any physical–chemical property that affects the retention.

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DECODING COMPLEX 2D SEPARATIONS

The 2D autocovariance function of the ordered chromatogram is " # nmax VT2 ðs2h =a2h þ 1Þ X 1 ðxbx kÞ2 ðyby kÞ2 exp cðx; yÞ ¼ 4s2x 4s2y 4psx sy mXY k¼0 ðnmax k1Þ

ð4:32Þ

where nmax þ 1 is the number of elements of the ordered series. The above autocovariance function shows a regular repetitive pattern of peaks. Since the distances between adjacent spots in the chromatogram are bx and by in the x and y directions, respectively, these distances and their harmonics will appear in the autocovariance function, yielding an ordered 2D spot series. Most often, real 2D chromatograms exhibit a composite ordered and disordered characteristic, that is, a series of disordered spots are superimposed over ordered spot sequences. When the chromatogram is derived from a mixture of several chemical families, a superficial look at the 2D separation map may give the impression of randomness. In that case, the autocovariance function, however, can resolve and help identify the hidden structured nature of the map. In Fig. 4.10, the superposition of 10 different ordered spot sequences is shown. All of them have different ax, ay phase values but the same ax, by frequencies. The symmetry of the spot series in the autocovariance function is obvious and the common frequency of the spot trains can be easily identified through the main spot train of the autocovariance function, which runs through the origin. The most intense, major spot train in the autocovariance function follows the direction of the spot trains in the separation map. The spot trains running along the x and y axes give the horizontal and vertical characteristic spot interdistances in the separation map. As an X-ray diffraction image map helps identify the lattice structure of a crystal, the autocovariance function of a 2D separation map may help recognize the chemical structure of complex mixtures. Another case where the spot trains of constant phases and random frequency values are superimposed is illustrated in Fig. 4.11. In the neighborhood of the origin, the frequency values can be easily identified. Symbol þ indicates the 10 frequency values for n ¼ 1 in Equation 4.27. When lines are drawn from the origin through these points, the spots corresponding to n ¼ 2 and n ¼ 3 can also be identified. Now the autocovariance function looks less ordered than before and the repetitive pattern is more difficult to recognize. Nevertheless, the directions along which the spots are repeated with constant distances can be identified and the superposition of chemical families is recognized. The applicability of the 2D autocovariance function method and the most relevant results obtained will be discussed in the next section.

4.5 APPLICATION TO 2D SEPARATIONS This section reviews the most recent results obtained by applying the abovedescribed methods to complex 2D separations. In particular, the case of 2D-PAGE

APPLICATION TO 2D SEPARATIONS

79

FIGURE 4.10 The autocovariance function of a complex map. (a) Density plot of 10 2D sequences (represented by Eq. 4.31). The structured map was obtained by assuming constant frequency values (bx ¼ 0.1 and by ¼ 0.1) and by changing the phase values (ax, ay)i to the following values: 1 (0.0445, 0.8182), 2 (0.0435, 0.7416), 3 (0.0232, 0.4456), 4 (0.0398, 0.6897), 5 (0.0554, 0.2403), 6 (0.0600, 0.0291), 7 (0.2585, 0.0411), 8 (0.4686, 0.0460), 9 (0.5264, 0.0301), 10 (0.8635, 0.0395). (b) 2D autocovariance function of map in (a) Reproduced from Marchetti et al. (2004) with permission from the American Chemical Society.

(2D-polyacrylamide gel electrophoresis) maps of protein mixtures is discussed. 2D PAGE is considered the classical and principal tool for protein separation—prior to mass spectrometry—to achieve the main goal of proteomics, that is, a comprehensive identification and quantification of every protein present in a complex biological sample that would allow analysis of an entire intact proteome (Wilkins et al., 1997; Righetti et al., 2001; Hamdan and Righetti, 2005). The major advantage of 2D-PAGE is that it enables simultaneous separation of thousands of unknown proteins, first by charge using isoelectric focusing (IEF) and

80

DECODING COMPLEX 2D SEPARATIONS

FIGURE 4.11 The autocovariance function of a structured map. (a) Density plot of 10 2D sequences (see Eq. 4.31). The structured map was simulated by assuming constant phase values (ax ¼ 0.05 and ay ¼ 0.05) and by changing the frequency values (bx, by)i to the following: 1 (0.1395, 0.2001), 2 (0.1932, 0.0542), 3 (0.1135, 0.0843), 4 (0.3568, 0.0549), 5 (0.0123, 0.2319), 6 (0.2873, 0.0102), 7 (0.0332, 0.1474), 8 (0.1512, 0.2576), 9 (0.3501, 0.1504), and 10 (0.2212, 0.2543). (b) 2D autocovariance function of map in (a); frequency position (bx, by)i; see (a) corresponding to correlation at n ¼ 1 (þ) (see Eq. 4.31); correlations at n ¼ 2 (~), correlation at n ¼ 3 (^); correlations belonging to the same series (dashed line). Reproduced from Marchetti et al. (2004) with permission from the American Chemical Society.

then by size using SDS–polyacrylamide gel electrophoresis. Since each cell or biological fluid can be formed by thousands of proteins present in a wide range of relative abundance and displaying great differences in structure and size, a comprehensive separation of all the proteins present in the sample is still far from being achieved and comigrating proteins in the same spot are in general present. At present, increasing accuracy and precision are achieved in image acquisition of the complex 2D gel and spot detection due to the development of computer-assisted analysis as well as in protein spot identification and quantitation due to the developing

APPLICATION TO 2D SEPARATIONS

81

of specific software programs (Melanie; PD Quest; Orengo et al., 2003). The explosion in proteomics research in recent years has brought a very large amount of generated information and data analysis algorithms to efficiently use the large amounts of data produced by each analytical run and solve the complexity of the proteome sample (Orengo et al., 2003; Farriol-Mathis et al., 2004; Stanislaus et al., 2004; Marengo et al., 2005). The statistical degree of overlapping (SDO) and 2D autocovariance function (ACVF) methods have been applied to 2D-PAGE maps (Marchetti et al., 2004; Pietrogrande et al., 2002, 2003, 2005, 2006a; Campostrini et al., 2005): the means for extracting information from the experimental data and their relevance to proteomics are discussed in the following. The procedures were validated on computersimulated maps. Their applicability to real samples was tested on reference maps obtained from literature sources. Application to experimental maps is also discussed. 2D maps with known separation properties were generated by computer calculations: each spot is considered as a point described by two position coordinates (pI and Mr of the center of the spot) and by a third coordinate describing spot intensity. The rejection algorithm (Pietrogrande et al., 2002; Marengo et al., 2005) was applied for independently generating the pI and Mr coordinates displaying the same distribution along the pI and Mr axes present in experimental maps: it was computed on the basis of the pI and Mr coordinates of 1956 identified spots in reference maps of human tissues retrieved from the SWISS-2D-PAGE database (Expasy). Intensity values were generated according to a random distribution, since it has been demonstrated to be the most probable distribution for a high number of components (Marchetti et al., 2004). Moreover, experimental reference maps of human tissues were studied: pI and Mr coordinates of identified spots were retrieved from the SWISS-2D-PAGE database, the values sx ¼ 0.009 pH and sx ¼ 0.0002 log Mr were assumed for spot dimension since they represent the standard case for experimental 2D-PAGE maps—normal sample loading of a tissue homogenate (ca. 1 mg total protein) and standard gel sizes (18 20 cm, IEF SDS-PAGE). As an example, the 2D-PAGE map of colorectal adenocarcinoma cell line (DLD1_HUMAN) (Demalte-Annessi et al., 1999) is reported in Fig. 4.1a. Looking into the details of the colorectal adenocarcinoma map, some regions exhibit the random pattern (disordered region in Fig. 4.1b, bottom histogram), while others show an order in spot position, since trains of protein spots are evident (ordered region in Fig. 4.1b, top histogram). The final result is that spot positions in the whole map are completely disordered, as the result of superimposition of many ordered and disordered sequences: by applying a c2 statistical test, the H0 hypothesis was accepted to verify that the SC positions in the real map display a uniformly random distribution (Pietrogrande et al., 2003, 2005). 4.5.1

Results from SMO Method

4.5.1.1 Estimation of the Separation Parameters The extension of the SDO procedure to 2D separations implies that the 2D map is divided into many strips considered as 1D separations on which computations are performed. Different

82

DECODING COMPLEX 2D SEPARATIONS

procedures can be used for division into strips: it can be performed along the pI or the Mr axis to obtain a fixed (10, second column in the table reported below) or a variable number of strips, by fixing the minimum number of SCs present in each strip (20, second column in the Table reported below) (Pietrogrande et al., 2002). On the basis of spot position and intensity, by assuming different critical inter0 distance Dx 0 values, experimental points can be obtained and fitted by a straight line the estimated number of (Eq. 4.13) whose slope represents a statistical estimation of m, single components. The values estimated for each 1D strip were added to obtain the total number of proteins (Pietrogrande et al., 2002). As an example, the data computed on colorectal adenocarcinoma cell line map (DLD1_HUMAN) (Demalte-Annessi et al., 1999; Pietrogrande et al., 2002). (Fig. 4.1a) are reported as follows. m 108 108

Strip number 10 3(20)

pﬃﬃﬃﬃ m

e%

CV%

101 10 105 10

5.6 2.8

2.0 1.9

m

Even if the number of proteins present in the map is low, the method provides a values with good accuracy—expressed as the relative helpful tool to estimate m vs. real m values, e%—and precision—as relative standard error of the estimated m deviation CV% on 50 runs—in particular, if the division procedure into variable number of strips is used (Pietrogrande et al., 2002). values estimated 4.5.1.2 Estimation of Spot Overlapping Degree Using the m from the SMO and 2D autocovariance function methods, the original SDO approach (Davis and Giddings, 1983, 1985) can be used to statistically estimate the overlapping degree present in the map, that is, number of spots formed by one, two, and three proteins (Pietrogrande et al., 2003). This information is useful for estimating the influence of different experimental conditions (strip dimension, detection system performance, pI range) on spot overlapping. The method (based on Eqs. 17–29) was applied to computer-generated 2D separations describing experimental 2D-PAGE gels. It was verified that the common situation of 2D-PAGE maps—ca. 1 mg total protein loaded and 18 20 cm, IEF SDS-PAGE gel sizes—is an overcrowded condition, where the singlets are the least abundant species. For example, in the case of 1500 proteins loaded, the singlets would be only 27% of the observed spots and this value falls to 14% for a total of 3000 polypeptide chains in the sample (Pietrogrande et al., 2003). Moreover, the method makes possible the prediction of the influence of different experimental conditions (strip dimension, detector system performance, pI range) on the overlapping degree. This information is useful for estimating the degree of error associated with identification and quantitation of each protein and for setting up experimental strategies, which will increase the resolution and separation performance (Pietrogrande et al., 2003). For example, the most overcrowded case

APPLICATION TO 2D SEPARATIONS

83

of a 3000 protein mixture, the dramatic situation of broad-range immobilized pH gradient (IPG) (pH 3–10 gradient), in 7 cm long strips—in which only 5% of the proteins will appear on the map as pure spots—can be significantly improved using longer strips—39 cm long to obtain 29% of pure spots—or detector devices with higher resolution—s ¼ 43%. An extraordinary improvement in spot purity is obtained if narrow-range strips—1 pH unit is covered between acid and basic zones—are used, that is, from 33% of singlets for 7 cm strips to 69% and 75% for 18 and 39 cm strips, respectively. Recently, the SMO and SDO models were, for the first time, applied to an experimental 2D-PAGE map and validated by data independently obtained by mass spectrometry (MS) analysis (Campostrini et al., 2005). The studied sample was a neuroblastoma xenograft implanted in mice, submitted to 2D-PAGE separation and analyzed by MS in a quadrupole—time-of-flight (TOF) mass spectrometer (2D map in Fig. 4.12). The spot purity was experimentally measured by MS analysis on 74 studied spots: 52 spots were found to be singlets (71% of the spots), 14 (19%) were doublets, 6 (8%) were triplets, and 1 each were quadruplets and quintuplets. The overlapping degree was also estimated using the SDO model: both the methods based on addition of 1D strips (Eqs 4.17–4.20) and on the whole 2D map (Eqs. 4.21–4.24) yielded the same results, since a low saturation is present in the map. The estimated values show an excellent agreement with experimental MS data

FIGURE 4.12 Experimental 2D-PAGE map of a neuroblastoma xenograft implanted in mice. The circles indicate the spots identified by MS analysis. Reproduced from Campostrini et al. (2005) with permission from Wiley-VCH Verlag GmbH.

84

DECODING COMPLEX 2D SEPARATIONS

suggesting that the SDO model is a powerful and robust method to accurately predict the overlapping degree present in a map. 4.5.1.3 Identification of the Ordered Pattern The SDO method can also be a powerful tool for identifying the presence of an ordered structure in the spot positions, even if it is hidden in the spot overcrowding and might escape detection. In fact, the specific feature of the experimental points, that is, deviation from linearity in Equation 4.13, can be related to properties of the separation pattern. A general relationship has been derived, showing that the function describing the interdistance distribution f(x) can be obtained by computing the first derivative of the relationship between the quantity 1=yobs and the critical interdistance values 0 Dx 0 (Pietrogrande et al., 2002):

f ðxÞ ¼ y

d

1 yobs

Dx0 0

ð4:33Þ

As an example, the case of an ordered region of the colorectal adenocarcinoma map (DLD1_HUMAN; Fig. 4.1b) is reported. Spot positions in an ordered region were projected into the pI separation axis and the histogram of pI interdistance distribution was computed (top inset in Fig. 4.1b). The order is singled out by the presence of some maxima in the histogram, corresponding to the interdistance values repeated in the map, for example, 0.04 and 0.06 pI values. This ordered separation pattern (deviation from linearity in Eq. 4.13) can also be correctly estimated by computation of Equation 4.33 (plot reported in the top inset in Fig. 4.1b). The obtained results show a good correspondence between the real (histogram in the inset in Fig. 4.1b) and the estimated distribution (continuous line in the inset in Fig. 4.1b). These results are particularly relevant, since such constant interdistances may be related to the chemical properties of the proteins present in the sample. For example, they may be consistent with protein isoforms differing in a constant variation of the number of ionogenic groups in the molecule suggesting the presence of co- and posttranslational modifications (Wilkins et al., 1997; Righetti et al., 2001; Mann and Jensen, 2003; Farriol-Mathis et al., 2004; Hamdan and Righetti, 2005). 4.5.2

Results from 2D Autocovariance Function Method

4.5.2.1 Estimation of the Separation Parameters The 2D autocovariance function method was validated on computer-generated maps describing experimental 2D-PAGE maps and applied to some reference 2D maps retrieved from the SWISS-2D-PAGE database (Marchetti et al., 2004; Pietrogrande et al., 2005): a hepatoblastoma-derived cell line (HEPG2_HUMAN) (Sanchez et al., 1995, 1997), a colorectal adenocarcinoma cell line (DL-1) (DLD1_HUMAN) (Demalte-Annessi et al., 1999) and a human plasma (PLASMA_HUMAN) (Blatter et al., 1993; Eckerskorn et al., 1997). The elliptical spot shape represents experimental conditions, that is,

APPLICATION TO 2D SEPARATIONS

85

sx ¼ 0.009 pH and sx ¼ 0.0002 logMr , correspond to the standard case of a 18-cm strip of broad pH range (pH 3–7) with standard (1 mm) scanner resolution (Pietrogrande et al., 2003, 2005). Spot abundance (AM) was described by an exponential (E) distribution (first column in the following table) yielding s2h =a2h ¼ 1:0 (fifth column). The 2D autocovariance function was computed on the digitized map signal using Equation 4.28 and the separation parameters were estimated according to Equation 4.30. The results obtained are reported in the following table (Pietrogrande et al., 2005). AM E E E

m

sx

HEPG2 0.009 99 DL-1 0.009 108 PLASMA 0.009 626

sy

s2h =a2h

mest

sx,est

sy,est

s2m =a2m

0.0002

1

100 10 0.009

0.0002

0.99

0.0002

1

104 10 0.009

0.0002

0.99

0.0002

1

601 24 0.009

0.0002

0.97

A correct estimation of the number of proteins, m, present in the sample can be obtained within a bias of 4% (compare second and sixth columns), though the number of proteins considered is low and the theoretical model describing the distribution of SC positions is far from being thoroughly known. The most critical parameter in Equation 4.30 is the SC abundance dispersion ratio (s2h =a2h , fifth column in the table). As a consequence of SC spot overlapping, the experimentally accessible parameter is the maximum spot dispersion ratio (s2m =a2m , ninth column), which is introduced in Equation 4.30 to approximate the s2h =a2h value (compare fifth and ninth columns in the table). The results in the table also show that the 2D autocovariance function method gives a correct estimation of the mean spot shape for all the simulated maps (sx and sy in third to fourth and seventh to eighth columns) (Pietrogrande et al., 2005). It must be emphasized that the availability of the SMO and 2D autocovariance function methods as two independent statistical procedures to estimate the same parameter, m, the number of proteins, is a helpful tool to verify the reliability of the results obtained. In the case of the 2D PAGE map of colorectal adenocarcinoma cell line (DL-1) an excellent agreement was found between the values obtained from the SMO method—m ¼ 101 10 and m ¼ 105 10—and the 2D autocovariance function procedure—m ¼ 104 10 (Pietrogrande et al., 2006a). 4.5.2.2 Identification of Ordered Structures The great strength of the 2D autocovariance function in identifying ordered sequences was tested in the case of spot trains in 2D-PAGE maps. They are consistent with protein isoforms differing in a constant variation of the number of ionogenic groups in the molecule, as a consequence of co- and post-translational modifications (PTMs), such as glycosylation, phosphorylation and deamidation (Wilkins et al., 1997; Righetti et al., 2001; Hamdan and Righetti, 2005). Identification of protein PTM’s is very important information for proteomics, since it has been well established that PTMs

86

DECODING COMPLEX 2D SEPARATIONS

APPLICATION TO 2D SEPARATIONS

87

occur on almost all proteins and are of extreme biological importance, that is, they can regulate a variety of protein activities, such as enzymatic activity, ability to interact with other proteins, subcellular localization, and targeted degradation (Mann and Jensen, 2003; Farriol-Mathis et al., 2004). As an example, the case of spot trains showing a one-dimensional shift parallel to the pI axis is reported. This is a common feature in 2D gels due to protein isoforms submitted to PTMs yielding a change in amino acid charges with a consequent alteration in pI, while not necessarily in Mr (Wilkins et al., 1997; Righetti et al., 2001; Hamdan and Righetti, 2005). For studying this effect, a computer-simulated map was generated (Fig. 4.13a), where a train of eight spots in the pI range 5–6.4 pH, with a constant DpI of 0.2 pH at a constant log Mr value of 0.67 was superimposed onto a disordered map containing 200 SCs. The 2D autocovariance function was computed on the selected map region that contains the spot train (eight proteins) in addition to 53 proteins randomly located (4.5–7 pI and 0.65–0.68 log Mr values, enlarged inset in Fig. 4.13a). From the 2D autocovariance function, the number of proteins present in this map region, m, can be correctly estimated: m ¼ 53 7 for the original computer-generated map containing 53 SCs (blue line in inset in Fig. 4.13b) and m ¼ 62 8 for the map where the spot train was added (red line in inset in Fig. 4.13b). In the 2D autocovariance function plot (Fig. 4.13b) well defined deterministic cones are evident along the DpI axis at values DpH 0.2, 0.4, 0.6 pH: they are related to the constant interdistances repeated in the spot trains. This behavior is more clearly shown by the intersection of the 2D autocovariance function with the DpI separation axis. The inset in Fig. 4.13b reports the 2D autocovariance function plots computed on the same map with (red line) and without (blue line) the spot train. A comparison between the two lines shows that the 2D autocovariance function peaks at 0.2, 0.4, 0.6 DpH (red line) clearly identifying the presence of the spot train singling out this ordered pattern from the random complexity of the map (blue line, from map without the spot train). The difference between the two lines identifies the contribution of the two components to the complex separation: the blue line corresponds to the random separation pattern present in the map; the red line describes the order in the 2D map due to the superimposed spot train. The high sensitivity of the 2D autocovariance function method in detecting order is noted; in fact it is able to detect the presence of only sevenfold repetitiveness hidden in a random pattern of 200 proteins (Pietrogrande et al., 2005).

3 FIGURE 4.13 Identification of a train of spots by the 2D autocovariance function method. (a) Simulated map formed by a train of eight spots in the pI range 5–6.4, with a constant DpI of 0.2 pH and at a constant log Mr value of 0.67, in addition to 200 SCs randomly located. Enlarged detail: selected 0.65–0.68 log Mr region of the map containing a train of eight spots. (b) Plot of the 2D autocovariance function computed on the selected map region containing a train of eight spots. Enlarged detail: 2D-EACVF values over the pI separation axis: comparison between the 2D EACVF computed on the map with the train of spots (red line) and the map without the spot train (blue line). Reproduced from Pietrogrande et al. (2005) with permission from Wiley-VCH Verlag GmbH. (See color plate.)

88

DECODING COMPLEX 2D SEPARATIONS

4.6 CONCLUDING REMARKS At present, increasing accuracy and precision are achieved in separation performance and signal acquisition for complex 2D separations. However, the complexity of the plethora of data obtained requires proper signal processing procedures for a complete extraction of all of the analytical information. The mathematical–statistical methods reviewed here have proven to be powerful tools for the extraction of the most relevant information on the separation: sample complexity, separation performance, overlapping extent, and identification of ordered patternspresentinspotpositionsrelatedtochemicalcompositionofthecomplexsample. Moreover, the two procedures display different and complementary properties so that each of them is the method of choice to obtain specific information on the 2D separations. The SMO procedure is an unique tool to quantitatively estimate the degree of peak overlapping present in a map as well as to predict the influence of different experimental conditions on peak overlapping. The strength of the 2D autocovariance function method lies in its ability to simply single out ordered retention pattern hidden in the complex separation, which can be related to information on the chemical composition of the complex mixture. These statistical methods give a comprehensive description of the whole separation and therefore can also be used as tools to investigate separation properties for quality assurance procedures. Further developments of this topic are in progress, both in developing theoretical models and in applying the procedures to real experimental cases. In particular, the methods can be extended to hyphenated techniques to investigate the complex signals obtained from mass spectrometry detection (Pietrogrande et al., 2006b).

ACKNOWLEDGMENTS This work was supported by the Italian University and Scientific Research Ministry (grants 2003039537_005 and 2005032388_004) and by the University of Ferrara, Italy; it was also supported by grant T 048887 from the Hungarian National Science Foundation (OTKA).

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Davis, J.M., Giddings, J.C. (1983). Statistical theory of component overlap in multicomponent chromatograms. Anal. Chem. 55, 418. Davis, J.M., Giddings, J.C. (1985). Statistical method for estimation of number of components from single complex chromatograms: application to experimental chromatograms. Anal. Chem. 57, 2178. Demalte-Annessi, I., Sanchez, J.-C., Hoogland, C., Rouge, V., Binz, P.-A., Appel, R.D., Hochstrasser, D.F. (1999). Submitted to the SWISS-2D-PAGE database. Dondi, F., DuhaleKahie, Y., Lodi, G., Remelli, M., Reschiglian, P., Bighi, C. (1986). Evaluation of the number of components in multicomponent liquid chromatograms of plant extracts. Anal. Chim. Acta. 191, 261. Dondi, F., Bassi, A., Cavazzini, A., Pietrogrande, M.C. (1998). A quantitative theory of the statistical degree of peak overlapping in chromatography. Anal. Chem. 70, 766. Eckerskorn, C., Strupat, K., Schleuder, D., Hochstrasser, D.F., Sanchez, J.-C., Lottspeich, F., Hillenkamp, F. (1997). Analysis of proteins by direct-scanning infrared-MALDI mass spectrometry after 2D-PAGE separation and electroblotting. Anal. Chem. 69, 2888. Expasy,http://www.expasy.ch. Farriol-Mathis, N., Garavelli, J.S., Boeckmann, B., Duvaud, S., Gasteiger, S.E., Gateau, E., Veuthey, A.-L., Bairoch, A. (2004). Annotation of post-translational modifications in the Swiss-Prot knowledge base. Proteomics 4, 1537. Felinger, A., Pasti, L., Dondi, F. (1990). Fourier analysis of multicomponent chromatograms. Theory and models. Anal. Chem. 62, 1846. Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edition. John Wiley & Sons Ltd, New York. Giddings, J.C. (1995). Sample dimensionality: a predictor of order–disorder in component peak distribution in multidimensional separation. J. Chromatogr. A 703, 3. Guiochon, G. (2005). Private communication. Hamdan, H., Righetti, P.G. Proteomics Today. Wiley, Hoboken, 2005. Mann, M., Jensen, O.L. (2003). Proteomic analysis of post-translational modifications. Nature Biotech. 21, 255. Marchetti, N., Felinger, A., Pasti, L., Pietrogrande, M.C., Dondi, F. (2004). Decoding twodimensional complex multicomponent separations by autocovariance function. Anal Chem 76, 3055. Marengo, E., Robotti, E., Antonucci, F., Cecconi, D., Campostrini, N., Righetti, P.G. (2005). Numerical approaches for quantitative analysis of two-dimensional maps: A review of commercial software and home-made systems. Proteomics 5, 654. Melanie II, Geneva BioInformatics, GeneBio S.A., http://www.genebio.com. Orengo, C.A., Jones, D.T., Thorton, J.M. (2003). Bioinformatics. Genes, Proteins and Computers. BIOSOxford, UK pp. 245–271. Oros, F.J., Davis, J.M. (1992). Comparison of statistical theories of spot overlap in twodimensional separations and verification of means for estimating the number of zones. J. Chromatogr. A 591, 1. PD Quest, Bio-Rad Laboratories Inc., http://www.biorad.com. Pietrogrande, M.C., Dondi, F., Felinger, A., Davis, J.M. (1995). Statistical study of peak overlapping in multicomponent chromatograms: importance of the retention pattern. Chemometr. Intell. Lab. Syst. 28, 239.

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Pietrogrande, M.C., Marchetti, N., Dondi, F., Righetti, P.G. (2002). Spot overlapping in two-dimensional polyacrylamide gel electrophoresis separations: a statistical study of complex protein maps. Electrophoresis 23, 283. Pietrogrande, M.C., Marchetti, N., Dondi, F., Righetti, P.G. (2003). Spot overlapping in twodimensional polyacrylamide gel electrophoresis maps: relevance to proteomics. Electrophoresis 24, 217. Pietrogrande, M.C., Marchetti, N., Dondi, F. Righetti, P.G. (2006a). Decoding 2D-PAGE complex maps: relevance to proteomics. J. Chromatogr. B 833, 51. Pietrogrande, M.C., Marchetti, N., Tosi, A., Dondi, F., Righetti, P.G. (2005). Decoding twodimensional polyacrylamide gel electrophoresis complex maps by autocovariance function: a simplified approach useful for proteomics. Electrophoresis 26, 2739. Pietrogrande, M.C., Zampolli, M.G., Dondi, F. (2006b). Identification and quantification of homologous series of compound in complex mixtures: autocovariance study of GC/MS chromatograms. Anal Chem. 78, 2579. Righetti, P.G., Stoyanov, A., Zhukov, M.Y. (2007). The Proteome revisited: Theory and Practice of All Relevant Electrophoretic Steps. Elsevier, Amsterdam. pp. 275-378. Sanchez, J.-C., Appel, R.D., Golaz, O., Pasquali, C., Ravier, F., Bairoch, A. Hochstrasser, D.F. (1995). Inside SWISS-2D PAGE database. Electrophoresis 16, 1131. Sanchez, J-C., Wirth, P., Jaccoud, S., Appel, R.D., Sarto, C.Wilkins, M.R., Hochstrasser, D.F. (1997). Simultaneous analysis of cyclin and oncogene expression using multiple monoclonal antibody immunoblots. Electrophoresis 18, 638. Schoenmakers, P., Marriott, P, Beens, J. (2003). Nomenclature and conventions in comprehensive multidimensional chromatography. LC-GC Europe 16, 335. Stanislaus, R., Jiang, L.H., Swartz, M., Arthur, J. Almeida, J.S. (2004). An XML standard for the dissemination of annotated 2D gel electrophoresis data complemented with mass spectrometry results. BMC Bioinformatics 5, 9. Wilkins, M.R., Williams, K.L., Hochstrasser, D.F. editors. (1997). Proteome Research: New Frontiers in Functional Genomics. Springer, New York.

PART II COLUMNS, INSTRUMENTATION AND METHODS DEVELOPMENT

5 INSTRUMENTATION FOR COMPREHENSIVE MULTIDIMENSIONAL LIQUID CHROMATOGRAPHY Robert E. Murphy Kroungold Analytical, Inc., Encinitas, CA 92024, USA

Mark R. Schure Theoretical Separation Science Laboratory, Rohm and Haas Co., Springhouse, PA 19477-0904, USA

5.1 INTRODUCTION Multidimensional liquid chromatography (MDLC) is a high performance separation system that is gaining popularity for the separation of complex samples including synthetic polymers and biomolecules. The two-dimensional version of MDLC, 2DLC, has gained popularity in recent years as a column-based separation method with much higher resolution, selectivity, and peak capacity than single-column separation methods. In these systems, sequential finite-volume aliquots from the first column (i.e., first dimension) are sampled by the second column (i.e., second dimension) typically by using sampling loops and valves. These multidimensional chromatographic systems, which operate primarily with liquid-phase samples, have been reviewed previously and these references are listed in Chapter 1 of this book. In this chapter, we will review the instrumentation requirements for comprehensive 2DLC systems and for the simpler systems that make deliberate cuts of the first dimension Multidimensional Liquid Chromatography: Theory and Applications in Industrial Chemistry and the Life Sciences, Edited by Steven A. Cohen and Mark R. Schure. Copyright 2008 John Wiley & Sons, Inc.

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FIGURE 5.1 Heart-cut 2DLC where a zone from the first-dimension separation is reinjected onto a second-dimension column for improved resolution of three analytes coeluting on the first-dimension column.

and divert the zone to another column—the so-called “heart cutting’’ mode of operation. In the heart-cutting mode of operation, one or several discrete zones are collected from the first-dimension column and reinjected into the second-dimension separation system. The resulting data are one or more individual one-dimensional datasets and are useful for resolving fused peaks from specific region(s) of the first-dimension separation system. An example of zone reinjection is shown in Fig. 5.1; clearly, the second column provides the selectivity for the three peaks that the first column did not have. Heart cutting can be implemented in the simplest of cases with a fraction collector (or even just a bottle) where the zone of interest or a segment of the zone is collected. The collected sample is then injected into an LC system with different column selectivity. This mode of operation has a long history starting with the isolation of plant chlorophyll extracts using adsorption columns (Manning and Strain, 1943). These early developments of the heart-cutting methodology are summarized by Dixon et al. (2006) in their review of MDLC. However, the modernversion of heart cutting appears to be described using high performance liquid chromatography (HPLC) equipment by Huber et al. (1973) where a valve is used to divert the column effluent into another column system with separate pump and detector(s). This configuration is shown in Fig. 5.2 utilizing a six-port valve. The simplest way to conceptualize the comprehensive 2DLC experiment is to collect fractions from a single column (in this case the first dimension or first column). All of the collected fractions are then injected individually into a second column (the second dimension). The resulting chromatograms are then put into the rows of a matrix. An eight-port or a ten-port valve with dual matching sample loops is generally used for automated comprehensive 2DLC so that solute is injected repetitively into the

HEART-CUTTING VERSUS COMPREHENSIVE MODE

Pump 1

Pump 1

Column 1

Column 1

Position A

95

Position B Waste

Waste

Loop

Loop

Pump 2

Pump 2 Column 2

Column 2

Detectors

Detectors

FIGURE 5.2 Heart-cut 2DLC with a six-port valve. In position A, the loop is being loaded with a sample from column 1, and column 2 is being equilibrated. In position B, the loop is injected onto column 2 where separation takes place with subsequent detection.

second-dimension separation system. Alternatively, columns can be used in place of sample loops. The specific choice of parameters used for the 2DLC experiment is discussed in the next chapter along with method development procedures. The detector signals from the second dimension column are formed into a matrix A and the matrix is visualized and analyzed. The Aij matrix is the amplitude of a singlechannel detector where i is the row number and represents the sequential injection number for the second-dimension analysis. The subscript j is the column number and designates the detector amplitude at time tj. It is assumed that the detector sampling rate is constant from injection to injection so that tj is a constant in each row. Furthermore tjþ1 tj is a constant throughout the chromatogram. The matrix is visualized through contours of constant amplitude or concentration or through a two-dimensional peak presentation. This presentation mode will be discussed in detail below.

5.2 HEART-CUTTING VERSUS COMPREHENSIVE MODE The number of samples or heart cuts injected into the second-dimension column defines whether the technique is referred to as heart cutting or comprehensive. We arbitrarily define heart-cut 2DLC as sampling less than 10 injections into the second dimension, and comprehensive 2DLC as sampling more than 10 injections into the second dimension with repetitive sampling over the length of the first-dimension experiment. Moreover, heart-cut 2DLC generally samples certain portions of interest

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TABLE 5.1 Classification and Examples of Two-Dimensional Liquid-Phase Separation Techniques Number of fractions sampled into second dimension Technique example Typically < 10 (heart cut)

Heart-cut LC Cross-fractionation LC/LC (mixed mode)

Typically > 10 and repetitive (comprehensive)

LC/LC (six-port valve) LC/CE Thermal FFF/GPC LC/LC (eight-port valve) Gel electrophoresis/LC CE/GE TLC/TLC IEF/PAGE

References Majors (1980) Augenstein and Stickler (1990) Pasch et al. (1992) Balke and Patel (1980) Chapter 11 Holland (1995) Lemmo and Jorgenson (1993) Venema et al. (1997) Erni and Frei (1978) Bushey and Jorgenson (1990) Rose and Opiteck (1994) Liu and Sweedler (1996) Rezanka (1996) Celis and Bravo (1984)

in the first dimension separation, whereas comprehensive 2DLC samples the entire first dimension into the second dimension. The difference in sampling also results in different data presentations. Heart-cut 2DLC is presented as stacked plots of each injection into the second dimension, and comprehensive 2DLC can be represented as a contour or projection. Table 5.1 lists several heart-cut and comprehensive techniques. Heart-cut 2DLC is very common and has great application for the increased resolution of one or several components from the first dimension (Augenstein and Stickler, 1990; Majors, 1980; Pasch et al., 1992; and Dixon et al., 2006). Heart-cut 2DLC for the analysis of polymers is often referred to as “cross-fractionation’’ (Balke and Patel, 1980). Protein digest analysis with MS/MS identification has been called “multidimensional protein identification technology’’ or “MUDPIT.’’This is described in detail in Chapter 11. In comprehensive 2DLC, partial sampling is used when the entire volume cannot be injected into the second dimension (Holland and Jorgenson, 1995; Venema et al., 1997) or when capillary electrophoresis (CE) is utilized in the second dimension (Lemmo and Jorgenson, 1993). 2DLC utilizing an eight-port or ten-port valve is commonly used for sampling the entire volume of the first dimension (Erni and Frei, 1978; Bushey and Jorgenson, 1990a). Gel electrophoresis (GE) was coupled to HPLC in a comprehensive mode (Rose and Opiteck, 1994), as well as CE and GE (Liu and Sweedler, 1996). Two-dimensional thin layer chromatography (2DTLC) (Rezanka, 1996) and polyacrylamide gel electrophoresis (PAGE)/isoelectric focusing (IEF) (Celis and Bravo, 1984) are also considered comprehensive since the entire sample is subjected to both dimensions noting that all two-dimensional planar techniques are comprehensive by definition.

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Far more information can be obtained by using comprehensive sampling as opposed to the heart-cutting method as higher resolution is typically obtained across the whole range of first-dimension elution (Erni and Frei, 1978; Bushey and Jorgenson, 1990b; Kilz et al., 1995; Murphy et al., 1998a). In some respect, heart cutting is useful when it is known where the zone is overlapped a priori but this case is rarely known for samples where composition is variable. The instrumentation used to implement this comprehensive sampling is the main topic of this chapter. The most common comprehensive mode uses a sampling valve so that second dimension elution can begin as soon as a sampling loop has stored the necessary amount of first column solute. In this case, there is no need for storing the effluent from the first column; it is continuously allocated to a sampling loop with subsequent injection into the second-dimension column. Schoenmakers et al. (2003) define a two-dimensional separation as comprehensive if 1. every part of the sample is subjected to two different separations; 2. equal percentages (either 100% or lower) of all sample components pass through both columns and eventually reach the detector; 3. the separation (resolution) obtained in the first dimension is essentially maintained. These authors clarify these criteria but the essential operation is that the comprehensive separation takes a one-dimensional data representation and through the use of a second separation mechanism converts this to a two-dimensional presentation of the data, as seen in most of the chapters of this book. Topics which will be presented in this chapter include the hardware, software, automation, valve and column configurations, and integration used in comprehensive 2DLC. Aspects of the 2DLC experiment in conjunction with multichannel detectors such as UV diode array optical detectors and mass spectrometers are discussed along with the handling of the data, which is expected to expand in scope in the future as chemometric methods are more widely used for data analysis. 5.3 CHROMATOGRAPHIC HARDWARE 5.3.1

Valves

Many different types of valves are used to control the collection and injection of stored column effluent. We will review most of the valve systems that have been used in 2DLC and will highlight the advantages of these systems where possible. Note that there is an endless combination of these configurations. We note one excellent review (Shalliker and Gray, 2006) that contains details of valve configurations with a good level of detail. When elution chromatography is used in both dimensions, the valve configurations are similar for the different column combinations. However, when CE is utilized as the second dimension, other types of interfaces not based on valves have been implemented with unique advantages. These and the microfluidic implementation of sampling systems for chip-based two-dimensional separations will be discussed below.

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5.3.1.1 2DLC with Partial Sampling or Heart-Cut Mode A six-port valve is generally used for the partial sampling of the first dimension separation. Fig. 5.2 shows the valve configuration for heart cutting parts of zones for further analysis in a second column. This setup can be used for sampling the first dimension one or several times depending on the run time in the second dimension and the problem being solved. The most common application of this valve is for analyzing one fraction that is unresolved in the first dimension, shown in Fig. 5.1. Operation with heart-cut 2DLC is not as complex as comprehensive 2DLC since the sampling is less frequent and most often done manually. Fig. 5.3 displays a schematic for the HPLC configuration using two six-port valves along with the steps employed for the quantitative analysis of antibodies in serum

FIGURE 5.3 2DLC configuration and sequence utilizing a protein A affinity column in the first dimension and an SEC column in the second dimension. In step 1, the sample is injected onto the affinity column and the first-dimension separation takes place while the SEC column is being equilibrated. In step 2, valve 1 moves to position 2 and a fraction of the affinity separation is collected into the loop. In step 3, valve 1 moves back to position 1 and the collected sample is injected onto the SEC column for MS analysis. In step 4, after the protein elutes from the SEC column valve 2 is switchedtoposition2andtheSECcolumneffluentissenttowastetoavoidsaltsfromenteringtheMS.

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FIGURE 5.4 Chromatograms of 2DLC (affinity/SEC/MS). Bottom trace is affinity separation with UV detection and 2 min fraction specified. Middle trace is MS total ion chromatogram showing protein elution and salts diverted to waste. Top trace in MS extracted ion chromatogram of protein of interest.

(Murphy et al., 2005). The system utilizes affinity chromatography in the first dimension for the retention of antibodies, and the six-port valve is used to capture the antibody fraction in the loop during elution, then transferred to a size exclusion chromatography (SEC) column. The SEC column separates the proteins from the lower mass salts so that mass spectrometry (MS) can be used as a detector without introducing deleterious effects due to the presence of salt. The resulting chromatograms are shown in Fig. 5.4. This system allows for the automated analysis of serum and is used for both quantitative and qualitative analyses. There are many more applications of 2DLC with a six-port valve and a list of biomedical applications was previously reviewed (Somsen and deJong, 2002). 5.3.1.2 2DLC with Complete Sampling or Comprehensive Mode Table 5.2 lists several comprehensive 2DLC methods with the corresponding columns in each dimension, valves, and detectors utilized. Many 2DLC systems use 8-port valves, 10-port valves, or fraction collectors to retain the entire sample or increase the concentration for the second dimension detection. An eight-port valve with matching sample loops is typically used for the coupling and repetitive sampling of the first-dimension separation system when the comprehensive mode of operation is utilized, as shown in Fig. 5.5. This valve configuration can be used for the heart-cut mode where only a portion of the sample from column 1 enters column 2, or it can be used in the comprehensive mode where the total sample from column 1 enters column 2 depending on the sampling rate. This was the valve configuration used by Erni and Frei (1978) in the first comprehensive 2DLC

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TABLE 5.2 Sampling First dimension

Two-Dimensional Liquid Separation Examples with Comprehensive Second dimension

Detection

Separation

Valve

References Bushey and Jorgenson (1990a) Isobe et al. (1991) Holland and Jorgenson (1995) Kilz et al. (1995) Opiteck et al. (1997)

CEC

SEC

UV

Protein mix

8-port

AEC

RPLC

UV

Brain extracts

6-port

AEC

RPLC

LIF

Protein digest

6-port

NPLC

SEC

UV, RI

6-port

CEC

RPLC

UV, MS

SEC

RPLC

UV

Copolymer blends Protein mix and E.coli lysate E.coli lysate

RPLC

SEC

ELS

Polymer mix

NPLC

RPLC

ELS

AEC/CEC

RPLC

UV

Surfactant oligomers Protein mix

HILIC

RPLC

UV, MS

CF

RPLC

MS

CEC

RPLC

UV

RPLC

RPLC

UV

Protein mix, cell lysates Breast cancer cell lysates Protein digest Maize metabolites

8-port

2 4-port

Opiteck et al. (1998) 8-port Murphy et al. (1998a) 8-port Murphy et al. (1998b) 8-port Unger et al. (2000) 8-port Murphy (2001) Fraction Chong et al. collector (2001) 10-port Stoll and Carr (2005) 2 6-port Stoll et al. (2006)

AEC: anion-exchange chromatography; CEC: cation-exchange chromatography; CF: chromatofocusing; HILIC: hydrophilic interaction chromatography; NPLC: normal-phase liquid chromatography; SEC: sizeexclusion chromatography; RPLC: reversed-phase liquid chromatography; ELS: evaporative light scattering; LIF: laser-induced fluorescence; MS: mass spectrometry; RI: refractive index; UV: ultraviolet.

instrument and in the first automated comprehensive 2DLC instrument (Bushey and Jorgenson, 1990a). Instead of using sample loops in the eight-port valve, several groups utilize matching columns to alternate sampling in the second dimension (Opiteck et al., 1997; Wagner et al., 2002). This can be difficult, however, because the elution volume of the columns must be exactly matched or adjacent rows in the 2D matrix will be offset and scaled due to variation between columns. This can be partially compensated for in software but it places severe demands on matching columns. Figure 5.6 shows the configuration of an eight-port valve with dual columns in the

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FIGURE 5.5 2DLC with eight-port valve and two sample loops. In position A, loop 1 (L1) is being loaded with sample from column 1, and loop 2 (L2) is being injected onto column 2. In position B, L2 is being loaded with sample from column 1, and L1 is being injected onto column 2.

second dimension. Sample loops are not necessary for 2DLC under a variety of conditions as highly aqueous solvent conditions can be used to load hydrophobic solutes into reversed-phase columns with subsequent elution under gradient conditions (Unger et al., 2000). The most common case of this is where components elute from an ion-exchange column into a reversed-phase column in the second dimension and these systems are commonly utilized in protein and peptide separations (Unger et al., 2000; Liu et al., 2002) and in other applications, for example, the separation of

Pump 1

Pump 1

Position A

Position B Column 1

Column 1

C2-B C2-A

Pump 2

Detectors Waste

Pump 2

Detectors Waste

FIGURE 5.6 2DLC with eight-port valve and two columns in second dimension. In position A, column 2-A is being loaded with sample from column 1, and column 2-B is being analyzed utilizing pump 2. In position B, column 2-B is being loaded with sample from column 1, and column 2-A is being analyzed.

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complex surfactant mixtures (Haefliger, 2003). In these cases, a 10-port valve is most commonly used, as discussed below. All of the valves used in 2DLC are two-position valves as they alternate between two positions (A and B in Figs. 5.2, 5.5 and 5.6). Two-position 10- and 12-port valves may also be used similarly to six- and eight-port valves and the differences in application with these are now discussed. It is commonly assumed in 2DLC instruments that employ sample loops that there is complete mixing within the sample loop prior to flushing the loop volume to the second column. However, it was discovered (Van der Horst and Schoenmakers, 2003) that if the fluids in the two sample loops are pumped into the second column in different order (e.g., in the forward direction in loop 1 followed by the reverse direction in loop 2) the resulting 2D chromatograms show irregularities. This is typically the case with an eight-port valve and will depend on loop volume and the inherent broadness of the component zones. This can be corrected by allowing the fluid in the loop to be pumped in the same direction for both sample loops by using a 10-port valve. Figure 5.7 shows the 10-port valve configuration that allows the flow to enter and exit the sample loops in the same direction as they were filled. In fact, 10-port valves are becoming more popular for 2DLC. One commercial 2DLC system, the Perceptive Biosystems Integral 100Q workstation, utilized multiple 10-port pneumatically-driven twoposition valves for column switching and other functions. This unit is no longer manufactured. Higher performance systems where gradient elution is conducted at high speed (Stoll and Carr, 2005; Stoll et al., 2006) utilize 10-port valves. It is also common with 10-port valves to dispense with the use of the sample loops and put two Pump 1 Position A

Pump 1 Position B

Column 1

Pump 2

Column 1

Pump 2

L1

L2

Column 2

Detectors

L1

L2

Waste Column 2

Waste

Detectors

FIGURE 5.7 2DLC with 10-port valve and two sample loops. In position A, loop 1 (L1) is being loaded with sample from column 1, and loop 2 (L2) is being injected onto column 2. In position B, L2 is being loaded with sample from column 1, and L1 is being injected onto column 2.

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second-dimension columns in place of the sample loops. This is a much more difficult experiment to balance but does have advantages while the columns are new and remain balanced. This configuration typically requires very narrow bore firstdimension columns to be used to minimize the volume of effluent that will be flushed into the next dimension. A 12-port valve was used for the periodic sampling of the first column onto multiple second-dimension columns for the 2DLC analysis of aromatic amines and other species (Venkatramani and Zelechonok, 2003). The utility of the 12-port valve is that two columns can be utilized in the second dimension and flow is kept constant through both columns. This configuration requires three sample loops for implementation. The output of the second-dimension columns are connected so that both columns continuously feed the detector. Multiposition valves direct the flow to multiple ports that may be useful for multiple columns in the second dimension. This approach eliminates or greatly reduces the constraints of slowing down the first dimension to get an adequate sampling rate. A potential configuration of using multiple columns in the second dimension is shown in Fig. 5.8. This system would need eight pumps and eight detectors to match the eight columns and would allow for every eighth sample to enter the same column, thus increasing the sampling rate over using one column in the second dimension. As with the dual-column arrangement shown in Fig. 5.6, the adjacent rows may not match perfectly, resulting in increased variability and compensation may be necessary via software. However, this approach may be useful if configured with parallel-column chromatography systems that are now being offered commercially.

First Dimension

Second Dimension

Detectors

FIGURE 5.8 2DLC with multipositional valve. Effluent for the first dimension column can be directed to one of eight columns in the second dimension for increased sampling rate.

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5.4 CE INTERFACES 5.4.1

Gated Interface for HPLC–CE

Because of the small volumes encountered in CE, implementing CE as a second dimension is difficult if a valve is used. More efficient, lower volume unions have been utilized in a number of cases. The main types of these interfaces include optical gating and flow gating, which are discussed below. Electrical gating is described in detail in Chapter 15. Fraction collection is also used, as discussed in Chapter 16, although this takes longer and is a less efficient method than the other comprehensive 2D schemes. Chip-based separation systems typically use some form of electrical gating and these systems will be discussed below. The first instrument to couple LC and CE in a comprehensive multidimensional separation system was described by Bushey and Jorgenson (1990b). This system used a six-port valve and utilized a unique timing diagram for applying the voltage to the CE system. It used a valve that was synchronized so as to utilize electroinjection of the zone into the CE system. Further applications of the electromigration method with valves include protein separations by SEC/CE (Lemmo and Jorgenson, 1993a). Optically gated interfaces (OGI) for diverting solute into a CE column from a firstdimension chromatography column were pioneered by Monnig et al. (1991a and 1991b). The principle of this type of interface is as follows. The components in the mixture to be analyzed are tagged with a fluorescent molecule and continuously introduced into the capillary inlet. Near the capillary inlet, a relatively high powered laser is utilized to photodegrade the fluorescent tag. When the tag is degraded, the solute becomes undetectable to a fluorescence detector, which is placed near the capillary outlet. The sample zone is generated by a transient blocking of the laser beam that opens a small width “fluorescent enabled’’ region in the solute mixture, which is separated within the CE column. In this way, the second-dimension column can be just an extension of the first-dimension column; no special tee or valve is needed for this dimension with the possible exception of having a splitting union so that column volumes may be matched to a smaller diameter capillary used for CE. This mode of operation is known to yield very narrow injection widths. However, this form of sample introduction to the CE capillary is limited to fluorescence detection and does not work for detection by mass spectrometry. Applications of OGI for the reversed-phase liquid chromatography (RPLC)/CE analysis of peptides have been described (Moore and Jorgenson, 1995a) as has the SEC/RPLC/CE separation of peptides (Moore and Jorgenson, 1995b)—a threedimensional separation system. The zone broadening mechanics of the OGI system have been studied (Moore and Jorgenson, 1993). Because of the sampling requirements for comprehensive MDLC, as discussed in Chapter 2, CE with an OGI in the third dimension is ideal because the total run time of the third-dimension separator must be very fast. Otherwise, the preceding dimension separator must be slowed down to allow proper sampling. OGI techniques have been used extensively in single capillary systems where a narrow injection width is needed along with a simplified interface. These techniques are summarized by Hapuarachchi et al. (2006). There are many advantages to this

CE INTERFACES

105

Stainless steel tubing

µA 1/16” channel

RPLC capillary

CZE capillary

1/16” o.d. teflon sleeve

Lexan

PEEK tubing

FIGURE 5.9 The flow gating interface from Hooker and Jorgenson (1997). The cross-flow of buffer prevents LC effluent from electromigrating onto the CE capillary until an injection is desired. This figure is used by permission of the American Chemical Society.

configuration including measuring certain chemicals within complex mixtures in real time in a small volume measurement system. Microdialysis interfaces that employ OGI have been described (Tao et al., 1998; Thompson et al., 1999). This type of interface is also very useful for 1D CE systems that are implemented on-chip through microfluidic systems, as described below. The flow gating interface (FGI) is another type of interface that couples columns to CE shown in Fig. 5.9. This interface is especially useful for very small volumes of column effluent, which would be impractical to store in a sample loop, as in the normal valve configuration. In the FGI, a cross flow of buffer is used in a unique interface to divert solute into the capillary for CE analysis. This system was originally described by Lemmo and Jorgenson (1993b). A revised design was published by Hooker and Jorgenson (1997). The FGI has been interfaced to other separation systems, for example, microdialysis, with subsequent CE detection; see Lada and coworkers (Lada and Kennedy, 1996, 1997; Lada et al., 1997). Multidimensional LC–LC and LC–CE separations were recently reviewed for biological molecules (Evans and Jorgenson, 2004). 5.4.2

Microfluidic Valves for On-Chip Multidimensional Analysis

Chip-based systems that employ microfluidics have become a popular research area and a number of systems are now available commercially that utilize electrophoresis-

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based separations. Implementing MDLC systems on a chip can be difficult. However, 1D chip-based systems that employ electrophoresis for biochemical analysis have been used, which employ optical gating (Pittman et al., 2003; Roddy et al., 2003), flow-through sampling (Chen et al., 2002), and microdialysis sampling (Huynh et al., 2004). For implementing 2D systems on-chip, a number of approaches can be followed. One approach to implementing multidimensional separation systems on-chip is to use micellar electrokinetic chromatography (MEKC) as the first dimension, followed by CE in the second dimension (Ramsey et al., 2003). This allows some degree of orthogonality as the MEKC separation mechanism is sensitive to the degree of hydrophobicity of the solute. The second-dimension separation mechanism is sensitive to the charge and size of the solute. Gating of the solute into the electrophoresis channel is done by electrical switching. Detection is on-chip by laser-induced fluorescence. An image of this chip system is shown in Fig. 5.10. Note that MEKC is a popular alternative to a packed-column RPLC system due to the difficulty in packing columns on the chip. However, it is not unreasonable to think that monolithic columns will soon be integrated into the chip. Regardless, the MEKC system has a reversed-phase like behavior with respect to retention mechanism but is somewhat limited to samples that are soluble in MEKC buffer solutions that tend to be aqueous. For most samples of biological origin, this is not typically a problem. The use of electrically-gated solute injection into the electrophoresis system simplifies the chip design as electrical connections are easy to implement as compared to the microfluidics part of the chip. Voltage waveform manipulation via hardware and software are relatively easy to control and implement. Other combinations of first and second dimensions on chip-based separation systems are becoming more common. A particularly interesting combination is IEF/PAGE. This is an analog of the planar system used by biochemists for decades. However, IEF does not typically allow for fluorescent tags that are essential for laserinduced fluorescent detection; one of the most common on-chip detection systems. These and other first- and second-dimension combinations are given in a recent paper that discusses the implementation of MEKC in the first dimension and GE in the second (Shadpour and Sopor, 2006). This is a very powerful system. Again, sampling of the first dimension is by electrical switching, which is the most convenient method for on-chip systems.

5.5 COLUMNS AND COMBINATIONS There are many combinations of separations techniques and methods of coupling these techniques currently employed in MDLC systems. Giddings (1984) has discussed a number of the possible combinations of techniques that can be coupled to form twodimensional systems in matrix form. This matrix includes column chromatography, field-flow fractionation (FFF), various types of electrophoresis experiments, and more. However, many of these matrix elements would be difficult if not impossible to reduce to practice.

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107

FIGURE 5.10 The image of a microchip used in 2D MEKC-CE described in Ramsey et al. (2003). Note that injections are made at valve 1 (V1) for the first-dimension MEKC separation. Solute is sampled into the CE system at V2. Other designations are provided in the original reference. This figure is used by permission of the American Chemical Society.

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We have categorized the various two-dimensional separation systems by the number of fractions and amount of effluent sampled into the second dimension (Table 5.1). This discussion will be restricted to two-dimensional systems, although most of the principles will apply to more than two coupled separation systems. Furthermore, we limit our discussion here to dimensions that utilize column chromatography. Our discussion of 2DGE is mentioned in passing although we recognize that a great deal of the methodology and analytical outcomes apply jointly to the planar techniques that have been available for years prior to coupling columns with automated valves. The choice of columns used for 2DLC is based upon experience with the sample and resolution required. The HPLC column descriptors of selectivity, resolution, peak capacity, sample capacity, degree of sample recovery, and speed of separation have been discussed previously (Unger et al., 2000). Columns with higher peak capacity and sample capacity (IEC, HIC, NPLC, and RPLC) are preferred in the first dimension, and higher speed columns (SEC and RPLC) are better in the second dimension. This is discussed in detail in Chapters 2 and 6. For optimum resolution in two dimensions, the columns selected need to be orthogonal. Orthogonality is discussed in Chapters 2, 3, and 12. If two particular chemical functionalities are to be separated, then the ease of interpretation of the resulting data is more important than the optimal resolution. For some separations, a mixed mode of separation is observed, and having two dimensions facilitates the study of the differences in separation mechanisms. Overall, there are many combinations of 2DLC separation systems, and as long as the two mobile phase solvents are miscible, they can be coupled for improved resolution. Temperature is another variable to be considered to speed up the separation in the second dimension and will be discussed below. The variety of columns that are utilized in 2DLC to separate complex mixtures is exemplified in Table 5.2. Column selectivity in 2DLC was recently reviewed by Jandera (2006). 5.5.1

Column Systems, Dilution, and Splitting

The scale (capillary or standard bore) of chromatography systems used in 2DLC can vary widely and is dependent on the amount of sample available, detection system, and HPLC availability in the laboratory. Since the chromatography is taking place in two dimensions, zone broadening and dilution are taking place on both columns. Zone broadening leads to sample dilution. The dilution effect has been studied (Schure, 1999) in the context of MDLC and certain combinations of columns are more beneficial in minimizing dilution. This dilution effect is also discussed in Chapter 2. In industrial laboratories, where samples are not typically volume limited, analytical-scale HPLC systems are routinely used. To combine two analytical HPLC systems would involve standard valves with loop volumes from 10 to 1000 mL, and control of each system. If a six-port valve is utilized, then sampling a portion of the effluent from the first to second dimension would allow lower volumes to be injected into the second dimension, but with a loss in sensitivity. A system employing an eight-port valve in the comprehensive mode with analytical flow rates has been used to separate polymers and surfactants (Murphy et al., 1998a, 1998b). A fast second-dimension column flow rate is generally used to accommodate the higher sampling rate. When a narrow range of

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analyte properties are examined, then isocratic techniques can be used in the second dimension. In Chapter 18, isocratic reversed-phase HPLC is used to separate the alkyl portion of a complex surfactant mixture at 1-min intervals during a normal phase gradient experiment. In samples where there is a broad range of analyte properties, such as biological samples, gradient HPLC is needed in both dimensions to improve resolution of the entire sample. Chong et al. (2001) have used gradient chromatofocusing in combination with gradient reversed-phase HPLC for the mapping of protein samples from breast cancer cell lysates. Holland and Jorgenson (1995) coupled capillary anion-exchange and reversed-phased HPLC columns for the analysis of tryptic digests of the contents of a single cell. Recent work by Carr and coworkers (Stoll et al., 2006) has shown the utility in using fast gradient separations in 2DLC that have been facilitated by working at elevated temperature. The use of elevated temperatures in 2DLC is important, especially in the second dimension because of the importance in speeding this dimension up. Thus, the column diameters chosen for the two dimensions are determined by the amount of sample available and will dictate the flow rate ranges available to use. In split-flow systems, where only a portion of the first-dimension effluent is injected into the second dimension, the choice of column size is unlimited and the two methods can be developed independently. In comprehensive systems where the entire sample from the first dimension is injected into the second dimension, the flow rates are generally lower in the first dimension to accommodate the lower injection volumes into the second dimension. For example, for a 1-mm ID column in the first dimension with a flow rate of 50 mL/min and a sampling rate of 1 min, 50 mL could be injected onto the second dimension. A 50-mL injection onto a 4.6-mm ID column flowing at 1 mL/min should be accommodated fairly well based upon its composition. In Chapter 6, the first dimension column diameters are estimated based upon the injection volume and sampling rate into the second dimension. 5.6 DETECTION Detection in 2DLC is the same as encountered in one-dimensional HPLC. Avariety of detectors are presented in Table 5.2. The choice of detector is dependent on the molecule being detected, the problem being solved, and the separation mode used for the second dimension. If MS detection is utilized, then volatile buffers are typically used in the second-dimension separation. Ultraviolet detection is used for peptides, proteins, and any molecules that contain an appropriate chromophore. Evaporative light scattering detection has become popular for the analysis of polymers and surfactants that do not contain UV chromophores. Refractive index (RI) detection is generally used with size exclusion chromatography for the analysis of polymers. 5.7 COMPUTER HARDWARE AND SOFTWARE The equipment used in 2DLC is the same as in HPLC with the addition of valves and integrated control of both HPLC systems. Most chromatography data systems allow

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control over one HPLC system, but few can control two HPLC systems and the repetitive sampling for a 2DLC system. Thus, for 2DLC the control of each HPLC system is done by selecting one system as the main system and the second one as a slave to the first. This allows initiation control of the second HPLC system by contact closure such as during valve actuation. If mass spectrometry detection is used, this method of control is utilized since MS systems can currently control one HPLC. There are some integrated 2DLC systems on the market that can control two HPLC systems and valves. Computer hardware and software used in 2DLC generally take care of three critical operations. These include real-time control of valves and sequencing functions such as autosampler control, formatting the time series data into a 2D data matrix, and analyzing the data. These will be described in some detail. The control of valves and sequencing of autosamplers and injectors is typically a slow speed operation for 2DLC. For example, valves must be controlled on the chromatographic timescale so that switching times on the order of tens of seconds to minutes is not uncommon. However, the sequencing time control must be highly accurate to achieve reproducible chromatograms. Data formatting is accomplished by taking one or more detector signals, for example ultraviolet and refractive index detector signals, and formatting the data to be in matrix format. The computer program must manipulate 2D chromatograms including scaling, chromatogram subtraction, signal processing, and moments analysis. For multiwavelength detectors and mass spectrometer detectors, the software should also allow individual spectral analysis functions such as selection of wavelengths or mass to charge ratios for display. More details of these aspects will be covered below. 5.7.1

Software Development

There are a few commercially available systems that are loosely or tightly integrated for 2DLC. There are commercial systems for multidimensional gas chromatography as the interface between columns can be accomplished without valves using thermal modulation (Marriott, 2002) and the thermal modulator is tightly coupled to the software. Most scientists who work on developing 2DLC systems write their own software and valve sequencing systems. The software for these systems can be developed in higher level languages with graphical icon-oriented systems; one example of this is National Instrument’s LabVIEW system (National Instruments, 2007), which is popular for real-time instrument control on both Microsoft Windows-based operating systems and on Apple’s Mac OS X-based computers. Other popular languages include Microsoft’s Visual Basic and Visual Cþþ programming languages that feature good integration and graphical user interfaces (GUI’s) on Windows-based systems. For the examples used in this chapter, we utilize software that is available commercially (Kroungold, 2007) and is an “add-on’’ for existing chromatography systems. This is one of a few commercially available packages that allow scientists to add columns, pumps, valves, and sample loops to make their own 2DLC systems.

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FIGURE 5.11 Timing diagram for comprehensive 2DLC with either a two-position valve (bottom) or four-position valve (top). Repetitive sampling of the first dimension at each time (T1, T2, T3, T4, . . .,) results in an injection onto the second-dimension column.

5.7.2

Valve Sequencing

The sequence control of valves, autosamplers, fraction collectors, and other devices is typically performed under program control although dedicated hardware controllers can be used here. Valves can be controlled by standard signals such as RS-232 serial lines, a USB (universal serial bus) port, digital signals such as those from TTL voltages (transistor–transistor logic where signal levels are less than 5 V) and relay closures. A good reference to these terms and details are available in a well-known electronics book (Horowitz and Hill, 1989). The program must utilize a clock for an accurate time base and the clock is usually the internal clock of the computer that can be accurate to better than a millisecond. The timing diagrams for 2DLC are shown in Fig. 5.11 for two-position and multiposition valves. A two-position valve has the option of positions A or B, whereas the four-position valve has positions A, B, C, and D available. For twoposition valves, the valve is initially in position A at the beginning of the experiment and the duration interval of the first segment is denoted as T1. During T1, effluent from the first dimension column fills the first sample loop. At the end of the T1 interval, the position of the valve is changed to position B and the solute contained in sample loop 1 is separated on the second-dimension column. Sample loop 2 is now filled with the effluent from the first column as the elution of the second column takes place into the detector. This procedure is repeated until the entire first-column effluent is sampled (i.e., all of the injected sample components have eluted from the first-dimension column) or the experiment is stopped. This timing plan is simple but alternative schemes can be easily accommodated. Two common modifications include provision for a predelay period occurring prior to commencement of the experiment and letting a second valve shunt a portion of the

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second-dimension effluent that may contain salt to waste. Both of these will now be explained. Elution in the second dimension need not be executed until solute is present in the first sample loop. This predelay period allows the first-dimension column void solvent to be dumped to waste collection prior to the arrival of the first retained component. After some initial time, the regular sampling interval T1 ¼ T2 ¼ T3 . . . is started. Mass spectrometers that use electrospray ionization (ESI) do not function well if the eluent contains low volatility salts. This is a major concern when an ion-exchange column is used as a first-dimension column and the salt concentration is used to modulate the retention in this column. In this case, another valve can be connected between the second-dimension column and the detector so that any salt from the second-dimension elution process that is either unretained or weakly retained can be diverted prior to feeding zones to the mass spectrometer. These parameters are typically entered into a program that accepts the parameters as shown in Fig. 5.12. As shown here, the user enters the total run time, seconddimension time (equal to T1 ¼ T2 ¼ T3 . . .), any initial delay time, and operation information regarding triggering and valve states. Once the unit is triggered through a user’s start signal (keyboard initiated, contact sense initiated from an autosampler, data acquisition module, etc.) the software presents a real-time view of the current valve position as shown in Fig. 5.13. The sequence of repetitive valve switching continues until the data acquisition cycles are completed or the user interrupts the sequence. At that time data analysis is initiated by the user. In automatic operation mode, which is one of the user-selected modes in the software, the autosampler triggers another data acquisition and valve switching cycle so that a tray of samples can be readily analyzed via 2DLC.

FIGURE 5.12 User interface for the simplest of the 2DLC modes with no fraction collection or salt diversion. Figure courtesy of Kroungold Analytical (2007).

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FIGURE 5.13 Real-time display of valve sequencing. Figure courtesy of Kroungold Analytical (2007).

5.7.3

Data Format and Storage

If the data system is tightly integrated with the 2DLC system, the native data format can be used to convert the successive one-dimensional data vectors into a matrix form for visualization and analysis. However, there are currently few commercially available 2D systems that have tight integration of data acquisition and analysis. Consequently, the data are stored as raw data that can be easily imported into external software in a number of ways. An especially easy way to do this is to export the raw data into a spreadsheet program like Microsoft Excel using the comma-separated variable (CSV) data format and then use a graphics program that can read CSV files. This has its limits though because data files can get very large in certain cases, which we will now discuss. Chromatography equipment manufacturers and users have embraced a series of exchange formats and data standards that expedite this process. The standard for this was formerly known as the AIA (Analytical Instrument Association) format. Approximately 20 years ago, the AIA developed a set of standards for chromatography and mass spectrometry data. The standard is now called the ANDI/netCDF Chromatography Data Interchange format and is available as standards documents from ASTM International (formerly the American Society for Testing and Materials) as the E 1948-98 standard guide for analytical data interchange protocol for chromatographic data and the E 1947-98 standard specification of analytical data interchange protocol for chromatographic data. A similar set of standards exists for mass spectrometry data. These include the E 2077-00 standard specification for analytical data interchange protocol for mass spectrometric data and the E 2078-00 standard

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guide for analytical data interchange protocol for mass spectrometric data. These systems use the NetCDF packaging standard (NetCDF, 2007) software that can package (encapsulate) or unpack data. This system has been utilized extensively for both chromatography and mass spectrometry data exchange and is a de facto standard for getting data between third party software and an instrument vendor’s proprietary data formats. One packaging format that lacks the efficiency of storage that NetCDF offers but is gaining universal acceptance as a data structure encapsulation standard is the eXtensible Markup Language or XML (Benz and Durant, 2003). Currently, there are activities for both chromatography and mass spectrometry in this area in the project called AnIML (Analytical Information Markup Language), which is now driven by the ASTM subcommittee E13.15. In the case of mass spectrometry data, there is an XML data standard, mzXML, which is used for proteomics data. Furthermore, the mzData format has been touted among equipment manufacturers for proteomics-related data. Information about these formats is easily found on the World Wide Web. However, there does not appear to be a standard yet for multidimensional chromatographic data. The size of the datasets for 2DLC depends on the detector. For a single wavelength detector such as an evaporative light scattering detector (ELSD), we assume the following: there are 80 samples of the first-dimension effluent with 3 min runs in the second dimension. With the signal sampled 10 times a second, the resulting number of data values is 144,000. If the data are sampled by a 20-bit analog-to-digital converter this could be stored in 8 bytes of character data and this would give files that were on the order of 1.2 megabytes in length. If a UV–vis multiwavelength detector, for example, a photodiode array detector, is utilized with a range of 200–450 nm and with a resolution of 0.1 nm, this dataset would comprise 2500 amplitude values at each time slice. If sampled twice a second, the entire 2DLC dataset would contain 2500 28,800 ¼ 72,000,000 or 72 million numbers. If this is kept as a binary file with 4 bytes per data storage value, the file is of the order of 288 megabytes in length. Others have reported file sizes of approximately 100 megabytes for multiwavelength detectors when 100 wavelengths are stored with sampling rates of 80 samples per second and with run times on the order of an hour. In this case, some file compression may also be utilized to reduce file size. Higher sampling rates and higher spectral resolution will increase the file size into the gigabyte range. If the file is to be stored in character format instead of binary, the file size would double or triple. The problem here is not as critical for data storage length, as desktop computers often have mass storage capabilities larger than 300 gigabytes. However, the time needed to read these large datasets becomes a critical parameter when routine handling and analysis of these datasets is to take place. Furthermore, these data cannot be completely memory resident when large datasets greater than perhaps 500 megabytes are present. This is often the case for 2DLC when using mass spectrometry and the data must be partitioned into memory, which leads to far more complex analysis software. In many cases, data can be obtained offline and analyzed in a comprehensive manner from a fraction collector. For example, LC and gas chromatography (GC) can be utilized to form a 2D experiment — LC/GC. This is quite useful as the retention

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FIGURE 5.14 Software to read individual files from an offline GC instrument into the 2D software. Figure courtesy of Kroungold Analytical (2007).

mechanism of LC is different than the GC step. This is a powerful combination when components are sufficiently volatile to be analyzed by GC. The GC detector is typically a flame ionization detector or a mass spectrometer. The GC experiment is run on every LC collected fraction. After data acquisition, the data are converted to the AIA format and then a program is used to collimate the individual data files back into a two-dimensional format. Such a program for doing this is shown in Fig. 5.14. Note that the GC step here need not be fast; in the case of off-line techniques, the seconddimension speed requirement can often be relaxed and run times up to 15 min can be accommodated for an overnight analysis of 60 collected fractions.

5.8 ZONE VISUALIZATION 5.8.1

Contour Visualization

There are two common means of visualizing comprehensive 2DLC data. One is to draw the data as contours with component zones showing as spots, much like 2DGE, and the other is to show zones as a two-dimensional amplitude plot where peaks are shown. A small variation on contour plotting is to map the data value to a color or gray scale and present the color or gray scale as a pixel. The contour plot and pixel mapping are more convenient as these modes of plotting do not tend to hide or exclude features due to high amplitude peaks in the vicinity of low amplitude peaks, although visualizing any low amplitude contour or pixel map is difficult when high amplitude and low amplitude regions are adjacent. The contour drawing capabilities of most graphics software finds the regions of constant peak amplitude through interpolation. Typically, 10–20 contour amplitudes are picked from maximum to minimum. This interpolation can be sophisticated and may include a noise minimizing basis function. However, the use of this filtering may sometimes distort the data presentation. Because 2DLC data are typically performed

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on matrices with sizes larger than 60 rows (first dimension) and 100 columns (second dimension), another approach is to simply map the peak amplitude into a color map and deposit the color as a pixel or in a small rectangular region in the plot. The user typically has control on the color mapping function that gives the most esthetically pleasing plot. This technique is commonly used and provides a fast computational approach to data presentation. However, interpolation is often used and the interpolation function, while appearing to reduce noise, can also artificially broaden the 2D chromatogram and must be used with care. Plotting the logarithm of the amplitude with contours, mapped pixels, or peak plotting can help distinguish large and small adjacent peaks. Taking the logarithm of the amplitude is essentially a compression operation. However, small values of noise in the data tend to become amplified in this approach and more sophisticated techniques may then be necessary for noise suppression, such as gating the noise floor. Gray-scale contour plots are useful and one is shown in Fig. 5.15. These data are quite typical of the zone profiles from a 2DLC experiment and appear similar to the data from 2DGE when digitized in a densitometer after the proper staining treatment. Color contours and pixel maps are useful as they help display more of the zone shape. Other examples of gray and color contouring and pixel mapping of data are plentiful throughout this text.

FIGURE 5.15 A typical gray-scale contour plot of the 2DLC separation of tryptic digest peptides from BSA. From Stoll and Carr (2005). Reprinted with permission of the American Chemical Society.

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2D Peak Presentation

A good example of a 2D peak plot where the amplitude is directly visualized showing peaks is given in Fig. 15.7 of Chapter 15. In this plot, peaks are shown at an angle projection of the two independent variable axes. One problem, besides obscuring data with this approach, is that it is difficult to determine the retention time on each axis; the scientist must be creative in the interpolation of the peak maxima back to the axes. However, aside from esthetic preferences, the data are essentially the same as the contour or direct pixel representation. An important point to note is that since the data are contained in digital form, it is easy to obtain peak maxima information through programmed analysis. Hence, the visualization step is largely a cosmetic step that should be viewed as an information-conveying process, rather than as a substitution for experimental parameter analysis. 5.8.3

Zone Visualization in Specific Chemical (pI) Regions

A number of variations to contour, pixel mapping and peak plotting exist and these tend to depend on the type of column used in the first or second dimension. For example, as shown in Fig. 1.2 in Chapter 1, chromatofocusing can be used to exploit the pI dependence of proteomics samples as one of the dimensions. Hence, one dimension shows a pI range and the other shows the temporal zone production in gradient elution RPLC. By plotting the peak amplitude as a function of color, it is easy to see the individual zones and these can be compared by examination of neighboring pI regions as well as the evolution as a function of gradient modifier or of time although the two do not need to be linearly related. This approach is extremely powerful and can be further elaborated by substituting specific mass-to-charge (m/z) ratio amplitudes from a mass spectrometer. Visualizing analytical chemical concentration data from multivariate instrumentation has for many years been a fruitful area for more research and 2DLC fuels this need. Chemometric analysis may provide a better handle on where (i.e., what regions of multidimensional space) to visualize the data. 5.8.4

External Plotting Programs

Plotting data with two independent variables can be very subjective as to the most pleasing form of information display. Therefore, it is common, once the data are assembled into matrix form, to use an external plotting and graphics program to provide customization to the data plot. In addition, these types of packages can be used to distinguish and label individual peak features. Many PC graphics packages, including the plotting software inside Microsoft Excel, can be used for this purpose. As mentioned previously, for spreadsheet-based plotting systems, data can be imported as a CSV data file where data are separated by commas on a row-by-row basis. Other packages can be used that will add color coding to the data amplitude on the contours and/or peaks and some packages will accept more sophisticated data formats in binary. One example of these software packages was Transform that was sold as a general data analysis package for data packaged in matrix form or as a linear dataset

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that could be converted to matrix form. It was used by Bushey and Jorgenson (1990a) in their original publication on 2DLC. These features are now standard in high performance visualization software. Other plotting software is contained within more elaborate mathematics and analysis integrated systems including MATLAB (2007) and Mathematica (2007). The software used to create Fig. 5.15 was done by programming within the MATLAB package. 5.8.5

Difference Plots

One useful visualization technique is to show the difference between two 2DLC chromatograms. The peak shape reproducibility must be high as difference measurements are inherently noisy operations, as opposed to integration measurements that tend to smooth data irregularities. An interesting paper entitled “Comparative visualization for comprehensive two-dimensional gas chromatography’’ (Hollingsworth et al., 2006) shows difference plots from two-dimensional gas chromatographic data. Difference plots have been used routinely in 2DLC polymer analysis and now are used in proteomics applications. 5.8.6

Multi-channel Data

Interesting data presentations can be made with UV–visible multi-wavelength datasets and mass spectral datasets. In these cases, there are many interesting possibilities. For example, one can show the integrated signal (e.g., the total ion current in the case of mass spectrometry) as a function of the 2D retention. Alternatively, one can show the signal at a particular wavelength or m/z ratio in the 2D space. Or one can show the selective signals as a function of the highest masses or the most intense wavelength or m/z ratio. These presentations are typically dependent on the application, for example, top-down or bottom-up proteomics (see Chapter 13) or polymer analysis (Chapter 17). The presentation mode of the dataset can be utilized to explore the data and see interesting features. In the case of mass spectrometry, we show the data in Fig. 5.16 of an analysis of a protein sample mixture. The user clicks on a peak of interest in the 2D chromatogram and the mass spectrum appears in the graph below the colored pixel map plot. This is the amplitude, as a function of the m/z ratio, of the mass spectrometer at that location in the 2D chromatogram. If the user clicks on a part of the mass spectrum of interest or enters a mass range to view, the 2DLC contours are then shown as in Fig. 5.17. In this way one can check if other peaks with a specific m/z ratio are present. This software also has an automated browser mode that displays the 2D chromatogram repeatedly at different m/z ratios within a specified range. Much like 1D LC/MS, the data presentation has many possibilities because the data are part of a multivariate dataset. This means that there are multiple independent variables; besides the two retention time variables in 2DLC, one also has the m/z ratio so that 2DLC/MS has three independent variables. This can be challenging to visualize; the problem increases dramatically when MS/MS systems are utilized, for example, as in proteomics studies.

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FIGURE 5.16 Mass spectrometry data and the corresponding mass spectrum of a selected spot. Figure courtesy of Kroungold Analytical (2007). (See color plate.)

5.9 DATA ANALYSIS AND SIGNAL PROCESSING 2DLC in many ways parallels 2DGE when it comes to signal processing and data analysis. Many of the same operations, from zone subtraction to the determination of adjacent resolution, are quite similar and software is available for these operations in 2DGE. One of the data operations researched for 2DGE is the similarity in electropherograms. This can also be determined in 2DLC using the same type of methodology. One important difference, however, is that in 2DGE or any type of planar detection system, the signal quality and dynamic range of optical densitometry detectors are much smaller than with the types of detectors found in column chromatography and in 2DLC. Advanced processing methods using Fourier-based techniques such as autocovariance methods, described in Chapter 4, have not found their way into commercially available systems. There is no reason that these cannot be developed further for measuring the similarity between 2DLC chromatograms and estimating the number of peaks present in the 2DLC chromatograms. Chemometric analysis of 2DLC data is further discussed in Chapter 2.

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FIGURE 5.17 The 2DLC contours for m/z ¼ 1301 with a window width of 2 m/z units. Figure courtesy of Kroungold Analytical (2007).

Moment analysis is one of the simplest types of analysis and is useful for measuring the performance of the chromatography. Moments can be used to measure the same things that are measured in 1D chromatographic systems; these include the first, second, and third moments, which are more accurate than the related peak maximum, peak width, and peak asymmetry. In 2D, however, these values each have a component in each dimension and this can be easily determined in software-based measurement systems.

5.10 FUTURE PROSPECTS 2DLC has great potential for the analysis of complex samples from industrial or biological origin. It provides a powerful tool for the analysis of membrane proteins (Lohaus et al., 2007) that cannot be detected in 2D gels. When combined with high resolution mass spectrometry, 2DLC has the resolution to analyze and identify approximately 1300 proteins in human plasma (Jin et al., 2005). There is no doubt that the instrumentation will evolve and will provide an excellent chromatographic platform for in-depth analysis that single-column methods cannot provide. Although

REFEREN