ADVANCES IN CHEMICAL PHYSICS VOLUME XLlll EDITORIAL BOARD C. J. BALLHAUSENK,o benhaven Universitets Fysisk-Kemiske Institut, Kemisk Laboratorium IV, Kobenhaven, Denmark J. J. M. BEENAKKERR,i jksuniversiteit te Leiden, Kamerlingh Onnes Laboratory, Leiden, Netherlands RICHARDB . BERNSTEIND, epartment of Chemistry, Columbia University, New York, New York, U.S.A. H. HAKEN,I nstitut fur Theoretische und Angewandte Physik der Technischen Hochschule, Stuttgart, Germany Yu L. KLIMONTOVITCMHo,s cow State University, Moscow, USSR RYOGOK UBO,D epartment of Physics, University of Tokyo, Tokyo, Japan M. MANDEL,C hemie-Complex der Rijks-Universiteit, Wassenaarseweg, Leiden, Netherlands PETERM AZUR, Institute Lorentz voor Theoretische Natuurkunde, Nieuwsteeg, Leiden, Netherlands GREGOIRNEIC OLIS,P ool de Physique, Faculte de Sciences, Universite Libre de Bruxelles, Bruxelles, Belgium S. ONO, Institute of Physics, University of Tokyo (College of General Education), Tokyo, Japan MICHAELP HILPO~IB,M Research Center, San Jose, California, U.S.A. J. C. POLANYI, Department of Chemistry, University of Toronto, Toronto, Ontario, Canada YVESP OMEAUC, ommissariat a 1'Energie Atomique, Centre #Etudes nucleares de Saclay, Division de la Physique, Gig-sur-Yvet te, France B. PULLMANIn,s titut de Biologie, Physico-Chimique, Universitk de Paris, Paris, France C. C. J. ROOTHAAND, epartments of Physics and Chemistry, The University of Chicago, Chicago, Illinois, U.S.A. IANR oss, Department of Chemistry, Australian National University, Canberra, Australia A.C.T. JOHNR oss, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. R. SCHECTERD, epartment of Chemical Engineering, University of Texas at Austin, Austin, Texas, USA. I. SHAVIIT,B attelle Memorial Institute, Columbus, Ohio, U.S.A. JAN STECKII, nstitute of Physical Chemistry of the Polish Academy of Sciences, Warsaw, Poland GEORGSEZ ASZ, General Electric Coporate R & D, Zurich, Switzerland KAZUHISATO MITA,D epartment of Physics, Faculty of Science, Kyoto University, Kyoto, Japan M. V. VOLKENSTEIINn,s titute of Molecular Biology, Academy of Science, MOSCOW, USSR E. BRIGHT WILSON,D epartment of Chemistry, Harvard University, Cambridge, Massachusetts, U.S.A. Advances in CHEMICAL PHYSICS EDITED BY I. PRIGOGINE University of Brussels Brussels, Belgium and University of Texas Austin, Texas AND STUART A. RICE Department of Chemistry and The James Franck Institute The University of Chicago Chicago, Illinois VOLUME XLm AN INTERSCIENCE@ PUBLICATION JOHN WILEY & SONS NEW YORK CHICHESTERBR ISBANE TORONTO 0 0 An Interscience@P ublication Copyright 8 1980 by John Wiley & Sons, Inc. AU rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. Librpry of Conpm Catalog Number: 58-9935 ISBN 047145741-X Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 CONTRIBUTORS TO VOLUME XLIII BRUCEL . CLARKED, epartment of Chemistry, University of Alberta, Edmonton, Alberta, Canada PETERH . RICHTERD,e partment of Chemistry, Massachusetts Institute of Technol- ogy, Cambridge, Massachusetts ITAMAR F’ROCACCIA, Department of Chemical Physics, Weizmann Institute of Science, Rehovot, Israel JOHN ROSS, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts INTRODUCTION Few of us can any longer keep up with the flood of scientific literature, even in specialized subfields. Any attempt to do more, and be broadly educated with respect to a large domain of science, has the appearance of tilting at windmills. Yet the synthesis of ideas drawn from different subjects into new, powerful, general concepts is as valuable as ever, and the desire to remain educated persists in all scientists. This series, Advances in Chemical Physics, is devoted to helping the reader obtain general information about a wide variety of topics in chemical physics, which field we interpret very broadly. Our intent is to have experts present comprehen- sive analyses of subjects of interest and to encourage the expression of individual points of view. We hope that this approach to the presentation of an overview of a subject will both stimulate new research and serve as a personalized learning text for beginners in a field. ILYA PRIGOGINE STUARTA . RICE vii CONTENTS STABILITOYF COMPLERXEA CTION NETWORKS by Bruce L. Clarke 1 CHEMICAINLS TABILITIES by Peter H. Richter, Itamar Procaccia, and John Ross 217 269 AUTHOR INDEX SUBJECTI NDEX 273 ix ADVANCES IN CHEMICAL PHYSICS VOLUME XLlll Advances In Chemical Physics, VolumeX LIII Edited by I. Prigogine, Stuart A. Rice Copyright © 1980 by John Wiley & Sons, Inc. STABILITY OF COMPLEX REACTION NETWORKS BRUCE L . CLARKE Department of Chemistry University of Alberta Edmonton. Alberta. Canada CONTENTS I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 7 A . The Major Problems . . . . . . . . . . . . . . . . . . . . 7 . B Summary of Previous Work on Stability Theorems for Chemical Networks . . . . . . . . . . . . . . . . . . . . . 10 . . . . . . . . . . . . . . . . . C Stoichiometric Network Analysis 12 I1. The Network and Its Accessible States . . . . . . . . . . . . . . . . 1 6 A . TheNetwork . . . . . . . . . . . . . . . . . . . . . . . 16 B . The Concentration Polyhedron . . . . . . . . . . . . . . . . . 1 8 C . The Current Cone and Polytope . . . . . . . . . . . . . . . . 2 0 D . The Extent of Reaction Polyhedron . . . . . . . . . . . . . . . 2 5 E . Convex Parameters for the Steady State . . . . . . . . . . . . . . 2 7 F . TheFquilibriumSimplex . . . . . . . . . . . . . . . . . . . 3 3 G . TheNonlinearEquationsofMotionandMultipleSteadyStates . . . . . 3 5 H . physical Limitations to the Accessible States . . . . . . . . . . . . 4 0 111 . Elementary Stability Analysis . . . . . . . . . . . . . . . . . . . 43 A . TheLincarkdDynamics . . . . . . . . . . . . . . . . . . . 4 3 B . Stability of Systems and Networks . . . . . . . . . . . . . . . . 4 7 C. Stability Proofs Using Hurwitz Determinants . . . . . . . . . . . . 53 D . Stability Proofs Using Lyapunov Functions . . . . . . . . . . . . 6 0 E . Theorems on Matrix Stability . . . . . . . . . . . . . . . . . 6 2 . . . . . . . . . . . . . . IV Networks Having Simple Lyapunov Functions 66 A . MixingStability . . . . . . . . . . . . . . . . . . . . . . 66 B . Some Fundamental Topological Concepts for Reaction Networks . . . . . 7 3 C . ComplexBalancedNetworks . . . . . . . . . . . . . . . . . 7 8 D . Trees and Sign Stability . . . . . . . . . . . . . . . . . . . 8 5 . E Mass Action Knot-Tree Skeleton Networks and Semidirected Species-Tree Networks . . . . . . . . . . . . . . . . . . . . 87 F. Thermodynamics and Lyapunov Functions . . . . . . . . . . . . . 9 4 . G Summary of the Lyapunov Functions and Some Speculations on the Road Ahead . . . . . . . . . . . . . . . . . . . . . 101 V . A Diagrammatic Approach to Instability in Stoichiometric Networks . . . . .1 03 A. Motivation . . . . . . . . . . . . . . . . . . . . . . . 103 B . Diagrams . . . . . . . . . . . . . . . . . . . . . . . * 107 1 2 B. L. CLARKE C. Current Diagrams and Current Matrix Diagrams. . . . . . . . . . 114 D. Diagrammatic Stability Polynomials . . . . . . . . . . . . . . 119 E. Theorems on Semistability . . . . . . . . . . . . . . . . . 125 F. Current Cycles . . . . . . . . . . . . . . . . . . . . . 126 G. Instability in Realistic Chemical Networks . . . . . . . . . . . 138 H. A Ubiquitous Source of Instability in Oxidation-Reduction Networks. . . 149 I. Critical Current Cycles, Extinction, and StoichiometricE xplosions. . . . 156 VI. Estimating Stability and the Bifurcation Set Using Exponent Polytopes . . . . 166 A. Introduction to the Problem . . . . . . . . . . . . . . . . . 166 B. Basic Ideas of the Exponent Polytope Method. . . . . . . . . . . 170 C. Practical Aspects of Using the Exponent Polytope for Polynomials in Many Variables . . . . . . . . . . . . . . . . . . . . 178 D. How the Face Structures of the Exponent Polytopes of the Stability Polynomials Are Related to Stoichiometric Network Stability . . 187 E. Hurwitz Nonmixing Networks: How Instability Occurs . . . . . . . 193 F. Hurwiiz Nonmixing Networks: On the Difficulty of Proving Stability. . . 199 G. The Transition from Network Stability to Sign Stability . . . . . . . 203 H. Networks with Diffusion . . . . . . . . . . . . . . . . . . 205 VII. Summary 207 Acknowledgments 213 References 213 INDEX OF EXAMPLES 11.1. Oregonator, E, pseudoreactions 11.2. -y for CH,, C2H6, C2H4,C 2H2 e, 11.3. and II, when r = 3, d = 1 11.4. E for = (1,1, - 1, - 1) 11.5. E of Oregonator 11.6. cp,, for unimolecular network using Hill's diagrams 11.7. Nonlinear equations of the Oregonator 11.8. Schlogl model steady states 11.9. Proof that the Oregonator has a single steady state 111.1. S and M for the Oregonator 111.2. Oregonator with unspecified reaction orders-sign stability 111.3. The Routh-Hurwitz theorem 111.4. Stability polynomials for a network 111.5. Sign stability of the Oregonator lv.l. E of Oregonator is mixing stable IV.2. Michaelis-Menten is not mixing stable but is constrained mix- ing asymptotically stable IV.3. Complexes, complex vectors, interactants, and knots of the Oregonator IV.4. A complex cycle (extreme CB network) is not always an extreme subnetwork