Progress in Theoretical Biology. Volume 5 PDF
Preview Progress in Theoretical Biology. Volume 5
Advisory Board FUMIO OOSAWA, Nagoya University, Nagoya, Japan ERNEST C. POLLARD, Pennsylvania State University, University Park, Pennsylvania OTTO SCHMITT, University of Minnesota, Minneapolis, Minnesota Contributors to This Volume Anita Babcock Stephen Grossberg Jan Hirschfeld Hans G. Othmer Vadim A. Ratner Rüstern N. Tchuraev John J. Tyson Andrew Wohlgemuth Advisory Board FUMIO OOSAWA, Nagoya University, Nagoya, Japan ERNEST C. POLLARD, Pennsylvania State University, University Park, Pennsylvania OTTO SCHMITT, University of Minnesota, Minneapolis, Minnesota Contributors to This Volume Anita Babcock Stephen Grossberg Jan Hirschfeld Hans G. Othmer Vadim A. Ratner Rüstern N. Tchuraev John J. Tyson Andrew Wohlgemuth Progress in Theoretical Biology Edited by ROBERT ROSEN Department of Physiology and Biophysics Faculty of Medicine Dalhousie University Halifax, Nova Scotia Canada and FRED M. SNELL Center for Theoretical Biology State University of New York at Buffalo Amherst, New York Volume 5 ® ACADEMIC PRESS New York San Francisco London 1978 A Subsidiary of Harcourt Brace Jovanovich, Publishers COPYRIGHT © 1978, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED, NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. Ill Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1 7DX LIBRARY OF CONGRESS CATALOG CARD NUMBER: 66-30154 ISBN 0-12-543105-8 PRINTED IN THE UNITED STATES OF AMERICA Contributors Numbers in parentheses refer to the pages on which the authors 1 contributions begin. ANITA BABCOCK,* Department of Biophysics, State University of New York at Buffalo, Amherst, New York 14226 (63) STEPHEN GROSSBERG, Department of Mathematics, Boston University, Boston, Massachusetts 02215 (183, 233) JAN HIRSCHFELD, State Institute for Blood Group Serology, Statens Rättskemiska Laboratorium, S-581 85, Linköping, Sweden (129) HANS G. OTHMER, Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903 (1) VADIM A. RATNER, Institute of Cytology and Genetics of the U.S.S.R. Academy of Sciences, Siberian Department, Novosibirsk 630090, U.S.S.R. (81) RUSTEM N. TCHURAEV, Institute of Cytology and Genetics of the U.S.S.R. Academy of Sciences, Siberian Department, Novosibirsk 630090, U.S.S.R. (81) JOHN J. TYSON,t Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14214 (1) ANDREW WOHLGEMUTH, Department of Mathematics, University of Maine at Orono, Orono, Maine 04473 (1) * Present address: Laboratory of Theoretical Biology, National Institutes of Health, Bethesda, Maryland 20014. tPresent address: Department of Biology, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061. IX Preface Theoretical and mathematical biology continue to be perhaps the most actively developing areas in all of the biological sciences. At the present time, there are four international journals devoted entirely to the theory and modeling of biological systems, each with long waiting lists. There are a half dozen other international journals dealing with specialized aspects of biology, and a half dozen others concerned with the theory of organized systems (of which organisms are the most conspicuous examples). A large number of articles concerned with modeling and theory are to be found in established biological journals which, not long ago, were exclusively devoted to experiments. There are few areas in science that can boast such a rate of growth over the past decade, or such a level of continuing accomplishment. The present volume of Progress in Theoretical Biology is intended to reflect this development, while emphasizing in particular the interface between genetics, morphogenesis, and behavior which has characterized the past several volumes in this serial publication. The article by Professors Tyson and Othmer is concerned with the detailed dynamic behaviors exhibited by cellular control circuits; this area is central to any attempt to understand intracellular behavior, and also to the understanding of the role of the cell as a morphogenetic and physiologi cal unit. Dr. Babcock's article deals with somewhat analogous problems at the level of genetic induction and repression, but employs an entirely different mathematical framework (random networks). The contribution of Professors Ratner and Tchuraev continues their detailed exploration of the properties of the genome and its relation to biological activity. Professors Hirschfeld and Wohlgemuth are concerned with one of the most interesting of physiological systems, the immunity system. They propose a drastic reinterpretation of the data relating immunologie specificity to genetics. Finally, two contributions by Professor Stephen Grossberg deal in depth with the basic problems of memory in behav ioral and developmental biology. In sum, then, it is hoped that this collection of working articles will convey some of the character and excitement that characterize contem porary theoretical and mathematical biology. xi Contents of Previous Volumes Volume 1 Chemical Evolution Melvin Calvin Biological Self-Replicating Systems Harold J. Morowitz Quantitative Aspects of Goal-Seeking Self-Organizing Systems Hans Bremermann Statistical Thermodynamics of Polymerization and Polymorphism of Protein Fumio Oosawa and Sugie Higashi The Role of Models in Theoretical Biology Walter R. Stahl Author Index—Subject Index Volume 2 Living Aggregates of Nonliving Parts: A Generalized Statistical Mechanical Theory Karl Kornacker Theoretical Methods in Systematic and Evolutionary Studies G. F. Estabrook Waves, Pulses, and the Theory of Neural Masses Walter J. Freeman Design for Autonomous Chemical Growth under Different Environmental Constraints Otto E. Ròssler Cooperative Processes in Biological Systems Narenda S. Goel Problems of Organization of Motor Systems Peter H. Greene Author Index—Subject Index xiii XIV CONTENTS OF PREVIOUS VOLUMES Volume 3 Ecosystem Patterns in Randomly Fluctuating Environments Robert M. May Classical and Instrumental Learning by Neural Networks Stephen Grossberg The Genetic Language V. A. Ratner Psychophysical Discrimination Alejandro B. Engel A Linear Systems Analysis of the Calcium Cycle in a Forested Watershed Ecosystem Jack B. Waide, Julia E. Krebs, Sandra P. Clarkson, and Eileen M. Setzler Subject Index Volum 4 e Theoretical Aspects of Genetic Complementation Vadim A. Ratner and Sergey N. Rodin An Allosteric Enzyme Model with Positive Feedback Applied to Glycolytic Oscillations A. Goldbeter and G. Nicolis Biological Observables Sorin Comorosan Structure, Stability, and Efficiency of Ecosystem Mumay Tansky Adaptation John H. Holland Subject Index PROGRESS IN THEORETICAL BIOLOGY, VOLUME 5 The Dynamics of Feedback Control Circuits in Biochemical Pathways John J. Tyson* Department of Mathematics, State University of New York at Buffalo, Buffalo, New York Hans G. Othmer Department of Mathematics, Rutgers University, New Brunswick, New Jersey I. Introduction A. Regulation of Enzyme Synthesis 2 B. Regulation of Enzyme Activity 4 II. Kinetic Equations 7 III. Inducible Systems A. Multiplicity and Stability of Steady States . .. 13 B. Global Stability Results 17 IV. Repressible Systems A. Local Stability Results 19 B. Global Stability Results 23 C. Small-Amplitude Periodic Solutions . . .. 26 D. Periodic Solutions in the Large 32 E. Fourier Approximation of Periodic Solutions . 34 V. Discussion A. Hysteresis Effects 38 B. Applications of Negative Feedback 41 C. Spatially Nonuniform Systems 45 Appendix A. Existence of Oscillatory Solutions for Negative Feedback Loops 49 * Present address: Department of Biology, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061. 1 Copyright® 1978 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-543105-8 2 JOHN J. TYSON AND HANS G. OTHMER Appendix B. Periodic Solutions of Negative Feedback Loops with Step Function Nonlinearity . . .. 53 Notation 59 References 60 I. Introduction A. REGULATION OF ENZYME SYNTHESIS Bacteria are able to use any of a variety of substances as their sole source of carbon because they can synthesize the enzymes needed to catabolize these substances. Usually these enzymes are produced only in the presence of their substrate or one of its analogues, and for this reason they are said to be inducible. In contrast, the enzymes in the biosynthetic pathways leading to essential metabolites such as amino and nucleic acids are synthesized in the cell when there is no external source of the metabolite. When the metabolite is available in the external growth medium, synthesis of these anabolic enzymes is re pressed. The operon model of Jacob and Monod (1961) provides the most commonly used framework for the analysis of enzyme induction and repression. In this model, each structural gene that codes for an enzyme or protein is linked with an operator gene that serves to regulate initiation of transcription. When a repressor molecule is bound to the operator gene, transcription is blocked. The binding of a repressor molecule to the operator is in turn modulated by a so-called effector molecule. In the case of inducible enzymes, the repressor is bound to the operator in the absence of effector, and transcription is blocked. When the substrate for such an enzyme is present, an effector molecule (usually the substrate, an analogue of it, or a product of it) can bind with repressor and thereby preclude binding of the latter to the operator. This permits transcription of the structural gene. The function of the effector is to provide an alternate kinetic pathway for repressor; this process can be modeled as a pair of competing reactions (Yagil and Yagil, 1971): R+ pS— RS , K,= RS /RSP P P R + 0^± OR, K = OR/RO 2 Here R = repressor, O = operator, and S = effector. Here and hereafter we shall use the same symbol for a chemical species and its concentra tion. We assume that these reactions occur quickly and are therefore always in equilibrium. Furthermore, we assume that the binding of
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