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Unformatted text preview: A GRAPHTHEORETIC ANALYSIS OF CHEMICAL REACTION NETWORKS I. Invariants, Network Equivalence and Nonexistence of Various Types of Steady States ∗ Hans G. Othmer Department of Mathematics University of Utah Salt Lake City, Utah 84112 1981 * Based on Rutgers University Course Notes, 1979 1 Contents 1 Abstract 2 2 Introduction 2 3 How Stoichiometry and Network Structure are Reflected in the Dynamical Equations. 4 3.1 The Graph Associated with a Reaction Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 Some Basic Concepts from Graph Theory and Convex Analysis. . . . . . . . . . . . . . . . . . . . . 7 4 Reaction Invariants 11 4.1 General Rate Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2 Mass Action and Related Types of Rate Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.3 Compactness of the Reaction Simplex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5 Dynamical Equivalence of Networks. 21 6 Necessary Conditions for the Existence of a Steady State. 24 6.1 The Relationship Between Local and Global Deficiency. . . . . . . . . . . . . . . . . . . . . . . . . 24 6.2 Nonexistence of Balanced Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6.3 Conditions under which S 2 = φ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.4 A Flow Chart For Determining Whether Any Steady State Can Exist. . . . . . . . . . . . . . . . . . . 32 7 Discussion 32 References 35 1 Abstract The dynamical behavior of a chemicallyreacting system is determined by the stoichiometry of the reactions, the structure of the graph underlying the network, and the reaction phenomenology embodied in the rate laws. Herein we develop a new approach to the analysis of networks, using graphtheoretic techniques, that separates the individual influences to the extent possible, and facilitates the analysis of their interaction. We show how the reaction invariants are related to the stoichiometry and the network structure and give sufficient conditions under which the reaction simplex is not compact. The notion of dynamical equivalence of networks is defined and three types of equivalence transformations are introduced. In particular, it is shown that a network of positive deficiency, a term defined later, is dynamically equivalent to one with zero deficiency. In our approach, the steady states of any network fall into three classes, and conditions are given under which each of these classes is empty. 2 Introduction The objective in a qualitative analysis of a dynamical system described by an evolution equation of the form ˙ u = F ( u,p ) , where u is the state and p is a parameter vector, is to predict the qualitative evolution of u for different initial conditions, and to determine how the evolution depends on the parameters. The basic problem is the same, irrespective of whether the equation arises from a problem in chemical reaction dynamics, ecological interactions between species,...
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This note was uploaded on 09/12/2011 for the course MAP 4305 taught by Professor Deleenheer during the Summer '06 term at University of Florida.
 Summer '06
 DeLeenheer

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