Bruce L. Clarke

Stoichiometric network analysis

Stoichiometric network analysis is a systematic, general approach to the qualitative, nonlinear dynamics of chemical reaction mechanisms and other systems with stoichiometry. The advantage of a qualitative approach is that no rate constants are needed to determine qualitative features of the dynamics. If one is interested in stability, the approach yields inequalities among the steady-state concentrations and the rate of flow through sequences of important reactions. These parameters are often the ones most easily measured experimentally. By comparing such experiments with the inequalities derived from stoichiometric network analysis, one can often prove that certain mechanisms cannot account for oscillations or other types of observed dynamics.The approach covers far more than stability. The existence of steady states of zero concentration has an interesting mathematics and applies to chemical evolution. The folding of the manifold of steady states can be found by direct calculation and plays a role in switching enzymes on and off. The approach leads to theorems showing that some steady states are globally attracting or, possibly, that a region containing chaos or an oscillation is globally attracting. The subject of sensitivity analysis has been reformulated in this context. Algorithms that apply many of the theoretical results to chemical networks have been developed and combined into a computer program package.

Complete set of steady states for the general stoichiometric dynamical system

The complete set of steady states for rate constants in the range 0?ki<∞ and concentrations in the range 0?Xi<∞ is given explicitly in parametric form for the general chemical reaction system. The only assumptions are that the stoichiometries are real numbers, and the reaction rates are proportional to functions of class Ck, k≳0; the functions are assumed to be positive in the interior of the domain. Hence, these results apply far more generally than just to chemical systems and should be valid for many ecological and economic models as well. The set of steady states in the interior of the domain is in general a simply connected differentiable manifold M of dimension n+r−d, where n = number of species, r = number of reactions, and d = rank ν (ν = stoichiometric matrix). The full set of steady states M* consists of those in the interior (M) plus a frequently very complicated set of steady states lying in the boundary. M* is the union of a set of differentiable manifolds but is not itself a differentiable m...

Bistability in chemical reaction networks: Theory and application to the peroxidase–oxidase reaction

Starting with a comprehensive list of elementary steps in the mechanism of the peroxidase–oxidase (PO) reaction, we extract the part of the mechanism essential to the experimentally observed bistability. A general systematic method is used to sort out the mechanism. First the extreme currents are found and the structure of the current polytope is determined. Then conditions for the existence of multiple steady states are used to identify the dominant extreme currents that are essential for bistability. A clue to the cause of bistability came from applying stoichiometric network analysis to the much simpler classical substrate‐inhibition enzyme mechanism. Three extreme currents are essential for bistability. These correspond to (i) a reversible flux of the inhibiting substrate, (ii) the catalytic cycle, and (iii) an inhibition pathway coupled to the catalytic cycle. The same three elements are found in the PO mechanism. Analysis of the model containing these elements shows that bistability requires less fe...

Theorems on chemical network stability

The stability of steady states of a chemical reaction system is considered within a diagrammatic formulation of the problem. The system’s stability depends upon the kinds of cycles that can be constructed from a set of arrows. The following theorems are proven. (1) A chemical network has no unstable steady states if the set of cycles which can be constructed contains only certain 2−cycles or is empty; (2) an m−cycle, which passes through a reactant whole self−vertex is exactly cancelled by autocatalysis destabilizes the network in certain restrictive circumstances; and (3) a 3−cycle as in (2) destabilizes the network under broader circumstances. The restrictive circumstances of the second theorem do not appear to be capable of being broadened in general because of complexities that can be understood within the full diagrammatic formulation of the problem.

Graph theoretic approach to the stability analysis of steady state chemical reaction networks

A form of graph theory is developed which makes it possible to write the Routh‐Hurwitz stability conditions for any network of chemical reactions as sums of graphs. These sums, which must all be positive for stability, can contain negative terms only through two mechanisms: first, by having an odd number of certain types of cycles in formally positive graphs, or second, by having an even number of these cycles and at least one cycle in formally negative graphs. The set of graphs in each stability inequality may be represented as a set of points which defines a convex polytope in a higher dimensional space. For large parameter values only the vertices of this polytope affect the stability of the network. For each vertex corresponding to a graph which is a negative term in a stability inequality there is a convex coneshaped contribution to the networks unstable region. For large parameter values, this region is the union of the interiors of these convex cones in parameter space whose boundaries are hyperpl...

Method for deriving Hopf and saddle-node bifurcation hypersurfaces and application to a model of the Belousov-Zhabotinskii system

Chemical mechanisms with oscillations or bistability undergo Hopf or saddle‐node bifurcations on parameter space hypersurfaces, which intersect in codimension‐2 Takens–Bogdanov bifurcation hypersurfaces. This paper develops a general method for deriving equations for these hypersurfaces in terms of rate constants and other experimentally controllable parameters. These equations may be used to obtain better rate constant values and confirm mechanisms from experimental data. The method is an extension of stoichiometric network analysis, which can obtain bifurcation hypersurface equations in special (h,j) parameters for small networks. This paper simplifies the approach using Orlando’s theorem and takes into consideration Wegscheider’s thermodynamic constraints on the rate constants. Large realistic mechanisms can be handled by a systematic method for approximating networks near bifurcation points using essential extreme currents. The algebraic problem of converting the bifurcation equations to rate constant...

Stability of topologically similar chemical networks

Theorems are proven which support the view that chemical reaction network topology determines which networks can have unstable steady states and when the steady states are unstable. These theorems make it possible to determine the stability of large networks in certain circumstances by determining the stability of simpler networks with identical topology. Chemical networks are classified as ’’qualitatively vertex stable’’ (QVS), ’’qualitatively vertex marginally stable’’ (QVM), and ’’qualitatively vertex unstable’’ (QVU). Roughly speaking, QVS networks always have stable steady states, QVM networks have linearized dynamics which is marginally stable for all rate constants and constrained concentrations, and QVU networks have unstable steady states for specific parameter ranges. The sense in which the preceding sentence is rough is discussed in the text. It is shown that the linear steady state stability analysis problems for topological similar networks are closely related. A geometrical and graph theoret...

Stability analysis of a model reaction network using graph theory

The stability of a model chemical reaction network which is unstable with respect to both homogeneous and inhomogeneous perturbations far from equilibrium is analyzed using a graph theoretic approach. Equations for the boundary of stability and the frequency and wavelength of the unstable perturbation at the boundary are obtained by interpreting the dominant graphs in the stability inequality geometrically. It is found that perturbations within a broad of wave vectors become unstable together at the point of marginal stability and that usually across this band ω(k) ∝ k−1. Of the hundreds of graphs in the stability inequality only a very few play a role in the instability. Most of these are abnormal cycle graphs through the autocatalytic vertex and the reminder are maximum overlap graphs from the diagonal term of the Hurwitz determinant. The three destabilizing graphs give insight into the normal modes associated with the three unstable regions. Comparison of this analysis with an earlier computer analysis...

General method for simplifying chemical networks while preserving overall stoichiometry in reduced mechanisms

A method is given for reducing the number of intermediates in reaction networks without altering the stoichiometric constraints. When a set of intermediates is to be eliminated, the reactions involving the intermediates must be replaced with the overall reactions of the extreme currents of the subnetwork whose only intermediates are the species to be eliminated. This method is automated easily for use in stability analysis and dynamical simulations.

Stability of the bromate–cerium–malonic acid network. I. Theoretical formulation

The steady state stability problem for the detailed mechanism of the bromate–cerium–malonic acid system (Belousov–Zhabotinskii system) is set up mathematically. Justification for examining a reduced network with four independent reactants and eight reactions is based on recently developed theorems on the stability of topologically similar chemical networks. The reduced network’s possible steady state currents are found to lie in a convex cone whose cross section is a three dimensional ’’current polytope’’ Πc with seven vertices—each vertex being one of the extremal ’’framing current loops’’ of the network. The number of steady states is proven to be either one or three and in several special cases is proven to be one. The graphs of possible arrows are then constructed for each of the framing current loops and the feedback cycles which can destabilize are discussed. The network is proven to be stable in each of several simplices of a decomposition of Πc. The stability problem is subdivided into units which...