Martin Feinberg

Chemical reaction network structure and the stability of complex isothermal reactors. I: The deficiency zero and deficiency one theorems

Abstract The dynamics of complex isothermal reactors are studied in general terms with special focus on connections between reaction network structure and the capacity of the corresponding differential equations to admit unstable behavior. As in some earlier work, the principal results rely on a classification of reaction networks by means of an easily computed non-negative integer index called the deficiency. This index often provides nontrivial information about the kind of dynamics that can be expected. Part of the previously reported Deficiency Zero Theorem is substantially generalized by the Deficiency One Theorem. The foundation is laid for a companion article containing a theory of multiple steady states generated by reaction networks of deficiency one.

The existence and uniqueness of steady states for a class of chemical reaction networks

My purpose here is to draw some general relationships between the structure of a chemical reaction network and the nature of the set of equilibrium states for the corresponding system of nonlinear ordinary differential equations. An example will illustrate the way in which a system of ordinary differential equations is induced by a chemical reaction network: We consider a closed* vessel containing a fluid mixture in which chemical reactions occur among species labeled A, B, C, D and E. The mixture is stirred continuously. The stirring is sufficiently effective as to render the temperature and the molar concentrations of all species uniform throughout the mixture. Moreover, the temperature and the volume of the mixture are kept time-invariant by external means. On the other hand, the composition of the mixture may experience temporal variation resulting from the occurrence of chemical reactions. We suppose that the chemical reactions occurring within the mixture are reasonably well reflected in the diagram

Multiple Equilibria in Complex Chemical Reaction Networks: I. The Injectivity Property

The capacity for multiple equilibria in an isothermal homogeneous continuous flow stirred tank reactor is determined by the reaction network. Examples show that there is a very delicate relationship between reaction network structure and the possibility of multiple equilibria. We suggest a new method for discriminating between networks that have the capacity for multiple equilibria and those that do not. Our method can be implemented using standard computer algebra software and gives answers for many reaction networks for which previous methods give no information.

Structural Sources of Robustness in Biochemical Reaction Networks

Steady As She Blows A fundamental characteristic of many biological control networks is the capacity to maintain the concentration of a particular component at steady state within a narrow range, in spite of variations in the amounts of other network components that might change as a result of environmental variables in the state of a cell. In a mathematical analysis, Shinar and Feinberg (p. 1389) reveal the essential requirements of a network robust to perturbation. Using this method, the sources of robustness in two bacterial systems—one that functions in osmoregulation and another that controls carbon flux in metabolism—were explained. Models of metabolic regulation show how the stability of specific components is maintained within a varying environment. In vivo variations in the concentrations of biomolecular species are inevitable. These variations in turn propagate along networks of chemical reactions and modify the concentrations of still other species, which influence biological activity. Because excessive variations in the amounts of certain active species might hamper cell function, regulation systems have evolved that act to maintain concentrations within tight bounds. We identify simple yet subtle structural attributes that impart concentration robustness to any mass-action network possessing them. We thereby describe a large class of robustness-inducing networks that already embraces two quite different biochemical modules for which concentration robustness has been observed experimentally: the Escherichia coli osmoregulation system EnvZ-OmpR and the glyoxylate bypass control system isocitrate dehydrogenase kinase-phosphatase–isocitrate dehydrogenase. The structural attributes identified here might confer robustness far more broadly.

MULTIPLE EQUILIBRIA IN COMPLEX CHEMICAL REACTION NETWORKS: II. THE SPECIES-REACTION GRAPH ∗

For mass action kinetics, the capacity for multiple equilibria in an isothermal homogeneous continuous flow stirred tank reactor is determined by the structure of the underlying network of chemical reactions. We suggest a new graph-theoretical method for discriminating between complex reaction networks that can admit multiple equilibria and those that cannot. In particular, we associate with each network a species-reaction graph, which is similar to reaction network representations drawn by biochemists, and we show that, if the graph satisfies certain weak conditions, the differential equations corresponding to the network cannot admit multiple equilibria {\em no matter what values the rate constants take}. Because these conditions are very mild, they amount to powerful (and quite delicate) necessary conditions that a network must satisfy if it is to have the capacity to engender multiple equilibria. Broad qualitative results of this kind are especially apt, for individual reaction rate constants are rare...

Chemical reaction network structure and the stability of complex isothermal reactors—II. Multiple steady states for networks of deficiency one

Abstract The following problem is considered: Given a deficiency one network, determine whether there exist rate constants for it such that the corresponding isothermal mass action differential equations admit multiple positive steady states. A procedure is given to make this determination for any deficiency one network, no matter how intricate, so long as it satisfies certain weak structural conditions. When there do exist rate constants that give rise to multiple steady states, such rate constants can in fact be exhibited. If multiple steady states are observed in a laboratory reactor with poorly understood chemistry, the theory provides sensitive means to screen candidates for the operative chemical mechanism. Even when measurements of the steady state compositions are fragmentary, the theory will sometimes indicate that a candidate network which does admit multiple steady states will nevertheless be unable to account for the particular measurements made. A catalytic CFSTR is considered as an example.

Dynamics of open chemical systems and the algebraic structure of the underlying reaction network

Abstract While there has been a longstanding interest in stability of non-isothermal reactors there has only recently developed a comparable interest in the dynamics of open isothermal reactors with complex chemical reaction networks. In the recent literature there has been paid particular attention to the study of biological reaction systems which might exhibit sustained oscillations (biological clocks) or bistability (biological switches). Results are presented which bear upon the relationship between the algebraic structure of the underlying reaction network and the extent to which reactors might give rise to such “exotic” dynamics.

Optimal reactor design from a geometric viewpoint—I. Universal properties of the attainable region

Abstract A geometric framework for studying optimal reactor design is developed. For a given feed and a prescribed kinetics (perhaps involving many reactions), focus is on the full set of product composition vectors that can be produced in principle by means of all possible steady-state designs that employ only reaction and mixing (including designs that transcend current imagination). This set, called the attainable region by F. J. M. Horn, carries the full range of outcomes available to the designer. Of special importance are its extreme points , for these determine the region completely, and reactor optima are often realized there. Although the attainable region is not generally discernible in advance, one can nevertheless prove that it has certain universal properties, which, in turn, provide information about qualitative designs that provide access to the extreme points. Despite the vast spectrum of designs the attainable region is intended to embrace, two theorems suggest that its extreme points will always be accessible by means of classical elementary reactor types taken in simple combination. These results suggest that any reactor product that is realizable can, in fact, be realized by parallel operation of those canonical reactor building blocks that give rise to the extreme points. This paper lays the groundwork for additional theory, in which special properties of reactors that access the extreme points will be studied in some detail.

Multiple steady states for chemical reaction networks of deficiency one

This article is intended as an addendum to The Existence and Uniqueness of Steady States for a Class of Chemical Reaction Networks [F1]. Readers are presumed to be familiar with the terminology and notation used there. My purpose here is to provide theory for determining when a deficiency-one reaction network, taken with mass-action kinetics, has the capacity to admit multiple positive equilibria. That is, I am interested in means to answer the following question: For a given deficiency-one network, do there exist rate constants such that the corresponding mass-action differential equations admit multiple positive equilibria (within a stoichiometric compatibility class)? Recall that the Deficiency-One Theorem [F1] already provides a negative answer for certain deficiency-one networks:

Necessary and sufficient conditions for detailed balancing in mass action systems of arbitrary complexity

Abstract Detailed balancing will not generally obtain in a mass system unless the rate constants are suitably well orchestrated. Simple necessary and sufficient conditions are given for detailed balancing in mass action systems of arbitrary complexity.