Multiple Equilibria in Complex Chemical Reaction Networks: Semiopen Mass Action Systems

Gheorghe Craciun, Martin Feinberg
<span title="">2010</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="" style="color: black;">SIAM Journal on Applied Mathematics</a> </i> &nbsp;
In two earlier articles, we provided sufficient conditions on (mass action) reaction network structure for the preclusion of multiple positive steady states in the context of what chemical engineers call the continuous flow stirred tank reactor. In such reactors, all species are deemed to be present in the effluent stream, a fact which played a strong role in the proofs. When certain species are deemed to be entrapped within the reactor, the questions that must be asked are more subtle, and the
more &raquo; ... mathematics becomes substantially more difficult. Here we extend results of the earlier papers to semiopen reactors and show that very similar results obtain, provided that the network of chemical reactions satisfies certain weak structural conditions; weak reversibility is sufficient but not necessary. Introduction. In two earlier papers [1], [3] , we developed means to determine whether a given (mass action) chemical reaction network has the capacity to exhibit multiple positive steady states in the context of what chemical engineers call the (isothermal) continuous flow stirred tank reactor (CFSTR). Some of those results are reviewed in [4] with special focus on biochemistry. When we say that a network has the capacity to admit multiple positive steady states, we mean that there are certain combinations of parameter values (e.g., kinetic rate constants, reactant supply rates) such that, for the network, the corresponding isothermal CFSTR mass action differential equations admit at least two distinct rest points at which all species concentrations are positive. (Among mass action networks generally, this is far less common than might be supposed.) In the absence of an overarching theory, determination of a network's capacity for multiple steady states is difficult, for one is confronted with a large system of polynomial equations in the species concentrations, in which many parameters appear. Nevertheless, the aforementioned articles provide means to assert for quite broad classes of reaction networks-including highly complex ones-that multiple positive steady states are impossible, regardless of parameter values. The test provided in the first article is largely computational, while the test provided in the second is tied to subtle aspects of a reaction network's structure as revealed in its speciesreaction graph. In fact, a theorem in [3] ensures that multiple positive steady states are impossible for a particular network unless the species-reaction graph for the network *
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="">doi:10.1137/090756387</a> <a target="_blank" rel="external noopener" href="">fatcat:sr37jqjxd5a4vaodf66co2lb5m</a> </span>
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