Models of biological phenomena often include complicated networks of chemical reactions. I have made some comments on this in a previous post. From a mathematical point of view this leads to large systems of ordinary differential equations (or possibly reaction diffusion equations) depending on a large number of parameters, whose precise values are often not known. At first sight it seems hopelessly complicated to prove general theorems about the dynamics of solutions of these systems. Surprisingly something can be done and it is interesting to enquire why. These ideas are associated with the name ‘chemical reaction network theory’. What are the special features of these systems which allow something to be done? The first property is that they are sparse. This means that the right hand side of the evolution equation for a given quantity only depends on a few of the other unknowns. The second is that with the choice of mass action or Michaelis-Menten kinetics, which are the simplest and most common ways of modelling the reactions, the problem of finding stationary solutions reduces to finding real solutions of systems of polynomial equations. This opens up the possibility of applying tools from (real) algebraic geometry. It is also the case that there is quite a lot of linear structure in the coefficients which allows the application of sophisticated techniques of linear algebra. In practise the first question which people would like to answer is whether there are biologically reasonable stationary solutions and if so how many. In particular, is there more than one (multistability)? The dynamics of these systems do often turn out to be relatively simple. It is also often the case that the qualitative behaviour depends only weakly (if at all) on the choice of parameters. I do not know to what extent this is a selection effect, i.e. that the examples which get into the literature are preferentially those which have these simple properties.
This theory started with a 1972 paper by the chemical engineers Fritz Horn and Roy Jackson (Arch. Rat. Mech. Anal. 47, 81). It seems that in later years the one who contributed most to the development of the subject was Martin Feinberg. For these systems it is possible to introduce a non-negative integer called the deficiency and the case where is zero is the simplest one. The fact that something can be said about the dynamics of solutions in this case which goes beyond statements about the existence of stationary solutions is due to the existence of a Lyapunov function. The source of this function is explained in the paper of Horn and Jackson. In certain cases it is closely related to the Helmholtz free energy which is required to decrease by thermodynamic arguments. The deficiency zero theorem says that the dynamical behaviour is determined by a property called weak reversibility. To explain this property I need some basic concepts. The unknowns in the dynamical system corresponding to a chemical reaction network are the concentrations of the chemical species involved, i.e. the different chemical substances. A complex is a formal linear combination of species with positive integer coefficients. Each reaction in the network corresponds to an ordered pair of complexes, which are the left hand side and right hand side of the reaction. A network is called weakly reversible if whenever there is a concatenation of reactions leading from a complex to a complex there is also one from to . When if a network is not weakly reversible then the corresponding dynamical system with mass action kinetics has no stationary solutions where all concentrations are positive. It also has no periodic solutions. If it is weakly reversible there is exactly one such stationary solution which is a global attractor for the system. More precisely, there are invariant manifolds called stoichiometric compatibility classes and the statements about uniqueness and stability refer to a given class of this type. Note that, in particular, a deficiency zero system can never have more than one relevant stationary solution (within a stoichiometric compatibility class).
There is a generalization of the deficiency zero theorem called the deficiency one theorem. To explain this it is necessary to introduce the notion of linkage classes. Given a network a graph can be defined whose nodes are complexes and where there there are edges joining these nodes corresponding to reactions. The connected components of the resulting graph are the linkage classes. Each component defines a network of its own and so has a natural definition of deficiency. It can be shown that the sum of the deficiencies of the linkage classes is no greater that the deficiency of the whole network. I will not describe all the hypotheses of the deficiency one theorem here. I will just mention that they include the conditions that no linkage class has a deficiency greater than one and that the sum of the deficiencies of the linkage classes is equal to the deficiency of the whole network.