Elementary flux modes

Elementary flux modes are a tool which can be used to take a reaction network of deficiency greater than one and produce subnetworks of deficiency one. If it can be shown that one of these subnetworks admits multistationarity then it can sometimes be concluded that the original network also does so. There are two conditions that need to be checked in order to allow this conclusion. The first is that the network under consideration must satisfy the condition $t=l$. The second is that genericity conditions must be satisfied which consist of requiring the non-vanishing of the determinants of certain matrices. If none of the subnetworks admit multistationarity then no conclusion is obtained for the full network. The core element of the proof is an application of the implicit function theorem. These conclusions are contained in a paper of Conradi et. al. (Proc. Nat. Acad. Sci. USA 104, 19175).

One of the ways of writing the condition for stationary solutions is $Nv(c)=0$, where as usual $N$ is the stoichiometric matrix and $v(c)$ is the vector of reaction rates. Since $v(c)$ is positive this means that we are looking for a positive element of the kernel of $N$. This suggests that it is interesting to look at the cone $K$ which is the intersection of the kernel of $N$ with the non-negative orthant. According to a general theorem of convex analysis $K$ consists of the linear combinations with non-negative coefficients of a finite set of vectors which have a mininum (non-zero) number of non-zero components. In the case of reaction networks these are the elementary flux modes. Recalling that $N=YI_a$ we see that positive vectors in the kernel of the incidence matrix $I_a$ are a special type of elementary flux modes. Those which are not in the kernel of $I_a$ are called stoichiometric generators. Each stoichiometric generator defines a subnetwork where those reaction constants of the full network are set to zero where the corresponding reactions are not in the support of the generator. It is these subnetworks which are the ones mentioned above in the context of multistationarity. The application of the implicit function theorem involves using a linear transformation to introduce adapted coordinates. Roughly speaking the new coordinates are of three types. The first are conserved quantities for the full network. The second are additional conserved quantities for the subnetwork, complementing those of the full network. Finally the third type represents quantities which are dynamical even for the subnetwork.

Here are some simple examples. In the extended Michaelis-Menten description of a single reaction there is just one elementary flux mode (up to multiplication by a positive constant) and it is not a stoichiometric generator. In the case of the simple futile cycle there are three elementary flux modes. Two of these, which are not stoichoimetric generators correspond to the binding and unbinding of one of the enzymes with its substrate. The third is a stoichoimetric generator and the associated subnetwork is obtained by removing the reactions where a substrate-enzyme complex dissociates into its original components. The dual futile cycle has four elementary flux modes of which two are stoichiometric generators. In the latter case we get the (stoichiometric generators of the) two simple futile cycles contained in the network. Of course these are not helpful for proving multistationarity. Another type of example is given by the simple models for the Calvin cycle which I have discussed elsewhere. The MA system considered there has two stoichiometric generators with components $(3,6,6,1,3,0,1)$ and $(3,5,5,1,3,1,0)$. I got these by working backwards from corresponding modes for the MM-MA system found by Grimbs et. al. This is purely on the level of a vague analogy. I wish I had a better understanding of how to get this type of answer more directly. Those authors used those modes to prove bistability for the MM-MA system so that this is an example where this machinery produces real results.