## The species-reaction graph

In the study of chemical reaction networks important features of the networks are often summarised in certain graphs. Probably that most frequently used is the species graph (or interaction graph), which I discussed in a previous post. The vertices are the species taking part in the network and the edges are related to the non-zero entries of the Jacobian matrix of the vector field defining the dynamics. Since the Jacobian matrix depends in general on the concentrations at which it is evaluated there is a graph (the local graph) for each set of values of the concentrations. Sometimes a global graph is defined as the union of the local graphs. A sign can be attached to each edge of the local graph according to the sign of the corresponding partial derivative. In the case, which does occur quite frequently in practise, that the signs of the partial derivatives are independent of the concentrations the distinction between local and global graphs is not necessary. In the general case a variant of the species graph has been defined by Kaltenbach (arXiv:1210.0320). In that case there is a directed edge from vertex $i$ to vertex $j$ if there is any set of concentrations for which the corresponding partial derivative is non-zero and instead of being labelled with a sign the edge is labelled with a function, namely the partial derivative itself.

Another more complicated graph is the species-reaction graph or directed species-reaction graph (DSR graph). As explained in detail by Kaltenbach the definition (and the name of the object) are not consistent in the literature. The species-reaction graph was introduced in a paper of Craciun and Feinberg (SIAM J. Appl. Math. 66, 1321). In a parallel development which started earlier Mincheva and Roussel (J. Math. Biol. 55, 61) developed results using this type of graph based on ideas of Ivanova which were little known in the West and for which published proofs were not available. In the sense used by Kaltenbach the DSR graph is an object closely related to his version of the interaction graph. It is a bipartite graph (i.e. there are two different types of vertices and each edge connects a vertex of one type with a vertex of the other). In the DSR graph the species define vertices of one type and the reactions the vertices of the other type. There is a directed edge from species $i$ to reaction $j$ if species $i$ is present on the LHS of reaction $j$. There is a directed edge from reaction $i$ to species $j$ if the net production of species $j$ in reaction $i$ is non-zero. The first type of edge is labelled by the partial derivative of flux $j$ with respect to species $i$. The second type is labelled by the corresponding stoichiometric coefficient. The DSR graph determines the interaction graph. The paper of Soliman I mentioned in a recent post uses the DSR graph in the sense of Kaltenbach.

A type of species-reaction graph has been used in quite a different way by Angeli, de Leenheer and Sontag to obtain conditions for the montonicity of the equations for a network written in reaction coordinates.