My paper “A computational approach to steady state correspondence of regular and generalized mass action systems” was just accepted in Bulletin of Mathematical Biology. A preprint of the paper is available here [1].

The paper furthers the theory and application of the network correspondence process known as network translation, which was introduced in my previous paper “Translated Chemical Reaction Networks” (available here [2]). For an example of how network translation works, consider following networks (regular on left, generalized on right):

The network on the left corresponds to the classic Lotka-Volterra system, where X_{1} corresponds to the prey and X_{2} corresponds to the predator. The generalized network on the right has the same governing mass action dynamics with the caveat that the kinetic complexes (dotted in graph) generate the monomials in the mass action equations rather than the stoichiometric complexes. The correspondence between the networks can be represented by the following translation scheme:

It is known that a well-structured generalized network can simplify the analysis of the steady states of the original mass action system. A particularly important property is weak reversibility, i.e. the property that, given a path from one node in the reaction graph to another, there must be a path back (as in the generalized network on the right). But this network structure is generally unknown. We do not know *a priori* what the generalized network should look like, or even that there is one!

This paper presents a partial answer to the question of how to computationally perform this translation process. The algorithm is implemented in a mixed-integer linear programming (MILP) framework and is capable of further implementing a technical result (Theorem 1 in the paper) which holds for certain more complicated networks.

Too good to be true? Can we now determine the steady state properties of any mass action system imaginable by simply plugging it into a computer program? Our enthusiasm should be tempered by a few theoretical and practical limitations:

- The program is capable of determining the necessary connections (i.e. reactions) in the translation but still requires a predetermined set of complexes to serve as the nodes in the graph. For large networks it can be very challenging to determine a reasonable guess as to the form of these complexes. Clearly, for wide-scale implementation, algorithmic determination of these complexes is required.
- The program can currently check for valid translations for a fixed set of rate constants in the original network, or for a stochastically determined set of rate constants. It would be useful to check, however, for well-structured translations within the set of all mass action systems with the same original structure (i.e. for
*all*rate constants). Leaving these parameters as decision variables, however, destroys the linearity of the algorithm, which makes implementation more difficult. - Many biochemical reaction systems do not have simple translations; in general, the process is messy. For example, consider the Wnt signaling pathway considered here [3]. It is easy to determine a network translation, but things become “crowded” in such a way that the steady states are not immediately discernible from the translated network structure (at least in the way required of the paper). Recent work, however, has suggested that the steady state properties may be related to the translation, but not in any obvious way—rather, we have to consider a system of subgraphs of the translated network. Stay tuned!

Nevertheless, the paper is a significant step towards the wide-scale implementation of network translation as a tool for investigating dynamical properties of mass action systems.

### References:

[1] M.D. Johnston. *A computational approach to steady state correspondence of regular and generalized mass action systems*, accepted in Bull. Math. Biol.

[2] M.D. Johnston. *Translated chemical reaction networks*, Bull. Math. Biol. 76(5), 1081-1116,2014.

[3] E. Gross, H.A. Harrington, Z. Rosen, and B. Sturmfels. *Algebraic Systems Biology: A Case Study for the Wnt Pathway*, available on the arXiv at arxiv:1502.03188