Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates

Abstract

This paper derives new results for certain classes of chemical reaction networks, linking structural to dynamical properties. In particular, it investigates their monotonicity and convergence under the assumption that the rates of the reactions are monotone functions of the concentrations of their reactants. This is satisfied for, yet not restricted to, the most common choices of the reaction kinetics such as mass action, Michaelis-Menten and Hill kinetics. The key idea is to find an alternative representation under which the resulting system is monotone. As a simple example, the paper shows that a phosphorylation/dephosphorylation process, which is involved in many signaling cascades, has a global stability property. We also provide a global stability result for a more complicated example that describes a regulatory pathway of a prevalent signal transduction module, the MAPK cascade.

Intermediates and Generic Convergence to Equilibria

“…The approach to generic and global convergence to equilibria in [3] is based upon the monotone systems theory of M. W. Hirsch [8,17,9]. The reader familiar with that theory will likely notice the connection, although most of the details have been deliberately hidden in our presentation by framing all concepts and results pertaining to monotonicity directly in terms of the graphical conditions given in [3].…”

“…This paper is organized as follows. In Section 2 we introduce our basic notation and working definition of reaction network, then review the graphical conditions for generic and global convergence to equilibria of [3] and state Theorem 1. In Section 3 we describe a systematic procedure to obtain a reduced reaction network by successively removing intermediates from a given network.…”

“…The reader familiar with that theory will likely notice the connection, although most of the details have been deliberately hidden in our presentation by framing all concepts and results pertaining to monotonicity directly in terms of the graphical conditions given in [3]. This paper is organized as follows.…”

“…In what follows, we will show that connectivity and the positive loop property of the R-graph are always invariant under the successive removal of intermediates under the assumptions of [3], meaning that the reduced network has them if and only if the original one does also, as illustrated in the example above. Thus, the conditions for the original model can be checked in the often times much simpler reduced model.…”

“…In [3], sufficient graphical conditions for a reaction network to exhibit generic and global convergence to equilibria (within each stoichiometric compatibility class) were given. The technique consists of checking that the R-graph of the network is such that every simple loop has an even number of negative edges, a property known as the positive loop property, and that there exists a directed path between any two reaction vertices in the directed SR-graph.…”