Symbolic Dynamics of Biochemical Pathways as Finite States Machines

We discuss the symbolic dynamics of biochemical networks with separate timescales. We show that symbolic dynamics of monomolecular reaction networks with separated rate constants can be described by deterministic, acyclic automata with a number of states that is inferior to the number of biochemical species. For nonlinear pathways, we propose a general approach to approximate their dynamics by finite state machines working on the metastable states of the network (long life states where the system has slow dynamics). For networks with polynomial rate functions we propose to compute metastable states as solutions of the tropical equilibration problem. Tropical equilibrations are defined by the equality of at least two dominant monomials of opposite signs in the differential equations of each dynamic variable. In algebraic geometry, tropical equilibrations are tantamount to tropical prevarieties, that are finite intersections of tropical hypersurfaces.