Symbolic dynamics of biochemical pathways as finite states machines

TL;DR

This paper explores how biochemical pathways can be modeled using finite state machines, providing a framework for understanding their dynamics through symbolic and tropical geometric methods.

Contribution

It introduces a novel approach to approximate nonlinear biochemical network dynamics with finite state machines using tropical equilibrations.

Findings

Monomolecular networks with separated rates are described by deterministic, acyclic automata.

Metastable states can be computed as solutions to tropical equilibration problems.

The approach offers a new way to analyze complex biochemical pathways.

Abstract

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.