The structure of MESSI biological systems

TL;DR

This paper introduces MESSI systems, a general framework for biological networks involving enzyme-substrate modifications, and provides mathematical tools to analyze their steady states and multistationarity.

Contribution

The paper defines MESSI systems, proves their conservation under mass-action kinetics, and offers explicit methods for analyzing steady states and multistationarity, especially for systems with toric steady states.

Findings

MESSI systems encompass many biological modification networks.

Under mass-action kinetics, MESSI systems are conservative.

An algorithm is provided to determine multistationarity in MESSI systems with toric steady states.

Abstract

We introduce a general framework for biological systems, called MESSI systems, that describe Modifications of type Enzyme-Substrate or Swap with Intermediates, and we prove general results based on the network structure. Many post-translational modification networks are MESSI systems. For example: the motifs in [Feliu and Wiuf (2012a)], sequential distributive and processive multisite phosphorylation networks, most of the examples in [Angeli et al. (2007)], phosphorylation cascades, two component systems as in [Kothamachu et al. (2015)], the bacterial EnvZ/OmpR network in [Shinar and Feinberg (2010)], and all linear networks. We show that, under mass-action kinetics, MESSI systems are conservative. We simplify the study of steady states of these systems by explicit elimination of intermediate complexes and we give conditions to ensure an explicit rational parametrization of the variety of steady states (inspired by [Feliu and Wiuf (2013a, 2013b), Thomson and Gunawardena (2009)]). We define an important subclass of MESSI systems with toric steady states [P\'erez Mill\'an et al. (2012)] and we give for MESSI systems with toric steady states an easy algorithm to determine the capacity for multistationarity. In this case, the algorithm provides rate constants for which multistationarity takes place, based on the theory of oriented matroids.