Sign-sensitivities for reaction networks: an algebraic approach

TL;DR

This paper develops an algebraic framework to analyze how the signs of concentration sensitivities in reaction networks depend on parameters and steady states, providing criteria to determine sign changes without explicit calculations.

Contribution

It introduces a closed-form formula for sign sensitivities, proposes a sign-based criterion for dependency analysis, and discusses the implications for systems with multiple steady states.

Findings

Sign sensitivities can be characterized algebraically.

A criterion to determine sign dependence without explicit sensitivity computation.

Analysis of stable versus multiple steady states in reaction networks.

Abstract

This paper presents an algebraic framework to study sign-sensitivities for reaction networks modeled by means of systems of ordinary differential equations. Specifically, we study the sign of the derivative of the concentrations of the species in the network at steady state with respect to a small perturbation on the parameter vector. We provide a closed formula for the derivatives that accommodates common perturbations, and illustrate its form with numerous examples. We argue that, mathematically, the study of the response to the system with respect to changes in total amounts is not well posed, and that one should rather consider perturbations with respect to the initial conditions. We find a sign-based criterion to determine, without computing the sensitivities, whether the sign depends on the steady state and parameters of the system. This is based on earlier results of so-called injective networks. Finally, we address systems with multiple steady states and the restriction to stable steady states.