Local and global robustness at steady state

TL;DR

This paper investigates the conditions under which the steady states of certain autonomous ODE systems, especially those from biochemical networks, exhibit robustness, linking local zero sensitivity to global absolute concentration robustness (ACR).

Contribution

It formalizes the relationship between local zero sensitivity and global ACR, providing criteria to identify ACR in biochemical reaction network models.

Findings

ACR implies zero sensitivity.

A criterion for local ACR in biochemical systems.

Identification of when local and global robustness properties differ.

Abstract

We study the robustness of the steady states of a class of systems of autonomous ordinary differential equations (ODEs), having as a central example those arising from (bio)chemical reaction networks. More precisely, we study under what conditions the steady states of the system are contained in a parallel translate of a coordinate hyperplane. To this end, we focus mainly on ODEs consisting of generalized polynomials, and make use of algebraic and geometric tools to relate the local and global structure of the set of steady states. Specifically, we consider the local property termed zero sensitivity at a coordinate $x_i$, which means that the tangent space is contained in a hyperplane of the form $x_i=c$, and provide a criterion to identify it. We consider the global property termed absolute concentration robustness (ACR), meaning that all steady states are contained in a hyperplane of the form $x_i=c$. We clarify and formalise the relation between the two approaches. In particular, we show that ACR implies zero sensitivity, and identify when the two properties do not agree, via an intermediate property we term local ACR. For families of systems arising from modelling biochemical reaction networks, we obtain the first practical and automated criterion to decide upon (local) ACR.