Local and global robustness at steady state

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 xi, which means that the tangent space is contained in a hyperplane of the form xi = 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 xi = c. We clarify and formalize 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 modeling biochemical reaction networks, we obtain the first practical and automated criterion to decide upon (local) ACR.

Sign‐sensitivity of metabolic networks: Which structures determine the sign of the responses

“…Mathematically, there is no need to impose such restriction, and we can freely consider real s j m , sj m ∈ R ≥0 . If a metabolite m appears at the left hand side of (10) with nonzero coefficient, then we say that m is an input of reaction j. Conversely, if m appears on the right hand side with nonzero coefficient, we call m an output of reaction j. Naturally, metabolic systems are open systems, exchanging chemicals with the outside environment via inflows and outflows.…”

“…In this section we present four examples to illustrate our results and some consequences of our analysis. To help visual intuition, we provide also a graphical representation, where we consider the metabolites as vertices and the reactions as directed hyperarrows, inheriting the orientation from the reaction orientation (10). It is one common representation, extensively used in chemistry, biology, and mathematics.…”

“…For a first few attempts to establish such a bridge, see. 10,11 Moreover, in Reference 12 Shinar and co-authors were able to derive quantitative bounds on the entries of the sensitivity matrix for reaction fluxes, in a mass-action kinetics context and for a regular class of networks. To our knowledge, only few contributions further address the signs of the sensitivity responses.…”

“…However, the precise mathematical connections between ACR and zero sensitivity are still to be investigated. For a first few attempts to establish such a bridge, see 10,11 . Moreover, in Reference 12 Shinar and co‐authors were able to derive quantitative bounds on the entries of the sensitivity matrix for reaction fluxes, in a mass‐action kinetics context and for a regular class of networks.…”