Foundations of Static and Dynamic Absolute Concentration Robustness

TL;DR

This paper explores the concept of Absolute Concentration Robustness (ACR) in reaction networks, introducing dynamic ACR to better capture empirical robustness and providing conditions for its occurrence in complex balanced networks.

Contribution

It introduces the concept of dynamic ACR, linking robustness to long-term dynamics, and establishes necessary and sufficient conditions for dynamic ACR in complex balanced reaction networks.

Findings

Dynamic ACR better captures empirical robustness than static ACR.

Necessary and sufficient conditions for dynamic ACR are derived for complex balanced networks.

The study connects long-term system behavior with robustness properties.

Abstract

Absolute Concentration Robustness (ACR) was introduced by Shinar and Feinberg as robustness of equilibrium species concentration in a mass action dynamical system. Their aim was to devise a mathematical condition that will ensure robustness in the function of the biological system being modeled. The robustness of function rests on what we refer to as empirical robustness -- the concentration of a species remains unvarying, when measured in the long run, across arbitrary initial conditions. Even simple examples show that the ACR notion introduced in Shinar and Feinberg (here referred to as static ACR) is neither necessary nor sufficient for empirical robustness. To make a stronger connection with empirical robustness, we define dynamic ACR, a property related to long-term, global dynamics, rather than only to equilibrium behavior. We discuss general dynamical systems with dynamic ACR properties as well as parametrized families of dynamical systems related to reaction networks. We find necessary and sufficient conditions for dynamic ACR in complex balanced reaction networks, a class of networks that is central to the theory of reaction networks.