Foundations of Static and Dynamic Absolute Concentration Robustness (Part I of Dynamic ACR Quadrilogy)

Absolute Concentration Robustness (ACR) was introduced by Shinar and Feinberg [1] as a way to define species concentration robustness in mass action dynamical systems. The idea 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 variable remains unvarying, when measured in the long run, across arbitrary initial conditions. While there is a positive correlation between ACR and empirical robustness, ACR is neither necessary nor sufficient for empirical robustness, a fact that can be noticed even in simple biochemical systems. To develop a stronger connection with empirical robustness, we define dynamic ACR, a property related to dynamics, rather than only to equilibrium behavior, and one that guarantees convergence to a robust value. We distinguish between wide basin and narrow basin versions of dynamic ACR, related to the size of the set of initial values that do not result in convergence to the robust value. We give numerous examples which help distinguish the various flavors of ACR as well as clearly illustrate and circumscribe the conditions that appear in the definitions. We discuss general dynamical systems with ACR properties as well as parametrized families of dynamical systems related to reaction networks. We discuss connections between ACR and complex balance, two notions central to the theory of reaction networks. We give precise conditions for presence and absence of dynamic ACR in complex balanced systems, which in turn yields a large body of reaction networks with dynamic ACR.