  Absolute Concentration Robustness (ACR) was introduced by Shinar and Feinberg
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.
