  It has recently been shown that structural conditions on the reaction
network, rather than a 'fine-tuning' of system parameters, often suffice to
impart 'absolute concentration robustness' on a wide class of biologically
relevant, deterministically modeled mass-action systems [Shinar and Feinberg,
Science, 2010]. We show here that fundamentally different conclusions about the
long-term behavior of such systems are reached if the systems are instead
modeled with stochastic dynamics and a discrete state space. Specifically, we
characterize a large class of models that exhibit convergence to a positive
robust equilibrium in the deterministic setting, whereas trajectories of the
corresponding stochastic models are necessarily absorbed by a set of states
that reside on the boundary of the state space, i.e. the system undergoes an
extinction event. If the time to extinction is large relative to the relevant
time-scales of the system, the process will appear to settle down to a
stationary distribution long before the inevitable extinction will occur. This
quasi-stationary distribution is considered for two systems taken from the
literature, and results consistent with absolute concentration robustness are
recovered by showing that the quasi-stationary distribution of the robust
species approaches a Poisson distribution.
