  Recent research in both the experimental and mathematical communities has
focused on biochemical interaction systems that satisfy an "absolute
concentration robustness" (ACR) property. The ACR property was first discovered
experimentally when, in a number of different systems, the concentrations of
key system components at equilibrium were observed to be robust to the total
concentration levels of the system. Followup mathematical work focused on
deterministic models of biochemical systems and demonstrated how chemical
reaction network theory can be utilized to explain this robustness. Later
mathematical work focused on the behavior of this same class of reaction
networks, though under the assumption that the dynamics were stochastic. Under
the stochastic assumption, it was proven that the system will undergo an
extinction event with a probability of one so long as the system is
conservative, showing starkly different long-time behavior than in the
deterministic setting. Here we consider a general class of stochastic models
that intersects with the class of ACR systems studied previously. We consider a
specific system scaling over compact time intervals and prove that in a limit
of this scaling the distribution of the abundances of the ACR species converges
to a certain product-form Poisson distribution whose mean is the ACR value of
the deterministic model. This result is in agreement with recent conjectures
pertaining to the behavior of ACR networks endowed with stochastic kinetics,
and helps to resolve the conflicting theoretical results pertaining to
deterministic and stochastic models in this setting.
