Finite Time Distributions of Stochastically Modeled Chemical Systems with Absolute Concentration Robustness

Abstract

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.

A hidden integral structure endows Absolute Concentration Robust systems with resilience to dynamical concentration disturbances

“…This is caused by the fact that all the molecules of B can be consumed by the reaction B → A, before the occurrence of a reaction A + B → 2B. Robustness at finite time intervals of some stochastically modeled ACR systems is recovered, but only in a multiscale limit sense [4]. Moreover, it is shown in [3] that absolute concentration robustness of the deterministic model does not necessarily imply an extinction event in the corresponding stochastic model, but the connection is still largely unexplored.…”

A review of the deterministic and diffusion approximations for stochastic chemical reaction networks

“…Firstly, both approximations are derived on a finite time horizon. It is well known that deterministic equations may fail to catch the limiting distribution of the corresponding stochastic model when time goes to infinity as it happens in the Example presented in Section 4.2, and for all the chemical systems with absolute concentration robustness [1,3]. At the same time, they can capture such asymptotic behaviour correctly in case of the complex balanced stochastic systems [10].…”

Low Variance Couplings for Stochastic Models of Intracellular Processes with Time-Dependent Rate Functions

“…The most widely used representation is usually called the random time change representation, and was developed by Thomas Kurtz [25]. It has been utilized widely for both the development of computational methods [1,2,3,6,8,14,22,23,24,30] and for analytical purposes [4,5,7,13]. For this representation, we start with independent unit-rate Poisson processes Y k (one for reach reaction channel) and define the process X as the solution to…”