Finite time distributions of stochastically modeled chemical systems with absolute concentration robustness

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

Finite time distributions of stochastically modeled chemical systems with absolute concentration robustness

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. Follow-up 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

Stationary distributions and condensation in autocatalytic CRN

We investigate a broad family of non weakly reversible stochastically modeled reaction networks (CRN), by looking at their steady-state distributions. Most known results on stationary distributions assume weak reversibility and zero deficiency. We first give explicitly product-form steady-state distributions for a class of non weakly reversible autocatalytic CRN of arbitrary deficiency. Examples of interest in statistical mechanics (inclusion process), life sciences and robotics (collective decision making in ant and robot swarms) are provided. The product-form nature of the steady-state then enables the study of condensation in particle systems that are generalizations of the inclusion process.Comment: 25 pages. Some typos corrected, shortened some part

Limits for Stochastic Reaction Networks

Reaction systems have been introduced in the 70s to model biochemical systems. Nowadays their range of applications has increased and they are fruitfully used in different fields. The concept is simple: some chemical species react, the set of chemical reactions form a graph and a rate function is associated with each reaction. Such functions describe the speed of the different reactions, or their propensities. Two modelling regimes are then available: the evolution of the different species concentrations can be deterministically modelled through a system of ODE, while the counts of the different species at a certain time are stochastically modelled by means of a continuous-time Markov chain. Our work concerns primarily stochastic reaction systems, and their asymptotic properties. In Paper I, we consider a reaction system with intermediate species, i.e. species that are produced and fast degraded along a path of reactions. Let the rates of degradation of the intermediate species be functions of a parameter N that tends to infinity. We consider a reduced system where the intermediate species have been eliminated, and find conditions on the degradation rate of the intermediates such that the behaviour of the reduced network tends to that of the original one. In particular, we prove a uniform punctual convergence in distribution and weak convergence of the integrals of continuous functions along the paths of the two models. Under some extra conditions, we also prove weak convergence of the two processes. The result is stated in the setting of multiscale reaction systems: the amounts of all the species and the rates of all the reactions of the original model can scale as powers of N. A similar result also holds for the deterministic case, as shown in Appendix IA. In Paper II, we focus on the stationary distributions of the stochastic reaction systems. Specifically, we build a theory for stochastic reaction systems that is parallel to the deficiency zero theory for deterministic systems, which dates back to the 70s. A deficiency theory for stochastic reaction systems was missing, and few results connecting deficiency and stochastic reaction systems were known. The theory we build connects special form of product-form stationary distributions with structural properties of the reaction graph of the system. In Paper III, a special class of reaction systems is considered, namely systems exhibiting absolute concentration robust species. Such species, in the deterministic modelling regime, assume always the same value at any positive steady state. In the stochastic setting, we prove that, if the initial condition is a point in the basin of attraction of a positive steady state of the corresponding deterministic model and tends to infinity, then up to a fixed time T the counts of the species exhibiting absolute concentration robustness are, on average, near to their equilibrium value. The result is not obvious because when the counts of some species tend to infinity, so do some rate functions, and the study of the system may become hard. Moreover, the result states a substantial concordance between the paths of the stochastic and the deterministic models

Discrepancies between extinction events and boundary equilibria in reaction networks

Reaction networks are mathematical models of interacting chemical species that are primarily used in biochemistry. There are two modeling regimes that are typically used, one of which is deterministic and one that is stochastic. In particular, the deterministic model consists of an autonomous system of differential equations, whereas the stochastic system is a continuous-time Markov chain. Connections between the two modeling regimes have been studied since the seminal paper by Kurtz (J Chem Phys 57(7):2976–2978, 1972), where the deterministic model is shown to be a limit of a properly rescaled stochastic model over compact time intervals. Further, more recent studies have connected the long-term behaviors of the two models when the reaction network satisfies certain graphical properties, such as weak reversibility and a deficiency of zero. These connections have led some to conjecture a link between the long-term behavior of the two models exists, in some sense. In particular, one is tempted to believe that positive recurrence of all states for the stochastic model implies the existence of positive equilibria in the deterministic setting, and that boundary equilibria of the deterministic model imply the occurrence of an extinction event in the stochastic setting. We prove in this paper that these implications do not hold in general, even if restricting the analysis to networks that are bimolecular and that conserve the total mass. In particular, we disprove the implications in the special case of models that have absolute concentration robustness, thus answering in the negative a conjecture stated in the literature in 2014