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Abstract Continuous-time Markov chains are frequently used to model the stochastic dynamics of (bio)chemical reaction networks. However, except in very special cases, they cannot be analyzed exactly. Additionally, simulation can be computationally intensive. An approach to address these challenges is to consider a more tractable diffusion approximation. Leite and Williams (Ann. Appl. Prob.29, 2019) proposed a reflected diffusion as an approximation for (bio)chemical reaction networks, which they called the constrained Langevin approximation (CLA) as it extends the usual Langevin approximation beyond the first time some chemical species becomes zero in number. Further explanation and examples of the CLA can be found in Anderson et al.( SIAM Multiscale Modeling Simul.17, 2019). In this paper, we extend the approximation of Leite and Williams to (nearly) density-dependent Markov chains, as a first step to obtaining error estimates for the CLA when the diffusion state space is one-dimensional, and we provide a bound for the error in a strong approximation. We discuss some applications for chemical reaction networks and epidemic models, and illustrate these with examples. Our method of proof is designed to generalize to higher dimensions, provided there is a Lipschitz Skorokhod map defining the reflected diffusion process. The existence of such a Lipschitz map is an open problem in dimensions more than one. 

Abstract Continuous-time Markov chains are frequently used as stochastic models for chemical reaction networks, especially in the growing field of systems biology. A fundamental problem for these Stochastic Chemical Reaction Networks (SCRNs) is to understand the dependence of the stochastic behavior of these systems on the chemical reaction rate parameters. Towards solving this problem, in this paper we develop theoretical tools called comparison theorems that provide stochastic ordering results for SCRNs. These theorems give sufficient conditions for monotonic dependence on parameters in these network models, which allow us to obtain, under suitable conditions, information about transient and steady-state behavior. These theorems exploit structural properties of SCRNs, beyond those of general continuous-time Markov chains. Furthermore, we derive two theorems to compare stationary distributions and mean first passage times for SCRNs with different parameter values, or with the same parameters and different initial conditions. These tools are developed for SCRNs taking values in a generic (finite or countably infinite) state space and can also be applied for non-mass-action kinetics models. When propensity functions are bounded, our method of proof gives an explicit method for coupling two comparable SCRNs, which can be used to simultaneously simulate their sample paths in a comparable manner. We illustrate our results with applications to models of enzymatic kinetics and epigenetic regulation by chromatin modifications.