Central limit theorems and diffusion approximations for multiscale Markov chain models

Abstract

Ordinary differential equations obtained as limits of Markov processes appear in many settings. They may arise by scaling large systems, or by averaging rapidly fluctuating systems, or in systems involving multiple time-scales, by a combination of the two. Motivated by models with multiple time-scales arising in systems biology, we present a general approach to proving a central limit theorem capturing the fluctuations of the original model around the deterministic limit. The central limit theorem provides a method for deriving an appropriate diffusion (Langevin) approximation.There are many proofs for theorems like these. In particular, results of both types can be proved using the martingale central limit theorem (Theorem A.1). For example, in the first case, there is typically a sequence of functions F N such that

A multiscale approximation in a heat shock response model of E. coli

“…Since $Z_{23}^{N,2}\left(t\right)$ is the species number scaled by N , we expect that r N = N 1/2 and the error between the scaled species numbers and their limit is of order $N_0^{-1/2}$. For a detailed approach to derive r N and U ( t ), see more about the central limit theorem in [14]. The fact that all components but the first one in the diffusion term in the equation for U ( t ) are zero supports the idea that noise is dominantly determined by the error between $Z_{23}^{N,2}\left(t\right)$ and $Z_{23}^2\left(t\right)$.…”

“…In the late stage of time period of order 10,000 sec, we study the error between the scaled species numbers and their limit analytically using the central limit theorem derived in [14] and show that the error is of order 10 −1 .…”

How spatial heterogeneity shapes multiscale biochemical reaction network dynamics

“…The above result holds, regardless of whether the fast fluctuating subsystem has some conserved quantities (as long as we assume that all molecular types combining into a single conserved quantity have the same movement equilibrium), as can be seen in the following classical example of enzymatic kinetics (see [21,26] for other results on the multiscale stochastic fluctuations in this model).…”

“…The dynamics of fast species follows a fast fluctuating Markov chain with dominant rates of O(N). The dynamics of slow types will be well approximated by a system of ordinary differential equations and diffusion fluctuations of size 1= ffiffiffiffi N p (theorem 2.11 of [21] and examples of [19]). For the derivation of the differential equation driving the slow species, a quasisteady-state assumption is used.…”

“…Recently, a number of intermediate or two-regime methods have been developed in order to reduce the computational complexity of problems [16][17][18] employing a compartment approach and approximations. Analytical results based on rigorous approximations provide important model reduction tools and can be used to significantly simplify the necessary steps in simulating stochasticity of the process [19][20][21][22]. In addition, they yield general comparisons for the average behaviour of the process under different scenarios for the speed of molecular motility [23].…”