Error Bound for Piecewise Deterministic Processes Modeling Stochastic Reaction Systems

Abstract

Biological processes involving the random interaction of d species with integer particle numbers are often modeled by a Markov jump process on N d 0 . Realizations of this process can, in principle, be generated with the classical stochastic simulation algorithm proposed in [19], but for very reactive systems this method is usually inefficient. Hybrid models based on piecewise deterministic processes offer an attractive alternative which can decrease the simulation time considerably in applications where species with rather low particle numbers interact with very abundant species. We investigate the convergence of the hybrid model to the original one for a class of reaction systems with two distinct scales. Our main result is an error bound which states that, under suitable assumptions, the hybrid model approximates the marginal distribution of the discrete species and the conditional moments of the continuous species up to an error of O`M −1´w here M is the scaling parameter of the partial thermodynamic limit.

Approximation and inference methods for stochastic biochemical kinetics—a tutorial review

“…In some cases, this is possible, however. For example, error bounds for some hybrid methods have been derived [213,214,208]. In [215] the convergence of different types of hybrid methods to the exact system have been studied, and in [184] criteria have been developed for the convergence of a CME to discrete-deterministic hybrid approximations.…”

The relative contributions of infectious and mitotic spread to HTLV-1 persistence

“…Thus the total number of possible reactions C ∈ ℕ at time t is C = 3S(t) . Following the formulation given in [26, 27], let X ( t ) = (( X i ( t )) i ∈ S ( t ) ) T be the state vector at time t of all clones. X(t) is a random variable in that consists of the random variables X i ( t ) ∈ ℕ 0 = ℕ ⋃{0} of the frequencies x i (t) of clones i = 1,…, S max , where S max is chosen to always be larger than S(t) for all t .…”

“…X(t) is a random variable in that consists of the random variables X i ( t ) ∈ ℕ 0 = ℕ ⋃{0} of the frequencies x i (t) of clones i = 1,…, S max , where S max is chosen to always be larger than S(t) for all t . The state vector X(t) evolves through a Markov jump process that depends only on the current state , and its evolution is given by where V c and α c respectively denote the stoichiometric vector and propensity function of reaction c [26, 27]. Equation (2) states that the population X(t) at time t is equal to the initial population y 0 plus the sum of the changes induced by all reactions.…”

Pathwise Error Bounds in Multiscale Variable Splitting Methods for Spatial Stochastic Kinetics

“…In the first category, tentative analysis of specific examples are found in [4], while [17,19] are of more general character and based on averaging techniques, and conditional expectations, respectively. A related analysis in the sense of meansquare convergence for operator splitting techniques is found in [11].…”