Abstract
Two multiscale algorithms for stochastic simulations of reaction–diffusion processes are analysed. They are applicable to systems which include regions with significantly different concentrations of molecules. In both methods, a domain of interest is divided into two subsets where continuous-time Markov chain models and stochastic partial differential equations (SPDEs) are used, respectively. In the first algorithm, Markov chain (compartment-based) models are coupled with reaction–diffusion SPDEs by considering a pseudo-compartment (also called an overlap or handshaking region) in the SPDE part of the computational domain right next to the interface. In the second algorithm, no overlap region is used. Further extensions of both schemes are presented, including the case of an adaptively chosen boundary between different modelling approaches.
Highlights
Stochastic models of well-mixed chemical systems are traditionally formulated in terms of continuous-time Markov chains, which can be simulated using the Gillespie stochastic simulation algorithm (SSA) (Gillespie 1977) or its equivalent formulations (Gibson and Bruck 2000; Cao et al 2004; Klingbeil et al 2011)
We have introduced two multiscale algorithms coupling the stochastic partial differential equations (SPDEs) and the Markov chain model, which provide good approximations to the solutions obtained by the Markov chain model applied in the entire spatial domain
The spatial chemical Langevin equation was applied to the Gray–Scott model, and its pattern formation was compared to the ones obtained by the reaction–diffusion master equation and partial differential equations (PDEs) (Ghosh et al 2015)
Summary
Stochastic models of well-mixed chemical systems are traditionally formulated in terms of continuous-time Markov chains, which can be simulated using the Gillespie stochastic simulation algorithm (SSA) (Gillespie 1977) or its equivalent formulations (Gibson and Bruck 2000; Cao et al 2004; Klingbeil et al 2011). In the thermodynamic limit (of large populations), compartment-based models lead to reaction–diffusion partial differential equations (PDEs) which are written in terms of spatio-temporal concentrations of chemical species. This property can be exploited to design multiscale (hybrid) algorithms which use the compartment-based Markov chain model in a subset of the simulated system and apply reaction–diffusion PDEs in other parts (Kalantzis 2009; Ferm et al 2010; Yates and Flegg 2015; Spill et al 2015; Harrison and Yates 2016). K = 1, 2, ..., K − 1), random processes counting the numbers of times that one molecule of the ith species in the kth compartment diffuses to the (k − 1)th compartment
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