Abstract
Cellular reactions have a multi-scale nature in the sense that the abundance of molecular species and the magnitude of reaction rates can vary across orders of magnitude. This diversity naturally leads to hybrid models that combine continuous and discrete modeling regimes. In order to capture this multi-scale nature, we proposed jump-diffusion approximations in a previous study. The key idea was to partition reactions into fast and slow groups, and then to combine a Markov jump updating scheme for the slow group with a diffusion (Langevin) updating scheme for the fast group. In this study we show that the joint probability density function of the jump-diffusion approximation over the reaction counting process satisfies a hybrid master equation that combines terms from the chemical master equation and from the Fokker–Planck equation. Inspired by the method of conditional moments, we propose a efficient method to solve this master equation using the moments of reaction counters of the fast reactions given the reaction counters of the slow reactions. For each time point of interest, we then solve a set of maximum entropy problems in order to recover the conditional probability density from its moments. This finally allows us to reconstruct the complete joint probability density over all reaction counters and hence obtain an approximate solution of the hybrid master equation. Finally, we show the accuracy of the method applied to a simple multi-scale conversion process.
Highlights
Reactions are inherently discrete and stochastic in nature [12,15,16]
We present the hybrid master equation for jump-diffusion approximation, which models systems with multi-scale nature
Fast reactions are modeled using diffusion approximation, while Markov chain representation is employed for slow reactions
Summary
Reactions are inherently discrete and stochastic in nature [12,15,16]. Ignoring stochastic fluctuations and the integer nature of molecule counts can result in inappropriate models which especially fail for small reaction compartments as encountered in cell biology. Hybrid methods separate reactions and/or species into different groups of reactions and/or species, and they use the diffusion or the deterministic modeling approach to describe the dynamics of fast reactions and/or species with high copy numbers, while a discrete Markov chain representation is utilized for slow reactions and/or species with low copy numbers. In this paper, based on this representation, we present the hybrid master equation (HME), which is the evolution equation for the joint probability density function of the jump diffusion approximation over the reaction counting process. It is possible to use different numerical methods to approximate the solution of the HME [35], we use the strategy proposed in [25].To solve the HME, we obtain the evolution equation for the marginal probability of slow reactions and the evolution equations for the conditional moments of the fast reactions given slow reactions [25]. E j , ej denote (R − L) × 1, L × 1, unit vectors with 1 in the j-th component and 0 in other coordinates
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