Abstract
Stochastic modelling is critical for studying many biochemical processes in a cell, in particular when some reacting species have low population numbers. For many such cellular processes the spatial distribution of the molecular species plays a key role. The evolution of spatially heterogeneous biochemical systems with some species in low amounts is accurately described by the mesoscopic model of the Reaction-Diffusion Master Equation. The Inhomogeneous Stochastic Simulation Algorithm provides an exact strategy to numerically solve this model, but it is computationally very expensive on realistic applications. We propose a novel adaptive time-stepping scheme for the tau-leaping method for approximating the solution of the Reaction-Diffusion Master Equation. This technique combines effective strategies for variable time-stepping with path preservation to reduce the computational cost, while maintaining the desired accuracy. The numerical tests on various examples arising in applications show the improved efficiency achieved by the new adaptive method.
Highlights
Stochastic modelling and simulation are crucial tools for studying important biological processes at the level of a single cell, when some molecules are in low copy number.[1,2,3] The random fluctuations due to small population numbers of certain biochemically reacting species have been observed experimentally.[4,5] Mathematically, the dynamics of these biochemical systems may be modelled using Markov processes.[6]
We illustrate the advantages of the new variable time-stepping strategy for the spatial tauleaping method on three models of heterogeneous biochemical systems of practical interest
We present below a 2-D model of a cyclic adenosine monophosphate activation of protein kinase A (PKA).[42]
Summary
Stochastic modelling and simulation are crucial tools for studying important biological processes at the level of a single cell, when some molecules are in low copy number.[1,2,3] The random fluctuations due to small population numbers of certain biochemically reacting species have been observed experimentally.[4,5] Mathematically, the dynamics of these biochemical systems may be modelled using Markov processes.[6]. We describe a new adaptive time-stepping technique for the tau-leaping method for approximating the exact solution of a heterogeneous stochastic discrete model of biochemical kinetics, the Reaction-Diffusion Master Equation. Variable time-steppping schemes employing integral and proportional-integral control in the numerical solution of commutative Stratonovich stochastic differential equations with multiple Wiener processes were first developed by Burrage & Burrage[47] and were recently considered for numerically solving commutative Itô stochastic differential equations.[48] Some ideas from these strategies are generalized below to efficiently select the steps in the tau-leaping method for the RDME.
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