Abstract
The simulation of stochastic reaction–diffusion systems using fine-grained representations can become computationally prohibitive when particle numbers become large. If particle numbers are sufficiently high then it may be possible to ignore stochastic fluctuations and use a more efficient coarse-grained simulation approach. Nevertheless, for multiscale systems which exhibit significant spatial variation in concentration, a coarse-grained approach may not be appropriate throughout the simulation domain. Such scenarios suggest a hybrid paradigm in which a computationally cheap, coarse-grained model is coupled to a more expensive, but more detailed fine-grained model, enabling the accurate simulation of the fine-scale dynamics at a reasonable computational cost. In this paper, in order to couple two representations of reaction–diffusion at distinct spatial scales, we allow them to overlap in a ‘blending region’. Both modelling paradigms provide a valid representation of the particle density in this region. From one end of the blending region to the other, control of the implementation of diffusion is passed from one modelling paradigm to another through the use of complementary ‘blending functions’ which scale up or down the contribution of each model to the overall diffusion. We establish the reliability of our novel hybrid paradigm by demonstrating its simulation on four exemplar reaction–diffusion scenarios.
Highlights
Many biological and physical systems are inherently multiscale in nature [1,2,3,4,5,6]
The first two problems are simulations of pure diffusion with different initial conditions, demonstrating that the fluxes over the interface of the hybrid model are consistent with the expected behaviour of the finer-scale representation in each hybrid model
A simple comparison of the expected densities at time t = 1 shows that the maximum magnitude of the partial differential equations (PDEs) relative error with respect to the compartment-based model is roughly 3 × 10−2, demonstrating that the size of the relative error we find between our hybrid method and the solution of the fully compartment-based simulations is of an appropriate order or magnitude, as it is similar to the difference in the concentration when comparing the equilibrium profile of the full PDE to the fully compartment-based method, adjusted for the specific voxel size
Summary
Many biological and physical systems are inherently multiscale in nature [1,2,3,4,5,6]. The finest representation we consider is a Brownian dynamics model at the microscopic scale [49,50,51,52] In these models, the trajectories of all particles are simulated (typically using a discrete fixed time-step paradigm) in continuous space [49,53,54,55]. These methods can be extremely computationally intensive They do, provide a comprehensive and accurate individual representation capable of incorporating stochasticity into particle positions and interaction times. Implementing a fine-scale individual-based representation in regions in which low-copy-number effects are of paramount importance can give significant improvements in accuracy in comparison to coarser models.
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