Smith Cameron A, Yates Christian A
Centre for Mathematical Biology, Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK.
R Soc Open Sci. 2018 Aug 1;5(8):180920. doi: 10.1098/rsos.180920. eCollection 2018 Aug.
Reaction-diffusion systems are used to represent many biological and physical phenomena. They model the random motion of particles (diffusion) and interactions between them (reactions). Such systems can be modelled at multiple scales with varying degrees of accuracy and computational efficiency. When representing genuinely multiscale phenomena, fine-scale models can be prohibitively expensive, whereas coarser models, although cheaper, often lack sufficient detail to accurately represent the phenomenon at hand. Spatial hybrid methods couple two or more of these representations in order to improve efficiency without compromising accuracy. In this paper, we present a novel spatial hybrid method, which we call the auxiliary region method (ARM), which couples PDE- and Brownian-based representations of reaction-diffusion systems. Numerical PDE solutions on one side of an interface are coupled to Brownian-based dynamics on the other side using compartment-based 'auxiliary regions'. We demonstrate that the hybrid method is able to simulate reaction-diffusion dynamics for a number of different test problems with high accuracy. Furthermore, we undertake error analysis on the ARM which demonstrates that it is robust to changes in the free parameters in the model, where previous coupling algorithms are not. In particular, we envisage that the method will be applicable for a wide range of spatial multi-scales problems including filopodial dynamics, intracellular signalling, embryogenesis and travelling wave phenomena.
反应扩散系统用于描述许多生物和物理现象。它们对粒子的随机运动（扩散）及其之间的相互作用（反应）进行建模。此类系统可以在多个尺度上进行建模，具有不同程度的准确性和计算效率。在表示真正的多尺度现象时，精细尺度模型的成本可能过高，而较粗粒度的模型虽然成本较低，但往往缺乏足够的细节来准确表示手头的现象。空间混合方法将两种或更多种这些表示方式结合起来，以提高效率而不影响准确性。在本文中，我们提出了一种新颖的空间混合方法，我们称之为辅助区域方法（ARM），它将反应扩散系统的基于偏微分方程（PDE）和基于布朗运动的表示方式结合起来。界面一侧的数值偏微分方程解通过基于隔室的“辅助区域”与另一侧基于布朗运动的动力学相结合。我们证明，该混合方法能够高精度地模拟许多不同测试问题的反应扩散动力学。此外，我们对ARM进行了误差分析，结果表明它对模型中自由参数的变化具有鲁棒性，而以前的耦合算法则不具备这一特性。特别是，我们设想该方法将适用于广泛的空间多尺度问题，包括丝状伪足动力学、细胞内信号传导、胚胎发生和行波现象。
