Department of Mathematics, Imperial College London, London, SW7 2AZ, UK.
J Math Biol. 2024 Nov 4;89(6):58. doi: 10.1007/s00285-024-02160-2.
There are many processes in cell biology that can be modeled in terms of particles diffusing in a two-dimensional (2D) or three-dimensional (3D) bounded domain containing a set of small subdomains or interior compartments , (singularly-perturbed diffusion problems). The domain could represent the cell membrane, the cell cytoplasm, the cell nucleus or the extracellular volume, while an individual compartment could represent a synapse, a membrane protein cluster, a biological condensate, or a quorum sensing bacterial cell. In this review we use a combination of matched asymptotic analysis and Green's function methods to solve a general type of singular boundary value problems (BVP) in 2D and 3D, in which an inhomogeneous Robin condition is imposed on each interior boundary . This allows us to incorporate a variety of previous studies of singularly perturbed diffusion problems into a single mathematical modeling framework. We mainly focus on steady-state solutions and the approach to steady-state, but also highlight some of the current challenges in dealing with time-dependent solutions and randomly switching processes.
细胞生物学中有许多过程可以用在二维（2D）或三维（3D）有界域中扩散粒子的方式来建模，该有界域包含一组小的子域或内部隔室（奇异微扰扩散问题）。该域可以表示细胞膜、细胞质、细胞核或细胞外体积，而单个隔室可以表示一个突触、一个膜蛋白簇、一个生物凝聚物或一个群体感应细菌细胞。在这篇综述中，我们使用匹配渐近分析和格林函数方法的组合来解决 2D 和 3D 中一般类型的奇异边值问题（BVP），其中在每个内部边界上施加不均匀的 Robin 条件。这使我们能够将奇异微扰扩散问题的各种先前研究纳入单个数学建模框架。我们主要关注稳态解和稳态的方法，但也强调了处理时变解和随机切换过程的一些当前挑战。
