SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page B132-B166, January 2021.
Mass-conservative reaction-diffusion systems have recently been proposed as a general framework to describe intracellular pattern formation. These systems have been used to model the conformational switching of proteins as they cycle from an inactive state in the cell cytoplasm, to an active state at the cell membrane. The active state then acts as input to downstream effectors. The paradigm of activation by recruitment to the membrane underpins a range of biological pathways, including G-protein signalling, growth control through Ras and PI 3-kinase, and cell polarity through Rac and Rho; all activate their targets by recruiting them from the cytoplasm to the membrane. Global mass conservation lies at the heart of these models, reflecting the property that the total number of active and inactive forms and total number of targets remains constant. Here we present a conservative arbitrary Lagrangian Eulerian (ALE) finite element method for the approximate solution of systems of bulk-surface reaction-diffusion equations on an evolving two-dimensional domain. Fundamental to the success of the method is the robust generation of bulk and surface meshes. For this purpose, we use a moving mesh partial differential equation (MMPDE) approach. Global conservation of the fully discrete finite element solution is established independently of the ALE velocity field and the time step size. The developed method is applied to model problems with known analytical solutions; these experiments indicate that the method is second-order accurate and globally conservative. The method is further applied to a model of a single cell migrating in the presence of an external chemotactic signal.

中文翻译：

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二维域上质量守恒反应扩散方程的守恒有限元ALE格式

SIAM科学计算杂志，第43卷，第1期，第B132-B166页，2021年1月。
最近已经提出了质量保守反应扩散系统作为描述细胞内模式形成的一般框架。这些系统已用于模拟蛋白质的构象转换，因为它们从细胞质中的非活性状态循环到细胞膜的活性状态。然后，活动状态充当下游执行器的输入。通过募集到膜而激活的范例支持了一系列生物途径，包括G蛋白信号传导，通过Ras和PI 3-激酶的生长控制以及通过Rac和Rho的细胞极性。所有这些都通过将它们从细胞质募集到膜来激活它们的靶标。全球质量守恒定律是这些模型的核心，反映出以下性质：活动和不活动形式的总数以及目标总数保持不变。在这里，我们提出了一个保守的任意拉格朗日欧拉（ALE）有限元方法，用于求解二维域上体表面反应扩散方程组的近似解。该方法成功的基础是大量生成体网格和曲面网格。为此，我们使用移动网格偏微分方程（MMPDE）方法。建立完全离散的有限元解的全局守恒与ALE速度场和时间步长无关。所开发的方法应用于具有已知解析解的模型问题。这些实验表明该方法是二阶准确的，并且在全局上是保守的。该方法进一步应用于在外部趋化信号存在下迁移的单个细胞的模型。