Peter Medawar Building for Pathogen Research, University of Oxford, Oxford, United Kingdom.
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom.
Rep Prog Phys. 2022 Oct 5;85(10). doi: 10.1088/1361-6633/ac906b.
Computational methods are now recognized as powerful and complementary approaches in various applied sciences such as biology. These computing methods are used to explore the gap between scales such as the one between molecular and cellular. Here we present recent progress in the development of computational approaches involving diffusion modeling, asymptotic analysis of the model partial differential equations, hybrid methods and simulations in the generic context of cell sensing and guidance via external gradients. Specifically, we highlight the reconstruction of the location of a point source in two and three dimensions from the steady-state diffusion fluxes arriving to narrow windows located on the cell. We discuss cases in which these windows are located on the boundary of a two-dimensional plane or three-dimensional half-space, on a disk in free space or inside a two-dimensional corridor, or a ball in three dimensions. The basis of this computational approach is explicit solutions of the Neumann-Green's function for the mentioned geometry. This analysis can be used to design hybrid simulations where Brownian paths are generated only in small regions in which the local spatial organization is relevant. Particle trajectories outside of this region are only implicitly treated by generating exit points at the boundary of this domain of interest. This greatly accelerates the simulation time by avoiding the explicit computation of Brownian paths in an infinite domain and serves to generate statistics, without following all trajectories at the same time, a process that can become numerically expensive quickly. Moreover, these computational approaches are used to reconstruct a point source and estimating the uncertainty in the source reconstruction due to an additive noise perturbation present in the fluxes. We also discuss the influence of various window configurations (cluster vs uniform distributions) on recovering the source position. Finally, the applications in developmental biology are formulated into computational principles that could underly neuronal navigation in the brain.
计算方法现在被认为是生物学等各种应用科学中强大且互补的方法。这些计算方法用于探索分子和细胞之间的尺度差距。在这里，我们介绍了涉及扩散建模、模型偏微分方程渐近分析、混合方法和模拟的最新进展，这些方法在外部梯度介导的细胞感应和导向的一般情况下都有应用。具体来说，我们从到达位于细胞上的窄窗口的稳态扩散通量中，突出了重建二维和三维中一个点源位置的方法。我们讨论了这些窗口位于二维平面或三维半空间的边界、自由空间中的圆盘、二维走廊内或三维球体内的情况。这种计算方法的基础是所提到的几何形状的 Neumann-Green 函数的显式解。这种分析可用于设计混合模拟，其中布朗路径仅在与局部空间组织相关的小区域中生成。在该感兴趣区域的边界处生成出口点，从而仅隐式地处理该区域之外的粒子轨迹。通过避免在无限域中显式计算布朗路径，大大加速了模拟时间，并且可以生成统计信息，而无需同时跟踪所有轨迹，这一过程很快就会变得非常耗费计算资源。此外，这些计算方法还用于重建点源，并估计由于通量中存在的加性噪声扰动而导致的源重建中的不确定性。我们还讨论了各种窗口配置（簇与均匀分布）对恢复源位置的影响。最后，将发育生物学中的应用转化为潜在的神经元在大脑中导航的计算原理。
