Mem3DG：用离散微分几何模拟三维膜机械化学动力学
Mem3DG: Modeling Membrane Mechanochemical Dynamics in 3D using Discrete Differential Geometry
摘要：
生物膜采用对细胞功能至关重要的不同形态。许多研究使用计算建模来了解各种机械化学因素如何影响膜形状转变。与基于近似的方法（例如，有限元法），离散网格模型类提供了更大的灵活性，以模拟复杂的物理和三维形状;其制定产生一个有效的算法，同时保持坐标自由的几何描述。然而，在离散上下文中的几何定义的模糊性，导致缺乏共识的离散网格模型是理论上和数值上最优的，双射的能量和力的离散和光滑的几何理论之间的关系仍然有待建立。我们解决这个问题，并提出了一个可扩展的框架，$\texttt{Mem3DG}$，用于建模的三维机械化学动力学膜的基础上离散微分几何（DDG）三角网格。DDG的形式主义解决了不一致性，并提供了一个统一的观点，如何将光滑和离散的能量和力。为了证明这一点，使用$\texttt{Mem3DG}$对一系列机械化学复杂性不断增加的示例进行建模：恢复经典的形状转变，例如1）双凹面圆盘、哑铃和波纹状，以及2）球形平片膜上的球形芽;研究膜力学与蛋白质流动性的耦合如何共同影响相和形状转变。随着膜超微结构的高分辨率3D成像变得越来越容易获得，我们设想Mem 3DG将作为一个端到端的工具，在用户指定的机械化学条件下模拟逼真的细胞几何形状。
英文摘要：
Biomembranes adopt varying morphologies that are vital to cellular functions. Many studies use computational modeling to understand how various mechanochemical factors contribute to membrane shape transformations. Compared to approximation-based methods (e.g., finite element method), the class of discrete mesh models offers greater flexibility to simulate complex physics and shapes in three dimensions; its formulation produces an efficient algorithm while maintaining coordinate-free geometric descriptions. However, ambiguities in geometric definitions in the discrete context have led to a lack of consensus on which discrete mesh model is theoretically and numerically optimal; a bijective relationship between the terms contributing to both the energy and forces from the discrete and smooth geometric theories remains to be established. We address this and present an extensible framework, $\texttt{Mem3DG}$, for modeling 3D mechanochemical dynamics of membranes based on Discrete Differential Geometry (DDG) on triangulated meshes. The formalism of DDG resolves the inconsistency and provides a unifying perspective on how to relate the smooth and discrete energy and forces. To demonstrate, $\texttt{Mem3DG}$ is used to model a sequence of examples with increasing mechanochemical complexity: recovering classical shape transformations such as 1) biconcave disk, dumbbell, and unduloid and 2) spherical bud on spherical, flat-patch membrane; investigating how the coupling of membrane mechanics with protein mobility jointly affects phase and shape transformation. As high-resolution 3D imaging of membrane ultrastructure becomes more readily available, we envision Mem3DG to be applied as an end-to-end tool to simulate realistic cell geometry under user-specified mechanochemical conditions.
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