Modeling Membrane Dynamics in 3D using Discrete Differential Geometry

Abstract

Biomembranes adopt varying morphological configurations that are vital to cellular functions. Many studies use computational modeling to understand how various mechanochemical factors contribute to membrane shape transformations. Compared to traditional 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 expressive coordinate-free geometric descriptions. However, ambiguities in geometric definitions in the discrete context have led to a lack of consensus on which discrete mesh based 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, 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. Mem3DG is designed to facilitate models in tandem with and mimicking experimental studies. It also supports the use of realistic membrane ultrastructure from 3D imaging as an input. To demonstrate, 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. While the first two examples serve as validation, the later examples provide a blueprint for extending Mem3DG to model a system of interest. 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. We hope that Mem3DG will be a useful tool to help advance the mission of computational membrane mechanobiology.