Mem3DG: Modeling Membrane Mechanochemical Dynamics in 3D using Discrete Differential Geometry

TL;DR

Mem3DG is a flexible, geometry-based computational framework that models 3D membrane shape changes driven by mechanochemical factors, unifying smooth and discrete geometric theories for accurate simulations.

Contribution

We introduce Mem3DG, a novel discrete differential geometry-based framework that resolves geometric ambiguities and unifies smooth and discrete membrane energy and force calculations.

Findings

Successfully modeled classical membrane shapes like biconcave and unduloid forms.

Simulated spherical bud formation on various membrane geometries.

Analyzed effects of membrane mechanics and protein mobility on shape transformations.

Abstract

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.