  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.
