Mem3DG

Mem3DG models the mechanochemical dynamics of biomembranes in three dimensions using Discrete Differential Geometry (DDG) on triangulated meshes.

Key Features:

• Discrete Mesh Models: Uses discrete mesh models on triangulated meshes as a coordinate-free geometric representation for 3D membrane mechanics, contrasted with approximation-based methods like finite element analysis.
• Discrete Differential Geometry (DDG) Formalism: Implements DDG on triangulated meshes to formulate energies and forces consistently in the discrete setting.
• Unifying Framework: Establishes a bijective relationship between terms contributing to discrete and smooth geometric theories of energy and forces to resolve ambiguities in geometric definitions.
• Extensible Physics: Extends the discrete framework to incorporate interfacial line tension, surface-bulk adsorption, protein lateral diffusion, and curvature-dependent protein aggregation.
• Modeling Classical Shape Transformations: Simulates classical membrane shapes including biconcave disks, dumbbells, unduloids, and spherical buds on flat-patch membranes and examines mechanochemical-driven phase and shape transformations.
• Mechanics–Protein Coupling: Models coupling between membrane mechanics and protein mobility to investigate curvature-dependent aggregation and lateral diffusion effects.
• Connection to Smooth Theory: Connects discrete membrane mechanics to smooth theory, including comparisons to the Helfrich Hamiltonian and identification of fundamental geometric invariants derived from discrete energy.

Scientific Applications:

• Mechanochemical Simulation of Cell Geometries: Simulates realistic cell membrane geometries under specified mechanochemical conditions.
• Interpretation of 3D Imaging: Bridges high-resolution 3D imaging of membrane ultrastructure with mechanochemical simulations to compare and interpret experimental observations.
• Study of Membrane Bending Processes: Provides quantitative insight into membrane bending relevant to nutrient trafficking and organelle morphology.
• Investigation of Phase and Shape Transformations: Examines how mechanochemical factors drive phase separation and morphological transitions in membranes.

Methodology:

Computations use Discrete Differential Geometry applied to triangulated meshes, derivation of discrete energies and forces with a bijective mapping to smooth geometric terms, simulation of classical membrane shapes, extensions to include interfacial line tension, surface-bulk adsorption, protein lateral diffusion, and curvature-dependent protein aggregation, and comparison of discrete energy invariants to the Helfrich Hamiltonian.