Remeshing flexible membranes under the control of free energy

Cell membranes are flexible and often undergo large-scale morphological changes during processes like mitosis, protrusion and retraction, or vesicle fusion. Mathematical modeling of cell membranes depends on a representation of the free-form surface by discrete meshes. During morphological changes, these meshes must be adjusted under the minimization of the total free energy. Current methodology for meshing is limited in one of two ways: 1) Free energy-dependent methods have no restriction on the mesh geometry. The resulting irregular meshes cause artifacts in follow-up models of morphodynamics. 2) Geometry-dependent methods maintain mesh quality but violate the physics of free energy minimization. To fill this gap, we regulate mesh geometries via a free-energy-determined remeshing process: adding and removing mesh elements upon morphological changes based on barrier crossings in a double-barrier potential between neighboring vertices in the meshes. We test the method's robustness by reproducing the morphodynamics of red blood cells and vesicle fusions; and we demonstrate the method's adaptability by simulating the formation of filopodia, lamellipodia and invaginations. Finally, we use the method to study a mechanical decoupling effect of two connected membrane tethers that has been recently observed experimentally, but has not been mechanistically explained in the context of a complete membrane surface. We propose a biophysical model that strengthens the decoupling effect and broadens the original interpretation of the experiment. The method is developed in C/Matlab and distributed via https://github.com/DanuserLab/biophysicsModels. Author summary Many cellular functions require morphological features such as tubulations, protrusions, and invaginations. These features emerge from membranes reshaped by curvature-inducing proteins, osmotic pressure, and other mechanical factors. To elucidate these morphodynamics mathematical models describing the relation between morphology, mechanics, and molecular compoisition are indispensable. The key to such models is discretizing the membrane. One popular discretization relies on meshing membranes into fully connected triangles. When necessary, the mesh configuration is updated via remeshing operations. These include adjusting the number of meshes, rearranging the mesh-to-mesh connectivity, and keeping the individual triangles nearly equilateral. Such flexibility and geometry is necessary accuracy wise. Current remeshing algorithms lack geometrical consistency or rely on user-defined rules that defy physical laws. Here, we propose a new algorithm that marries the needs for geometrical and physical accuracy in a double-barrier potential V-in. The two barriers confine the edges in the triangles to ensure their geometrical consistency. Large enough mechanical perturbations cause barrier crossings that trigger remeshing to restore the perturbed geometry. This physics-based algorithm is robust and more efficient than rule-based algorithms. We implement the algorithm to simulate distinct morphologies and to examine how lipid diffusion controls the mechanical coupling of distant locations on the membrane.

Automated summary

Cell membranes are flexible and often undergo large-scale morphological changes, which can be modeled using mathematical representations with discrete meshes. Current meshing methodologies have limitations in either maintaining mesh geometry or physics of free energy minimization. A new method is proposed here, where mesh geometries are regulated via a free-energy-determined remeshing process to address this gap, with robustness demonstrated through reproducing morphodynamics and adaptability shown by simulating various cellular processes. The method aims to provide a more accurate and physics-based approach for studying membrane dynamics.