Biomembranes undergo complex, non-axisymmetric deformations governed by Kirchhoff–Love kinematicsand revealed by a three-dimensional computational framework

Abstract

Biomembranes play a central role in various phenomena like locomotion of cells, cell-cell interactions, packaging and transport of nutrients, transmission of nerve impulses, and in maintaining organelle morphology and functionality. During these processes, the membranes undergo significant morphological changes through deformation, scission, and fusion. Modelling the underlying mechanics of such morphological changes has traditionally relied on reduced order axisymmetric representations of membrane geometry and deformation. Axisymmetric representations, while robust and extensively deployed, suffer from their inability to model-symmetry breaking deformations and structural bifurcations. To address this limitation, a three-dimensional computational mechanics framework for high fidelity modelling of biomembrane deformation is presented. The proposed framework brings together Kirchhoff–Love thin-shell kinematics, Helfrich-energy-based mechanics, and state-of-the-art numerical techniques for modelling deformation of surface geometries. Lipid bilayers are represented as spline-based surface discretizations immersed in a three-dimensional space; this enables modelling of a wide spectrum of membrane geometries, boundary conditions, and deformations that are physically admissible in a three-dimensional space. The mathematical basis of the framework and its numerical machinery are presented, and their utility is demonstrated by modelling three classical, yet non-trivial, membrane deformation problems: formation of tubular shapes and their lateral constriction, Piezo1-induced membrane footprint generation and gating response, and the budding of membranes by protein coats during endocytosis. For each problem, the full three-dimensional membrane deformation is captured, potential symmetry-breaking deformation paths identified, and various case studies of boundary and load conditions are presented. Using the endocytic vesicle budding as a case study, we also present a ‘phase diagram’ for its symmetric and broken-symmetry states.

Mem3DG: Modeling membrane mechanochemical dynamics in 3D using discrete differential geometry

“…To ensure that such an approximation can remain valid throughout a trajectory, we have incorporated algorithmic solutions to adaptively maintain an isotropically well-resolved discrete geometry. This is achieved by two operations: 1) mesh regularization using local force constraints, which are common in FEM ( 78 , 82 , 83 , 85 ) ( Appendix E.2 ); and 2) mesh mutations such as decimating, flipping, and collapsing edges. Beyond regularization, these local force constraints can also be used to model underlying physics of a problem of interest.…”

“…Solvers of membrane shape in 3D have also been developed and can be categorized into three groups: 1) phase field or level set methods ( 73 – 77 ), 2) finite element method (FEM) ( 78 – 85 ), and 3) discrete surface mesh models ( 60 , 86 – 99 ). These methods and others, reviewed in detail by Guckenberger et al ( 100 ), differ in the strategy used to discretize the membrane domain and compute the relevant derivatives.…”

“…First, the numerical evaluation of smooth geometric measurements on arbitrary manifolds in an FEM framework requires non-intuitive tensor algebra to translate the shape equation in coordinate where it is ready to be solved. After this formulation, solving the shape equation can require the use of high-order function basis such as the C 1 conforming FEM based on subdivision scheme ( 78 , 79 ) or isogeometric analysis (IGA) ( 81 – 83 , 85 ), which adds code complexity and run-time cost. Extending an FEM framework to incorporate new physics, topological changes, or interfaces with other models requires advanced mathematical and coding skills.…”

Modelling membrane curvature generation using mechanics and machine learning

“…We found this solver to be amenable to our problem since it is relatively robust and fast, allowing us to rapidly generate a large dataset. However, other methods may yield convergent solutions where we could not find them using this solver (for instance, see [55][56][57]) The solver performed better when the initial mesh (between 0 and 1) was squared so that the density of points near the centre (which corresponds to the centre of the patch) was larger. The mesh was then resized to the total patch area.…”

Mem3DG: modeling membrane mechanochemical dynamics in 3D using discrete differential geometry

“…While these methods are suitable for idealized shapes, these assumptions are not consistent with the membrane shapes found in biology are and thus not general enough for advancing the field. Solvers of membrane shape in 3D have also been developed and can be categorized into three groups: 1) phase field or level set methods (73)(74)(75)(76)(77), 2) finite element method (FEM) (78)(79)(80)(81)(82)(83)(84)(85), and 3) discrete surface mesh models (60,(86)(87)(88)(89)(90)(91)(92)(93)(94)(95)(96)(97)(98)(99). These methods and others, reviewed in detail by Guckenberger et al (100), differ in the strategy used to discretize the membrane domain and compute the relevant derivatives.…”