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

Debabrata Auddya; Xiaoxuan Zhang; Rahul Gulati; Ritvik Vasan; Krishna Garikipati; Padmini Rangamani; Shiva Rudraraju

doi:10.1098/rspa.2021.0246

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

Proc Math Phys Eng Sci. 2021 Nov;477(2255):20210246. doi: 10.1098/rspa.2021.0246. Epub 2021 Nov 10.

Authors

Debabrata Auddya¹, Xiaoxuan Zhang², Rahul Gulati¹, Ritvik Vasan³, Krishna Garikipati^{2

4

5}, Padmini Rangamani³, Shiva Rudraraju¹

Affiliations

¹ Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, WI 53706, USA.
² Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA.
³ Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093, USA.
⁴ Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA.
⁵ Michigan Institute for Computational Discovery and Engineering, University of Michigan, Ann Arbor, MI 48109, USA.

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.

Keywords: FEM; Kirchhoff-Love; biomembranes; endocytosis; isogeometric analysis; non-axisymmetric.

Associated data

figshare/10.6084/m9.figshare.c.5672457