Modelling cell motility and chemotaxis with evolving surface finite elements

Abstract

We present a mathematical and a computational framework for the modelling of cell motility. The cell membrane is represented by an evolving surface, with the movement of the cell determined by the interaction of various forces that act normal to the surface. We consider external forces such as those that may arise owing to inhomogeneities in the medium and a pressure that constrains the enclosed volume, as well as internal forces that arise from the reaction of the cells' surface to stretching and bending. We also consider a protrusive force associated with a reaction -diffusion system (RDS) posed on the cell membrane, with cell polarization modelled by this surface RDS. The computational method is based on an evolving surface finite-element method. The general method can account for the large deformations that arise in cell motility and allows the simulation of cell migration in three dimensions. We illustrate applications of the proposed modelling framework and numerical method by reporting on numerical simulations of a model for eukaryotic chemotaxis and a model for the persistent movement of keratocytes in two and three space dimensions. Movies of the simulated cells can be obtained from http://homepages.warwick.ac.uk/ maskae/CV_Warwick/Chemotaxis.html.

Numerical Preservation of Velocity Induced Invariant Regions for Reaction–Diffusion Systems on Evolving Surfaces

“…In many applications the spatial domain is a curved surface rather than a flat domain, which may be time-dependent. Among the applications of RDSs on surfaces we mention brain growth [26], cell migration [3], chemotaxis [12], developmental biology [28], electrodeposition [24] and phase field modeling [42]. The growing interest toward PDEs on evolving surfaces has stimulated the development of several numerical methods for such problems, among which we mention (but not limited to) embedding methods [2], kernel methods [18], implicit boundary integral methods [5,35], surface finite element methods (SFEM) [10] and some of their recent variations and extensions [13,16,17,20,23,40].…”

Analysis and Simulations of Coupled Bulk-surface Reaction-Diffusion Systems on Exponentially Evolving Volumes

“…The coupling of bulk and surface dynamics through the use of partial differential equations (PDEs) in multi-dimensions is prevalent in many cellular biological systems and fluid dynamics [3,5,8,15,28,29,32,33,35,36] as well as in other exotic areas of solid mechanics such as topological insulator thin films [9] and in large-and small-scale atmospheric and coupled ocean-atmosphere models for air-sea interactions [4]. In the areas of cellular and developmental biology, in many of these applications and processes, the formation of heterogeneous distributions of chemical substances emerge through symmetry breaking of morphological instabilities [17].…”

A Physiologically Required G Protein-coupled Receptor (GPCR)-Regulator of G Protein Signaling (RGS) Interaction That Compartmentalizes RGS Activity

“…Most GPCRs share a common signaling mechanism of coupling to and activating G proteins, which propagate the signaling reaction (1,2). GPCRs couple to a heterotrimeric G protein complex consisting of a nucleotide binding G␣ subunit and a G␤␥ dimer.…”

“…Some cellular mechanisms contributing to RGS membrane targeting have previously been identified. Plasma membrane recruitment of RGS proteins can occur as a result of G protein activation (11,12), enhanced expression of specific unactivated G␣ subunits or GPCRs (1,2,13), intrinsic transmembrane-spanning regions (3,4,14), post-translational lipid modifications (5,15), or electrostatic interactions with membrane lipids (6,16) or via scaffolding proteins (6,7). Specific domains, namely the disheveled, Egl-10, and pleckstrin (DEP) domains within some RGS proteins, can interact directly with internal loop regions (8,17) and the intracellular C-terminal tail of GPCRs to promote selectivity of RGS activity at the plasma membrane (9,18).…”