A coupled bulk-surface model for cell polarisation

• Published in: Journal of theoretical biology Vol. 481; pp. 119 - 135
• Main Authors: Cusseddu, D., Edelstein-Keshet, L., Mackenzie, J.A., Portet, S., Madzvamuse, A.
• Format: Journal Article
• Language: English
• Published: England Elsevier Ltd 21.11.2019
• Subjects: Asymptotic and local perturbation theory; Bulk-surface finite elements; Bulk-surface wave pinning model; Cell polarisation; Coupled bulk-surface semilinear partial differential equations; Reaction-diffusion systems; Coupled bulk-surface semilinear partial differential equations; Bulk-surface wave pinning model; Reaction-diffusion systems; Cell polarisation; Bulk-surface finite elements; Asymptotic and local perturbation theory
• ISSN: 0022-5193; 1095-8541
• DOI: 10.1016/j.jtbi.2018.09.008

•We present a three-dimensional bulk-surface model for cellular polarisation based on a conceptual single GTPase protein cycle.•Membrane-bound active and cytosolic inactive GTPase interact through boundary conditions at the surface of the domain.•Positive feedback causes propagation of GTPase activity which gets finally pinned by the depletion of inactive GTPase.•Parameter regions for polarisation are described for domains with different volume/surface ratio.•Three dimensional simulations are presented on simple and complex geometries. Several cellular activities, such as directed cell migration, are coordinated by an intricate network of biochemical reactions which lead to a polarised state of the cell, in which cellular symmetry is broken, causing the cell to have a well defined front and back. Recent work on balancing biological complexity with mathematical tractability resulted in the proposal and formulation of a famous minimal model for cell polarisation, known as the wave pinning model. In this study, we present a three-dimensional generalisation of this mathematical framework through the maturing theory of coupled bulk-surface semilinear partial differential equations in which protein compartmentalisation becomes natural. We show how a local perturbation over the surface can trigger propagating reactions, eventually stopped in a stable profile by the interplay with the bulk component. We describe the behavior of the model through asymptotic and local perturbation analysis, in which the role of the geometry is investigated. The bulk-surface finite element method is used to generate numerical simulations over simple and complex geometries, which confirm our analysis, showing pattern formation due to propagation and pinning dynamics. The generality of our mathematical and computational framework allows to study more complex biochemical reactions and biomechanical properties associated with cell polarisation in multi-dimensions.