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Modelling fluid deformable surfaces with an emphasis on biological interfaces

Published online by Cambridge University Press:  10 June 2019

Alejandro Torres-Sánchez
Affiliation:
LaCàN, Universitat Politècnica de Catalunya – BarcelonaTech, 08034 Barcelona, Spain
Daniel Millán
Affiliation:
LaCàN, Universitat Politècnica de Catalunya – BarcelonaTech, 08034 Barcelona, Spain CONICET and Facultad de Ciencias Aplicadas a la Industria, Universidad Nacional de Cuyo, 5600 San Rafael, Argentina
Marino Arroyo*
Affiliation:
LaCàN, Universitat Politècnica de Catalunya – BarcelonaTech, 08034 Barcelona, Spain Institute for Bioengineering of Catalonia, The Barcelona Institute of Science and Technology, 08028 Barcelona, Spain
*
Email address for correspondence: marino.arroyo@upc.edu

Abstract

Fluid deformable surfaces are ubiquitous in cell and tissue biology, including lipid bilayers, the actomyosin cortex or epithelial cell sheets. These interfaces exhibit a complex interplay between elasticity, low Reynolds number interfacial hydrodynamics, chemistry and geometry, and govern important biological processes such as cellular traffic, division, migration or tissue morphogenesis. To address the modelling challenges posed by this class of problems, in which interfacial phenomena tightly interact with the shape and dynamics of the surface, we develop a general continuum mechanics and computational framework for fluid deformable surfaces. The dual solid–fluid nature of fluid deformable surfaces challenges classical Lagrangian or Eulerian descriptions of deforming bodies. Here, we extend the notion of arbitrarily Lagrangian–Eulerian (ALE) formulations, well-established for bulk media, to deforming surfaces. To systematically develop models for fluid deformable surfaces, which consistently treat all couplings between fields and geometry, we follow a nonlinear Onsager formalism according to which the dynamics minimizes a Rayleighian functional where dissipation, power input and energy release rate compete. Finally, we propose new computational methods, which build on Onsager’s formalism and our ALE formulation, to deal with the resulting stiff system of higher-order partial differential equations. We apply our theoretical and computational methodology to classical models for lipid bilayers and the cell cortex. The methods developed here allow us to formulate/simulate these models in their full three-dimensional generality, accounting for finite curvatures and finite shape changes.

Type
JFM Papers
Copyright
© 2019 Cambridge University Press 

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Torres-Sánchez supplementary movie 1

Remeshing during the relaxation dynamics of an inextensible monolayer with bending energy.

Download Torres-Sánchez supplementary movie 1(Video)
Video 6 MB

Torres-Sánchez supplementary movie 2

Relaxation dynamics of a density disturbance of 25% in the outer monolayer of a lipid bilayer modelled with the Seifert-Langer model. The density disturbance drives flows and also shape changes before relaxing to an equilibrium.

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Video 467 KB

Torres-Sánchez supplementary movie 3

Self-polarization of a compressed and non-adherent cell leading to cell migration. An initial density disturbance and a sufficiently large contractile activity lead to cortical flows and shape changes that result in the self-polarization of the cell, with a steady state in which a continuous flow from the front to the rear of the cell is sustained. Friction with the confining plates leads to cell migration.

Download Torres-Sánchez supplementary movie 3(Video)
Video 1 MB
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