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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 Reynoldsnumber interfacial hydrodynamics, chemistry, and geometry, and govern importantbiological processes such as cellular traffic, division, migration, or tissuemorphogenesis. To address the modelling challenges posed by this class ofproblems, in which interfacial phenomena tightly interact with the shape anddynamics of the surface, we develop a general continuum mechanics andcomputational framework for fluid deformable surfaces. The dual solid-fluidnature of fluid deformable surfaces challenges classical Lagrangian or Euleriandescriptions of deforming bodies. Here, we extend the notion of ArbitrarilyLagrangian-Eulerian (ALE) formulations, well-established for bulk media, todeforming surfaces. To systematically develop models for fluid deformablesurfaces, which consistently treat all couplings between fields and geometry,we follow a nonlinear Onsager formalism according to which the dynamicsminimize a Rayleighian functional where dissipation, power input and energyrelease rate compete. Finally, we propose new computational methods, whichbuild on Onsager's formalism and our ALE formulation, to deal with theresulting stiff system of higher-order of partial differential equations. Weapply our theoretical and computational methodology to classical models forlipid bilayers and the cell cortex. The methods developed here allow us toformulate/simulate these models for the first time in their fullthree-dimensional generality, accounting for finite curvatures and finite shapechanges.

Modelling fluid deformable surfaces with an emphasis on biological interfaces