[Paper Review] A phase-field model for active contractile surfaces

Sebastian Aland, Claudia Wohlgemuth|arXiv (Cornell University)|Jun 29, 2023

Solidification and crystal growth phenomenaMaterials Science3 citations

TL;DR

This paper presents a novel phase-field model for active contractile surfaces that couples viscous hydrodynamics in the bulk fluid to surface-mediated active contraction via a diffuse-interface approach. The method enables topologically accurate simulations of cell division without remeshing, validated through linear stability and demonstrated in polarized migration and ring formation.

ABSTRACT

The morphogenesis of cells and tissues involves an interplay between chemical signals and active forces on their surrounding surface layers. The complex interaction of hydrodynamics and material flows on such active surfaces leads to pattern formation and shape dynamics which can involve topological transitions, for example during cell division. To better understand such processes requires novel numerical tools. Here, we present a phase-field model for an active deformable surface interacting with the surrounding fluids. The model couples hydrodynamics in the bulk to viscous flow along the diffuse surface, driven by active contraction of a surface species. As a new feature in phase-field modeling, we include the viscosity of a diffuse interface and stabilize the interface profile in the Stokes-Cahn-Hilliard equation by an auxiliary advection velocity, which is constant normal to the interface. The method is numerically validated with previous results based on linear stability analysis. Further, we highlight some distinct features of the new method, like the avoidance of re-meshing and the inclusion of contact mechanics, as we simulate the self-organized polarization and migration of a cell through a narrow channel. Finally, we study the formation of a contractile ring on the surface and illustrate the capability of the method to resolve topological transitions by a first simulation of a full cell division.

Motivation & Objective

• To develop a numerical framework for simulating active deformable surfaces undergoing large shape changes and topological transitions, such as cell division.
• To overcome limitations of grid-based surface methods by introducing a phase-field approach that avoids remeshing and enables complex interface dynamics.
• To couple surface contractility, viscous hydrodynamics in bulk fluids, and surface concentration dynamics in a single diffuse-interface formulation.
• To enable simulation of contact mechanics and confined migration, such as cell movement through narrow channels.
• To validate the model against linear stability analysis and demonstrate its capability in nonlinear, large-deformation regimes.

Proposed method

• A phase-field function φ is introduced to implicitly represent the cell membrane and cortex, with φ = 1 in the intracellular domain and φ = 0 in the extracellular domain, smoothly transitioning across a diffuse interface.
• The model couples the Cahn-Hilliard equation for phase separation with a Stokes-Cahn-Hilliard system that includes surface viscosity and active contraction forces via a contractile species concentration c.
• An auxiliary advection velocity is introduced to stabilize the interface profile in the Stokes-Cahn-Hilliard equation, ensuring robustness and preventing spurious oscillations.
• The surface hydrodynamics are modeled using a projection operator PΓ onto the diffuse interface, enabling consistent surface gradient and divergence operators.
• The momentum equation includes contributions from bulk viscous stresses, surface tension, and active forces from the contractile species, with the active force term proportional to ∇σ(c) and modulated by a Peclét number Pe.
• Axisymmetric geometry is used to reduce computational cost, with careful treatment of rotational symmetry to preserve cell volume and avoid artificial fluid dissolution.

Experimental results

Research questions

• RQ1 Can a phase-field model accurately simulate the formation of a contractile ring on a deformable active surface?
• RQ2 How does the inclusion of diffuse-interface viscosity and auxiliary advection stabilize the interface during large deformations and topological transitions?
• RQ3 Can the model simulate polarized cell migration through confined geometries with contact mechanics?
• RQ4 Does the phase-field approach correctly reproduce linear stability results from sharp-interface models for spherical surfaces?
• RQ5 Can the model resolve full cell division with topological transition from one cell to two daughter cells without remeshing?

Key findings

• The phase-field model successfully reproduces linear stability results from previous sharp-interface models for spherical geometries, validating its accuracy in the linear regime.
• The method enables stable simulation of large deformations and topological transitions, including the first phase-field simulation of a full cell division process with spontaneous ring formation and fission.
• The inclusion of an auxiliary advection velocity stabilizes the diffuse interface profile in the Stokes-Cahn-Hilliard equation, preventing unphysical oscillations and improving numerical robustness.
• The model captures self-organized polarization and migration of a cell through a narrow channel, including contact with walls, demonstrating its capability in confined environments.
• The diffuse-interface formulation naturally handles complex interface dynamics without remeshing, enabling simulations of evolving surfaces with changing topology.
• The model preserves cell volume accurately by avoiding rotational symmetry in the chemical potential operator Δμ, preventing artificial fluid influx or dissolution.