A numerical approach for the dynamics of active viscoelastic surfaces

The dynamics of active viscoelastic surfaces play an important role in biological systems. One prominent example is the actin cortex, a thin bio-polymer sheet underneath the outer membrane of biological cells which combines active molecular force generation with viscoelastic behavior characterized by elastic properties at short timescales and viscous properties at longer timescales. We consider a surface Maxwell model within dominant rheology and an additional active term to model the dynamics of the actin cortex. This captures both shear and dilational surface dynamics. We propose a monolithic numerical approach based on the surface finite-element method (SFEM), validate the results for special cases, and experimentally demonstrate convergence properties. Moreover, imposing a ring-shaped region of an enhanced active stress mimicking the contractile ring during cytokinesis, we observe different types of emergent patterns and shape dynamics depending on the viscoelastic properties. While viscous surfaces show a ring, which slips to one side of the surface, viscoelasticity provides a stabilization mechanism of the ring, thus, forming a requirement for subsequent cell division. This study provides an example that viscoelastic properties are key ingredients to understand biological materials.