Cell motility, a crucial element in life processes, has predominantly been described through numerical models of complex cytoskeleton dynamics. In this talk, I shall present an analytical moving-boundary model of a cell under external confinement, which has been highlighted as an important factor influencing the speed and shape of motile cells. External confinement, together with the presence of an internal passive fluid, impose physical constraints that greatly simplify the system hydrodynamics. This model reveals that a basic coupling between the passive viscous flow and an active cytoplasmic chemical is sufficient to obtain a large variety of deformation and motility patterns that closely resemble experimental observations.

More precisely here I will introduce a minimal hydrodynamic model of polarization, migration, and deformation of a biological cell confined between two parallel surfaces. In this depiction, the cell cytoplasm is driven out of equilibrium by the actin-myosin cytoskeleton. The cytoplasm is modeled as a passive viscous droplet in the Hele-Shaw flow regime. It contains a diffusive solute which actively transduces the applied cytoskeleton force.

While fairly simple and analytically tractable, this quasi-2D model predicts a range of compelling dynamic behaviours. A linear stability analysis of the system reveals that solute activity first destabilizes a global polarization-translation mode, prompting cell motility through spontaneous-symmetry-breaking. At higher activity, the system crosses a series of Hopf bifurcations leading to coupled oscillations of droplet shape and solute concentration profiles. At the nonlinear level, we find traveling-wave solutions associated with unique polarized shapes resembling experimental observations. Altogether, this model offers an analytical paradigm of active deformable systems in which Stokes hydrodynamics are coupled to diffusive force-transducers.