  The cell cortex is a thin layer beneath the plasma membrane that gives animal
cells mechanical resistance and drives most of their shape changes, from
migration, division to multicellular morphogenesis. It is mainly composed of
actin filaments, actin binding proteins, and myosin molecular motors.
Constantly stirred by myosin motors and under fast renewal, this material may
be well described by viscous and contractile active-gel theories. Here, we
assume that the cortex is a thin viscous shell with non-negligible curvature
and use asymptotic expansions to find the leading-order equations describing
its shape dynamics, starting from constitutive equations for an incompressible
viscous active gel. Reducing the three-dimensional equations leads to a
Koiter-like shell theory, where both resistance to stretching and bending rates
are present. Constitutive equations are completed by a kinematical equation
describing the evolution of the cortex thickness with turnover. We show that
tension and moment resultants depend not only on the shell deformation rate and
motor activity but also on the active turnover of the material, which may also
exert either contractile or extensile stress. Using the finite-element method,
we implement our theory numerically to study two biological examples of drastic
cell shape changes: osmotic shocks and cell division. Our work provides a
numerical implementation of thin active viscous layers and a generic
theoretical framework to develop shell theories for slender active biological
structures.
