  Active force generation by actin-myosin cortex coupled to the cell membrane
allows the cell to deform, respond to the environment, and mediate cell
motility and division. Several membrane-bound activator proteins move along it
and couple to the membrane curvature. Besides, they can act as nucleating sites
for the growth of filamentous actin. Actin polymerization can generate a local
outward push on the membrane. Inward pull from the contractile actomyosin
cortex can propagate along the membrane via actin filaments. We use coupled
evolution of fields to perform linear stability analysis and numerical
calculations. As activity overcomes the stabilizing factors such as surface
tension and bending rigidity, the spherical membrane shows instability towards
pattern formation, localized pulsation, and running pulsation between poles. We
present our results in terms of phase diagrams and evolutions of the coupled
fields. They have relevance for living cells and can be verified in experiments
on artificial cell-like constructs.
