  Many processes in eukaryotic cells, including cell motility, rely on the
growth of branched actin networks from surfaces. Despite its central role the
mechano-chemical coupling mechanisms which guide the growth process are poorly
understood, and a general continuum description combining growth and mechanics
is lacking. We develop a theory that bridges the gap between mesoscale and
continuum limit and propose a general framework providing the evolution law of
actin networks growing under stress. This formulation opens an area for the
systematic study of actin dynamics in arbitrary geometries. Our framework
predicts a morphological instability of actin growth on a rigid sphere, leading
to a spontaneous polarization of the network with a mode selection
corresponding to a comet, as reported experimentally. We show that the
mechanics of the contact between the network and the surface plays a crucial
role, in that it determines directly the existence of the instability. We
extract scaling laws relating growth dynamics and network properties offering
basic perspectives for new experiments on growing actin networks.
