Asters, Spirals and Vortices in Mixtures of Motors and Microtubules: The Effects of Confining Geometries on Pattern Formation

We model the effects of confinement on the stable self-organized patterns obtained in the non-equilibrium steady states of mixtures of molecular motors and microtubules. In experiments [Nedelec et al. Nature, 389, 305 (1997); Surrey et al., Science, 292, 1167 (2001)] performed in a quasi-two-dimensional confined geometry, microtubules are oriented by complexes of motor proteins. This interaction yields a variety of patterns, including arangements of asters, vortices and disordered configurations. We model this system via a two-dimensional vector field describing the local coarse-grained microtubule orientation and two scalar density fields associated to molecular motors. These scalar fields describe motors which either attach to and move along microtubules or diffuse freely within the solvent. Transitions between single aster, spiral and vortex states are obtained as a consequence of confinement, as parameters in our model are varied. We also obtain other novel states, including ``outward asters'', distorted vortices and lattices of asters on confined systems. We calculate the steady state distribution of bound and free motors in aster and vortex configurations of microtubules and compare these to our simulation results, providing qualitative arguments for the stability of different patterns in various regimes of parameter space. We also study the role of crowding or ``saturation'' effects on the density profiles of motors in asters, discussing the role of such effects in stabilizing single asters.