Recent experiments have shown that a system of microtubules and molecular
motors is capable of sustaining a variety of large-scale two-dimensional
structures (asters, vortices, and other topological defects). Underlying
this phenomenon is a multiscale process, where nonlinear interactions on a
microscopic scale result in the emergence of coherent structures on the
macroscopic scale. For the study of this process, we propose a mathematical
framework based on a master equation on the mesoscopic scale. The equation
describes the evolution of the density of microtubules as a function of
position and angular orientation. The interaction kernel is determined by
the molecular interactions on the microscopic scale. Macroscopic
spatiotemporal structures are identified through dimension reduction from
the mesoscopic scale. All three scales interact strongly, resulting in
macroscopically observable patterns of self-organization.

Recently we have developed mathematical analysis and simulations to address
different aspects of the problem. At the mesoscopic level we have developed
an approach at analytically solving the master equation in the homogeneous
2D case.

At the microscopic case we have investigated the effects of flexibility of
filaments on their collision properties that control the kernel of the
master equation at the mesoscopic level. We have shown that flexibility
enhances the inelasticity of interaction and that motor dwelling may be
responsible for complete alignment of tubules. Furthermore, we have
investigated interaction of much softer filaments, such as actin, which is
responsible for the cytoskeleton formation. Interacting actin pairs
exhibited an Euler-type buckling instability for sufficient motor stengths.
It is conjectured that the buckled states are responsible for cross-linking
of actin into cytoskeletal networks and, thereby, for the rheological
properties of these networks.