  In this study we formulate a theoretical approach, based on a Boltzmann-like
kinetic equation, to describe pattern formation in two-dimensional mixtures of
microtubular filaments and molecular motors. Following the previous work by
Aranson and Tsimring [Phys. Rev. E {\bf 74}, 031915 (2006)] we model the
motor-induced reorientation of microtubules as collision rules, and devise a
semi-analytical method to calculate the corresponding interaction integrals.
This procedure yields an infinite hierarchy of kinetic equations that we
terminate by employing a well-established closure strategy, developed in the
pattern-formation community and based on a power-counting argument. We thus
arrive at a closed set of coupled equations for slowly varying local density
and orientation of the microtubules, and study its behaviour by performing a
linear stability analysis and direct numerical simulations. By comparing our
method with the work of Aranson and Tsimring, we assess the validity of the
assumptions required to derive their and our theories. We demonstrate that our
approximation-free evaluation of the interaction integrals and our choice of a
systematic closure strategy result in a rather different dynamical behaviour
than was previously reported. Based on our theory, we discuss the ensuing phase
diagram and the patterns observed.
