Kinetic theory of pattern formation in mixtures of microtubules and molecular motors

TL;DR

This paper develops a kinetic theory based on a Boltzmann-like equation to analyze pattern formation in mixtures of microtubules and molecular motors, providing a more accurate and systematic approach than previous models.

Contribution

It introduces a new semi-analytical method for calculating interaction integrals and employs a systematic closure strategy, leading to different dynamical predictions compared to earlier work.

Findings

Different dynamical behaviors observed with the new method.

Systematic closure strategy improves model accuracy.

Phase diagram and pattern formation analyzed.

Abstract

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.