Self-organisation in mixtures of microtubules and motor proteins
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Self-organisation of mixtures of biological polymers and molecular motors provides a fascinating manifestation of active matter. Microtubules re-oriented by the molecular motors can form far-from-equilibrium cell-scale structures, such as the mitotic spindle apparatus. It is believed that different motor types favour formation of distinct patterns: clustering motors control the formation of spindle poles and asters, while microtubule-sliding motors organise antiparallel bundles presenting in the spindle central part. The link between individual microscopic motor-induced interactions of filaments and the macroscopic dynamics at cell-size scales is poorly understood. Here we enhance our understanding of this problem and 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. In the first part of the thesis, we derive hydrodynamic equations that govern the collective behaviour of microtubules in the presence of clustering motors. We build on a kinetic method developed earlier by Aranson and Tsimring and model the motor-induced reorientation of microtubules as collision rules. The procedure of coarse-graining yields a set of equations for local density and orientation of the microtubules. We study its behaviour by performing a linear stability analysis and direct numerical simulations. We discuss the observed patterns including asters and chaotic stripe-like structures and consider the ensuing phase diagram. In the second part of this study, we consider molecular motors which can push apart antiparallel microtubules and cluster parallel ones. Using the developed approach, we obtain a set of equations for the microtubular density, orientation, and tensor of alignment. Through numerical simulations, we show that this model generically creates either stable stripes with the antiparallel arrangement of filaments inside them or an ever-evolving pattern where stripes periodically form, rotate, self-extend and then split up. We derive a minimal model which displays the same instability as the full model and clarifies the underlying mechanism. We argue that our minimal model unifies various previous observations of chaotic behaviour in the dry active matter into a general universality class. Finally, we discuss obtained models, compare them to identify common features, and offer the directions of future advances.