Self-organisation in mixtures of microtubules and motor proteins
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Date
06/07/2019Author
Maryshev, Ivan
Metadata
Abstract
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.