Abstract

Slender filaments such as microtubules are ubiquitous in nature, driving fluid flow at the microscopic scale. Molecular motors like dynein and kinesin translocate along microtubules, causing a range of both steady and time-dependent behaviours. The coordinated motion of microtubules can lead to phenomena like cytoplasmic streaming and ciliary beating, generating fluid flows on larger scales. In this talk we provide a comprehensive overview of the emerging dynamics of the most fundamental model that captures the effect of molecular motors on a single filament; the follower force model, whereby a compressive force is imposed at the filament tip. We vary both the strength of this force and the slenderness of the filament to explore the resulting state space, using a filament model built on Kirchoff’s rod theory and which employs unit quaternions and implicit time integration to handle the development of the filament’s local frame over time [1]. Employing a Jacobian-Free Newton-Krylov method, we establish both steady and time-periodic solutions to the model, as well as new, quasi-periodic solutions. We classify and fully characterize the bifurcations yielding different states and analyse their stability. In doing so, we provide a clear picture of the full bifurcation diagram for the fundamental model of microtubule-motor protein complexes.

References:

[1] S. F. Schoeller, A. K. Townsend, T. A. Westwood and E. E. Keaveny, Methods for suspensions of passive and active filaments, Journal of Computational Physics, 424 (2021)