Bifurcation Analysis of Active Filament Models

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. 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.