Curvature-Sensitive Kinesin Binding Can Explain Microtubule Ring Formation and Reveals Chaotic Dynamics in a Mathematical Model

Simon P. Pearce, Matthias Heil, Oliver E. Jensen, Gareth Wyn Jones, Andreas Prokop

Microtubules are filamentous tubular protein polymers which are essential for a range of cellular behaviour, and are generally straight over micron length scales. However, in some gliding assays, where microtubules move over a carpet of molecular motors, individual microtubules can also form tight arcs or rings, even in the absence of crosslinking proteins. Understanding this phenomenon may provide important explanations for similar highly curved microtubules which can be found in nerve cells undergoing neurodegeneration. We propose a model for gliding assays where the kinesins moving the microtubules over the surface induce ring formation through differential binding, substantiated by recent findings that a mutant version of the motor protein kinesin applied in solution is able to lock-in microtubule curvature. For certain parameter regimes, our model predicts that both straight and curved microtubules can exist simultaneously as stable steady states, as has been seen experimentally. Additionally, unsteady solutions are found, where a wave of differential binding propagates down the microtubule as it glides across the surface, which can lead to chaotic motion. Whilst this model explains two-dimensional microtubule behaviour in an experimental gliding assay, it has the potential to be adapted to explain pathological curling in nerve cells.