How to curl a microtubule
Microtubules (MTs) are one of the main components of cells, and are essential for many biological functions. As the stiffest cytoskeletal polymer, they are generally seen to be very straight over cellular length-scales. However, in areas of neurodegeneration highly curved MTs are seen with radius of curvature of a micron. Similarly curved MT rings are also sometimes seen in vitro, where MTs are moved over a surface by the motor protein kinesin, amongst other MTs translocating as rigid rods. Recent evidence suggests that some microtubule-associated proteins such as kinesin are able to sense and alter MT curvature, and so we model MTs as inextensible rods with a preferred curvature, which is controlled by the differential binding of the surface-bound kinesin. We find that there exist parameter regimes wherein metastable rings can form, and hence offer this differential binding as an explanation for these highly curved MTs seen both in vitro and in vivo. For certain parameter regimes, this model predicts that both straight and curved MTs 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 MT as it glides across the surface, which can lead to chaotic motion via a period-doubling bifurcation. I will also briefly mention the use of the compound matrix method to calculate the Evans function for solving eigenvalue boundary-value problems, and present a Mathematica package to facilitate the calculation of this.