Modelling plant cortical microtubules : elastic curves on smooth surfaces

The self-organization of microtubule polymers along the inner surface (cortex) of the plant cell membrane is an essential element in facilitating directional cell growth. The key questions are: what gives rise to the ordering and orientation of MT patterns? Mathematical and computational modelling has proven successful in providing insights: the process is distilled into a system of interacting curves on a two-dimensional surface. There has been interest in the role of cell geometry in this process. My work revisits a common assumption, that microtubule shapes are described by geodesics. More realistically, microtubules are relatively rigid filaments and should seek to minimize bending, resulting in elastic curves. My model of elastic curves on cylindrical cells shows that the curvature influence on microtubules should orient microtubules in directions opposite to what is biologically favourable. I investigate and model potential biological processes involved with overcoming this curvature influence. Beyond the cylinder, solving for elastic curves on general surfaces has remained mathematically challenging. Utilizing methods in differential geometry and numerical optimization, I generalize this model to other geometries: I solve for elastic curves on various surfaces to predict bifurcations and diverse curves resulting from non-local curvature sensing. This opens the field to realistic models on complex geometries.