Modelling Plant Cortical Mictrotubules: Curvature Sensing from Bending Energy

Within plant cells, the self-organization of cortical microtubules (MTs) into ordered arrays is an important process for directional growth. There is growing evidence that cortical MTs respond to cell shape and/or mechanical stresses in the cell wall, requiring in silico models on complicated surfaces to provide a complete understanding. Most models assume that MTs are directionally persistent, following geodesics of the surface. This ignores the expected tendency of these elastic filaments to minimize curvature. Our recent model incorporated minimization of MT curvature in cylindrical cells and found curvature to be significant in biasing the array organization. Here, we generalize to a larger class of surfaces, studying individual microtubule shapes to provide insights into the role of geometric cues and highlight differences with previous models that use the geodesic assumption. We first show that geodesic models, including current models with finite persistence lengths, exhibit an invariance across certain geometries, leading to biophysically counterintuitive results. Incorporating curvature minimization, we show the difficulties imposed by high-curvature cell edges, elucidating potential new roles of proteins in helping microtubules traverse edges. Lastly, we show that geometries with competing curvature cues result in diverse curves previously not considered. These results provide geometric intuition for how various cell geometries affect individual cortical microtubules, helping us to better understand the processes required for the establishment of microtubule arrays in broad contexts such as: bundles spanning adjacent cell faces in prism-like root and leaf epidermis cells, protruding geometries of trichome cells, and rounded surfaces such as confined protoplasts.