Cytrynbaum, E. | Search bioRxiv

bioRxiv2026

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.

Developmental Biology↗

bioRxiv2019

Mechanical constraints influence microtubule orientation in plant cells

In growing plant cells, parallel ordering of microtubules (MTs) along the inner surface of the cell membrane influences the direction of cell expansion and thereby plant morphology. For correct expansion of organs that primarily grow by elongating, such as roots and stems, MTs must bend in the high-curvature direction along the cylindrically shaped cell membrane in order to form the required circumferential arrays. Previous studies, which have recapitulated the self-organization of these arrays, ignored MT mechanics and assumed MTs follow geodesics of the cell surface. Here, we show, through analysis of a derived Euler-Lagrange equation, that an elastic MT constrained to a cylindrical surface will deflect away from geodesics and toward low curvature directions to minimize bending energy. This occurs when the curvature of the cell surface is relatively high for a given anchor density. In the limit of infinite anchor density, MTs always follow geodesics. We compare our analytical predictions to measured curvatures and anchor densities and find that the regime in which cells are forming these cortical arrays straddles the region of parameter space in which arrays must form under the antagonistic influence of this mechanically induced deflection. Although this introduces a potential obstacle to forming circumferentially orientated arrays that needs to be accounted for in the models, it also raises the question of whether plants use this mechanical phenomenon to regulate array orientation. The model also constitutes an elegant generalization of the classical Euler-bucking instability along with an intrinsic unfolding of the associated pitchfork bifurcation.\n\nSIGNIFICANCE STATEMENTThe cortical microtubule array in plant cells is a dynamic structure whose organization influences cell function, growth, and plant morphology. During cell growth, the array must be highly ordered and perpendicular to the axis of expansion. In other instances, it can be disordered or axially oriented. Experimental and modelling studies have outlined the importance of various microtubule-microtubule interactions in the ordering and orienting process. We explore the influence of cell shape on the bending of microtubules that has been largely ignored to date. We find that cell shape can induce microtubule bending that steers them away from high-curvature directions. This bending force can act as an obstacle to proposed mechanisms for array organization and also as a mechanism for regulating reorientation of the array.

cell biology↗