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The self-organization of ordered cortical microtubule arrays plays an important role in the development of plant cells. This is observed to emerge from a combination various factors such as microtubule-microtubule interactions, nucleation, and localization of microtubule-associated proteins. Distilling this process into the interaction of one-dimensional bodies on the two-dimensional cortex, quantitative models have been proposed to emulate array formation. These have assisted in understanding the importance of each aspect in addition to identifying potential avenues of experimentation. Until recently, the direct mechanical influence of cell geometry on the constrained microtubule trajectories have been largely ignored in computational models. Modelling microtubules as thin elastic rods constrained on a surface, it has been found that microtubules shapes may differ significantly from the previously assumed geodesics. Restricting to cylinders, we formulate a model incorporating the mechanics of curvature-induced deflection and examine its implications. First, we introduce a minimal anchoring process necessary to describe the geometry of individual microtubules as they grow and become fixed to the cortex. Curvature-induced deflection is governed by the extent to which microtubules can explore the surface curvature, as prescribed by the segment lengths between anchoring. This, in turn, can be modulated by the anchoring kinetics proposed. We implement this model as an event-driven simulation in Python, accommodating for the curvilinear shapes prescribed by the anchoring process. In doing so, ambiguities between differing models are identified. Although these are not resolved here, these provide a possible explanation to the reported conflicting results among differing models. Based on some preliminary simulations using our model, we find that the curvature mechanics provides strong influence for longitudinal alignment with realistic anchoring rates. In particular, catastrophe-inducing edges are unable to overcome this influence to reproduce the observed transverse arrays during the cell elongation phase. This simulation provides the opportunity for further exploration into mechanical influences on array formation and their regulation by the anchoring process. |
