Math Biology Seminar: Eric Cytrynbaum

In growing plant cells, parallel ordering of microtubules (MTs) influences the direction of cell expansion. Models of MT growth in the plane and on polyhedral surfaces have shown that growing-MT encounters lead to the formation of ordered arrays. The polyhedral surfaces models assume that when a MT crosses an edge, it emerges on the adjacent face at the same angle with the edge as the incident angle (i.e. following geodesics). This assumption ignores the MT mechanics - an elastic rod constrained to a rigid surface ought to deflect away from a geodesic when such a deflection decreases its energy. Here, we show this principle for a growing elastic rod on a cylindrical surface with one end clamped. We write down an energy functional that accounts for the bending energy of the rod and derive the associated Euler-Lagrange equation getting a two-variable boundary value problem. Minima and their stability can be found analytically in some cases. The system has a locus of saddle-nodes with a pitchfork in the symmetric case. In general, growing rods deflect away from high curvature directions and toward the flat axial direction, as expected. A rod growing circumferentially continues to grow circumferentially until a critical length (the pitchfork) after which it buckles up or down the cylindrical wall. Our results indicate that, for consistency with observations, the growing tip of MTs ought to be no longer than the radius of curvature of the cell.