# A buckling instability and its influence on microtubule orientation in plant cells

## TL;DR

This paper investigates how buckling instability influences microtubule orientation in plant cells, revealing that mechanical forces can cause microtubules to deviate from geodesic paths, affecting cell growth patterns.

## Contribution

The study introduces an analytical model showing how elastic microtubules on cylindrical surfaces deflect due to curvature, impacting array formation and plant cell morphology.

## Key findings

- Microtubules tend to deflect away from geodesics at high curvature.
- Array formation is influenced by the balance between curvature and anchor density.
- The buckling model generalizes classical Euler instability to biological structures.

## Abstract

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. Computational 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.