# Conformal growth of Arabidopsis leaves

## TL;DR

Arabidopsis leaf growth can be accurately modeled using a conformal map, which maintains leaf flatness through isotropic expansion and a simple linear gradient of growth rates.

## Contribution

A new model for leaf growth using conformal maps that achieves high accuracy and explains flatness maintenance.

## Key findings

- Tracking data can be reproduced with 90% accuracy using a conformal map.
- Adding nonlinear terms to a linear map improves accuracy to 89%.
- A linear gradient of growth rates could be generated by a diffusible signal.

## Abstract

I show that Arabidopsis leaf growth can be described with high accuracy by a conformal map, where expansion is locally isotropic (the same in all directions) but the amount of expansion can vary with position. Data obtained by tracking leaf growth over time can be reproduced with almost 90% accuracy by such a map. At 7-8 days, 67% of leaf growth is accounted for by uniform expansion of the whole leaf, but the best fitting conformal map, having 89% accuracy, is achieved by adding nonlinear terms to this linear map. Growth according to a conformal map has the property of maintaining the flatness of a leaf. To understand how such a map is generated, one can mathematically subtract the effects of tissue expansion and calculate the underlying distribution of growth rate. This turns out to be a simple linear gradient, which could be generated by a diffusible signal.