Pattern formation in auxin flux

TL;DR

This paper analyzes the stability of auxin flux patterns in plant tissues, revealing that critical states are composed of isolated building blocks and establishing convergence properties of the transport process.

Contribution

It provides a detailed stability analysis of a leading model of polar auxin flux, introducing Lyapunov functions to prove convergence to critical configurations.

Findings

Critical auxin flux states are composed of isolated building blocks.

Lyapunov functions are established, proving convergence of transport processes.

The model's stability analysis offers insights into plant morphogenesis mechanisms.

Abstract

The plant hormone auxin is fundamental for plant growth, and its spatial distribution in plant tissues is critical for plant morphogenesis. We consider a leading model of the polar auxin flux, and study in full detail the stability of the possible equilibrium configurations. We show that the critical states of the auxin transport process are composed of basic building blocks, which are isolated in a background of auxin depleted cells, and are not geometrically regular in general. The same model was considered recently through a continuous limit and a coupling to the von Karman equations, to model the interplay of biochemistry and mechanics during plant growth. Our conclusions might be of interest in this setting, since, for example, we establish the existence of Lyapunov functions for the auxin flux, proving in this way the convergence of pure transport processes toward the set of critical configurations.