Scaling Relations for Auxin Waves

TL;DR

This paper investigates a mathematical model for auxin wave formation in plants, deriving scaling laws for wave speed and width, and providing explicit wave profiles to understand biological parameter effects.

Contribution

It introduces a family of traveling wave solutions for auxin transport, with explicit profiles and scaling relations, extending existing mathematical methods to biological wave phenomena.

Findings

Derived scaling relations for auxin wave speed and width.

Verified theoretical results with numerical simulations.

Provided explicit wave profiles linking biological parameters to wave characteristics.

Abstract

We analyze an 'up-the-gradient' model for the formation of transport channels of the phytohormone auxin, through auxin-mediated polarization of the PIN1 auxin transporter. We show that this model admits a family of travelling wave solutions that is parameterized by the height of the auxin-pulse. We uncover scaling relations for the speed and width of these waves and verify these rigorous results with numerical computations. In addition, we provide explicit expressions for the leading-order wave profiles, which allows the influence of the biological parameters in the problem to be readily identified. Our proofs are based on a generalization of the scaling principle developed by Friesecke and Pego to construct pulse solutions to the classic Fermi-Pasta-Ulam-Tsingou model, which describes a one-dimensional chain of coupled nonlinear springs.