Nonlinear self-adapting wave patterns

TL;DR

This paper introduces a novel traveling wave pattern that can adapt to the size of the physical system, especially in systems with zero wavevector instabilities, exemplified by the Min protein system in E. coli.

Contribution

It presents a new class of nonlinear wave patterns capable of self-adapting to system size, expanding understanding of wave behavior in symmetry-breaking biological systems.

Findings

Traveling waves can deform as system size increases without adding nodes.

The Min system exhibits such adaptive wave patterns due to its symmetry and conservation laws.

The pattern behavior differs from traditional Turing instabilities with finite wavenumber bands.

Abstract

We propose a new type of traveling wave pattern, one that can adapt to the size of physical system in which it is embedded. Such a system arises when the initial state has an instability that extends down to zero wavevector, connecting at that point to two symmetry modes of the underlying dynamical system. The Min system of proteins in E. coli is such as system with the symmetry emerging from the global conservation of two proteins, MinD and MinE. For this and related systems, traveling waves can adiabatically deform as the system is increased in size without the increase in node number that would be expected for an oscillatory version of a Turing instability containing an allowed wavenumber band with a finite minimum.