Wavelength selection of rippling patterns in myxobacteria

TL;DR

This paper develops an analytical approach to determine the wavelength of rippling patterns in myxobacteria, revealing how internal signaling influences pattern formation and stability, extending previous numerical studies.

Contribution

It derives an evolution equation for pattern wavenumber in a mathematical model of bacterial rippling, providing analytical insights into pattern selection and stability.

Findings

Derived an evolution equation for reversal point density.

Validated the pattern selection rule through numerical solutions.

Identified conditions leading to pattern destruction and phase transitions.

Abstract

Rippling patterns of myxobacteria appear in starving colonies before they aggregate to form fruiting bodies. These periodic traveling cell density waves arise from the coordination of individual cell reversals, resulting from an internal clock regulating them, and from contact signaling during bacterial collisions. Here we revisit a mathematical model of rippling in myxobacteria due to Igoshin et al.\ [Proc. Natl. Acad. Sci. USA {\bf 98}, 14913 (2001) and Phys. Rev. E {\bf 70}, 041911 (2004)]. Bacteria in this model are phase oscillators with an extra internal phase through which they are coupled to a mean-field of oppositely moving bacteria. Previously, patterns for this model were obtained only by numerical methods and it was not possible to find their wavenumber analytically. We derive an evolution equation for the reversal point density that selects the pattern wavenumber in the weak signaling limit, show the validity of the selection rule by solving numerically the model equations and describe other stable patterns in the strong signaling limit. The nonlocal mean-field coupling tends to decohere and confine patterns. Under appropriate circumstances, it can annihilate the patterns leaving a constant density state via a nonequilibrium phase transition reminiscent of destruction of synchronization in the Kuramoto model.