# The N-flagella problem: Elastohydrodynamic motility transition of multi-flagellated bacteria

## TL;DR

This paper presents a theoretical analysis of how multi-flagellated bacteria transition from non-motile to swimming states, revealing that the most unstable mode leads to linear swimming combined with rotation, consistent with experimental observations.

## Contribution

The study analytically solves the elastohydrodynamic stability problem for multi-flagellated bacteria, identifying the dominant motility mode and its relation to bacterial morphology.

## Key findings

- Most unstable mode causes linear swimming with rotation
- Up to 6 unstable modes related to rigid-body motion
- Results align with experimental bacterial swimming behavior

## Abstract

Peritrichous bacteria such as Escherichia coli swim in viscous fluids by forming a helical bundle of flagellar filaments. The filaments are spatially distributed around the cell body to which they are connected via a flexible hook. To understand how the swimming direction of the cell is determined, we theoretically investigate the elastohydrodynamic motility problem of a multi-flagellated bacterium. Specifically, we consider a spherical cell body with a number N of flagella which are initially symmetrically arranged in a plane in order to provide an equilibrium state. We analytically solve the linear stability problem and find that at most 6 modes can be unstable and that these correspond to the degrees of freedom for the rigid-body motion of the cell body. Although there exists a rotation-dominated mode that generates negligible locomotion, we show that for the typical morphological parameters of bacteria the most unstable mode results in linear swimming in one direction accompanied by rotation around the same axis, as observed experimentally.