A study of bacterial flagellar bundling

Abstract

Certain bacteria, such as Escherichia coli (E. coli) and Salmonella typhimurium (S. typhimurium), use multiple flagella often concentrated at one end of their bodies to induce locomotion. Each flagellum is formed in a left-handed helix and has a motor at the base that rotates the flagellum in a corkscrew motion. We present a computational model of the flagellar motion and their hydrodynamic interaction. The model is based on the equations of Stokes flow to describe the fluid motion. The elasticity of the flagella is modeled with a network of elastic springs while the motor is represented by a torque at the base of each flagellum. The fluid velocity due to the forces is described by regularized Stokeslets and the velocity due to the torques by the associated regularized rotlets. Their expressions are derived. The model is used to analyze the swimming motion of a single flagellum and of a group of three flagella in close proximity to one another. When all flagellar motors rotate counterclockwise, the hydrodynamic interaction can lead to bundling. We present an analysis of the flow surrounding the flagella. When at least one of the motors changes its direction of rotation, the same initial conditions lead to a tumbling behavior characterized by the separation of the flagella, changes in their orientation, and no net swimming motion. The analysis of the flow provides some intuition for these processes.

Synchronization and bundling of anchored bacterial flagella

“…Note that the tail part of the bundle has a larger separation than the middle part, which is consistent with previous studies. 5,35 The average distance at the tail region is about 1.5-3 times larger than the smallest distance in the tightly bundled middle region. The larger distances at the bundle end are determined by the force balance between the mechanical force, specifically excluded-volume interactions, opposing wrapping, and hydrodynamic interactions promoting bundling.…”

“…However, the time for synchronization due to direct hydrodynamic interactions quickly increases with increasing distance. This time has been estimated theoretically to increase as d 5 , within the hydrodynamic far-field approximation, for a related system of two rigid dumbbells with their midpoints fixed by stiff springs.…”

“…4 Numerical investigations using Stokes equations for fluids and various models for a flagellum, such as bead-spring models, provide insight into the bundling process and the run-andtumble dynamics. 5,10,15,35 The propulsion dynamics, the flow field, and the polymorphic transitions of a flagellum have been investigated by an elastic network model in ref. 10.…”

The Orientation of Swimming Biflagellates in Shear Flows

“…It employs regularized forces to approximate a boundary integral formulation of the Stokes equations. Recently, it has been used to study the hydrodynamics of bacteria such as E. coli (Flores et al, 2005) and Bacillus subtilis (Cisneros et al, 2007), and the nematode Turbatrix (Cortez et al, 2004). The beauty of the method lies in its relative simplicity, with computations restricted to the boundary of the microorganism, from which the flow at any point in the fluid can be obtained.…”

The physics of flagellar motion of E. coli during chemotaxis

“…It was Srigiriragu and Powers (2005) who proposed a coarsegrained continuum model of flagella that incorporated their structural organization. Flores et al (2005) also incorporated the structural details by considering the flagellum as a helical hollow tube with material points connected to each other through elastic springs. This method incorporates both the elastic and hydrodynamic behavior of flagellar propulsion.…”