Dynamic properties of bacterial flagellar motors

Abstract

TETHERED bacteria¹ rotate at the angular velocity at which the torque generated by the flagellar motor² is balanced by the torque due to the viscous drag. In general, M = bηΩ, where M is the torque, η is the viscosity, Ω is the angular velocity, and b is a coefficient which depends on the size and the shape of the cell, the position of the axis of rotation, and the distance between the cell and the wall. For a sphere of radius a (not too close to the wall) M = 8πη a³Ω (ref. 3). Viscous forces are so large in comparison with inertial forces⁴ that Ω will change with M virtually instantaneously; any discontinuities in the one will be evident in the other. Consider a cell of radius a and uniform density ρ rotating at an angular velocity Ω₀; if its motor is suddenly disengaged, Ω will decay exponentially to 0 with a time constant ρ a²/15η, and the cell will stop in Ωρ a²/15_η radians. For Escherichia coli this is less than a millionth of a revolution. The cell also is subject to rotational diffusion, but this will be evident only if the coupling between the flagellum and the body of the cell is fluid. The root-mean-square deviation in the angular position is (2Dt)^½, where D is the rotational diffusion constant and t is the time. For a cell which can rotate freely, D = kT/bη, where k is Boltzmann's constant and T is the absolute temperature.