Simulations of the Whirling Instability by the Immersed Boundary Method
Sookkyung Lim and Charles S. Peskin

How do bacteria move? Many species of bacteria swim in an aqueous environment by means of rotating flagella. E. Coli is a peritrichous bacterium with typically 6 flagella distributed about the cell. The motility of E. coli derives from the use of flagella which consist of a helical filament driven by a rotary motor at the cell surface. The rotation of the flagellar motor has the ability to switch direction so that both clockwise(CW) and counterclockwise(CCW) rotation of the flagellum may occur. The energy to drive the motor is obtained from the proton motive force.

When the motors turn CCW (when viewed looking towards the cell), the filaments rotate parallel in a concerted bundle that pushes the cell body steadily forward, and the cell is said to "run". The rotation rate of the filaments within the bundle is around 100Hz. During a run, all of the motors have to rotate CCW. When one or more of the flagellar motors abruptly change direction of rotation from CCW to CW, the flagellar filaments work independently, and the cell body moves erratically with little net displacement; the cell is then said to "tumble". The effect of tumbling is to randomize the direction of the next run. These two modes alternate. The cell runs and tumbles, executing a three-dimensional random walk.

Although the ultimate goal is to simulate the two modes of motility (running and tumbling) of E.coli, we begin with a simple question which concerns the whirling instability of a rotating elastic filament.

The 3-D computational model we are dealing with here is an elastic and neutrally bouyant filament having micro-architecture motivated by bacterial flagella. It is a flexible cylindrical tube with two layers, corresponding to the outer surface and inner surface of the hollow flagellum. It has a filament outer diameter of 23nm throughout the length and a hollow central channel with a diameter of about 8.9nm. There is a motor at the bottom. We use the Immersed Boundary (IB) method to study the interaction between the elastic filament and the surrounding viscous fluid as governed by the incompressible Navier-Stokes equations at a very low but nonzero Reynolds number.

A surprising conclusion of our study is the existence of a sharp transition between two drastically different motions : twirling and overwhirling. In twirling, the filament configuration is straight, but in overwhirling it is bent through more than 180^o . We have seen this transition occur as a consequence of change in the spinning frequency of only 2%. Presumably, the transition is sharp, and an arbitrarily small change in frequency across the critical frequency would suffice. This large change in amplitude as a consequence of a small change in a bifurcation parameter is the hallmark of a subcritical bifurcation.

We have confirmed the subcritical nature of this bifurcation by demonstrating another of its characteristic features : bistability at spinning frequencies just below the critical frequency. At such a subcritical frequency, we have shown that small amplitude initial bends relax to twirling, whereas larger amplitude initial bends grow into overwhirling. Thus twirling and overwhirling coexist as possible stable dynamical states at subcritical spinning frequencies.

[1] Sookkyung Lim and Charles S. Peskin, Simulations of the whirling instability by the immersed boundary method, SIAM. J. Sci. Comput. Vol. 25, No. 6, pp. 2066-2083, 2004

[2] Sookkyung Lim and Charles S. Peskin, Subcritical bifurcation of a rotating elastic filament in a viscous fluid by the immersed boundary method, Proceedings of the Second M.I.T. Conference on Computational Fluid and Solid Mechanics