Hydrodynamics and direction change of tumbling bacteria

The bacterium {\it Escherichia coli} swims by rotating several helical flagellar filaments, which are gathered in a bundle behind the cell during `runs' wherein the cell moves steadily forward. In between runs, the cell undergoes quick `tumble' events, during which at least one flagellum reverses its rotation direction and separates from the bundle, resulting in erratic motion in place. Alternating between runs and tumbles allows cells to sample space by stochastically changing their propulsion direction after each tumble. Statistically this change of direction is not uniform, with a distribution skewed towards smaller angles with an average of about 67$^\circ$, first measured by Berg and Brown in 1972. In the present work we develop a theoretical approach to model the angular distribution of swimming {\it E. coli} cells during tumbles. We first use past experimental imaging to construct a kinematic description of the dynamics of the flagellar filaments. We then use low-Reynolds number hydrodynamics to compute the consequences of the kinematic model on the force and torque balance of the cell, and deduce the change in orientation. Numerical simulations of our model are in good agreement with experimental observations.