Stochastic Model for Direction Changes of Swimming Bacteria

Abstract

In this work we introduce a stochastic model to describe directional changes in the movement of swimming bacteria. We use the probability density function (PDF) of turn angles, measured on tumbling E. coli wild-type, to build a Langevin equation for the deflection of the bacterial body swimming in isotropic media. We solved analytically this equation by means of the Green function method and show that three parameters are sufficient to describe the movement: a characteristic time, the steady-state solution and a control parameter. We conclude that the tumble motion, which is manifested as abrupt turns, is primarily caused by the rotational boost generated by the flagellar motor and complementarily by the rotational diffusion introduced by noise. We show that, in the tumble motion, the deflection is a non-stationary stochastic processes during times where the tumble occurs. By tuning the control parameter our model is able to explain small turns of the bacteria around their centers of mass along the run. We show that the deflection during the run is an Ornstein-Uhlenbeck process, which for typical run times is stationary. We conclude that, along the run, the rotational boosts do not exist or are neglectable and that only the rotational diffusion remains. Thus we have a single model to explain the turns of the bacterium during the run or tumble movements, through a control parameter that can be tuned through a critical value that can explain the transition between the two turn behaviours. This model is also able to explain very satisfactory all available statistical experimental data, such as PDFs and average values of turning angles and times, of both run and tumble motions.