epl draft Sedimentation , trapping , and rectification of dilute bacteria

The run-and-tumble dynamics of bacteria, as exhibited by E. coli, offers a simple experimental realization of non-Brownian, yet diffusive, particles. Here we present some analytic and numerical results for models of the ideal (low-density) limit in which the particles have no hydrodynamic or other interactions and hence undergo independent motions. We address three cases: sedimentation under gravity; confinement by a harmonic external potential; and rectification by a strip of ‘funnel gates’ which we model by a zone in which tumble rate depends on swim direction. We compare our results with recent experimental and simulation literature and highlight similarities and differences with the diffusive motion of colloidal particles. Recent years have seen an upsurge of activity at the interface between physics and biology. In many cases, relatively wellestablished physics tools have been used to address problems of major interest to biologists. In other cases problems that might appear arcane to some (though not all) biologists have nonetheless posed interesting new physics questions. For instance, the motion of E. coli and similar bacteria involves a series of roughly straight-line ‘runs’ punctuated by rapid changes of direction (‘tumbles’) [1]. This run-and-tumble dynamics can, in idealized form, serve as a paradigm for non-Brownian diffusive motion, and be used to explore some central concerns of nonequilibrium dynamics, such as the origins of phase separation in systems without detailed balance [3, 4]. E. coli is a unicellular organism with external flagellar filaments, which form a bundle at one pole of the cell [1]. When these filaments synchronously rotate counterclockwise, the organism swims in trajectory that is basically straight (with small effects of rotational Brownian motion) [5]. A stochastic intracellular event results in a change of activity [6,7] so that one or more filaments start rotating clockwise; the organism then tumbles [8, 9]. The tumble is of short duration; once all filaments start rotating counterclockwise again, a new run begins. Supposing the runs to have fixed direction and speed v, and the tumbles to occur instantaneously and randomly at rate α (with each tumble fully randomizing the direction of motion), it is easily shown that in free space such run-and-tumble organisms obey the diffusion equation at large length and time scales, with diffusivity D = v/αd where d is the dimensionality. Yet, because bacteria are not Brownian particles close to thermal equilibrium, one cannot write the usual Einstein relations D = kTμ. Here μ is the mobility, which controls the mean velocity v = μF of a particle subject to an external force F; whenever this force is conservative (F = −∇U ), the Einstein relation ensures that, for truly Brownian particles, the equilibrium mean density ρ obeys the Boltzmann distribution, ρ ∝ exp[−μU/D] = exp[−U/kT ]. As will be seen from two of the examples below (sedimentation and the harmonic trap), forced run-and-tumble particles show more complicated behaviour than do Brownian ones. Indeed, even when forces are conservative and a flux-free steadystate density is recovered, this is not in general a Boltzmann distribution for the potential U , even if D is rescaled at will: there is no general ‘effective temperature’ concept. This was discussed briefly in [3] which however focussed on the case of interacting particles. In the present work we explore the noninteracting limit in more detail, presenting a range of results that include an exact steady-state calculation of the sedimentation decay length in 3D. (The interactions that we omit include hydrodynamics [10], which may bring important changes even at modest density.) Oddly, we have found very little discussion of external force fields in the previous literature on noninteracting run-and-tumble particles [11, 12], much of which assumes rotational symmetry of the set of possible particle velocities. This assumption is inspired by models of bacterial chemotaxis, which for good biological reasons take the tumble rate, and not the swim speed, to be a function of swim direction. But external forces, such as gravity, break this symmetry and thus require separate treatment. As well as external force fields, we consider below the effect of asymmetric obstacles, such as funnel gates [13], arguing that these can be modelled by a zone in which there is effective asymmetry in the tumbling rates for particles moving in oppop-1 ar X iv :0 90 3. 32 47 v1 [ co nd -m at . s ta tm ec h] 1 8 M ar 2 00 9 J. Tailleur M. E. Cates site directions. We show that this viewpoint provides a semiquantitative account of the rectification experiments of [13]. Effective temperature: We first consider noninteracting runand-tumble particles in 1D with symmetric tumbling rates α and swim speeds v. At times long compared to α−1 and length scales large compared to ` ≡ v/α (the run-length) the motion is diffusive, with diffusivity D = v/α. In the presence of an external potential U , the speeds of leftand right-going bacteria become vL,R = v ± μ∇U . The diffusion-drift equation for the probability density P becomes [3, 11] ∂tP = −∇·J J = − 2 − μ(∇U) α ∇P − μ∇U ( 1− 2μ∆U α )