Statistical Mechanics of Interacting Run-and-Tumble Bacteria

Abstract

We consider self-propelled particles undergoing run-and-tumble dynamics (as exhibited by E. coli) in one dimension. Building on previous analyses at drift-diffusion level for the one-particle density, we add both interactions and noise, enabling discussion of domain formation by 'self-trapping', and other collective phenomena. Mapping onto detailed-balance systems is possible in certain cases.PACS numbers: 87.10.Mn, 87.17.Jj Several species of bacteria, including Escherichia coli, perform self-propulsion by a sequence of 'runs' -periods of almost straight-line motion at near-constant speed (v) -punctuated by sudden and rapid randomizations in direction, or 'tumbles', occurring stochastically with rate α. It is no surprise that the resulting class of random walk gives a diffusive relaxation of the number density at large scales [1]. The resulting diffusion constant ∼ v 2 /α, is vastly larger than that of non-swimming particles undergoing pure thermal motion at room temperature. Therefore, apart from, e.g., the upper limit it imposes on the duration of a straight run (set by rotational diffusion), true Brownian motion can usually be ignored.Because bacterial diffusion is not thermal, the steadystate probability density cannot be written as, with H a Hamiltonian, even for a single diffuser. The physicist's intuition can easily be led astray: for instance, Refs.[2, 3, 4] address models (of chemotaxis) comprising noninteracting particles in 1D, with no external forces, but v(x), α(x) functions of position x. Instead of a uniform density, as would arise with any force-free detailed-balance dynamics, one findsHere we extend previous analyses of run-and-tumble motion to the many-particle level, addressing the roles of noise and interactions. These determine, for instance, the dynamic correlator of run-and-tumble bacteria, which is measurable by light scattering at low density [5] and at higher density, in principle, by particle-tracking microscopy [6]. Additionally, particles for which v, α depend on the local density (either via thermodynamic interactions such as depletion [7], or kinetic effects such as collision-induced tumbles) could show collective phenomena such as domain-formation or flocking. Such effects have previously been addressed within models where a self-propelled particle responds vectorially to the velocity of its neighbors, by direct sensing or passive hydrodynamics [8,9,10]. Below, we shall find, for run-and-tumble dynamics, similar effects in even simpler cases when only the speed of a particle is density-dependent.In making the transition from a single particle to many, most bacterial modelling approximates the number density by a simple replacement ρ = N p [3,4]. But even for noninteracting particles, ρ (unlike p) is a fluctuating quantity, and a full statistical mechanics must compute noise terms for ρ. As seen below, these are not ad-hoc, but follow from the run-and-tumble dynamics directly.To allow relatively rigorous progress we work in 1-D throughout. For d > 1, although good descriptions...

Light-activated self-propelled colloids

“…Increasing the surface fraction of particles, this simple mechanism leads to a dynamic phase transition from a gas phase of hot colloids [10] to a dense state, resulting from the 'traffic jam' of the persistent self-propelled particles [46,[50][51][52][53][54][55][56][57]. The emergence of arrested phase owing to density-dependent mobility has been discussed theoretically in the context of bacteria by Tailleur & Cates [58]. The role of the self-trapping mechanism for the emergence of clustering was shown in the recent and remarkable experiments by Buttinoni et al [59], where they used 'large' 4 μm self-propelled carbon-coated Janus colloids, which self-propel under illumination in a near-critical water-lutidine mixture [60], and for which the caps can be optically resolved, indicating the direction of self-propulsion.…”

Freezing and phase separation of self-propelled disks

“…The dynamics of a homogeneous fluid of self-propelled particles has been described by an effective continuum theory where motility suppression is incorporated in a density-dependent propulsion speed v(ρ) 12,15,16,34 . The mean-field model applies when particles experience many collisions before their directed motion becomes uncorrelated by rotational noise, i.e., for ζ >> 1 26 .…”

Dynamics of self-propelled particles under strong confinement

“…When the persistence length of active motion becomes comparable to the mean free path, uniquely active effects arise that transcend the thermodynamically allowed behaviors of equilibrium systems, including giant number fluctuations and spontaneous flow [3,14,[16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Importantly, a sufficient active persistence length is the only requirement for macroscopic manifestations of activity, as revealed by athermal phase separation of nonaligning, repulsive self-propelled particles [31][32][33][34][35][36][37][38][39][40][41].When boundaries and obstacles are patterned on the scale of the active correlation length, they dramatically alter the dynamics of the system, and striking macroscopic properties emerge [42][43][44][45][46][47][48][49]; for example, ratchets and funnels drive spontaneous flow in active fluids [42][43][44][45][46]. This effect has been used to direct bacterial motion [50] and harness bacterial power to propel microscopic gears …”