Minimal continuum theories of structure formation in dense active fluids

Abstract

Self-sustained dynamical phases of living matter can exhibit remarkable similarities over a wide range of scales, from mesoscopic vortex structures in microbial suspensions and motility assays of biopolymers to turbulent largescale instabilities in flocks of birds or schools of fish. Here, we argue that, in many cases, the phenomenology of such active states can be efficiently described in terms of fourth-and higher-order partial differential equations. Structural transitions in these models can be interpreted as Landau-type kinematic transitions in Fourier (wavenumber) space, suggesting that microscopically different biological systems can share universal long-wavelength features. This general idea is illustrated through numerical simulations for two classes of continuum models for incompressible active fluids: a Swift-Hohenberg-type scalar field theory, and a minimal vector model that extends the classical Toner-Tu theory and appears to be a promising candidate for the quantitative description of dense bacterial suspensions. We discuss how microscopic symmetry-breaking mechanisms can enter macroscopic continuum descriptions of collective microbial motion near surfaces, and conclude by outlining future applications.

The Aquatic Dance of Bacteria

“…week ending 31 MAY 2013 228102-3 [28] shows that when 0 Þ 0, > 0, v 0 > 0, À 2 > 0, and À 0 < 0, this is one of the simplest vector models to describe phenomenologically the formation of jets and turbulent vortices in quasi-incompressible active suspensions. Very recently, the 2D version of Eq.…”

“…We now examine how these data compare to predictions of a theory of active fluids introduced recently [7,28]. This minimal continuum model assumes that, at high concentrations, the bacterial flow due to swimming and advection can be described by a single velocity field vðt; xÞ and a pressure pðt; xÞ.…”

“…One approach to remedy this problem is to characterize collective turbulent dynamics of bacteria [17,18] and other low Reynolds number swimmers, just as in high Reynolds number fluid turbulence, in terms of kinetic energy, mean squared vorticity (enstrophy) and spatiotemporal correlation functions, and to compare with an appropriate longwavelength theory (i.e., Navier-Stokes-type equations). We present such an analysis here, measuring collective behavior in dense suspensions of the bacterium Bacillus subtilis in comparison to predictions of a (fourth-order) continuum model for bacterial flow [7,28]. [3,4,29,30], freestanding films [5,8,27,31], on surfaces [6,32,33], or quasi-2D microfluidic chambers [7] focused separately on the bacterial and fluid components, leaving uncertain how accurately passive tracers [34,35] reflect collective bacterial dynamics.…”

“…This minimal continuum model assumes that, at high concentrations, the bacterial flow due to swimming and advection can be described by a single velocity field vðt; xÞ and a pressure pðt; xÞ. The theory accounts, phenomenologically, for the most relevant physical effects: incompressibility, local jet-formation by polar alignment, nematic interactions, and stress-induced instability, represented by higher-order spatial derivatives arising from a longwavelength expansion of the effective stress-tensor [28]. The field dynamics is governed by the incompressibility condition r Á v ¼ 0 and…”

“…The parameter 0 describes advection and nematic interactions, and 1 an active pressure contribution [28]. For pusher swimmers [37] like B. subtilis, general considerations of hydrodynamic [44] and nematic stresses [28,45] suggest that 0 ! 1 and 1 ' ð 0 À 1Þ=3 !…”