Enhanced diffusion of nonswimmers in a three-dimensional bath of motile bacteria

We show, using differential dynamic microscopy, that the diffusivity of nonmotile cells in a three-dimensional (3D) population of motile E. coli is enhanced by an amount proportional to the active cell flux. While nonmotile mutants without flagella and mutants with paralyzed flagella have quite different thermal diffusivities and therefore hydrodynamic radii, their diffusivities are enhanced to the same extent by swimmers in the regime of cell densities explored here. Integrating the advective motion of nonswimmers caused by swimmers with finite persistence-length trajectories predicts our observations to within 2%, indicating that fluid entrainment is not relevant for diffusion enhancement in 3D.

Figure 1

The ISF from fluorescence DDM of a sample at $\varphi =0.1\%$ with WT cells and motA mutants at a number ratio of 7:3 plotted against (a) $\tau$ and (b) $q^2\tau$. In each case, $q$ increases from red to purple in a rainbow scale in the range $0.62<q<4.46$ $\mu$m${}^{-1}$. The inset to (b) plots the fitted diffusivity of the nonmotile motA (N2) cells as a function of $q$.

Figure 2

(a) Effective diffusivity $D_{\mathrm{N}2}$ of motA (circles) and fliF (squares) in suspensions of E. coli AB1157 containing no glucose (red, $\overline{v}=13$–16 $\mu$m/s) or containing glucose (black, $\overline{v}\lesssim 25$ $\mu$m/s), while $\alpha$ is varied at a fixed total cell density $\varphi =0.1\%$, with best-fit lines. (b) The same data plotted as diffusivity enhancement $\Delta D=D_{\mathrm{N}2}-D_{0,\mathrm{N}2}$ for both motA and fliF data sets with and without glucose vs the active particle flux, $J_A=\overline{v}{n}_A=\overline{v}\alpha \varphi /V$. Green and blue points: data taken by varying $\varphi$ or by varying both $\varphi$ and $\alpha$. Equation (1) fitted through all points gives $\beta =7.1\pm 0.4\mu {\text{m}}^4$. Dashed line: prediction of [14]. All error bars give $\pm$ standard deviation.

Figure 3

Schematic of an “elementary scattering event” between a swimmer and a tracer. See text for definition of symbols.