Characterization and Control of the Run-and-Tumble Dynamics of Escherichia Coli

We characterize the full spatiotemporal gait of populations of swimming Escherichia coli using renewal processes to analyze the measurements of intermediate scattering functions. This allows us to demonstrate quantitatively how the persistence length of an engineered strain can be controlled by a chemical inducer and to report a controlled transition from perpetual tumbling to smooth swimming. For wild-type E. coli, we measure simultaneously the microscopic motility parameters and the large-scale effective diffusivity, hence quantitatively bridging for the first time small-scale directed swimming and macroscopic diffusion.

Figure 1

Engineered strain NZ1. (a) Scheme of the regulation: cheZ expression driven by Plac/ara-1 is suppressed by the LacI suppressor. Exogenously adding IPTG induces cheZ expression by reducing LacI suppression. (b) Cells are expected to tumble continuously at low IPTG concentration and to enter a smooth swimming state at high IPTG concentration.

Figure 2

Theoretical ISFs for a suspension comprising a fraction $\alpha$ of RT bacteria and $1-\alpha$ of diffusing cells, for several wave numbers $k$. We consider different fractions of run time $p_R={\tau }_R/({\tau }_R+{\tau }_T)$, using ${\tau }_R=1\mathrm{s}$ and ${\tau }_T=0,0.1,0.5\mathrm{s}$, with $\overline{v}=15\mathrm{\mu }\mathrm{m}{\mathrm{s}}^{-1}$, ${\sigma }_v=4.5\mathrm{\mu }\mathrm{m}{\mathrm{s}}^{-1}$, $\mathrm{D}=0.3\mathrm{\mu }{\mathrm{m}}^2{\mathrm{s}}^{-1}$, and $\alpha =0.95$. For smooth swimmers with fixed speed $v_0$, $f(k,\tau )=\mathrm{sinc}(v_{0}k\tau )\exp (-D{k}^2\tau )$. When $p_R=1$, this leads to clear oscillations of $f(k,\tau )$ at small length scales (red arrow). As the tumbling rate ${\tau }_T^{-1}$ increases (black arrow), these oscillations are smeared out and the diffusive dynamics of the tumbling bacteria eventually leads to a diffusive plateau around $f\sim 1-p_R$. At large but finite times, depending on the value of $k$, the diffusive cells may not have decorrelated, leading to a nonzero plateau of $f(k,\tau )$ (blue arrow).

Figure 3

Engineered E. coli strain NZ1. (a) ISFs for IPTG concentrations 25 and $150\mathrm{\mu }\mathrm{M}$. The ISFs are shifted vertically and gray dotted lines correspond to $f=0$. Symbols and lines represent experiments and fits to the theory, respectively, and different colors correspond to different wave numbers $k$. (b) Comparison of the ISFs for different IPTG concentrations and wave numbers. (c) Average speed $\overline{v}$, run time ${\tau }_R$, and persistence length ${\ell }_p$, as a function of IPTG concentration. Red and blue symbols correspond to two biological replicates. The error is estimated from successive measurements of the ISF using the same sample. (d) Independent measurements of the number of CheZ mRNA per cell at various IPTG concentrations (left panel) allow correlating the persistence length ${\ell }_p$ and the number of CheZ mRNA per cell (right panel).

Figure 4

Measurements of the ISFs for a WT E. coli suspension (symbols). Fits to Eq. (2) (lines) lead to $\overline{v}=15.95\mathrm{\mu }\mathrm{m}{\mathrm{s}}^{-1}$, ${\sigma }_v=5.78\mathrm{\mu }\mathrm{m}{\mathrm{s}}^{-1}$, $D=0.24\mathrm{\mu }{\mathrm{m}}^2{\mathrm{s}}^{-1}$, $\alpha =0.96$, ${\tau }_R=2.39\mathrm{s}$, and ${\tau }_T=0.38\mathrm{s}$. The ISFs are shifted vertically as in Fig. 3.

Figure 5

Measured ISFs of WT E. coli at $k=0.0126\mathrm{\mu }{\mathrm{m}}^{-1}$ and $k=0.0151\mathrm{\mu }{\mathrm{m}}^{-1}$ (symbols) fitted against the ISF of diffusive particles (lines), giving effective diffusivities $D_{\mathrm{eff}}=192\mathrm{\mu }{\mathrm{m}}^2{\mathrm{s}}^{-1}$ and $D_{\mathrm{eff}}=178\mathrm{\mu }{\mathrm{m}}^2{\mathrm{s}}^{-1}$, respectively. The ISFs are shifted vertically as in Fig. 3.