Quantitative characterization of run-and-tumble statistics in bulk bacterial suspensions

We introduce a numerical method to extract the parameters of run-and-tumble dynamics from experimental measurements of the intermediate scattering function. We show that proceeding in Laplace space is unpractical and employ instead renewal processes to work directly in real time. We first validate our approach against data produced using agent-based simulations. This allows us to identify the length and time scales required for an accurate measurement of the motility parameters, including tumbling frequency and swim speed. We compare different models for the run-and-tumble dynamics by accounting for speed variability at the single-cell and population level, respectively. Finally, we apply our approach to experimental data on wild-type Escherichia coli obtained using differential dynamic microscopy.

Figure 1

Schematic of the run-and-tumble motion of a particle with mean run and tumbling times, ${\tau }_R$ and ${\tau }_T$, respectively. The particle moves at velocity $v$ during the run phase. $P_R$ and $P_T$ are the probabilities of the swimmer to be in a run or tumbling phase. $T$ and $R$ denote the probabilities to start tumbling or running, respectively.

Figure 2

ISFs, $f_{RT}(k,\tau )$, for our RT model. (a), (b) Intrinsic speed variability model for different fractions of run times $p_R={\tau }_R/({\tau }_R+{\tau }_T)$. In (a) we vary the tumbling time with parameter values ${\tau }_R=1$ s and ${\tau }_T=0,0.1,0.5\text{s}$, and in (b) we vary the run time, ${\tau }_T=0.1$ s and ${\tau }_R=2,1$ s. The other motility parameters are $\overline{v}=15\mu \text{m}{\text{s}}^{-1}, {\sigma }_v=4.5\mu \text{m}{\text{s}}^{-1}$, and $D=0.3\mu {\text{m}}^2{\text{s}}^{-1}$. (c) Comparison of the ISF for the model with speed fluctuations at the single cell ($S$; solid line) and population ($P$; dashed line) level using identical parameters (${\tau }_R=1$ s, ${\tau }_T=0.1\text{s}, \overline{v}=15\mu \text{m}{\text{s}}^{-1}, {\sigma }_v=4.5\mu \text{m}{\text{s}}^{-1}$, and $D=0.3\mu {\text{m}}^2{\text{s}}^{-1}$).

Figure 3

(a) Snapshot of a RT simulation with the low magnification (see Appendix pp2). (b), (c) Validation of theoretical predictions with simulations for nine parameter sets with $\overline{v}=17\mu \text{m}{\mathrm{s}}^{-1}, {\sigma }_v=4.3\mu \text{m}{\text{s}}^{-1}, D=0.3\mu {\text{m}}^2{\text{s}}^{-1}, \alpha =0.9$, varying the mean run and tumbling times, ${\tau }_R\in [0.5,1.5]\text{s}$ and ${\tau }_T\in [0.1,0.5]\text{s}$. (b) Parameter estimates obtained by fitting theoretical predictions of the ISF to agent-based simulations. The estimates are compared with the true parameters for the nine data sets. They were extracted from a global fit including wave numbers $k\in [0.15,2.21]\mu {\text{m}}^{-1}$. Dark gray regions correspond to fitted parameters within $\pm 10\%$ of the true values (light gray corresponds to $\pm 20\%$). (c) ISF, $f(k,\tau )$, for ${\tau }_R=1.25$ s, ${\tau }_T=0.2$ s (data set 8) and different wave numbers $k$. Theoretical predictions and simulations correspond to lines and symbols, respectively. The ISFs are shifted w.r.t. the $y$ axis and the gray dotted lines indicate $y=0$.

Figure 4

Parameter estimates obtained by fitting theoretical predictions of the ISF to agent-based simulations with speed variability at the population level. Parameters are the same as in Fig. 3: $\overline{v}=17\mu \text{m}{\text{s}}^{-1}, {\sigma }_v=4.3\mu \text{m}{\text{s}}^{-1}, D=0.3\mu {\text{m}}^2{\text{s}}^{-1}, \alpha =0.9$. The mean run and tumbling times are varied within ${\tau }_R\in [0.5,1.5]\text{s}$ and ${\tau }_T\in [0.1,0.5]\text{s}$. (a) Fitting in lag time $\tau$ with numerical inverse Laplace transform of theoretical ISF. (b, c) Fitting in Laplace time $s$ using the numerical Laplace transform of the data, with $s$ ranging from (b) $1/{\tau }_{\max }$ to $1/{\tau }_{\min }$ and (c) $10/{\tau }_{\max }$ to $1/{\tau }_{\min }$. The estimates are compared with their true values for the nine data sets. Outliers with an error such that ($\text{fits/true}-1)>1$ are not shown. We used a global fit including wave numbers $k\in [0.15,2.21]\mu {\text{m}}^{-1}$. Dark gray regions correspond to fitted parameters within $\pm 10\%$ error of the true values (light gray corresponds to $\pm 20\%$).

Figure 5

ISFs for different wave numbers $k$. Symbols represent experimental results for E. coli bacteria (same as in Fig. 3 of Ref. [31]), and lines are the theoretical predictions obtained by considering speed variability at population level. The data are fitted to the numerical inverse of Eq. (18) including wave numbers $k\in [0.04,1.89]\mu {\mathrm{m}}^{-1}$.

Figure 6

ISFs obtained from simulations for parameters taken in the regime where the two models are distinguishable. We use the parameters measured for wild-type E. coli [31] except for an extremely large ${\sigma }_v$: $\overline{v}=15.95\mu \text{m}{\text{s}}^{-1}, {\sigma }_v=15\mu \text{m}{\text{s}}^{-1}, D=0.24\mu {\text{m}}^2{\text{s}}^{-1}, \alpha =0.96, {\tau }_R=2.39\text{s}, {\tau }_T=0.38\text{s}$. Dots represent simulation data. Solid curves are the fits to the right model, and the dashed curves are fits to the wrong model. We note that the deviation is more significant at small $k$ values. The inset shows the mean relative error ${\sum }_i\left|e_i\right|/6$ of the fitted parameters, where $e_i=(p_{i,\mathrm{fit}}-p_{i,\mathrm{input}})/p_{i,\mathrm{input}}$, and $\mathbf{p}=(\overline{v},{\sigma }_v,{\tau }_R,{\tau }_T,D,\alpha )$.