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

Swimming is key for many micro-organisms to survive [1][2][3][4][5][6][7][8].Such 'active matter' is necessarily far from thermal equilibrium [9][10][11][12] and displays peculiar transport properties, which enable foraging [13] and escaping from harm [14].The flagellated bacterium Escherichia coli is a model system for active matter experiments [3,[15][16][17][18][19][20][21].Much is known about its genetics, biochemistry, and ultrastructure, but relating this knowledge to the emerging phenotype, for instance to predict the three-dimensional (3D) pattern of locomotion (or gait) of a swimming population, remains a challenge.The bacterium's 'run and tumble' (RT) dynamics [2] alternates between persistent motion along the cell's axis and sudden changes of direction.While the motion of isolated flagella has been studied in detail using in vitro single-motor experiments [22,23], a quantitative characterization of the full 3D gait of swimming populations of multi-flagellated bacteria has been out of reach so far.This stems, in particular, from the need for measurements over length scales ranging from the order of the short-time runs (∼ 1 − 10 µm) to far beyond the persistence length ( 10 2 µm).Assuming exponentially distributed run and tumble durations, the RT dynamics is predicted to lead to a large-scale diffusion [2,[24][25][26], but this claim has seldom been demonstrated experimentally, and the underlying assumptions have recently been ques-tioned [22,27].The accurate characterization of RT dynamics will therefore fill an important gap.
At the same time, while many aspects of the motility of E. coli have been brought under direct experimental control [2,4,18,[28][29][30][31][32][33][34][35], including the ability to regulate its run speed by light [36,37], there is currently only limited scope to fine-tune its overall gait compared to synthetic swimmers [38][39][40][41][42][43] because the bacterium's tumble dynamics is difficult to control independently.The aforementioned lack of good methods to quantify its RT dynamics contributes to these difficulties, which in turn limits the use of E. coli as a model organism for fundamental active matter research.
In this letter, we report a full characterization of the 3D gait of E. coli, which enables us to demonstrate how the RT dynamics of an engineered strain can be quantitatively tuned from perpetual tumbling to smooth swimming as the concentration of a chemical inducer is varied.These cells can be used in future work under conditions in which their persistence length can be predicted a priori once experimental conditions are specified.
We characterize a bacterium's displacement ∆r(τ ) in Fourier space using the intermediate scattering function (ISF): Analyzing the ISF with a renewal theory then allows us to extract microscopic dynamical parameters such as the particle speed and the run and tumble durations.
To measure the ISF, we extend conventional differential dynamic microscopy (DDM) [44] to collect data encompassing both short-length-scale directed-swimming and large-length-scale diffusive regimes.DDM allows us to work in bulk fluid to minimize hydrodynamic interactions with surfaces [45][46][47][48][49].It also circumvents the need of single-cell tracking, which requires customized Lagrangian [1,2,50,51] or holographic [52,53] microscopy and is limited by the need for low cell concentration, statistical accuracy, and short trajectories.Our data confirm to order of magnitude previous measurements of E. coli motility [1], albeit with a significantly larger run time.We find large length-scale diffusive behavior and compare the extracted diffusivity, D eff , with a theoretical prediction based on the microscopic motility parameters.
The predicted D eff is robust against experimental complexities, but speed fluctuations contribute ∼ 10% of its value.Below, we focus on the biophysical implications of our results, and detail our method elsewhere [54].Bacterial strain.We engineered the NZ1 strain by deleting the cheZ gene in E. coli K12 and adding the inducible plasmid Plac/ara-1-cheZ [17] (Fig. 1a).Deleting cheZ suppresses the transition from clock-wise to counter-clock-wise flagella rotation, so that cells tumble permanently.The plasmid restores expression of cheZ at a rate dependant on the concentration of Isopropyl β-d-1-thiogalactopyranoside (IPTG).It is expected, though never yet confirmed, that tuning the concentration of IPTG during the growth of this strain allows the control of RT dynamics (Fig. 1b).Bacteria were cultured and re-suspended carefully in motility buffer [3] to ensure a very high final motile fraction, α 95% [55].
ISF measurement and analysis.To characterize the gait of the NZ1 strain at different IPTG concentrations, we first measured its ISF by DDM.We then fitted it to the calculated ISF of a well-established model of RT bacteria [1,25,[56][57][58] that is modified to account for recently-observed intrinsic fluctuations of the propulsion speed [59].In this model, bacteria run in quasi straight lines at speed v until they enter a tumbling phase, at rate τ −1 R , during which they fully randomize their orientations.They resume swimming at rate τ −1 T with a new swim speed, sampled from a Schulz distribution p(v) characterized by a mean velocity v and a standard deviation σ v [44].(For an alternative way to account for swimming-speed fluctuations, see Ref. [54].)In addition, cells diffuse translationally with diffusivity D during both run and tumble phases.There is also a fraction 1 − α of non-motile cells that undergo Brownian motion only, also with diffusivity D [60].The ISF for a non-interacting E. coli suspension predicted by this model reads: where f RT (k, τ ) is the ISF of RT bacteria [54].Measuring this ISF for a wide range of k and τ values then allows disentangling the contributions of diffusion, swimming, and tumbling to the dynamics (Fig. 2).Fitting Eq. ( 2) to data finally yields the set of kinetic parameters {v, σ v , τ R , τ T , α, D}.
To measure the ISF experimentally, we imaged cells in sealed capillaries on a fully-automated inverted brightfield microscope with a sCMOS camera.A full characterization of RT dynamics requires accessing length scales much greater than the persistence length p in all directions.This necessitated a large depth of field at low k to ensure that bacteria remain in view over large distances  in 3D.We thus consecutively recorded movies at 2× and 10× magnifications and extracted the corresponding ISFs for k < 0.9 µm −1 and k ≥ 0.9 µm −1 using DDM [44,61], which are then fitted to our renewal theory using a numerical protocol described elsewhere [54].
IPTG-induced transition from tumbling to swimming.We grew suspensions of the NZ1 strain at several IPTG concentrations and measured the ISFs.Representative data over approximately four decades in time and two decades in length are shown in Fig. 3a.Oscillations typical of persistent swimmers (Fig. 2) are seen most clearly at the higher IPTG concentration and high k values.
In more detail, Fig 3b compares the ISFs at a given k for three IPTG concentrations.In the absence of IPTG, diffusion dominates and the oscillations are absent.At IPTG=25µM, oscillations are seen for k = 0.66µm −1 and k = 0.1µm −1 (corresponding to length scales of ∼ 2π/k ≈ 10 and 60 µm respectively), while data at the smallest k shows a smooth decay: the RT dynamics becomes effectively diffusive on such a large scale.At the highest IPTG concentration, 150µM, oscillations are seen at all scales, showing a strong enhancement of the persistence length.To our knowledge, this is the first demonstration of the quantitative tuning of the 3D gait of E. coli by varying external conditions.
Our protocol also allows us to quantify phenotypic heterogeneity.We repeated such measurements for 8 IPTG concentrations, using two biological replicates and typi-cally 3 to 4 successive measurements of f (k, τ ) per replicate and IPTG concentration.The fitted kinetic parameters are then averaged for each replicate and plotted in Fig. 3c.The small error bars show that successive measurements of f (k, τ ) at each condition and for each biological replicate yielded consistent results.The observed variability between replicates (compare red and blue data points) therefore quantifies the degree of phenotypic heterogeneity in a clonal population.
Increasing IPTG leads to a robust increase of the persistence length, p = vτ R , which translates into oscillations in the ISF, Figs.3a-b.Capturing quantitatively such fine-grained features requires very good statistics, a narrow speed distribution, and a low fraction of nonmotile cells, a challenge that is met by the experimental protocol described in [55].Since there is little variability in the average speed (v 23.5 µm s −1 at all finite IPTG concentrations), the two orders of magnitude increase in p is the result of a dramatic increase in the run duration.
Given the high speed of our NZ1 strain and the long run durations at high IPTG concentrations, the extraction of τ R requires sampling large length scales, at which we find that the tumbling times τ T cannot be reliably measured.We have tested the robustness of our results with respect to τ T by fixing it at τ T = 0.1, 0.2, 0.3 s and extracting the remaining kinetic parameters.We find that the average velocity of the bacteria remains unaffected and the systematic increase of the persistence length persists (Fig. S1 [55]).Finally, measurements of the evolution of single motor statistics of WT E. coli in different environments have been reported before [23], Fig. 3d shows for the first time the measured correlation between p and the number of CheZ mRNA in the cell.
Wild-type E. coli. Figure 4 shows the measured ISFs for a dilute suspension of WT E. coli over approximately four decades in time and two decades in length.The ISFs display an intermediate plateau at large k, which is a signature of the diffusive motion of both the small non-motile fraction ( 5%) and of the tumbling cells.The plateau disappears at low k and long times, which reflects the randomization of the swimming direction.
The WT data are well fitted by our renewal theory [54] at all k (Fig. 4, solid lines).We find that 96 ± 0.1% of the bacteria swim at a mean speed v = 16 ± 0.1 µm s −1 with standard deviation σ v = 5.78 ± 0.13 µm s −1 (errors are obtained by Jackknife resampling method [62]).
Finally, our data allow us to probe a range of length and time scales large enough to bridge short-scale directed swimming and large-scale diffusive motion.This is an important challenge since recent experiments have questioned the experimental relevance of exponentiallydistributed run and tumble durations [22,27] .The ISF of purely diffusive particles, f (k, τ ) = A exp(−k 2 D eff τ ), where A is a constant, gives a good account of our large-scale data (shown for two values of k in Fig. 5).Averaging over our two smallest values of k leads to D eff = 185 ± 7 µm 2 s −1 .In an RT model with exponentially distributed run and tumble durations, Using parameters from fitting the measured ISF (caption, Fig. 4), we find D th eff = 198 ± 11 µm 2 s −1 , which is remarkably close to the measured value.Interestingly, the σ 2 v term arising from velocity fluctuations contributes to ∼ 10% of the value of D th eff .Note that our measured D eff is three orders of magnitude larger than the fitted single-particle diffusivity, D = 0.24±0.01µm 2 s −1 , which highlights the ability of our protocol to provide information on active-particle dynamics over a large spatiotemporal range.
Conclusion.-By characterizing the dynamics of E. coli over a wide range of length and time scales, we demonstrated for the first time how the tumbling dynamics of an engineered strain can be quantitatively tuned independently of the swimming speed.Furthermore, we have characterized to a high statistical accuracy the full 3D gait of WT E. coli in bulk suspensions, from smallscale persistent motion to large-scale diffusion.We have shown that a microscopic RT model with exponentially distributed RT durations describes both regimes.The use of more realistic distributions [27] can be accommodated in our approach, but is unlikely to change significantly any of our conclusions.
Our work lays the foundation for the high-throughput study of the swimming gait of a variety of microorganisms, such as the RT pattern of Bacillus subtilis [59] or the run-reverse motion of marine bacteria [63] and archaea [64], using standard microscopy.More generally, the ability of microorganisms to respond to chemical gradients, i.e., chemotaxis, is a vital part of their foraging and survival strategy.Our method in combination with spatiotemporally-resolved DDM [20] opens the way to the high-throughput study of such response at the population level.
Most of the experimental, imaging, and analysis protocols are either standard or can be found in previous publications [1][2][3][4][5][6].To make the reading as self-content as possible, we nevertheless briefly describe them below, as well as specify the culture conditions and biological protocols used in our study.The numerical protocol and its validation are presented in a joint submission [7].

A. Samples
We used two strains of E. coli K12: a wild-type AB1157, which have been described elsewhere [2], and the engineered strain (NZ1) obtained from AMB1655.We cultured both strains using standard protocols described elsewhere [2].Briefly, an overnight culture was obtained by inoculating a single colony into 10 mL of Luria broth followed by incubation at 30 • C/200 rpm for 16 to 18 h.Then this was inoculated into 35 mL of Tryptone broth (TB) medium (1:100 dilution), which was incubated for 4 h (30 • C/200 rpm) to obtain a late-exponential-phase culture.Cells were then harvested and concentrated by gentle filtration (0.45 µm Immobilon filters, Millipore) and resuspended in motility buffer (pH 7.0, 6.2 mM K2HPO4, 3.8 mM KH2PO4, 67 mM NaCl, and 0.1 mM).Here, we used a single filtration step to avoid damage to the flagella, which yields suspensions (2-4 mL at ≈ 5 × 10 8 cells/ml) with higher motile fraction (> 95%) and a narrower swimming speed distribution than for multiple filtration steps [2].For the engineered strain NZ1, we added IPTG at a given concentration in the range (0-400 µM) in TB.

B. Imaging
Results from computer simulations [7] suggested one needs to access a wide range of wave numbers k corresponding to length scales ranging from the bacterial size to over the run length (i.e.∼ 100 µm) for a reliable analysis.Thus, a large depth of field is required at low k values.We consecutively recorded movies (512 pxl ×512 pxl) using * C.K. and Y.Z.contributed equally.; ckurzthaler@pks.mpg.de† C.K. and Y.Z.contributed equally.; yfzhao2021@suda.edu.cn‡ julien.tailleur@univ-paris-diderot.fr§ vincent.martinez@ed.ac.uk two magnifications (10×/0.3,8000 images at 100 fps and 2×/0.06,32000 images at 50 fps), an inverted bright-field microscope (Nikon Ti E), and a sCMOS camera (Hamamatsu Orca Flash 4.0 v2, using 2 × 2 binning).We restricted the condenser aperture to ≈ 5 mm diameter so that both 10× and 2× imaging could be performed using the same illumination conditions, which yields a depth of field of about 300 µm for the lowest magnification.Due to the large (in 3D) required for reliable analysis, we used glass capillaries with a large height (1 mm) and focused in the middle.Note that the depth of field for 10× objective is too small for the required large length-scales.Finally, the sampling volumes corresponding to high and low magnifications were estimated to ≈ 10 −5 and 3 × 10 −3 mL (assuming a depth of field of ≈ 15 and 300 µm, respectively) corresponding to ≈ 10 4 and 10 6 cells in the field of view, respectively, for a suspension with ≈ 5 × 10 8 cells/mL.The data for the mutants are fitted using k ∈ [0.036, 1.42]µm −1 and for the data of the wild type we included k ∈ [0.04, 1.89]µm −1 .

C. Differential Dynamic Microscopy (DDM)
We then analyzed each movie using DDM to extract the ISFs, following [1,5].Briefly, we computed the differential image correlation function (DICF), g(k, τ ), i.e., the power spectrum of the difference between pairs of images separated by time τ , by g(k, τ , where I(k, t) is the Fourier transform of the image I(r, t) and • t denotes an average over time t.Under suitable imaging conditions and for isotropic motion, the DICF yields the ISF [1,3,8] with • k denoting average over k and A(k) and B(k) the signal amplitude and instrumental noise, respectively.These coefficients are obtained from the plateau of g(k, τ ) at long and short times, where the ISF approaches f (k, τ → ∞) → 0 and f (k, τ → 0) → 1, respectively.

II. ENGINEERED E. COLI (NZ1)
A. Fitting results The fitting estimates are shown in Fig. S.1 by the solid symbols.To test the robustness of our results under the fluctuations in the estimates of D and τ T , we fix τ T at 0.1, 0.2, 0.3 s, and D at 0.3 µm 2 s −1 and extract the remaining parameters.We find that the fraction of moving bacteria, the average velocity, and the average run time remain unaffected.The systematic increase of the persistence length persists.It suggests that the Brownian diffusion constant D cannot be reliably measured in the sample with few non-motile cells.The difficulty in estimating τ T is highlighted as well.

B. Estimation of CheZ mRNA per cell
We measure the number of CheZ mRNA per cell by reverse transcription of quantitative PCR as previously described [9].Briefly, Around 0.5 OD NZ1 cells induced at different concentrations of IPTG in TB was collected by centrifugation.Cell lysates were prepared and spiked with a known amount (around 10 ng) of external RNA.Total RNA was extracted using the RNeasy Mini Kit (QIAGEN) and 700 ng of the extracted RNA was treated by DNase I, Amp Grade (Invitrogen) to eliminate genomic DNA and plasmid contamination.Purified RNA was used for reverse transcription using PrimeScriptTM RT reagent Kit (TaKaRa) and the reverse transcribed DNA was then quantified by qPCR using qTOWER3 (Analytik Jena).The copy numbers of CheZ mRNA and the external RNA were calculated according to a standard curve of known quantities of the corresponding linearized plasmid.To estimate the CheZ mRNA copy numbers per cell, a recovery rate, which describes the ratio between the measured and initial amounts of mRNA, was calculated by dividing the measured output by the input amount of the spike mRNA.Then the measured amounts of CheZ mRNA was divided by the recovery rate to obtain its initial amounts per sample.This number was further divided by the number of cells present in the initial sample to calculate the number of CheZ mRNA copies per cell.The number of cells per sample was obtained from the measured absorbance at 600 nm (OD600).To correlate OD600 to the number of bacteria, the measured absorbance was plotted against the colony forming units (CFU) of

FIG. 1 .
FIG. 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.
FIG. 3. Engineered E. coli strain NZ1.(a) ISFs for IPTG concentrations 25 µM and 150 µM.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 wavenumbers k.(b) Comparison of the ISFs for different IPTG concentrations and wavenumbers.(c) Average speed v, run time τR, and persistence length 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 p and the number of CheZ mRNA per cell (right panel).

3 FIG. S. 1 .
FIG. S.1.The fitting estimates of the two replicates of NZ1 as a function of IPTG concentration.The estimates with fitting all the parameters are shown as solid symbols.The open symbols show the estimates with fixing τT at 0.1, 0.2, 0.3 s, or D at 0.3 µm 2 s −1 .