To directly investigate the ordering mechanism in a coherent migration group, we quantified the stochastic behaviors of bacterial run-and-tumble motions relative to the stable migrating group. To achieve this, we employed a microfluidic device that generated a stable propagating band of bacteria, as previously reported (Fu et al., 2018; Saragosti et al., 2011). Using aspartate (Asp) as the only chemoattractant to drive the migration of E. coli, we tracked a small fraction of fluorescent bacteria (JCY1) as representatives of the non-fluorescent wild-type cells (RP437) (see Materials and methods, Appendix 1—figure 1, Video 1). Because the group velocity $V_G=\langle \Delta x_i\left(t\right)/\Delta t\rangle$ was constant over time, ${\displaystyle V_G\sim 3.0\mu \mathrm{m}/\mathrm{s}}$ (Appendix 1—figure 2), we were able to map the tracks to a moving coordinate ${\displaystyle z=x-V_{G}t}$.

Video 1

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With the shifted tracks, we calculated the key statistics of the single-cell behaviors. We first checked that the average instantaneous velocity, ${\displaystyle V_I(z)=⟨\mathrm{\Delta }{x}_i(z)/\mathrm{\Delta }t⟩}$, was constant along the density profile $\rho \left(z\right)$ (Figure 1B). This result confirmed that the group migrates coherently. Then, we identified all the run states and tumble states of individual trajectories using a previously described computer assistant program (Dufour et al., 2016; Waite et al., 2016), to ensure that the tracks are separated into successive tumble-and-run events (see Materials and methods). Comparing the sample events initiated from the back (b), middle (m), and front (f) of the migration group, we observed that the runs in the front were longer but distributed more uniformly in terms of the directionality, whereas the runs in the back were shorter but more oriented toward the group migration direction (Figure 1C, Appendix 1—figure 3A, B). Quantitatively, the statistics of run length (and duration) displayed exponential distributions with the means for the direction of group migration longer than those for the opposite direction (Figure 1D, Appendix 1—figure 3C). The angular distribution of the run length confirmed the difference in directionality between cells in different spatial locations (Figure 1E, Appendix 1—figure 3D). We also confirmed that cells in the back showed greater directional persistence toward the migration direction (Appendix 1—figure 4A–C), as (Saragosti et al., 2011) reported previously. All these results suggested that the bacteria in the back run more effectively toward the direction of group migration than those in the front.

To quantify the spatial extent of the drift efficiency, we defined the expected drift velocity $V_D\equiv \frac{\langle l_R\cdot coscos{\theta }_R\rangle }{\langle {\tau }_R+{\tau }_T\rangle }$ , by the projection of the average run length along the migration direction over the average duration of runs and tumbles (Dufour et al., 2014). This quantity reflects the effective run speed of run-and-tumble events that start running on a given location relative to the group. The drift velocity was found to decrease from the back to the front, crossing the group velocity $V_G$ in the middle of the group (Figure 1F). This particular trend of $V_D\left(z\right)$ suggests a mean reverting behavior of bacteria in the group: the cells at the back drift faster than the group ($V_D\S amp;gt;V_G$), enabling the cells to catch up with the group; at the same time, the cells in the front drift slower than the group ($V_D\S amp;lt;V_G$), causing them to slow down and fall back (Figure 1G). Such mean reverting process also results in sub-diffusion of individuals relative to the group (Appendix 1—figure 3F), for which the mean square displacement (MSD) is constrained over time (Appendix 1—figure 3G). The slope of the spatial extent of $V_D\left(z\right)$, ${\displaystyle -r=-0.05\mathrm{m}\mathrm{i}{\mathrm{n}}^{-1}}$ , quantifies the speed at which individuals revert to its center.

We noted that the spatial profile of the expected drift velocity $V_D\left(z\right)$ was different from the instantaneous velocity $V_I\left(z\right)$ . This is because the instantaneous velocity $V_I\left(z\right)$ defines the average speed of cells in a given time interval $dt$, which reflects the positional shift of a group of cells at a given position. However, the expected drift velocity $V_D\left(z\right)$ defines the average speed of run-and-tumble events that start tumbling at a given position. Since the run duration is explicitly modulated by the gradient of chemoattractant $g\left(z\right)$ and is dependent on the chemotactic ability $\chi$ , $V_D\left(z\right)=\chi g\left(z\right)$ represents the drift velocity driven by the external stimuli (Celani and Vergassola, 2010; de Gennes, 2004; Dufour et al., 2014; Si et al., 2012).

To understand how the spatially structured profile of the drift velocity $V_D$ impacts on the group migration, we first adopted a Langevin-type model that describes bacterial motions as an active Brownian particle driven by the expected drift velocity $V_D$ and a random force: $dx=V_{D}dt+ϵdW$ (Berg, 2004). In this model, the run-and-tumble random motions are considered as a Gaussian random force $ϵdW$, while the cell motions are imposed to a deterministic force $V_D$ (Rosen, 1973; Rosen, 1974). The strength of Gaussian noise can be estimated by the effective diffusion coefficient $ϵ=\sqrt{2D}$ , while the drift velocity is determined by the product of the perceived gradient $g\left(z\right)$ and the chemotactic ability $\chi$, $V_D=\chi g\left(z\right)$ . Such a stochastic description of bacterial motions has been proven equivalent to the classic Keller-Segel (KS) model that described the population dynamics of bacterial chemotactic group migration (Keller and Segel, 1971a, Rosen, 1973). In the moving coordinate, ${\displaystyle z=x-V_{G}t}$ , this Langevin-type model specifies that $dz=V_D\left(z\right)dt-V_{G}dt+ϵdW$. Thereby, the cell motions relative to the migrating group are modulated by two effective forces: one generated by $V_D\left(z\right)$, which pushes the cells to catch up with the wave; and another generated by $-V_G$ , which drags the cells to fall behind the wave. These two ‘forces’ constrain the random motions of individuals in an effective potential well $U\left(z\right)$ (Figure 2A).

Figure 2

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To estimate the spatial profile of the effective driven force $V_D\left(z\right)$, we analyzed the KS model with moving ansatz (supplementary text). Assuming the density profile $\rho \left(z\right)$ directly measured from experiments (Figure 1B), we can deduce the chemoattractant concentration profile $S\left(z\right)$ (Appendix 1—figure 4D), as well as the perceived gradient $g\left(z\right)$ (Figure 2B). Since the perceived gradient $g\left(z\right)$ is almost linear in the main part of the wave profile, we approximated it by $g\left(z\right)\approx g_0+g_{1}z$ (Figure 2B, dashed line), which allows us to transform the stochastic model to an OU-type equation:

By this simplified model, we obtained a clear picture how individual behaviors are regulated relative to the group: cells are imposed to a driving force linearly dependent on their relative positions in the group, $F\left(z\right)=\chi g_{1}z+\left(\chi g_0-V_G\right)$ . This suggests that the random motions of bacteria are constrained in a parabolic moving potential well, $U\left(z\right)=\frac{1}{2}\chi \left|g_1\right|z^2+\left(\chi g_0-V_G\right)z+U_0$ (Figure 2A), where $U_0$ is set by $U\left(\infty \right)=0$. The position that minimizes $U\left(z\right)$ is also the balanced position, $z_0=-\frac{g_0}{g_1}-\frac{V_G}{\chi \left|g_1\right|}$ , where the driving force is null $F\left(z_0\right)=0$. Behind the balanced position, cells experience a pushed force to catch up the group. Cells would start to fall back once they exceed the balanced position, where the driving force becomes negative. Therefore, the decreasing profile of the driving force enable cells to perform mean reverting behaviors around the balanced position. In addition, the expected rate that cells tend to revert back to the balanced position is defined by the slope of the spatially dependent driving force, $r=\chi \left|g_1\right|$ (Figure 2B).

Given the knowledge of individual behaviors, we studied the spatial distribution of population on the group migration. The OU model (Equation 1) which describes the single-cell stochastic motions has an equivalent form, known as the Fokker-Planck equation (Equation S19) which describes the spatial-temporal dynamics of cell density distribution. This population model provides a traveling wave solution with the mean position around $z_0$ and standard deviation $\sigma =\frac{ϵ}{\sqrt{2\chi \left|g_1\right|}}$ .

From the solution, we noted that the effective driving force, as well as the drift velocity, has a negative slope ($g_1\S amp;lt;0$). The decreasing profile of the drift velocity suggests that the leading front of the group migrates as a pushed wave front (Gandhi et al., 2016; van Saarloos, 2003). As a key feature of the pushed wave, the leading front of the group drops parabolically, which is much faster than that of diffusion front. This further leads to a tight density profile of group migration for a single phenotype population.

To address whether this pushed wave front still holds in presence of phenotypic diversity, we further examined the above analysis with cells of diverse chemotactic abilities ${\chi }_i$ imposed to the same moving perceived gradient $g\left(z\right)$ (Figure 2B). Given a monotonically decreasing profile of perceived gradient $g\left(z\right)$ , the driving force that each phenotypic individual experiences exhibits the same spatial dependency with the slopes depend on the intrinsic phenotypic properties of each phenotype $r_i={\chi }_i\left|g_1\right|$ . This monotonic dependency means that the balanced positions $z_0$ of the diverse phenotypes are orderly arranged. By the stochastic Langevin-type model with phenotypic diversity in chemotactic ability, we first confirmed that each phenotypic population migrates at a constant speed $V_G$ , following the moving gradient $g\left(z\right)$ (see supplementary text). The density profiles of cells with different ${\chi }_i$ follow the same shape but are spatially orderly aligned (Figure 2C). Under the same moving gradient $g\left(z\right)$, the driving force ${\chi }_{i}g\left(z\right)$ is phenotype-dependent, so that the bottom position of the potential well, $z_{0,i}=-\frac{g_0}{g_1}-\frac{V_G}{{\chi }_i\left|g_1\right|}$, is also spatially arranged according to ${\chi }_i$ (Figure 2D). As predicted by the OU model, the balanced positions $z_{0,i}$ of different phenotypes increase with their chemotactic abilities ${\chi }_i$ (Figure 2E, black line), while the standard deviation (${\sigma }_i$) of the density profiles decreases with ${\chi }_i$ (Figure 2E, blue line). We also confirmed the ordered structure of phenotypes by a particle-based model of the Langevin-type dynamics coupled with chemoattractant consumption (Appendix 1—figure 5). Therefore, under the spatially decreasing driving force, cells with phenotypic diversity perform the same type of mean reverting processes with spatially ordered mean positions.

The spatial order of phenotypes does not directly promise a compact group migration with phenotypic diversity. By close examination of the density profile, we found that each phenotypic subpopulation propagates as a pushed wave front. We further calculated the total density profile of the entire migratory group with Gaussian distribution of chemotactic abilities under a decreasing linear gradient. Simulation reveals a pushed wave front for the combination of these subpopulations with different chemotactic abilities (Figure 2F, insert). In addition, we checked that the width of the entire group maintains in a converged width over long time, suggesting that the pushed wave profile enables the migratory population with diversity to keep in a tight shape (Figure 2F).

We examined the migration profiles under other forms of perceived gradient $g\left(z\right)$ : a constant perceived gradient and a spatially increasing perceived gradient. In the first case, individual bacteria of identical phenotype follow the diffusion process relative to the group, where the standard deviation of the population increasing with time by $\sigma =2\sqrt{Dt}$ (Figure 2G, dashed line). Each phenotype subpopulation is expected to have a constant drift velocity over space. However, as the drift velocity would vary by the chemotactic ability, each subpopulation migrates in different group speeds, suggesting a compact group migration of diverse population cannot be maintained in a constant perceived gradient (Figure 2G). In the latter case, when imposed to a spatially increasing driving force (a pulled wave, by definition), individual cells display super-diffusion processes. The density profile easily diverges in this case (Figure 2H). Therefore, we concluded that the pushed wave front, driven by a decreasing shape of driving force, enables a spatially ordered and compact pattern of phenotypes while on the move.

Although the simplified OU-type model (Equation1) represents a key aspect of the ordering mechanism of phenotypes, it does not detail the signaling processes of bacterial chemotaxis, such as receptor amplification, adaptation, and motor responses (Sumpter, 2010), and it cannot predict bacterial run-and-tumble behavior. To consolidate the proposed mechanism underlying the emergence of spatial orders from the individual random motions, we further performed agent-based simulations integrated with the chemotactic pathway, multi-flagella competition, and boundary effect in three dimensions (3D) (Dufour et al., 2014; Jiang et al., 2010; Sneddon et al., 2012). In the agent-based model, the attractant dynamics governed by diffusion and bacterial consumption is described by a reaction-diffusion equation (Equation S2) as previous multiscale models (Erban and Othmer, 2005; Xue and Othmer, 2009). We constructed a population with multiple phenotypes defined by different chemotactic abilities ${\chi }_i$ , where ${\chi }_i$ was varied by changing the receptor gain $N$ (for details, see supplementary text). Since the receptor gain $N$ only affects the amplification factor by which a cell responds to the gradient, the variation in bacterial motility $ϵ$ is unchanged. As a result, a dense band of migrating cells that follow a self-generated moving chemoattractant gradient via consumption were recaptured as experiments (Appendix 1—figure 6A). The phenotypes were spatially ordered as $\chi$ varies (Appendix 1—figure 6B), and the velocity profile of each phenotype decreases from the back to front (Appendix 1—figure 6C) as predicted by the OU-type model.

To better analyze the simulations, we simplified the simulation with a non-consumable attractant profile $S\left(z\right)$ moving with constant speed $V_G$ (Appendix 1—figure 4D). Using this simplified model, we first checked that the mean positions of the density profiles of cells with different receptor gains $N$, as well as their peaks, were orderly aligned with respect to chemotactic ability ${\chi }_i$ .

As an important advantage of the agent-based simulations, the model allowed us to analyze single-cell behavior during the ordered group migrations. For each phenotype $i$, the expected drift velocity $V_{D,i}\left(z\right)$ decreased along the density profile (Figure 3A). Consistent with the ordered structure of the density profiles, the intersection between $V_{D,i}\left(z\right)$ and $V_G$ exhibited the same order of chemotactic ability ${\chi }_i$ (Appendix 1—figure 7). As the reversion rate $r_i=\left|\frac{d{V}_{D,i}\left(z\right)}{dz}\right|$ showed a positive correlation to ${\chi }_i$ , cells with lower receptor gain $N$ (resulting in a smaller $\chi$) experienced a weaker reverting force toward the center (Figure 3A insert). Thus, the effective moving potential, $U_i\left(z\right)$ , which constrains the cells around the mean positions sorted by their chemotactic abilities, becomes flat for cells with lower chemotaxis ability $\chi$ (Figure 3B; Long, 2019). As a result, cells of each phenotype perform sub-diffusion, whereby the MSD along the migration coordinate relative to the group was bound at a level negatively correlated to $\chi$ (Figure 3C). The width of the density distribution, as an effect of the reversion force, decreased with the reversion rate, as an approximate linear function. Using this agent-based model, we further obtained similar results for populations of different ${\chi }_i$ through adaptation time $\tau$, or basal CheY protein level $Y_{p0}$ , which determined the basic tumble bias $T{B}_0$ (Dufour et al., 2014; Jiang et al., 2010; Sneddon et al., 2012; Appendix 1—figure 8).

Figure 3

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To verify the model predictions on the individual behaviors of different phenotypes, we experimentally measured the trajectories of cells with different chemotactic abilities during the group migration. Specifically, we altered the chemotaxis abilities of cells by titrating the expression level of Tar, which is under the control of a small molecule inducer aTc (Sourjik and Berg, 2004; Zheng et al., 2016) (see Materials and methods and Figure 4A). The variations in the expression of Tar would lead to different receptor gains in response to the Asp gradient (Adler, 1966b; Adler, 1969; Sumpter, 2010), but the tumble bias and growth rate would not change. Using the migration speed of the bacterial range expansion to quantify the chemotaxis ability of the titrated strains, we found that the chemotaxis ability increases with aTc concentration (see Materials and methods and Appendix 1—figure 9).

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The Tar-titrated cells labeled with yellow fluorescent protein (YFP; strain JCY20) were added to the wild-type population at a ratio of 1 in 400. Within the wild-type population, 1 in 50 cells was labeled with red fluorescent protein (RFP) (strain JCY2). As the Tar-titrated strain constituted a small portion of the pre-mixed population, we considered that the density profile of the population would be invariant to different levels of induction of Tar. The premixed population could generate collective group migration, similar to the wild-type population (Appendix 1—figure 9). The trajectories of the YFP-labeled cells were tracked to represent the behavior of cells with different chemotactic abilities, while the profile of wild-type cells with RFP was also measured to characterize the density distribution of the entire migratory population.

By comparing the statistics of cells with different Tar expression levels, we found that the expected drift velocity $V_{D,i}\left(z\right)$ followed the same decreasing pattern from back to front (Figure 4B). More importantly, as the Tar expression level (chemotactic ability) increased, the slope of the decreasing pattern increased, which was consistent with the model prediction shown in Figure 3A. The intersections between $V_{D,i}\left(z\right)$ and $V_G$ , as well as the peak positions and mean positions of each Tar-titrated density profile (Figure 4C), shifted toward the front as the chemotactic ability increased (as measured by the migration rate on agar plates; Cremer et al., 2019; Liu et al., 2019). The $V_D$ cross point was always behind the peak position and the mean position (Figure 4C), suggesting that cells were leaking behind. Moreover, the width of each Tar-titrated density profile (defined by $2{\sigma }_i$) decreased as the reversion rate $r_i$ increased (Figure 4D), consistent with the model results in Figure 3C. Thus, as the OU-type model predicts, the width of the density profile is controlled by the reversion rate determined by the chemotactic ability ${\chi }_i$ .

The wild-type strain E. coli (RP437) and its mutants were used in this study, where all plasmids were kindly provided by Dr Chenli Liu. Specifically, the Tar-titratable strain was constructed by recombineering according to previous research (Zheng et al., 2016). Specifically, the DNA cassette of the Ptet-tetR-tar feedback loop was amplified and inserted into the chromosomal attB site by recombineering with the aid of plasmid pSim5. The tar gene at the native locus was seamlessly replaced with the aph gene by using the same recombineering protocol. To color-code the strains, we use plasmids with chloramphenicol-resistant gene carrying yfp under constitutive promoter (for JCY1 strain) and pLambda-driven mRFP1 plasmids maintained by kanamycin (for JCY2). To color-code Tar-titratable strain (JCY20), a plasmid carrying yfp chloramphenicol-resistant gene was transformed into constructed Tar-titratable strain. The tar-only strain (UU1624) was modified from RP734 and was kindly provided by Prof. Johan Sandy Parkinson.

For bacterial culture, the M9 supplemented medium was used. The preparation of the M9 supplemented medium follows the recipe in previous study (Fu et al., 2018): 1×M9 salts, supplemented with 0.4% (v/v) glycerol, 0.1% (w/v) casamino acids, 1.0 mM magnesium sulfate, and 0.05% (w/v) polyvinylpyrrolidone-40. 1×M9 salts were prepared to be 5×M9 salts stock solution: 33.9 g/L Na₂HPO₄, 15 g/L KH₂PO₄, 2.5 g/L NaCl, 5.0 g/L NH₄Cl.

For migration experiments in the micro-channel, the M9 motility buffer was used. The recipe was:1×M9 salts, supplemented with 0.4% (v/v) glycerol, 1.0 mM magnesium sulfate, and 0.05% (w/v) polyvinylpyrrolidone-40, 0.1 mM EDTA, 0.01 mM methionine, and supplemented with 200 µM aspartic acid.

For the migration rate measurements, the M9 amino acid medium with 0.2% (w/v) agar was used to prepare swim plate (Liu et al., 2019). The recipe was: 1× M9 salts, supplemented with 0.4% (v/v) glycerol, 1× amino acid, 200 µM aspartic acid, 1.0 mM magnesium sulfate, and 0.05% (w/v) polyvinylpyrrolidone-40. 1× amino acid were prepared to be 5× amino acid stock solution: 4 mM alanine, 26 mM arginine (HCl), 0.5 mM cysteine (HCl·H₂O), 3.3 mM glutamic acid (K salt), 3 mM glutamine, 4 mM glycine, 1 mM histidine (HCl·H₂O), 2 mM isoleucine, 4 mM leucine, lysine, 1 mM methionine, 2 mM phenylalanine, 2 mM proline, threonine, 0.5 mM tryptophane, 1 mM tyrosine, 3 mM valine. All experiments were carried out at 30°C. Plasmids were maintained by 50 µg/mL kanamycin or 25 µg/mL chloramphenicol.

The bacteria from frozen stock were streaked onto the standard Luria-Bertani agar plate with 2% (w/v) agar and cultured at 37°C overnight. Three to five separate colonies were picked and inoculated in 2 mL M9 supplemented medium for overnight culture with corresponding antibiotics to maintain plasmids. The overnight culture was diluted by 1:100 into 2 mL M9 supplemented medium the next morning. For Tar titration strains, related aTc were added in this step. When the culture OD600 reaches 0.2–0.25, it was then diluted into pre-warmed 15 mL M9 supplemented medium so that the final OD600 was about 0.05 (Liu et al., 2019; Zheng et al., 2020; Zheng et al., 2016).

Bacteria were washed with the M9 motility buffer and were re-suspended in fresh M9 motility buffer to concentrate cell density at OD600 about 1.0. Then, the wild-type strain and fluorescent strain were mixed with ratio of 400:1 before loaded in the microfluidic chamber (Fu et al., 2018; Saragosti et al., 2011). For Tar titration experiments, the wild-type strain (RP437) was mixed with two fluorescent strains (JCY2 and JCY20) by 400:8:1.

The microfluidic devices were fabricated with the same protocol and the same design as previous research (Bai et al., 2018; Fu et al., 2018), except that the capillary channel was designed longer than that of previous ones. The size of the main channel was 20 mm×0.6 mm× 0.02 mm and only one gate at the end of the channel was kept (Appendix 1—figure 1A).

Fluorescent cells were mixed with non-fluorescent cells by 1:400 for cell tracking in the dense band. Sample of mixed cells with density OD600 ≈1.0 was gently loaded into the microfluidic device and then the device was spun for 15 min at 3000 rpm in a 30°C environmental room so that almost 100,000-150,000 cells were placed to the end of the channel. After spinning, the microfluidic device was placed on an inverted microscope (Nikon Ti-E) equipped with a custom environmental chamber set to 50% humidity and 30°C.

The microscope and its automated stage were controlled by a custom MATLAB script via the μManager interface (Edelstein et al., 2014; Fu et al., 2018). About 30 min after the sample loading, a 4× objective (Nikon CFI Plan Fluor DL4X F, NA 0.13, WD 16.4 mm , PhL) was placed in the wave front for imaging. The fluorescent bacteria, seen as randomly picked samples of the migrating group, were captured continuously with frame rate of 9 fps in 10 min until they leave the view. Typical tracks are longer than 300 s. Time-lapsed images were acquired by a ZYLA 4.2MP Plus CL10 camera (2048 × 2048 arrays of 6.5 µm×6.5 µm pixels) at 9 frames/s (fps) . An LED illuminator (0034R-X-Cite 110LED) and an EYFP block (Chroma 49003; Ex: ET500/X 20, Em: ET535/30 m) compose the lightening system.

For the Tar titration experiments, the channel was first scanned with 10× objective (CFI Plan Fluor DL 10× A, NA 0.30, WD 15.2 mm, PH-1) enlighten by an LED illuminator (0034R-X-Cite 110LED) through the RFP block (Chroma 49005, Ex: ET545/X 30, Em: ET620/60 m) and EYFP block channels for seven neighbored views around the migration group. These images were further combined to two large pictures of the RFP strains and YFP strains. The channel was scanned twice, respectively before and after the 10 min tracking of fluorescent Tar-titrated cells.

The acquired movie was first analyzed with the U-track software package to identify bacteria and to get their trajectories (Jaqaman et al., 2008). Then the tracks were labeled by run state and tumble state by a custom MATLAB package (Waite et al., 2016) using a previously described clustering algorithm (Dufour et al., 2016).

The group velocity $V_G$ was calculated by averaging the frame-to-frame velocity (${\displaystyle dt\approx 0.11\mathrm{s}}$) over all tracks and all time. The cell number for the first frame over a spatial bin of ${\displaystyle \mathrm{\Delta }x=60\mu \mathrm{m}}$ and a channel section a=12,000 µm² were calculated to get the density profile $\rho \left(x,t=0\right)=\frac{\sum i\left(x,t\right)}{a\cdot dx}$ . The peak position of the first frame ($x_{peak}\left(t=0\right)$) was then determined by the maximum of $\rho (x,t=0)$. The position of each bacterium ($x_i\left(t\right)$) was transformed to moving coordinate position $z_i$ by the group velocity $V_G$ and origin of the axis on the density peak by $z_i=x_i\left(t\right)-V_{G}t-x_{peak}\left(t=0\right)$ . Given the relative position of each cell, we recalculated the density profile in moving coordinate $\rho \left(z\right)=\frac{\sum i\left(z\right)}{a\cdot dx}$ . The width of the density profile was defined by two times the standard deviation of relative positions $2\sigma =2\sqrt{\frac{1}{n-1}{\sum }_{i=1}^n{\left(z_i-\langle z\rangle \right)}^2}$ . The spatial distribution of the instantaneous velocities $\langle V_I\left(z\right)\rangle$ was calculated by averaging the velocity in spatial bin of ${\displaystyle \mathrm{\Delta }z=240\mu \mathrm{m}}$.

A tumble-run event is the minimal element of bacterial behavior. The typical spatial scale of a tumble-run event is about 20 μm, which is much smaller than the spatial bin size chosen in this study (240 µm). The spatial distributions of run time $\langle {\tau }_R\left(z\right)\rangle$, tumble time $\langle {\tau }_T\left(z\right)\rangle$, and run length $\langle l_R\left(z\right)\rangle$ were calculated by averaging the related values of all the events with tumbling position ($z_T$) located in each spatial bin ($z$). As the displacement of tumble is small, the tumbling position is approximately the starting position of runs. For each tumble-run event, we have the vector linking starting position and end position of the run. The running angle ${\theta }_R$ is then defined by the angle between run direction and the group migration direction. One can easily deduce all the other quantities with the formulations in Table 1.

Growth rates of Tar-titrated strains were calculated from exponential fitting ($R^2\S amp;gt;0.99$) over measured curves of cell density (OD600) with respect to time. A 250 mL flask with 20 mL M9 supplement medium were used. All measurements were performed in a vibrator of rotation rate of 150 rpm at 30 °C. OD600 was measured by a spectrophotometer reader every 25 min. Each strain has been measured for at least three times.

The semi-solid agar plate was illuminated from bottom by a circular white LED array with a light box as described previously (Liu et al., 2011; Liu et al., 2019; Wolfe and Berg, 1989) and was imaged at each 2 hr by a camera located on the top. As bacteria swimming in the plate forms ‘Adler ring’, we used the first maximal cell density from the edge to define the moving edge of bacterial chemotaxis. The migration rate was then calculated from a linear fit over the data of edge positions in respect to time ($R^2\S amp;gt;0.99$).

Details of the theoretical models and numerical simulations were presented in the appendix notes. In which, the Langevin equation was deduced and solved numerically with a particle-based simulation; the approximated OU-type equation and its traveling wave solution were deduced; an agent-based simulation of bacterial with chemotaxis pathway was performed following previous works (Dufour et al., 2014; Jiang et al., 2010; Sneddon et al., 2012).

The bacterial chemotactic migration in a capillary was first modeled by Keller and Segel in 1971 (Keller and Segel, 1971a). In this model, the bacterial cell division was neglected as the nutrition in the capillary was poor. In 1D, the dynamics of cell density $\rho \left(x,t\right)$ is a combination of diffusion and chemotaxis:

where $D$ and $\chi$ respectively represent the diffusion coefficient and chemotactic ability of bacteria, and $g\left(x,t\right)$ represents the perceived gradient of chemoattractant.

The dynamics of the chemoattractant $S(x,t)$ follows a reaction-diffusion-type equation with a diffusion coefficient $D_s$ and consumption with a constant rate $k$:

From the chemoattractant profile $S(x,t)$, we can get $g\left(x,t\right)$ by:

with $K_{on},K_{off}$ were dissociation constants for active and inactive conformation of the receptor, respectively.

Following Rosen, 1973, one can deduce the Fokker-Planck equation associated to the KS model that describes the probability $P=P(x,y;t)$ to observe of a cell which initiates from position $y$ at position $x$ and time $t$. The related equation writes:

Equation S4 is equivalent to the Langevin dynamics as:

where $x_i$ represents the position of the i-th bacteria and $g\left(x_i,t\right)$ is the perceived gradient of each bacteria at time $t$. $dW$ is a Wiener process that describes a random walk with noise level $ϵ=\sqrt{2D}$ .

As previously proofed, the expected drift velocity equals the production of chemotactic ability $\chi$ and the perceived gradient $g$, $V_D=\chi g$ (Waite et al., 2016). So that, the Langevin-type Equation S5 can be reformed to:

The initial condition of the chemoattractant was ${\displaystyle S\left(x,0\right)=S_0=200\mu \mathrm{M}}$. In the stochastic model, the density profile $\rho \left(x,t\right)$ was calculated by the sum of cell number ($\sum i(x,t)$) in certain location $[x,x+\Delta x]$ and time $t$ over volume $a\cdot \Delta x$, where $a$ is the capillary section area and $dx$ is the spatial bin size:

The Langevin-type model in the absolute coordinate $x$ (Equation S5) was simulated with a customized code using MATLAB. In each case, 100,000 particles were simulated in a 1D space where ${\displaystyle x\in \left[0,20\right]\mathrm{m}\mathrm{m}}$. The absorption boundaries were applied for ${\displaystyle x=0\mathrm{m}\mathrm{m}}$ and ${\displaystyle x=20\mathrm{m}\mathrm{m}}$, so that particles stay at the boundary until they leave them actively (${\displaystyle x\left(x>0\mathrm{m}\mathrm{m}\right)=0}$ and ${\displaystyle x\left(x>20\mathrm{m}\mathrm{m}\right)=20\mathrm{m}\mathrm{m}}$). Initially all particles were placed at ${\displaystyle x=0\mathrm{m}\mathrm{m}}$ and their positions were calculated according to Equation S7 at each time step of $\Delta t=0.05s$ for 60 min. The random noise follows a Gaussian distribution generated by MATLAB function ‘randn.m’ with amplitude multiplied by $\sqrt{2}ϵ\sqrt{\Delta t}$ .

Unlike the particles that change their position continuously, the spatial profile of cell density $\rho \left(x,t\right)$ was first calculated on a space mesh of grid ${\displaystyle \mathrm{\Delta }x=10\mu \mathrm{m}}$, and then linearly interpolated to a finer mesh of grid ${\displaystyle \mathrm{\Delta }x=0.1\mu \mathrm{m}}$. The chemoattractant $S\left(x,t\right)$ and perceived gradient $g\left(x,t\right)$ were deduced accordingly. At each time point, following Equation S7, cells located on grid $\left[x,x+\Delta x\right]$ were summed and were divided over $a\Delta x$ to get the density profile. In order to avoid fast consumption near $S=0$, a saturation effect of consumption rate was added to Equation S3 using $S/(S+K_s)$.

where ${\displaystyle K_s=0.1\mu \mathrm{M}}$, of which the value was small and does not affect the linear relation in most range of $S\S amp;gt;K_s$ .

Then the chemoattractant profile $S\left(x,t\right)$ was calculated from $\rho \left(x,t\right)$ according to Equation S8, using a fourth-order Runge-Kutta method from an initial condition of $S\left(x,0\right)=S_0$ . The perceived gradient $g\left(x,t\right)$ was further calculated from $S\left(x,t\right)$ on Equation S3. Having the discrete $g\left(x,t\right)$ over grid of ${\displaystyle \mathrm{\Delta }x=0.1\mu \mathrm{m}}$, the perceived gradient of each particle $g\left(x_i\right)$ was assigned by detecting the particle position over the spatial mesh grid.

Following the above instructions, density profile and peak positions with bacteria of ${\displaystyle \chi =0.11\mathrm{m}{\mathrm{m}}^2\mathrm{m}\mathrm{i}{\mathrm{n}}^{-1}}$ and $D=\chi /22$ were plotted in Appendix 1—figure 5A. The group migration reached steady state after 20 min simulation. The peak velocity was then defined by linear fitting of peak positions for ${\displaystyle t\in \left[20,60\right]\mathrm{m}\mathrm{i}\mathrm{n}}$, which was found to be ${\displaystyle V_p=3.03\mu \mathrm{m}/\mathrm{s}}$.

To simulate a bacterial group of multiple phenotypes, we constructed 100,000 cells with their chemotactic ability $\chi$ follows a Gaussian distribution with the average chemotactic ability ${\displaystyle ⟨\chi ⟩=0.11\mathrm{m}{\mathrm{m}}^2\mathrm{m}\mathrm{i}{\mathrm{n}}^{-1}}$ and s.d. of 0.03 mm²min^-1 (Appendix 1—figure 5B) while the diffusion coefficient $D$ was fixed to $\langle \chi \rangle /22$ for all cells. The consumption rate of each bacterium was also set constant of ${\displaystyle 4.6\cdot {10}^{-9}\mu \mathrm{M}/\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}/\mathrm{s}}$. The group of multi-phenotypes migrates slower than single phenotype with peak velocity ${\displaystyle V_p=2.90\mu \mathrm{m}/\mathrm{s}}$ (Appendix 1—figure 5A). The bacterial positions after 20 min simulation were further transformed to the moving coordinate $z_i=x_i-V_{p}t$. They were subgrouped according to their chemotactic ability $\chi$ to six subgroups. And the corresponding average density profiles of each subgroups were plotted in Appendix 1—figure 5D with colors defined in Appendix 1—figure 5C. The density profile of subgroups was found ordered according to their average chemotactic ability $\chi$. The time-shifted profiles, $S\left(z\right)$ and $g\left(z\right)$, were shown in Appendix 1—figure 5E,F.

The simulations of bacteria with consumption confirmed that coherent group migration and ordered spatial structures can emerge from a system of active particles modulated by gradient of consumable chemoattractant.

As the density profile was found stable during the experimental time scale (30 min), we are allowed to use a moving coordinate $z=x-V_{G}t$ in searching for a traveling wave solution. Equation S2, Equation S5, Equation S7 become:

Given a stable density profile measured experimentally, we can integrate Equation S10 to get a stable profile of chemoattractant $S\left(z\right)$. As the chemoattractant was unconsumed at $z=\infty$ and it is completely consumed as the wave passes $z=-\infty$, the boundary condition of $S\left(z\right)$ can be written as $S\left(\infty \right)=S_0$ and $S\left(-\infty \right)=0$. We get:

where $M\left(z\right)={\int }_z^{\infty }\rho \left(z\right)dz$ is the accumulated number of cells up to position $z$. We then used the experimentally measured cell density profile together with the group speed $V_G$ to calculate the moving attractant profile $S\left(z\right)$ (Appendix 1—figure 5A). The perceived gradient $g\left(z\right)$ in the moving coordinate was then calculated by:

In a stable migration group, the $g\left(z\right)$ deceases from back to front over the density profile. Suppose the gradient does not steepen toward the back, then cells with lower chemotactic abilities will fall behind and see shallower gradient, drifting even slower and causing the wave to spread out. Since the gradient is formed by local consumption, the front of the wave will have less cells and become shallower, while the back of the wave will accumulate more cells and become steeper. This will automatically readjust the gradient steepness until the back becomes steep enough to allow cells will lower chemotactic abilities to catch up, stabilizing the gradient and the traveling wave.

The chemoattractant profile $S\left(z\right)$ can be deduced by an experimentally measured stable density profile $\rho \left(z\right)$ in the moving coordinate. With the experimental mesh grid ${\displaystyle \mathrm{\Delta }x=240\mu \mathrm{m}}$, we plot the average $S\left(z\right)$ over three biological replicates in Appendix 1—figure 4D (circles). In order to apply this profile into the particle-based simulation, we first expanded the space range of $S\left(z\right)$ to ${\displaystyle x\in \left[-4800,4800\right]\mu \mathrm{m}}$ by setting ${\displaystyle S\left(z<-2400\mu \mathrm{m}\right)=0}$ and ${\displaystyle S\left(z>2400\mu \mathrm{m}\right)=S_0}$ . Next, we used a cubic spline data interpolation to construct a profile of $S\left(z\right)$ with much finer spatial mesh grid ${\displaystyle \mathrm{\Delta }x=0.001\mu \mathrm{m}}$ (Appendix 1—figure 4D, line). Therefore, a $g\left(z\right)$ profile on finer meshes was calculated accordingly (Figure 2A). The $g\left(z\right)$ profile was then translated to an absolute coordinate by $g\left(x,t\right)=g\left(z\right)+V_{G}t$, with ${\displaystyle V_G=3.0\mu \mathrm{m}/\mathrm{s}}$.

Using the preset moving gradient $g\left(x,t\right)$ , we can simulate the particle motions without calculation of $\rho (x,t)$ and $S\left(x,t\right)$ . We used the same method as described in Section 2 to simulate particle motions, and at each time point $t$, we assigned $g\left(x_i\right)$ by indexing bacterial position on $g\left(x,t\right)$ , with spatial precision of ${\displaystyle \mathrm{\Delta }x=0.001\mu \mathrm{m}}$.

For bacteria of four different $\chi$ ranging from 0.15 to 0.3 mm²min^-1 , we simulated 10,000 particles for each phenotype with the same noise level ${\displaystyle ϵ=15\mathrm{m}\mathrm{m}\cdot \mathrm{m}\mathrm{i}{\mathrm{n}}^{-0.5}}$ . During simulation of 60 min, bacteria of all phenotypes move as a group with peak positions linearly increasing with time. The fitted slope of all peaks equals the preset velocity of the moving gradient $V_G$ . In the moving coordinate $z$, the density profiles of all phenotypes were ordered (Figure 2A), and they were clearly ordered by $\chi$. The peak positions and mean positions of the density profile increase with $\chi$ (Figure 2C), while the width (defined by $\sigma$) decreases with $\chi$ (Figure 2D).

The perceived gradient $g\left(z\right)$ deduced from experimental data shows linear decreasing trend in the major range of density profile ${\displaystyle -0.4\mathrm{m}\mathrm{m}<\mathrm{z}<0.4\mathrm{m}\mathrm{m}}$. It can be then linearly fitted by $g\left(z\right)\approx g_{1}z+g_0$ with ${\displaystyle g_1\approx -2.1\mathrm{m}{\mathrm{m}}^{-2}}$ and ${\displaystyle g_0\approx 1.2\mathrm{m}{\mathrm{m}}^{-1}}$ . Subscribe this linear approximation in Equation S7, we get:

Now, we compare Equation S10 to the standard OU model given by $dx=-\lambda xdt+ϵdW$. Equation S9 can be taken as an OU-type process with $-\chi g_1$ as the reversion rate $\lambda$ and the mean position deviated from origin to $z_0$:

where the mean position $z_0$ is the steady-state position of its ensemble average $\langle z\rangle$:

The original OU model describes an active particle motion under a linear force field around $x=0$. Since Equation S10 can be taken as a generalized OU model, we can thus consider our bacteria in the migration group as an active particle under an effective force field of $\chi g_{1}z+\left(\chi g_0-V_G\right)$ . Integrating this force field over $z$, we get an effective potential well $U\left(z\right)$ :

The $U_0$ is an integral constant controlled by boundary condition, which was set by $U\left(\infty \right)=0$ in this article. The effective potential well $U\left(z\right)$ generated by $U\left(z\right)=\int \left(\chi g\left(z\right)-V_G\right)dz$ from the particle-based simulation and Equation S17 are compared in Figure 2B.

Using a variable substitution of $\tilde{x}=z-(\chi g_0-V_G)/g_1$ , and applying the general solution of the standard OU model of $\tilde{x}\left(t\right)={\tilde{x}}_0{e}^{-\lambda t}+{\int }_0^tϵdW\cdot e^{-\lambda \left(t-t^{`}\right)}d{t}^{`}$ (Cherstvy et al., 2018), we get the mean square displacement: $MSD\equiv \langle {\left(z\left(t\right)-z_0\right)}^2\rangle ={\langle z_0\rangle }^2{\left(1-e^{-\chi g_{1}t}\right)}^2+\frac{{ϵ}^2}{2\chi g_1}\left(1-e^{-2\chi g_{1}t}\right)$ . The standard distribution $\sigma$ of the density distribution is the limit of MSD:

The trend of both mean positions (Figure 2C) and width (Figure 2D) was successfully predicted by analytical solutions of the OU-type model Equation S15 and Equation S18.

Moreover, the OU-type model Equation S14 has an equivalent Fokker-Planck equation that describes the probability density function of particles on $z$, $P(z,t)$:

At $t\to \infty$, the probability density function reaches the steady stat $P\left(z\right)$, where the density profile follows $0=\frac{d}{dz}\left(\left(\chi g_1{z}_i+\left(\chi g_0-V_G\right)\right)P\left(z\right)\right)+D\frac{d^{2}P\left(z\right)}{d{z}^2}$ . Solving this equation with boundary conditions of ${\displaystyle P\left(z=\pm \mathrm{\infty }\right)=0}$ and ${\displaystyle \frac{\mathrm{\partial }P\left(z=\pm \mathrm{\infty }\right)}{\mathrm{\partial }z}=0}$ , we get the steady stat density profile $P\left(z\right)=e^{-\frac{1}{D}(\frac{1}{2}\chi g_1{z}^2+\left(\chi g_0-V_G\right)z+C_0)}$ with $C_0$ is the integral constant that allows ${\int }_{-\infty }^{\infty }P\left(z\right)dz=1$. The exponent of this density profile is position-dependent that drops in parabolic form as $z\to \infty$. Compare to the front of Fisher wave, where the exponent of the density profile decreases linearly with $z$ ($P\left(z\right)=e^{-Cz}$), we know that the density profile of a chemotaxis migration wave is more compact than a Fisher wave.

In order to simulate more details of bacterial group migration in a capillary, we use a previously established chemotactic pathway integrated agent-based simulation. This model integrated bacterial chemotactic signaling pathway to determine the switch rate of motors, the conformation changes of flagella, as well as the competition of multiple flagella.

We first simulated 80,000 cells evenly distributed in four phenotypes of different receptor gain $N$ with consumption of chemoattractant. The cells were simulated as described in a previous paper (Saragosti et al., 2011), while the chemoattractant field S was calculated at each time step (${\displaystyle \mathrm{\Delta }t=0.01\mathrm{s}}$) according to Equation S8. In a 3D channel of 0.2 mm×0.01 mm× 10 mm, space was discretized to grid of 2.5 µm×2.5 µm× 2.5 µm . The evolution of the attractant during each time step was calculated by the sum of the consumption of every cell in the neighbor grids weighted by their effective diffusion distance. The consumption rate was fitted for reasonable migration speed. Detailed methods were described in a previous paper (Xue and Othmer, 2009). The perceived attractant of each cell was then linearly interpolated from the closest grids of the attractant field.

Simulation starts by setting all cells in one end of the channel (${\displaystyle x=0}$), randomized in y and z directions and setting uniform chemoattractant concentration ${\displaystyle S_0=200\mu \mathrm{M}}$ within the channel. Flip boundary condition was used for all three directions. After 20 min simulation, the bacterial population from stable migration groups (Appendix 1—figure 6A) tracks with cellular run-tumble stat was recorded from 30 to 32 min with time step of 0.1 s. Using the group velocity $V_G$ of this period as the speed of moving coordinate $z=x-V_{G}t$, density distribution of subgroups of different receptor gain $N$ was plotted to show spatial ordering of subgroups (Appendix 1—figure 6B). The tracks were analyzed with exactly the same method as the experimental tracks (see Materials and methods). The expected drift velocity $V_D\left(z\right)$ of all subgroups decreases from back to front and was also spatially ordered (Appendix 1—figure 6C).

The particle-based simulations (Section 4) have already shown that the chemoattractant consumption can be decoupled from bacterial motions. To simplify the analysis, we used the same non-consumable moving gradient $g\left(x,t\right)$ described in Figure 2A. Using the preset gradient, we simulated 100,000 bacteria of four evenly distribution phenotypes of different receptor gain $N$ or adaptation time $\tau$ or basal CheY-p level $Y_{p0}$ . The simulation was performed in 3D with reflection boundary on ${\displaystyle x=0\mathrm{m}\mathrm{m}\mathrm{\&}\mathrm{x}=20\mathrm{m}\mathrm{m}}$ and with free boundaries in the other two directions ($y,z$). Initially all cells were placed on ${\displaystyle x=2\mathrm{m}\mathrm{m},\mathrm{y}=\mathrm{z}=0\mathrm{m}\mathrm{m}}$. In each time point with step $\Delta t=0.01s$, the internal states of every bacterium were updated according to their perceived gradient indexed by $g\left(x_i\right)$ with spatial precision ${\displaystyle \mathrm{\Delta }x=0.001\mu \mathrm{m}}$. And then the bacterial moving stat (run or tumble) was updated accordingly, which results in movement to the next time point. Most parameters used in the simulation follow previous paper (Saragosti et al., 2011), while the changed parameters were the upper limit of chemoattractant sensing ${\displaystyle K_{on}=1000\mu \mathrm{M}}$ (Fu et al., 2018); the lower limit of chemoattractant sensing ${\displaystyle K_{off}=1\mu \mathrm{M}}$ (Si et al., 2012); the diffusion coefficient of chemoattractant ${\displaystyle D_s=1000\mu {\mathrm{m}}^2/\mathrm{s}}$ (Fu et al., 2018); the run speed of bacteria ${\displaystyle v_0=30\mu \mathrm{m}/\mathrm{s}}$ (Keller and Segel, 1971b); the adaptation time ${\tau }_M=1s$ (Waite et al., 2016); the motor number was set 5 as in Sneddon et al., 2012; while the basic CheY-P level $Y{p}_0$ = ${\displaystyle 2.77\mu \mathrm{M}}$, which corresponds to $T{B}_0\approx 0.27$. After 40 min of simulation, bacterial tracks were recorded with 10 fps for 20 min.

In order to compare the agent-based simulation results to the analytical OU-type model, we need to estimate the chemotactic ability $\chi$ from the parameters of chemotaxis pathway. Considering an simple case where perceived gradient of chemoattractant $g$ is a fixed value, the averaged drift velocity of a group of bacteria $V_D$ can be estimated by the parameters of bacterial chemotaxis pathway (Dufour et al., 2014): $V_D\approx \frac{{\tau }_{R0}^{`}}{1+{\tau }_{R0}/{\tau }_M}\frac{\left(1-T{B}_0\right)v_0^{2}Ng}{d}$ . In this equation, ${\tau }_{R0}$ is the run time in homogenous environment, ${\tau }_{R0}^{`}=d{\tau }_{R0}/df$ with $f$ is the free energy of the receptor, ${\tau }_M$ is the adaptation time, $d$ is the dimension (e.g. d is 3 in this case), $T{B}_0$ is the tumble bias in homogenous environment, and $N$ is the receptor gain.

On the other hand, given the expression of $V_D$ , comparing the two expressions of expected drift velocity in fixed gradient, we can deduce:

Using the expression Equation S20, we can deduce the required receptor gain $N$ from $\chi$, which is the case shown in the main text in Figure 3.

To investigate the bacterial behavior in moving gradient, we have simulated bacteria with $\chi$ ranging from 0.15 to 0.30 corresponding to receptor gain $N$ ranging from 14.5 to 25.4. Related results were shown in Figure 3 and discussed in the main text.

Receptor gain is not the only parameter that affects the chemotactic ability. We have also varied the adaptation time ${\tau }_M$ and basal CheY-p level $Y_{p0}$ to investigate the effect of these parameters on bacterial behaviors with a moving chemoattractant profile of larger slope of perceived gradient, ${\displaystyle g_1=-2.5\mathrm{m}{\mathrm{m}}^{-2}}$ . For both cases, the density profile $\rho \left(z\right)$ spatially ordered according to the variant parameter, and the expected velocities for call parameters $V_d\left(z\right)$ decrease from back to front. These results show that bacteria perform mean reversion flow for all these parameters and were spatially ordered according to phenotypes.

For the case of basic CheY-p level $Y_{p0}$ ranging from 2.70 µM to 2.85 µM, the basic tumble bias $T{B}_0$ varies from 0.07–0.19, and the chemotactic ability $\chi$ decreases with $Y_{p0}$ from 0.16 to 0.25 mm²min^-1 . The mean positions of subgroups increase with $\chi$ so that it decreases with $Y_{p0}$ . Unlike the receptor gain $N$, the basic tumble bias also increases the diffusion coefficient $D$. Analogized to the OU-type model, varying $Y_{p0}$ changes the reversion rate $\chi g$ and the noise level $ϵ=\sqrt{2D}$ at the same time. Thus, the reversion rate decreases with $Y_{p0}$ and the width of the density profile $\sigma$ increases with the reversion rate $r$ (Appendix 1—figure 8A–D).

For the case of adaptation time ${\tau }_M$ ranging from 1 to 10 s, mean positions and width of the density profile show non-monotonic dependence on ${\tau }_M$ . This phenomenon shows that adaptation time has a trade-off effect on bacterial chemotaxis. As Frankel et al., 2014 have discussed, the increasing adaptation time enforces chemotactic ability, while it weakens the sensitivity of bacteria to the chemoattractant gradient changes. Such dual effect was also reflected on the reversion rate defined by the slope of $V_D$ (Appendix 1—figure 8E–H).

As discussed in the main text, the reversion flow of individuals in the migrating group relies on a decreasing driving force from back to front. In the case of bacterial chemotaxis, such decreasing force comes from the gradient of perceived signal of chemoattractant, which is consumed by all the agents. Such mechanism can be generalized to the systems of trail-following migration (e.g. ants), where the signal is secreted by agents instead of being consumed. In this case, the signal concentration $S_s(x,t)$ decreases from back to front, because more agents have passed the back than front, thus have secreted more signal on the back. Such dynamics can be modeled by similar Langevin equations and signal accumulations as discussed in Section 1:

where ${\displaystyle \gamma =3\ast {10}^{-13}\mu \mathrm{M}\cdot {\mathrm{s}}^{-1}\cdot \mathrm{c}\mathrm{e}\mathrm{l}{\mathrm{l}}^{-1}}$ is the secretion rate of the signal $S_s$ and $\beta =0.001s^{-1}$ is the decay rate of it. Other terms and parameters are identical to equations in Section 1.

Then, we solve Equation S21 numerically by applying 10⁶ particles with normally distributed response ability ${\chi }_i$ of mean value of 1 and s.d. of 0.3 (Appendix 1—figure 10C). Results show that a coherent migrating group is formed spontaneously with constant migration speed ∼6.7 µm/s (Appendix 1—figure 10B). In the moving coordinate, the ‘effective global force’ $S_s\left(z\right)$ decreases from the back to front as predicted. The particles with different response ability ${\chi }_i$ are ordered arranged (Appendix 1—figure 10D), while particles with larger χ locate more in the front.

Appendix 1—figure 1

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Appendix 1—figure 2

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Appendix 1—figure 3

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Appendix 1—figure 4

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Appendix 1—figure 5

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Appendix 1—figure 6

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Appendix 1—figure 7

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Appendix 1—figure 8

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Appendix 1—figure 9

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Appendix 1—figure 10

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The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

The authors acknowledge C Liu, for sharing E. coli strains and plasmids; S Huang for help with the microfluidics; T Emonet for help with single-cell tracking and agent-based simulations; T Hwa, F Jin, L Luo, J Long for insightful discussions. This work was financially supported by the National Key R&D Program of China (Grant No. 2018YFA0903400), National Natural Science Foundation of China (Grant No. 32071417), CAS Interdisciplinary Innovation Team (Grant No. JCTD-2019-16), Guangdong Provincial Key Laboratory of Synthetic Genomics (Grant No. 2019B030301006), Strategic Priority Research Program of Chinese Academy of Sciences (XDPB1803) to XF, National Natural Science Foundation of China (Grant Nos. 11804355 and 31770111) to YB.

© 2021, Bai et al.

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