Abstract
Flagellated bacteria move collectively in a swirling pattern on agar surfaces immersed in a thin layer of viscous “swarm fluid”, but the role of this fluid in mediating the cooperation of the bacterial population is not well understood. Herein, we use gold nanorods (AuNRs) as single particle tracers to explore the spatiotemporal structure of the swarm fluid. We observed that individual AuNRs are transported in a plane of ~2 μm above the motile cells. They can travel for long distances (>700 μm) in a 2D plane at high speed (often >50 μm2/s) without interferences from bacterial movements. The particles are apparently lifted up and transported by collective mixing of the small vortices around bacteria during localized clustering and de-clustering of the motile cells, exhibiting superdiffusive and non-Gaussian characteristics with alternating large-step jumps and confined lingering. Their motions are consistent with the Lévy walk (LW) model, revealing efficient transport flows above swarms. These flows provide obstacle-free highways for long-range material transportations, shed light on how swarming bacteria perform population-level communications, and reveal the essential role of the fluid phase on the emergence of large-scale synergy. This approach is promising for probing complex fluid dynamics and transports in other collective systems.
Introduction
Bacterial swarming is collective migrations of flagellated cells across agar surfaces with swirling patterns1–4. During swarming, fast moving bacteria are trapped in a thin layer of viscous fluid called “swarm fluid”5–8. The layer of fluid is only micrometers-thick and even thinner at the swarm edge, and has Reynold number as low as 10−5. The swarm fluid can trap surfactants, modify the surface tension of liquids, supports the flagella operation, and carry nutrients or other signaling molecules9–11. A fundamental challenge is to understand the relationships between bacteria cells and the fluid medium5,12,13. Biologically, bacteria can only “sense” changes in the adjacent fluid environment to coordinate their behavior. For instance, in quorum sensing, the cells respond to the accumulations of signaling molecules dispersed in the fluid through gene regulations14,15. The bacteria also react to chemical gradients to control their short-time run lengths through rotating the flagellar16,17. Theoretical work such as the Vicsek Model18,19 is based on the assumption of collisions and alignments of single bacterium with its neighbors in short-range. To consider hydrodynamic interactions between motile cells in the context of large-scale collective dynamics5,20–24, some physical models treated the swarm fluid as a continuum entangling the bacteria phase and fluid together20,24, in which the bacterial community are treated as discrete individual self-propelled particles surrounded by an incompressible and inseparable fluid20,25.
Yet experimental evidences based on single particle tracking (SPT) have shown that the fluid environment is complicated and heterogeneous. SPT has long been used as a powerful tool to investigate complex system26,27. Previous studies have used micro-particles to track the dynamics of bacterial fluid28,29. Wu et al. revealed an intensive matter transfer flow (rate v= 8 μm/s) in the leading edge of the swarm using 1~2 μm micro-bubbles30. Zhang et al. found 0.2 μm MgO particles dropped on the liquid-air interface (±20 μm from the swarm front) only diffused normally within a small region (~ 4 μm2, v= 0.9 μm/s) as the swarm front approaches31. Be’er et al. found 0.5 μm MgO particles exhibited superdffusive (v= ~ 9 μm/s) on the upper surface (within 100 μm from the leading edge)32. However, since the thickness of the fluid layer is comparable to the dimension of the particles, micron-sized tracers could either have obvious collisions with the motile cell bodies or be trapped at the liquid-air interface, making the tracers’ motions incapable of reflecting the real motions of the pure fluid medium.
Here, using Bacillus subtilis as a model swarming system33, we introduced 40 × 84 nm gold nanorods (AuNRs) as tracers into the upper region of swarm fluid at near the edge of the swarming bacteria monolayer. Have good photostability and low cytotoxicity34–36, plasmonic AuNRs have been used as tracers for long-duration observation in biological studies with high temporal and spatial resolution. We observed that the nanotracers move continuously in 2D at ~2 μm above the bacteria layer, advected by the collective vortices resulting from dynamic clustering of bacteria. They could travel rapidly for a total distance of ~800 μm over the top of hundreds of bacterial cells, while having no direct physical contact with any of them. Their trajectories could be best described by a non-Gaussian superdiffusive model, Lévy walk. Compared to random Brownian motion, Lévy walks could lead to highly efficient long-range transport in complex fluid37–39. Therefore, mediated by the swarm fluid, the highly active bacteria community create a dynamically well-organized flow transport network above their own bodies, possibly providing a long-range communication avenue for the population. These discoveries may help to understand the information exchange in the bacterial population and the swarm dynamics. This single nanoparticle tracking method could be potentially applied to probe the fluid dynamics and matter transport in other collective systems.
Experimental results
Gold nanorods are lifted above the swarming bacteria
To study bacteria swarming, we chose the wild-type Bacillus subtilis 3610 strain as the model system33. After ~ 2 h inoculation the bacterial colony begins swarming on the wet agar surface and expands outward in swirling patterns, forming a dendrite-like edge pattern (Figure 1a). The characteristic scale of the swirls is about 10 μm and they last for about 0.2 s. To monitor the bacteria fluid by SPT, we used monodispersed 40 nm × 84 nm AuNRs (Supplementary Fig. 1). They were modified with SH-PEG moieties to make them electrically neutral, so that to prevent their adhesion to the negative charged bacterial cell surface. To introduce the AuNRs into the fluid without causing disturbances, we used a micro-sprayer to atomize the AuNR solution at about 20 mm above the biofilm. By producing an aerosol, the nanotracers naturally fall onto the surfactant fluid surface with minimum interference to the bacterial community. The gravity makes the AuNRs penetrate into the liquid rather than float on the air-liquid interface.
For imaging, we used an inverted darkfield microscope with a 20X long working distance objective and a color CMOS camera (Supplementary Fig. 2). The probing area is near the edge of the monolayer of swarming bacteria where the cells are most active. Under circular oblique illumination, the broadband absorption of the individual bacteria and the plasmonic scattering from single AuNRs allow them to be imaged simultaneously (Figure 1b). The former appears dark and the latter appears red since the plasmonic scattering maximum of the AuNR is 650 nm. Interestingly, different from the 3D Brownian motion in a passive homogenous solution, i.e. being out-of-focus stochastically, the single AuNRs are seemingly sliding on a 2D plane, with high lateral moving rates but little axial movement (Supplementary Movie 1). We could generally observe dozens of AuNRs in one frame and most of the single AuNRs could be tracked in focus continuously for tens of seconds until they run out of the boundaries of the view field. With the particles keep moving in-and-out, a considerable number of particles can be seen even after 1 hour of their addition.
Close examination of the single AuNR motions indicates that they move just like single bacterium but are relatively independent. They walk, jump or run stochastically among the bacteria dynamic clusters at fast but varying speeds. At any given time, however, the motion of the single AuNRs could hardly be attributed to the physical contact such as dragging, pushing or collision from a single bacterium. Although most bacteria appear to be locally clustered together, many AuNR trajectories are straight and elongated (Supplementary Movie 2) covering several hundred bacteria. Some particles are even able to cross half of the field of view, traveling a net distance of ~150 μm and a total distance of ~ 760 μm. Comparing the sequence of consecutive positions of single AuNRs with nearby single bacteria indicates that the trajectories of nanotracers and cells often cross but do not affect each other (Figure 1c). Therefore, the AuNRs must be transported in a thin 2D layer above the bacteria monolayer. By optical slicing using a reflective laser confocal microscope, the axial distance between the two layers is determined to be ~ 2 μm.
Control experiments indicate that the elevation of the single AuNRs results from the dynamic interactions between the nanotracers and the active bacteria. On one hand, not many tracers could exhibit similar behaviors as the 40 × 84 nm AuNRs. Particles of lower density such as 0.5 μm diameter polystyrene microbeads (Supplementary Fig.1, Movie 3) stayed afloat on the air/liquid interface and were almost motionless. Heavier particles such as 120 nm Au nanospheres (Supplementary Fig.1, Movie 4) swiftly penetrated the entire liquid and bacteria layer, sticking to and even sinking into the agar gel. On the other hand, only the swarming bacteria could raise and drive the nanotracers in the fluid. The AuNRs perform Brownian motions both in the culture medium having moving cells but without swarming (diffusion coefficient D= 3.6 μm2/s) and in the culture suspensions where the bacteria are filtered out of the medium (D= 2.5 μm2/s) (Supplementary Fig. 3). Moreover, if the bacteria are killed under UV light illumination and stop swarming (Supplementary Movie 5), the AuNRs could only undergo slow Brownian diffusion (D= 0.08 μm2/s) in the upper viscous solution, and no fast, long-distance transport could be observed (Supplementary Fig. 3).
The AuNRs are advected by the collective flow of the swarms
With no physical contact between the bacteria bodies and the nanotracers, the driving force for the AuNR moving in a 2D fluid layer above the colony can only come from the advection that is generated by the collective fluidic flow from the swirling cell populations. The bacterial swarm fluid has low Reynolds number, hence the motion of the particles almost completely depends on the local force7. Since the AuNRs appear moving stochastically, the local flow fields must be spatially heterogeneous and change frequently, which can only result from the dynamic clustering of multiple bacteria near the particle. Zhang et. al. has reported that bacterial colonies could spontaneously form tightly packed clusters40. Within the spatial scale of the clusters, bacterial movements are strongly coordinated, manifesting as a high degree of correlation between cell body orientation, direction of movement, and velocity. Moreover, the dynamic clusters could evolve, collide, fuse and then re-split into small clusters. Since the axial rotation of bacterium and its flagella swinging would generate a small vortex around its body, the dynamic clustering of multiple bacteria would inevitably result in collision, fusion and splitting of their small vortices, and provide the forces to drive the AuNRs. Therefore, rather than being carried by individual bacteria, the AuNRs are “hitchhiking” on top of dynamically formed bacteria clusters (Figure 1d, e). Notably, these cluster “vehicles” are not stable or long lasting. Whenever encountering “traffic jam”, a cluster will swiftly split and recombine with other individual cells. Meanwhile, the AuNR “passenger” will stop transiently and being transferred to another new “vehicle”, leading to their speed or direction variation. Hence, the transport of the single AuNRs is the result of synergistic effect of local bacterial vortices and is realized through heterogeneous hydrodynamic flow above the bacterial layer.
Particle image velocity (PIV) analyses41 were performed to qualitatively investigate the relationships between the two-dimensional AuNR motions and the swirls created by the swarms. From the local orientation and distribution patterns of the flow field velocity vectors, we noticed two localized, instant flow field patterns associated with the nanotracers’ motions. One is the “sticking mode” (Mode A, Figure 2a), where the nanotracers are trapped temporarily or “lingering” in a small area. The magnitude of the velocity vectors of the flow field at the AuNR position is close to 0, resulting from that the rotation of the bacterial swirls around the particle cancel out each other. The other is the “jetting mode” (Mode B, Figure 2b), where the nanotracers “jump” large distances in regions between the sticking area. The sum of the flow field velocity vectors and the joint forces of the adjacent bacterial swirls all point to the same direction. Sequential evolution of the velocity vector patterns indicates that the motion of the single AuNRs is alternating rapidly between the two states (Supplementary Movie 6, 7). The “jump-and-linger” mode of the nanotracers appears similar to the “run-and-tumble” mode of the bacterium, making the single AuNRs behave just like active single bacteria42, though with different underlying mechanisms.
The motions of AuNRs are superdiffusive and non-Gaussian
The phenomenological observation of the AuNR transports in the fluid layer above the bacteria community is further verified through single particle tracking (SPT) analysis. Hundreds of AuNR trajectories are obtained with time duration ranging from 0.02 s to 40 s and traveling distances ranging from 0.3 μm to 800 μm. To get the basic features of the movement, the speed, step size, the motion direction and the difference in motion direction of the AuNRs are examined. We find that the AuNRs move at a speed about 16 μm/s in average (range from 0.03 μm/s to 146 μm/s) and the average step size is about 1.5 μm (0.016 s). As can be seen from Figure 3a and 3b, the single AuNRs at any given time could randomly point to any direction, but the distribution of the variation of the particle directions over a constant time period Δt is centered at zero degree from Δt = 1 to Δt = 60 (0.016 ~ 0.96 s), indicating that the particles tend to keep their directions within short time durations. The mean square displacement (MSD) analyses show that the AuNRs are performing superdiffusion with the ensemble-averaged scaling exponent α of 1.38 and a diffusion coefficient of ~ 21 μm2/s (Figure 3c), nearly 300 times larger than normal diffusion in the pure surfactant liquid. The probability density functions (PDFs) show that the distributions of the scaled displacements at different time intervals deviate from the Gaussian. They are both arrow-headed (with a large number of displacements close to zero) and heavy-tailed (decaying more slowly than Gaussian at long displacements) (Figure 3d), suggesting a high probability for the AuNRs to move either at very small steps or at large steps.
Careful observations of representative AuNR trajectories indicate that there is a weak negative correlation (with a coefficient of −0.33) between the local speed of the particle and the variation degree of the particle direction (Figure 3e, f). The particles tend to keep the original direction when moving fast and change their directions of motion more frequently when the movements slow down, leading to the trajectory appears to be made up of many large jumps with a lot of short pauses in between. To identify the local regions with different dynamic properties, the instant motion states in the trajectory are revealed by trajectory segmentation using temporal slicing windows. By calculating the local scaling exponent from the MSDs, we found that there exist two states, superdiffusive state (jumps) with α > 1 and Brownian state (lingering) with α ≤ 1 within the same trajectory (Figure 3g). To qualitatively estimate the time scale of the motion states, we tune the size of temporal slicing windows. The sub-trajectories are mostly superdiffusive when the window size is longer than 10 frames. However, when the window size is shortened to 10 frames or less, the local α values exhibits the largest variation (Figure 3h) in the range of α ≤ 1, indicating that the Brownian-like movement is more prominent in a time window of less than 0.2 s, which is consistent with the characteristic time of the bacterial swirls.
Long-range transportations of nanotracers can be described by Lévy walk
Taken together, from both PIV and SPT analysis, we can develop such a picture: during bacteria swarming, individual bacteria create small vortices around their bodies in the surfactant fluid; the coordination of local vortices creates a spatiotemporally heterogeneous flow field above the bacterial layer, allowing both advected rapid transporting and elevated transient trapping of single nanotracers. Interestingly, while the alternating frequency of the AuNR motion state, or the characteristic time of the bacterial cluster formation and dispersion, is not likely to be greater than ~ 0.2 s, the particles have a tendency to keep their walking direction much longer than that. That is, the stochastic switches between the rapid jump and Brownian rest of AuNRs eventually result in their transportation with continually reinforcing directionality.
It has been reported that such directionality persistence, characterized by alternating jump and lingering with superdiffusive and non-Gaussian statistics, often fits with the Lévy process. In particular, the Lévy walk (LW) model describes that the particles walk large distances at a constant speed with local turnings steps. The walking distances mathematically converge to the power-law distribution with no characteristic scales37. To test if the single AuNRs were doing LW, “turning points” in trajectories are defined with an instant turning angle larger than a certain threshold43 (Supplementary Fig. 4, Note 1). Figure 4a shows a typical long trajectory with a total time of ~ 40 s and total travel distance of 760 μm marked with turning points in an angular speed threshold of 60 rad/s. The distances between two consequent turns are considered as a “flight”. We found that the particle moves at a constant speed of ~16 μm/s through the cumulated length of “flights” as a function of time (Figure 4b). The obtained flight-lengths can be fitted with a power law distribution in the tail by using maximum likelihood estimation. In addition, when choosing different turning-angle-threshold (30, 45, 60 rad/s), the power law index has a variation from −2.38 ~ −2.69 (Figure 4c, Supplementary Note 2). The possibility of matching exponential decays is ruled out by comparing with the Akaike’s information criterion weights44. The power-law model is favored with the AIC weights of 0.99. Another evidence is that the power law tail of the velocity correlation function decays as Δ t−δ with δ = 0.50 in average45–47 (Figure 4d, Supplementary Note 3). According to the previous study, δ should equal to 2 − α, when α = 1.38 is the scaling superdiffusive exponent calculated from MSD. Therefore, we can conclude that the motion behavior of the AuNRs is consistent with the LW model. It has been reported that the motion of single bacteria in B. subtilis colony is also LW, however, the statistics is slight different48. The superdiffusion exponent of the active bacteria is 1.66, indicating more directionality than the nanotracers. Moreover, to get the power law index value of −2.5, the turning-angle-thresholds of the bacteria (5, 10, 15 rad/s) are much smaller than those of the nanotracers, indicating the AuNRs were making turns more dramatically. Therefore, the LW behavior of the AuNRs is similar to the single bacteria motions in the swarm but statistically different.
Discussion and Summery
In bacteria swarming systems, how the multicellular collective patterns emerge from a unicellular structure and how information exchange is performed in a swiftly proliferating and expanding population are still a critical issue to explore. With continuous short-range collisions and alignments, it is hard to imagine the existence of certain special “messenger” bacteria capable of carrying and spreading the information rapidly across a wide space. It must be the viscous fluid environment of the swarms that not only enables the self-propelled bacterial cells to spin their flagella to generate thrusts, but also serves as the medium to carry the nutrients and signals from one location to the other in the colony, e.g. the signaling molecules involved in quorum sensing. A previous study deposited MgO microparticles on the surface of the swarm fluid, and superdiffusive trajectories were obtained32. Nevertheless, probably due to the large size, no long-range transportation of the microtracers was observed that was associated with the coordination mechanism at large spatial scales.
In this work, the nanotracers are far smaller than single bacteria in size and are lifted above the swarming bacteria. They are neither being pushed by nor “riding” on any single bacterium. Rather, the AuNRs are traveling in 2D in the surfactant fluid layer due to the local synergy of the vortices around the bacteria, and exhibit non-Gaussian superdiffusive characteristics with alternating jumps and lingering, consistent with LW statistics (Figure 5a, b). Under the Lévy motion mode, the AuNR tracers appear no longer passive and could actively and efficiently travel over long distances across multiple bacterial clusters (Figure 5b). Compared with the LW statistics of individual bacteria showed by Ariel et al.48,49, the LW of single AuNRs have smaller super-diffusion exponents and larger truncating angular speeds, indicating that the AuNRs are moving more random and less directed than the bacteria in the swarms. This could be attributed to the fact that the bacteria are consuming energy and driven by active flagella rotations, but the AuNRs are passively transported by the swarm fluid. Moreover, the bacteria and the AuNRs are spatially divided into two phases with different environments. The single bacteria move in densely packed populations and often encounter physical hindrance from others over short distance. However, the AuNRs are traveling in an obstacle-free fluid layer above the motile cells. The two similar but different LW transports suggest that the collective movement of high-density bacteria cells produced simultaneously two distinct biological mechanisms. The LWs of the single bacteria may have advantages in foraging or avoiding the hazards, which are associated with continuous adaptation to local environments or even “conflict” with neighbors48,49. On the other hand, the LWs of AuNR tracers reveal efficient, long-range transportations in the upper fluid medium, which could facilitate communications such as circulations of metabolites and nutrients, fluid mixings for oxygen dissolution, and transports of signaling molecules over long distances. In other words, the two LWs represent the simultaneous near-range competitions and large-scale cooperation between individuals within the same bacteria community, respectively. Interestingly, from the viewpoint of a single bacterium, it should only be able to sense the change of its surroundings and develop its own path through the gaps between neighbors, but would not “foresee” the formation of LW “highways” for efficient transportations and collaborations. Nevertheless, the collective motions of all the individuals lead to the emergence of a long-range communication LW network that “link” their activities altogether.
Lévy patterns have been identified in a wide range of organisms, from cells, birds, to even human hunters37,44,50. Most studies consider LW an optimal search strategy being evolved during adaptive interactions with heterogeneous environments. The LW of passive AuNRs, however, is the result of collective local mixing of vortices around the swarming bacterial bodies. Since the vortices have no clear physical boundaries and are short-lived due to collisions and mixings with each other, our results support a recent hypothesis that the LW can emerge spontaneously and naturally from innocuous responses to the surroundings51. Since individual bacteria are also performing LW, to our knowledge, this is the first report of a second Lévy network in a complex living system. The collective behaviors of bacteria produce fluid flows that are physically and functionally separated from the cell bodies. The coincidences and mutual reinforcements of two Lévy networks, possibly one for optimizing the near-range foraging and the other for optimizing the long-range communications, apparently lead to higher-level surviving and adaptation to the environment. Similar mechanisms may constitute the underlying physics of the emergence of large-scale coordination in nature such as flocking of birds, schooling of fishes and collective spreading of cancer cells. Nevertheless, by using nanoparticle tracers, we expect that new collective properties on bacterial and other collective motion systems could be discovered.
Experimental methods
Bacteria strain and colony growth
Wild-type B. subtilis strain 3610 (purchased from China General Microbiological Culture Collection center) is a gram-positive bacterium with a rod-shaped body and multiple flagella, which generates a propelling force in the direction of its body. They swim with a mean speed about 10 ~ 30 μm/s in a thin liquid film about micrometers thick on the substrate. In our experiments, the bacteria have mean dimensions of 0.7 μm × 4 μm. Colonies were cultured on soft LB agar substrates (10 g/L tryptone, 5 g/L yeast extract, 0.5% agar, all purchased from Bacto Difco, 5 g/L NaCl, purchased from Beijing Chemical Works). For inoculation, 5 μL of B. subtilis overnight culture (37 °C, 200 rpm, OD 650 = 0.5) is placed on the center of the agar. The inoculated gel is stored in an incubator at 30 °C. After a lag time of 2 h, the colony starts to expand outward with a speed about 1 cm / hour. After about 2 h, when a colony reached a radius about 2 cm, it could be observed under microscope at the edge of colony where the bacteria almost swarms in a monolayer.
Tracer materials and the microspray technology
PEG-modified 40×84 nm gold nanorods were purchased from NanoSeedz and were diluted 20 times before use. Carboxyl modified polystyrene microspheres (0.5 μm) were purchased from Dae technique. Gold nanospheres having size about 120 nm were synthesized using a method reported previously52. For microspray without disturbing the bacteria colony, we added 50 μL of the chosen particle solution into a high-frequency vibrating atomizer (5W, 5V) and then sprayed it out to a direction parallel to the surface of the colony. The small microdroplets would slowly fall onto the surface of the colony. After that, we immediately took the specimen to the microscope for observation.
Imaging technique and data analysis
The specimen was placed under an inverted microscope (Nikon Ellipse Ti-U) equipped with a 100 W halogen tungsten lamp, a dark-field air condenser, a 20X long working distance objective lens and an Olympus DP74 color CMOS camera with imaging rate of 60 fps. To localize the precise z-position of the AuNR nanotracer layer relative to the bacteria layer, a confocal microscope equipped with a 20X long working distance objective lens and a 647 nm laser (Nikon A1) was utilized. Single nanoparticle tracking was performed using Image J and a code written in IDL. Other data analysis was performed using MATLAB and Origin. The segmentation of trajectory using the “angle method”43, the quantification of its power-law characteristics, and the evaluation of the power-law model are described in detail in the Supplementary notes. The flow field was measured by using a GUI-based software called PIVlab based on discrete Fourier transform (DFT)53.
Author contributions
Y. H. designed the experiments and wrote the paper; J. F. designed and performed the experiments, analyzed the data, and wrote the paper; Z. Z. analyzed the data and wrote the paper; X. W. performed the experiments and analyzed the data. J. X. analyzed the data.
Conflict of interest statement
The authors declare no competing interests.
Supplementary notes
Supplementary Note 1: The angle method
The long trajectory of the AuNR is segmented to several parts by the “angle method” modified from Turchin1, which is mainly based on the identification of the reorientation events during single particle walking. As illustrated in the figure above, multiple steps experienced in n time intervals are aggregated into a single “flight” segment, if the turning angle θ between the starting position and the end position is smaller than a predefined angle threshold value θmax. To determine the threshold value θmax, rather than stick to those reported in the literature, we test a wide range of angles from small to large and choose those angles that allow the resulting segment-lengths to converge to the power-law statistics (see discussions below). For the long single AuNR trajectories obtained in this study, it is discovered that whether the θmax is too small (e.g. 5° or 10°) or too large (e.g. >100°), the “flight” segments are either over sampled or under sampled, leading to deviations during the fitting.
Supplementary Note 2: Lévy walk models and the power law fitting
To characterize the power law statistics of the Lévy walk model, we used a continuous distribution2,3, where α is a constant parameter of the distribution known as the exponent or scaling parameter. The scaling parameter typically lies in the range 1 < μ ≤ 3, although there are occasional exceptions. Since the distribution diverges at zero, so there must be a lower bound (xmin > 0) on the power-law behavior. The approach uses log-likelihood fitting methods from the tail started at the lower bound xmin that is determined in an automated way by a software developed by Aaron et al.2 For each possible choice of xmin, the index value a is estimated via the method of maximum likelihood, and the Kolmogorov-Smirnov goodness-of-fit statistic D is calculated. The selected estimation of xmin is the value that gives a minimum D over all values of xmin. The selection of the power law model over the exponential model is justified using the Akaike weights that calculated the relative likelihoods of either model. See the detailed expression form and calculating methods in the reference article4.
Supplementary Note 3: The auto-correlation function of velocity for LWs
Let v(t) denotes the velocity at time t of a particle following a LW. In the original LW model5, particles move at constant speed between random reorientations. We defined the velocity auto-correlation function6 as where 〈·〉 denotes averaging over all times t in independent samples of infinite trajectories. Assuming waiting times between reorientation events have a density ψ(τ) with a power-law tail, ψ(τ) ~ τ−β+1, then we have C(Δt)~t−(β−1)3,5,7. Here, α=1.38, β = 3 − α =1.62 (β is the Levy stable parameter with α + β = 3). Thus, the C(t) should have a tail with about 0.6. We note that because of the “jump-and-linger” two-state motion of the AuNRs, it is normal to have a small deviation.
Supplementary movies
Supplementary Movie 1. Individual AuNRs move on the upper layer of the swarming bacteria. The single AuNRs appear as red points. The scale bar is 15 μm.
Supplementary Movie 2. Trajectories of the single particle tracking results of multiple AuNRs. The scale bar is 10 μm.
Supplementary Movie 3. Polystyrene microbeads (0.5 μm) adhere on the air/liquid surface of the fluid layer of swarming bacteria.
Supplementary Movie 4. Gold nanospheres (120 nm) sink into the bottom of the bacteria fluid layer, rocking and colliding with the bacteria locally in a small area.
Supplementary Movie 5. AuNRs move in a Brownian way in the fluid layer after the bacteria have been exposed to the UV light illumination for a long period of time and stopped swarming.
Supplementary Movie 6. Particle image velocity (PIV) vectors overlaid on the raw image. The velocity vectors are shown in green arrows and the interpolations are shown in orange. The yellow circle marks the position of the AuNR being carried across the field. The moving single bacterium appears in long brown rod. The scale bar is 10 μm.
Supplementary Movie 7. PIV mapping overlaid on the PIV vectors, where the clockwise spin is shown in blue shades and the counter clockwise spin are shown in red shades. The scale bar is 10 μm.
Acknowledgements
The authors are grateful to Dr. Shuchun Liu (Institute of Microbiology, Chinese Academy of Sciences) for providing the bacteria strains. This work was supported by the National Natural Science Foundation of China with grant numbers of 21425519, 91853105 and 11574222.