Abstract
Ranging from subcellular organelle biogenesis to embryo development, the formation of self-organized structures is a hallmark of living systems. Whereas the emergence of ordered spatial patterns in biology is often driven by intricate chemical signalling that coordinates cellular behaviour and differentiation1,2,3,4, purely physical interactions can drive the formation of regular biological patterns such as crystalline vortex arrays in suspensions of spermatozoa5 and bacteria6. Here we discovered a new route to self-organized pattern formation driven by physical interactions, which creates large-scale regular spatial structures with multiscale ordering. Specifically we found that dense bacterial living matter spontaneously developed a lattice of mesoscale, fast-spinning vortices; these vortices each consisted of around 104–105 motile bacterial cells and were arranged in space at greater than centimetre scale and with apparent hexagonal order, whereas individual cells in the vortices moved in coordinated directions with strong polar and vortical order. Single-cell tracking and numerical simulations suggest that the phenomenon is enabled by self-enhanced mobility in the system—that is, the speed of individual cells increasing with cell-generated collective stresses at a given cell density. Stress-induced mobility enhancement and fluidization is prevalent in dense living matter at various scales of length7,8,9. Our findings demonstrate that self-enhanced mobility offers a simple physical mechanism for pattern formation in living systems and, more generally, in other active matter systems10 near the boundary of fluid- and solid-like behaviours11,12,13,14,15,16,17.
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The data supporting the findings of this study are included within the paper and its Supplementary Information.
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The custom codes used in this study are available from the corresponding author on request.
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Acknowledgements
We thank A. Mukherjee and C. M. Schroeder (University of Illinois at Urbana-Champaign) and H. Berg (Harvard University) for generous gifts of bacterial strains. We thank Y. Wang (The Chinese University of Hong Kong) for help with computing resources and R. Zhang (HKUST) for helpful discussions on numerical simulations. This work was supported by the Ministry of Science and Technology of China (no. 2020YFA0910700), the Research Grants Council of Hong Kong SAR (RGC ref. nos. 14307821,14307822, RFS2021-4S04 and CUHK Direct Grants) and the National Natural Science Foundation of China (NSFC no. 31971182). Y.W. acknowledges support from New Cornerstone Science Foundation through the Xplorer Prize.
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H.X. discovered the phenomena, designed the study, performed experiments, developed the model, performed simulations and analysed and interpreted the data. Y.W. conceived the project, designed the study and analysed and interpreted the data. Y.W. and H.X. wrote the paper.
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Extended data figures and tables
Extended Data Fig. 1 Vortex lattice pattern at different length scales.
(a) Vortex lattice pattern at centimeter scale. Upper: phase-contrast image; lower: vorticity field associated with the phase-contrast image, computed based on the collective velocity field. The length and width of the panel is ~1.4 cm and 3.3 mm, respectively. The upper panel is a composite image obtained by stitching a sequence of 5 images taken at connected smaller windows; for each window, a short video lasting ~10 s was taken to compute the collective velocity field via PIV (Methods). The vortices appeared everywhere in the image except at the upper region; this region is close to the edge of the suspension film, where the fluid film has a greater thickness. The vortices at the rightmost region in the phase-contrast image (upper panel) are not apparent because the cell densities inside and outside the vortices are similar, but they can be visualized in the vorticity field (lower panel). Scale bar, 1 mm. (b–d) Phase-contrast image (panel c), collective speed distribution (panel d), and vorticity distribution (panel e) of the vortex lattice pattern shown in main text Fig. 1a–c. Panels b-d share the same scale bar (500 µm). (e–g) Enlarged view of the area enclosed by the box in panels b-d. The arrows in panel f represent the velocity direction of the local collective velocity. Scale bar, 200 µm.
Extended Data Fig. 2 Spatial correlation of the vorticity field and motion bias of isolated S. marcescens cells.
(a) Spatial correlation of the vorticity field in a quasi-2D dense bacterial active fluid. This panel is associated with main text Fig. 1a–c (Methods). Colorbar provided to the right is in arbitrary unit. Scale bar, 200 µm. (b) Motion bias of isolated S. marcescens cells swimming near a solid substrate. This panel shows the probability distribution of the signed curvature of single-cell trajectories near a solid substrate, with the mean curvature being −0.023 µm−1. The signed curvature of cell trajectories was computed based on 1-s segments of cell trajectories (positive: CCW; negative: CW). To obtain the distribution, cells were extracted from dense0 S. marcescens suspensions that displayed the large-scale ordered vortex lattice, diluted to an appropriate density, and deposited on freshly made 0.6% LB agar surface to form a quasi-2D dilute bacterial suspension drop. S. marcescens cells in the prepared suspension drop were tracked in fluorescence microscopy, while the environmental temperature was maintained at 30 °C with a custom-built temperature-control system (Methods).
Extended Data Fig. 3 Characterization of cell density distribution in the vortex lattice pattern.
(a) Probability distributions of fluorescence intensity inside (red) and outside (blue) the vortices in a representative vortex lattice pattern. Fluorescence intensity is a measure of cell density because cells were labelled by GFP. The averaged density difference between inside and outside vortices of the vortex lattice pattern obtained from the fluorescence intensity distributions is ~15%. (b) Spatial distribution of the fluorescence intensity in the vortex lattice pattern analyzed in panel a. The cell density inside vortices is slighter lower than or comparable to the regions outside the vortices. Scale bar, 200 µm. (c) Enlarged view of the areas enclosed by the boxes in panel b. Each box contains a vortex. Left (blue box in panel b): the cell density inside the vortex is slightly lower than outside; right (red box in panel b): the cell density inside the vortex is similar to outside. (d) Vorticity fields corresponding to the regions shown in panel c. Color bar represents the magnitude of vorticity while the arrows represent the direction of local collective velocity vectors. The vorticity fields for the two vortices are similar, suggesting that cell density difference is not necessary for vortex formation. Scale bars in panel c and d, 50 µm. (e) Fluorescence intensity plotted along the straight lines across the two vortices in panel c. The red and blue plots in panel e were plotted along the lines with the respective color in panel c. Data shown in the plots were normalized by the mean fluorescence intensity in regions outside the vortices.
Extended Data Fig. 4 Behavior of quasi-2D bacterial active fluids below the critical cell density for developing ordered vortex lattice.
(a,c) Spatial distribution of instantaneous vorticity computed based on collective velocity field of bacterial active fluids (panel a: 2.0 × 1010 cells/mL; panel c, 3.7 × 1010 cells/mL). The vorticity fields are plotted in the same manner as in main text Fig. 1c. Both panels show disordered spatial distribution of vortices. Panels a and c share the same scale bar (500 µm). (b,d) Apparent single-cell speed plotted against local polar order (as a proxy of local collective active stress) in bacterial active fluids (panel b: 2.0 × 1010 cells/mL; panel d, 3.7 × 1010 cells/mL). In panel b, single-cell speed is almost independent of local polar order (i.e., the mobility enhancement coefficient β ≈ 0). In panel d, single-cell speed is weakly correlated with local polar order (β ≈ 0.48). Error bars in panel b and d indicate standard deviation (N > 50 single-cell trajectories for each data point).
Extended Data Fig. 5 Suppression of bacterial motility by violet light illumination.
(a) Collective speed averaged over the entire field of view was plotted as a function of time. The vortex lattice was illuminated by violet light at T = 35.5 s (violet dashed line) and the motility of all cells in the entire field of view was suppressed by violet light illumination. This figure is associated with Video S3. (b) Tuning single-cell motility of S. marcescens via violet-light illumination. Single-cell motility refers to the intrinsic speed cells in an isolated environment; it is different from the apparent single-cell speed measured in dense suspensions where cell’s motion is affected by the mechanical environment. To measure single-cell motility, cells were extracted from dense S. marcescens suspensions that displayed the large-scale ordered vortex lattice, diluted to an appropriate density, and deposited on freshly made 0.6% LB agar surface to form a quasi-2D dilute bacterial suspension drop. S. marcescens cells in the prepared suspension drop were tracked in fluorescence microscopy through a 20x objective lens, and starting from T = 35.5 s the cells were continuously illuminated by 406 nm violet light (Methods). The environmental temperature was maintained at 30 °C with a custom-built temperature-control system (Methods). The speed of an isolated cell (i.e., the motility of the cell) at a specific time T was computed based on its trajectory tracked from (T-0.5) s to (T + 0.5) s; the speeds of isolated cells computed from (T-12.5) s to (T + 12.5) s were then averaged and taken to be the mean of single-cell motility at time T. Before violet-light illumination, the temporal variation of the speed of isolated cells is ~17%, suggesting that cells prepared from the same overnight culture should have a similar self-propulsion force. Data presented in this figure show that the average collective speed measured in the dense suspension (panel a) is proportional to single-cell motility (panel c) measured in isolated environment during violet light illumination. Therefore, the average collective speed is an appropriate proxy of single-cell motility.
Extended Data Fig. 6 Sporadic vortices in quasi-2D dense E. coli suspensions and in numerical simulations.
(a,b) Time-averaged collective velocity field (panel a) and vorticity field (panel b) experimentally measured in a dense E. coli suspension (Methods) showing sporadic and disordered vortices. Isolated E. coli cells have a mean speed of ~10 µm/s (versus ~25 µm/s for isolated cells of S. marcescens). Data presented in the two panels were averaged over a duration of 10 s. In panel a, arrows and colormap represent collective velocity directions and magnitude, respectively, with the colorbar provided to the right, in unit of µm/s. Panels a,b share the same scale bar, 500 µm. Also see Video S4. (c,d) Time-averaged particle speed spatial distribution (panel c) and vorticity field (panel d) showing sporadic and disordered vortices in particle-based simulation with a low particle activity (simulation parameters: f0 = 7 and β = 1.2). Data were averaged over a duration of 10 time units. Colormap in panel c represents particle speed, with the colorbar provided to the right. (e,f) Time-averaged collective velocity field (panel e) and vorticity field (panel f) showing sporadic and disordered vortices in continuum modeling with a low activity (simulation parameters: |S| = 2.6 and β = 2.0). Data were averaged over a duration of 10 time units and plotted in the same manner as panels a,b. The simulation parameter sets in panels c,d and e,f were chosen from the boundary between the active turbulence state and the vortex lattice state in Fig. 4c and Fig. 4f, respectively.
Extended Data Fig. 7 Emergence of vortex lattice pattern in the particle-based simulation.
Time sequence of spatial distributions of instantaneous collective speed (panel a), local polar order (panel b), and vorticity (panel c) during the emergence of ordered vortex lattice in the particle-based simulation. The magnitude of particle speed, local polar order and vorticity is indicated by the colorbars provided to the right of each panel. The vorticity field shown in panel c was plotted in the same manner as in main text Fig. 1c. The time stamp in each sub-panel indicates the elapsed time units in the simulation. Simulation parameters: particle activity f0 = 10 and mobility enhancement coefficient β = 1.2. This figure is associated with main text Fig. 4d,e.
Extended Data Fig. 8 Triplet-distribution function in particle-based (panel a) and continuum (panel b) simulations.
Solid lines were obtained by fitting the data into a sum of three Gaussian functions (Methods). Simulation parameters were identical to those used in main text Fig. 4d,e and Fig. 4g,h, respectively: particle activity f0 = 10 and mobility enhancement coefficient β = 1.2 in panel a; activity |S| = 3.5 and mobility enhancement coefficient β = 2.0 in panel b.
Extended Data Fig. 9 Characterization of signatures for the transition to stable vortex lattice state in particle-based and continuum modeling.
Panels a, c, e, g, i: results from particle-based simulations; panels b, d, f, h, j: results from continuum simulations. (a,b) Probability distributions of normalized particle speed (panel a) and of collective speed (panel b) during the emergence of vortex lattice pattern (in comparison with Fig. 3c). The colors represent time units elapsed in the simulations. Simulation parameters: f0 = 10 and β = 1.2 in panel a; |S| = 3.5 and β = 2.0 in panel b. (c,d) Peak ratio in collective speed distribution (Methods) plotted against β (in comparison with Fig. 3e). (e,f) Peak ratio in collective speed distribution plotted against the f0 (panel e) and against |S| (panel f) (in comparison with Fig. 3f). (g,h) Steady-state autocorrelation time of collective velocity (as a measure of vortex lifetime; Methods) plotted against β (in comparison with Fig. 3d). (i,j) Steady-state autocorrelation time of collective velocity plotted against f0 (panel i) and against |S| (panel j). In panel c-f, the solid circles represent data from bimodal speed distributions, while the empty circles correspond to zero peak ratio and represent data from unimodal speed distributions (i.e., the higher-speed peak has a height of zero). Simulation parameters: f0 = 10 and β from 0 to 1.0 for panels c,g; f0 from 2 to 10 and β = 0.8 for panels e,i; |S| = 3.5 and β from 0 to 2 for panels d,h; |S| from 2 to 3.5 and β = 2.0 for panels f,j. (k) Mean radius of vortices is negatively correlated with the mean angular velocity of particles \((\Omega ={\left\langle {\omega }_{i}\right\rangle }_{i})\) in a simplified particle-based model, where the time evolution of particle angular velocity ωi (Eq. [3] in Methods) is replaced by drawing its value from a distribution derived from Extended Data Fig. 2b. Error bars represent the standard deviation, N = 20. Simulation parameters: f0 = 10 and β = 1.2.
Extended Data Fig. 10 Emergence of collective motion patterns in the continuum model.
This figure shows the time sequences of vorticity field in numerical simulations of the continuum model with different mobility enhancement coefficient (β = 0, 0.5, 1.0 and 2.0 for panels a, b, c and d, respectively; activity |S| = 3.5 for all panels). The time stamp in each sub-panel indicates the elapsed time units in the simulation. The vorticity fields are plotted in the same manner as in main text Fig. 1c.
Supplementary information
Supplementary Information
Supplementary sections: 1, motivation for the modelling; 2, supplementary modelling results; 3, comparison between our 24 phenomena and other vortex patterns observed experimentally.
Supplementary Video 1
Ordered vortex lattice in quasi-2D dense bacterial active fluids consisting of S. marcescens cells. Left, phase-contrast video. Middle, spatial distribution of instantaneous collective velocity associated with the phase-contrast video. Arrows represent velocity direction; the colour map at right indicates velocity magnitude (μm s−1). Right, spatial distribution of instantaneous vorticity associated with the phase-contrast video. Vorticity value indicated on the colour bar to the right is normalized by the mean of absolute vorticity over the entire field. Positive and negative values of vorticity correspond to counterclockwise and clockwise rotation, respectively. This video is played at 50 frames s−1 with real elapsed time indicated in the time stamp. All panels share the same scale bar, 500 μm. Associated with Fig. 1a–c.
Supplementary Video 2
Emergence of ordered vortex lattice in quasi-2D dense bacterial active fluids consisting of S. marcescens cells. Left, phase-contrast video. Middle, spatial distribution of instantaneous collective velocity associated with the phase-contrast video. Arrows represent velocity direction; the colour map at right indicates velocity magnitude (μm s−1). Right, spatial distribution of instantaneous vorticity associated with the phase-contrast video and plotted in the same manner as for the right panel of Supplementary Video 1. This video consists of three sequential subvideos (each lasting 50 s) representing different stages of vortex lattice development. This video is played at 50 frames s−1 with real elapsed time indicated in the time stamp. All panels share the same scale bar, 200 μm. Associated with Fig. 3a.
Supplementary Video 3
Motility deactivation causes dissolution of vortex lattice pattern in quasi-2D dense active fluids comprising S. marcescens. Left, phase-contrast video. Middle, spatial distribution of instantaneous collective velocity associated with the phase-contrast video. Arrows represent velocity direction; the colour map at right indicates velocity magnitude (μm s−1). Right, spatial distribution of instantaneous vorticity associated with the phase-contrast video and plotted in the same manner as for the right panel of Supplementary Video 1. The vortex lattice was illuminated by violet light at T = 35.5 s, and the motility of all cells in the entire field of view was suppressed by violet light illumination (Extended Data Fig. 5). This video is played at 50 frames s−1 with real elapsed time indicated in the time stamp. All panels share the same scale bar, 200 μm.
Supplementary Video 4
Sporadic vortices in quasi-2D dense E. coli suspensions. Approximate cell density was 8 × 1,010 ml−1. Left, phase-contrast video. Middle, spatial distribution of instantaneous collective velocity associated with the phase-contrast video. Arrows represent velocity direction; the colour map at right indicates velocity magnitude (μm s−1). Right, spatial distribution of instantaneous vorticity associated with the phase-contrast video and plotted in the same manner as in Fig. 1c. This video is played at ten frames s−1 with the real elapsed time indicated in the time stamp. All panels share the same scale bar, 500 μm. Associated with Extended Data Fig. 6a,b. Note that the field of view is large (roughly 3 × 3 mm2) and the collective motion of cells in the phase-contrast video is invisible to the human eye.
Supplementary Video 5
Emergence of vortex lattice in particle-based simulation. Left, spatial distribution of instantaneous particle speed in the simulation, with the colour map at right indicating velocity magnitude. Middle, spatial distribution of instantaneous vorticity associated with particle speed distribution and plotted in the same manner as for the right panel of Supplementary Video 1. Right, spatial distribution of local polar order in the simulation, with the colour map at right indicating the magnitude of order. Simulation parameters: particle activity f0 = 10 and mobility enhancement coefficient β = 1.2. This video is played at 20 frames s−1 with the elapsed simulation time units indicated in the time stamp. Associated with Fig. 4d,e and Extended Data Fig. 7.
Supplementary Video 6
Emergence of hexagonal vortex lattice in the continuum model. Left, spatial distribution of instantaneous velocity in a numerical simulation of the continuum model. Arrows represent velocity direction; colour map at right indicates velocity magnitude. Right, spatial distribution of instantaneous vorticity associated with velocity distribution and plotted in the same manner as for the right panel of Supplementary Video 1. Simulation parameters: activity |S| = 3.5 and mobility enhancement coefficient β = 2.0. This video is played at 20 frames s−1 with simulation time units indicated in the time stamp. Associated with Fig. 4g,h and Extended Data Fig. 10d.
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Xu, H., Wu, Y. Self-enhanced mobility enables vortex pattern formation in living matter. Nature 627, 553–558 (2024). https://doi.org/10.1038/s41586-024-07114-8
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DOI: https://doi.org/10.1038/s41586-024-07114-8
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