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# Autonomous waves and global motion modes in living active solids

## Abstract

Elastic active matter—also called an active solid—consists of self-propelled units embedded in an elastic matrix and it resists deformation. This shape-preserving property and the intrinsic non-equilibrium nature make active solids an attractive potential component for self-driven devices, but their mechanical properties and emergent behaviour remain poorly understood. Here, using a biofilm-based living active solid, we observe self-sustained elastic waves with wave properties not seen in passive solids, such as power-law scaling of wave speed with activity. Under isotropic confinement, the active solid develops two topologically distinct global motion modes that can be selectively excited, with a step-like frequency jump at the transition between the two modes. Our findings reveal spatiotemporal order in elastic active matter and may guide the development of solid-state adaptive or living materials.

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## Data availability

The data supporting the findings of this study are included within the paper and its Supplementary Information. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

## Code availability

The custom codes used in this study are available from the corresponding author upon request.

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## Acknowledgements

We thank Y. Li and W. Zuo for building the image acquisition and microscope stage temperature-control systems, K. Gibbs (Harvard University) for providing the bacterial strains, S. Liu for helpful discussions, S. Liu for assistance in experiments and L. Xu (CUHK) for assistance with bulk rheology measurement. Y.H. and R.Z. thank Q. Wang and J. Tian for fruitful discussions. This work was supported by the Research Grants Council of Hong Kong SAR (RGC ref. no. RFS2021-4S04, 14303918, 14306820, to Y.W.; ref. no. 16300221, to R.Z.), the Ministry of Science and Technology Most China (2020YFA0910700, to Y.W.), and CUHK Direct Grants (to Y.W.).

## Author information

Authors

### Contributions

H.X. discovered the phenomena, designed the study, performed experiments, performed simulations, and analysed and interpreted the data. Y.H. and R.Z. developed the theory and improved the simulations. Y.W. conceived the project, designed the study, and analysed and interpreted the data. R.Z. and Y.W. supervised the study. Y.W. wrote the first draft and all authors contributed to the revision of the manuscript.

### Corresponding author

Correspondence to Yilin Wu.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature Physics thanks Jing Yan, Thomas Angelini and Cristian Huepe for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data

### Extended Data Fig. 1 Proteus mirabilis colonies at the early stage of biofilm development with prominent production of extracellular polymer matrix.

(a) Phase-contrast image of a circular disk-shaped P. mirabilis colony grown for 14 hr at 37 °C after inoculation with overnight culture (Methods). Scale bar, 500 μm. (b) Fluorescent image of extracellular amyloid fibrils matrix labelled by Thioflavin T (Methods) in the P. mirabilis biofilm shown in panel a. The outer rim of P. mirabilis colonies at this development stage is mostly occupied by immotile cells that have transitioned to the sessile state but expressed little extracellular matrix. The width of this outer immotile rim varies from tens to hundreds of µm across different colonies; in the case of panel a, the immotile rim ranges from radius R = ~860 µm (measured from the colony centre) to R = 1154 µm (i.e, the colony edge), spanning a width of ~300 µm. The inner region of the colonies with Thioflavin T fluorescence (that is, the biofilm region; enclosed by the dashed circle in panels a,b) is where we choose to study and refer to as early-stage biofilm or bacterial active solid, such as the fields in main text Figs. 1c,d and 3a. The immotile outer rim of the colony serves as the lateral spatial confinement for the bacterial active solid. Scale bar, 500 μm. (c) Enlarged view of the centre of panel b. Scale bar, 100 μm. (d) Phase-contrast images of oval-shaped P. mirabilis colonies with various values of the eccentricity. Scale bars, 500 μm. The field in main text Fig. 2a corresponds to the stripe region enclosed by the dashed rectangle in panel d.

### Extended Data Fig. 2 Dynamic moduli of the bacterial active solid derived from early-stage P. mirabilis biofilms.

Storage (G’; red) and loss (G”; blue) shear modulus of the bacterial active solid as a function of strain were measured by bulk rheometry with a cone-plate rheometer in shear-strain-amplitude sweep mode (at constant frequency) (Methods). The oscillation frequency and the temperature of the rheometer were set at 1 Hz and 37 °C, respectively. Data are presented as mean + /-S.D. (N = 3). The solid and dashed lines are guides to the eye.

### Extended Data Fig. 3 Spatial distribution of local oscillation frequency in bacterial active solids.

Panels a and b shows the frequency distribution in circular disk-shaped bacterial active solids undergoing global oscillatory translation and oscillatory rotation, respectively, with the colour bars indicating the magnitude of local oscillation frequency (unit: Hz). Scale bars, 500 μm. The frequency at the centre of panel b is absent because the velocity there is vanishing. Panel c shows the frequency distribution in a rectangular region of an oval-shaped bacterial active solid that displays the transverse standing wave, with the colour bar indicating the magnitude of local oscillation frequency (unit: Hz). The longer side of the selected region is parallel to the major axis of symmetry of the bacterial active solid. Scale bar, 200 μm.

### Extended Data Fig. 4 Tuning single-cell speed of P. mirabilis via violet-light illumination.

To obtain this plot, cells were extracted from the P. mirabilis-based bacterial active solids that were undergoing either global oscillatory translation or global oscillatory rotation; the extracted cells were mixed with 0.02% Tween 20 and deposited on 0.6% LB agar surface, forming a quasi-2D dilute bacterial suspension drop. P. mirabilis cells in the prepared quasi-2D dilute bacterial suspension drop were continuously illuminated by 406 nm violet light starting from T = 0 s while being tracked in phase-contrast microscopy (Methods); note that for single-cell tracking, a 20x objective lens was used. The speed of an individual cell at a specific time T was computed based on its trajectory tracked from (T-0.5) s to (T + 0.5) s; the single-cell speeds computed from (T-25) s to (T + 25) s were then averaged and taken to be the mean cell speed at T. Data shown in the plot was normalized by the mean speed at T = 0 s (that is, the free-swimming speed of cells without blue light illumination; V0 = 26.7 ± 6.9 µm/s; mean±S.D., N = 2500). Data are presented as mean + /-S.D. (N = 2500).

### Extended Data Fig. 5 Emergent global motion modes in modelled active solid under isotropic lateral confinement.

(a) Schematic diagram of the bead–spring model for active solid under two-dimensional isotropic lateral confinement (Methods). The model consists of N = 511 self-propelled particles (black solid circles). Every nearest-neighbour pair of particles is connected by an interparticle spring with spring constant kb (red). The particles also experience elastic forces due to substrate adhesion and lateral spatial confinement (see main text) via a restoring spring (green) and a boundary spring (blue) with spring constant ks and kr, respectively. The three spring constants together determine the system’s local elasticity. In simulations the interparticle spring constant kb was used as a proxy for the system’s local elasticity, with the ratios between kb, ks and kr fixed. (b,c) Representative trajectories of particles in the modelled active solid that underwent global oscillatory translation (panel b) and oscillatory rotation (panel c). Most particles (except those very near the centre or the boundary) followed periodically oscillating quasi-circular trajectories (at relatively high activity; panel b; Supplementary Video 9) or quasi-linear concentric trajectories (at relatively low activity; panel c; Supplementary Video 10) with highly synchronized phases (insets of panel b,c), in the same manner as the motion of matrix-embedded cells in the experiments undergoing global oscillatory translation (main text Fig. 1a) or rotation (main text Fig. 1b), respectively. Black dot in each panel indicates the centre of the simulation domain. Scale bars represent 1/3 of the interparticle distance at equilibrium and colour map indicates time. Insets: Oscillation phases of individual particle’s velocity components plotted in the same way as in insets of Fig. 1a,b. Simulation parameters:v0 = 15 (panel b) or v0 = 3 (panel c), kb = 12,kr = 0.6, and ks = 0.044. (d,e) Temporal dynamics of spatially averaged particle velocity in the modelled active solid in global oscillatory translation mode (panel d) or oscillatory rotation mode (panel e). The velocity was averaged over all particles in the simulation and then decomposed as Cartesian (yellow and blue traces) and polar-coordinate components (red: tangential or azimuthal component; green: radial component). The spatially averaged particle velocity in the two emergent modes was characterized by distinct temporal dynamics in Cartesian or polar coordinates similar to that found in the experiments (main text Fig. 1e,f). Simulation parameters are identical to those used in panels b,c.

### Extended Data Fig. 6 Elasticity dependence of the global motion modes in active solids under isotropic lateral confinement.

(a) Oscillation frequency of global motion modes in modelled active solids as a function of kb (fixing v0 = 4). Colour of data points indicates the mode of global motion (blue: oscillatory translation; red: oscillatory rotation). Data are presented as mean + /-S.D. (N=100 simulation runs). (b) Temperature dependence of P. mirabilis single-cell speed. P. mirabilis cells in quasi-2D dilute bacterial suspension drops (prepared in the same manner as described in the caption of Extended Data Fig. 4) were tracked in fluorescent microscopy while the environmental temperature was varied from 24 °C to 50 °C with a custom-built temperature-control system (Methods). As shown in the plot the speed of cells only changed slightly in this temperature range (up to ~15%). The mean speed of cells at a specific temperature was computed based on 1-s segments of cell trajectories tracked in a 200-s time window. Data shown in the plot was normalized by the mean speed at temperature 24 °C. Data are presented as mean + /-S.D. (N = 2500). (c,d) Transition of global motion modes in bacterial active solids controlled by temperature. Panel c shows the temperature dependence of oscillation frequency in the bacterial active solid during mode transition. Colour of data points indicates the mode of global motion (blue: oscillatory translation; red: oscillatory rotation). Panel d shows the temporal dynamics of spatially averaged collective velocity during transition from the oscillatory rotation mode to the oscillatory translation mode following the decrease of temperature. The spatially averaged collective velocity was decomposed as Cartesian components (yellow and blue traces; upper part of panel d) and polar-coordinate components (red: tangential or azimuthal component, green: radial component; lower part of panel d). Data in panels c,d were from a representative experiment (>5 replicates).

### Extended Data Fig. 7 Self-sustained transverse standing waves in modelled active solids under anisotropic lateral confinement.

This figure is associated with main text Fig. 4d. Simulation parameters:v0 = 10, kb = 4, kr = 0.2, and ks = 0.015. (a) Schematic diagrams of the bead–spring model for active solid under two-dimensional anisotropic lateral confinement (Methods). The elliptical confinement (left) mimics the anisotropic lateral confinement geometry of oval-shaped bacterial active solids used in main text Fig. 2. Rectangular lateral confinement (right) produces similar simulation results. (b) Time sequence of particle velocity field in a modelled active solid under elliptical lateral confinement that displays the transverse standing wave. The longer side of the rectangular domain shown here is in parallel to the major axis of the elliptical confinement. T denotes the period of oscillation. Arrows represent velocity direction and colour map indicates velocity magnitude. (c) Spatial distributions of local oscillation frequency and phase associated with panel b. The local oscillation frequency is homogeneous in space (upper part of the panel, with the colour bar indicating the magnitude of frequency). The phase of parallel (vx; parallel to the major axis) and transverse (vy; perpendicular to the major axis) component of particle velocity is denoted as φx (lower) and φy (upper), respectively. Scale bar under panel c is shared by panel b and represents 7.7 times of the interparticle distance at equilibrium. (d) Averaged amplitude distribution of vy along major axis of the modelled active solid. Data are presented as mean + /-S.D. (N=100 simulation runs). Inset: temporal evolution of vy profile in panel b along the major axis of the modelled active solid, with colours representing time (blue: 0; red: T/4; yellow: T/2; green: 3 T/4). (e) Spatiotemporal autocorrelation of vx along the major axis of the modelled active solid (that is, the abscissa of the figure); the pattern of periodic, horizontal lanes with high correlation is similar to that seen in experiment (main text Fig. 2e). Colour map at right side indicates the autocorrelation magnitude (a.u.).

### Extended Data Fig. 8 Emergent global motion modes in bacterial active solids derived from S. marcescens biofilms.

Temporal dynamics of spatially averaged collective velocity in the global oscillatory translation mode (panel a) and oscillatory rotation mode (panel b). The spatially averaged collective velocity was decomposed as Cartesian (yellow and blue traces) and polar-coordinate components (red: tangential or azimuthal component; green: radial component). In the oscillatory translation mode, the polar-coordinate components are negligible; in the oscillatory rotation mode, both the radial and the Cartesian components are negligible.

### Extended Data Fig. 9 Coexistence of two global motion modes in bacterial active solids under isotropic lateral confinement.

(a,b) Temporal dynamics of spatially averaged collective velocity in circular disk-shaped bacterial active solids where the oscillatory translation and oscillatory rotation modes co-existed with identical frequencies ($${f}_{t}={f}_{r}=0.15$$ Hz; panel a) or with the frequency of the oscillatory translation mode doubled ($${f}_{t}=0.14$$ Hz, $${f}_{r}=0.07$$ Hz; panel b). The velocity was decomposed as Cartesian (yellow and blue traces) and polar-coordinate components (red: tangential component; green: radial component). (c,d) Panels c and d display representative trajectories of mass elements in bacterial active solids analysed in panels a and b, respectively. In each panel the trajectories were obtained by integrating the spatially averaged collective velocity over a $$50\,\mu m\times 50\,\mu m$$ domain located ~ 500 µm from the centre of the disk-shaped bacterial active solid (black dot) at different polar angles. The trajectories were brought close to the centre for better visualization, and thus the scale bars (panel c, 10 μm; panel d, 20 μm) apply to the trajectories only. Colour map indicates time (unit: s).

### Extended Data Fig. 10 Coexistence of two global motion modes in modelled active solid under isotropic lateral confinement.

(a,b) Temporal dynamics of collective velocity of the system where oscillatory translation and oscillatory rotation modes co-existed with identical frequencies ($${f}_{t}={f}_{r}=0.10$$; panel a) or with the frequency of the oscillatory translation mode doubled ($${f}_{t}=0.13$$, $${f}_{r}=0.07$$; panel b). The collective velocity was averaged over all particles in the simulation and then decomposed as Cartesian (yellow and blue traces; upper part of each panel) and polar-coordinate components (red: tangential or azimuthal component, green: radial component; lower part of each panel). Simulation parameters: panel a, $${v}_{0}=5,{k}_{b}=12$$; panel b, $${v}_{0}=9,{k}_{b}=16$$; the ratios between kb, ks and kr are fixed (Methods). (c,d) Panels c and d display the trajectory of representative particles in the simulations analysed in panels a and b, respectively. In each panel the particle was chosen at ~5 times of the equilibrium interparticle distance from the centre of the circular simulation domain (black dot) at different polar angles. The trajectories were brought close to the centre for better visualization, and thus the scale bars (indicating 1/3 interparticle distance at equilibrium) apply to the trajectories only. Colour map indicates time.

## Supplementary information

### Supplementary Information

Supplementary Text and Figs. 1–3.

### Supplementary Video 1

Emergent global motion in the form of oscillatory translation in a circular disk-shaped P. mirabilis biofilm. Left: phase-contrast videos taken at different fields of a P. mirabilis biofilm undergoing oscillatory translation. The dashed line indicates the distance between the centre of adjacent fields (500 µm). Middle: the centre region of the biofilm. Right: fluorescence videos of GFP-labelled cells embedded in the biofilm taken at the same fields as shown in the corresponding phase-contrast videos. The GFP-labelled cells served as tracers of local mass elements. The phase-contrast and fluorescence videos at a specific field were recorded simultaneously; the videos at different fields were recorded at different times but their starting frames were aligned in phase when assembling the videos. The videos are played at 50 frames per second with the real elapsed time indicated in the time stamp. Scale bar at the lower right corner, 100 μm.

### Supplementary Video 2

Enlarged view of the oscillatory translation mode in a circular disk-shaped P. mirabilis biofilm. The entire field of view is located at the centre of a P. mirabilis biofilm undergoing oscillatory translation. Shown in this fluorescence video are GFP-labelled cells embedded in the biofilm that served as tracers of local mass elements. The video is played at 50 frames per second with the real elapsed time indicated in the time stamp. Scale bar, 20 µm.

### Supplementary Video 3

Emergent global motion in the form of oscillatory rotation in a circular disk-shaped P. mirabilis biofilm. Left: phase-contrast videos taken at different fields of a P. mirabilis biofilm undergoing oscillatory rotation. The dashed line indicates the distance between the centre of adjacent fields (500 µm). Middle: the centre region of the biofilm. Right: fluorescence videos of GFP-labelled cells embedded in the biofilm taken at the same fields as shown in the corresponding phase-contrast videos. The GFP-labelled cells served as tracers of local mass elements. The phase-contrast and fluorescence videos at a specific field were recorded simultaneously; the videos at different fields were recorded at different times but their starting frames were aligned in phase when assembling the videos. The videos are played at 50 frames per second with the real elapsed time indicated in the time stamp. Scale bar at the lower right corner, 100 μm.

### Supplementary Video 4

Enlarged view of the oscillatory rotation mode in a circular disk-shaped P. mirabilis biofilm. The entire field of view is located at the centre of a P. mirabilis biofilm undergoing oscillatory rotation. Shown in this fluorescence video are GFP-labelled cells embedded in the biofilm that served as tracers of local mass elements. The video is played at 50 frames per second with the real elapsed time indicated in the time stamp. Scale bar, 20 µm.

### Supplementary Video 5

Collective motion of mass elements in the circular disk-shaped bacterial active solid undergoing global oscillatory translation. Mass elements (solid dots in black) are represented by virtual Lagrangian particles, whose positions are reconstructed based on the collective velocity field measured by optical flow analysis on large-scale phase-contrast images (Methods). The video is played at 50 frames per second with the real elapsed time indicated in the time stamp. Scale bar, 200 μm. It is associated with Fig. 1c.

### Supplementary Video 6

Collective motion of mass elements in the circular disk-shaped bacterial active solid undergoing global oscillatory rotation. Mass elements (solid dots in black) are represented by virtual Lagrangian particles, whose positions are reconstructed based on the collective velocity field measured by optical flow analysis on large-scale phase-contrast images (Methods). The video is played at 50 frames per second with the real elapsed time indicated in the time stamp. Scale bar, 200 μm. It is associated with Fig. 1d.

### Supplementary Video 7

Self-sustained transverse standing waves in bacterial active solid under anisotropic lateral confinement. This video shows the temporal dynamics of collective velocity field associated with Fig. 2a. The collective velocity field was measured by optical flow analysis on phase-contrast images (Methods). Top: collective velocity component perpendicular to the major axis of the bacterial active solid (that is, transverse component. Middle: collective velocity component parallel to the major axis of the bacterial active solid. Bottom: collective velocity. In all panels the arrows indicate velocity direction and the colour map indicate velocity magnitude (unit: μm s−1). The video is played at 50 frames per second with the real elapsed time indicated in the time stamp.

### Supplementary Video 8

Transition of global motion modes in bacterial active solids controlled by bacterial activity. A circular disk-shaped bacterial active solid derived from early-stage P. mirabilis biofilms was initially undergoing global oscillatory translation. Stating from T = 50 s, it was continuously illuminated by 406 nm violet light and as the cell activity was decreased by violet light, the motion of the bacterial active solid transited to oscillatory rotation mode. The entire field of view is located at the centre of the biofilm. This phase-contrast video is played at 50 frames per second with the real elapsed time indicated in the time stamp. The activity (that is, the average collective speed) of the active solid is shown below the time stamp. Scale bar, 200 μm.

### Supplementary Video 9

Emergent global motion in the form of oscillatory translation in modelled active solid under isotropic lateral confinement. It is associated with Extended Data Fig. 5b,d. Active particles in the model are represented by ellipsoids. Arrows associated with the ellipsoids indicate the instantaneous velocity direction of the active particles, and the colour of each ellipsoid as well as its arrow indicates velocity magnitude of the active particle. Simulation parameters: v0 = 15, kb = 12, kr = 0.6 and ks = 0.044.

### Supplementary Video 10

Emergent global motion in the form of oscillatory rotation in modelled active solid under isotropic lateral confinement. It is associated with Extended Data Fig. 5c,e. Active particles in the model are represented by ellipsoids. Arrows associated with the ellipsoids indicate the instantaneous velocity direction of the active particles, and the colour of each ellipsoid as well as its arrow indicates velocity magnitude of the active particle. Simulation parameters: v0 = 3, kb = 12, kr = 0.6 and ks = 0.005. Note that ks chosen here is smaller than the one used in Supplementary Video 9 to produce a sufficiently large amplitude in the oscillatory rotation mode, such that the mode is better visualized.

### Supplementary Video 11

Self-sustained transverse standing waves in modelled active solids under anisotropic lateral confinement. It is associated with Fig. 4d and Extended Data Fig. 7. Active particles in the model are represented by ellipsoids. Arrows associated with the ellipsoids indicate the instantaneous velocity direction of the active particles, and the colour of each ellipsoid as well as its arrow indicates velocity magnitude of the active particle. Simulation parameters: v0 = 10, kb = 4, kr = 0.2 and ks = 0.015.

### Supplementary Video 12

Self-sustained travelling wave in a modelled active solid in unconfined 2D space. This video associated with Fig. 4e shows the spatiotemporal dynamics of the phase of local oscillation in the self-sustained travelling wave. Colour map indicates phase magnitude. Simulation parameters: v = 0, kb = 4, ks = 0.015 and kr = 0.

### Supplementary Video 13

Emergent global motion in S. marcescens early-stage biofilm. S. marcescens biofilms undergoing oscillatory translation mode (left) and oscillatory rotation mode (right) were imaged in fluorescence. Shown in this fluorescence video are cells labelled by a green fluorescence protein that served as tracers of local mass elements of the biofilm. This video is played at 20 frames per second with the real elapsed time indicated in the time stamp. Scale bar, 20 μm.

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Xu, H., Huang, Y., Zhang, R. et al. Autonomous waves and global motion modes in living active solids. Nat. Phys. 19, 46–51 (2023). https://doi.org/10.1038/s41567-022-01836-0

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• DOI: https://doi.org/10.1038/s41567-022-01836-0

• ### Bacteria surfing the elastic wave

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