## Abstract

Topological defects determine the structure and function of physical and biological matter over a wide range of scales, from the turbulent vortices in planetary atmospheres, oceans or quantum fluids to bioelectrical signalling in the heart^{1,2,3} and brain^{4}, and cell death^{5}. Many advances have been made in understanding and controlling the defect dynamics in active^{6,7,8,9} and passive^{9,10} non-equilibrium fluids. Yet, it remains unknown whether the statistical laws that govern the dynamics of defects in classical^{11} or quantum fluids^{12,13,14} extend to the active matter^{7,15,16} and information flows^{17,18} in living systems. Here, we show that a defect-mediated turbulence underlies the complex wave propagation patterns of Rho-GTP signalling protein on the membrane of starfish egg cells, a process relevant to cytoskeletal remodelling and cell proliferation^{19,20}. Our experiments reveal that the phase velocity field extracted from Rho-GTP concentration waves exhibits vortical defect motions and annihilation dynamics reminiscent of those seen in quantum systems^{12,13}, bacterial turbulence^{15} and active nematics^{7}. Several key statistics and scaling laws of the defect dynamics can be captured by a minimal Helmholtz–Onsager point vortex model^{21} as well as a generic complex Ginzburg–Landau^{22} continuum theory, suggesting a close correspondence between the biochemical signal propagation on the surface of a living cell and a widely studied class of two-dimensional turbulence^{23} and wave^{22} phenomena.

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## Data availability

All data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

## Code availability

The algorithms and simulations codes are described in the Methods and Supplementary Information.

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## Acknowledgements

We thank M. Zwierlein, M. Kardar, H. Kedia and N. Goldenfeld for helpful discussions. This research was supported by a Sloan Research Fellowship (N.F.), a National Science Foundation CAREER Award (N.F.), a James S. McDonnell Foundation Complex Systems Scholar Award (J.D.) and the MIT Solomon Buchsbaum Research Fund (J.D.).

## Author information

### Affiliations

### Contributions

N.F. and J.D. designed and supervised the research. T.H.T. and M.T. performed experiments. J.L., T.H.T. and M.T. analysed the data. P.W.M. performed simulations. All authors discussed the experiment and simulation results and wrote the paper.

### Corresponding authors

Correspondence to Jörn Dunkel or Nikta Fakhri.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Additional information

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

## Extended data

### Extended Data Fig. 1 Visualization of phase determination at single pixel level.

**a**, Fluorescence intensity at each individual pixel, *I*(*t*) is oscillatory. The gray dotted line is the mean intensity \(\bar I\left( t \right)\). **b**, Phase from oscillation can be determined by plotting the two dimensional phase portrait of *I*(*t* + *τ*) versus *I*(*t*). See Methods for more details.

### Extended Data Fig. 2 Statistical characterization of defect dynamics.

**a**, Mean squared displacement of defects plotted as a function of lag time *τ*. Dotted line has a slope of 1. **b**, Defect charge imbalance, quantified using the metric (*N*_{p}−*N*_{m})/*N*_{t}, is plotted as a function of time. Here, *N*_{p}, *N*_{m}, and *N*_{t} denotes the number of +1, -1, and total defect respectively. **(c,d)**, Fraction of creation (on, **c**) and annihilation (off, **d**) events normalized by total defect number *N*_{on}/*N*_{t} and *N*_{on}/*N*_{t} plotted as a function of time for all 11 states. **e**, Distribution of +1 and -1 defects lifetime for all 11 states. **f**, Relative fractions of persistent dipole (purple), transient dipole (blue), cluster (orange) and free vortex (yellow) for all 11 states identified according to the clustering algorithm as described in Methods.

### Extended Data Fig. 3 Extracting characteristic wavelength and wave number from steady state patterns.

**a**, Normalized intensity cross-correlation curves for 11 patterns. Shaded region: Standard error throughout time frames. **b**, Characteristic wavelength inversely correlates with defect density. Error bar: Standard deviation of quantities throughout time frames. Dashed line is the best linear fit. **c**, Example 2D energy spectrum from one pattern snapshot. Heat map color was scaled logarithmically for optimal visualization. **d**, Normalized 1D energy spectrum used to calculate the characteristic wave number shown in Fig. 1i (inset).

### Extended Data Fig. 4 Time series of effective kinetic energy and effective enstrophy.

**a**, Effective kinetic energy *E*(*t*) plotted as a function of time. **b**, Effective enstrophy Ω(*t*) plotted as a function of time.

### Extended Data Fig. 5

Streamlines of phase velocity field are deformed during an annihilation event.

### Extended Data Fig. 6 Defect speed and Voronoi tessellation distributions.

**a**, Defect speed statistics for 11 states, as plotted in Fig. 3a. **b**, Voronoi tessellation area statistics for 11 states, as plotted in Fig. 3b. **c**, Normalized area distribution of Poisson Voronoi cells (blue dashed), hyperuniform Voronoi cells (orange) on a sphere, and defect Voronoi cells (black, same as in Fig. 3b). **d**, and **e**, Voronoi area distribution for Poisson (**d**) and hyperuniform points (**e**) for point density Λ = 100, 200,…, 1000 obtained by averaging *N* = 10000 and *N* = 1000 realizations respectively. Insets: Snapshot of a Poisson Voronoi distribution (**d**) and hyperuniform Voronoi distribution (**e**).

### Extended Data Fig. 7 Helmholtz-Onsager point vortex simulation at high and low energies.

Example of Voronoi cells (left) and phase fields (right) for additional energy values, as in Fig. 3g, corresponding to the red (high energy) and blue (low energy) lines in Fig. 3c. In the high energy regime, like-charge vortices cluster, while opposite charge vortices form pairs in the low energy regime.

### Extended Data Fig. 8 Morphologies of spirals in CGL parameter space.

Snapshots of spiral reaction diffusion waves taken at steady state for various values of *b* an *c*. Simulations here were done on a 500 ×500 grid, but in all other respects are the same as described in the Methods section. Images are from time *T* = 400. Insets correspond to experimentally observed morphologies, matched according to the procedure described in Methods.

### Extended Data Fig. 9 Speed statistics for defects interacting pairwise with a Bessel-form potential.

**a**, Probability distribution of normalized defect speed at three different energy levels (sub-critical, critical, super-critical). **b**, Mean defect scalar velocities as a function of total energy, scaled by the energy difference compared to the critical state. Error bars correspond to variance in speed for individual defects.

### Extended Data Fig. 10 Mean statistics of vortex speed and area for experimental and simulated states.

**a**, Mean vortex speeds for each experimental state used to normalize speed (Extended data Fig 6a) to obtain Fig. 3a. Error bars correspond to variance of speed distribution. **b**, Mean Voronoi tessellation areas for each experimental state used to normalize Voronoi tessellation areas (Extended data Fig 6b) to obtain Fig. 3b. **c**, Mean speed of vortices in point-vortex model as a function of deviation from critical energy. Error bars correspond to variance of speed distribution. **d**, Mean Voronoi tessellation areas used to normalize main text Fig. 3d. Error bars show variance of single particle area distribution. **e**, Mean vortex speeds for CGL simulations as a function of parameter b and c. Error bars correspond to variance in speed for individual vortices. **f**, Mean Voronoi tessellation areas for CGL simulations in Fig. 3i: Error bars correspond to variance in domain size for an individual vortex.

## Supplementary information

### 41567_2020_841_MOESM3_ESM.avi

Time-lapse video of Rho-GTP waves of Ect2 GEF overexpression states 1, 2, 6 and 9. Rho-GTP is visualized using fluorescent reporter GFP-rGBD and imaged using confocal microscopy at a frame rate of 10–12 s and pixel resolution of 0.58–0.62 µm per pixel.

### 41567_2020_841_MOESM4_ESM.avi

Time-lapse video of Rho-GTP waves of Ect2 GEF overexpression states 1, 2, 6 and 9. Rho-GTP is visualized using fluorescent reporter GFP-rGBD and imaged using confocal microscopy at a frame rate of 10–12 s and pixel resolution of 0.58–0.62 µm per pixel.

### 41567_2020_841_MOESM5_ESM.avi

Time-lapse video of Rho-GTP waves of Ect2 GEF overexpression states 1, 2, 6 and 9. Rho-GTP is visualized using fluorescent reporter GFP-rGBD and imaged using confocal microscopy at a frame rate of 10–12 s and pixel resolution of 0.58–0.62 µm per pixel.

### 41567_2020_841_MOESM6_ESM.avi

### 41567_2020_841_MOESM7_ESM.avi

Time-lapse video of Rho-GTP waves shown concurrently with its phase field, defect trajectories, vorticity field and streamlines during an annihilation event. The defect pair disappears at *t* = 0 s. Scale bar, 5 µm.

### 41567_2020_841_MOESM8_ESM.mov

Video showing the dipole pair dynamics in State 2. Persistent dipole pairs are indicated by solid black line while transient dipole pairs are indicated by dotted black line. Defects in +1 and −1 clusters are connected by red and blue lines respectively.

### Supplementary Information

Theoretical derivations and references.

### Supplementary Video 1

### Supplementary Video 2

### Supplementary Video 3

### Supplementary Video 4

### Supplementary Video 5

Time-lapse video of Rho-GTP waves shown concurrently with its phase field, defect trajectories, vorticity field and streamlines during an annihilation event. The defect pair disappears at *t* = 0 s. Scale bar, 5 µm.

### Supplementary Video 6

Video showing the dipole pair dynamics in State 2. Persistent dipole pairs are indicated by solid black line while transient dipole pairs are indicated by dotted black line. Defects in +1 and −1 clusters are connected by red and blue lines respectively.

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Tan, T.H., Liu, J., Miller, P.W. *et al.* Topological turbulence in the membrane of a living cell.
*Nat. Phys.* (2020). https://doi.org/10.1038/s41567-020-0841-9

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