Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

# Topological turbulence in the membrane of a living cell

## 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 heart1,2,3 and brain4, and cell death5. Many advances have been made in understanding and controlling the defect dynamics in active6,7,8,9 and passive9,10 non-equilibrium fluids. Yet, it remains unknown whether the statistical laws that govern the dynamics of defects in classical11 or quantum fluids12,13,14 extend to the active matter7,15,16 and information flows17,18 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 proliferation19,20. 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 systems12,13, bacterial turbulence15 and active nematics7. Several key statistics and scaling laws of the defect dynamics can be captured by a minimal Helmholtz–Onsager point vortex model21 as well as a generic complex Ginzburg–Landau22 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 turbulence23 and wave22 phenomena.

This is a preview of subscription content, access via your institution

## Relevant articles

• ### Emergence of active turbulence in microswimmer suspensions due to active hydrodynamic stress and volume exclusion

Communications Physics Open Access 03 March 2022

• ### Spatiotemporal development of coexisting wave domains of Rho activity in the cell cortex

Scientific Reports Open Access 30 September 2021

• ### Bulk-surface coupling identifies the mechanistic connection between Min-protein patterns in vivo and in vitro

Nature Communications Open Access 03 June 2021

## Access options

\$32.00

All prices are NET prices.

## 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.

## References

1. Gray, R. A., Pertsov, A. M. & Jalife, J. Spatial and temporal organization during cardiac fibrillation. Nature 392, 75–78 (1998).

2. Witkowski, F. X. et al. Spatiotemporal evolution of ventricular fibrillation. Nature 392, 78–82 (1998).

3. Christoph, J. et al. Electromechanical vortex filaments during cardiac fibrillation. Nature 555, 667–672 (2018).

4. Huang, X. Y. et al. Spiral wave dynamics in neocortex. Neuron 68, 978–990 (2010).

5. Saw, T. B. et al. Topological defects in epithelia govern cell death and extrusion. Nature 544, 212–216 (2017).

6. Sanchez, T., Chen, D. T., DeCamp, S. J., Heymann, M. & Dogic, Z. Spontaneous motion in hierarchically assembled active matter. Nature 491, 431–434 (2012).

7. Giomi, L. Geometry and topology of turbulence in active nematics. Phys. Rev. X 5, 031003 (2015).

8. Nishiguchi, D., Aranson, I. S., Snezhko, A. & Sokolov, A. Engineering bacterial vortex lattice via direct laser lithography. Nat. Commun. 9, 4486 (2018).

9. Musevic, I., Skarabot, M., Tkalec, U., Ravnik, M. & Zumer, S. Two-dimensional nematic colloidal crystals self-assembled by topological defects. Science 313, 954–958 (2006).

10. Scheeler, M. W., van Rees, W. M., Kedia, H., Kleckner, D. & Irvine, W. T. M. Complete measurement of helicity and its dynamics in vortex tubes. Science 357, 487–490 (2017).

11. Eyink, G. L. & Sreenivasan, K. R. Onsager and the theory of hydrodynamic turbulence. Rev. Mod. Phys. 78, 87–135 (2006).

12. Anderson, B. P. Resource article: experiments with vortices in superfluid atomic gases. J. Low Temp. Phys. 161, 574–602 (2010).

13. Fetter, A. L. Vortices and dynamics in trapped Bose–Einstein condensates. J. Low Temp. Phys. 161, 445–459 (2010).

14. Bradley, A. S. & Anderson, B. P. Energy spectra of vortex distributions in two-dimensional quantum turbulence. Phys. Rev. X 2, 041001 (2012).

15. Dunkel, J. et al. Fluid dynamics of bacterial turbulence. Phys. Rev. Lett. 110, 228102 (2013).

16. Wu, K. T. et al. Transition from turbulent to coherent flows in confined three-dimensional active fluids. Science 355, eaal1979 (2017).

17. Lechleiter, J., Girard, S., Peralta, E. & Clapham, D. Spiral calcium wave-propagation and annihilation in Xenopus laevis oocytes. Science 252, 123–126 (1991).

18. Lechleiter, J. D. & Clapham, D. E. Spiral waves and intracellular calcium signaling. J. Physiol. 86, 123–128 (1992).

19. Etienne-Manneville, S. & Hall, A. Rho GTPases in cell biology. Nature 420, 629–635 (2002).

20. Bement, W. M. et al. Activator–inhibitor coupling between Rho signaling and actin assembly makes the cell cortex an excitable medium. Nat. Cell Biol. 17, 1471–1483 (2015).

21. Onsager, L. Statistical hydrodynamics. Nuovo Cim. 6, 279–287 (1949).

22. Aranson, I. S. & Kramer, L. The world of the complex Ginzburg–Landau equation. Rev. Mod. Phys. 74, 99–143 (2002).

23. Kraichnan, R. H. & Montgomery, D. Two-dimensional turbulence. Rep. Prog. Phys. 43, 547–619 (1980).

24. Mermin, N. D. The topological theory of defects in ordered media. Rev. Mod. Phys. 51, 591–648 (1979).

25. Moffatt, H. K. Degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, 117–129 (1969).

26. Kleckner, D. & Irvine, W. T. M. Creation and dynamics of knotted vortices. Nat. Phys. 9, 253–258 (2013).

27. Paulose, J., Chen, B. G. G. & Vitelli, V. Topological modes bound to dislocations in mechanical metamaterials. Nat. Phys. 11, 153–156 (2015).

28. Keber, F. C. et al. Topology and dynamics of active nematic vesicles. Science 345, 1135–1139 (2014).

29. Fyfe, D. & Montgomery, D. High-beta turbulence in two-dimensional magnetohydrodynamics. J. Plasma Phys. 16, 181–191 (2009).

30. Zwierlein, M. W., Abo-Shaeer, J. R., Schirotzek, A., Schunck, C. H. & Ketterle, W. Vortices and superfluidity in a strongly interacting Fermi gas. Nature 435, 1047–1051 (2005).

31. Gauthier, G. et al. Giant vortex clusters in a two-dimensional quantum fluid. Science 364, 1264–1267 (2019).

32. Kawaguchi, K., Kageyama, R. & Sano, M. Topological defects control collective dynamics in neural progenitor cell cultures. Nature 545, 327–331 (2017).

33. Epstein, I. R. & Pojman, J. A. An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns, and Chaos (Oxford Univ. Press, 1998).

34. Halatek, J. & Frey, E. Rethinking pattern formation in reaction–diffusion systems. Nat. Phys. 14, 507–514 (2018).

35. Petrov, V., Gaspar, V., Masere, J. & Showalter, K. Controlling chaos in the Belousov–Zhabotinsky reaction. Nature 361, 240–243 (1993).

36. Jakubith, S., Rotermund, H. H., Engel, W., Vonoertzen, A. & Ertl, G. Spatiotemporal concentration patterns in a surface-reaction: propagating and standing waves, rotating spirals, and turbulence. Phys. Rev. Lett. 65, 3013–3016 (1990).

37. Hildebrand, M., Bar, M. & Eiswirth, M. Statistics of topological defects and spatiotemporal chaos in a reaction-diffusion system. Phys. Rev. Lett. 75, 1503–1506 (1995).

38. Totz, J. F., Rode, J., Tinsley, M. R., Showalter, K. & Engel, H. Spiral wave chimera states in large populations of coupled chemical oscillators. Nat. Phys. 14, 282–285 (2018).

39. Coullet, P., Gil, L. & Lega, J. Defect-mediated turbulence. Phys. Rev. Lett. 62, 1619–1622 (1989).

40. Sawai, S., Thomason, P. A. & Cox, E. C. An autoregulatory circuit for long-range self-organization in Dictyostelium cell populations. Nature 433, 323–326 (2005).

41. Reeves, M. T., Anderson, B. P. & Bradley, A. S. Classical and quantum regimes of two-dimensional turbulence in trapped Bose–Einstein condensates. Phys. Rev. A 86, 053621 (2012).

42. Neely, T. W. et al. Characteristics of two-dimensional quantum turbulence in a compressible superfluid. Phys. Rev. Lett. 111, 235301 (2013).

43. Marchetti, M. C. et al. Hydrodynamics of soft active matter. Rev. Mod. Phys. 85, 1143 (2013).

44. Lee, H. Y., Yahyanejad, M. & Kardar, M. Symmetry considerations and development of pinwheels in visual maps. Proc. Natl Acad. Sci. USA 100, 16036–16040 (2003).

45. Billam, T. P., Reeves, M. T., Anderson, B. P. & Bradley, A. S. Onsager–Kraichnan condensation in decaying two-dimensional quantum turbulence. Phys. Rev. Lett. 112, 145301 (2014).

46. Yu, X. Q., Billam, T. P., Nian, J., Reeves, M. T. & Bradley, A. S. Theory of the vortex-clustering transition in a confined two-dimensional quantum fluid. Phys. Rev. A 94, 023602 (2016).

47. Bray, A. J. Velocity distribution of topological defects in phase-ordering systems. Phys. Rev. E 55, 5297–5301 (1997).

48. Angheluta, L., Jeraldo, P. & Goldenfeld, N. Anisotropic velocity statistics of topological defects under shear flow. Phys. Rev. E 85, 011153 (2012).

49. Aguareles, M., Chapman, S. J. & Witelski, T. Motion of spiral waves in the complex Ginzburg–Landau equation. Phys. D 239, 348–365 (2010).

50. Dritschel, D. G. & Boatto, S. The motion of point vortices on closed surfaces. Proc. R. Soc. A 471, 20140890 (2015).

51. Rica, S. & Tirapegui, E. Dynamics of defects in the complex Ginzburg–Landau equation. Phys. D 61, 246–252 (1992).

52. Skaugen, A. & Angheluta, L. Velocity statistics for nonuniform configurations of point vortices. Phys. Rev. E 93, 042137 (2016).

53. Boffetta, G. & Ecke, R. E. Two-dimensional turbulence. Annu. Rev. Fluid Mech. 44, 427–451 (2012).

54. Benink, H. A. & Bement, W. M. Concentric zones of active RhoA and Cdc42 around single cell wounds. J. Cell Biol. 168, 429–439 (2005).

55. Hunter, J. D. Matplotlib: a 2D graphics environment. Comput. Sci. Eng. 9, 90–95 (2007).

56. Valani, R. N., Groszek, A. J. & Simula, T. P. Einstein–Bose condensation of Onsager vortices. New J. Phys. 20, 053038 (2018).

57. Ashbee, T. Point Vortex Dynamics Simulation (MATLAB Central File Exchange, accessed 31 August 2018); https://www.mathworks.com/matlabcentral/fileexchange/49103-point-vortex-dynamics-simulation

58. Townsend, A., Wilber, H. & Wright, G. B. Computing with functions in spherical and polar geometries I. The sphere. SIAM J. Sci. Comput. 38, C403–C425 (2016).

## 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

Authors

### 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.

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 (NpNm)/Nt, is plotted as a function of time. Here, Np, Nm, and Nt 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 Non/Nt and Non/Nt 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

### Supplementary Information

Theoretical derivations and references.

### Supplementary Video 1

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.

### Supplementary Video 2

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.

### Supplementary Video 3

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.

### Supplementary Video 4

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.

### 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.

## Rights and permissions

Reprints and Permissions

Tan, T.H., Liu, J., Miller, P.W. et al. Topological turbulence in the membrane of a living cell. Nat. Phys. 16, 657–662 (2020). https://doi.org/10.1038/s41567-020-0841-9

• Accepted:

• Published:

• Issue Date:

• DOI: https://doi.org/10.1038/s41567-020-0841-9

• ### Topological active matter

• Suraj Shankar
• Anton Souslov
• Vincenzo Vitelli

Nature Reviews Physics (2022)

• ### Emergence of active turbulence in microswimmer suspensions due to active hydrodynamic stress and volume exclusion

• Kai Qi
• Elmar Westphal
• Roland G. Winkler

Communications Physics (2022)

• ### Forced and spontaneous symmetry breaking in cell polarization

• Pearson W. Miller
• Daniel Fortunato
• Stanislav Shvartsman

Nature Computational Science (2022)

• ### Bulk-surface coupling identifies the mechanistic connection between Min-protein patterns in vivo and in vitro

• Fridtjof Brauns
• Grzegorz Pawlik
• Cees Dekker

Nature Communications (2021)