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Stochastic nonlinear dynamics of confined cell migration in two-state systems

A Publisher Correction to this article was published on 18 March 2019

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Abstract

Migrating cells in physiological processes, including development, homeostasis and cancer, encounter structured environments and are forced to overcome physical obstacles. Yet, the dynamics of confined cell migration remains poorly understood, and thus there is a need to study the complex motility of cells in controlled confining microenvironments. Here, we develop two-state micropatterns, consisting of two adhesive sites connected by a thin constriction, in which migrating cells perform repeated stochastic transitions. This minimal system enables us to obtain a large ensemble of single-cell trajectories. From these trajectories, we infer an equation of cell motion, which decomposes the dynamics into deterministic and stochastic contributions in position–velocity phase space. Our results reveal that cells in two-state micropatterns exhibit intricate nonlinear migratory dynamics, with qualitatively similar features for a cancerous (MDA-MB-231) and a non-cancerous (MCF10A) cell line. In both cases, the cells drive themselves deterministically into the thin constriction; a process that is sped up by noise. Interestingly, however, these two cell lines have distinct deterministic dynamics: MDA-MB-231 cells exhibit a limit cycle, while MCF10A cells show excitable bistable dynamics. Our approach yields a conceptual framework that may be extended to understand cell migration in more complex confining environments.

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Fig. 1: Experimental set-up of two-state micropatterns with MDA-MB-231 cells.
Fig. 2: Statistics of the hopping process (MDA-MB-231).
Fig. 3: Deterministic and stochastic contributions to the equation of motion for MDA-MB-231 cells with and without constriction, and MCF10A cells with constriction (L = 35 µm).
Fig. 4: Nonlinear deterministic dynamics of the cell migration (L= 35 µm).

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Data availability

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

Change history

  • 18 March 2019

    In the version of this Article originally published online, in the ‘Journal peer review information’ statement, the reviewer Henrik Flyvbjerg was mistakenly not included; the statement has now been updated accordingly.

References

  1. Koser, D. E. et al. Mechanosensing is critical for axon growth in the developing brain. Nat. Neurosci. 19, 1592–1598 (2016).

    Article  Google Scholar 

  2. Vedula, S. R. K. et al. Forces driving epithelial wound healing. Nat. Phys. 10, 683–690 (2014).

    Article  Google Scholar 

  3. Friedl, P. & Wolf, K. Tumour-cell invasion and migration: diversity and escape mechanisms. Nat. Rev. Cancer 3, 362–374 (2003).

    Article  Google Scholar 

  4. Park, J. A. et al. Unjamming and cell shape in the asthmatic airway epithelium. Nat. Mater. 14, 1040–1048 (2015).

    Article  ADS  Google Scholar 

  5. Wu, P.-H., Giri, A., Sun, S. X. & Wirtz, D. Three-dimensional cell migration does not follow a random walk. Proc. Natl Acad. Sci. USA 111, 3949–3954 (2014).

    Article  ADS  Google Scholar 

  6. Even-Ram, S. & Yamada, K. M. Cell migration in 3D matrix. Curr. Opin. Cell Biol. 17, 524–532 (2005).

    Article  Google Scholar 

  7. Green, B. J. et al. Pore shape defines paths of metastatic cell migration. Nano Lett. 18, 2140–2147 (2018).

    Article  ADS  Google Scholar 

  8. Paul, C. D., Mistriotis, P. & Konstantopoulos, K. Cancer cell motility: lessons from migration in confined spaces. Nat. Rev. Cancer 17, 131–140 (2017).

    Article  Google Scholar 

  9. Wolf, K. et al. Physical limits of cell migration: control by ECM space and nuclear deformation and tuning by proteolysis and traction force. J. Cell Biol. 201, 1069–1084 (2013).

    Article  Google Scholar 

  10. Przibram, K. Über die ungeordnete Bewegung niederer Tiere. Pflügers Arch. Physiol. 153, 401–405 (1917).

    Article  Google Scholar 

  11. Fürth, R. Die Brownsche Bewegung bei Berücksichtigung einer Persistenz der Bewegungsrichtung. Mit Anwendungen auf die Bewegung lebender Infusorien. Z. Phys. 2, 244–256 (1920).

    Article  ADS  Google Scholar 

  12. Gail, M. H. & Boone, C. W. The locomotion of mouse fibroblasts in tissue culture. Biophys. J. 10, 980–993 (1970).

    Article  ADS  Google Scholar 

  13. Selmeczi, D., Mosler, S., Hagedorn, P. H., Larsen, N. B. & Flyvbjerg, H. Cell motility as persistent random motion: theories from experiments. Biophys. J. 89, 912–931 (2005).

    Article  Google Scholar 

  14. Metzner, C. et al. Superstatistical analysis and modelling of heterogeneous random walks. Nat. Commun. 6, 7516 (2015).

    Article  Google Scholar 

  15. Singhvi, R. et al. Engineering cell shape and function. Science 264, 696–698 (1994).

    Article  ADS  Google Scholar 

  16. Chen, C., Mrksich, M., Huang, S., Whitesides, G. M. & Ingber, D. E. Geometric control of cell life and death. Science 276, 1425–1428 (1997).

    Article  Google Scholar 

  17. Thery, M. et al. Anisotropy of cell adhesive microenvironment governs cell internal organization and orientation of polarity. Proc. Natl Acad. Sci. USA 103, 19771–19776 (2006).

    Article  ADS  Google Scholar 

  18. Maiuri, P. et al. Actin flows mediate a universal coupling between cell speed and cell persistence. Cell 161, 374–386 (2015).

    Article  Google Scholar 

  19. Schreiber, C., Segerer, F. J., Wagner, E., Roidl, A. & Rädler, J. O. Ring-shaped microlanes and chemical barriers as a platform for probing single-cell igration. Sci. Rep. 6, 26858 (2016).

    Article  ADS  Google Scholar 

  20. Prentice-Mott, H. V. et al. Directional memory arises from long-lived cytoskeletal asymmetries in polarized chemotactic cells. Proc. Natl Acad. Sci. USA 113, 1267–1272 (2016).

    Article  ADS  Google Scholar 

  21. Caballero, D., Voituriez, R. & Riveline, D. Protrusion fluctuations direct cell motion. Biophys. J. 107, 34–42 (2014).

    Article  ADS  Google Scholar 

  22. Mahmud, G. et al. Directing cell motions on micropatterned ratchets. Nat. Phys. 5, 606–612 (2009).

    Article  Google Scholar 

  23. Chang, S. S., Guo, W. H., Kim, Y. & Wang, Y. L. Guidance of cell migration by substrate dimension. Biophys. J. 104, 313–321 (2013).

    Article  ADS  Google Scholar 

  24. Kushiro, K. & Asthagiri, A. R. Modular design of micropattern geometry achieves combinatorial enhancements in cell motility. Langmuir 28, 4357–4362 (2012).

    Article  Google Scholar 

  25. Albert, P. J. & Schwarz, U. S. Dynamics of cell shape and forces on micropatterned substrates predicted by a cellular Potts model. Biophys. J. 106, 2340–2352 (2014).

    Article  ADS  Google Scholar 

  26. Camley, B. A. & Rappel, W. J. Velocity alignment leads to high persistence in confined cells. Phys. Rev. E 89, 062705 (2014).

    Article  ADS  Google Scholar 

  27. Bi, D., Lopez, J. H., Schwarz, J. M. & Manning, M. L. Energy barriers and cell migration in densely packed tissues. Soft Matter 10, 1885–1890 (2014).

    Article  ADS  Google Scholar 

  28. Czirók, A., Schlett, K., Madarász, E. & Vicsek, T. Exponential distribution of locomotion activity in cell cultures. Phys. Rev. Lett. 81, 3038–3041 (1998).

    Article  ADS  Google Scholar 

  29. Takagi, H., Sato, M. J., Yanagida, T. & Ueda, M. Functional analysis of spontaneous cell movement under different physiological conditions. PLoS One 3, e2648 (2008).

    Article  ADS  Google Scholar 

  30. Li, L., Cox, E. C. & Flyvbjerg, H. ‘Dicty dynamics’: Dictyostelium motility as persistent random motion. Phys. Biol. 8, 046006 (2011).

    Article  ADS  Google Scholar 

  31. Pedersen, J. N. et al. How to connect time-lapse recorded trajectories of motile microorganisms with dynamical models in continuous time. Phys. Rev. E 94, 062401 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  32. Van der Pol, B. On relaxation-oscillations. Phil. Mag. 2, 978–992 (1926).

    Article  Google Scholar 

  33. Stephens, G. J., Johnson-Kerner, B., Bialek, W. & Ryu, W. S. Dimensionality and dynamics in the behavior of C. elegans. PLoS Comput. Biol. 4, e1000028 (2008).

  34. Dieterich, P., Klages, R., Preuss, R. & Schwab, A. Anomalous dynamics of cell migration. Proc. Natl Acad. Sci. USA 105, 459–463 (2008).

    Article  ADS  Google Scholar 

  35. Potdar, A. A., Jeon, J., Weaver, A. M., Quaranta, V. & Cummings, P. T. Human mammary epithelial cells exhibit a bimodal correlated random walk pattern. PLoS One 5, e9636 (2010).

    Article  ADS  Google Scholar 

  36. Mak, M., Reinhart-King, C. A. & Erickson, D. Microfabricated physical spatial gradients for investigating cell migration and invasion dynamics. PLoS One 6, e20825 (2011).

    Article  ADS  Google Scholar 

  37. Kraning-Rush, C. M., Carey, S. P., Lampi, M. C. & Reinhart-King, C. A. Microfabricated collagen tracks facilitate single cell metastatic invasion in 3D. Integr. Biol. 5, 606–616 (2013).

    Article  Google Scholar 

  38. Berzat, A. & Hall, A. Cellular responses to extracellular guidance cues. EMBO J. 29, 2734–2745 (2010).

    Article  Google Scholar 

  39. Nakanishi, J. et al. Spatiotemporal control of migration of single cells on a photoactivatable cell microarray. J. Am. Chem. Soc. 129, 6694–6695 (2007).

    Article  Google Scholar 

  40. Oswald, L., Grosser, S., Smith, D. M. & Käs, J. A. Jamming transitions in cancer. J. Phys. D 50, 483001 (2017).

    Article  ADS  Google Scholar 

  41. Bi, D., Lopez, J. H., Schwarz, J. M. & Manning, M. L. A density-independent rigidity transition in biological tissues. Nat. Phys. 11, 1074–1079 (2015).

    Article  Google Scholar 

  42. Le Berre, M. et al. Geometric friction directs cell migration. Phys. Rev. Lett. 111, 198101 (2013).

    Article  ADS  Google Scholar 

  43. Neve, R. M. et al. A collection of breast cancer cell lines for the study of functionally distinct cancer subtypes. Cancer Cell. 10, 515–527 (2006).

    Article  Google Scholar 

  44. Segerer, F. J., Thüroff, F., Piera Alberola, A., Frey, E. & Rädler, J. O. Emergence and persistence of collective cell migration on small circular micropatterns. Phys. Rev. Lett. 114, 228102 (2015).

    Article  ADS  Google Scholar 

  45. Camley, B. A., Zhao, Y., Li, B., Levine, H. & Rappel, W. J. Periodic migration in a physical model of cells on micropatterns. Phys. Rev. Lett. 111, 158102 (2013).

    Article  ADS  Google Scholar 

  46. Segerer, F. J. et al. Versatile method to generate multiple types of micropatterns. Biointerphases 11, 011005 (2016).

    Article  Google Scholar 

  47. Schneider, C. A., Rasband, W. S. & Eliceiri, K. W. NIH Image to ImageJ: 25 years of image analysis. Nat. Methods 9, 671–675 (2012).

    Article  Google Scholar 

  48. Siegert, S., Friedrich, R. & Peinke, J. Analysis of data sets of stochastic systems. Phys. Lett. A 243, 275–280 (1998).

    Article  ADS  MathSciNet  Google Scholar 

  49. Ragwitz, M. & Kantz, H. Indispensable finite time corrections for Fokker–Planck equations from time series data. Phys. Rev. Lett. 87, 254501 (2001).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We thank E. Frey, F. Brauns, G. Gradziuk, D. Lubensky, P. Ronceray, K. Bassler and N. Wingreen for useful comments, C. Leu for the preparation of wafers and A. Reiser for providing the transfection protocol. This work was supported by grants from the German Science Foundation (DFG) through the Collaborative Research Center (SFB) 1032 (projects B01 and B12). D.B.B. is supported by a DFG fellowship within the Graduate School of Quantitative Biosciences Munich and by the Joachim Herz Stiftung.

Author information

Authors and Affiliations

Authors

Contributions

A.F., C.S., P.J.F.R. and J.O.R designed experiments; A.F. and C.S. performed experiments; D.B.B., A.F. and C.S. analysed data. D.B.B. and C.P.B. developed the theoretical model. D.B.B., A.F., C.S., J.O.R. and C.P.B. wrote the manuscript.

Corresponding authors

Correspondence to Joachim O. Rädler or Chase P. Broedersz.

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Competing interests

The authors declare no competing interests.

Additional information

Journal peer review information: Nature Physics thanks Henrik Flyvbjerg, Jonas Pedersen, Ulrich Schwarz and the other anonymous reviewer(s) 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.

Supplementary information

Supplementary Information

Supplementary Figures 1–20 and Supplementary References 1–7.

Reporting Summary

Supplementary Video 1

Single MDA-MB-231 cells transitioning repeatedly between the square adhesion sites of the two-state micropattern. Transitions are usually preceded by the formation of a protrusion along the bridge. The cell nucleus is fluorescently labelled to allow automated tracking of cell positions. The bridge length is L = 16 µm. Scale bar, 25 µm.

Supplementary Video 2

Single MDA-MB-231 cells transitioning repeatedly between the square adhesion sites of the two-state micropattern. Transitions are usually preceded by the formation of a protrusion along the bridge. The cell nucleus is fluorescently labelled to allow automated tracking of cell positions. The bridge length is L = 35 µm. Scale bar, 25 µm.

Supplementary Video 3

Single MDA-MB-231 cells transitioning repeatedly between the square adhesion sites of the two-state micropattern. Transitions are usually preceded by the formation of a protrusion along the bridge. The cell nucleus is fluorescently labelled to allow automated tracking of cell positions. The bridge length is L = 56 µm. Scale bar, 25 µm.

Supplementary Video 4

Exemplary field of view of MDA-MB-231 cells migrating on two-state micropatterns of the same bridge length (L = 35 µm). All cells perform transitions between the square adhesion sites. Not all micropatterns are occupied, which is due to the low cell seeding density used to ensure single-cell occupancy. Cell nuclei are labelled for semi-automated detection of cell positions. Scale bar, 25 µm.

Supplementary Video 5

Single MDA-MB-231 cells transfected with LifeAct-GFP to visualize actin on two-state micropatterns of bridge length L = 35 µm. The outline of the underlying micropattern is drawn as a reference up to scale. Actin hotspots are visible at the tip of the transition-mediating lamellipodium, as well as during the dynamic exploration of the square adhesion sites. Actin fibres reorganize dynamically.

Supplementary Video 6

Single MDA-MB-231 cells transfected with LifeAct-GFP to visualize actin on two-state micropatterns of bridge length L = 35 µm. The outline of the underlying micropattern is drawn as a reference up to scale. Actin hotspots are visible at the tip of the transition-mediating lamellipodium, as well as during the dynamic exploration of the square adhesion sites. Actin fibres reorganize dynamically.

Supplementary Video 7

Single MDA-MB-231 cell on a stripe micropattern without constriction of total length 103 µm. The cell moves back and forth, repolarizing on contact with the pattern’s borders. When the cell is positioned in the middle of the pattern, quick changes in the direction of lamellipodia formation can be seen. The cell nucleus is fluorescently labelled to allow automated tracking of cell positions.

Supplementary Video 8

Sparsely seeded MDA-MB-231 cells freely migrating on a homogeneous fibronectin-coated 2D surface. Cells move randomly on the surface. Cell nuclei are fluorescently labelled for automated cell tracking. Scale bar, 100 µm.

Supplementary Video 9

Single MCF10A cell transitioning repeatedly between the square adhesion sites of the two-state micropattern. Transitions are usually preceded by the formation of a protrusion along the bridge. Several times, protrusions along the bridge are formed that do not lead to a transition. The cell nucleus is fluorescently labelled to allow automated tracking of cell positions. Bridge length L = 35 µm.

Supplementary Video 10

Single MDA-MB-436 cell transitioning between the square adhesion sites of the two-state micropattern. The cell nucleus is fluorescently labelled to allow automated tracking of cell positions. Bridge length L = 35 µm.

Supplementary Video 11

Single MDCK cell transitioning between the square adhesion sites of the two-state micropattern. The cell nucleus is fluorescently labelled to allow automated tracking of cell positions. Bridge length L = 35 µm.

Supplementary Video 12

Single HuH7 cell transitioning between the square adhesion sites of the two-state micropattern. The cell nucleus is fluorescently labelled to allow automated tracking of cell positions. Bridge length L = 35 µm.

Supplementary Video 13

Single A549 cell transitioning between the square adhesion sites of the two-state micropattern. The cell nucleus is fluorescently labelled to allow automated tracking of cell positions. Bridge length L = 35 µm.

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Brückner, D.B., Fink, A., Schreiber, C. et al. Stochastic nonlinear dynamics of confined cell migration in two-state systems. Nat. Phys. 15, 595–601 (2019). https://doi.org/10.1038/s41567-019-0445-4

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