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Forced and spontaneous symmetry breaking in cell polarization

Nature Computational Science volume 2, pages 504–511 (2022)Cite this article

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An Author Correction to this article was published on 17 October 2022

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Abstract

How does breaking the symmetry of an equation alter the symmetry of its solutions? Here, we systematically examine how reducing underlying symmetries from spherical to axisymmetric influences the dynamics of an archetypal model of cell polarization, a key process of biological spatial self-organization. Cell polarization is characterized by nonlinear and non-local dynamics, but we overcome the theory challenges these traits pose by introducing a broadly applicable numerical scheme allowing us to efficiently study continuum models in a wide range of geometries. Guided by numerical results, we discover a dynamical hierarchy of timescales that allows us to reduce relaxation to a purely geometric problem of area-preserving geodesic curvature flow. Through application of variational results, we analytically construct steady states on a number of biologically relevant shapes. In doing so, we reveal non-trivial solutions for symmetry breaking.

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Fig. 1: Dynamics of cell polarization.
Fig. 2: Pattern formation over a hierarchy of timescales.
Fig. 3: Geometric effects on cap localization.
Fig. 4: Polarity selection on asymmetric surfaces.

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Data availability

Source data for all quantitative results are available with this manuscript and on Zenodo50.

Code availability

The code for our scheme is open source and an implementation in MATLAB (R2021a) is available on both GitHub26 and Zenodo27. This code is licensed under the MIT license.

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References

  1. Heath, T. L. et al. The Thirteen Books of Euclid’s Elements (Courier, 1956).

  2. Gross, P. et al. Guiding self-organized pattern formation in cell polarity establishment. Nat. Phys. 15, 293–300 (2019).

    Article  Google Scholar 

  3. Mietke, A., Jemseena, V., Kumar, K. V., Sbalzarini, I. F. & Jülicher, F. Minimal model of cellular symmetry breaking. Phys. Rev. Lett. 123, 188101 (2019).

    Article  MathSciNet  Google Scholar 

  4. Brauns, F. et al. Bulk–surface coupling identifies the mechanistic connection between Min-protein patterns in vivo and in vitro. Nat. Commun. 12, 3312 (2021).

    Article  Google Scholar 

  5. Zhu, M. et al. Developmental clock and mechanism of de novo polarization of the mouse embryo. Science 370, eabd2703 (2020).

    Article  Google Scholar 

  6. Chen, W., Nie, Q., Yi, T.-M. & Chou, C.-S. Modelling of yeast mating reveals robustness strategies for cell–cell interactions. PLoS Comput. Biol. 12, e1004988 (2016).

    Article  Google Scholar 

  7. Rappel, W.-J. & Edelstein-Keshet, L. Mechanisms of cell polarization. Curr. Opin. Syst. Biol. 3, 43–53 (2017).

    Article  Google Scholar 

  8. Tostevin, F., Wigbers, M., Søgaard-Andersen, L. & Gerland, U. Four different mechanisms for switching cell polarity. PLoS Comput. Biol. 17, e1008587 (2021).

    Article  Google Scholar 

  9. Goryachev, A. B. & Leda, M. Many roads to symmetry breaking: molecular mechanisms and theoretical models of yeast cell polarity. Mol. Biol. Cell 28, 370–380 (2017).

    Article  Google Scholar 

  10. Thalmeier, D., Halatek, J. & Frey, E. Geometry-induced protein pattern formation. Proc. Natl Acad. Sci. USA 113, 548–553 (2016).

    Article  Google Scholar 

  11. Diegmiller, R., Montanelli, H., Muratov, C. B. & Shvartsman, S. Y. Spherical caps in cell polarization. Biophys. J. 115, 26–30 (2018).

    Article  Google Scholar 

  12. Bäcker, J. P. & Röger, M. Analysis and asymptotic reduction of a bulk-surface reaction-diffusion model of Gierer-Meinhardt type. Commun. Pure Appl. Math. 21, 1139 (2022).

    MathSciNet  MATH  Google Scholar 

  13. Gamba, A., Kolokolov, I., Lebedev, V. & Ortenzi, G. Universal features of cell polarization processes. J. Stat. Mech. Theory Exp. 2009, P02019 (2009).

    Article  Google Scholar 

  14. Cusseddu, D., Edelstein-Keshet, L., Mackenzie, J. A., Portet, S. & Madzvamuse, A. A coupled bulk–surface model for cell polarisation. J. Theor. Biol. 481, 119–135 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  15. Geßele, R., Halatek, J., Würthner, L. & Frey, E. Geometric cues stabilise long-axis polarisation of PAR protein patterns in C. elegans. Nat. Commun. 11, 539 (2020).

    Article  Google Scholar 

  16. Rätz, A. & Röger, M. Symmetry breaking in a bulk–surface reaction–diffusion model for signalling networks. Nonlinearity 27, 1805 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  17. Trogdon, M. et al. The effect of cell geometry on polarization in budding yeast. PLoS Comput. Biol. 14, e1006241 (2018).

    Article  Google Scholar 

  18. Golubitsky, M., Stewart, I. & Schaeffer, D. G. Singularities and Groups in Bifurcation Theory: Volume II (Applied Mathematical Sciences Vol. 69, Springer, 2012).

  19. Merilees, P. E. The pseudospectral approximation applied to the shallow water equations on a sphere. Atmosphere 11, 13–20 (1973).

    Article  Google Scholar 

  20. Orszag, S. A. Fourier series on spheres. Mon. Weather Rev. 102, 56–75 (1974).

    Article  Google Scholar 

  21. 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).

    Article  MathSciNet  MATH  Google Scholar 

  22. Goryachev, A. B. & Pokhilko, A. V. Dynamics of cdc42 network embodies a Turing-type mechanism of yeast cell polarity. FEBS Lett. 582, 1437–1443 (2008).

    Article  Google Scholar 

  23. Otsuji, M. et al. A mass conserved reaction–diffusion system captures properties of cell polarity. PLoS Comput. Biol. 3, e108 (2007).

    Article  MathSciNet  Google Scholar 

  24. Mori, Y., Jilkine, A. & Edelstein-Keshet, L. Wave-pinning and cell polarity from a bistable reaction–diffusion system. Biophys. J. 94, 3684–3697 (2008).

    Article  Google Scholar 

  25. Sharma, V. & Morgan, J. Global existence of solutions to reaction–diffusion systems with mass transport type boundary conditions. SIAM J. Math. Anal. 48, 4202–4240 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  26. Fortunato, D. Spectral methods for reaction–diffusion on axisymmetric surfaces. GitHub https://github.com/danfortunato/surface-diffusion (2022).

  27. Fortunato, D. danfortunato/surface-diffusion (v1.0.0). Zenodo https://doi.org/10.5281/zenodo.6762738 (2022).

  28. Ihrig, E. & Golubitsky, M. Pattern selection with O(3) symmetry. Physica D 13, 1–33 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  29. Henry, M., Hilhorst, D. & Muratov, C. B. A multiple scale pattern formation cascade in reaction–diffusion systems of activator–inhibitor type. Interfaces Free Bound. 20, 297–336 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  30. Camley, B. A., Zhao, Y., Li, B., Levine, H. & Rappel, W.-J. Crawling and turning in a minimal reaction–diffusion cell motility model: coupling cell shape and biochemistry. Phys. Rev. E 95, 012401 (2017).

    Article  Google Scholar 

  31. Vanderlei, B., Feng, J. J. & Edelstein-Keshet, L. A computational model of cell polarization and motility coupling mechanics and biochemistry. Multiscale Model. Simul. 9, 1420–1443 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  32. Brauns, F., Weyer, H., Halatek, J., Yoon, J. & Frey, E. Wavelength selection by interrupted coarsening in reaction–diffusion systems. Phys. Rev. Lett. 126, 104101 (2021).

    Article  Google Scholar 

  33. Ritoré, M. & Sinestrari, C. Mean Curvature Flow and Isoperimetric Inequalities (Springer, 2010).

  34. Rubinstein, J. & Sternberg, P. Nonlocal reaction–diffusion equations and nucleation. IMA J. Appl. Math. 48, 249–264 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  35. Ros, A. The isoperimetric problem. Glob. Theory Minim. Surf. 2, 175–209 (2001).

    MathSciNet  MATH  Google Scholar 

  36. Morgan, F. & Johnson, D. L. Some sharp isoperimetric theorems for Riemannian manifolds. Indiana Univ. Math. J. 49, 1017–1041 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  37. Ritoré, M. Constant geodesic curvature curves and isoperimetric domains in rotationally symmetric surfaces. Commun. Anal. Geom. 9, 1093–1138 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  38. Fonda, P., Rinaldin, M., Kraft, D. J. & Giomi, L. Interface geometry of binary mixtures on curved substrates. Phys. Rev. E 98, 032801 (2018).

    Article  Google Scholar 

  39. Wigbers, M. C. et al. A hierarchy of protein patterns robustly decodes cell shape information. Nat. Phys. 17, 578–584 (2021).

    Article  Google Scholar 

  40. Zhang, Z., Zwick, S., Loew, E., Grimley, J. S. & Ramanathan, S. Mouse embryo geometry drives formation of robust signaling gradients through receptor localization. Nat. Commun. 10, 4516 (2019).

    Article  Google Scholar 

  41. Tan, T. H. et al. Topological turbulence in the membrane of a living cell. Nat. Phys. 16, 657–662 (2020).

    Article  Google Scholar 

  42. Deneke, V. E., Melbinger, A., Vergassola, M. & Di Talia, S. Waves of cdk1 activity in S phase synchronize the cell cycle in Drosophila embryos. Dev. Cell 38, 399–412 (2016).

    Article  Google Scholar 

  43. Liu, J. et al. Topological braiding and virtual particles on the cell membrane. Proc. Natl Acad. Sci. USA 118, e2104191118 (2021).

    Article  Google Scholar 

  44. Miller, P. W., Stoop, N. & Dunkel, J. Geometry of wave propagation on active deformable surfaces. Phys. Rev. Lett. 120, 268001 (2018).

    Article  Google Scholar 

  45. Baker, R. E. & Maini, P. A mechanism for morphogen-controlled domain growth. J. Math. Biol. 54, 597–622 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  46. Gomez, D., Iyaniwura, S., Paquin-Lefebvre, F. & Ward, M. Pattern forming systems coupling linear bulk diffusion to dynamically active membranes or cells. Philos. Trans. R. Soc. A 379, 20200276 (2021).

    Article  MathSciNet  Google Scholar 

  47. Kirk, B. S., Peterson, J. W., Stogner, R. H. & Carey, G. F. libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations. Eng. Comput. 22, 237–254 (2006).

    Article  Google Scholar 

  48. Calvo, M., de Frutos, J. & Novo, J. Linearly implicit Runge–Kutta methods for advection–reaction–diffusion equations. Appl. Numer. Math. 37, 535–549 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  49. Miller, P. W. & Dunkel, J. Gait-optimized locomotion of wave-driven soft sheets. Soft Matter 16, 3991–3999 (2020).

    Article  Google Scholar 

  50. Miller, P. Source data for ‘Forced and spontaneous symmetry breaking in cell polarization’. Zenodo https://doi.org/10.5281/zenodo.6774314 (2022).

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Acknowledgements

We thank Rocky Diegmiller, Boris Slepchenko, Martin Golubistky and Matteo Novaga for helpful discussions, and Lucy Reading-Ikkanda for assistance with graphical design. This work was supported by NIH grant R01 GM134204 to S.S.

Author information

Author notes

  1. These authors contributed equally: Pearson W. Miller, Daniel Fortunato.

Authors and Affiliations

  1. Center for Computational Biology, Flatiron Institute, New York, NY, USA

    Pearson W. Miller & Stanislav Shvartsman

  2. Center for Computational Mathematics, Flatiron Institute, New York, NY, USA

    Daniel Fortunato & Leslie Greengard

  3. Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, USA

    Cyrill Muratov

  4. Dipartimento di Matematica, Università di Pisa, Pisa, Italy

    Cyrill Muratov

  5. Courant Institute, New York University, New York, NY, USA

    Leslie Greengard & Stanislav Shvartsman

  6. Department of Molecular Biology, Princeton University, Princeton, NJ, USA

    Stanislav Shvartsman

  7. Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ, USA

    Stanislav Shvartsman

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Contributions

S.S., L.G. and C.M. designed the research. P.W.M. performed analytical and numerical studies. D.F. developed the numerical method and software. All authors discussed the results and wrote the paper.

Corresponding authors

Correspondence to Cyrill Muratov, Leslie Greengard or Stanislav Shvartsman.

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Nature Computational Science thanks Johannes Borgqvist, Anotida Madzvamuse and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Primary Handling Editor: Ananya Rastogi, in collaboration with the Nature Computational Science team.

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Supplementary Information

Supplementary Discussion, Figs. 1–4 and Table 1.

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Supplementary Video 1

Simulation of cell polarization on a variety of surfaces.

Supplementary Video 2

Different initial conditions on an egg-shaped surface.

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Numerical source data.

Source Data Fig. 3

Numerical source data.

Source Data Fig. 4

Numerical source data.

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Miller, P.W., Fortunato, D., Muratov, C. et al. Forced and spontaneous symmetry breaking in cell polarization. Nat Comput Sci 2, 504–511 (2022). https://doi.org/10.1038/s43588-022-00295-0

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