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Forced and spontaneous symmetry breaking in cell polarization

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

This article has been updated


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.

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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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.

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Authors and Affiliations



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 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.

Source data

Source Data Fig. 2

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).

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