Clustering versus sorting: a mass-conserving reaction-diffusion model of planar polarity puncta
Abstract
Planar cell polarity is preceded by the clustering of polarity proteins into discrete, low-turnover membrane subdomains (puncta), yet the minimal interactions that nucleate puncta, set their number, and segregate opposite orientations remain unclear. We address these questions with a mass-conserving reaction-diffusion model in which two diffusible monomers bind reversibly across a cell-cell junction into trans-complexes of two orientations, with feedback entering only through concentration-dependent rates. Above a critical density, the uniform state undergoes a long-wavelength mass-redistribution instability rather than a finite-wavelength Turing bifurcation. The form of the feedback then selects between two morphologies: broad mesas fixed by a Maxwell construction under saturating feedback, and narrow mass-limited spikes under unbounded feedback. For spikes we obtain closed-form expressions exhibiting a clean separation of amplitude (mass and feedback), width (the complex diffusion length), and spacing (the monomer screening length). Within a fast-monomer reduction we prove, for any number and arrangement of puncta, that like-oriented arrays coarsen, so multiplicity is metastable and kinetically determined. The second monomer reservoir introduces a second screening length that rate-limits competition by the harmonic mean of the monomer diffusivities, and the spectrum of a punctum remains free of oscillatory ("blinking") instabilities throughout. Finally, sign-definite cross-modulation of turnover converts clustering into orientation sorting - the mutual exclusion of orientations along a single contact: mass redistribution sets puncta number, and the sign of the cross-coupling determines whether orientations segregate. Puncta number and orientation sorting are thus governed by mathematically separable ingredients.
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BibTeX
@misc{hossein2026clustering,
title = {{Clustering versus sorting: a mass-conserving reaction-diffusion model of planar polarity puncta}},
author = {Sk Raj Hossein and Eman Alwani and Andreas Buttenschön and Alexander G. Fletcher},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29679},
eprint = {2608.29679},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29679},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Clustering versus sorting: a mass-conserving reaction-diffusion model of planar polarity puncta
AU - Sk Raj Hossein
AU - Eman Alwani
AU - Andreas Buttenschön
AU - Alexander G. Fletcher
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29679
UR - https://arxiv.org/abs/2608.29679
AB - Planar cell polarity is preceded by the clustering of polarity proteins into discrete, low-turnover membrane subdomains (puncta), yet the minimal interactions that nucleate puncta, set their number, and segregate opposite orientations remain unclear. We address these questions with a mass-conserving reaction-diffusion model in which two diffusible monomers bind reversibly across a cell-cell junction into trans-complexes of two orientations, with feedback entering only through concentration-dependent rates. Above a critical density, the uniform state undergoes a long-wavelength mass-redistribution instability rather than a finite-wavelength Turing bifurcation. The form of the feedback then selects between two morphologies: broad mesas fixed by a Maxwell construction under saturating feedback, and narrow mass-limited spikes under unbounded feedback. For spikes we obtain closed-form expressions exhibiting a clean separation of amplitude (mass and feedback), width (the complex diffusion length), and spacing (the monomer screening length). Within a fast-monomer reduction we prove, for any number and arrangement of puncta, that like-oriented arrays coarsen, so multiplicity is metastable and kinetically determined. The second monomer reservoir introduces a second screening length that rate-limits competition by the harmonic mean of the monomer diffusivities, and the spectrum of a punctum remains free of oscillatory ("blinking") instabilities throughout. Finally, sign-definite cross-modulation of turnover converts clustering into orientation sorting - the mutual exclusion of orientations along a single contact: mass redistribution sets puncta number, and the sign of the cross-coupling determines whether orientations segregate. Puncta number and orientation sorting are thus governed by mathematically separable ingredients.
ER -