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arXiv · 1403.5318

Stripe to spot transition in a plant root hair initiation model

V. F. Breña-Medina·D. Avitabile·A. R. Champneys·M. J. Ward
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Abstract

A generalised Schnakenberg reaction-diffusion system with source and loss terms and a spatially dependent coefficient of the nonlinear term is studied both numerically and analytically in two spatial dimensions. The system has been proposed as a model of hair initiation in the epidermal cells of plant roots. Specifically the model captures the kinetics of a small G-protein ROP, which can occur in active and inactive forms, and whose activation is believed to be mediated by a gradient of the plant hormone auxin. Here the model is made more realistic with the inclusion of a transverse co-ordinate. Localised stripe-like solutions of active ROP occur for high enough total auxin concentration and lie on a complex bifurcation diagram of single and multi-pulse solutions. Transverse stability computations, confirmed by numerical simulation show that, apart from a boundary stripe, these 1D solutions typically undergo a transverse instability into spots. The spots so formed typically drift and undergo secondary instabilities such as spot replication. A novel 2D numerical continuation analysis is performed that shows the various stable hybrid spot-like states can coexist. The parameter values studied lead to a natural singularly perturbed, so-called semi-strong interaction regime. This scaling enables an analytical explanation of the initial instability, by describing the dispersion relation of a certain non-local eigenvalue problem. The analytical results are found to agree favourably with the numerics. Possible biological implications of the results are discussed.

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V. F. Breña-Medina, D. Avitabile, A. R. Champneys, M. J. Ward. 2015-03-24. Stripe to spot transition in a plant root hair initiation model. https://doi.org/10.1137/140964527

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