Mathematical Modeling of Plant Root Hair Initiation: Dynamics of Localized Patches

A mathematical analysis is undertaken of a Schnakenberg reaction-diffusion system in one dimension with a spatial gradient governing the active reaction. This system has previously been proposed as a model of the initiation of hairs from the root epidermis Arabidopsis, a key cellular-level morphogen...

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Bibliographic Details
Published in:SIAM journal on applied dynamical systems 2014-01, Vol.13 (1), p.210-248
Main Authors: Bren͂a--Medina, V., Champneys, A. R., Grierson, C., Ward, M. J.
Format: Article
Language:English
Subjects:
Applied mathematics
Biological effects
Dynamical systems
Dynamics
Eigenvalues
Hair
Hypotheses
Hysteresis
Mathematical analysis
Mathematical models
Morphogenesis
Parameters
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Summary:A mathematical analysis is undertaken of a Schnakenberg reaction-diffusion system in one dimension with a spatial gradient governing the active reaction. This system has previously been proposed as a model of the initiation of hairs from the root epidermis Arabidopsis, a key cellular-level morphogenesis problem. This process involves the dynamics of the small G-proteins, Rhos of plants, which bind to form a single localized patch on the cell membrane, prompting cell wall softening and subsequent hair growth. A numerical bifurcation analysis is presented as two key parameters, involving the cell length and the overall concentration of the auxin catalyst, are varied. The results show hysteretic transitions from a boundary patch to a single interior patch, and to multiple patches whose locations are carefully controlled by the auxin gradient. The results are confirmed by an asymptotic analysis using semistrong interaction theory, leading to closed form expressions for the patch locations and intensities. A close agreement between the numerical bifurcation results and the asymptotic theory is found for biologically realistic parameter values. Insight into the initiation of transition mechanisms is obtained through a linearized stability analysis based on a nonlocal eigenvalue problem. The results provide further explanation of the recent agreement found between the model and biological data for both wild-type and mutant hair cells. [PUBLICATION ABSTRACT]
ISSN:1536-0040
1536-0040
DOI:10.1137/120902264