  We study pattern formation in a 2-D reaction-diffusion (RD) sub-cellular
model characterizing the effect of a spatial gradient of a plant hormone
distribution on a family of G-proteins associated with root-hair (RH)
initiation in the plant cell Arabidopsis thaliana. The activation of these
G-proteins, known as the Rho of Plants (ROPs), by the plant hormone auxin, is
known to promote certain protuberances on root hair cells, which are crucial
for both anchorage and the uptake of nutrients from the soil. Our mathematical
model for the activation of ROPs by the auxin gradient is an extension of the
model of Payne and Grierson [PLoS ONE, 12(4), (2009)], and consists of a
two-component Schnakenberg-type RD system with spatially heterogeneous
coefficients on a 2-D domain. The nonlinear kinetics in this RD system model
the nonlinear interactions between the active and inactive forms of ROPs. By
using a singular perturbation analysis to study 2-D localized spatial patterns
of active ROPs, it is shown that the spatial variations in the nonlinear
reaction kinetics, due to the auxin gradient, lead to a slow spatial alignment
of the localized regions of active ROPs along the longitudinal midline of the
plant cell. Numerical bifurcation analysis, together with time-dependent
numerical simulations of the RD system are used to illustrate both 2-D
localized patterns in the model, and the spatial alignment of localized
structures.
