An optimization problem for the fundamental eigenvalue λ 0 of the Laplacian in a planar simply-connected domain that contains N small identically-shaped holes, each of radius ε 1, is considered. The boundary condition on the domain is assumed to be of Neumann type, and a Dirichlet condition is imposed on the boundary of each of the holes. As an application, the reciprocal of the fundamental eigenvalue λ 0 is proportional to the expected lifetime for Brownian motion in a domain with a reflecting boundary that contains N small traps. For small hole radii ε, a two-term asymptotic expansion for λ 0 is derived in terms of certain properties of the Neumann Green's function for the Laplacian. Only the second term in this expansion depends on the locations x i , for i = 1,. .. , N, of the small holes. For the unit disk, ring-type configurations of holes are constructed to optimize this term with respect to the hole locations. The results yield hole configurations that asymptotically optimize λ 0. For a class of symmetric dumbbell-shaped domains containing exactly one hole, it is shown that there is a unique hole location that maximizes λ 0. For an asymmetric dumbbell-shaped domain, it is shown that there can be two hole locations that locally maximize λ 0. This optimization problem is found to be directly related to an oxygen transport problem in skeletal muscle tissue, and to determining equilibrium locations of spikes to the Gierer-Meinhardt reactiondiffusion model. It is also closely related to the problem of determining equilibrium vortex configurations within the context of the Ginzburg-Landau theory of superconductivity.
Abstract. 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 twocomponent 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.1. Introduction. We examine the effect of a spatially-dependent plant hormone distribution on a family of proteins associated with root hair (RH) initiation in a specific plant cell. This process is modeled by a generalized Schnakenberg reaction-diffusion (RD) system on a 2-D domain with both source and loss terms, and with a spatial gradient modeling the spatially inhomogeneous distribution of the plant hormone auxin. This system is an extension of a model proposed by Payne and Grierson in [33], and analyzed in a 1-D context in the companion articles [4,6]. The new goal of this paper, in comparison with [4,6], is to analyze 2-D localized spot patterns in the RD system (1.2), and how these 2-D patterns are influenced by the spatially inhomogeneous auxin distribution.We now give a brief description of the biology underlying the RD model. In this model, an on-and-off switching process of a small G-protein subfamily, called the Rho of Plants (ROPs), is assumed to occur in a RH cell of the plant Arabidopsis thaliana. ROPs are known to be involved in RH cell morphogenesis at several distinct stages (see [13,23] for details). Such a biochemical process is believed to be catalyzed by a plant hormone called auxin (cf. [33]). Typically, auxin-transport models are formulated to study polarization events between cells (cf. [14]). However, little is known about specific details of auxin flow within a cell. In [4,6,33] a simple transport process is assumed to govern the auxin flux through a RH cell, which postulates that auxin diffuses much faster than ROPs in the cell, owing partially to the in-and out-pum...
A generalized 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 coordinate. Localized stripe-like solutions of active ROP occur for high enough total auxin concentration and lie on a complex bifurcation diagram of single-and multipulse solutions. Transverse stability computations, confirmed by numerical simulation show that, apart from a boundary stripe, these one-dimensional (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 twodimensional (2D) numerical continuation analysis is performed that shows that the various stable hybrid spot-like states can coexist. The parameter values studied lead to a natural, singularly perturbed, so-called semistrong interaction regime. This scaling enables an analytical explanation of the initial instability by describing the dispersion relation of a certain nonlocal eigenvalue problem. The analytical results are found to agree favorably with the numerics. Possible biological implications of the results are discussed.
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 twocomponent 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.
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