Models of Vascular Pattern Formation in Leaves.

The vegetative hormone Auxin is involved in vascular tissues formation throughout the plant. Trans-membrane carrier proteins transporting auxin from cell to cell and distributed asymmetrically around each cell give to auxin a polarized movement in tissues, creating streams of auxin that presume future vascular bundles. According to the canalization hypothesis, auxin transport ability of cells is thought to increase with auxin flux, resulting in the self-enhancement of this flux along auxin paths. In this study we evaluate a series of models based on canalization hypothesis using carrier proteins, under different assumptions concerning auxin flux formation and carrier protein dynamics. Simulations are run on a hexagonal lattice with uniform auxin production. A single cell located in the margin of the lattice indicates the petiole, and acts as an auxin sink. The main results are: (1) We obtain branching auxin distribution patterns. (2) The type of self-enhancement described by the functional form of the carrier proteins regulation responding to the auxin flux intensity in different parts of a cell, has a strong effect on the possibility of generating the branching patterns. For response functions with acceleration in the increase of carrier protein numbers compared to the auxin flux, branching patterns are likely to be generated. For linear or decelerating response functions, no branching patterns are formed. (3) When branching patterns are formed, auxin distribution greatly differs between the case in which the number of carrier proteins in different parts of a cell are regulated independently, and the case in which different parts of a cell compete for a limited number of carrier proteins. In the former case, the auxin level is lower in veins than in the surrounding tissue, while in the latter, the auxin is present in greater abundance in veins. These results suggest that canalization is a good candidate for describing plant vein pattern formation. ****** Formation of the vascular system in plant leaves can be explained by the canalization hypothesis which states that veins are formed in an initially homogeneous field by a self-organizing process between the plant hormone auxin and auxin carrier proteins. Previous models of canalization can generate vein patterns with branching but fail to generate vein patterns with closed loops. However, closed vein loops are commonly observed in plant leaves and are important in making them robust to herbivore attacks and physical damage. Here we propose a new model which generates a vein system with closed loops. We postulate that the ‘‘flux bifurcator'' level is enhanced in cells with a high auxin flux and that it causes reallocation of auxin carriers toward neighbouring cells also having a high bifurcator level. This causes the auxin flux to bifurcate, allowing vein tips to attach to other veins creating vein loops. We explore several alternative functional forms for the flux bifurcator affecting the reallocation of efflux carriers and examine parameter dependence of the resulting vein pattern.J'étudie la formation du système vasculaire des feuilles des plantes à l'aide de modèles mathématiques. L'hypothèse de canalisation d'une phytohormone, l'auxin, stipule que l'auto activation de son transport entre les cellules crée des chemins préférentiels qui se différencieront plus tard en système vasculaire. J'entreprends une analyse numérique de modèles de canalisation sur une grande matrice et parviens à créer des motifs branchés dans lesquels circule l'auxin. Une analyse de stabilité d'un modèle simplifié nous éclaire sur les raisons de la formation de ces motifs et l'impossibilité de créer un réseau réticulé. La majorité des plantes ayant un système vasculaire réticulé, je modifie le modèle de façon à obtenir ce type de réseau. En ajoutant une variable biologiquement plausible je parviens à créer un réseau réticulé dans lequel l'auxin circule uniformément. Enfin, je discute des relations entre la formation du système vasculaire et de la spirale de phyllotaxie