Self-organization of the vascular system in plant leaves: inter-dependent dynamics of auxin flux and carrier proteins.

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.