The leaf venation pattern of plants shows remarkable diversity and species-specificity. However, the mechanism underlying the pattern formation and pattern diversity remains unclear. We developed a mathematical model that is based on the positive feedback regulation between plant hormone auxin and its efflux carrier. This system can generate auxin flow pathways by self-organization from an almost homogeneous state. This result explains a well-known experimental phenomenon referred as to "polar auxin transport." The model can produce diverse leaf venation patterns with spatial regularity under similar conditions to those of leaf development, that is, in the presence of leaf expansion and auxin sink. Final venation patterns are strikingly affected by leaf shape and leaf expansion. These results indicate that the positive feedback regulation between auxin and its efflux carrier is a central dynamic in leaf venation pattern formation. The diversity of leaf venation patterns in plant species is probably due to the differences of leaf shape and leaf expansion pattern.
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