Long-distance hormone transport via the phloem.

Several key plant hormones are synthesised in the shoot and are advected within the phloem to the root tip. In the root tip, these hormones regulate growth and developmental processes, and responses to environmental cues. However, we lack understanding of how environmental factors and biological parameters affect the delivery of hormones to the root tip. In this study, we build on existing models of phloem flow to develop a mathematical model of sugar transport alongside the transport of a generic hormone. We derive the equations for osmotically driven flow in a long, thin pipe with spatially varying membrane properties to capture the phloem loading and unloading zones. Motivated by experimental findings, we formulate solute membrane transport in terms of passive and active components, and incorporate solute unloading via bulk flow (i.e. advection with the water efflux) by including the Staverman reflection coefficient. We use the model to investigate the coupling between the sugar and hormone dynamics. The model predicts that environmental cues that lead to an increase in active sugar loading, an increase in bulk flow sugar unloading or a decrease in the relative root sugar concentration result in an increase in phloem transport velocity. Furthermore, the model reveals that such increases in phloem transport velocity result in an increase in hormone delivery to the root tip for passively loaded hormones.