A simplest steady-state Munch-like model of phloem translocation, with source and pathway and sink

Abstract

In the 80 years since its introduction by Münch, the pressure-driven mass-flow model of phloem translocation has become hegemonic, and has been mathematically modelled in many different fashions but not, to our knowledge, by one that incorporated the equations of hydrodynamics with those of osmosis and slice-source and slice-sink boundary conditions to yield a system that admits of an analytical steady-state solution for the sap velocity in a single sieve tube. To overcome this situation, we drastically simplified the problem by: (i) justifying a low Peclet number idealisation in which transverse variations could be neglected; (ii) justifying a low viscosity idealisation in which axial pressure drops could be neglected; and (iii) assuming a sink of strength sufficient to lower the photosynthate concentration at the extreme distal end of the sieve tube to levels at which it became unimportant. The resulting ordinary nonlinear second-order differential equation in sap velocity and axial position was of a generalised Liénard form with a single forcing parameter; and this is reason enough for the lack of a known analytic solution. However, since the forcing parameter was very large, it was possible to deduce approximate second-order solutions for behavior in the source, sink and transport regions: the sap velocity is zero at the slice-source, climbs with exponential rapidity to a plateau, maintains this plateau over most of the sieve tube, and then drops with exponential rapidity to zero at the slice-sink.