Mathematical Modelling of the Phloem: The Importance of Diffusion on Sugar Transport at Osmotic Equilibrium

Abstract

Plants rely on the conducting vessels of the phloem to transport the products of photosynthesis from the leaves to the roots, or to any other organs, for growth, metabolism and storage. Transport within the phloem is due to an osmotically-generated pressure gradient and is hence inherently nonlinear. Since convection dominates over di↵usion in the main bulk flow, the e↵ects of di↵usive transport have generally been neglected by previous authors. However, di↵usion is important due to boundary layers that form at the ends of the phloem, and at the leaf-stem and stem-root boundaries. We present a mathematical model of transport which includes the e↵ects of di↵usion. We solve the system analytically in the limit of high Munch number which corresponds to osmotic equilibrium, and numerically for all parameter values. We find that the bulk solution is dependent on the di↵usion-dominated boundary layers. Hence, even for large Péclet number, it is not always correct to neglect di↵usion. We consider the cases of passive and active sugar loading and unloading. We show that for active unloading the solutions diverge with increasing Péclet. For passive unloading the convergence of the solutions is dependent on the magnitude of loading. Di↵usion also permits the modelling of an axial e✏ux of sugar in the root zone which may be important for the growing root tip and for promoting symbiotic biological interactions in the soil. Therefore, di↵usion is an essential mechanism for transport in the phloem and must be included to accurately predict flow.

Wet-tip versus dry-tip regimes of osmotically driven fluid flow

“…used their phloem flow model to establish a biomechanical length limit for plant needles around the observed 5 cm 20 . Among these theoretical studies, some considered the bi-directionally coupled osmolyte-flow system but had to apply simplifying assumptions for solving the model, i.e., either negligible hydrodynamic pressure compared to the osmotic pressure, thereby uncoupling the problem 21–23 , or considering a limit case of small solute secretion rate 4 and, additionally, diffusion dominating over advection 19 , or replacing the mixed set of boundary conditions by pure velocity boundary conditions and not solving for the pressure profile explicitly 12–18 .…”

A Proposed Drought Response Equation Added to the Münch-Horwitz Theory of Phloem Transport

“…Goeschl et aI. 1976, Thompson and Holbrook 2003, Hölttä et al 2006, Jensen et al 2011, Payvendi et al 2014. Mathematically consistent models must include equations for at least five dependent variables, (1) the rates of solute Unloading [jUi] in sink areas, (2) Osmotic Influx and Efflux [jWi] of water through the sieve tube membrane, (3) turgor Pressure [Pi], (4) transport Speed (i.e.…”

The Application of Contrast Media for In Vivo Feature Enhancement in X-Ray Computed Tomography of Soil-Grown Plant Roots

“…The absence of visible root tips in the scan volumes of interest makes it difficult to assess the end fate of the medium. Under normal conditions, photosynthesates are unloaded from phloem at sinks (i.e., root tips), a process which, along with the osmotic transport of fluid between xylem and phloem, can occur along the entire root length, with the result that the axial velocity profile of the phloem is nonconstant and nonlinear in both space and time (Payvandi et al, 2014 b ). This is one among a number of reasons that the end fate of iodinated contrast molecules is difficult to predict.…”