A time-dependent mathematical expression of the münch hypothesis of phloem transport

A time-dependent mathematical expression of the Münch, osmotically driven mass flow hypothesis of phloem transport is presented. The dependent variables include concentration of solutes, pressure, velocity of phloem sap, osmotic flux of water, and concentration dependent unloading of solutes. The model meets conservation requirements during all iterations, and responds realistically to changes in independent variables. Given the same set of independent variables the time-dependent model converges to the same values as the closed-form steady-state model of Goeschl et al. (1976) regardless of the initial conditions.     

Münch,morphology, microfluidics – our structural problem with the phloem

The sieve tubes of the phloem are enigmatic structures. Their role as channels for the distribution of assimilates was established in the 19th century, but their sensitivity to disturbations has hampered the elucidation of their transport mechanisms and its regulation ever since. Ernst Münch's classical monograph of 1930 is generally regarded as the first coherent theory of phloem transport, but the ‘Münchian’ pressure flow mechanism had been discussed already before the turn of the century. Münch's impact rather rested on his simple physical models of the phloem that visualized pressure flow in an intuitive way, and we argue that the downscaling of such models to realistic, low‐Reynolds‐number sizes will boost our understanding of phloem transport in this century just as Münch's models did in the previous one. However, biologically meaningful physical models that could be used to test predictions of the many existing mathematical models would have to be designed in analogy with natural phloem structures. Unfortunately, the study of phloem anatomy seems in decline, and we still lack basic quantitative data required for evaluating the plausibility of our theoretical deductions. In this review, we provide a subjective overview of unresolved problems in angiosperm phloem structure research within a functional context.     

Experimental tests of the Münch–Horwitz theory of phloem transport: effects of loading rates*

Abstract Extended square-wave tracer kinetics using ¹¹CO₂ were used to measure the speed of transport and activity level (proportional to concentration) in the phloem at high and low loading rates in six species of plants. In all cases, increased loading rates resulted in increased concentration. In most cases speed also increased, however, in two cases speed was lower and tracer activity was much higher at the higher loading rate. All the responses are consistent with the Münch Horwitz theory of phloem transport, depending upon the equation used to represent the unloading mechanism as described in a previous paper (Goeschl & Magnuson, 1986). For example, the latter two cases are consistent with the assumption that the unloading rate was limited by a process with saturable kinetics (enzyme-like).     

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

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 diffusion in the main bulk flow, the effects of diffusive transport have generally been neglected by previous authors. However, diffusion 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 effects of diffusion. We solve the system analytically in the limit of high Münch number which corresponds to osmotic equilibrium and numerically for all parameter values. We find that the bulk solution is dependent on the diffusion-dominated boundary layers. Hence, even for large Péclet number, it is not always correct to neglect diffusion. 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. Diffusion also permits the modelling of an axial efflux 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, diffusion is an essential mechanism for transport in the phloem and must be included to accurately predict flow.     

Physiological implications of the Münch-Horwitz theory of phloem transport: effects of loading rates*

Abstract Predicted effects of phloem loading rates on the five profiles of unloading rate, osmotic water flux, pressure, transport speed and concentration, in hypothetical sieve tubes with different sink properties, were calculated using the steady-state mathematical expression of the Münch hypothesis of phloem transport. The prediction that increased loading rates always increases the concentration, and generally increase the speed of translocates through the sieve tube, is emphasized since these parameters are accessible for experimental testing. This particular prediction contrasts with a previous prediction (Tyree, Christy, & Ferrier, 1974), that where concentration was held constant at the loading end, concentration along the rest of the sieve tube would decrease, while speed would increase greatly. Where the unloading mechanism was assigned saturable (enzyme-like) kinetics, increased loading rates (in the range well below the V_max of the sink) caused both transport speed and concentration to increase. However, as loading rates approached the V_max of the sinks, speed reached a maximum and then declined, and concentration increased substantially. This was particularly true at very high values of Km, e.g. > 0.1 mol cm^?3.     

A closed‐form solution for steady‐state coupled phloem/xylem flow using the Lambert‐W function

A closed‐form solution for steady‐state coupled phloem/xylem flow is presented. This incorporates the basic Münch flow model of phloem transport, the cohesion model of xylem flow, and local variation in the xylem water potential and lateral water flow along the transport pathway. Use of the Lambert‐W function allows this solution to be obtained under much more general and realistic conditions than has previously been possible. Variation in phloem resistance (i.e. viscosity) with solute concentration, and deviations from the Van't Hoff expression for osmotic potential are included. It is shown that the model predictions match those of the equilibrium solution of a numerical time‐dependent model based upon the same mechanistic assumptions. The effect of xylem flow upon phloem flow can readily be calculated, which has not been possible in any previous analytical model. It is also shown how this new analytical solution can handle multiple sources and sinks within a complex architecture, and can describe competition between sinks. The model provides new insights into Münch flow by explicitly including interactions with xylem flow and water potential in the closed‐form solution, and is expected to be useful as a component part of larger numerical models of entire plants.     

Phloem-localized, proton-coupled sucrose carrier ZmSUT1 mediates sucrose efflux under the control of the sucrose gradient and the proton motive force

The phloem network is as essential for plants as the vascular system is for humans. This network, assembled by nucleus- and vacuole-free interconnected living cells, represents a long distance transport pathway for nutrients and information. According to the Münch hypothesis, osmolytes such as sucrose generate the hydrostatic pressure that drives nutrient and water flow between the source and the sink phloem (Münch, E. (1930) Die Stoffbewegungen in der Pflanze, Gustav Fischer, Jena, Germany). Although proton-coupled sucrose carriers have been localized to the sieve tube and the companion cell plasma membrane of both source and sink tissues, knowledge of the molecular representatives and the mechanism of the sucrose phloem efflux is still scant. We expressed ZmSUT1, a maize sucrose/proton symporter, in Xenopus oocytes and studied the transport characteristics of the carrier by electrophysiological methods. Using the patch clamp techniques in the giant inside-out patch mode, we altered the chemical and electrochemical gradient across the sucrose carrier and analyzed the currents generated by the proton flux. Thereby we could show that ZmSUT1 is capable of mediating both the sucrose uptake into the phloem in mature leaves (source) as well as the desorption of sugar from the phloem vessels into heterotrophic tissues (sink). As predicted from a perfect molecular machine, the ZmSUT1-mediated sucrose-coupled proton current was reversible and depended on the direction of the sucrose and pH gradient as well as the membrane potential across the transporter.     

Concentration-dependent Unloading as a Necessary Assumption for a Closed Form Mathematical Model of Osmotically Driven Pressure Flow in Phloem

Previous attempts to model steady state Münch pressure flow in phloem (Christy and Ferrier. [1973]. Plant Physiol. 52: 531-538; and Ferrier et al. [1974]. Plant Physiol. 54: 589-600) lack sufficient equations, and results were produced which do not represent correct mathematical solutions. Additional equations for the present closed form model were derived by assuming that unloading of a given solute is dependent upon the concentration of that solute in the sieve tube elements. Examples of linear and enzymic type unloading mechanisms are given, although other concentration-dependent mechanisms could be substituted. A method for a numerical solution is outlined, and proof of convergence is presented along with some representative data and the speed of computer calculations. The model provides the minimal set of equations for describing the Münch pressure flow hypothesis as it might operate in plants.     

Role of Concentration-dependent Unloading in Mathematical Models of Münch Transport

It has been erroneously claimed (Goeschl et al. [1976] Plant Physiol. 58: 556-562) that concentration-dependent unloading is required for a correct mathematical solution for Münch phloem transport. Although it may prove to be physiologically correct, concentration-dependent unloading is not a mathematical necessity. Furthermore, its use in a mathematical model may not be desirable, because there is an infinite family of solutions for any given unloading distribution, irrespective of whether concentration-dependent unloading is assumed. Some illustrative numerical results are presented.     

Modeling xylem and phloem water flows in trees according to cohesion theory and Münch hypothesis

Water and solute flows in the coupled system of xylem and phloem were modeled together with predictions for xylem and whole stem diameter changes. With the model we could produce water circulation between xylem and phloem as presented by the Münch hypothesis. Viscosity was modeled as an explicit function of solute concentration and this was found to vary the resistance of the phloem sap flow by many orders of magnitude in the possible physiological range of sap concentrations. Also, the sensitivity of the predicted phloem translocation to changes in the boundary conditions and parameters such as sugar loading, transpiration, and hydraulic conductivity were studied. The system was found to be quite sensitive to the sugar-loading rate, as too high sugar concentration, (approximately 7 MPa) would cause phloem translocation to be irreversibly hindered and soon totally blocked due to accumulation of sugar at the top of the phloem and the consequent rise in the viscosity of the phloem sap. Too low sugar loading rate, on the other hand, would not induce a sufficient axial water pressure gradient. The model also revealed the existence of Münch “counter flow”, i.e., xylem water flow in the absence of transpiration resulting from water circulation between the xylem and phloem. Modeled diameter changes of the stem were found to be compatible with actual stem diameter measurements from earlier studies. The diurnal diameter variation of the whole stem was approximately 0.1 mm of which the xylem constituted approximately one-third.     

Phloem Unloading within the Seed coat of Pisum sativum Observed using Surgically Modified Seeds

Phloem unloading in pea seed coats was observed by removingthe embryos from developing seeds and washing the attached coatswith a weakly buffered solution. The quantity of labelled photosynthateappearing in the washing solution varied immediately when thesolute concentration was changed, and is shown to be an osmoticresponse. This response is predicted by the Münch theoryof phloem transport with concentration dependent unloading.Respiratory inhibitors and the sulphydryl modifying reagentPCMBS had a slow effect upon the washout of tracer, which arrivedwithin the seed coat prior to inhibitor application, but completelystopped any washout of tracer arriving after its application.This time-course suggests that the inhibitors were not directlyinhibiting unloading, but preventing further tracer from enteringthe region of unloading within the seed coat. Phloem unloadingwithin the seed coats of Pisum appears to be passive and notdependent upon a PCMBS-sensitive carrier. Key words: Pisum sativum, seeds, phloem unloading     

Real-time imaging of phloem unloading in the root tip of Arabidopsis

Confocal laser scanning microscopy (CLSM) has been used to image phloem transport and unloading in the root tip of Arabidopsis. The fluorescent probe 5(6) carboxyfluorescein (CF) was ester loaded into a single cotyledon and the entire seedling placed within an observation chamber under the microscope. Translocation of CF to the root tip was rapid, followed by unloading into discrete concentric files of cells. The position of the prominent unloading ‘zone’ corresponded precisely with that of the two protophloem files of sieve elements, demonstrating a functional role of these cells in symplastic sieve-element unloading. Symplastic transport following unloading was confined to the elongating zone of the root with little basipetal transport to more mature cells. Following photobleaching of the unloading zone, phloem transport was restored immediately into the protophloem sieve elements, followed rapidly by lateral, symplastic sieve-element unloading. The results demonstrate that phloem transport processes can now be imaged in real time, and non-invasively, within an intact plant system.     

Application of a single-solute non-steady-state phloem model to the study of long-distance assimilate transport

A mass-balanced, finite-difference solution to Münch's osmotically generated pressure-flow hypothesis is developed for the study of non-steady-state sucrose transport in the phloem tissue of plants. Major improvements over previous modeling efforts are the inclusion of wall elasticity, nonlinear functions of viscosity and solute potential, an enhanced calculation of sieve pore resistance, and the introduction of a slope-limiting total variation diminishing method for determining the concentration of sucrose at node boundaries. The numerical properties of the model are discussed, as is the history of the modeling of pressure-driven phloem transport. Idealized results are presented for a sharp, fast-moving concentration front, and the effect of changing sieve tube length on the transport of sucrose in both the steady-state and non-steady-state cases is examined. Most of the resistance to transport is found to be axial, rather than radial (via membrane transport), and most of the axial resistance is due to the sieve plates. Because of the sieve plates, sieve tube elasticity does not provide a significant enhancement to conductivity at high pressure, as previously suspected. The transit time of sucrose through a sieve tube is found to be inversely proportional to the square of the sieve tube's length; following that observation, it is suggested that 20 1-m-long sieve tubes could transport sucrose 20 times faster than a single 20 m sieve tube. Short sieve tubes would be highly sensitive to differentials between loading and unloading rate, and would require close cooperation with adjacent companion cells for proper function.     

Universality of phloem transport in seed plants

Since Münch in the 1920s proposed that sugar transport in the phloem vascular system is driven by osmotic pressure gradients, his hypothesis has been strongly supported by evidence from herbaceous angiosperms. Experimental constraints made it difficult to test this proposal in large trees, where the distance between source and sink might prove incompatible with the hypothesis. Recently, the theoretical optimization of the Münch mechanism was shown to lead to surprisingly simple predictions for the dimensions of the phloem sieve elements in relation to that of fast growing angiosperms. These results can be obtained in a very transparent way using a simple coupled resistor model. To test the universality of the Münch mechanism, we compiled anatomical data for 32 angiosperm and 38 gymnosperm trees with heights spanning 0.1-50 m. The species studied showed a remarkable correlation with the scaling predictions. The compiled data allowed calculating stem sieve element conductivity and predicting phloem sap flow velocity. The central finding of this work is that all vascular plants seem to have evolved efficient osmotic pumping units, despite their huge disparity in size and morphology. This contribution extends the physical understanding of phloem transport, and will facilitate detailed comparison between theory and field experiments.     

Revisiting the Münch pressure-flow hypothesis for long-distance transport of carbohydrates: modelling the dynamics of solute transport inside a semipermeable tube

A mathematical model of the Münch pressure-flow hypothesis for long-distance transport of carbohydrates via sieve tubes is constructed using the Navier-Stokes equation for the motion of a viscous fluid and the van't Hoff equation for osmotic pressure. Assuming spatial dimensions that are appropriate for a sieve tube and ensuring suitable initial profiles of the solute concentration and solution velocity lets the model become mathematically tractable and concise. In the steady-state case, it is shown via an analytical expression that the solute flux is diffusion-like with the apparent diffusivity coefficient being proportional to the local solute concentration and around seven orders of magnitude greater than a diffusivity coefficient for sucrose in water. It is also shown that, in the steady-state case, the hydraulic conductivity over one metre can be calculated explicitly from the tube radius and physical constants and so can be compared with experimentally determined values. In the time-dependent case, it is shown via numerical simulations that the solute (or water) can simultaneously travel in opposite directions at different locations along the tube and, similarly, change direction of travel over time at a particular location along the tube.     

Ion Transport and the Transpiration Stream

In a long-term experiment with maize grown at different humidities, Tanner and Beevers (1990) demonstrated that the amount of water lost by the plants in transpiration (plus guttation) could be reduced by a factor of three without any adverse effect on growth. As a consequence, the authors questioned the importance of the transpiration stream in supplying the shoot with minerals, arguing that there are other causes of mass flow in the xylem (such as Münch counterflow from phloem to xylem, and water consumed by growing sink tissues) that may, in the limit, be capable on their own of providing the shoot with minerals. This hypothesis is discussed here in the light of recent work on xylem water relations. It is shown to involve the incorrect premise that, if transpiration were required for long-distance ion transport, plants should grow less well at high humidity. Instead, solute flux to the shoot can be demonstrated by experiment to remain constant over a wide range of transpiration rates, since the concentration of solutes in the xylem sap varies inversely with transpiration rate. Independent evidence suggests that the non-transpirational component of mass flow in the xylem is small and is unlikely to be able to provide the shoot adequately with minerals in the absence of transpiration. A simple corollary of this view is that plant growth should be reduced at very low transpiration rates, a prediction that should be testable at sufficiently high humidities under carefully controlled conditions.     

Water Flow Across the Sieve-Tube Boundary: Estimating Turgor and Some Implications For Phloem Loading and Unloading. I. Theory

A mathematical model of water and sucrose transport across thesieve tube boundary is presented, based on conservation of matterand the phenomenological equations for plasmodesmatal transportbetween the sieve elements and their associated cells. Plasmodesmataltransport coefficients are discussed. In parts II–IV,the equations developed here are used to assess: (i) the estimationof phloem turgor gradients from osmotic pressure gradients;(ii) plasmodesmatal transport of water and sucrose between thesieve elements and adjacent cells; and (iii) the plausibilityof symplastic and apoplastic phloem loading and unloading insome primary sources and sinks. A list of symbols is given inAppendix 1 of this paper Phloem, turgor, osmotic pressure, loading, unloading, plasmodesmata, Munch hypothesis     

The effect of alternative prey on the dynamics of imperfect Batesian and Müllerian mimicries

Both Batesian and Müllerian mimicries are considered classical evidence of natural selection where predation pressure has, at times, created a striking similarity between unrelated prey species. Batesian mimicry, in which palatable mimics resemble unpalatable aposematic species, is parasitic and only beneficial to the mimics. By contrast, in classical Müllerian mimicry the cost of predators' avoidance learning is shared between similar unpalatable co-mimics, and therefore mimicry benefits all parties. Recent studies using mathematical modeling have questioned the dynamics of Müllerian mimicry, suggesting that fitness benefits should be calculated in a way similar to Batesian mimicry; that is, according to the relative unpalatability difference between co-mimics. Batesian mimicry is very sensitive to the availability of alternative prey, but the effects of alternative prey for Müllerian dynamics are not known and experiments are rare. We designed two experiments to test the effect of alternative prey on imperfect Batesian and Müllerian mimicry complexes. When alternative prey were scarce, imperfect Batesian mimics were selected out from the population, but abundantly available alternative prey relaxed selection against imperfect mimics. Birds learned to avoid both Müllerian models and mimics irrespective of the availability of alternative prey. However, the rate of avoidance learning of models increased when alternative prey were abundant. This experiment suggests that the availability of alternative prey affects the dynamics of both Müllerian and Batesian mimicry, but in different ways.     

A Mathematical Treatment of Munch's Pressure-Flow Hypothesis of Phloem Translocation

The steady state solutions of two mathematical models are used to evaluate Münch's pressure-flow hypothesis of phloem translocation. The models assume a continuous active loading and unloading of translocate but differ in the site of loading and unloading and the route of water to the sieve tube. The dimensions of the translocation system taken are the average observed values for sugar beet and are intended to simulate translocation from a mature source leaf to an expanding sink leaf. The volume flow rate of solution along the sieve tube, water flow rate into the sieve tube, hydrostatic pressure, and concentration of sucrose in the sieve tube are obtained from a numerical computer solution of the models. The mass transfer rate, velocity of translocation, and osmotic and hydrostatic pressures are consistent with empirical findings. Owing to the resistance to water flow offered by the lateral membranes, the hydrostatic pressure generated by the osmotic pressure can be considerably less than would be predicted by the solute concentration. These models suggest that translocation at observed rates and velocities can be driven by a water potential difference between the sieve tube and surrounding tissue and are consistent with the pressure-flow hypothesis of translocation.