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.     

Linking phloem function to structure: Analysis with a coupled xylem-phloem transport model

We carried out a theoretical analysis of phloem transport based on Münch hypothesis by developing a coupled xylem-phloem transport model. Results showed that the maximum sugar transport rate of the phloem was limited by solution viscosity and that transport requirements were strongly affected by prevailing xylem water potential. The minimum number of xylem and phloem conduits required to sustain transpiration and assimilation, respectively, were calculated. At its maximum sugar transport rate, the phloem functioned with a high turgor pressure difference between the sugar sources and sinks but the turgor pressure difference was reduced if additional parallel conduits were added or solute relays were introduced. Solute relays were shown to decrease the number of parallel sieve tubes needed for phloem transport, leading to a more uniform turgor pressure and allowing faster information transmission within the phloem. Because xylem water potential affected both xylem and phloem transport, the conductance of the two systems was found to be coupled such that large structural investments in the xylem reduced the need for investment in the phloem and vice versa.     

Thermodynamic analysis of the interaction of the xylem water and phloem sugar solution and its significance for the cohesion theory

The cohesion theory explains water transport in trees by the evaporation of water in the leaves (transpiration), which in turn generates the tension required for sap ascent, i.e. the flow of pure water from the soil through the root system and the non-living cells of the tree (xylem tracheids) up to the leaves. Only a small part of this water flow entering the leaves is used in photosynthesis to produce sugar solution, which is transported from the leaves through the living cells (phloem) to everywhere in the tree where it is needed and used. The phloem sieves are connected to the xylem tracheids by water transparent membranes, which means that the upflow of pure water and downflow of sugar solution interact with each other, causing the osmotic pressure in the sugar solution (Münch model). In this paper we analyse this interaction with a thermodynamic approach and we show that some open questions in the cohesion theory can then perhaps be better understood. For example, why under a quite high tension the water can flow in the xylem mostly without any notable cavitation, and how the suction force itself depends on the cavitation. Minimizing Gibbs energy of the system of xylem and phloem, we derive extended vapor pressure and osmotic pressure equations, which include gas bubbles in the xylem conduits as well as the cellulose-air-water interface term. With the aid of the vapor pressure equation derived here, we estimate the suction force that the cavitation controlled by the phloem sugar solution can generate at high moisture contents. We also estimate the suction force that the transpiration can generate by moisture gradient at low moisture contents. From the general osmotic pressure equation we derive an equation for calculating the degree of cavitation with different sugar solution concentrations and we show the conditions under which the cavitation in the xylem is totally avoided. Using recent field measurement results for a Scotch pine, the theory is demonstrated by showing its predictions for possible amounts of cavitation or embolism from morning hours to late afternoon.     

Consequences of insufficient equations in models of the Münch hypothesis of phloem transport

Abstract The suggestion of concentration dependent unloading as a necessary assumption for a closed-form mathematical expression of the Münch hypothesis of phloem transport (Goeschl et at., 1976) has been recently questioned (Ferrier & Christy, 1977). Also questioned was whether previous models, known to have one less equation than dependent variables, have produced erroneous or misleading results. A review of Münch's original concepts and recent comments by several researchers indicate that an equation describing the exit of solutes from sieve tubes is essential for consistency with the Münch hypothesis. Evidence is presented to show that previous mathematical predictions and interpretations regarding ‘runaway velocities’ have resulted in part from mathematical inconsistencies and characteristics of the numerical solution procedures. Likewise, a proposal that different steady-state velocities and concentrations of transport at a given loading rate might be determined by ‘previous history’ of the system can be explained on the basis of implied relationships hidden in models with insufficient equations. Finally it is concluded that concentration dependent unloading in models of the Münch hypothesis will not detract, but rather will contribute to their usefulness in experimental testing and other research regarding phloem transport.     

Generalized Münch coupling between sugar and water fluxes for modelling carbon allocation as affected by water status

A model of within-plant carbon allocation is proposed which makes a generalized use of the Münch mechanism to integrate carbon and water functions and their involvement in growth limitations. The plant is envisioned as a branched network of resistive pathways (phloem and xylem) with nodal organs acting as sources and sinks for sucrose. Four elementary organs (leaf, stem, fruit, root) are described with their particular sink functions and hydraulic attributes. Given the rates of photosynthesis and transpiration and the hydraulic properties of the network as inputs, the model calculates the internal fluxes of water and sucrose. Xylem water potential (Psi), phloem sucrose concentration (C) and turgor pressure (P) are calculated everywhere in the network accounting for osmotic equilibrium between apoplasm and symplasm and coupled functioning of xylem and phloem. The fluxes of phloem and xylem saps are driven by the gradients of P and Psi, respectively. The fruit growth rate is assumed as turgor pressure dependent. To demonstrate its ability to address within-plant competition, the model is run with a simple-branched structure gathering three leaves, eight stem segments, three competing growing fruits and one root. The model was programmed with P-Spice, a software specifically designed for simulating electrical circuits but easily adaptable to physiology. Simulations of internal water fluxes, sucrose concentrations and fruit growth rates are given for different conditions of soil water availability and hydraulic resistances (sensitivity analysis). The discussion focuses on the potential interest of this approach in functional--structural plant models to address water stress-induced effects.     

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.     

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.     

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.     

Time lags for xylem and stem diameter variations in a Scots pine tree

Diameter variations in the xylem and whole stem (i.e. over bark) stem of a Scots pine (Pinus sylvestris L.) tree were measured at four heights over a 23 d period at 5 min intervals. Cross‐correlation analysis was used to calculate time lags between the measurements. Xylem diameter measurements at the different heights had time lags varying from 10 to 50 min, measurements at the lower heights lagging behind the most. This result was in good agreement with the cohesion theory of transpiration. For the whole stem diameter measurements, the treetop lagged behind all other heights and the shortest lags were midway along the stem. Changes in whole stem diameter always lagged behind those of xylem stem diameter (30–110 min), and at all heights. The considerable differences in the behaviour of xylem and whole stem diameter support the Münch hypothesis of phloem flow. Time lags calculated separately for the shrinkage (morning) and swelling (afternoon) periods indicated shorter time lags during the swelling periods. The non‐destructive methods used show promise in the simultaneous study of flow dynamics of xylem and phloem in trees.     

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.     

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.     

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.     

Phloem flow and sugar transport in Ricinus communis L. is inhibited under anoxic conditions of shoot or roots

Anoxic conditions should hamper the transport of sugar in the phloem, as this is an active process. The canopy is a carbohydrate source and the roots are carbohydrate sinks. By fumigating the shoot with N₂ or flooding the rhizosphere, anoxic conditions in the source or sink, respectively, were induced. Volume flow, velocity, conducting area and stationary water of the phloem were assessed by non‐invasive magnetic resonance imaging (MRI) flowmetry. Carbohydrates and δ¹³C in leaves, roots and phloem saps were determined. Following flooding, volume flow and conducting area of the phloem declined and sugar concentrations in leaves and in phloem saps slightly increased. Oligosaccharides appeared in phloem saps and after 3 d, carbon transport was reduced to 77%. Additionally, the xylem flow declined and showed finally no daily rhythm. Anoxia of the shoot resulted within minutes in a reduction of volume flow, conductive area and sucrose in the phloem sap decreased. Sugar transport dropped to below 40% by the end of the N₂ treatment. However, volume flow and phloem sap sugar tended to recover during the N₂ treatment. Both anoxia treatments hampered sugar transport. The flow velocity remained about constant, although phloem sap sugar concentration changed during treatments. Apparently, stored starch was remobilized under anoxia.     

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.     

MRI of long-distance water transport: a comparison of the phloem and xylem flow characteristics and dynamics in poplar, castor bean, tomato and tobacco

We used dedicated magnetic resonance imaging (MRI) equipment and methods to study phloem and xylem transport in large potted plants. Quantitative flow profiles were obtained on a per-pixel basis, giving parameter maps of velocity, flow-conducting area and volume flow (flux). The diurnal xylem and phloem flow dynamics in poplar, castor bean, tomato and tobacco were compared. In poplar, clear diurnal differences in phloem flow profile were found, but phloem flux remained constant. In tomato, only small diurnal differences in flow profile were observed. In castor bean and tobacco, phloem flow remained unchanged. In all plants, xylem flow profiles showed large diurnal variation. Decreases in xylem flux were accompanied by a decrease in velocity and flow-conducting area. The diurnal changes in flow-conducting area of phloem and xylem could not be explained by pressure-dependent elastic changes in conduit diameter. The phloem to xylem flux ratio reflects what fraction of xylem water is used for phloem transport (Münch's counterflow). This ratio was large at night for poplar (0.19), castor bean (0.37) and tobacco (0.55), but low in tomato (0.04). The differences in phloem flow velocity between the four species, as well as within a diurnal cycle, were remarkably small (0.25-0.40 mm s(-1)). We hypothesize that upper and lower bounds for phloem flow velocity may exist: when phloem flow velocity is too high, parietal organelles may be stripped away from sieve tube walls; when sap flow is too slow or is highly variable, phloem-borne signalling could become unpredictable.     

Simultaneous measurement of water flow velocity and solute transport in xylem and phloem of adult plants of Ricinus communis over a daily time course by nuclear magnetic resonance spectrometry

A new method for simultaneously quantifying rates of flow in xylem and phloem using the FLASH imaging capabilities of nuclear magnetic resonance (NMR) spectrometry was applied in this study. The method has a time resolution of up to 4 min (for the xylem) and was used to measure the velocity of flows in phloem and xylem for periods of several hours to days. For the first time, diurnal time course measurements of flow velocities and apparent volume flows in phloem and xylem in the hypocotyl of 40‐d‐old Ricinus communis L were obtained. Additional data on gas exchange and the chemical composition of leaves, xylem and phloem sap were used to assess the role of leaves as sinks for xylem sap and sources for phloem. The velocity in the phloem (0·250 ± 0·004 mm s^?1) was constant over a full day and not notably affected by the light/dark cycle. Sucrose was loaded into the phloem and transported at night, owing to degradation of starch accumulated during the day. Concentrations of solutes in the phloem were generally less during the night than during the day but varied little within either the day or night. In contrast to the phloem, flow velocities in the xylem were about 1·6‐fold higher in the light (0·401 ± 0·004 mm s^?1) than in the dark (0·255 ± 0·003 mm s^?1) and volume flow varied commensurately. Larger delays were observed in changes to xylem flow velocity with variation in light than in gas exchange. The relative rates of solute transport during day and night were estimated on the basis of relative flow and solute concentrations in xylem and phloem. In general, changes in relative flow rates were compensated for by changes in solute concentration during the daily light/dark cycle. However, the major solutes (K⁺, NO₃^?) varied appreciably in relative concentrations. Hence the regulation of loading into transport systems seems to be more important to the overall process of solute transport than do changes in mass flow. Due to transport behaviour, the chemical composition of leaves varied during the day only with regard to starch and soluble carbohydrates.