TY - JOUR
AU - Jensen, Kaare, H
AB - Abstract Xylem and phloem are the two main conveyance systems in plants allowing exchanges of water and carbohydrates between roots and leaves. While each system has been studied in isolation for well over a century, the coupling and coordination between them remains the subject of inquiry and active research and frames the scope of the review here. Using a set of balance equations, hazards of bubble formation and their role in shaping xylem pressure and its corollary impact on phloem pressure and sugar transport are featured. The behavior of an isolated and freely floating air bubble within the xylem is first analyzed so as to introduce key principles such as the Helmholtz free energy and its links to embryonic bubble sizes. These principles are extended by considering bubbles filled with water vapor and air arising from air seeding. Using this framework, key results about stability and hazards of bubbles in contact with xylem walls are discussed. A chemical equilibrium between phloem and xylem systems is then introduced to link xylem and osmotic pressures. The consequences of such a link for sugar concentration needed to sustain efficient phloem transport by osmosis in the loading zone is presented. Catastrophic cases where phloem dysfunction occurs are analyzed in terms of xylem function and its vulnerability to cavitation. A link between operating pressures in the soil system bounded by field capacity and wilting points and maintenance of phloem functioning are discussed as conjectures to be tested in the future. Introduction Plants are Earth’s primary solar energy collectors, and their combined mass exceeds all other life forms on the planet. To achieve this impressive feat, they rely on a range of physio-chemical mechanisms. Among these are long-distance transport of water and assimilates. The focus of this review is characterizing basic biophysical processes that are necessary for long-distance transport in vascular land plants and elucidating some of the fundamental limitations they impose on plant life. Plants evolved in and were for the longest part of their history restricted to aquatic habitats. In such a habitat, water was ubiquitous and sunlight and carbon dioxide, the other two ingredients of photosynthesis, were consistently available. The adaptation to a terrestrial lifestyle prompted the development of distinct subaerial and aboveground organs. Roots were assigned the extraction of water and (micro-)nutrients from the soil system while the green parts became exclusively responsible for the assimilation of water, carbon dioxide and sunlight into carbohydrates. Clearly, such a spatial differentiation of functions makes transport systems for the distribution of water and carbohydrates within the plant body a necessity. Thus, colonization of land necessitated engineering solutions that resulted in the development of xylem and phloem, the two main conveyance systems in plants. Because the exchange of water and carbohydrates occurs mainly between roots and leaves (i.e., the main suppliers and consumers of water and carbohydrates), it is not surprising that xylem and phloem functioning are partially coupled. The physical and physiological mechanisms responsible for their functioning and partial coupling frame the compass of the work here and are summarized in Figure 1. Figure 1. View largeDownload slide Impact of xylem conditions on phloem transport. Photoassimilate transport in the phloem is powered by differences in osmolyte concentration cp between source (⁠ x=0 ⁠) and sink (x=L) ⁠, which generates a pressure drop Δpp=pp(0)−pp(L) in the phloem. The solute concentration required to maintain constant transport depends on the xylem tension ps (Eq. (29)). Figure 1. View largeDownload slide Impact of xylem conditions on phloem transport. Photoassimilate transport in the phloem is powered by differences in osmolyte concentration cp between source (⁠ x=0 ⁠) and sink (x=L) ⁠, which generates a pressure drop Δpp=pp(0)−pp(L) in the phloem. The solute concentration required to maintain constant transport depends on the xylem tension ps (Eq. (29)). Perhaps the most striking difference between xylem and phloem functioning is that the former is driven by the purely physical mechanisms of transpiration, which functions for some extended time even with dead trees, while the latter requires energy input from living plant cells and water supply from the xylem. In terms of energy demand, xylem water transport is reasonably ‘cheap’ for the plant as no active pumping is required. The downside is that xylem water is in a thermodynamically metastable state of tension and water transport from the soil to the leaves is prone to hydraulic failure. It may come as a surprise that plants still rely on this mode of water transport in the xylem despite some 400 million years of evolution. Metastability also complicates even the most delicate experiments as it is often difficult to a priori discern whether an experimental result describes the functioning of the system or its collapse into a stable state, due to disturbances caused by the measurement act. Hence, reliance on theoretical description of the xylem system and its interaction with the phloem must complement measurements so as to push the understanding of the two transport systems as far as possible. Proceeding in this spirit, the work here first considers the functioning, dysfunction and repair of the xylem water transport system, which is a necessary first step to inquire about phloem transport. The implications of this physical perspective for the partial coupling between xylem and phloem is then evaluated. Xylem water transport Plants require a reliable supply of water because (i) water is inevitably lost from plant tissues by transpiration, (ii) water flow distributes dissolved nutrients such as nitrogen, phosphorus or potassium within the plant, (iii) the phloem transport exchanges water with the xylem system and (iv) the process of photosynthesis does require small amounts water as raw material (though this amount is far smaller than transpirational losses). The problem of water supply became acute when land plants began to grow higher than a few centimeters (leading to a race upwards). Water must be elevated against the gravitational force, which is made more difficult by the increasing conduit lengths that lead to increased frictional losses of energy. To a lesser extent, increasing height above the ground also resulted in the main foliage encountering dryer atmospheric states. Hence, the taller a plant grows, the greater becomes the difference between the atmospheric humidity close to its leaves and the nearly saturated humidity within its leaves. Since this difference is presumed to be a ‘driving force’ for transpiration (analogous to Dalton’s 1802 evaporation law), taller plants transpire more water than short plants, necessitating a more elaborate water transporting system (Raven 1984, 1993, 2003, Raven and Edwards 2004, 2014). Conducting bundles were present in early land plants with an upright growth habit as indicated by macroscopic remains of plants with cooksonioid (after Cooksonia, an extinct group of primitive land plants) or rhyniophytic organization as early as the Late Silurian. These plants consisted of dichotomously branching and leafless axes with Rhynia gwynne-vaughanii or Aglaophyton major being iconic examples (see Figure 2). However, the water-conducting cells of these early land plants were diverse: A. major, for example, did not possess ‘genuine’ xylem but rather a cell type more similar to moss hydroids. During the Devonian, size and complexity of plants increased rapidly and trees were well established during the Late Devonian, among them Archaeopteris as the first tree with secondary xylem similar to that of conifers (Lucas et al. 2013). Figure 2. View largeDownload slide Reconstruction of early land plants (a) Aglaophyton major and (b) Rhynia gwynne-vaughanii (Lower Devonian, 410 Ma) from the Rhynie cherts near Aberdeen (Scotland). Rhynia is one of the earliest known vascular plants. Height: about 20 cm. (c) Transverse section through a Rhynia axis showing xylem (x), ‘phloem’ (p), inner cortex (ic), outer cortex (oc), epidermis (e) and cuticle (c), scale bar = 1 mm. Images (a) and (b) redrawn after (Kenrick and Crane 1997), image (c) by courtesy of the Department of Palaeobotany, University of Münster. Figure 2. View largeDownload slide Reconstruction of early land plants (a) Aglaophyton major and (b) Rhynia gwynne-vaughanii (Lower Devonian, 410 Ma) from the Rhynie cherts near Aberdeen (Scotland). Rhynia is one of the earliest known vascular plants. Height: about 20 cm. (c) Transverse section through a Rhynia axis showing xylem (x), ‘phloem’ (p), inner cortex (ic), outer cortex (oc), epidermis (e) and cuticle (c), scale bar = 1 mm. Images (a) and (b) redrawn after (Kenrick and Crane 1997), image (c) by courtesy of the Department of Palaeobotany, University of Münster. Short history of the cohesion-tension theory Modern attempts to understand water transport in plants date back to about 400 years ago when William Harvey (1628) and Marcello Malpighi (1661) drew analogies with blood circulation in animals (Scott 1927). In 1727, Stephen Hales could show (Floto 1999) that water flow in trees is upwards from the soil to the leaves. An important step towards the cohesion-tension theory’s development was Heinrich Cotta’s identification of sapwood as the transporting plant tissue (Cotta 1806), the result of a prize competition organized by the Imperial Academy of Scientists in Erlangen in the year 1798. The question for the driving force of water movement became the subject of heated debates between ‘vitalists’ and ‘physicists’. The first group contended the idea that living cells should play a major role in the transport mechanism, although they were neither able to identify these cells nor to suggest a detailed transport mechanism. The ‘physicists’ view that only dead cells are involved, hence, the mechanism of water transport must rely directly on physical effects, was confirmed by an experiment conducted by Eduard Strasburger (1891). Strasburger immersed the roots of a tree into a concentrated solution of copper sulfate (CuSO4). Although the CuSO4 killed all living cells in its way almost instantly, it moved upwards into the leaves (which became evident when they were killed, too). Moreover, the uptake of the solution and the process of transpiration from the dead leaves went on for several weeks afterwards. Shortly after these seminal experiments, Dixon and Joly (1894) showed that water is lifted up against gravity even if the leaves of a detached twig is put into a pressure vessel and kept there under three bars overpressure while the lower end of the twig is put into a water reservoir and kept under atmospheric pressure. That water can withstand high tensional forces was contradictory to everyday experiences back then, given the inability of water to resist shear stresses. Experiments by Berthelot (1850) and Jäger (1892) showed that water can withstand high tensional forces though some debate followed on whether such experiments actually measured the cohesive forces of water or the adhesive force of water on solid tubes. The last missing element of the cohesion-tension theory was provided more or less independently by John Joly and Henry Horatio Dixon, both of Trinity College Dublin, and by Eugen Askenasy of Heidelberg University in 1894 (afterwards they quarreled about priority): they found that in the leaf parenchyma where transpiration takes place, a curved water/air interface develops providing the surface tension that counteracts the weight of the water column and pulls it up against gravity (Brown 2013). With Dixon’s book from 1914 (‘Transpiration and the Ascent of Sap in Plants’ (Dixon 1914)) the cohesion-tension theory found its final form. Accounts of historical and recent developments of the cohesion-tension theory can be found in Tyree and Sperry (1989) and Tyree and Zimmermann (2002). For a detailed description see Brown (2013). Outline of the cohesion-tension theory Plant water transport in vascular plants relies on two basic principles: (i) transpiration in the leaves provides the driving force that allows water flow to overcome the gravitational pull and frictional forces between water molecules and the vessel, and (ii) the cohesion of water is sufficient to maintain water molecules bound together during the transport phase to leaves. To illustrate the coordination of these two principles, imagine a long chain (or rather a three-dimensional network) of water molecules whose upper end is fastened to the parenchyma tissue of a plant leaf where transpiration, the transformation of liquid water into water vapor, occurs. From there, the chain hangs down through twigs, trunks, stem and roots into soil water. Each chain link adheres to its neighbors by electromagnetic forces (van-der-Waals forces) that are established due to the highly inhomogeneous and different distributions of positive and negative electric charges within the water molecule (another example of the biological fruitfulness of the physical ‘anomaly’ of water, as compared with other liquids). Every water molecule that evaporates in the leaves draws up the entire chain for about one molecular diameter. Pushing this analogy further, the higher up in the chain a link (=water molecule) is, the higher the load it has to carry, because the latter consists of the weight of all molecules below. This is why a chain that is hanging down can be expected to break at its uppermost link. Thus, the tension in the water chain increases towards the crown of the tree. When the chain is drawn up by evaporation or loss of water molecules to the atmosphere, friction between the water molecules and the vessel walls adds to the tensile forces within the water molecules forming the chain. The following calculation gives an ‘order of magnitude’ estimate of the stresses acting upon the water in a plant conduit in a hydrostatic state. A water column of cross section A hanging down a distance h exerts a force G=ρAhg onto its attachment (with ρ=103kg/m3=densityofwater ⁠, g≈10m/s2=gravitational acceleration at the surface of the earth). With trees not exceeding a maximum height (⁠ hmax≈120m ⁠), a lower limit of the magnitude of the stress that the uppermost molecular layers of the water column within the conduit of a tree must be able to resist is given by pg=GA=ρgh⪅ρghmax=1.2MPa (1) This estimate is crude because it only accounts for the weight of the water column and neglects additional resistances due to friction. In fact, realistic stresses may be higher by an order of magnitude (see Larcher 2003, Hopkins and Huner 2004, Taiz 2006). Experimentally, it is difficult to distinguish resistance against shear stress from resistance against tension stress because it is not easy to apply only one of the two stress varieties to liquid water. Hence, measurements of the breaking strength of water against tension stress may be more informative about the quality of the experimental setup than reporting the true value of breaking strength of water. Nonetheless, such experiments can provide a reliable lower limit. From the literature, values of the breaking strength of water range between 50 MPa and 100 MPa and this range seems to be (i) reliable and (ii) in accordance with theoretical considerations (Temperley and Trevena 1994). Hence, even if realistic stresses are much higher than the 1.2 MPa estimated above for the water in a tree conduit, they appear well below the lower limit of the breaking strength of water. Inside a tree, juxtaposition of this physical limit of breaking strength of water may be an overestimate. Values of breaking strength have been measured or derived for pure water that contains insignificant quantities of molecular species other than water (for example no air bubbles). The reason for this restriction is found in the radial characteristics of the intermolecular force between water molecules: for distances smaller than an equilibrium separation req ⁠, two water molecules exchange a repulsive force (whose large strength accounts for the small compressibility of water). For separations greater than req ⁠, the molecules attract each other. The attractive force has a maximum at a separation rmax ⁠. If the separation is increased further, the attraction decreases and becomes eventually effectively zero. When water is contaminated by other molecular species, some water molecules become farther separated from one another than they were before (in pure water) because molecules of the contaminating species have entered the space between them. If this effect shifts the mean separation between adjacent water molecules to values much beyond rmax ⁠, the attractive force between these water molecules decreases. It directly follows that the effective breaking strength of the whole network of water molecules (or the chain) is reduced. If the breaking strength is reduced far enough, this effect becomes catastrophic. It should be noted, however, that solitary molecules that are completely surrounded by water molecules (macroscopically speaking, they are dissolved in water) do not usually represent a threat to the integrity of the network of water molecules. Soil particles or air bubbles, however, may become dangerous if their diameter is large enough. To minimize this threat, the roots of (vascular) plants have developed excellent filter systems able to keep these contaminants out of the flow path (see, e.g., Sitte et al. 2002). Interrelation with phloem transport Photosynthesis in plant leaves converts light into chemical energy that is stored in sugar molecules for later use in metabolism, growth and reproduction. Sugars are exported from the leaf by bulk liquid flow under positive pressure through phloem sieve elements, which form a microfluidic tube network linking distal parts of the plant (Stroock et al. 2014). The flow is driven by osmotic pressure differences between sugar sources in leaves and sinks at sites where sugars are consumed, as shown in Figure 1. Phloem sap contains a plethora of osmolytes, but the primary determinant of phloem pressure is the concentration of sugars and small ions (Jensen 2018). Phloem transport thus relies on the osmotic exchange of water with the xylem system, and the conditions in the xylem have a direct impact on phloem flow (Sevanto 2018). Briefly, when incident photosynthetic photon flux density is >0, photosynthesis commences in leaves, the byproduct of which are sugars. Assimilated sugars that accumulate in the mesophyll are transported to companion cells and subsequently to sieve elements in the so-called loading zone (i.e., leaf). The loading is achieved by molecular diffusion (i.e., passive and common in angiosperm and gymnosperm) or polymer trapping and apoplastic pumping (i.e., active and common in herbaceous species) mechanisms (Jensen et al. 2016). The production rate of sucrose in the loading zone can be approximated by αp(βfcAl) ⁠, where fc is the photosynthetic rate that is linked to the transpirational losses from leaves (quantified by a water-use efficiency), αp(∈[0,1]) and β are species-specific loading efficiency and the number of sucrose molecules produced from one assimilated CO2 molecule (⁠ β=1/12 for sucrose only), respectively. The αp=1 when all sucrose molecules produced from fc enter the loading zone. The total water potential in the phloem loading zone includes turgor pressure (pp) ⁠, the aforementioned hydrostatic pressure pg given by Eq. (1) and the osmotic potential (Π) that increases with increasing sugar concentration (due to fc ⁠) and is presumed to be the dominant term. If chemical and thermal equilibrium exist between the phloem and xylem shown in Figure 1, then ps+pg≈pp+pg−Π ⁠, where ps is the xylem pressure in the most distal part (i.e., vicinity of the leaf) assuming negligible sugar concentration in the xylem. The ps is impacted by all the hazards associated with formation of air bubbles in the xylem arising from the metastable state of water whereas Π is dictated by sugar accumulation arising from finite fc ⁠. This review is primarily focused on the hazard of bubble formation and their role in shaping ps as well as their subsequent impact on Π ⁠. Hazards to water transport Basics and preliminaries As noted earlier, one of the main hazards to water transport is its metastable state, which is prone to bubble formation and subsequent embolism. In fact, Haberlandt already noted (in 1914) that ‘Vessels and tracheids normally contain both, air and water, the relative amounts of the two substances vary according to the season and the time of day’. There is now recognition that formation of gas emboli are common occurrences in the xylem of many plant species. This section reviews basic concepts on bubble stability within a xylem vessel. A gas bubble immersed in a xylem fluid and filled with air and water vapor may experience the following balance situations: A balance of mechanical forces: (a) the randomly moving gas molecules inside the bubble act as an outward directed force that tries to expand the bubble; (b) the surface tension of the gas/liquid interface tries to contract the bubble; (c) the pressure ps of the xylem water contributes to the expanding force for ps<0 and to the contracting force for ps>0 ⁠. A balance with respect to the exchange of water molecules with the bubble’s surroundings: if the water vapor partial pressure within the bubble differs from the water vapor saturation pressure, water molecules either condensate at the bubble’s gas/liquid interface or vaporize from there into the bubble. A balance with respect to the exchange of air molecules with the bubble’s surroundings: the number of air molecules remains constant only if a ‘diffusional equilibrium’ prevails between the air partial pressure within the bubble and the concentration of air molecules dissolved in the xylem water. Otherwise, air molecules from within the bubble either dissolve at the bubble’s gas/liquid interface in the xylem water and diffuse towards regions of lower air molecule concentration, or else a diffusional current through the xylem water delivers air molecules to the bubble surface and into the bubble. Since the exchange process between water vapor and water is faster than the exchange of air by a factor of about 106 and since the dissolution of a bubble filled merely with water vapor of an initial radius of 1 μm immersed in water with ps=−1MPa lasts about 2.9×10−10s ⁠, in what follows it is assumed that bubbles are balanced with respect to the first two situations but not necessarily with respect to the third. Depending on whether a bubble consists merely of water vapor or contains additionally air molecules, two effects that are potentially hazardous for the conductance of the xylem vessels and the xylem potential must be considered: spontaneous cavitation air seeding. Spontaneous cavitation Spontaneous cavitation arises due to statistical fluctuations (caused, for instance, by Brownian movement) of the energy of a group of neighboring water molecules in liquid state. If these molecules acquire enough kinetic energy to ‘cut’ the network of intermolecular binding forces defining the liquid state and to become a cluster of gaseous molecules, then an embryonic bubble filled with water vapor forms within the liquid. Whether this embryonic bubble disappears soon afterwards because its constituents are being ‘resorbed’ into the liquid surrounding the bubble or whether it grows because more water molecules vaporize into it depends (i) on its initial radius R0 and (ii) on the tension ps<0 (=negative water pressure) existing in the water column. The reversible free energy (referred to as the Helmholtz free energy in thermodynamics) associated with the spontaneous formation of an embryonic bubble with radius R is given by (see Debenedetti 1996, Oertli 1971, Pickard 1981) W(R)=4πγR2−4π3(pw−ps)R3 (2) (γ: surface tension of air/water interface, pw ⁠: pressure of water vapor within the bubble, ps ⁠: water pressure) and depicted in Figure 3a. Figure 3. View largeDownload slide Energy of bubble formation W(R) plotted against bubble radius R. For illustration, water pressure in the conduit is set to ps=−1MPa ⁠. Equilibrium between expanding and contracting forces is realized only at bubble radii R representing extrema of W(R) ⁠. Maxima indicate unstable, minima stable equilibrium. (a) Bubble filled only with water vapor. (b) Bubble filled with water vapor and air. Curves are plotted for the following numbers of air molecules: na,1=7.5×10−20mol ⁠, ncrit=7.5×10−19mol and na,3=9×10−19mol ⁠. Figure 3. View largeDownload slide Energy of bubble formation W(R) plotted against bubble radius R. For illustration, water pressure in the conduit is set to ps=−1MPa ⁠. Equilibrium between expanding and contracting forces is realized only at bubble radii R representing extrema of W(R) ⁠. Maxima indicate unstable, minima stable equilibrium. (a) Bubble filled only with water vapor. (b) Bubble filled with water vapor and air. Curves are plotted for the following numbers of air molecules: na,1=7.5×10−20mol ⁠, ncrit=7.5×10−19mol and na,3=9×10−19mol ⁠. The further development of the gas bubble is essentially controlled by forces obtained from Eq. (2) by differentiation: ∂W/∂R=8πγR−4πpwR2+4πpsR2 (3) The first term on the right-hand side represents the contracting surface force exerted by the liquid/gas-interface at the bubble surface. The second term represents the expanding gas pressure of water vapor and air molecules. The third term represents the xylem water pressure; for ps>0 it has a contracting effect on the gas bubble, for ps<0 an expanding effect. A bubble of radius R that neither expands nor contracts because the respective forces are at equilibrium is characterized by ∂W/∂R=0 ⁠. In molecular terms, the number of water vapor molecules vaporizing into the bubble equals the number of molecules condensing into the surrounding liquid at equilibrium. In the case ∂W/∂R>0 ⁠, condensation dominates and the bubble disappears. For ∂W/∂R<0 ⁠, vaporization is the dominating effect and the bubble expands further. To locate the extrema of W(R) given in Eq. (2), the radius R satisfying ∂W/∂R=0 must be determined and it must be checked that W(R) is a maximum, not minimum. The solutions for ∂W/∂R=0 are R=0 (representing an artifact) and R=2γpw−ps=:Rvap (4) The second derivative of W(R) ⁠, evaluated at Rvap ⁠, amounts to ∂2W/∂R2=8πγ−8π(pw−ps)R|R=Rvap=−8πγ<0 (5) Hence, R=Rvap represents a maximum of W(R) corresponding to an unstable equilibrium of forces (whereas a positive value would indicate a minimum of W(R) and a stable equilibrium as may occur when setting R=0 ⁠). In other words, if the radius R of a spontaneously emerging vapor bubble obeys the relation R>Rvap ⁠, the bubble is unstable and expands (in principle indefinitely). For R<Rvap ⁠, the bubble vanishes (to a stable R=0 ⁠). Hence, the stability of an embryonic bubble depends on the value of Rvap at its inception. In turn, Rvap depends on the reciprocal of the water pressure (or, rather, tension) ps in the bulk xylem. Thus, the more negative the water pressure is, the smaller is the bubble radius dividing the regimes of bubble collapse and expansion. For example, under a water pressure of ps=−0.01MPa ⁠, all bubbles with a radius smaller than Rvap≈10.94μm will collapse immediately. For the more negative pressure ps=−1MPa this radius amounts to the much smaller value Rvap≈0.143μm (see Figure 3 and the discussion in Oertli 1971). Put differently, a bubble of initial radius R≈5μm collapses (without doing harm to the xylem system) under a water pressure of ps=−0.01MPa but expands until embolism under ps=−1MPa ⁠. The actual bubble growth phase requires consideration of other factors such viscous forces, inertia and unsteadiness that are represented by the so-called Rayleigh–Plesset equation (briefly considered later on). Irrespective of the details of bubble growth, this analysis suggests that a water column under tension constitutes a metastable system in the following sense: if the water column is ‘disturbed’ by the introduction of an embryonic bubble of radius R<Rvap ⁠, then the bubble dissolves. For R>Rvap ⁠, however, the bubble inflates until the water column collapses into a stable state characterized by a positive water pressure ps>0 ⁠. Based on arguments from statistical physics and thermodynamics, it has been shown (Maris and Balibar 2000, Oertli 1971, Pickard 1981) that for the negative pressures to be expected in plant conduits spontaneous cavitation plays only a minor role in causing embolism (as compared with the air seeding process, see below) because the tension values occurring in the xylem are not high enough to permit a significant formation rate of expanding bubbles. Oertli (1971) calculates that in a xylem system consisting of 1litre=10−3m3 water and being under a pressure ps=−127MPa an embolism occurs (on average) every 100years ⁠. If ps increases only slightly (meaning a decrease in |ps| ⁠) the time span without embolism increases tremendously. Put differently, the probability of a spontaneous bubble formation is for ps>−100MPa (meaning |ps|<100MPa ⁠) essentially zero. Trees function far away from this value. Air seeding Air seeding can occur when a group of ‘air’ molecules find their way into a water conduit. Since a completely impermeable material does not exist in biological (nor in artificial) tissues, tiny leaks will always and everywhere be present. Larger cracks and openings in wood may be produced by the wind bending twigs of the tree as a whole, or by animals such as woodpeckers or beetles in search of food. To describe this mixture of water vapor and air, Eq. (2) is now extended by introducing a term representing the na air molecules (Shen et al. 2002, 2003) resulting in (cf. Figure 3b) W(R)=−4π3(pw−ps)R3+4πγR2+3ℛTnalog(RmaxR) (6) where ℛ is the gas constant, and two arbitrary constants have been chosen such that (i) in the absence of air molecules (i.e., na=0 ⁠, that is, in the case of a water vapor bubble) the formation energy W vanishes for R=0 ⁠, and (ii) the contribution of the air molecules to the formation energy W(R) (the last term in Eq. (6)) vanishes for R=Rmax ⁠. Rmax:=2γ−ps (7) denotes the maximum equilibrium radius that a gas bubble immersed in water of negative pressure ps<0 can attain (see Konrad and Roth-Nebelsick 2003; a rigorous derivation of Eq. (6) can be found in Debenedetti (1996)). The analysis of the compressing and expanding forces acting upon a bubble will be based upon Eq. (6). For these calculations, the following two assumptions are made: pw≈pwsat (⁠ pwsat ⁠: saturation vapor pressure that only varies with temperature). The exchange of water molecules between a bubble and surrounding liquid is a very intensive one (for a quantitative estimate of the exchange intensity see Adamson and Gast (1997)). Therefore, the water vapor partial pressure in the bubble readjusts to the water vapor saturation pressure almost immediately. na≈const. ⁠: this assumption is only valid on very short time scales associated with bubble formation considered here. The following observations corroborate its plausibility: (i) the low solubility of air in water, implying that only a small fraction of the air molecules in the bubble’s interior are lost to the surrounding liquid within the considered time scale, and, (ii) the slowness of diffusional transport in liquids: a rough estimate — employing the relation (traveltime)≈(distance)2/Dair (where Dair≈2.1×10−9m2/s ⁠) — shows that dissolved air molecules diffusing away from a bubble need about half a day to cover a distance of 1 cm. Proceeding similarly as above, the bubble radii where equilibrium of forces prevails are found from ∂W/∂R=8πγR−{4π(pw−ps)R2+3ℛTnaR}=0 (8) and their stability is assessed by evaluating the sign of ∂2W/∂R2 at the equilibrium point(s). Equation (8) represents a cubic equation with respect to R. Its structure suggests that it has one, two or no real and positive solutions depending on the values of the quantities γ ⁠, na ⁠, ps and pw ⁠. The crucial quantity to be explored here is na ⁠, the number of air molecules in the bubble, and its relation to ncrit:=128πγ381ℛT1(pw−ps)2 (9) the maximum number of air molecules that a bubble can accommodate. For na>ncrit ⁠, no solutions exist. For na=ncrit one solution exists, and for 0≤na<ncrit two solutions of ∂W/∂R=0 exist. These two solutions are located at the radii Rstab=Rvap3(1−2cos[13arccos(1−2nancrit)+π3])Rinst=Rvap3(1+2cos[13arccos(1−2nancrit)]) (10) where Rvap=2γ/(pw−ps) ⁠. Air–water vapor bubbles show a more complex behavior than pure water vapor bubbles because their behavior depends on the number na of air molecules in the bubble (i.e., a new control variable in the stability analysis): Evaluating ∂2W/∂R2 ⁠, the bubble at radius Rstab is in stable equilibrium whereas at radius Rinst ⁠, it is in an unstable equilibrium (meaning it will show a tendency to expand). Eqs (10) imply Rstab≤Rinst ⁠. The equality is realized for na=ncrit when both Rstab and Rinst (the two equilibrium solutions collide) approach the quantity Rcrit:=4γ3(pw−ps)=23Rvap (11) For na→0 we find Rstab→0 and Rinst→Rvap ⁠. For na>ncrit both Rstab and Rinst become undefined, meaning that the bubble expands without limits (see Figures 3b and 4). The bubble behavior is now illustrated by Figure 3b. A bubble that contains na,3>ncrit air molecules fulfills the condition ∂W/∂R<0 for all R>0 ⁠. Equation (8) implies then that the bubble can only expand. If the bubble contains na,1<ncrit air molecules, its future depends also on its initial radius R: in the case R>Rinst it expands, in the case 0<R<Rinst the bubble expands or contracts towards Rstab ⁠. Table 1 gives a few values for Rcrit and ncrit ⁠. Figure 4, derived from Eq. (8), illustrates the interdependence of initial radius and air content in a stability diagram formed by the (na,R)-plane (the two control variables). It reveals several aspects of embolism potential of an embryonic bubble: The ability of an air–water vapor bubble to seed embolism increases with its air content (with the water vapor pressure being constant and only dependent on temperature) (see Figure 4): a bubble of initial radius R2 containing n1 air molecules contracts towards the stable radius Rstab ⁠, a bubble of the same initial radius containing n4 air molecules (with n4>n1 ⁠), however, bursts. Because of pw≪|ps| (the water vapor pressure pw is usually much smaller than the absolute value of the water pressure ps ⁠) the quantities ncrit ⁠, Rinst ⁠, Rstab and Rvap are approximately proportional to the reciprocal of ps (see Eqs (9), (10) and (4)). Thus, if ps decreases (i.e., becomes more negative), the stability region that represents all bubbles (of radius R and containing na air molecules) drifting to stable equilibrium shrinks. (In Figure 4 the stability regions are the green areas left and below the solid curves Rinst ⁠.) Hence, bubbles that were within the stability region at a given water pressure may become unstable because the borderline between stability and instability has wandered across their (fixed) location in the (na,R)-plane, due to the drop in ps ⁠. Immediate collapse of an air–water vapor bubble is impossible. Unlike water vapor molecules, air molecules cannot instantly condense to the liquid state (at least not under temperatures and pressures compatible with living plants). Hence, a stable air–water vapor bubble can only disappear by dissolution through diffusion of air molecules from the bubble into the surrounding water. Whether or not this diffusion will occur depends on the concentration of air molecules already dissolved in the surrounding water. This dissolution and its associated dynamics will be discussed in section ‘Diffusional equilibrium and dissolution of floating bubbles’. Figure 4. View largeDownload slide Regions of stability and instability of bubbles of radius R, filled with water vapor and na air molecules (cf. Eq. (8)). (a) Stability diagram for a given water pressure ps,1 ⁠. (b) Stability diagram for a water pressure ps,2 with ps,2<ps,1 (i.e., ps,2 is more negative than ps,1 ⁠). Only bubbles lying on the curves Rstab (broken line) and Rinst (solid line) are in stable or unstable equilibrium. Bubbles represented by other (na,R)-values either contract or expand (indicated by broken arrows). Bubbles in the green regions (left and below the solid curves Rinst ⁠) are contracting or expanding towards the stable radii Rstab lying on the dotted curves. Bubbles characterized by (na,R)-values outside the stability regions expand. If ps decreases, the region of stability shrinks. Hence, bubbles that are in equilibrium under water pressure ps,1 (such as those inhabited by n3 air molecules in part (a) of the figure) fall outside of the equilibrium region if ps drops to ps,2<ps,1 (see part (b) of the figure). Figure 4. View largeDownload slide Regions of stability and instability of bubbles of radius R, filled with water vapor and na air molecules (cf. Eq. (8)). (a) Stability diagram for a given water pressure ps,1 ⁠. (b) Stability diagram for a water pressure ps,2 with ps,2<ps,1 (i.e., ps,2 is more negative than ps,1 ⁠). Only bubbles lying on the curves Rstab (broken line) and Rinst (solid line) are in stable or unstable equilibrium. Bubbles represented by other (na,R)-values either contract or expand (indicated by broken arrows). Bubbles in the green regions (left and below the solid curves Rinst ⁠) are contracting or expanding towards the stable radii Rstab lying on the dotted curves. Bubbles characterized by (na,R)-values outside the stability regions expand. If ps decreases, the region of stability shrinks. Hence, bubbles that are in equilibrium under water pressure ps,1 (such as those inhabited by n3 air molecules in part (a) of the figure) fall outside of the equilibrium region if ps drops to ps,2<ps,1 (see part (b) of the figure). Table 1. Values of Rcrit and ncrit calculated from Eqs (9) and (11) for the conduit water pressures ps given in the first row. ps −1 MPa −10−1 MPa −10−2 MPa Rcrit 0.096 μm 0.93 μm 7.29 μm ncrit 7.4 × 10−19 mol 7.0 × 10−17 mol 4.3 × 10−15 mol ps −1 MPa −10−1 MPa −10−2 MPa Rcrit 0.096 μm 0.93 μm 7.29 μm ncrit 7.4 × 10−19 mol 7.0 × 10−17 mol 4.3 × 10−15 mol Table 1. Values of Rcrit and ncrit calculated from Eqs (9) and (11) for the conduit water pressures ps given in the first row. ps −1 MPa −10−1 MPa −10−2 MPa Rcrit 0.096 μm 0.93 μm 7.29 μm ncrit 7.4 × 10−19 mol 7.0 × 10−17 mol 4.3 × 10−15 mol ps −1 MPa −10−1 MPa −10−2 MPa Rcrit 0.096 μm 0.93 μm 7.29 μm ncrit 7.4 × 10−19 mol 7.0 × 10−17 mol 4.3 × 10−15 mol Conclusion: spontaneous cavitation vs air seeding The appearance of an embryonic bubble does not necessarily lead to vessel embolism. For the xylem sap pressures typical for trees, the emergence of bubbles containing only water vapor is extremely unlikely, as discussed in section ‘Spontaneous cavitation'. Bubbles containing water vapor and air molecules dissolve, according to section ‘Diffusional equilibrium and dissolution of floating bubbles' quite rapidly, provided the bubbles are small enough and do not contain too many air particles. In sufficiently big bubbles, however, the expansion tendency of the gas particles inevitably outweighs the surface tension and the bubble will either grow and fill the vessel or split up into daughter bubbles, which may or may not dissolve. A detailed and explicitly time-dependent exposition of these processes can be found in Konrad and Roth-Nebelsick (2003). Safety measures against embolism Experimental evidence of embolism Cavitation and embolism can be demonstrated indirectly or by direct observation using imaging techniques. Indirect measurement of embolism is mostly conducted by producing vulnerability curves (VC) that express the loss of hydraulic conductivity of a stem or a stem segment against xylem pressure. Different techniques are applied: air dehydration (‘natural’ drying of a sample or a whole potted plant), air injection (application of positive gas pressure to the sample, which should provide the same amount of embolism as the corresponding negative pressure) and centrifugation (xylem tension generated by spinning of the stem segment in a centrifuge) (Venturas et al. 2017). Conductivity is mainly measured by driving water through a sample with a defined pressure gradient. Also, xylem tension is difficult to measure because it is in a metastable state sensitive against any disturbance. The most ‘practical’ way to measure xylem tension is by using the pressure chamber. Here, the sample is exposed to an increasing positive pressure that is monitored with one end of the sample sticking out of the device. Tension is then determined as being identical with respect to the absolute value to that positive pressure necessary to drive liquid to the surface of the cut end (Vogt 2001). Another method is the pressure probe that records the tension directly. Here, a fine capillary is brought into a conduit and the ‘pull’ of the xylem tension is recorded (Wei et al. 1999). This method is, however, also invasive and can disturb the metastable state of xylem tension. Although numerous and often consistent results were collected over the years based on these methods, there are still methodological debates (Cochard et al. 2013, Rockwell et al. 2014, Venturas et al. 2017). Various artifacts and problems may occur, particularly with invasive techniques, as may be expected when measuring water in a metastable state. Methods that are based on cutting samples potentially allow bubbles (and other impurities) to enter the sample or water oversaturated with air is pressed into the xylem, which can bias the results (Wheeler et al. 2013, Wang et al. 2014). A critical discussion of invasive approaches has been provided by Cochard et al. (2013). A non-invasive method is stem psychrometry (Vogt 2001), but attaching the brittle sensors firmly to functional xylem and isolating the contact zone from external influences is quite challenging. Another non-invasive technique is to record acoustic emissions emitted during the cavitation event, which has its own pitfalls (Vergeynst et al. 2014). Direct observation of cavitation and embolism is possible with new imaging techniques such as high-resolution magnetic resonance imaging (Holbrook et al. 2001, Clearwater and Clark 2003) or high-resolution X-ray tomography (synchrotron- or desktop-based) (Brodersen et al. 2010, Cochard et al. 2015, Ryu et al. 2016) and neutron radiography (Tötzke et al. 2013). Another new tool was recently introduced by Brodribb et al. (2017) based on light transmission. The various experimental problems are accentuating the debate on xylem refilling. Refilling must represent a repair mechanism of dysfunctional conduits thereby recovering hydraulic conductivity. The original concept was that positive pressure was necessary for xylem refilling due to, for example, thorough re-wetting of the soil by extensive rain (Vogt 2001) or root pressure during spring (Sperry et al. 1987). Some years ago, evidence appeared for a ‘novel’ type of refilling with conduit repair occurring under xylem tension (Salleo et al. 1996, Hacke and Sperry 2003, Stiller et al. 2005). These findings appeared to contradict conventional models on xylem functioning and prompted discussions on the possible mechanisms (Holbrook and Zwieniecki 1999, Tyree et al. 1999, Konrad and Roth-Nebelsick 2003, Vesala et al. 2003). Basically, living cells and particularly the phloem are suggested to be involved (Salleo et al. 1996, Nardini et al. 2011). The feasibility of this refilling model depends largely on the question of whether the dissolution of the ‘embolizing’ lumen bubbles can be completed before the hydraulic isolation of the embolized vessel fails due to the dissolution of the pit bubbles. Therefore, the temporal behavior of both spherically symmetric lumen bubble and of axially symmetric pit bubble is of central significance to embolism repair and to the general dynamics of bubbles in xylem conduits. The dynamics of spherically symmetric bubbles and the temporal development of bubbles in cone-shaped pits has been explored in Konrad and Roth-Nebelsick (2003). By no means is this debate settled and the aforementioned refilling and/or the degree to which conduits can be refilled under negative tension (Venturas et al. 2017) remain open questions. Possible experimental artifacts biasing determinations of xylem tension and embolism/repair continue to emerge. For instance, recent experiments on Laurus nobilis, a ‘classic’ species for which the novel refilling was demonstrated early, provided inconsistent results (Cochard et al. 2015, Nardini et al. 2017). However, deactivating or interrupting the phloem by, for example, girdling, can obviously block refilling, an indication that metabolic processes of living cells are involved in xylem repair (Salleo et al. 1996, Trifilò et al. 2014). New imaging techniques as described above allow for directly observing refilling of conduits. For Vitis vinifera, it was indeed shown that the water entering empty conduits occurs through vessel-parenchyma pits and therefore must originate from adjacent living cells (Brodersen et al. 2018). High-resolution observations of refilling conduits indicate a long lifetime of small bubbles seated within the pit region. Moreover, the bubbles persisting during xylem refilling appear to be above the critical radius with respect to the measured xylem tension. Hence, these bubbles should have expanded thereby preventing conduit refill. A possible explanation is that the xylem of V. vinifera is segmented and that different xylem pressure values can exist at different xylem locations (i.e., non-uniform pressure distribution). These current controversies and unexpected results demonstrate that despite the availability of new experimental techniques, the process of embolism and embolism repair is not fully resolved. The most reliable way to circumvent the problems and precariousness that trouble attempts to obtain VCs experimentally is — probably — to refine the theoretical model of conduit embolism to such a degree that it can be used to derive VCs indirectly from experiments based upon ‘stable’ and unambiguously measurably parameters, defined in the framework of the soil–plant–atmosphere continuum. Safety measures provided by conduit architecture Various anatomical features of the xylem can be understood as improving safety against embolism. Their purpose is (i) to avoid embolism altogether, (ii) to minimize its consequences by isolating embolised conduit segments of the water transport pathway and (iii) to reestablish conduit functionality. There are two types of conduits: tracheids and vessels. Whereas tracheids are elongated single cells, vessels are formed by series of cells whose end walls are perforated or even removed. The latter conduit type can be long, particularly in ring-porous species. However, all conduit types are shorter than the entire transport pathway, and water has to cross from conduit to conduit on its ascent (Figure 5). The transport pathway is therefore segmented, and this segmentation and the structure of the interconnecting pits are essential for transport safety. The role of pits is to allow water to enter and leave conduits and to isolate a cavitated conduit. There are two kinds of pits that are permeable for water but not for gas after a cavitation has occurred. Figure 5. View largeDownload slide (a) Re-routing of water flow in a conduit with torus-margo pits (typical for gymnosperms) and with (b) membrane pits (typical for angiosperms). In torus-margo pits, a pressure difference in adjacent conduits caused by embolism pushes the torus against the pit opening, the pit closes. Membrane pits possess membranes with tiny openings. In case of embolism, highly curved gas/water interfaces develop in these openings, effectively closing the pit. (c) Adjacent conduits of Laurus nobilis. The conduits are cut open along their longitudinal axes. The pits can be seen in cross section (right arrow) and top view (left arrow). (d) Detail of (c). Illustrations (a) and (b) redrawn from Taiz (2006). Figure 5. View largeDownload slide (a) Re-routing of water flow in a conduit with torus-margo pits (typical for gymnosperms) and with (b) membrane pits (typical for angiosperms). In torus-margo pits, a pressure difference in adjacent conduits caused by embolism pushes the torus against the pit opening, the pit closes. Membrane pits possess membranes with tiny openings. In case of embolism, highly curved gas/water interfaces develop in these openings, effectively closing the pit. (c) Adjacent conduits of Laurus nobilis. The conduits are cut open along their longitudinal axes. The pits can be seen in cross section (right arrow) and top view (left arrow). (d) Detail of (c). Illustrations (a) and (b) redrawn from Taiz (2006). When an embryonic bubble with R>Rcrit and/or n>ncrit expands within a conduit, the mean distances between the (liquid) water molecules decrease, the water pressure increases. Hence, water molecules are pushed through the pit openings into adjacent conduit elements that are still intact and are under (negative) ‘operating pressure’. The two pit types achieve hydraulic isolation of the dysfunctional conduit segment in different ways: If a torus-margo pit operates in ‘normal mode’ the flow velocity is rather low (gymnosperms have maximum flow velocities of 1 m h–1 to 2 m h–1), see Larcher (2003) and the flow rate Q through a pit opening (see Figure 6d–f) attains moderate values below a certain critical flow rate Qcrit ⁠. The two forces acting on the torus — one originates from the flowing water pushing the (impermeable) torus, the other one is exerted by the elastic (and permeable) margo — are in stable equilibrium (see Chapman et al. (1977)) and the torus occupies a somewhat deflected (depending on the flow rate Q) but stable position within the pit. In ‘embolism mode’, the water rushes much more quickly through the pit opening; Q exceeds Qcrit ⁠, equilibrium between the two forces becomes impossible (see Chapman et al. (1977)) and a stable position for the torus does not exist any more. The torus is pushed against the pit border, the pit closes and remains closed due to the water pressure difference between dysfunctional and intact conduit element (Figure 6f). Figure 6. View largeDownload slide Differences and similarities between membrane pits (a–c) and torus-margo pits (d–f). The membranes in membrane pits possess many tiny pores through which the xylem water flows. During embolism, strongly curved gas/water interfaces develop in these pores (typical radii: 0.005 –0.02 μm). In torus-margo pits, water flows through the permeable and elastic margo. (b) Pit membrane of Sambucus nigra, top view. Scanning electron microscope (SEM) picture of ethanol-dried material. (c) Pit of Laurus nobilis, cross-section. SEM picture of ethanol-dried material. (e) Pit with torus, top view, Abies alba. SEM picture of ethanol-dried material. (f) Top view of pits of Pinus sylvestris. The two pits on the left show closed pits, with the tori pressed to the pit aperture, whereas the two pits on the right show the pit aperture (pit membranes were removed during preparation). The relief of the two tori indicates the pit apertures. This effect is due to the still hydrated state of the tori, made possible by the CryoSEM technique. Additionally, the upper torus is positioned ‘off-center’. This possibly illustrates a reason for air-seeding: tori not covering the aperture completely. During embolism, the flowing water pushes the impermeable torus against the pit opening closing effectively the aperture. Illustrations (a) and (d) are redrawn from Esau (1960). Figure 6. View largeDownload slide Differences and similarities between membrane pits (a–c) and torus-margo pits (d–f). The membranes in membrane pits possess many tiny pores through which the xylem water flows. During embolism, strongly curved gas/water interfaces develop in these pores (typical radii: 0.005 –0.02 μm). In torus-margo pits, water flows through the permeable and elastic margo. (b) Pit membrane of Sambucus nigra, top view. Scanning electron microscope (SEM) picture of ethanol-dried material. (c) Pit of Laurus nobilis, cross-section. SEM picture of ethanol-dried material. (e) Pit with torus, top view, Abies alba. SEM picture of ethanol-dried material. (f) Top view of pits of Pinus sylvestris. The two pits on the left show closed pits, with the tori pressed to the pit aperture, whereas the two pits on the right show the pit aperture (pit membranes were removed during preparation). The relief of the two tori indicates the pit apertures. This effect is due to the still hydrated state of the tori, made possible by the CryoSEM technique. Additionally, the upper torus is positioned ‘off-center’. This possibly illustrates a reason for air-seeding: tori not covering the aperture completely. During embolism, the flowing water pushes the impermeable torus against the pit opening closing effectively the aperture. Illustrations (a) and (d) are redrawn from Esau (1960). Membrane pits (see Figure 6a–c) possess membranes with numerous tiny pores (typically with a radius of rp≈0.005μm to 0.02 μm) through which the conduit water flows in ‘normal mode’ (see Choat et al. (2003)). During embolism, the embryonic gas bubble inflates — perhaps also releasing daughter bubbles — until it touches eventually the pit membrane. In the membrane, pores develop strongly curved gas/water interfaces (‘menisci’) whose radii of curvature have probably similar values as the membrane pore radii. Hence, according to the Young–Laplace-Equation, the menisci can be expected to withstand pressure differences (related to the rp values just given) of about Δp=2γrp≈7MPa…28MPa (12) This should be sufficient to isolate an embolized conduit segment from its intact neighbors. The nature of pit membranes is still under debate. In particular, there is evidence that the pit membranes are covered by a hydrogel (Pesacreta et al. 2005, Lee et al. 2012). Diffusional equilibrium and dissolution of floating bubbles The dynamic behavior of bubbles of radius R is now briefly considered. Conventionally, the absence of mechanical equilibrium is described by the so-called Rayleigh–Plesset (RP) equation, which is derived from the Navier–Stokes equation for an incompressible fluid flow assuming spherical symmetry (Rayleigh 1917, Plesset 1949). It is given by p−psρ=Rd2Rdt2+32(dRdt)2+4νRdRdt+2γρR (13) where p=pa+pw is the total gas pressure in the bubble (air plus saturated water vapor) assumed to be uniform, ps is, as before, the liquid water pressure far from the bubble and can vary in time t, ρ and ν are the density and kinematic viscosity of water far from the bubble location, γ is the surface tension of the bubble, and R(t) is a time-varying bubble radius. The RP equation (Eq. (13)) assumes isothermal conditions so that temperature differences between the gas and liquid phases are not significant (i.e., the bubble-liquid system is in thermal but not necessarily mechanical or air mass equilibrium). At mechanical equilibrium, the RP equation can be simplified by setting dR/dt=0 and the Young–Laplace equation is recovered. As noted earlier, the time scale associated with exchanges of air molecules between an isolated bubble and the surrounding water is assumed to be sufficiently slow compared with those associated with mechanical forces described by the RP. For example, the molecular diffusion coefficient of air in water is on the order of 10−9 m2 s–1 whereas the kinematic viscosity of water ν/ρ=10−6m2s-1 ⁠. This contrast results in an air diffusional time scale that is 103 times larger than the viscous time scales in the RP equation. Hence, for purposes of determining time scales of air molecules diffusing into or out of a cavity in water, the forces acting on the bubble are presumed to have attained mechanical equilibrium sufficiently fast though corrections to it can be carried out using the RP equation if such refinements are warranted (Plesset and Prosperetti 1977, Brennen 2013). With this background, the dynamic behavior of a floating bubble exchanging air molecules with the surrounding liquid is now considered assuming both mechanical and thermal equilibrium. This exchange rests upon two physical effects (see Figure 7): At the bubble’s air/water interface, air particles constantly either dissolve into or escape from the water. As a consequence of this intensive exchange, the (partial) pressure pa of the air inside the bubble and the concentration CR of the dissolved air particles in the liquid in the near vicinity of the bubble are proportional to each other and given by Henry’s Law: CR=kHpa (14) The value of the constant kH depends on the gas and liquid species involved. If the concentration CR of dissolved gas particles close to the bubble deviates from the value Cd farther away in the liquid, diffusional currents, directed from areas of higher to areas of lower concentration, arise. Figure 7. View largeDownload slide Diffusional fluxes of air molecules between bubble and xylem water. pa designates the partial pressure of air molecules in the bubble and CR is the concentration of air molecules that dissolve in water close to the bubble, according to Henry’s Law, CR=kHpa ⁠. The fictitious pressure pd represents the concentration Cd of air molecules dissolved in the xylem water at distance d from the bubble. It is defined in analogy to Henry’s Law via Cd=kHpd ⁠. pa<pd causes a diffusional flux j into the bubble. For pa>pd ⁠, air molecules leave the bubble and dissolve in the xylem water. Figure 7. View largeDownload slide Diffusional fluxes of air molecules between bubble and xylem water. pa designates the partial pressure of air molecules in the bubble and CR is the concentration of air molecules that dissolve in water close to the bubble, according to Henry’s Law, CR=kHpa ⁠. The fictitious pressure pd represents the concentration Cd of air molecules dissolved in the xylem water at distance d from the bubble. It is defined in analogy to Henry’s Law via Cd=kHpd ⁠. pa<pd causes a diffusional flux j into the bubble. For pa>pd ⁠, air molecules leave the bubble and dissolve in the xylem water. As the combined result of both processes, air particles are transported either out of and away from the gas bubble or into the opposite direction until a diffusional equilibrium situation between CR and Cd is attained. To facilitate this notion further and in analogy to Henry’s Law, a (fictitious) air pressure in water away from the bubble is defined as pd:=CdkH (15) Employing this definition the situation of diffusional (or exchange) equilibrium can be characterized by the equality pd=pa (16) The Young–Laplace- equation (derived from the RP equation by setting dR/dt=0 ⁠) is now used to link p to ps p=ps+2γR (17) Splitting the gas pressure into the partial pressures of water vapor and air according to p=pw+pa we find from Eq. (17) for bubbles in approximate mechanical equilibrium that pa=p−pw=ps−pw+2γR (18) Observing that this relation establishes a one-to-one correspondence between pa and R, Eq. (16) is used to define an equilibrium bubble radius Req:=2γpd+pw−ps (19) Here, Req defines an equilibrium with respect to the exchange of air particles between the bubble and the surrounding water whereas the radii Rstab and Rinst (defined in Eq. (10)) represent equilibria with respect to the mechanical forces that expand or contract the bubble. Combining Eqs. (10) and (18), the behavior of gas bubbles in mechanical equilibrium but not with respect to exchange of air particles between the bubble and surrounding water can now be understood (consult Figure 8): an air molecular exchange equilibrium exists if pa=pd is realized. (In the lower part of Figure 8, this case is indicated by the intersections between the broken, horizontal lines and the pa(R)-curve at (Req,1,pd,1) and (Req,2,pd,2) ⁠.) Depending on whether Req lies in the interval 0<Rcrit or in Rcrit<Rvap ⁠, the bubble goes through qualitatively different developments: Rcrit<Req (cf. Figure 8a): if the bubble starts at position A (to the left of Rcrit ⁠) the inequality pa>pd is valid, causing air particles to leave the bubble. This loss entails a contraction of the bubble (as can be seen from the na(R) diagram in the upper part of the figure), which raises in turn the bubble’s partial air pressure pa ⁠. This accelerates the particle loss and so on, until the bubble has dissolved. If the bubble starts at position C (between Rcrit and Req,2 ⁠), pa>pd causing air particles to leave the bubble, the bubble responds by an expansion according to the na(R) diagram. The bubble’s partial air pressure pa decreases whereupon smaller numbers of air particles leave the bubble and so on. This process slows down but continues until pa=pd is achieved. If the bubble starts at position D, pa<pd ⁠, air particles enter the bubble. The bubble reacts by a volume contraction. The bubble’s partial air pressure pa increases whereupon less air particles enter the bubble and so on until pa=pd is achieved. Req<Rcrit (cf. Figure 8b): if the bubble starts at position A the same reasoning as in Figure 8a applies. If the bubble starts at position B (between Req,1 and Rcrit ⁠), pa<pd ⁠. Now air particles from the surrounding water enter the bubble, which then reacts by an expansion according to the na(R) diagram. The bubble’s partial air pressure pa decreases whereupon still more air particles enter the bubble and so on. This process continues either until na exceeds ncrit (and the bubble bursts) or until all air particles within the xylem water have assembled in the bubble. If the bubble starts at position D, pa<pd causing air particles to enter the bubble. The bubble reacts by a contraction towards Rcrit where the bubble bursts. Figure 8. View largeDownload slide The behavior of gas bubbles that are in equilibrium with respect to mechanical forces but not with respect to exchange of air particles between bubble and surrounding water. Exchange equilibrium exists if pa=pd is realized (⁠ pa denotes the partial pressure of air molecules in the bubble, the fictitious pressure pd represents the concentration Cd of air molecules dissolved in the xylem water close to the bubble). In the lower parts of the figures, these cases are indicated by the intersections between the broken, horizontal lines and the pa(R)-curve at (Req,1,pd,1) and (Req,2,pd,2) ⁠. Depending on whether Req lies in the interval Rcrit < Rvap (subfigure (a)) or in 0 < Rcrit (subfigure (b)), the bubble goes through qualitatively different developments, explained in the text. Figure 8. View largeDownload slide The behavior of gas bubbles that are in equilibrium with respect to mechanical forces but not with respect to exchange of air particles between bubble and surrounding water. Exchange equilibrium exists if pa=pd is realized (⁠ pa denotes the partial pressure of air molecules in the bubble, the fictitious pressure pd represents the concentration Cd of air molecules dissolved in the xylem water close to the bubble). In the lower parts of the figures, these cases are indicated by the intersections between the broken, horizontal lines and the pa(R)-curve at (Req,1,pd,1) and (Req,2,pd,2) ⁠. Depending on whether Req lies in the interval Rcrit < Rvap (subfigure (a)) or in 0 < Rcrit (subfigure (b)), the bubble goes through qualitatively different developments, explained in the text. Obviously, equilibria with respect to the exchange of air particles related to a bubble radius Req lying within the interval 0<Req<Rcrit are unstable, similar equilibria within Rcrit<Req<Rvap are, however, stable. Hence, stable mechanical and exchange equilibria preclude each other: for 0<R<Rcrit ⁠, the mechanical equilibrium is stable and the exchange equilibrium is unstable. For Rcrit<R<Rvap ⁠, it is the other way round. The different time scales on which the processes connected with the two equilibria operate impose a clear hierarchy on their relevance: since the exchange of air particles is a very slow process, compared with the action of the expanding and contracting forces, a mechanically unstable bubble with Rinst(na) has negligible chances to settle down to a stable exchange equilibrium. It is much more probable that it either bursts or shrinks to the corresponding stable radius Rstab(na) where an unstable exchange equilibrium awaits it. Depending on the relation between pa and pd this state will finally develop into a bubble burst due to congestion (i.e., na>ncrit ⁠) or to complete bubble dissolution. These observations imply a condition that guarantees that bubble dissolution occurs: bubbles represented by position A in Figure 8 are in stable mechanical equilibrium and lose air to the surrounding water, expressed by the relations R<Rcrit and pa>pd ⁠, or, by the equivalent inequality Req>Rcrit (20) Upon insertion of the definitions in Eqs (11), (19) and (15) this condition can be reformulated as a relation between the concentration Cd of gas particles dissolved in the surrounding liquid, the (negative) pressure ps of this liquid, and the water vapor saturation pressure pw Cd=kHpd<kH2(pw−ps) (21) Solving this relation for ps yields ps<−2pd+pw (22) Since soil water and xylem sap are in contact with air at atmospheric pressure, it is reasonable to assume pd≈patmosphere≈10−1MPa ⁠. Neglecting the term pw on the right-hand side (at 25 °C we find pw≈~3167Pa<<10−1MPa≈pd ⁠) Eq. (22) reduces to ps⪅−2×10−1MPa=−2×patmosphere (23) Hence, if the condition of Eq. (23) is fulfilled, embryonic bubbles appearing in the stability region depicted in Figure 4 dissolve after a while in the surrounding water and do no harm. Implications of xylem functionality to phloem transport As earlier noted, sugars produced from photosynthesis are transported to distal parts of the plant by bulk flow through the phloem, an interconnected system of cellular conduits that link essentially all parts of the plant. In addition to sugar transport, the phloem plays a role in transport of signaling molecules. Sugars are exported from the leaf by bulk liquid flow through phloem sieve elements. Sieve elements are approximately cylindrical, of radius r = 1 μm to 10 μm and length l = 0.1 mm to 1 mm. The cellular conduits are connected end-to-end (and in some cases radially) by sieve plates, modified cell walls perforated by numerous enlarged plasmodesmata pores of radius rp = 0.1 μm to 1 μm (Jensen et al. 2012). The phloem conduits carry a sap that contains about 20% sugar by weight (Jensen et al. 2013), and the average flow speeds are in the range of v∼25μms-1 to 250 μm s–1 (~0.1 m h–1 to 1 m h–1), resulting in low-Reynolds-number flow conditions dominated by viscous effects (Jensen et al. 2012). The flow is driven by osmotic pressure differences between sources (i.e., leaf) and sinks (assigned here to roots though sinks can span other plant organs). Phloem sap contains numerous compounds, but as earlier noted, the primary determinant of phloem pressure are concentration of sugars and small ions (Jensen et al. 2016). Phloem transport relies on the osmotic exchange of water with the xylem system, and the conditions in the xylem have a direct impact on phloem flow (Figure 1) (Sevanto 2014, 2018, Stroock et al. 2014, Huang et al. 2018) as earlier noted. The cellular conduits — and the total conductive area — of the phloem, however, is significantly smaller than the xylem, hence the feedback appears to act primarily in one way: phloem flow is impacted by conditions in the xylem, but not vice versa. The impact of xylem pressure on phloem concentration, sap viscosity and carbohydrate allocation patterns are discussed here. Impact of xylem pressure on phloem solute concentration The impact of xylem pressure on solute concentration is considered as follows. Thermal and chemical equilibrium between phloem and xylem requires that the water potentials ψp:=pp−Πp+pg and ψs:=ps−Πs+pg of phloem and xylem, respectively, are equal. Here, pp (Pa) is the (positive) hydrostatic pressure in the phloem, ps (Pa) is the (negative) hydrostatic pressure in the xylem, pg is the gravitational potential (as in Eq. (1)) and Πp,s denotes the osmotic potential in the two tissues. Assuming dilute solutions and a negligible concentration in xylem sap, the equality ψp=ψs simplifies to pp−ℛTcp=ps (24) where cp (mol m–3) is the osmolyte concentration in the phloem, ℛ (J K mol–1) is the gas constant, and T (K) is temperature. For clarity, the van’t Hoff approximation for the osmotic pressure Π∼ℛTc is used, which is accurate in the low-concentration regime (cp< 500mol/m3) ⁠, and acceptable at higher concentrations. Applying Eq. (24) to the source (at x=0 in Figure 1) and to the sink (at x=L ⁠) the pressure difference Δpp between source and sink available to drive phloem flow is Δpp=ℛTΔcp+Δps (25) Note that xylem pressure drop Δps=ps(0)−ps(L) is negative whereas the difference Δcp=cp(0)−cp(L) of solute concentration in the phloem is positive: because sugars are unloaded in the sink region (x=L) ⁠, the solute concentration there is significantly lower than in the source tissue (x=0) ⁠, that is, cp(0)≫cp(L) and Δcp=cp(0)−cp(L)≈cp(0) ⁠. Due to the different signs of Δps and Δcp the pressure difference Δpp between source and sink can, at least in principle, become negative, which indicates that the phloem sap flows within the phloem network in the wrong direction, thus cutting parts of the plant from the supply with assimilates A similar ‘reversed flow’ may occur between phloem and xylem: in the source area, the water that is loaded in the phloem with assimilates is drawn from the xylem; therefore, around x=0 ⁠, the water potentials in phloem and xylem should be related according to ψp(0)≈pp(0)−ℛTcp(0)⪅p(0)≈ψs(0) (26) (with the approximations used in Eq. (24)). Failing this, such a ‘reversed flow’ is likely to cause impairment of the phloem system. The xylem may not benefit from this additional water subsidy originating from the phloem given its small volume compared with the xylem. Hence, the xylem pressure that leads to such flow reversal in the phloem may be used to finger-print severe drought conditions in the plant. The approximation cp(0)≫cp(L) simplifies Eq. (25) to Δpp=ℛTcp(0)+Δps (27) The flow rate Q in the phloem, and hence the rate of energy transport, is directly proportional to the pressure drop Δpp ⁠, i.e., Q=ΔppRp (28) where Rp is the hydraulic resistance in the phloem. Assuming that the organism strives to keep Q at a constant level to maintain metabolism implies that Δpp should remain unchanged. This leads to an expression for the phloem osmolyte concentration cp as a function of xylem pressure drop cp=Δpp−ΔpsℛT (29) A pressure drop of approximately Δpp∼1MPa is required to maintain phloem transport in a moderately sized tree (Jensen 2018). If xylem tension is negligible compared to Δpp ⁠, a concentration of cp=400molm-3 is required to maintain sap flow (Eq. (29), see also Figure 9a). In tall trees, the xylem pressure drop due to gravitational potential alone can reach Δps=−1.2MPa (Eq. (1)), corresponding to cp=880mol/m3 ⁠. Viscous friction to flow in the stem and soil further decreases the pressure in the xylem, with values reported in the range from Δps=−2MPa to −8 MPa (Stroock et al. 2014, Sevanto 2018). This corresponds to osmolarities in the range cp∼1000molm-3 to 3000 mol m–3. Such concentration interval is beyond most of the observed sugar concentrations in both passive and active loaders (Figure 9b and c). It can be surmised that other osmolytes contribute to phloem transport beyond sucrose. Figure 9. View largeDownload slide Impact of xylem conditions on phloem transport. (a) Osmolyte concentration cp required to drive phloem flow as function of xylem pressure drop Δps (Eq. (29), plotted for Δpp=1MPa and T=300K ⁠). The orange dot indicates the maximum tension Δps=−1.2MPa due to gravitational effects (Eq. (1)). (b) Observed phloem sugar concentration cp in passive loaders. (c) Observed phloem sugar concentration cp in active loaders. The vertical axes in (b) and (c) indicate the number of species in each horizontal bin. Data from Jensen et al. (2013). Figure 9. View largeDownload slide Impact of xylem conditions on phloem transport. (a) Osmolyte concentration cp required to drive phloem flow as function of xylem pressure drop Δps (Eq. (29), plotted for Δpp=1MPa and T=300K ⁠). The orange dot indicates the maximum tension Δps=−1.2MPa due to gravitational effects (Eq. (1)). (b) Observed phloem sugar concentration cp in passive loaders. (c) Observed phloem sugar concentration cp in active loaders. The vertical axes in (b) and (c) indicate the number of species in each horizontal bin. Data from Jensen et al. (2013). Impact of xylem pressure on phloem sap viscosity The preceding discussion highlighted the effect of xylem tension on phloem sap concentration: As xylem tension Δps increases, the osmolyte concentration cp must also increase to keep energy transport rates constant (Eq. (29)). Solute concentration, however, affects not only the osmotic potential Π ⁠, but also the viscosity η (Pa) of the solution (Lang 1978, Hölttä et al. 2006, Jensen et al. 2013, Sevanto 2018). For instance, the viscosity of 3000 mol m–3 sucrose solution is ∼1000 times greater than pure water (Figure 10a). By contrast, an osmotically equivalent solution of small ions (such as NaCl) is only ∼1.3 times more viscous (Figure 10b). Figure 10. View largeDownload slide Impact of xylem conditions on phloem sap viscosity. (a) Aqueous sugar solution viscosity η plotted as function of sucrose concentration cp ⁠. The viscosity increases with concentration. The effect is less pronounced for small ions such as the NaCl solution shown in panel (b). The limit cp=0 corresponds to pure water. The solution viscosity is strongly affected by the presence of large sugar molecules, less so by small ions. Data from Haynes (2014). Figure 10. View largeDownload slide Impact of xylem conditions on phloem sap viscosity. (a) Aqueous sugar solution viscosity η plotted as function of sucrose concentration cp ⁠. The viscosity increases with concentration. The effect is less pronounced for small ions such as the NaCl solution shown in panel (b). The limit cp=0 corresponds to pure water. The solution viscosity is strongly affected by the presence of large sugar molecules, less so by small ions. Data from Haynes (2014). The increase in viscosity η with phloem solute concentration cp has a direct negative impact on the transport rate Q (Eq. (28)). For a cylindrical tube of length L and radius a ⁠, the flow rate/pressure-drop relation (the Hagen–Poiseuille law) is, as before, Q=πa48ηLΔpp (30) i.e., the flow rate Q scales as 1/η ⁠. Even at moderate phloem sugar content, the assumption that equal pressures generate equal flows is thus violated (Eq. (28)), and unrealistically large sugar concentrations would be needed to drive sugar transport if this was the sole osmolyte. To further highlight the difference between the case where either sugars or other osmolytes drive transport, we proceed to consider the sugar current I=Qcs (mol s–1), where cs is the phloem sugar concentration at the source. If sugar is the main osmolyte responsible for transport, then cp=cs and the current is quadratic in cs ⁠, c.f. Eqs. (30) and (29): I∼cs2η(cs) (31) Here, a relatively small xylem tension is assumed for simplicity (Δps≪Δpp) and thus the phloem pressure drop is Δpp∼ℛTcp from Eq. (29). The transport rate I in Eq. (31) peaks at cs∼1200mol/m3 (Figure 11a), significantly above the mean sap sugar content observed in both active and passive loaders (Figure 9b and c). If, by contrast, a different osmolyte is driving transport, the pressure difference Δpp is independent of cs and the current scales linearly with cs ⁠: I∼csη(cs) (32) Figure 11. View largeDownload slide Impact of sap viscosity on sugar transport. (a) Sugar current I as a function of sap sugar concentration cs ⁠, assuming that sugar is the primary osmolyte (Eq. (31)). (b) Sugar current I plotted as a function of sap sugar concentration cs ⁠, assuming that sugar is not the primary osmolyte (Eq. (31)). Data from Haynes (2014). Figure 11. View largeDownload slide Impact of sap viscosity on sugar transport. (a) Sugar current I as a function of sap sugar concentration cs ⁠, assuming that sugar is the primary osmolyte (Eq. (31)). (b) Sugar current I plotted as a function of sap sugar concentration cs ⁠, assuming that sugar is not the primary osmolyte (Eq. (31)). Data from Haynes (2014). The transport rate in Eq. (32) peaks at cs∼800molm-3 (Figure 11b), relatively close to the mean sap sugar content observed in both active and passive loaders (Figure 9b and c). In summary, osmolyte concentration needed to drive phloem flow depends on the xylem water status. As xylem tension increases, the phloem concentration must increase dramatically (Eq. (29), Figure 9). The sharp increase in viscosity with sugar concentration makes it likely that other smaller solutes contribute to the osmolarity of phloem sap thereby increasing the concentration without increasing appreciably the viscosity (Figure 10). Impact of drought on xylem and phloem transport The flow rate Qs of xylem sap is, similar to in the phloem case, Eq. (28), proportional to the pressure drop Δps between x=0 and x=L (cf. Figure 1) Qs=−ΔpsRs (33) where Rs is the hydraulic resistance of the xylem and the minus sign ensures that upward flow in the xylem is counted positive. The assumption Πs≈0 of negligible osmotic potential in xylem sap implies Δps≈Δψs=ψs(0)−ψs(L)<0 for the water potential. Assuming further that (i) Rs is not affected appreciably by drought and (ii) the plant maintains ψs(0) approximately constant, changes in Δps are primarily due to the impact of drought on soil moisture, respective soil water potential ψs(L)≈ps(L) ⁠. Experimentally obtained vulnerability curves (Manzoni et al. 2014) and theoretical considerations based on models of the xylem pathway between soil and atmosphere (Manzoni et al. 2014, Huang et al. 2017, 2018) corroborate these assumptions: (i) usually, the operating pressure in the xylem rarely drops below the point, labeled ψ12 ⁠, where Rs increases by only 12%, and (ii) the plant maintains ψs(0) approximately constant at a value that sustains maximum transpiration rate without damaging the xylem. Studies conducted to contrast isohydric and anisohydric behavior (Kolb and Sperry 1999, Sperry et al. 2002, Schultz 2003, McDowell et al. 2008) reported that species with larger ψ12 tend to exhibit more isohydric behavior while anisohydric behavior occurs in species with smaller ψ12 ⁠. Thus, as long as soil water potential ψs(L) remains in the interval ψPWP⪅ψs(L)⪅ψFC (defined by the permanent wilting point, ψPWP≈−1.5MPa ⁠, and the water potential related to field capacity, typically ψFC≈−0.006MPa to −0.03 MPa, depending on soil composition and structure), most plants can sustain phloem function during drought (Sevanto et al. 2014, Sevanto 2018). It is to be noted that a naïve juxtaposition of phloem dysfunction to plant mortality cannot be realistic without other considerations (growth, respiration costs, and resistance to pests and diseases.) However, coupling of xylem and phloem transport as outlined above, mitigates the consequences of drought in two ways: If a drought sets in, the aforementioned assumptions imply that soil water potential, the difference Δps and, according to Eq. (33), xylem sap flow rate Qs decreases, that is, the plant economizes on water consumption from the soil as expected. As has been pointed out above, phloem transport breaks down if Δpp becomes negative, which can occur if Δps<0 on the right-hand side of Eq. (25) or (27) outweighs the (always positive) concentration term. The decrease of Δps caused by a progressive drought precludes such a development: if the plant production of assimilates is approximately unaffected by the drought (that is, cp(0)≈const. ⁠) the decrease of Δps causes an increase in Δpp ⁠, the driving force of phloem transport, and also an increase in the phloem flow rate Q ⁠, according to Eq. (28). Discussion and future outlook This review focused on the physical transport processes in the xylem and phloem systems as well as their coupling through xylem pressure. The goal was to mathematically describe and evaluate the biophysical basis of the main transport laws that can then be used (or upscaled) to devise coupled soil–xylem–leaf–phloem approaches. Attempts to carry out such an exercise are beginning to proliferate in the literature (see, e.g., Huang et al. 2018). There is a growing body of literature highlighting the need for accounting for simultaneous xylem and phloem constraints on stomatal conductance gs and corollary variables. This need is driven by immediate problems such as the effects of droughts, elevated atmospheric CO2 and temperature, or nutrient amendments on gas exchanges with the atmosphere and plant health (Mencuccini and Hölttä 2010, Nikinmaa et al. 2013, Sevanto et al. 2014, Hölttä et al. 2017, Huang et al. 2018). The xylem section here focused on connections between xylem water pressure, bubble dynamics and their stability, and key anatomical traits. The next immediate step is to establish upscaled equations from the aforementioned formulation that predict the shape of the so-called vulnerability curves (VC). The VC features loss in plant (or branch) hydraulic conductance with decreasing amount of water available in the soil or plant. Despite the significance of VCs, linkages between anatomical pit-level (microscopic) properties and stem-level (macroscopic) hydraulic response to drought remains a subject of inquiry and is virtually required when quantifying drought-induced mortality in trees or coupling xylem to phloem at a macroscopic level. The time is ripe for such an advance given the rapid development of measuring techniques for VCs and the capacity to image plant anatomical traits at unprecedented resolution. Theoretical challenges, partly highlighted in this review, are the multi-scaled nature of xylem transport. Size distributions of length and diameters of conduits range from 10−3 m to 10−1 m and 10−4 m to 10−2 m, respectively, whereas pit scale size distributions that include diameter of pores span 10−9 m to 10−7 m and that of the pit membranes containing them are three orders of magnitudes larger. Because hydraulic failure is impacted by processes acting on all these scales, up-scaling from bubble-scale to plant-level while accommodating all the anatomical traits requires novel theoretical and modeling tactics that are still in development. The work here also highlighted that xylem pressures have disproportionate impact on phloem transport (but not the converse) through multiple mechanisms, few of which have been discussed. The next steps are links between the carbon and water economies of plants that require combining the biochemical demand for atmospheric CO2 with gas transfer through stomates, liquid water transport in the soil–xylem hydraulic system and sucrose export in the phloem such as the ones covered here. A number of attempts (Hunt and Manzoni 2015, Huang et al. 2018) are currently underway to complete such coupling and are likely to benefit from the material covered here. The details of this coupling are best kept for a future review. Acknowledgments The authors also thank Stefano Manzoni for all the helpful comments and Birgit Binder, Entringen, for drawing Figures 2a, 2b, 5a, 5b, 6a and 6d. Conflict of interest None declared. Funding K.H.J. was supported by a research grant (13166) from VILLUM FONDEN. G.K. acknowledges support from the US National Science Foundation, Grant/Award Numbers: NSF‐AGS‐1644382, NSF‐DGE‐1068871, NSF‐EAR‐1344703, NSF‐IOS‐1754893. 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TI - Xylem functioning, dysfunction and repair: a physical perspective and implications for phloem transport
JF - Tree Physiology
DO - 10.1093/treephys/tpy097
DA - 2019-02-01
UR - https://www.deepdyve.com/lp/oxford-university-press/xylem-functioning-dysfunction-and-repair-a-physical-perspective-and-ZogAcgbeCL
SP - 243
VL - 39
IS - 2
DP - DeepDyve
ER -