Plant Physiology Preview. Published on April 23, 2015, as DOI:10.1104/pp.114.256602
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Running head: Bubble pressure in vessels after cavitation
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Author for correspondence:
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Melvin T. Tyree
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College of Forestry, Northwest A&F University, Yangling, Shaanxi 712100, China
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Email: [email protected]
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Telephone: +86 029 87080363
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Research area:
Ecophysiology and Sustainability
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Copyright 2015 by the American Society of Plant Biologists
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Studies on the tempo of bubble formation in recently cavitated vessels: A model to
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predict the pressure of air bubbles
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Yujie Wang, Ruihua Pan, Melvin T. Tyree*
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College of Forestry, Northwest A&F University, Yangling, Shaanxi 712100, China
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*
Author for correspondence.
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SUMMARY:
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An ideal gas model of air bubble dynamics in vessels brings new insights and understanding
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of the tempo of embolism formation in stems.
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Footnotes:
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1
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*
This research was supported by the 5-year 1000-talents research grant awarded to MTT.
Corresponding author: Melvin T. Tyree, e-mail: [email protected]
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ABSTRACT
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A cavitation event in a vessel replaces water with a mixture of water vapor and air. A
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quantitative theory is presented to argue that the tempo of filling of vessels with air has two
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phases: a fast process that extracts air from stem tissue adjacent to the cavitated vessels (<10
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s) and a slow phase that extracts air from the atmosphere outside the stem (>10 h). A model
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was designed to estimate how water tension near recently cavitated vessels causes bubbles in
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embolized vessel to expand or contract as tension (T) increases or decreases, respectively.
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The model also predicts that the hydraulic conductivity (kh) of a stem will increase as bubbles
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collapse. The pressure of air bubbles trapped in vessels of a stem can be predicted from the
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model based on fitting curves of kh vs T. The model was validated using data from six stem
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segments each of Acer mono Maxim. and a clonal hybrid 84K poplar (Populus alba L. ×
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Populus glandulosa Uyeki). The model was fitted to results with root mean square error <
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3%. The model provided new insight into the study of embolism formation in stem-tissue and
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helped quantify the bubble pressure immediately after the fast process referred to above.
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INTRODUCTION
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Vulnerability curves (VCs) have been viewed as a good measure of drought resistance of
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woody stems (Cochard et al., 2013). Increasing drought increases the xylem tension and
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eventually induces cavitation of the water in conduits when the tension exceeds a certain
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threshold (Sperry and Tyree, 1988; Sperry et al., 1996). A cavitated vessel first fills with
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water vapor and eventually fills with air at atmospheric pressure because of Henry’s law that
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describes gas equilibrium at the water/air interfaces. The time required for the progress
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mainly depends on the penetration rate of air into the recently cavitated vessel lumen via
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diffusion through the liquid phase.
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Previous studies were made about how fast bubbles disappear in embolized stems
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because of the solubility of air in water when water pressure exceeds atmospheric pressure,
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and the process takes 10 to 100 hours depending on conditions (Tyree and Yang, 1992; Yang
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and Tyree, 1992). The tempo of bubble disappearance was measured by following the rise in
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stem conductivity (kh) versus time. The theory in Yang and Tyree (1992) relied on the same
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principles used in this paper (Henry’s and Fick’s Law and the ideal gas law), but modeling
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and experiments were done at pressures between 1 and 3 times atmospheric rather than sub-
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atmospheric pressure (negative pressure). However, much less is known about tempo of
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bubble formation in recently cavitated vessels (Brodersen et al., 2013). If the progress of
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embolus formation takes several minutes then no changes in conductivity could be observed
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with available techniques, but if it takes hours then the tempo of bubbles can be studied by
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rapidly inducing cavitation with increasing Tension (T), and after cavitation-induction
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measuring the influence of T on stem hydraulic conductivity (kh) as T is reduced gradually to
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zero. If air bubbles are at a pressure (bubble pressure, ) lower than a threshold near
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atmospheric pressure, bubbles ought to collapse when T decreases according to the ideal gas
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law and Henry’s law (see theory below). The consequence of bubble collapse will be partial
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filling of vessels with water and the rest with air bubbles. The partial filling of water in a
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recently cavitated vessel ought to increase the lumen conductivity from zero and connect the
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embolized vessel to adjacent conductive vessels and hence ought to increase the conductivity
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of the stem by an additional flow pathway (Wheeler et al., 2005; Hacke et al., 2006). The
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vascular system of stems is a complicated network with vessels of different lengths,
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diameters and orientation (Evert, 2006), and the complex vessel network make the additional
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pathway possible. Therefore, bubble collapse could be detected through the impact of T on
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the kh of the stems in a way that is very similar to the methods used by Yang and Tyree (1992)
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but requires a more sophisticated centrifuge technique to induce embolism.
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Many studies have assumed that the bubble pressure in newly cavitated vessels ought to
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be near atmospheric pressure and no corrections for bubble pressure have been taken in
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measuring PLC (Percent Loss of Conductivity) when tension is lower than a critical threshold
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(Li et al., 2008; Wang et al., 2014a). As a result of bubble collapse, the measured kh under a
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mild tension should be higher than that under high tension (> 0.5 MPa). And the lower the
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initial bubble pressure is the more bubbles collapse with decreasing tension.
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The aim of this study is to construct a model that estimates average bubble pressure in
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partly embolized stems from the functional dependence of kh on T, and with this model we
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can further our understanding of the tempo of bubble formation in stems. In this paper we
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will argue that the tempo of bubble formation is in two phases: an initial rapid phase (seconds
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to minutes to complete) followed by a much slower phase (many hours to complete). Since
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there is no method for measuring the rapid phase, the rapid phase will be described
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theoretically below. Next a theory will be developed that allows the estimation of the pressure
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of air in recently formed bubbles in vessels during the slow phase. An experimental
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validation of the model will follow that will yield values of bubble pressure within the first 1
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to 2 h following the fast phase of embolism formation in vessels.
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THEORIES AND MODELS
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The theory of fast embolus equilibrium following cavitation events
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Water-filled vessel-lumina usually occupy about 10% (8.6% in Acer mono and 15.1% in
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Populus 84K) of the volume of wood. The remainder of the wood can be divided between air,
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water and solids distributed between living and dead cells in woody xylem tissue; how much
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of the volume is air depends on prior history. Immediately following the rapid cavitation of a
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fraction (α) of the water volume (vessels), the vessel lumina will be filled primarily with
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water vapor. Some simple calculations demonstrate that within a few seconds or minutes a
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cavitated vessel will reach a quasi-stable pressure below atmospheric pressure by drawing
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from air dissolved in the immediate vicinity of the cavitated vessel. Three mechanisms are
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considered for air entry into a newly cavitated vessel: (1) the air derived from the dissolved-
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air in the water of the cavitated vessel before cavitation event happened, (2) mass flow
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through the pit pore that seeds cavitation and (3) diffusion of air from surrounding tissue.
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Based on the equilibrium partial pressure of water vapor ‘above’ liquid water plus the
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solubility of air in water (Henry’s law), the first mechanism contributes about 3.2 kPa water
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vapor (at 298 K), 1.2 kPa N2 and 0.6 kPa O2 if the water in the embolized vessel is saturated
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with air and if all of it contributes to the embolus following the cavitation event. The N2 and
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O2 partial pressures have to be viewed as an upper bound estimates because some of the
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dissolved air may be drawn up into the transpiration stream before it can escape into the
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cavitated vessel.
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For the second mechanism we can suppose that the mechanism of cavitation is air
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seeding from an adjacent vessel already filled with air. Air seeding is supposed to occur
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through the largest pit membrane pore at the tension that is big enough to pull the air/water
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interface through a hypothetical pore of the pit membrane. If this pore remains filled with air
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following the cavitation event then the pore could deliver air by pneumatic flow of air from
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the vessel that seeds the cavitation to the newly cavitated vessel. Calculation based on this
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model of pneumatic flow reveals that the process would take days to completely fill the
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cavitated vessel and hence can be ignored (see Supplemental Appendix 1), and most
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researchers assume that the pore seeding embolism will fill with water within seconds after
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the cavitation event.
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The third mechanism requires us to invoke Henry’s law to calculate how much air might
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be delivered from endogenous sources (from within the stem excluding the cavitated vessels).
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Here we will assume no air comes from exogenous sources (from outside the stem). First we
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consider the final equilibrium from endogenous sources then we will approximate the time to
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reach this ‘early equilibrium’. It is worth noting that the air in fibers and extracellular spaces
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may also be a major source of air in this progress, however we followed the standard practice
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of flushing stems with degassed water to eliminate air in vessels and surrounding fibers.
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Solving Henry’s law for a void forming a volume fraction α of space surrounded by air-
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saturated water, yields the following relationship (see Supplemental Appendix 1):
1 1 (1)
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, where PG is the partial pressure of gas species “G” at equilibrium, is the partial pressure
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of N2 or O2 in the atmosphere, KG is the Henry’s Law constant of gas “G”, RT is the gas
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constant times Kelvin temperature, and
is the fraction of the water-volume that has
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cavitated. The final bubble pressure in the embolized vessels should be: మ ,
మ
మ
where మ and మ can be computed from Eq. 1 and మ is the saturated water vapor
and the theoretical
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pressure (3.2 kPa at 298 K). Fig. 1 shows the relationship between
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initial bubble pressure in stem. The curves in Fig. 1 are upper bounded estimates of the early
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equilibrium pressure (PG) because it is assumed the surrounding water was saturated with air
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as determined by . But in experiments presented below, stems were flushed with partly
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degassed solution so the amount of gas in surrounding tissue would have been below the
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saturation level ( ).
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The initial equilibration time was calculated using a simplified model for diffusion of gas
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from a cylindrical shell surrounding recently cavitated vessels. Let’s assume that α of the
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water volume instantly cavitated and that the cavitated vessels are evenly distributed
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throughout the stem. In a 1 cm segment of wood, each cavitated vessel of radius = r would
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draw air from a cylindrical shell of water of radius = R such that the volume of the water
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shell = 1 cm π(R2-r2)β cm3 of water, where β is the fraction of the wood volume that is water
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and 2R is the average distance between cavitated vessels. The maximum distance that air in
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solution would have to diffuse to the cell surface is x = R-r. The mean time, t, that it takes for
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half of the air-molecules to diffuse through water a distance x is proportional to the distance
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squared: x2 = 2DGt , where DG is the diffusion coefficient of gas species “G” in water (DG of
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O2 and N2 in water are 2.10E-5 and 1.88E-5, respectively). Using a typical value of vessel
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radius = 15 μm, α = 0.05 (5% of water embolized) and DG we can assign an estimated
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maximum distance of diffusion to be about x ≅ 55 μm and the time for > half the air to
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diffuse into the newly cavitated vessel to be ≅ 2 s so 99% equilibrium would occur in < 14 s
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(about 6 half times). A more rigorous solution of the problem produced a result that was quite
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close to this estimate (Supplemental Appendix 1). If a smaller percentage of vessels cavitate,
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then the diffusion distance, x, for endogenous gas would be increased making the t increase
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proportional to the square of x. For example if 15% of the vessel cavitated instead of 50%
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then x would increase by a factor of 3 and t would increase by a factor of 9.
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We argue in this paper that there should be a fast and a slow equilibrium phase. The fast
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phase is caused by localized equilibrium over a short x distance. By the time the fast
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equilibrium is completed, all subsequent Henry’s law equilibrium has to be satisfied by
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diffusion over a much longer distance. For a 1 cm diameter stem this distance is roughly 5
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mm (versus about 0.05 mm for the fast equilibrium) for air to diffuse from exogenous sources
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outside the stem. Since the time for equilibrium is proportional to square of x it follows that
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the slow equilibrium time will be (5/0.05)2 =104 times longer. So if the fast phase takes 10 s
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the slow phase will be 105 s or about 1 day.
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In conclusion, we expect the initial equilibration of bubble pressure to be quite fast (10
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to 100 s) depending on the percentage of vessels cavitated. In contrast, the time for
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exogenous sources of air (outside the stem) to diffuse into the stem should be much longer.
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The ‘back of the envelope’ calculation in the previous paragraph suggests this time is quite
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long. Some experimental validation about the time it might take can be gained from the time
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it takes air bubbles to dissolve in stems (Tyree and Yang, 1992). From Tables 2 and 3 in Tyree
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and Yang (1992), it can be seen that the time for all bubbles to dissolve when the applied
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pressure is 14 kPa is about 150 h and is about 15 h when the applied pressure is 150 kPa for
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stems about the size used in the cavitron. If we equate full recovery to about 3 half times as
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measured by Tyree and Yang (1992) then the time for recovery will be up to 16 h near
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atmospheric pressure. We anticipated similar half times for the reverse process of bubble
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formation. In the second paper in this series we measured the tempo experimentally and time
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needed was > 1 day. The experimental conclusions of the above experiments were in
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agreement with theoretical models based on Henry’s and Fick’s laws.
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Single vessel model without tension gradients
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The single vessel model could be an ideal gas law model provided the number of moles of
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gas in the bubble is constant. In this special case the bubble volume (Vb) or length (Lb), from
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absolute bubble pressure (Pb) and absolute water pressure (Pw) in or adjacent to the vessel.
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Assuming cylindrical geometry for vessels we can say that:
(2a)
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When the bubble completely fills the entire vessel it has a pressure = and it occupies the
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entire length of the vessel, Lv. As the bubble collapses to Lb < Lv the bubble pressure increases
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(Pb > ) according to the ideal gas relation in Eq. 2a. However we have already argued
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above that gases quickly equilibrate between a vacuum void and surrounding tissue, and that
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the equilibration time is generally <10 s (see appendix 1 for a more exact solutions). But the
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time required to do measurements of changes in kh resulting from changes in bubble length
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takes >> 100 s. Bubble pressure will equilibrate with water adjacent to the bubble and any
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increase in bubble pressure will result in the rapid movement of air from the bubble to the
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surrounding solution which would invalidate the application of the ideal gas law. So we have
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to appeal again the Henry’s law to explain changes in bubble size.
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Taking into account Henry’s law solubility together with the ideal gas law yields the
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following solution where the number of moles of water and air are conserved in the vicinity
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of the embolus:
1α
1
α
(2b)
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with the restriction that Lb≤Lv if the right terms yield a larger value of Lb. Eq. 2a is similar to
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2b except for the last term generated by Henry’s law considerations, where α = the initial
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fraction of embolized volume with initial bubble pressure,
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Henry’s law constant for air = value based on 20% O2 and 80% N2. Equation 2b is maybe
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wrong by 1 or 2 % because it does not account for the amount of air/water entering the vessel
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as the bubble collapses. The bubble collapse could be a little more if water enters with air
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dissolved below the equilibrium concentration for Pb or could collapse less if the water that
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enters is more saturated with air. But in most solutions the error will be just a few percent.
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and KA is the equivalent
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At final equilibrium, Pb should be higher than Pw because of the surface tension (γ) and
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contact angle (θ) between water/air interface and vessel wall of lumen with diameter Dv. In
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this paper, we assigned θ = 45º as the typical contact angle, which range between 42º and 55º
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in a various species (Zwieniecki and Holbrook, 2000). Hence the bubble pressure is always
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higher than water pressure by capillary pressure (PC):
4 cos (3a)
(3b)
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Tension (T) is typically reported relative to barometric pressure (Pbaro), whereas Pw is the
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absolute pressure; hence we have Pw = -T+Pbaro hence
(3c)
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The bubble pressure will change depending of changes in Pw (or T) and the number of moles
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of air in bubble. In this paper compute the bubble volume change is computed by assuming
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Henry’s law equilibration happens continuously between Pb and the surrounding water.
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The total resistance of a vessel (Rv) consists of lumen resistance (RL) and pit resistance
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(RP). The lumen resistance is proportional to the length of water in the lumen while the pit
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resistance is proportional to the reciprocal length of water (Hacke et al., 2006). So resistance
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of a vessel with length Lv and water length Lw will be:
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,
,
(4a)
(4b)
(4c)
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, where RL,0 and RP,0 represent the lumen resistance and pit resistance of a non-embolized
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vessel, respectively. We use the generally accepted and empirical approximation that lumen
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resistance is equal to pit resistance in any vessel regardless of vessel length, Lv (Hacke et al.,
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2006), hence when we assume that RL, 0 = RP, 0 and from Eqs (4a,b,c) we get:
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(5)
2 ,
, where Rv,0 = RL,0+RP,0. The value of can be substituted into (Eq. 5). If we want
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an expression for single vessel conductivity, (kh,s) then by definition of conductivity and
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resistance kh,s=Lv/Rv we have:
, 2
2 · ·
, ,
(6)
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Ultimately the value of kh,s depends on Pw or T. The conductivity of a stem segment can
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be equated to the sum of many kh,s values in series and parallel. When kh is measured in a
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conductivity apparatus (gravimetric method, Li et al. 2008) Pw is above Pbaro and it changes
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linearly with distance in the stem so embolized vessels at different locations in the stem
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would have only slightly different values of kh,s. In contrast if kh is measured in a cavitron
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then, T or Pw is a quadratic function of radial position in the stem relative to the axis of
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rotation in the centrifuge (Cochard, 2002; Cai et al., 2010; Hacke et al., 2015), so the location
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of the vessel affects the value of Pw and hence the value of kh,s changes more dramatically
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with position as shown in Fig. 2. In this paper kh of whole stems were measured in a
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centrifuge hence individual vessel kh,s values were modulated by changes in T near the vessel.
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Most readers can skip the detailed derivation of how kh of a whole stem depends on the
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collective behavior of individual kh,s values in the cavitron (shown in Supplemental Appendix
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2), because the most important thing is to realize that kh measured in a Cochard cavitron will
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change with the maximum tension at the center of rotation (Tc) at the axis of rotation in a
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cavitron and that other values of T in the stem will be a quadratic function of Tc. Another
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important thing to remember is that Tc is consistently more than the average tension ( )
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experience by vessels. The ratio of /Tc is always equal to 2/3.
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the
vessel lumen is fully embolized and the single vessel kh = 0, but as soon as T < The behavior of Eq. 6 is plotted in Fig. 3. When the tension, T >
the theoretical kh begins to rise because of bubble collapse. Each curve in Fig. 3 represents
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how kh will change with T from low to high initial bubble pressure, , in Henry’s law
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equilibrium with surrounding tissue; at the three lowest bubble pressures the bubble
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completely dissolves as T approaches 0. In contrast at high values the bubble is only
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partly dissolved in the surrounding tissue. The readers should note that the recovery of
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conductivity happens only if the restored water in embolized vessels is adjacent to functional
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vessels. In a highly embolized stem, say PLC > 90%, this assumption may not be true, but for
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simplicity we ignore this issue in the rest of this paper. Addressing this real case would
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require specific knowledge of the three-dimensional interconnectivity of vessels and such
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knowledge is not available.
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RESULTS
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Sensitivity Analysis of the Hydraulic Recovery Model (see appendix 2)
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Standard sensitivity analysis of the model output (kh) to the model parameters is needed to
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reveal which model parameters are most important in a ‘hydraulic recovery curve’, i.e., a plot
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of how much kh increases in response to a decline in the tension at the axis of rotation (Tc). As
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discussed above and in Supplemental Appendix 2, we know that four parameters affect the
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final output of the model: (1) average bubble pressure, (2) fraction of embolized vessels in
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the stem, (3) vessel length and (4) vessel diameter which determines the capillary pressure;
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the relationship between the model output and the parameters can be studied by keeping three
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of them constant while changing the fourth parameter.
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(1) Impact of the bubble pressure
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According to the fast embolus equilibrium model, when the stem is under high tension, say 1
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MPa in the center, cavitated vessels will be filled with mixture of water vapor and sub-
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atmospheric partial pressure of N2 and O2. The model predicts that the kh measured in a
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cavitron should increase as the centrifugal tension decreases because the bubbles will
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collapse in response to the rise of the absolute water pressure, Pw. The impact of initial
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bubble pressure on how the stem hydraulic conductivity kh changes with tension is shown in
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Fig. 4. Fig. 4 provides the theoretical basis for ‘measuring’ bubble pressure through curve
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fitting as explained in the results.
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According to the analysis in Fig. 1, it seems likely that the initial should be around 30
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kPa when about 5% of the water volume is embolized (about 50% PLC assuming vessels
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make up 10% of the water volume). Fig. 3 clearly shows that kh should be nearly independent
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of central tension (Tc) until Tc fell below 0.2 MPa. Then kh sharply increased with decreased
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Tc from 0.2 to 0.01 MPa. The tension axis is plotted on a log scale to display the data more
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clearly otherwise the plots would just show rapidly rising curves near the origin of the plot
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without much separation in the curves (plot not shown). The curves of kh versus Tc rose faster
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in Fig. 4 than Fig. 3 because in a cavitron, the vessels far from the rotation axis were at lower
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tension than near the axis of rotation, so the bubbles far from the axis start collapsing before
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the bubbles in vessels near the axis of rotation.
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(2) Impact of the fraction of embolized vessels (ε)
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Based on the argument in Fig. 1 (Eq. 1) it seems unlikely that ε (fraction of embolized vessels
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to the total number of vessels) could change without changing too since ε affects the
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fraction of embolized water (α). Fig. 5 shows the prediction assuming it were possible to
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change ε independently of . It is worth noting that there should be a fairly close correlation
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between ε and the PLC observed at any given Tc.
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(3) Impact of vessel length
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Fig. 6 demonstrated that the modeled value of kh was relatively independent of vessel length
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(Lv). This is probably because we assumed that lumen resistance and pit-membrane resistance
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was equal independent of the vessel length, which agrees with the general empirical results
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(Hacke et al., 2006).
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(4) Impact of Vessel Diameter
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Vessel diameter affects the capillary pressure in cavitated vessel hence it had some impact on
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the theoretical hydraulic recovery curve. According to Eq. 3a, PC is inversely proportional to
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vessel diameter. Fig. 7 demonstrated that the maximum error caused by PC is about 0.6% per
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kPa. Bubbles collapsed more with small diameter vessels than large diameter vessels because
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of the higher values of Pc.
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Experimental Results: Curve fitting to yield bubble pressure, 316
The hydraulic recovery model (Appendix 2) was used to fit the results of 6 Acer mono and 6
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Populus 84K stems as shown in Fig. 8 and Fig. 9. The fitting involved finding the bubble
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pressure before bubble collapse, , that minimized the root mean square error (rmsE) in the
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fitted line. The average rmsE was generally less than 3% as shown in Table I. The mean ± sd
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bubble pressures in Table 1 was: 54 ± 14 kPa. All the bubble pressures estimated were lower
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than an atmospheric pressure.
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The results of 12 fittings showed that > 70% of the kh change happened when the center
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tension was lower than 0.1 MPa. In term of absolute water pressure, a center tension of 0.1
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MPa corresponds to 0 kPa absolute water pressure. The kh value was stable when tension was
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higher than 0.3 MPa. On average, the kh recovered 31 ± 8 % and 68 ± 14 % (mean ± se) of
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theoretical maximum recovery in Acer and Populus respectively when we decreased the
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tension as shown in Fig. 8 and Fig. 9. The minimum value of Tc used for measuring kh was
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0.045 MPa. The tension profile is quadratic (Fig. 2) so a Tc = 0.045 corresponded to a mean
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tension of 0.030 MPa, because the mean was always 2/3 of Tc.
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Vessel length and vessel diameter distributions measured in Acer mono and Populus
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84K are shown in Fig. 10 and Fig. 11 respectively. Acer had significantly shorter vessels at
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2.851±0.030 cm (se, N=6) than Populus at 5.797±0.144 cm (se, N=6). The mean vessel
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diameters were also significantly less in Acer than Populus, 20.07 ± 0.09 µm (se, N=3575)
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versus 28.70 ± 0.11 µm (se, N=4307), respectively.
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DISCUSSION
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Impact of the model parameters and estimation of 340
Sensitivity analysis of our hydraulic recovery model revealed little impact of vessel length on
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the determination of kh. The estimated impact of using the mean vessel lengths above rather
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than a vessel length distribution biased our results by < 0.1% (calculations not shown).
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Vessel diameter distributions were as shown in Fig. 11, and from Fig. 7 we know that
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the maximum error caused by capillary pressure is 0.6% per kPa. Therefore using the
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arithmetic mean of diameter generated a 0.09% error on the result compared to using a vessel
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diameter distribution (calculation not shown).
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As shown in Fig. 4 and Fig. 5 the primary factors controlling hydraulic recovery curves
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were and ε. The value of ε was equated to the initial PLC at high central tension before
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349
the tension was reduced (segment under water excluded). As the hydraulic recovery model
350
predicted, the lower was the more bubbles collapsed and the more kh recovered in a stem
351
at constant ε values. As a result, the relative change of hydraulic conductivity was mainly
352
determined by . However it must be remembered that and ε are not totally independent
353
of each other (Fig. 1 and Eq. 1) after the fast phase of embolism formation.
354
Only the conductivity values measured in the cavitron were used to predict in stems.
355
Data used to fit kh versus tension curves covers a sufficient range of values to make accurate
356
predictions of (Tc from 0.032 MPa to 1.0 MPa). Within the tension range, kh recovered 31%
357
and 68% of the theoretical maximum recovery in Acer and Populus respectively. The 358
calculated for Populus was more accurate than for Acer; but the likely error was not enough
359
to alter the basic conclusion: When cavitation was produced in a cavitron over a period of 1 h,
360
relatively stable resulted that were significantly less than atmospheric pressure (100 kPa
361
for the elevation of the university lab) based on a t-test (p < 10-4 for both species).
362
The purpose of this paper was to present the hydraulic recovery model and show by
363
curve-fitting of data (kh versus tension) that values of initial bubble pressure can be derived
364
from the fitted curves. The values of bubble pressure were higher than anticipated by us (54
365
kPa Table 1 versus about 35 kPa Fig. 1), but the reason for this will be explained in the next
366
paper in this series where we measure the tempo of increase in .
367
368
The impact of bubble pressure on kh and vulnerability curves
369
The model predicted that hydraulic conductivity (kh) will increase whenever the tension at the
370
axis of rotation (Tc) is decreased below about 0.3MPa. The model results also predict that the
371
lower is the more kh depends on tension. The value of kh was most sensitive to decreases
372
in Tc between 0.2 to 0.01 MPa. The results of Acer mono and Populus 84K revealed that the
373
model was validated by our data. Fitting the model to experimental data allowed the
374
computation of the bubble pressure ( ) in embolized vessels. The hydraulic recovery model
375
is an ‘equilibrium model’ that indicates the final equilibrium, but it can be imagined that
376
water has to flow into the stem to refill the vessels as the bubbles collapse and hence we must
377
address the issue of equilibration time to get an accurate value of kh after a change in tension.
378
Bubble collapse in stems involves the absorption of water into stems, and the water
379
absorption will influence the kh value measured in centrifuge resulting in an over-estimation
380
of hydraulic conductivity. The over-estimation is a result of the assumption of steady-state,
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381
i.e., that all water that enters the high pressure end of the stem segment flows through to the
382
low pressure end. But if some of the water stays behind to fill part of the vessels (or other
383
rehydrating regions) then the flow is overestimated at any given pressure drop resulting in a
384
kh that is too high as we measured the influx flow in a cavitron. Therefore accurate
385
measurement of kh can be done only after this absorption process is completed. Preliminary
386
measurements suggested to us that a 30 min wait was long enough to achieve stable kh values
387
at the smallest Tc = 0.03 MPa but became much faster as Tc increased. The water absorption
388
also happens when the Sperry rotor is used to induce cavitation, and the standard protocol for
389
kh measurements in a conductivity apparatus is to correct measurements of kh for this
390
background absorption rate (Hacke et al., 2000; Torres-Ruiz et al., 2012). In the following
391
paper of this series, a model of water absorption in stem is presented giving a better view of
392
kinetics of water absorption in stems due to bubble collapse.
393
Cochard (2002) and Li et al. (2008) both observed no difference in kh measured in a
394
stem at the target TC vs the kh measured later at a smaller TC while stems were spinning in a
395
centrifuge using the original Cochard rotor or functionally similar rotor fabricated by
396
modifying a Sperry rotor (Li et al., 2008). This is also the personal experience of MTT when
397
testing his Cochard cavitron in 2006 (unpublished results). These observations contrast with
398
the results of this paper and deserve more analysis. In Li et al. (2008) no data were given so
399
the range of TC over which kh was measured was not given; in the earlier study (Fig. 4 in
400
Cochard 2002) it is clear that relative changes in kh were measured from TC = 1.8 to 0.1 MPa;
401
this range should have been sufficient to detect the beginning of a trend (compare Fig. 8 and
402
Fig. 9), but the SE/mean of Cochard’s early measurements were quite high (0.05 to 0.15) so
403
no trend was detectable. The experiment has to be continued down to TC = 0.03 MPa to
404
discern the trend and the trend has to be discerned in individual stem segments, because the
405
variance of the mean kh is so large that it obscures the relationship. Proving the trends
406
reported in this paper also requires that kh be measured with high precision on multiple
407
measurements of kh at a given TC (SE/mean < 0.02 and N> 10 is desirable). During the course
408
of our investigation techniques were improved; hence some of the later experiments in this
409
manuscript had higher precision values with an error closer to 0.004 in SE/mean kh (Wang et
410
al., 2014b).
411
In our experiments the minimum tension used for the measurement of kh ranged from
412
0.032 MPa to 0.057 MPa, but during gravity flow measurements the applied water pressure is
413
2 to 3 kPa above barometric pressure (= a tension of -0.002 to -0.003). Using our model we
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414
extended the theoretical calculations to include how positive pressure (negative tension)
415
would affect kh measured by gravity flow. The result of these model calculations are reported
416
in Table I (Sperry PLC). We compare the Sperry PLC to the average cavitron value (Cavitron
417
PLC) and the theoretical value near the axis of rotation (Central PLC). The deviation of the
418
theoretical ‘Sperry PLC’ values compared to the ‘Cavitron PLC’ values in Table I are quite
419
large and probably the worst case possible because Sperry method arrives at the PLC values
420
shown in Table I through various steps. In each step the sample is spun to a higher tension
421
then removed for measurement in a conductivity apparatus. After each kh measurement in a
422
conductivity apparatus more air saturated water is perfused through the stem restoring the
423
water-filled spaces near the vessels to a concentration near to a saturated value (Henry’s law).
424
So when the stem is spun again the quick equilibrium happens again and the bubble pressure
425
of each embolized vessel would increase. In contrast the model predicts what would happen
426
if a high PLC were induced in a stem segment in one step then moved immediately to a
427
conductivity apparatus. Or model predicts that the Cochard VC will be different from Sperry
428
VC, but the stepwise measurements in gravity method will make the bubble pressure rise to
429
atmospheric pressure faster. Besides, the bubble collapse affected VC by decreased PLC (y-
430
axis), but Tension (x-axis) is less affected because of the steep slope at P50. Therefore no
431
significant difference had been observed between two methods (Li et al., 2008).
432
In theory, the P50 (positive value) of a Cavitron VC is lower than that of a Sperry VC
433
(each points measured at equilibrium) because of the collapse of bubbles starting at a
434
pressure less than 101.3 kPa (standard atmospheric pressure at sea level). Even if the pressure
435
is 101.3 kPa there would be some collapse due to the impact of surface tension on PC and the
436
2 or 3% extra bubble compression resulting from gravity flow measurements at 2 or 3 kPa
437
above atmospheric pressure. But gravity flow and cavitron flow measurements are uncertain
438
because of uncertainty of temperature and stem rehydration. The measurement of kh changes
439
2.3% °C-1 because of the influence of the temperature dependence of viscosity. Also stems
440
dehydrated in a centrifuge and then returned to a lower tension or atmospheric pressure
441
tended to rehydrate for the first 30 to 60 minutes causing an overestimation / underestimation
442
of flow and kh unless care is taken to correct for rehydration affects. Readers should consult
443
Torres-Ruiz et al. (2012) and Wang et al. (2014b) for methods of dealing with these problems
444
in the gravity flow method and cavitron method of measuring kh, respectively.
445
446
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447
CONCLUSION
448
In conclusion, the model predicts that the bubble pressure in recently cavitated vessels affects
449
how much kh changes with TC in a cavitron. Vessel length and vessel diameter have little
450
impact on the shape of hydraulic recovery curves (Figs. 5 and 6). Hydraulic conductivity
451
increased due to the collapse of air bubbles in stem when the central tension decreased, and
452
the bubble pressure could be estimated by our model with an error less than 3%. It is also
453
worth noting that bubble collapse will not affect very much the value of P50 measured by
454
Sperry’s technique. This follows because bubble collapse influences most the y-value of a
455
vulnerability curve (PLC) but near P50 the curve is quite steep so a large change in PLC
456
results in a small change of the x-axis value (tension or negative pressure).
457
These models provides a new insight into the tempo of bubble pressure and bubble
458
formation in recently cavitated stems, i.e. that there should be a rapid rise in within a
459
minute of cavitation followed by a much slower rise >1 day. The next paper of this series will
460
more fully address the time needed for bubble pressure to rise in the slow phase of the
461
equilibration with air external to the stem.
462
463
464
MATERIALS AND METHODS
465
Branches (> 1 m long) of a hybrid Populus 84K (Populus alba × Populus glandulosa) and
466
Acer mono were sampled from and near Northwest A&F University campus, and segments
467
were excised while immersed in water. Segments with basal diameter of 5.5 to 7.5 mm were
468
trimmed to 27.4 cm in length under water. The segments were flushed with 10 mM KCl for
469
30 min under 200 kPa pressure (300 kPa absolute pressure). The liquid we used to flush the
470
stem was filtered by 0.02 µm filter and degased under 40 kPa (absolute pressure) with a
471
vacuum pump. Approximately 50% PLC was rapidly induced in a Cochard cavitron by
472
spinning to a tension of 2.2 MPa for Populus 84K and 4.2 MPa for Acer mono. Then the
473
central tension was decreased to yield an absolute pressure higher than a perfect vacuum for
474
at least 30 minutes (typically at 0.057 MPa in centrifuge tension at the axis of rotation) to
475
make sure that equilibrium was attained, i.e., that the period of water absorption had ceased.
476
The kh of the segments were measured from the low to high tension values shown in Table II.
477
Each tension was held for 3-10 minutes until the stems were ‘equilibrated’ (stable kh
478
measured). An improved regression method was used to calculate kh to obtain a 4-5 times
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Copyright © 2015 American Society of Plant Biologists. All rights reserved.
479
higher precision (Wang et al., 2014b). Preliminary experiments were done to determine how
480
long tension had to be held constant to get stable kh values. These results indicated that 30
481
minutes was enough at 0.045 MPa and 3-10 minutes was enough for tensions > 0.1 MPa.
482
Python(x,y) 2.7.5 was used to program the model and the data acquired of kh versus TC
483
was used to fit the model to estimate the average pressure of air bubbles in stem by least
484
square package in Python. The code for the model can be found in Supplemental Appendix 3.
485
Vessel parameters
486
Vessel lengths were measured by the “air injection method” described by Cohen et al.
487
(2003) and Wang et al. (2014a). Briefly, long shoots were cut with a sharp razor blade and
488
injected with compressed air at ∆ = 100 kPa (200 kPa absolute pressure) from the distal end;
489
air was collected from the basal end immersed in water. Stems segments (20-cm-long) were
490
sequentially excised until bubbles were observed emerging from the distal end, and the flow
491
rate of air bubbles (Q= ΔV/ΔT) was measured by water displacement. The stem was
492
progressively shortened and Q measured at each length as above. The air conductance of cut-
493
open vessels, C, was calculated from 494
into C=C0 e-kx, where C0 is the maximum conductance as x approaches 0 and k is an
495
extinction coefficient, and x is the length of stem. The k-values were computed from the slope
496
the curve of ln(C) versus x; and according to the theory of (Cohen et al. (2003)), the most
497
common vessel length, Lmode, equals to 1/k and mean vessel length is calculated by Lmean=
498
2Lmode. Six stems were measured to calculate to average vessel length of Acer mono and
499
Populus 84K, respectively.
∆
as described in Cohen et al. (2003) and fitted
500
Vessel diameters were measured on 25-μm-thick sections that were stained and
501
photographed using a microscope (Leica DM4000B, Nussloch, Germany) to take pictures.
502
Image analysis software (Win CELL 2012, Regent Instruments Canada Inc., Quebec City,
503
Canada) was used to measure the diameter of hundreds of vessels and arithmetic means were
504
used to calculate capillary pressure by Eq. 3a. Diameters were calculated from vessel area
505
assuming circular geometry. The sum of vessel areas divided by wood area containing the
506
vessels yielded values relative vessel fraction in both species.
507
508
509
ACKNOWLEDGEMENT
510
We acknowledge the precious help from Feng Feng in collecting samples and maintaining the
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511
flushing apparatus.
512
513
514
LITERATURE CITED
515
516
Brodersen CR, McElrone AJ, Choat B, Lee EF, Shackel KA, Matthews MA (2013) In
517
vivo visualizations of drought-induced embolism spread in Vitis vinifera. Plant
518
Physiol 161: 1820-1829
519
520
521
522
Cai J, Hacke U, Zhang S, Tyree MT (2010) What happens when stems are embolized in a
centrifuge? Testing the cavitron theory. Physiol Plant 140: 311-320
Cochard H (2002) A technique for measuring xylem hydraulic conductance under high
negative pressures. Plant Cell Environ 25: 815-819
523
Cochard H, Badel E, Herbette S, Delzon S, Choat B, Jansen S (2013) Methods for
524
measuring plant vulnerability to cavitation: a critical review. J Exp Bot 64: 4779-
525
4791
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527
528
529
530
531
532
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Cohen S, Bennink J, Tyree M (2003) Air method measurements of apple vessel length
distributions with improved apparatus and theory. J Exp Bot 54: 1889-1897
Evert RF (2006) Esau's Plant Anatomy: Meristems, Cells, and Tissues of the Plant Body:
Their Structure, Function, and Development, Third Edition. John Wiley \& Sons
Hacke UG, Sperry JS, Pittermann J (2000) Drought experience and cavitation resistance in
six shrubs from the Great Basin, Utah. Basic and Applied Ecology 1: 31-41
Hacke UG, Sperry JS, Wheeler JK, Castro L (2006) Scaling of angiosperm xylem
structure with safety and efficiency. Tree Physiol 26: 689-701
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Hacke UG, Venturas MD, MacKinnon ED, Jacobsen AL, Sperry JS, Pratt RB (2015)
535
The standard centrifuge method accurately measures vulnerability curves of long-
536
vesselled olive stems. New Phytol 205: 116-127
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Li Y, Sperry JS, Taneda H, Bush SE, Hacke UG (2008) Evaluation of centrifugal methods
538
for measuring xylem cavitation in conifers, diffuse- and ring-porous angiosperms.
539
New Phytol 177: 558-568
540
Sperry JS, Saliendra NZ, Pockman WT, Cochard H, Cruiziat P, Davis SD, Ewers FW,
541
Tyree MT (1996) New evidence for large negative xylem pressures and their
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measurement by the pressure chamber method. Plant Cell Environ 19: 427-436
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Sperry JS, Tyree MT (1988) Mechanism of water stress-induced xylem embolism. Plant
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544
Physiol 88: 581-587
545
Torres-Ruiz JM, Sperry JS, Fernandez JE (2012) Improving xylem hydraulic conductivity
546
measurements by correcting the error caused by passive water uptake. Physiol Plant
547
146: 129-135
548
549
Tyree MT, Yang S (1992) Hydraulic Conductivity Recovery versus Water Pressure in Xylem
of Acer saccharum. Plant Physiol 100: 669-676
550
Wang R, Zhang L, Zhang S, Cai J, Tyree MT (2014a) Water relations of Robinia
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pseudoacacia L.: do vessels cavitate and refill diurnally or are R-shaped curves
552
invalid in Robinia? Plant Cell Environ 37: 2667-2678
553
Wang Y, Burlett R, Feng F, Tyree M (2014b) Improved precision of hydraulic conductance
554
measurements using a Cochard rotor in two different centrifuges. Journal of Plant
555
Hydraulics 1: e-0007
556
Wheeler JK, Sperry JS, Hacke UG, Hoang N (2005) Inter-vessel pitting and cavitation in
557
woody Rosaceae and other vesselled plants: a basis for a safety versus efficiency
558
trade-off in xylem transport. Plant Cell Environ 28: 800-812
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Yang S, Tyree MT (1992) A Theoretical-Model of Hydraulic Conductivity Recovery from
560
Embolism with Comparison to Experimental-Data on Acer-Saccharum. Plant Cell
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Environ 15: 633-643
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Zwieniecki MA, Holbrook NM (2000) Bordered pit structure and vessel wall surface
properties. Implications for embolism repair. Plant Physiol 123: 1015-1020
564
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565
Captions
566
Figure 1. Relationship between the fraction of embolized water volume and partial pressure
567
in the embolized vessels. Dots line represents the partial pressure of nitrogen, dash line
568
represents that of oxygen and solid line represent the sum of partial pressure of oxygen,
569
nitrogen and saturated water vapor pressure (3.2 kPa at 298K). This figure ignores the 1%
570
contribution of argon to the gas mix in air.
571
572
Figure 2. Pressure distribution in a cavitron and the pattern of bubble collapse due to
573
pressure distribution. Panel A shows the absolute pressure distribution of three different
574
central tensions: 0.05, 0.10 and 0.20 MPa. Average tensions are always 2/3 of the central
575
tension. Panel B shows the bubble collapse in vessels located in different regions in stems.
576
577
Figure 3. This is the solution of Eq. 6: single vessel model showing how the conductivity of
578
a single vessel (kh) changes with tension. The y-axis is the ratio of an embolized vessel to the
579
kh of a water filled vessel. In this solution Pc = 7 kPa and was assigned values of 10 to
580
101.3 kPa as shown in the legend.
581
582
Figure 4. Prediction of hydraulic recovery curve is shown versus tension. Each curve is
583
based on a different initial values (10, 30, 50, 70 and 90 kPa) in the embolized vessels in
584
the theoretical stem; maximum hydraulic conductivity kmax was set to 1E-4 Kg m MPa-1 s-1
585
for each curve; average vessel length was 5 cm; capillary pressure was 12 kPa; the fraction of
586
embolized vessels (ε) was 50% in the embolized segment (regions b, c and d in Fig. S4,
587
namely the portion of the segment that is not immersed under water).
588
589
Figure 5. Prediction of hydraulic recovery curve with different values of ε. The relationship
590
between PLC and fraction of vessels cavitated is given by PLC/100 = ε. Each curve is based
591
on different ε (40%, 50% and 60%); maximum hydraulic conductivity kmax is 1E-4 Kg m
592
MPa-1 s-1; average vessel length is 5 cm; capillary pressure is 12 kPa; and bubble pressure, ,
593
is 50 kPa.
594
595
Figure 6. Prediction of hydraulic recovery curve with different average vessel lengths. Each
596
curve is based on different average vessel length of stem (Lv = 1, 3, 5, 7 and 9 cm as the
597
legend shows); maximum hydraulic conductivity kmax is 1E-4 Kg m MPa-1 s-1; average bubble
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Copyright © 2015 American Society of Plant Biologists. All rights reserved.
598
pressure ( is 50 kPa; capillary pressure is 12 kPa; and ε in the embolized segment is 50%.
599
600
Figure 7. Prediction of hydraulic recovery curve with different capillary pressures. PC is the
601
average capillary pressure (4, 8, 12 and 16 kPa for each curve); maximum hydraulic
602
conductivity kmax is 1E-4 Kg m MPa-1 s-1; average bubble pressure ( ) is 50 kPa; average
603
vessel length is 5 cm; and ε in the embolized segment is 50%. The vessel diameter that
604
corresponds to each PC depends on cosθ (θ=π/4). So the vessel diameters are 50.89, 25.44,
605
16.96, and 12.72 μm for PC values of 4, 8, 12, and 16 kPa, respectively.
606
607
Figure 8. Example of hydraulic recovery curve fittings of Acer mono. The solid circles are
608
the hydraulic conductivity measured in the cavitron and the solid lines are the best-fit curves.
609
The details of the curves are as shown in Table I.
610
611
Figure 9. Example of hydraulic recovery curve fittings of Populus 84K. The solid circles are
612
the hydraulic conductivity measured in the cavitron and the solid lines are the best-fit curves.
613
The details of the curves are as shown in Table I.
614
615
Figure 10. Vessel length distributions in Acer mono and Populus 84K. Vessel length
616
frequency was computed by using Px =k2xe-kx, where k was the extinction coefficient of the
617
stem and x is the vessel length. The values of k of Acer mono and Populus 84K were 1.0246
618
and 0.3113, respectively.
619
620
Figure 11. Vessel diameter distribution in Acer mono and Populus 84K. The average
621
diameters of Acer mono and Populus 84K were 20.07 and 28.70 μm (arithmetic mean),
622
respectively.
623
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Table I Results of hydraulic recovery curve fitting of 6 Acer mono stems and 6 Populus 84K
stems. kmax is the maximum hydraulic conductivity of stem; Cavitron PLC is the experimental
PLC measured in centrifuge; Sperry PLC is the PLC under atmospheric pressure predicted by
the model; Central PLC is the PLC at the central region of stem predicted by the model; ܲ௕‫ כ‬is
the average bubble pressure in embolized vessels; ERMS is the route mean square error of the
curve fitting; Tform is the time used to induce embolism in stem; and Trecovery is the total time
used to measure the curve.
Acer momo
stem 1 stem 2 stem 3 stem 4 stem 5 stem 6
-1 -1
kmax (E-5 Kg m MPa s )
1.57
5.02
4.28
4.69
3.28
5.96
Cavitron PLC (%)
39.55
57.00
65.88
66.08
72.60
82.70
Sperry PLC (%)
0.76
4.47
13.57
7.80
6.45
9.04
Central PLC (%)
42.66
61.49
71.07
71.28
78.32
89.21
ܲ௕‫כ‬
44.59
53.81
67.11
57.82
52.71
54.45
(kPa)
ERMS
1.77% 2.78% 1.60% 2.20% 4.58% 4.03%
Tform (h)
0.41
0.41
0.22
0.36
0.82
0.35
Trecovery (h)
0.75
0.53
0.82
0.48
0.65
0.82
Populus 84K
stem 1 stem 2 stem 3 stem 4 stem 5 stem 6
-1 -1
kmax (E-5 Kg m MPa s )
5.25
8.07
7.07
9.13
8.95
5.68
Cavitron PLC (%)
51.59
43.62
51.31
58.64
39.49
39.32
Sperry PLC (%)
16.12
0.92
21.95
9.16
0.65
2.54
Central PLC (%)
55.65
47.05
55.35
63.26
42.60
42.41
ܲ௕‫כ‬
57.76
72.45
34.45
37.25
77.44
38.48
(kPa)
ERMS
1.72% 1.82% 2.06% 2.18% 2.48% 2.59%
Tform (h)
0.56
0.20
0.47
0.53
0.26
0.49
Trecovery (h)
0.77
0.33
0.48
0.82
0.27
0.29
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Copyright © 2015 American Society of Plant Biologists. All rights reserved.
Table II Table of RPM (revolutions per minute) and TC (tension at the axis of rotation) in the
Cavitron we used. The maximum RPM used for Populus was 5000 and for Acer was 7000.
RPM TC (MPa) RPM TC (MPa) RPM TC (MPa)
500
0.022
1400
0.173
4000
1.415
600
0.032
1600
0.226
4500
1.791
700
0.043
1800
0.287
5000
2.211
800
0.057
2000
0.354
5500
2.675
900
0.072
2500
0.553
6000
3.184
1000
0.088
3000
0.796
6500
3.736
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120
O2
N2
100
All
Pressure (kPa)
80
60
40
20
0
0.00
0.05
0.10
0.15
α = fraction of embolized water volume
0.20
Figure 1. Relationship between the fraction of embolized water volume and partial pressure in the
embolized vessels. Dots line represents the partial pressure of nitrogen, dash line represents that of
oxygen and solid line represent the sum of partial pressure of oxygen, nitrogen and saturated water
vapor pressure (3.2 kPa at 298K). This figure ignores the 1% contribution of argon to the gas mix in
air.
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Copyright © 2015 American Society of Plant Biologists. All rights reserved.
1 5 0
A
0 .0 5 M P a
0 .1 0 M P a
0 .2 0 M P a
P re s s u re (k P a )
1 0 0
5 0
0
-5 0
-1 0 0
-0 .1 5
-0 .1 0
-0 .0 5
0 .0 0
0 .0 5
0 .1 0
0 .1 5
D is ta n c e to th e a x is ( m )
B
Figure 2. Pressure distribution in cavitron and the pattern of bubble collapse due to pressure distribution. Panel A shows the absolute pressure distribution of three different tensions: 0.05, 0.10 and 0.20
MPa. Panel B shows the bubble collapse in vessels that locates in different sites in stems.
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Copyright © 2015 American Society of Plant Biologists. All rights reserved.
10.0 kPa
20.0 kPa
30.0 kPa
40.0 kPa
50.0 kPa
60.0 kPa
70.0 kPa
80.0 kPa
90.0 kPa
101.3 kPa
1.0
Single Vessel kh
0.8
0.6
0.4
0.2
0.0
10-3
10-2
10-1
Tension (MPa)
100
Figure 3. This is the solution of Eq. 6: single vessel model showing how the conductivity of a single
vessel (kh ) changes with tension. The y-axis is the ratio of an embolized vessel to the kh of a water
filled vessel. In this solution Pc = 7 kPa and Pb∗ was assigned values of 10 to 101.3 kPa as shown in
the legend.
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1.0 1e 4
Pb
Pb
Pb
Pb
Pb
kh (Kg m MPa−1 s−1 )
0.9
= 10 kPa
= 30 kPa
= 50 kPa
= 70 kPa
= 90 kPa
0.8
0.7
0.6
0.5 -2
10
10-1
100
Tension (MPa)
101
Figure 4. Prediction of hydraulic recovery curve is shown versus tension. Each curve is based on
a different initial Pb∗ values (10, 30, 50, 70 and 90 kPa) in the embolized vessels in the theoretical
stem; maximum hydraulic conductivity kmax was set to 1E-4 Kg m MPa−1 s−1 for each curve; average
vessel length was 5 cm; capillary pressure was 12 kPa; the fraction of embolized vessels () was 50%
in the embolized segment (regions b, c and d in Fig. S4, namely the portion of the segment that is not
immersed under water).
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Figure 5. Prediction of hydraulic recovery curve with different values of �. The relationship between
PLC and fraction of vessels cavitated is given by PLC/100 = �. Each curve is based on different �
(40%, 50% and 60%); maximum hydraulic conductivity kmax is 1E-4 Kg m MPa−1 s−1 ; average vessel
length is 5 cm; capillary pressure is 12 kPa; and bubble pressure, Pb∗ , is 50 kPa.
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Figure 6. Prediction of hydraulic recovery curve with different average vessel lengths. Each curve
is based on different average vessel length of stem (Lv = 1, 3, 5, 7 and 9 cm as the legend shows);
maximum hydraulic conductivity kmax is 1E-4 Kg m MPa−1 s−1 ; average bubble pressure (Pb∗ ) is 50
kPa; capillary pressure is 12 kPa; and � in the embolized segment is 50%.
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Figure 7. Prediction of hydraulic recovery curve with different capillary pressures. PC is the average
capillary pressure (4, 8, 12 and 16 kPa for each curve); maximum hydraulic conductivity kmax is 1E-4
Kg m MPa−1 s−1 ; average bubble pressure (Pb∗ ) is 50 kPa; average vessel length is 5 cm; and � in
the embolized segment is 50%. The vessel diameter that corresponds to each PC depends on cosθ
(θ = π/4). So the vessel diameters are 50.89, 25.44, 16.96, and 12.72 µm for PC values of 4, 8, 12,
and 16 kPa, respectively.
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Figure 8. Example of hydraulic recovery curve fittings of Acer mono. The solid circles are the hydraulic conductivity measured in the cavitron and the solid lines are the best-fit curves. The details of
the curves are as shown in Table I.
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Figure 9. Example of hydraulic recovery curve fittings of Populus 84K. The solid circles are the
hydraulic conductivity measured in the cavitron and the solid lines are the best-fit curves. The details
of the curves are as shown in Table I.
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Figure 10. Vessel length distributions in Acer mono and Populus 84K. Vessel length frequency was
computed by using Px = k 2 xe−kx , where k was the extinction coefficient of the stem and x is the
vessel length. The values of k of Acer mono and Populus 84K were 1.0246 and 0.3113, respectively.
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Figure 11. Vessel diameter distribution in Acer mono and Populus 84K. The average diameters of Acer
mono and Populus 84K were 20.07 and 28.70 µm (arithmetic mean), respectively.
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