Journal of Theoretical Biology 2

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angiosperm conduit with a homogeneous pit membrane and a typical gymnosperm conduit with a torusmargo pit membrane structure.

which creates a pull of a continuous water column through

volumetric expansion energy to overcome the surfaceenergy needed to make the new gas/liquid interface, andthe bubble is able to grow (Brennen, 1995).

However, a gas bubble expanding in a xylem conduit has to

dynamics of the bubble growth inside a xylem conduit hasnot received much attention. The dynamics of bubblegrowth inside a xylem conduit has not been modeled

ARTICLE IN PRESSbefore, and the only observation of embolism dynamicsfrom the literature, as far as we know, is the study ofLewis et al. (1994), where full embolism in the tracheids of

0022-5193/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jtbi.2007.05.033

Corresponding author. Tel.: +44 131 650 5427; fax: +44 131 662 0478.E-mail address: teemu.holtta@ed.ac.uk (T. Holtta).the xylem (Zimmermann, 1983). Water is regularly in ameta-stable state, where its pressure has dropped belowsaturation vapor pressure (Nobel, 1991). Under theseconditions the water columns are vulnerable to cavitationby formation of gas bubbles from air seeding, i.e. airpenetration from adjacent conduits or air spaces throughlittle pores, or by actual phase transition through hetero-geneous nucleation (e.g. Pickard, 1981; Tyree, 1997). Theseprocesses induce a gas phase large enough for the

displace water volume in the conduit as it grows, and therelatively inelastic lignied walls of the conduit resist thevolumetric expansion of the conduit. At the same time, theoutow of liquid water out of the conduit is restricted bythe hydraulic resistance of the conduit lumen itself and thebordered pits through which water is exchanged withneighboring xylem conduits.The stability of gas bubbles in the xylem in connection to

cavitation has been modeled (e.g. Shen et al., 2002), but themembrane to the low pressure side of the pit chamber, was found to be possible while the emboli was still small. Concurrent with pit

aspiration, the high resistance to water ow out of the conduit through the cell walls or aspirated pits will make the embolism process

slow. In case of no pit aspiration and always for conduits with homogeneous pit membranes, embolism growth is more rapid but still

much slower than bubble growth in bulk water under similar water tension. The time needed for the embolism to ll a whole conduit was

found to be dependent on pit and cell wall conductance, conduit radius, xylem water tension, pressure rise in adjacent conduits due to

water freed from the embolising conduit, and the rigidity and structure of the pits in the case of margotorus type pit membrane. The

water pressure in the conduit hosting the bubble was found to occur almost immediately after bubble induction inside a conduit, creating

a sudden tension release in the conduit, which can be detected by acoustic and ultra-acoustic monitoring of xylem cavitation.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Embolism formation; Bordered pits; Bubble growth; Pit aspiration

1. Introduction

According to the cohesion-tension theory water ow inplants is driven by water evaporation at the leaf surfaces,

In a liquid under tension, unrestricted by solid bound-aries, a bubble above a critical size would growexplosively, in a fraction of a second, to ll the volumeof an individual xylem water conduit (Brennen, 1995).For conduits with torusmargo type pits pit membrane deection was also modeled and pit aspiration, the displacement of the pitA model of bubble growth lead

T. Holttaa,, T. VeaSchool of GeoSciences, Crew Building, University of

bDepartment of Physical Sciences, University ofcDepartment of Forest Ecology, University of H

Received 29 March 2007; received in revis

Available on

Abstract

The dynamics of a gas bubble inside a water conduit after a cav49 (2007) 111123

ng to xylem conduit embolism

lab, E. Nikinmaac

nburgh, West Mains Road, EH9 3JN Edinburgh, UK

inki, P.O. Box 64, FIN-00014 Helsinki, Finland

nki, P.O. Box 24, FIN-00014 Helsinki, Finland

orm 18 May 2007; accepted 24 May 2007

2 June 2007

ion event was modeled. A distinction was made between a typical

www.elsevier.com/locate/yjtbi

ARTICLE IN PRESSretiThuja occidentalis L. was found to occur in approximately5min after embolism induction. In this study we model thegrowth of the gas bubble inside a xylem water conduitfollowing cavitation. A distinction is made between aconduit with a homogeneous pit membrane typically foundin angiosperms and a conduit with a torusmargo pitfound in gymnosperms. For the later, pit aspiration and itseffect on embolism formation is also taken into account.Bordered pits are small circular regions in the conduit

wall in which the secondary wall is missing (Siau, 1984).Bordered pits consist of a pit chamber and a pit membranethrough which water ow among adjacent water conduitstakes place (Taiz and Zeiger, 1998). The pit membranestructure of gymnosperms and angiosperms is different.Angiosperms have usually homogeneous pit membranes,which must allow water passage through them and at thesame time prevent the passage of air by trapping airwatermeniscus by capillary action (Sperry and Hacke, 2004).Gymnosperm pit membrane has usually a more compli-cated structure (Hacke et al., 2004), in which the closingmembrane is made up of a thick central region, torus, and athin peripheral region, margo. The torus is relativelyimpermeable to water ow, while the margo is perforatedand much more permeable to water ow (Siau, 1984).The function of the pit membrane is to block the passage

of air from embolised conduits to water lled ones, thuspreventing the spreading of embolisms. The airwaterinterface between an embolised and a functioning conduitresides to the pores of the pit membrane (Zimmermann,1983). The pressure difference in the water and air phasesover the membrane exerts a force to stretch the microbrilstrands that are holding the membrane in place, and themembrane is deected to the low pressure side of the pitchamber (e.g. Petty, 1972, Hacke et al., 2004, Sperry andHacke, 2004). However, if the pressure difference over thepit membrane grows larger than the surface tension forcesneeded to maintain the liquidgas interface intact, air isseeded from the embolised conduit to the adjacent conduitand the embolism spreads. The size of largest individualpore of in the pit membranes separating a water lledconduit from an embolised is thought to determine themaximum pressure difference that can exist over themembrane without the induction of air-seeding (Tyree,1997).The pit membrane should also be displaced when

another type of force, other than that caused by surfacetension over the airwater interface in the membrane, isacting on it. Induction of a large pressure difference inliquid water between two adjacent conduits should alsocause pit membrane deection by creating a hydrostaticforce on the pit membrane. However, in normal transpira-tion driven water ow situation the pressure difference overthe membrane is far too small to displace the pit membraneconsiderably and cause pit aspiration (Gregory and Petty,1973). Bolton and Petty (1978) and Chapman et al. (1977)

T. Holtta et al. / Journal of Theo112have modeled the deection of torusmargo type gymnos-perm pit membrane in the presence of a much larger waterux between two conduits than that resulting fromtranspiration driven water ow. Both studies came to theconclusion that torusmargo type pits act like valves,permitting only moderate water uxes through them aswith large water uxes the torus blocks the pit opening andno water will ow through the pit. Sperry and Tyree (1990)found experimentally that the hydraulic conductivity ofgymnosperm wood samples decreased as the pressuregradient used to drive water through the samples increasedmuch above transpiration-induced values. Pit aspirationwas hypothesized as the reason for this drop in hydraulicconductivity. Also Hammel (1967) and Robson et al.(1988) proposed that a large liquid pressure differencebetween two conduits would cause pit aspiration of atorusmargo pit, but in the case of a partly frozen and anunfrozen xylem tracheid. As we show later in this study,large water pressure differences and temporarily largewater uxes will be developed between adjacent conduits inconnection to embolism formation. For gymnospermswith torusmargo type pits this could induce the closedvalve type behavior as described above, whereas forangiosperms with homogeneous pit membranes, pitmembrane deection would have only little effect on thefunctioning of the pit.The dynamics of a gas bubble inside a xylem water

conduit are calculated here using numerical methods tosolve the equations associated with the gas phase growth,water ow out of a conduit hosting the growing bubble,water pressure development, and also the transient pitconductance for the torusmargo structure pits. The timerequired for a gas bubble to completely ll an embolisingconduit is calculated for varying conduit structures andxylem tensions. The time-scale needed to drain a conduit ofwater is also interesting in view of the possible embolismand embolism relling cycles. Experimental studieshave shown that embolisms are frequently relled, andsome studies have even observed relling during relativelyhigh transpiration (Melcher et al., 2001; Tyree et al.,1999; Canny, 1997). If a conduit would not be fully drainedof water before relling commences, it could also beeasier and faster to rell. The dynamics of bubble growth isalso interesting from the perspective of the detection ofacoustic and ultra-acoustic sonic emissions, which areobserved concurrently with cavitation events (e.g. Milburnand Johnson, 1966; Tyree and Dixon, 1983). It is notcompletely clear what exactly gives rise to these emissions(Jackson and Grace, 1996).

2. Theory

Here we describe how the dynamics of the bubble/gasphase in an embolising conduit is described in its differentphases in the calculations. Fig. 1 depicts an outline of thedifferent phases of a bubble/gas phase growing in a conduitfor a tree with torusmargo type pit membranes. Fig. 1A

cal Biology 249 (2007) 111123shows the initial stage of the process where a critical sizebubble has been induced inside the conduit lumen and the

ARTICLE IN PRESSretiT. Holtta et al. / Journal of Theopits still remain un-aspirated. In Fig. 1B the bubble radiushas grown and the bubble is still spherical. The pitmembranes have been displaced to the sides of the adjacentconduits under tension (i.e. the pits are aspirated) dueto the water pressure difference between the conduits.

Fig. 1. (A) Expanding spherical gas bubble in water conduit with

torusmargo type pits. The pits are un-aspirated in the beginning of the

growth process, as the water pressure in the conduit hosting the bubble has

not yet risen high enough. Water ows out of all the pits connecting the

embolising conduit to adjacent conduits. (B) Expanding spherical gas

bubble in water conduit. The pits have become aspirated as the pressure

difference between the conduit hosting the bubble and adjacent conduits

has risen above a threshold value. (C) Spreading of the gas phase in a

water conduit treated as a capillary. The pits remain aspirated, as the

pressure difference between the conduit hosting the bubble and adjacent

conduits has risen. Water ows out only from the conduit wall area that is

covered with water.In Fig. 1C the gas phase has reached the radius of theconduit and is no longer spherical. The gas phase spreadstowards the tapered ends of the conduit. The pits remainaspirated. However, pit aspiration does not occur at all ifthe pressure difference over the pit membrane does notgrow high enough to overcome the elastic forces in themargo strands, or alternatively pit aspiration can alsooccur after the gas phase has reached the conduit diametersize. For angiosperms conduits with homogeneous pitmembranes, pit membrane deection to the aspiratedposition could also occur but there would not be animpermeable torus to seal the pit opening.

2.1. Induction of a critical size gas bubble

The beginning of an embolisation process, where a waterconduit in the xylem is eventually lled with air, is theinduction of a gas bubble past the critical size by actualphase transition through heterogeneous nucleation, butmore likely by air-seeding from an adjacent lumen or acrack in the conduit wall (e.g. Tyree, 1997). The reader canturn to e.g. Tyree (1997) and Steudle (2001) for moredetails about these processes. For the modeling presentedhere, the actual mechanism responsible for the past criticalsize bubble formation does not effect the results. Thegeneral term cavitation will be used in this study to refer toall of the above processes as this is customary, althoughonly heterogeneous nucleation can be considered cavitationin the strictest physical sense (Holtta et al., 2002). Theradius of the critical size bubble is determined by thepressure difference between the liquid and gas phase, andsurface tension of water according to the Laplace equation(Brennen, 1995):

RC 2g

Pg Pl, (1)

where RC is the radius of the critical size bubble, g is thesurface tension of water, Pg and Pl are the gas and liquidphase pressures. The radius RC is the same as the radius ofa maximum pore size in the pit membrane needed to causeair seeding.

2.2. Dynamics of a bubble past the critical size in the initial

phase where the bubble is spherical

The dynamics of a spherical bubble, in the absence ofthermal effects and incompressible liquid, can be describedby the RayleighPlesset equation (e.g. Brennen,1995):

Rd2R

dt2 32

dR

dt

2 Pg Pl

rL 4m

R

dR

dt 2grLR

, (2)

where R is the bubble radius, t is time, Pg is the gas pressureinside the bubble, Pl is the water pressure inside the lumenhosting the bubble, rL is water density, m is the dynamicviscosity of water, and g is the surface tension of water.

cal Biology 249 (2007) 111123 113The gas pressure inside the bubble (Pg) will always have avalue between saturation vapor pressure and atmospheric

equals the volumetric growth of the bubble and the second

ARTICLE IN PRESSretiterm is the water ow from the conduit hosting the bubbleto adjacent conduits. The water pressure inside theadjacent conduits, P0, does not remain constant, but variesas the water pushed from an embolising conduit will affectthe water balance of the adjacent conduits.

2.3. Growth of the gas phase after it has contacted lumen

walls

Eqs. (2)(4) describe bubble dynamics when the bubbleis small enough to remain spherical. However, when thebubble radius reaches the radius of the conduit, the gasbubble cannot be considered a spherical bubble anymore. Instead, the growth of the gas phase is now describedas spreading of a gas phase in a capillary, where thepressure exerted by the gas phase pushes water out ofthe conduit. Capillary analogy is used because the length ofthe conduit is very large compared to its diameter. Wemake an assumption that the gas bubble is initially in themiddle of the conduit. The spreading of the gas phase inthe conduit may now be obtained from the HagenPoiseuille equation

dlg

dt pr

4cond

8mlgDP, (5)

where lg is the distance of gas/liquid interface from thecenter of the conduit, and DP is the pressure differencebetween the liquid and gas phase in the conduit, and rcond isthe conduit radius. This pressure difference DP is writtenpressure, depending on the rate of air diffusion from thewater into the bubble. Air diffusion between the gas andwater is not modeled here, and the gas pressure inside thebubble Pg will be given a constant value of saturationvapor pressure, i.e. air diffusion is considered to be slowcompared to the time scale of the process.The change in water pressure in the conduit hosting the

bubble due to changes in conduit volume is calculated fromHookes law (e.g. Dainty, 1963).

dPl

dt 1VEr

dV

dt(3)

where Er is the volumetric elastic modulus of the conduit, Vis the conduit volume, and dV is the change in the conduitvolume. Two processes affect the change in conduitvolume: the change in gas bubble volume as bubble radiuschanges and the exchange of water with adjacent xylemconduits. dV is explicitly written

dV

dt 4pR2 dR

dt kPl P0 (4)

where P0 is the water pressure of the adjacent conduits, andk is the hydraulic conductance (m3 Pa1 s1) of the xylemconduit. The rst term on the right-hand side of Eq. (4)

T. Holtta et al. / Journal of Theo114DP Pg Pl 2g cos arcond

, (6)where is a is the contact angle between the liquid phaseand the conduit wall. The last term on the right-handside of the equation is the capillary pressure, and it isresisting the spreading of the gas phase. Here we assumethe conduit wall to be totally wettable, i.e. a is set to zero.Eq. (3) is now again used to calculate the relation betweenthe water pressure and conduit volume. The term dVis now written

dV

dt pr2cond

dlg

dt kPl P0Clg (7)

where rcond is the conduit radius, and variable C, which isdependent on lg, is introduced to be the fraction of the totalconduit wall area, which is covered with water. C is addedas water can only ow out of the conduit at locations wherethere is hydraulic connection with the conduit wall. Againthe rst term on the right-hand side of Eq. (7) equals thevolumetric growth of the bubble and the second term is thewater ow from the conduit hosting the bubble to adjacentconduits.

2.4. Calculation of the hydraulic resistance between the

conduit hosting the bubble and adjacent conduits

The hydraulic conductance k between the conduithosting the bubble and the adjacent conduits is calculatedfrom the conductance the pits and the cell wall connectedin parallel.

k kpNpits kcw (8)where kp is the hydraulic conductance of one pit, kcw is theconductance of the cell wall, and Npits is the number of pitsin the conduit (connected in parallel). The conductance ofthe cell wall will be important only if the pits are aspiratedas un-aspirated pits provide a parallel pathway with muchlower resistance.Pit conductance is constant for angiosperm pits as there

is no impermeable torus which position determines theresistance to water ow. In theory, the pit conductance ofangiosperm type pits should increase slightly whensubjected to a pressure difference over it as the pitmembrane and the pores will stretch (Sperry and Hacke,2004), but this is not taken into account here. Forgymnosperm pits with a torusmargo structure the situa-tion is different. The conductance of a gymnosperm pit canvary from that of an un-aspirated pit where the torus is inthe middle of the pit chamber to zero in a fully aspiratedstate. The position of the torus and the conductance to owis dependent on the force acting on the pit membrane andthe structure and rigidity of the pit membrane. Followingthe protocol of Bolton and Petty (1978), the conductanceof a gymnosperm pit can be broken down into severalconductances in series: that of the pit apertures, the poresin the margo, and the border-torus annulus. The gymnos-perm pit and its components are depicted in Fig. 2A. The

cal Biology 249 (2007) 111123parameters used to describe the geometry of the pit arepresented in Fig. 2B. The pit aperture conductance ka is

ARTICLE IN PRESS

go s

reticalculated from the HagenPoiseuille equation

ka pK4

8LKZ(9)

where LK and K are the length and radius of the aperture(see Fig. 2). The conductance of the pit membrane pores kmin the margo is calculated from the HagenPoiseuilleequation between water ow and pressure difference

4

Fig. 2. (A) A schematic illustration of a gymnosperm pit with a torusmar

torus annuli. (B) The parameters used to describe the geometry of the pit.

T. Holtta et al. / Journal of Theokm 8prpore npore

lpore 1:15rpZ(10)

where rpore is the pore radius, npore is the number of poresand lpore is the length of a pore, which is the same as themargo thickness.The conductance of pit apertures and pit membrane

pores will be constant even when the pit membrane isdeected. The conductance of the border-torus annulus willchange according to the distance of the torus from the pitopening H. The conductance of the border-torus annuli iscalculated to be the ow rate divided by the pressure dropover border-torus annuli from the following equation(Bolton and Petty, 1978):

P 3QZ lnM=K4pH3

0:771r Q4pKH

22 K

M

2" #(11)

where Q is the water ow rate, M is the torus radius.H decreases as the membrane is deected. When H goes tozero the, pit becomes fully aspirated and no water will owthrough the pit. The rst term on the right-hand side is theHagenPoiseuille equation modied for the geometry ofthe situation, while the second term gives the kinetic energycorrection. The kinetic energy correction is needed whenthe velocity of water ow will be very high. This will occurwhen the pit is near to aspiration but mass ow through thepit still remains high.Pit membrane deection is calculated from the force

acting pit membrane. In equilibrium, the force acting onthe membrane is balanced by the tension in the margostrands holding the pit membrane in place. Therefore thedensity, width and elastic modulus of the margo strandsholding the pit membrane in place, and the height andwidth of the pit chamber need to be known (e.g. Hacke

tructure. Symbols: (1) pit aperture, (2) pores in the margo and (3) border-

cal Biology 249 (2007) 111123 115et al., 2004; Bolton and Petty, 1978; Gregory and Petty,1973, 1972). The force acting on pit membrane is thepressure difference created by the ow over the pitmembrane pores and border-torus annulus, i.e. thepressure drop calculated from Eqs. (10) and (11), timesthe cross-sectional area of the torus (Bolton and Petty,1978). In case of no ow through the pits, i.e. when the pitsare aspirated, the force acting on the pit membrane is thehydrostatic pressure difference between adjacent conduitstimes the cross-sectional area of the torus. The formuladescribing the relationship between the pressure differenceover the pit membrane and pit membrane deection is(Petty, 1972)

DPpDm

2

2 nsEerAf siny, (12)

where DP is the pressure difference over the membrane, Dmis the torus diameter, ns is the number of margo strands,E is the strain of the margo strands, er is the elasticmodulus of the strands, Af is the cross-sectional area of thestrands, and y is the angle of deection of the pitmembrane from the middle of the pit chamber. By takinggeometrical considerations into account, the pressuredifference from Eq. (12) can be expressed as a solely as

ARTICLE IN PRESSretithe function of H (Petty, 1972)

DP 4nserAfpD2m

LM BH2

qLM

10@

1A BH

LM

, (13)

where LM is the margo strand length. The total con-ductance of a gymnosperm pit is calculated to be theconductances of the three different components used todescribe the pit (Eqs. (9)(11)) connected in series. The pitaperture and border-torus annuli conductance are calcu-lated twice, as there are two pit apertures and border-torusannuli for the ow to cross. For the border-torus annuli,the value of H is different for the upstream and down-stream components. For calculating the relation betweenthe water ux and pressure difference in the upstreamborder-torus annulus, the last term in brackets is sub-stituted by its reciprocal (Bolton and Petty, 1978).

2.5. Modeling the change in water balance of adjacent

conduits as a result of water freed from the embolising

conduit

Water freed from an embolising conduit will affect thewater content and water pressure of the conduits surround-ing embolising conduit (the term P0 in Eqs. (4) and (7)). Aswater is freed from the embolising conduit, it is pushedto adjacent conduits, and their water pressure will risebecause of this. This rise in water pressure will in turninduce water ow further away from the embolisingconduit. To estimate this effect on the embolism process,the transient water and pressure balances of conduitsdirectly adjacent to the embolising one and conduitswhich are at most N conduits distance away from theembolising conduit are also modeled. N is given thevalue 100. Increasing N beyond this does not have anynoticeable effects on the results. The embolising conduit isassumed to be in direct hydraulic contact with M adjacentconduits.Water inow to conduit i 1 equals one Mth of the

water ow from the embolising conduit

Q1;in kPl P1Clg

Mi 1, (14)

where Qi,in is the inow rate to conduit i and Pi is the waterpressure in conduit i, and M is the number of neighboringconduits. Water inow Qi;in to conduits i 41 is

Qi;in kPi Pi1 i41. (15)Water ow out of the conduit i (Qi,out) is equal to the

inow to conduit i+1 for ioNQi;out Qi1;in ioN (16)and

QN;out kPN Pbulk i N (17)

T. Holtta et al. / Journal of Theo116for i N. This means that water from the conduit N isconnected to an innite water volume reservoir with bulkxylem pressure Pbulk. Equation for mass conservation is

dmi

dt Qi;in Qi;out, (18)

where mi is the mass of water in conduit i. The consequentpressure changes are calculated from Hookes law

dPi

dt Er

1

rVdmi

dt, (19)

where r is the density of water. Here we do not consider theeffects of other xylem tissue such as bers and living cellssurrounding the embolised conduit. Tissue with elasticitysuch as living cells would be expected to show less changein pressure when its water content varies. For gymnospermwood the fraction of other tissue in the xylem is typicallysmall and the effect of the capacitive tissue would be low.

2.6. Solving the equations

The independent variables R, Pl, V, and P0 in Eqs. (2)(4),and (19) for the spherical bubble, or alternatively theindependent variables lg, Pl, V, and P0 in Eqs. (3), (5), (7),and (19) for the gas spreading in a capillary, are couplednon-linear differential equations which are solved numeri-cally. Initial values are given to them and then their timedevelopment is solved using the fourth-order RungeKuttamethod (Press et al., 1989) for the spherical bubble case andEuler method for the capillary case. The value of the timestep is chosen so that changing that value does not have anynoticeable inuence on the results. The time step used has tobe very small, especially in the beginning of the process. Forgymnosperms with torusmargo type pits, the transienttorus displacement and pit conductance are solved itera-tively during each time step by nding the correct torusdisplacement simultaneously while the pressure differenceover the torus is solved.

3. Parametrization

Parameters used in the base case model simulations arelisted in Table 1. The values for the base case have beenchosen to represent a typical gymnosperm earlywoodtracheid with a torusmargo structure and a typicalangiosperm vessel. The bulk xylem water pressure is givena value 1.0MPa, a typical value during transpiration.This is the initial value for the water pressure in theembolising conduit and in the adjacent conduits. Theradius and length of a gymnosperm conduit are givenvalues of 20 mm and 3mm, respectively. These are typicalvalues for earlywood gymnosperm tracheids (Lancashireand Ennos, 2002). The values given for angiosperm vesselradius and length are 50 mm and 3mm, respectively. Theinitial value for the bubble radius is calculated from Eq. (1)to be 1.46 107m. The elastic modulus of the conduit isgiven a value 750MPa (Peramaki et al., 2001). Saturation

cal Biology 249 (2007) 111123vapor pressure was given the value 0.003MPa, and surfacetension of water is given the value 0.073Nm1. These

ARTICLE IN PRESSretivalues are for water at 25 1C. The number of pits in agymnosperm conduit is estimated to be 100. According toUsta (2005), the number of pits per earlywood tracheidsvaries from 50 to 300. The number of pits in an angiospermconduit is calculated from the gymnosperm value, assum-ing the pit density per cell wall area is the same for both, tobe 250. The dimensions, surface area, elasticity, and

Table 1

Parameters used in the model

Initial xylem pressure

(Pl)

1.0MPa Estimated

Hydraulic conductance

of the cell wall (kcw)

8 1021m3 Pa1 s1 Estimated

Gymnosperm conduit

radius (rcond)

20 mm Lancashire andEnnos (2002)

Gymnosperm conduit

length (l)

3mm Lancashire and

Ennos (2002)

Number of pits in a

gymnosperm conduit

(Npits)

100 Usta (2005)

Angiosperm conduit

radius (rcond)

50 mm Estimated

Angiosperm conduit

length (l)

3mm Estimated

Angiosperm pit

conductance (kp)

1.67 1017m3 Pa1 s1 Estimated

Number of pits in a

angiosperm conduit

(Npits)

250 Estimated

Conduit elastic modulus

(Er)

750MPa Peramaki et al.

(2001)

T. Holtta et al. / Journal of Theohydraulic conductivity of the conduits adjacent to theembolising conduit are made equal to that of theembolising conduit. The number of conduits connected tothe embolising conduit M is given a value of 4.The cell wall conductance was given a value

8 1021m3 Pa1 s1. Only a few studies have been madeabout the hydraulic conductance through the xylem conduitcell wall (De Boer and Volkov, 2003). Wisniewski et al.(1987a, b) found that the primary and secondary walls ofvessel elements of various woody plant species, except partsthat were associated with the pit membranes, were imperme-able to lanthanum nitrate, which can penetrate capillaries assmall as 2 nm. This suggests that the cell walls have a verylow permeability to water (De Boer and Volkov, 2003).Petty and Palin (1981, 1983) measured the tangential andradial cell wall permeability of tracheids of the order of10211020m2. Assuming a cell wall thickness of 6mm andviscosity of pure water at 25 1C, this value can be convertedinto conductance, and is found to be over ten times largerthan our value. However, the method they used most likelyoverestimates the permeability (Aumann and Ford, 2002).Aumann and Ford (2002) used a value of 2.25 1019m s1for cell wall conductivity of tracheids. Assuming a cell wallthickness of 6mm, this value can be converted intoconductance, and is found to be approximately onethousandth of our value. No values for the conductance ofthe aspirated pits were found in the literature, although it isstated in many references that the torus is impermeable towater (e.g. Gregory and Petty, 1973; Pittermann et al.,2005). Therefore, the conductance of aspirated pits isassumed to be the same as that of the cell wall.Table 2 shows the parameters used for the gymnosperm

torusmargo type pit structure. Although there are manyindividual parameters in Table 2 describing pit structureand its elastic behavior, more important than the values ofthese individual parameters is their combined effect on pitconductivity and the force required for complete pitaspiration. Eqs. (911) and parameters in Table 2 for thedescription of a gymnosperm pit yield a value of1 1015m3 Pa1 s1 for the conductance of singleun-aspirated pit. This is similar to values reported in theliterature (e.g. Lancashire and Ennos, 2002). An angios-perm pit with a homogeneous pit membrane was given aconductance one-sixtieth of this (Pittermann et al., 2005).Similarly, the parameters give a pressure difference of0.19MPa required to aspirate the pit, which is inaccordance with pressure differences calculated for pitaspiration for gymnosperms by Bolton and Petty (1978)and Hacke et al. (2004). An initial time step of 1012 s was

Table 2

Parameters used for detailed description of gymnosperm pits, values are

from Bolton and Petty (1978)

Cross-sectional area of micro-bril strands (Af) 707 1018m2Number of micro-bril strands in margo (ns) 100

Elastic modulus of strands (er) 3Gpa

Maximum torus displacement (B) 2 106mPit membrane diameter (Dm) 17.0 106mTorus radius (M) 4.25 106mMargo strand length (LM) 4.25 106mPit aperture radius (K) 2.7 106mPit aperture length (LK) 1.5 106mNumber of pores in margo (npore) 200

Margo pore radius (rpore) 2.1 107mMargo pore thickness (lpore) 1.5 107m

cal Biology 249 (2007) 111123 117used, from where it is gradually raised to 108 s. Runningour numerical model on a fast Unix computer allowed usto use such a small time step. With a larger time step, thenumerical solution would become unstable.In the Results section we demonstrate the dynamics of

embolism growth for a typical gymnosperm pit with atorusmargo structure and a typical angiosperm conduitwith a homogeneous pit membrane using the modelpresented above. Also sensitivity analysis to the mostimportant parameters affecting embolism formationdynamics is done.

4. Results

4.1. Gymnosperms with a torusmargo pit structure

The dynamics of the bubble and the gas phase growthfor the base case parameterization with a gymnosperm

ARTICLE IN PRESSretiT. Holtta et al. / Journal of Theo118torusmargo structure is demonstrated in Fig. 3. Time iscalculated from the induction of the critical size bubble.Initially the bubble radius (Fig. 3A) grows rapidly as thepressure difference between the bubble (saturationvapor pressure) and the liquid water in the water conduit(Fig. 3B) is large and the pit is un-aspirated. As the waterpressure in the embolising conduit rises more rapidly thanin the adjacent conduits water ux between the embolisingconduit and the adjacent conduits (Fig. 3C) increases. Thepit membrane displacement (Fig. 3A) increases as the forcedeecting it grows until the value required for pitaspiration is reached and the pit becomes shut as itaspirates. The force acting on the torus (inset Fig. 3B) isalmost equal to the pressure difference between theembolising and adjacent conduits. The moment of pitaspiration is shown with an arrow. Following aspiration,hydraulic conductance to water ow out of the conduitdecreases by many orders of magnitude (Fig. 3C) andgrowth of the bubble slows dramatically. After reachingthe conduit diameter, at approximately 3 s, the gas phasespreads in the capillary (Fig. 3D). The water pressure inthe conduit in the capillary phase (Fig. 3B) remains nearlyconstant. The pressure is dictated by the gas phase and thecapillary pressures. As the water pressure in the adjacentconduit (Fig. 3B) remains essentially constant during thewhole process, the driving force for the water ow out ofthe conduit remains almost constant in the capillary phase.

Fig. 3. (A) Bubble radius (gray line) and torus displacement H (dark line) as a

with torusmargo pit structure. Time is calculated from the induction of a crit

conduit (gray line) and the adjacent conduits (dark line). (C) Water ux out of t

(gray line). (D) Relative volume of water in the conduit hosting the embolismcal Biology 249 (2007) 111123The pits remain aspirated as there is a hydrostatic pressuredifference over the torus which remains higher thanthreshold for pit aspiration. The water ow rate out ofthe conduit decreases gradually as the gas spreads becausethe fraction of the conduit wall covered with waterdecreases (Fig. 3D). Full embolism is achieved in about20min. The Reynolds number, estimated from the equa-tion for water ow through a pipe, reaches a maximumvalue of approximately 10 in the border pit annuli of thepits just before the pits aspirate (not shown), and thendrops abruptly. The ow is therefore likely to remainlaminar at all times. If turbulence would arise it would beminor and occur just prior to pit aspiration, and it wouldnot have signicant effect on the results.

4.2. Angiosperms with a homogeneous pit structure

The dynamics of embolism growth is very different for aconduit with a homogeneous angiosperm pit. Initially, thebubble radius (Fig. 4A) has the same dynamics as in thetorusmargo case, but continues fast growth unlike in theprevious case as pit conductivity does not drop due topit aspiration. Water pressures in the embolising conduit(Fig. 4B) rises initially very fast, but the pressure rise slowsdown as the water pressure in the adjacent conduits(Fig. 4B) rises much above bulk xylem pressure. Thewhole embolisation process is now much faster (Fig. 4C),

function of time for the base case parameters for a gymnosperm conduit

ical size bubble. The arrow marks pit aspiration. (B) Water pressure in the

he embolising conduit (dark line) and hydraulic conductance of the conduit

as a function of time.

ARTICLE IN PRESSretiT. Holtta et al. / Journal of Theooccurring in a fraction of a second. Similar dynamicswould be expected for a pit with a torusmargo structure ifthe pits would not aspirate. Although embolism formationis rapid, it is actually slow compared to bubble growth inbulk water under the same tension. The time needed for abubble to grow to the same volume as the base case xylemconduit in bulk water under the same tension can becalculated from Eq. (2) to be approximately 50 ms.

4.3. Sensitivity for parameters

The time required for the bubble to ll the conduitcompletely as a function of the bulk xylem water pressurefor the conduit with a torusmargo structure, i.e. the casepresented in Fig. 3, is illustrated in Fig. 5. The time neededfor complete embolism has a maximum at a xylem pressurewhere the developing water pressure difference between theembolising and adjacent conduits reaches the threshold forpit aspiration. The xylem water pressure at the maximumembolism time is about 0.2MPa in this case. At xylempressures higher than this, i.e. at lower water tensions, thepits will not aspirate and embolism is fast. At lower xylempressures than the one which causes the maximum inembolism time, the time needed for complete embolismincreases as the bulk xylem water pressure decreasesbecause then the pressure difference driving water out ofthe conduit is larger. In the case of a homogeneous pit

Fig. 4. (A) Bubble radius as a function of time for the base case parameters

pressure in the conduit (gray line) and the adjacent conduits (dark line). (C) Re

time.cal Biology 249 (2007) 111123 119membrane (not shown), embolism time always increaseswith increasing xylem pressure, i.e. with decreasing xylemtension.For a case with a torusmargo structure, i.e. the case

presented in Fig. 3, the cell wall conductance will determinethe resistance of water ow out of the embolising conduitafter aspiration has occurred and will therefore inuenceembolism time strongly (Fig. 6A). Embolism time isinversely proportional to the hydraulic conductance ofthe cell wall. Varying the hydraulic conductance of the pitsdid not affect the base case results very strongly. Loweringpit conductance did not have practically any effect on theresults. With increased pit conductance, the total embolismtime will be slightly shorter as it takes longer for pits toaspirate (Fig. 6B). For the homogeneous angiosperm pitscell wall conductance does not affect embolism dynamics,as long it remains much lower than the conductance of thepits. A lower pit hydraulic conductance will result in anincrease in the embolism time and a higher conductancewill decrease it (Fig. 6C). The decrease in embolism timewill become less pronounced as the conductance grows tovery high as then pressure distribution in the adjacentconduits is also affected strongly.The effect of various other parameter values on the

embolism dynamics was also tested. For conduits withtorusmargo type pits, decreasing the elasticity of the pits,for example by increasing the elastic modulus of the margo

for an angiosperm conduit with a homogeneous pit structure. (B) Water

lative volume of water in the conduit hosting the embolism as a function of

ARTICLE IN PRESSretiT. Holtta et al. / Journal of Theo120strands, would increase the threshold pressure for pitaspiration linearly. Less elastic pit membranes would needhigher bulk xylem tensions for pit aspiration. If pit densityis constant, wider conduits will take longer to embolise asthey have a larger volume to surface area ratio thannarrower conduits, i.e. more water has to ow out throughrelatively a smaller conduit wall area. The relationship

Fig. 5. Time needed for complete embolism as a function of the xylem wat

Fig. 6. (A) Time needed for complete embolism as a function of the cell wal

Values in the x-axis are shown relative to the base case values. (B) Time ne

gymnosperm conduit with torusmargo pit structure. (C) Time needed for co

conduit with a homogeneous pit structure.cal Biology 249 (2007) 111123between conduit diameter and embolism time is linear.Conduit length was found not to have any affect onembolism time, as the conduit volume to conduit wall arearatio remains constant with varying conduit length. If thenumber of functioning water conduits directly adjacent tothe embolising one were reduced along with surface areawhich the embolising conduit could exchange water with

er pressure for a gymnosperm conduit with torusmargo pit structure.

l conductance for a gymnosperm conduit with torusmargo pit structure.

eded for complete embolism as a function of the pit conductance for a

mplete embolism as a function of the pit conductance for an angiosperm

ARTICLE IN PRESSretiadjacent conduits, then embolism time increased byapproximately the same factor as the neighboring conduitsand surface area were reduced. Changing the elasticmodulus did not practically affect the growth rate, as longas it was not reduced to magnitudes much below reason-able. The past critical size bubbles were not found tocollapse in any situation as the conduit water pressure doesnot rise higher than the gas pressure at any time during thebubble growth process. Sub-critical size bubbles alwayscollapsed. These results are not shown in the gures.

5. Discussion

The principle aim of this study was to model the time-scale required for the gas phase induced by cavitation to llan entire xylem conduit, i.e. to fully embolise a conduit.Embolism formation was found to be slower from whatwould be expected for a bubble to grow in bulk water.While gas bubbles would grow very rapidly in bulk waterunder tension, the inelasticity of the conduit walls togetherwith hydraulic resistance of the conduit lumen andbordered pits will restrain the gas phase growth. As theconduit cannot expand much when the gas phase volume isgrowing, ow rate of water out of the conduit in turn isdetermined by the developing pressure difference betweenthe embolising and adjacent conduit and the hydraulicconductance of the water conduit. Although transpirationis not explicitly present in the model, transpiration rate willdetermine the xylem bulk water tension, and therefore thepull which drains water out of the embolising conduit. Fortorusmargo type gymnosperm pits, the time-scale ofthe process was found to be strongly dependent on thefunctioning of the bordered pits between conduits. If theforce exerted by the water ow out of the embolisingconduit on the pit membrane grows larger than the forceneeded to aspirate the pits, embolism formation willbecome much slower, as water from the conduit will beslowly drained through the cell wall. The time needed forcomplete embolism will then be determined by theconductance of the cell wall, bulk xylem pressure, andconduit radius. The force per unit area acting on the toruswas found to be nearly as large as the difference in waterpressure between the embolising and adjacent conduits.This is because almost the entire pressure drop on thepathway between the embolising and adjacent conduitsoccurs at the margo pits (when the ow rate is small) or theborder-torus annuli (when the ow rate is large) where it isapplicable to displace the torus. Unlike gymnospermearlywood tracheids, latewood tracheids would not beexpected to aspirate at the water uxes and water pressuredifferences caused by embolism as their structure andrigidity will resist aspiration up to much higher pressuredifferences (e.g. Petty, 1972; Siau, 1984).The water pressure in the conduit hosting the bubble was

found to reach its saturation value, i.e. a value where the

T. Holtta et al. / Journal of Theowater pressure is the sum of pressure of the gas phase andthe capillary pressure, already in the very beginning of theembolism process in the base case. This happens as thewater cannot ow out of the embolising conduit atthe same volumetric rate as the gas phase expands, sowater pressure in the conduit rises by many atmospheres inas little as few microseconds. The rapid tension release isalso transmitted to the conduit walls, and this is probablythe signal that is observed in acoustic and ultra-acousticmonitoring of cavitation events (e.g. Tyree and Dixon,1983). Water pressure in the adjacent conduits remainedclose to bulk xylem water pressure after pit aspirationoccurred. In this case the driving pressure differencefor water ow out of the conduit could be approximatedquite accurately to be the difference between saturationvapor pressure and bulk xylem pressure. However, thiswas shown not to be the case in the absence of pitaspiration, where water is emptied very fast out of theembolising conduit, and water pressure in the surroundingconduits can rise noticeably and affect the embolismdynamics. The water released from the embolising conduitand the consequent rise in water pressure of the adjacentconduits could also have a capacitive role during times ofwater decit (e.g Meinzer et al., 2001). This water freedfrom the embolising conduit to the adjacent tissue willsupply extra water, and might prevent the bulk waterpotential from falling to harmfully low levels during peakwater stress.Of the various parameters, the hydraulic conductance of

the conduit when the pits are aspirated, which practicallyreduces to being the hydraulic conductance of the cell wall,is probably the least well-known of the various parameters(De Boer and Volkov, 2003; Aumann and Ford, 2002), andwill affect the total embolism time very dramatically in thecase of pit aspiration for conduits with torusmargo typepits. Therefore the times predicted here for completeembolism in this case are not meant to be accurate, evento the order of magnitude. If we calculate the totalembolism time from the two extreme cell wall conductancevalues given in the literature, the high value given by Pettyand Pallin (1981, 1983) and the low value from Aumannand Ford (2002), the time scale for complete waterdrainage out of a conifer conduit with aspirated pits couldrange from tens of seconds to an approximately a month ifother parameters were to be as in the base case. However, ifeven one of the pits would remain un-aspirated, forexample due to a more rigid structure, then this one pitwould provide a much lower resistance pathway out of theembolising conduit, and the time-scale for embolismformation would be comparable to the homogeneous pitmembrane case.The dynamics described here for embolism formation are

for a case where a single conduit is embolising surroundedby other water conducting conduits connetected to thetranspiration stream. If also other conduits were toembolise during the time-scale described here, this wouldraise the water pressure in the adjacent conduits due to

cal Biology 249 (2007) 111123 121water release from the other embolising conduits. Theore-tically, this would decrease water ux out of the embolising

ARTICLE IN PRESSreticonduit and therefore decrease the likelihood of pitaspiration and slow down embolism formation in othercases. Whether the gas inside the bubble has atmosphericpressure, i.e. air is assumed to diffuse very fast to thebubble, or saturation vapor pressure, i.e. no air diffusion,has only a small effect on the bubble growth rate. Air lledgas phase grows slightly faster than a water vapor lledbecause the pressure gradient driving water out of theconduit is approximately 0.1MPa larger in the air lledcase. Our model might slightly overestimate the growthrate of gas phase because the whole conduit area coveredby liquid water is taken as the area that can exchange waterwith adjacent conduits in the model. If a part of this areawould not be able to exchange water, then water ow outof the embolising conduit would also be slower. Also thetapered ends of the conduits are not formulated in themodel. The capillary pressure resisting the bubble growthwould actually increase at the nal stages of the embolismprocess, as the capillary pressure is inversely proportionalto the conduit radius. The tapered ends of the conduitwould thus slightly hinder the spreading of the gas phase.For the torusmargo type pits, the long time possibly

required for complete embolism formation might haveconsequences on the ability to rell embolised conduits.Possibly relling could commence before all of the water isdrained out of the conduit. Hydraulic isolation fromadjacent conduits has been proposed to be a requisite forcomplete embolism relling of a conduit (Holbrook andZwieniecki, 1999). Zwieniecki and Holbrook (2000) showthat this could be accomplished by the air spaces in the pitswhich would be dissolved only at very end of the rellingprocess and hydraulic contact would then be re-achieved.Hydraulic isolation would also occur between the embolis-ing conduit and adjacent conduits as a result of the pitaspiration modeled in this study.

Acknowledgement

Helsingin Sanomain 100-vuotissaatio is acknowledgedfor the research funding.

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ARTICLE IN PRESST. Holtta et al. / Journal of Theoretical Biology 249 (2007) 111123 123

A model of bubble growth leading to xylem conduit embolismIntroductionTheoryInduction of a critical size gas bubbleDynamics of a bubble past the critical size in the initial phase where the bubble is sphericalGrowth of the gas phase after it has contacted lumen wallsCalculation of the hydraulic resistance between the conduit hosting the bubble and adjacent conduitsModeling the change in water balance of adjacent conduits as a result of water freed from the embolising conduitSolving the equations

ParametrizationResultsGymnosperms with a torus-margo pit structureAngiosperms with a homogeneous pit structureSensitivity for parameters

DiscussionAcknowledgementReferences