universality of phloem transport in seed plants

Universality of phloem transport in seed plantspce_2472 1065..1076

KÅRE HARTVIG JENSEN1*, JOHANNES LIESCHE2*, TOMAS BOHR1 & ALEXANDER SCHULZ2

1Department of Physics, Center for Fluid Dynamics, Technical University of Denmark, DTU Physics Building 309, DK-2800Kongens Lyngby, Denmark and 2Department of Plant Biology and Biotechnology, University of Copenhagen,Thorvaldsensvej 40, DK-1871 Frederiksberg C, Denmark

ABSTRACT

Since Münch in the 1920s proposed that sugar transport inthe phloem vascular system is driven by osmotic pressuregradients, his hypothesis has been strongly supported byevidence from herbaceous angiosperms. Experimental con-straints made it difficult to test this proposal in large trees,where the distance between source and sink might proveincompatible with the hypothesis. Recently, the theoreticaloptimization of the Münch mechanism was shown to lead tosurprisingly simple predictions for the dimensions of thephloem sieve elements in relation to that of fast growingangiosperms. These results can be obtained in a very trans-parent way using a simple coupled resistor model. To testthe universality of the Münch mechanism, we compiled ana-tomical data for 32 angiosperm and 38 gymnosperm treeswith heights spanning 0.1–50 m. The species studied showeda remarkable correlation with the scaling predictions. Thecompiled data allowed calculating stem sieve element con-ductivity and predicting phloem sap flow velocity. Thecentral finding of this work is that all vascular plants seem tohave evolved efficient osmotic pumping units, despite theirhuge disparity in size and morphology. This contributionextends the physical understanding of phloem transport,and will facilitate detailed comparison between theory andfield experiments.

Key-words: long-distance transport; Münch mechanism;phloem; scaling; sieve elements; sugar; trees.

INTRODUCTION

Vascular transport of photoassimilates in plants fromsource to sink takes place in sieve elements (SEs). Thesespecialized cells of the phloem form a continuous networkrunning throughout the plant. The most widely acceptedmechanism for phloem transport is the osmotic pressure-driven mass flow as proposed by Münch in the 1920s(Münch 1930). According to Münch, sugar produced in theleaves generates an osmotic pressure which drives a flow ofwater and sugar from source to sink, in accordance with thebasic needs of the plants (Fig. 1a).

There is considerable knowledge of phloem transport inherbaceous angiosperms, where a large number of studieshave contributed to our view on the mechanism of loading,translocation and unloading in this plant group (Holbrook& Zwieniecki 2005). Recent work involving translocationvelocity measurements (Windt et al. 2006; Jensen et al.2011), theoretical modelling and microfluidic model experi-ments (Jensen et al. 2009, 2011) have shown that the phloemvascular system of herbaceous angiosperms is geometricallyoptimized for rapid translocation, and that the Münchmechanism is sufficient to account for the observed trans-location rates (Jensen et al. 2011). A relevant question is,whether this is also universally the case in trees, in particu-lar in gymnosperms, which have not been considered so far.

All measurements of sap flow velocity in gymnospermtrees, except one (Willenbrink & Kollmann 1966), givea significantly slower speed compared with woodyangiosperms (Crafts & Crisp 1971). Differences in method-ology hinder the generalization of experimental results butdirect comparison with identical experimental set-upsshowed the same velocity difference (Thompson et al. 1979;Dannoura et al. 2011).Typical translocation velocities foundin angiosperm are of the order 1 m h-1 (Windt et al. 2006;Mullendore et al. 2010; Jensen et al. 2011), while observedvelocities in gymnosperms typically are two to five timesslower (Crafts & Crisp 1971; Thompson et al. 1979; Plainet al. 2009; Dannoura et al. 2011). Given the fact that thetallest trees are gymnosperms and that their SE anatomywith endoplasmic reticulum-obstructed sieve pores (Schulz1992) appears incompatible with Münch pressure flow(Turgeon 2010), some authors have speculated that thetransport process in gymnosperms may differ funda-mentally from that found in angiosperms (Crafts 1939;Kollmann 1975; Liesche, Martens & Schulz 2011), althoughindirect experimental evidence suggests otherwise (Münch1930; Watson 1980; Sevanto et al. 2003).

Despite the progress in theoretical modelling of Münchflow and agreement with experimental data from herba-ceous plants, fundamental questions about phloem trans-port in both angiosperm and gymnosperm trees remain.Experimental data on transport speed, osmotic potentialand conductivity are still scarce (Knoblauch & Peters 2010).The largest plants where phloem sap velocity was measuredwere a 4 m poplar [~0.7 m h-1 (Windt et al. 2006)] andseveral beech (0.22–1.21 m h-1), oak (0.36–1.02 m h-1) andpine trees (0.09–0.21 m h-1) of 8–10 m height (Dannoura

Correspondence: A. Schulz. e-mail: [email protected]

*These authors contributed equally to this work.

Plant, Cell and Environment (2012) 35, 1065–1076 doi: 10.1111/j.1365-3040.2011.02472.x

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© 2011 Blackwell Publishing Ltd 1065

et al. 2011); all species can easily reach heights of more than20 m. If the observed transport velocity is representative forall exemplars, then an increased osmotic potential and/orhigher sieve tube conductivity would be needed to offsetthe stem length effect. Despite the inaccessibility of thephloem to measurement of osmotic pressure and thereforelack of direct evidence (Millburn & Kallarackal 1989), it isnow assumed that phloem pressure does not scale withplant height (Turgeon 2010). The measurement of keyfeatures of the SE anatomy should allow estimation of the

conductivity and therefore answer the question if tall treeshave the potential to transport with similar velocitiesobserved in small trees and herbaceous plants.

METHODS

Theoretical analysis of the Münchpressure-flow mechanism

The most widely accepted mechanism for phloem transportis the osmotically driven pressure flow proposed by Münch

C

SE

SE

SE

R

F

CCPC

PC

CCStr

Str

C

C

SE

SE

R R

T

(b)

(c)

SugarWater

Water

Phloem

Source

Sink

a

at

ar

ltrans

ltrans

lsource

lsource

(d) (e)

(a)

Figure 1. Aspects of plant anatomy relevant to phloem transport. (a) Schematic sketch of sugar translocation in plants according to theMünch hypothesis. In the source leaves, sugar (black dots) produced by photosynthesis is delivered into the phloem. Because of osmosis,the high concentration of sugar creates a flow of water across the semipermeable cell membrane from the surrounding tissue into thephloem. This in turn pushes the water and sugar already present forward, thereby creating a bulk flow from sugar source to sugar sink. Atthe sink, for example, the root, removal of sugar from the phloem causes the water to leave the cells because the osmotic driving force isno longer present. The loading and unloading processes are indicated by curved arrows. (b) Macroscopic parameters of phloem transport.Stem length ltrans and leaf length lsource indicated for an angiosperm (left and top middle) and gymnosperm (right and bottom middle).(c) Schematic sketch of sieve element (SE) geometry. In cross section, angiosperm SEs (top) are typically circular with radius a, whilegymnosperm SEs are rectangular with tangential half width at and radial half width ar. (d) and (e) Cross sections of secondary phloemin the stem of mature trees. Stem phloem consists of the conducting SEs, the companion cells (CC) in case of angiosperms (d) andStrasburger cells (Str) in case of gymnosperms (e), axially arranged ray parenchyma cells (R), fibres (F), parenchyma cells (PC) andsometimes tannin cells (T). In most species, only a part of the current year’s phloem at the cambium (C) is functional. The arrowheadindicates a simple sieve plate, typical for angiosperm phloem. Scale bars = 20 mm; (d) secondary phloem of Robinia pseudoacacia adaptedfrom Evert (1984); (e) secondary phloem of Picea abies adapted from Schulz & Behnke (1987).

1066 K. Hartvig Jensen et al.

© 2011 Blackwell Publishing Ltd, Plant, Cell and Environment, 35, 1065–1076

in the 1920s (Münch 1930). According to Münch, sugarproduced in the leaves generates an osmotic pressure whichdrives a flow of water and sugar from source to sink, assketched in Fig. 1a. The quantitative description of thistranslocation process falls in two categories: one (1) whichuses solutions of the detailed equations governing fluid flowand solute transport (see, e.g. Thompson & Holbrook2003b; Jensen et al. 2009; Pickard & Abraham-Shrauner2009; Jensen et al. 2011); and another (2) which uses highlevel resistance models (see, e.g. Minchin, Thorpe & Farrar1993) to characterize the flow. Here, we use a type of resis-tor model to describe the flow. Note that Jensen et al.recently showed a direct correspondence between certaintype (1) and (2) models (Jensen et al. 2011).

The flow velocity u (for a list of symbols, see Table 1) atwhich the dissolved sugar is moving depends on the mag-nitude of the osmotic pressure difference Dp betweensource and sink and on the hydraulic resistance R of thetranslocation pathway

uA

pR

= 1 ∆(1)

where A is the cross-sectional area of the SE (Jensen et al.2011).The combined resistance experienced by the liquid asit moves along the phloem translocation pathway can bedivided in three parts corresponding to the resistance in theleaf (source), stem (translocation) and root (sink) regions,as sketched in Fig. 1a:

R R R R= + +source trans sink(2)

The magnitude of these three resistance componentsdepends on the macroscopic size of the plant, in particularleaf length and stem length (Fig. 1b), the microscopic geom-etry of the SEs, and on the material properties of the semi-permeable cell membrane and the sugar solution.

In this analysis, we ignore differences in the mode ofphloem loading and unloading of sugars, which might besymplasmic or apoplasmic, active or passive (Rennie &Turgeon 2009). In any case, source organs are identified byhigh sugar concentrations and sink organs by high sugarconsumption. This leads to an effective osmotic pressuredifference Dp between source and sink and water influx andefflux in source and sink, respectively (see Fig. 1a).

SEs are predominantly found in two different shapes:cylindrical, typical of angiosperms, and cuboidal, of rectan-gular cross section, often found in gymnosperm trees(Fig. 1c,d,e). Here, we examine the case of cylindrical SEs indetail and only state results for cuboids, which are studiedmore carefully in Appendix A. For cylindrical SEs, the stemresistance is approximately that of a cylindrical tube

Rla

transtrans= 8

4

ηπ

, where h is the viscosity of the liquid, ltrans is

the length of the stem, and a is the radius of the SE (Fig. 1c).The cross-sectional area is simply A = pa2. The number andsize of sieve pores connecting adjacent SEs are alsobelieved to play a role (Mullendore et al. 2010), and mayincrease the effective viscosity of the liquid significantly(Thompson & Holbrook 2003b), but for simplicity we willnot take this into account in the present analysis. As thelength scales for leaves and roots are smaller than the trans-location (stem) length, we assume that the resistance of the

Table 1. Nomenclature

Name Symbol Value Unit Reference

SE radius, effective osmotic radius a mSE half width ar mSE half height at mSE cross-sectional area A m2

Geometric factor G 16 (circular), 3 (rectangular)Conductivity k m2

Membrane permeability Lp 5 ¥ 10-14 m Pa–1 s–1 (Thompson & Holbrook 2003b)Sink length lsink mSource/leaf length lsource mStem/translocation length ltrans mNumber of pores in membrane NResistance R (Pa s)/m3

Osmotic pressure difference Dp 0.7 MPa (Turgeon 2010)Velocity u m s–1

Material factor V 16Lph mMaterial factor W 3Lph(1 + d)/(d - 0.63) mSE aspect ratio δ = a

at

r

!2Viscosity h 2 ¥ 10-3 Pa s (Thompson & Holbrook 2003b)Membrane thickness k mMembrane pore radius r mMembrane pore covering fraction fMembrane area W m2

SE, sieve element.

Universality of phloem transport in seed plants 1067


source and sink regions is dominated by the osmotic resis-

tance through surface area, that is RaL l

sourcep source

= 12π

and

RaL l

sinkp sink

= 12π

where Lp is the permeability of the

semipermeable membrane. In terms of osmotic water trans-port, the permeability of the plasma membrane is deter-mined by aquaporins (see Appendix B). From Eqns 1 and 2,we arrive at a simple expression for the velocity u as afunction of the geometric and material parameters of theproblem

u a l l lL p

a l lVl l

, , ,source trans sink

p

source sink

source tran

( ) =2

2

∆ ss sink source sinkl a l l+ +( )3 (3)

where we introduced the short-hand notation V = 16Lph.From Eqn 3 we recover several results found in the phloemliterature, for example, that the transit time of a single sugar

molecule tl

u= trans scales as ltrans

2 when ltrans is very large as

found numerically by Thompson and Holbrook (Thompson& Holbrook 2003a). It is apparent from Eqn 3 that thetranslocation velocity u has a maximum as a function of cellradius a for fixed source, translocation and sink lengthswhen a = a*, where

a Vl l ll l

Vl

l l* source trans sink

source sink

trans

source s

31

2 2=+

=+−

iink−1 (4)

At this value of the radius, the osmotic pumping mecha-nism is operating at its maximum capacity. Inserting a = a*into (3) gives the optimal speed

u a l l lL p

Kl l l( *, , , ) /source trans sink

ptrans source sink

21 3 1

∆= +− − −− −( )1 2 3/

(5)

where K = 3-122/3V-1/3. If we assume that the sink length lsink

(e.g. the length of the roots) is always larger than the sourcelength (the leaves), the largest velocity is actually foundwhen lsink >> lsource, where the right hand side of (5) givesKl ltrans source

−1 3 2 3/ / . In the case lsource = lsink the right hand side,however, gives practically the same result, being simply afactor 2-2/3 ª 0.8 lower, so the precise choice of the ratio ofthese length scales is unimportant. In the following, we thusassume that lsource = lsink and have for the optimum radiusthat

a L l l* p source trans3 16= η (6)

a result first found by Jensen et al. (2011).It is interesting to note that the optimality condition (4)

means that R R Rtrans source sink= +( )12 , that is, that optimality

sets the resistance through the stem to the mean resistanceof the source and the sink.

For a rectangular cell (see Appendix A), we find for theoptimized radius (when lsource = lsink)

a L l l* p source trans3 3= η (7)

where the effective osmotic ‘Münch’ radius of the cuboidal

cells is given by a aa a

a a= −

+

r

t r

t r

0 63 1 3. /

. Here, ar and at are the

half width and half height of the cells, respectively (seeFig. 1c). Combining Eqns 6 and 7 we can write a generalequation for the optimized ‘Münch’ radius a* valid in bothgeometries:

a GL l l* p source trans3 = η . (8)

Here, G is a geometric factor depending only on theshape of the cell (with the value 16 for cylindrical cells and3 for cuboidal cells). Under the assumption that plants areoptimized for rapid phloem transport, Eqn 8 puts a con-straint on the relative size of the various plant organs: cellradius a, stem length ltrans and leaf length lsource. The productof the membrane permeability Lp and liquid viscosity h isalso a length scale, related to the hydrodynamic size anddensity of the pores in the semipermeable membrane (seeAppendix B). In this way, Eqn 8 directly couples the mac-roscopic and microscopic structures of the plant.

Conductivity

Another equivalent formulation of the Münch flow Eqn 1can be given in terms of the hydraulic conductivity k:

uk p

l=

η∆trans

(9)

where there is an inverse relation between conductivity and

resistance, cf. Eqn 1, klAR

= η trans . With Eqns 3 and A4, the

conductivity can be calculated directly from Eqn 9 as

ku l

p= η trans

∆. We note that while it generally depends on the

geometric and material parameters of the problem, it doesnot depend on the pressure drop Dp, as u ! Dp.The conduc-tivity k gives a measure of how well the plant is able toconduct fluid flow, and is commonly used in quantitativestudies of transport both in phloem (see, e.g. Thompson &Holbrook 2003b or Mullendore et al. 2010) and in xylem(Becker, Tyree & Tsuda 1999). Compared with Eqn 1, ituses the average pressure drop per unit length Dp/ltrans

rather than the absolute pressure difference betweensource and sink Dp to characterize the flow, thus allowingfor a direct comparison of the hydraulic properties of plantsof different heights.

Experimental methods

SE radii of secondary phloem, leaf size and stem size (givenin Table 2) for 32 angiosperm species and 38 gymnospermspecies were obtained from the literature (Chang 1954a,b;Esau 1969; Schulz & Behnke 1987; Jensen et al. 2011) andfrom samples taken in the field. In case a literature sourcedid provide a value for SE radius but not for leaf and stemsize, average values for these two measures were derivedfrom online references.



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d

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k[m

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ms-1

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330

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(14.

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rg.

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280

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The 13 field samples were collected at the CharlottenlundArboretum, Denmark, on 30 April 2011 and at the Univer-sity of Copenhagen, Faculty of Life Sciences in Frederiks-berg, Denmark, on 27 May 2011.At a stem height of around1.3 m, the outer bark was removed in an area large enoughto cut out 1 cm by 2 cm samples of the inner bark, includingthe current-year phloem. Tree height, ltrans, of the sampledtree was measured as the distance between the ground andthe top of the tree (Fig. 1b). Average leaf length, lsource, wasdetermined by measuring the length of the leaf blade orneedle without petiole (Fig. 1b). All trees were mature.

The bark samples were cross-sectioned with a razor bladeand bright-field images of the inner bark were taken with aLeica SP5X confocal microscope. SE diameters (withoutcell wall) of the current-year phloem were determinedmanually with the help of LAS AF Lite (Leica, Wetzlar,Germany) or Velocity (Perkin Elmer,Wellesley, MA, USA)software.A minimum of 24 SEs were measured per plant ontwo to four sections. Only clearly differentiated SEs wereselected, as identified by the absence of visible cellularcontent, their specific shape and their size in relation to thatof other cell types (see Fig. 1d,e).

RESULTS

A comparison between the anatomical phloem data fromgymnosperm and angiosperm species given in Table 2reveals several features.As shown in Fig. 2, there appears tobe no clear correlation neither between stem size ltrans andleaf size lsource (Fig. 2a) nor between leaf size lsource and singlestem SE cross-sectional area A (Fig. 2c). Interestingly, asshown in Fig. 2b, the SE area A is seen to scale with plantheight ltrans in herbaceous plants. Regression analysis yieldsA l∝ ±

trans1 3 0 1. . (r2 = 0.90; N = 13; aRMA = 1.4 " 0.1), but we find

no increase in SE area in plants higher than 5 m. Here, wefollow Niklas (1994) and give scaling exponents obtained byleast square (aLS_ = 1.3 " 0.1) and reduced major axis(aRMA_ = 1.4 " 0.1) regressions for a dataset with N plantsand correlation coefficient r. SE area of angiosperm as wellas gymnosperm trees is seen to saturate above ltrans = 5 mnear the value A ! 10-9 m2 = 103 mm2.

Figure 2. Plots of anatomical phloem data for 32 angiospermand 38 gymnosperm species. (a) Leaf size lsource plotted as afunction of stem size ltrans. (b) Sieve element (SE) area A plottedas a function of stem size ltrans. (c) SE area A plotted as afunction of leaf size lsource. In (b), the SE area A saturates nearlstem = 5 m at the value = 10-9 m2 = 103 mm2. Dashed line in(b) through the data points for herbaceous species is theregression curve obtained from least square regression analysis,A l∝ ±

stem1 3 0 1. . (r2 = 0.90; N = 13; aRMA = 1.4 " 0.1). The conductivity

k was calculated from Eqns 3, 9 and A4. In the plot we assumethat the viscosity h and membrane permeability Lp do not scalewith plant height. Parameters used are Lp = 5 ¥ 10-14 m/(Pa s)and h = 2 ¥ 10-3 Pa s (Thompson & Holbrook 2003b). Symbollegend: angiosperm trees (open circle), angiosperm herbs (blackdot), angiosperm shrubs (grey dot), gymnosperm trees (opensquare), gymnosperm trees with scales (open triangle).

10−1 100 101 10210−3

10−2

10−1

100

ltrans [m]

l sour

ce[m

]

(a)

10−1 100 101 10210−12

10−11

10−10

10−9

10−8

ltrans [m]

A[m

2]

(b)

10−3 10−2 10−1 10010−12

10−11

10−10

10−9

10−8

lsource [m]

A[m

2]

(c)



A comparison between the experimental data and theoptimality prediction of Eqn 7 is shown in Fig. 3. Weobserve that many angiosperms as well as gymnosperms fallon the predicted scaling line, suggesting that both taxa areoptimized for rapid translocation in the phloem. Regressionyields lsourceltrans ∝ a2.7"0.2 (r2 = 0.69; N = 70; aRMA = 3.3 "0.2) for 32 angiosperms and 38 gymnosperms, close to thepredicted value 3, cf. Eqn 8. A similar scaling exponent(aLS = 2.6 " 0.3) was found by Jensen et al. (2011) for amuch smaller dataset (19 angiosperms, 1 gymnosperm).To quantify whether aLS = 2.7 " 0.2 and aRMA = 3.3 " 0.2differ significantly from the predicted value 3.0, we

calculate the test statistic t-values tLSLS

LS

= − =ασ

3 01 5

.. and

tRMARMA

RMA

= − =ασ

3 01 5

.. , see, for example, Taylor (1997).

The probability of obtaining an answer that differs from 3.0by t = 1.5 or more standard deviations is found from thenormal error integral to be 13.4%. We therefore concludethat there is insufficient evidence to indicate a differencebetween the obtained and predicted scaling exponents.These results indicate that even though, as discussed in theIntroduction, gymnosperms have significantly lower trans-location speeds than found in angiosperms, the size of theirSE is equally optimized for efficient translocation.

The conductivity k is plotted as a function of stem lengthltrans in Fig. 4. The figure indicates that the conductivityin herbaceous plants scales with the height of the plant,and regression yields k lherb trans∝ ±0 97 0 10. . , (r2 = 0.90; N = 13;aRMA = 1.02 " 0.10). Angiosperm and gymnosperm treesshow similar scalings but tend to have relatively lower

conductivities compared with herbaceous plants. If wecompare within trees, we find that gymnosperms have lowerconductivities than angiosperm trees of similar height.

DISCUSSION

Optimum velocity scaling law

The fact that the scaling relationship between the structuralparameters of phloem transport, leaf length, stem lengthand SE radius is the same in gymnosperms as inangiosperms (Fig. 3) is a strong indication that gymno-sperms employ the same basic mechanism for phloemtransport. Active transport facilitation, which has beenhypothesized to contribute to phloem transport in trees(Lang 1979; Aikman 1980) and especially gymnosperms(Kollmann 1975; Liesche et al. 2011), would likely havealtered the scaling relationship. For example, for the relaymechanism proposed by Lang (Lang 1979), where thetranslocation pathway is split into shorter, hydraulically iso-lated segments, one would expect to find narrower SEs ifthe osmotic pumping mechanism was optimized in a similarway to that described in the present manuscript.

Ernst Münch explicitly included gymnosperms in hisproposition of the pressure-flow hypothesis, providing evi-dence for the validity of the mechanism in this plant groupby relating seasonal stem growth to sugar transport capacityin conifers (Münch 1930). Other authors excluded a contri-bution by energy-dependent transporters along the stem bycooling experiments (Watson 1980) and demonstrated thedirect correlation of leaf carbohydrates with sink activityas expected for a system driven by hydrostatic pressure

10−6 10−5

10−2

10−1

100

101

102

a [m]

Gl so

urce

l tra

ns [m

2 ]

Figure 3. Plot of Glsourceltrans as a function of the effectiveosmotic radius a. The plants are hydraulically optimized for rapidtranslocation in the phloem if the points fall on the solid blackline (slope 3), as predicted by Eqn 8. The dashed line is theregression curve obtained from least square regression analysis,lsourceltrans ∝ a2.7"0.10(r2 = 0.69; N = 70; aRMA = 3.3 " 0.2) Symbollegend: angiosperm trees (open circle), angiosperm herbs (blackdot), angiosperm shrubs (grey dot), gymnosperm trees (opensquare), gymnosperm trees with scales (open triangle).

10−1 100 101 102

10−13

10−12

10−11

ltrans [m]

k [m

2 ]

Figure 4 Plot of the conductivity k as a function of stem lengthltrans.The dashed line through the data points for herbaceous species isthe regression curve obtained from least square regressionanalysis, k lherb trans∝ ±0 97 0 10. . (r2 = 0.90; N = 13; aRMA = 1.02 " 0.10).Symbol legend: angiosperm trees (open circle), angiosperm herbs(black dot), angiosperm shrubs (grey dot), gymnosperm trees(open square), gymnosperm trees with scales (open triangle).



potential (Sevanto et al. 2003).The anatomical optimizationof gymnosperm phloem for rapid transport, shown here,corroborates the assumption that gymnosperms employ theMünch mechanism for phloem transport.

Although the experimental data support the optimalityhypothesis, a number of species divert from the generalscaling behaviour. The scaling pre-factor Lp (Eqn 7) is theproduct of the membrane permeability Lp and sugar solu-tion viscosity h which in the present analysis has beenassumed equal for all species. These may, however, varyslightly among species (Thompson & Holbrook 2003b), inpart explaining the vertical deviations from the scaling law.In addition to this, a subgroup of gymnosperms with scale-like leaves lies below the predicted scaling line and alsodiverts from the other gymnosperms in the following analy-sis (Figs 3–5). This might be due to our definition of lsource asthe anatomical unit of a leaf or needle which does notnecessarily correspond to the physiological unit. In case ofthe scale-like leaves, which are usually less than 2 mm long,the physiological unit might consist of several individualscales. However, for the sake of consistency we did notchange the definition of lsource for scale-like leaves in ouranalysis.

All data examined in the present paper were taken frommature plants. It is an open question whether plants arehydraulically optimized during growth, moving along thesolid black line in Fig. 3, or if they have the same SE radiusin all phases of growth, thus moving along a vertical axis.The authors will address this in a future publication.

Flow velocity and conductivity in the phloem

The resistor model introduced in the present paper pro-vides a framework for understanding many qualitative andquantitative features of long-distance phloem transportobserved in plants. For example, the specific flow conduc-tivity k (Eqn 8) was found to scale with plant height ask lherb trans∝ ±0 97 0 10. . (r2 = 0.90; N = 13; aRMA = 1.02 " 0.10), for her-baceous angiosperms. Another important result was thatthe conductivity was significantly lower in gymnospermtrees compared with angiosperm trees of similar height(Fig. 4). This may in part explain why the observed translo-cation speeds in gymnosperms are slower than inangiosperms. The velocity u calculated from the conductiv-ity k (see Eqn 8) is plotted as a function of stem length ltrnas

in Fig. 5. Here, we assume that the pressure drop Dp andviscosity h do not scale with plant height (Turgeon 2010),and that the membrane permeability Lp does not varyamong the species. Our calculations predict that herbaceousspecies translocate with speeds of about 1 m h-1,angiosperm trees with speeds in the range 0.1–1 m h-1 andgymnosperm trees in the range 0.01–1 m h-1, mostly inagreement with earlier experimental results (Crafts & Crisp1971;Thompson et al. 1979). Consistent with our findings forthe conductivity, the flow speed predicted in gymnospermsis significantly lower than in angiosperm trees of compa-rable height.

Our results indicate that the slower phloem translocationis the result of the different SE anatomy, that is, the smallereffective radius, which generally reduces the conductivity incomparison with angiosperms of similar height (see Fig. 4).The sieve plate resistance, not considered here, might havean additional negative effect on sap flow velocity because ofthe narrower pore structure (Kollmann 1975; Schulz 1992).Gymnosperms are, however, as efficient as angiosperms interms of utilizing the full potential of the osmotic Münchmechanism, as evidenced by the range of sieve elementradii developed in evolution.

The slow phloem transport in gymnosperms might beoffset by a larger number of SEs to accommodate a suffi-cient volume (Münch 1930; Schulz 1990) at least in matureconifers which show seasonal growth comparable toangiosperm trees (Münch 1930; Bond 1989). Aspects ofcarbon partitioning such as low diurnal variation in export(Yang et al. 2002; Bansal & Germino 2009) might contributeto the absolute transport volumes of gymnosperms. The bigcarbon reserves in all plant organs of gymnosperms guar-antee a sufficient supply even when transport is slow(Ericsson & Persson 1980; Cranswick, Rook & Zabkiewicz1987; Webb & Kilpatrick 1993).

SE cross-sectional area

The relation between plant height and stem SE cross-sectional area was found to be profoundly differentbetween herbaceous plants and trees (Fig. 2b). In herba-ceous plants, the SE area was found to scale with plantheight as A lherb trans∝ ±1 3 0 1. . (r2 = 0.90; N = 13; aRMA = 1.4 " 0.1).

10−1 100 101 102

10−5

10−4

10−3

ltrans [m]

m [m

s–1]

1[m h–1]

0.1[m h–1]

0.01[m h–1]

Figure 5. Plot of calculated flow velocity u determined as afunction of stem length ltrans determined from Eqn 1. In the plot,we assume that osmotic pressure Dp, viscosity h and membranepermeability Lp do not scale with plant height. The solid linesindicate the location of 0.01, 0.1 and 1 m h-1 levels. Parametersused are Lp = 5 ¥ 10-14 ms-1 Pa-1, h = 2 mPa and Dp= 0.7 MPa(Thompson & Holbrook 2003b; Turgeon 2010). Symbol legend:angiosperm trees (open circle), angiosperm herbs (black dot),angiosperm shrubs (grey dot), gymnosperm trees (open square),gymnosperm trees with scales (open triangle).



The SE area in trees was found to be mostly larger than inherbs, but limited to values around 103 mm2 and appears tobe independent of plant height in both angiosperm andgymnosperm species.

The SE cross-sectional area might be limited by the effec-tiveness of SE maintenance. Mature SEs of angiosperms arefunctionally dependent on ontogenetically related compan-ion cells, to which they are connected via specialized plas-modesmata. With larger SE diameter, the interface mightget too small for efficient turnover of proteins and lipidsfrom the companion cells. Gymnosperm SEs might well beeven more limited with respect to the maximal cross-sectional area, as the Strasburger cells, they are associatedwith, have a much smaller contact interface with SEs thanangiosperm companion cells have (Schulz 1990).

In spite of this similarity in SE area, we find that theconductivity k is significantly smaller in gymnosperms thanangiosperms of similar height. The reason for this is thatcuboidal cells, often found in gymnosperm trees, offer largerhydraulic resistance to flow than a cylindrical cell of equalcross-sectional area A. For a square cross section (ar = at)the increase is 13%, while for rectangular cross section withar = 1/2at, the ratio typically found in the data examined inthe present paper, the increase is 39%.

General

The central finding of this work is that both gymnospermand angiosperm plants are geometrically optimized forrapid translocation in the phloem and that the flow conduc-tivity is significantly lower in gymnosperms compared withangiosperms of similar height.

The results demonstrate universal optimization of thephloem in seed plants for a transport compatible with theMünch mechanism and contribute to our understanding ofcarbon allocation in trees, especially in gymnosperms.

ACKNOWLEDGMENTS

Imaging data were collected at the Center for AdvancedBioimaging (CAB) Denmark, University of Copenhagen.This work was supported by the Danish National ResearchFoundation, Grant No. 74.

REFERENCES

Aikman D.P. (1980) Contractile proteins and hypotheses concern-ing their role in phloem transport. Canadian Journal of Botany.Journal Canadien De Botanique 58, 826–832.

Bansal S. & Germino M.J. (2009) Temporal variation of nonstruc-tural carbohydrates in montane conifers: similarities and differ-ences among developmental stages, species and environmentalconditions. Tree Physiology 29, 559–568.

Becker P., Tyree M.T. & Tsuda M. (1999) Hydraulic conductancesof angiosperms versus conifers: similar transport sufficiency atthe whole-plant level. Tree Physiology 19, 445–452.

Bond W.J. (1989) The tortoise and the hare – ecology ofangiosperm dominance and gymnosperm persistence. BiologicalJournal of the Linnean Society. Linnean Society of London 36,227–249.

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Received 6 September 2011; received in revised form 20 November2011; accepted for publication 30 November 2011

APPENDIX A: MATHEMATICAL MODEL

Rectangular SEs

For rectangular SEs, we have for the resistances in Eqn 2that

Ra a L l

sourcer t p source

=+( )

14

, (A1)

Rla a n

aa

n aa

transtrans

r tn odd

r

t

t

r

= −

∑34

11 192

3 5 5

ηπ

π,

tanh

−

−

−

1

3

134

1 0 63! ηla a

aa

trans

r t

r

t

. ,

(A2)

Ra a L l

sinkr t p sink

=+( )

14

, (A3)

The error introduced by the approximation made in

Eqn A2 depends on the aspect ratio δ = aa

t

r

. For d = 1 it is

13%, while for d = 2 it is down to 0.2%, see, for example,Bruus (2008). Given the data given in Table 2, we find thatd is approximately constant and take on values in the ranged ! 1.5 - 2. The shape of the sieve elements depends on theshape of the cambial initials which also give rise to thexylem tracheary elements (Esau 1969; Carlquist 1975). Wespeculate that this ratio of cell size is influenced by physi-ological and mechanical constrains to the xylem cells. Inaddition, divisions during sieve element differentiation asseen in many angiosperm species do not seem to becommon in gymnosperms (Esau 1969).

With the expression for the velocity given in Eqns A1–A3, we write the flow velocity u, cf. Eqn 3, as

u a l l lpL

a l lWl

r source trans sink

p

r source sink

sour

, , ,( )+( )

=δδ∆ 1

2

cce trans sink r source sinkl l a l l+ +( )3,

(A4)

where W = 3Lph(1 + d)/(d - 0.63). As in Eqn 4, this has amaximum when

aWl l l

l lr

source trans sink

source sink

*3 2=+

(A5)

Under the assumption that lsource = lsink this corresponds to

aL

l lrp

source trans*3 3 10 63

= +( )−η δ

δ .(A6)

Defining the effective osmotic radius as a a= −+( ) =r

δδ

0 631

1 3. /

aa a

a ar

t r

r t

−+

0 63 1 3. /

, this can be written as

a L l l* p source trans3 3= η . (A7)



APPENDIX B: THE PERMEABILITY LENGTH LphIn the expression for the optimized radius (Eqn 8), theproduct Lph of the membrane permeability and viscosity ofthe phloem occurs. This is a microscopic length scale havingto do with the structure of the semipermeable membrane.Let us think of the membrane as consisting of N pores of ahydrodynamic radius r and length k. With Poiseuille flow,the volume flux Qacross a given area W of the membranedriven by a pressure difference Dp0 is then

Q N p= πρη κ

4

08 w

∆ (B1)

where hw = 10-3 Pa s is the viscosity of the water that pen-etrates the membrane.The number of pores N is taken to beproportional to the area W:

N = φπρΩ

2 (B2)

where f is the covering fraction, that is, the fraction of themembrane surface area covered by pores. Thus,

Q p L p= =φ ρη κ

Ω Ω2

0 08 w

p∆ ∆ (B3)

from which we see that

Lpw

η φ ηη

ρκ

φ ρκ

= ≈2 2

8 4(B4)

Taking Lp ª 5 ¥ 10-14 m s-1 Pa-1 and h ª 2 ¥ 10-3 Pa s =2hw

we get Lph ª 10-16 m.If we use typical values for the membrane thickness

k ª 5 nm = 5 ¥ 10-9 m and the pore radii r ª 2 Å =2 ¥ 10-10 m, we must take f ª 0.5 ¥ 10-4.

Of course, for these atomic length scales the estimateusing Poiseuille flow is invalid. Instead, we can compare tothe estimates in the literature for the permeability of singleaquaporins. In Nielsen (2010), a typical value of the perme-ability coefficient for single channel aquaporin is given aspf ª 10-14 cm3 s-1 = 10-20 m3 s-1. This permeability can bewritten as

pRTV

Lfw

p

m Pa m s Pa

=

= × × × × × ×− − − −

πρ φ

φ π

2

20 2 8 14 1 14 10 1 3 10 5 10.

≈≈ × − −φ 10 24 3 1m s

(B5)

This is not far from f ª 10-4 as obtained above, showing thatour estimate (B4) actually has the right order of magnitude.



universality of phloem transport in seed plants

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