Dynamics of leaf gas exchange, xylem and phloem transport ...

PART OF A SPECIAL ISSUE ON FUNCTIONAL–STRUCTURAL PLANT MODELLING

Dynamics of leaf gas exchange, xylem and phloem transport, water potentialand carbohydrate concentration in a realistic 3-D model tree crown

Eero Nikinmaa1,*, Risto Sievanen2 and Teemu Holtta1

1Department of Forest Sciences, University of Helsinki, PO Box 27, Helsinki, 00014, Finlandand 2Finnish Forest Research Institute, Vantaa Research Unit, PO Box 18, Vantaa 01301, Finland

* For correspondence. E-mail [email protected]

Received: 9 December 2013 Returned for revision: 12 February 2014 Accepted: 12 March 2014

† Background and Aims Tree models simulate productivity using general gas exchange responses and structuralrelationships, but they rarely check whether leaf gas exchange and resulting water and assimilate transport anddriving pressure gradients remain within acceptable physical boundaries. This study presents an implementationof the cohesion–tension theory of xylem transport and the Munch hypothesis of phloem transport in a realistic3-D tree structure and assesses the gas exchange and transport dynamics.† Methods A mechanistic model of xylem and phloem transport was used, together with a tested leaf assimilation andtranspiration model in a realistic tree architecture to simulate leaf gas exchange and water and carbohydrate transportwithin an 8-year-old Scots pine tree. The model solved the dynamics of the amounts of water and sucrose solute in thexylem, cambium and phloem using a fine-grained mesh with a system of coupled ordinary differential equations.† Key Results The simulations predicted the observed patterns of pressure gradients and sugarconcentration. Diurnalvariation of environmental conditions influenced tree-level gradients in turgor pressure and sugar concentration,which are important drivers of carbon allocation. The results and between-shoot variation were sensitive to structuraland functional parameters such as tree-level scaling of conduit size and phloem unloading.† Conclusions Linking whole-tree-level water and assimilate transport, gas exchange and sink activity opens a newavenue for plant studies, as features that are difficult to measure can be studied dynamically with the model. Tree-levelresponses to local and external conditions can be tested, thus making the approach described here a good test-benchfor studies of whole-tree physiology.

Key words: Functional–structural plant modelling, functional–structural plant models, Scots pine, Pinus sylvestris,long-distance transport, turgor, xylem tension, unloading, structure, LIGNUM, phloem transport, 3-D model, treecrown, canopy gas exchange.

INTRODUCTION

The development of a tree crown is one of the principal means bywhich trees respond to their growing environment (Nikinmaa,1992). Plasticity of crown development determines the effi-ciency of light capture (Posada et al., 2012), nitrogen use(Mooney and Gulmon, 1979; Field, 1983) and tree-level waterrelations (Zimmermann, 1983). The connection between treestems and branches and their water transporting role has intri-gued scientists for hundreds of years [see Leonardo’s notes(MacCurdy, 2002)]. The study of regularities between woodyaxes and their functional role has a long history in botany [seePressler (1865) and Jaccard (1913), cited by Zimmermann,1983; Huber, 1928], and this line of thought has led to generaltheories about the build-up of trees (Shinozaki et al., 1964;West et al., 1999). The hydraulic limitation of tree size (Ryanand Yoder, 1997; Koch et al., 2004) has been considered as thefundamental boundary condition for the development andscaling of water-conducting tissue within a tree (West et al.,1999). Also, hydraulic architecture, which determines how tran-spiration translates into changes in water potential, has been con-sidered to be a fundamental design property of crown patterns(Zimmermann, 1983; Nikinmaa et al., 2003).

Tree development results from a complex interaction betweencarbohydrate sources and sinks that is mediated by the conduct-ive properties of tissue responsible for long-distance transport.Growth of new tissue requires assimilated carbohydrates forcell wall synthesis, but these compounds also have a role increating sufficient turgor pressure for cell wall extension(Holtta et al., 2010; Pantin et al., 2012). Transpiration of treecrowns is directly connected to hydraulic conductivity fromsoil to transpiring leaves. Assimilate transport in phloem isclosely connected to xylem transport (Ferrier et al., 1975;Boersma et al., 1991; Holtta et al., 2006; Hall and Minchin,2013). Low water potential in leaves will slow down phloemtransport or cause even a transient reversal of flow (Holttaet al., 2006). Together, the magnitude of negative pressure inxylem and the osmotic strength that assimilated sugars andother osmolites maintain in growing tissue determine theextent to which new tissue can expand (Holtta et al., 2010;Pantin et al., 2012). Because of these linkages, growth of treecrowns follows from an interesting interaction between existingwoody structure, its transport properties and the gas exchangebehaviour of leaves within the crown.

The environmental variables that drive photosynthesis andtranspiration vary a lot within tree crowns and canopies,

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influencing the attainable photosynthetic and transpiration rates.Leaf stomata, which control gas exchange with the exterior, havewell known responses to the main environmental variables: light,vapour pressure deficit and temperature (Landsberg, 1986).Stomata are also responsive to the water-conducting pathwaysof trees and soil water status (Buckley, 2005; Duursma et al.,2008). The optimal water use scheme describes well the environ-mental responses of leaf gas exchange (e.g. Hari and Makela,2003) and is compatible with the currently most widely usedgas exchange models (Medlyn et al., 2011). Nikinmaa et al.(2013) recently showed that such behaviour arises if treesattempt to control their leaf gas exchange to maximize assimi-late export from leaves. These findings thus suggest that gasexchange is linked to carbohydrate sink activity, enhancingthe links between growth, structure, transport and canopy gasexchange.

Tree crown architecture is responsive to even the speciesidentity of the tree’s neighbours (Lintunen and Kaitaniemi,2010), although the exact mechanisms behind such behaviourare not known. Functional–structural plant models have beendeveloped for describing the plasticity of crown shape in re-sponse to variation in the growing environment. In thesemodels the development of the tree crown results from initiation,extension and thickening of individual apical meristems and latershoots (Sievanen et al., 2000). These developmental processesdepend on the local environment, the position within the treetopology and often also on the availability of resources, suchas carbon and nitrogen, for example at an annual level(Prusinkiewicz and Lindermayer, 1990). The success of thesemodels depends on how well they predict resource capture andits dependency on environmental conditions and how wellgrowth rules, which are often determined empirically, capturethe growth process. Although connections seem to existbetween gas exchange, structure, transport and growth, veryfew models consider the interactions in resource capture andsolve growth as a local process in which growth results fromthe local conditions and resource transport from sources (Allenet al., 2005).

The linkages between leaf gas exchange, tree architecture andvascular structure and transport set strong boundary conditionsfor possible process rates within a given structure that have notbeen fully exploited in plant studies. For example, cavitation inxylem and viscosity build-up in phloem are feedforward-typeprocesses that reduce the hydraulic conductivity of the xylemand phloem, setting limits to maximal transport and gas ex-change rates for a given structure (Holtta and Nikinmaa, 2013).Although modelled separately in a number of papers, here wecombine for the first time the implementation of the cohesion–tension theory of xylem transport and the Munch hypothesis ofphloem transport and a tested leaf gas exchange model in arealistic 3-D tree structure and study the resulting water and as-similate transport. This makes it possible to study the distributionof water tension, turgor pressure and osmotic concentrationwithin the different crown axes in more detail than with previousmodels (e.g. Nikinmaa et al., 2013), because we can now makeassumptions about structural properties and driving gas ex-change rates that are as realistic as possible and can also testthese rates against observed patterns. We further study how thedifferent aspects of within-crown transport are influenced byvariation in xylem and phloem properties and sink strength and

discuss the significance of the results for functional–structuraltree modelling.

MATERIALS AND METHODS

We combined the xylem and phloem transport model of Holttaet al. (2006) and the assimilation and transpiration model SPPof Makela et al. (2006) with the 3-D tree model for Scots pine(Pinus sylvestris) trees (Sievanen et al., 2008) in the LIGNUMmodelling framework (Sievanen et al., 2010). The xylem andphloem transport model deals with the dynamics of water andsucrose transport and associated axial and radial concentrationand pressure gradients in and between xylem and phloem. TheSPP model evaluates the rates of photosynthesis of a Scotspine shoot as a function of radiation [photosynthetically activeradiation (PAR)], temperature and saturation deficit of watervapour. LIGNUM provides the tree hydraulic architecture andevaluates the environmental conditions of individual shoots. InLIGNUM, Scots pine trees consist of internodes that containheartwood, sapwood and possibly needles. For this applicationwe also consider the phloem layer [constant thickness 2 mm(Holtta et al., 2013)] (Fig. 1A). The transport model (Holttaet al., 2006) considers the amounts of water and sucrose solutein xylem, cambium and phloem using afine-grained (in compari-son with the size of the tree) mesh and solves their dynamics as asystem of coupled ordinary differential equations. We use thisapproach to suit tree level analyses and formulate the transportmodel as ordinary differential equations of water pressure in

SapwoodHeartwood

Phloem

FI G. 1. (A) An internode in LIGNUM consisting of heartwood (brown),sapwood (yellow), phloem (green) and needles. The internode and sizes of heart-wood and sapwood were derived as the result of growth processes in LIGNUM(e.g. pipe model and senescence of sapwood) and the thickness of phloem is2 mm. (B) The sapling tree used in the calculations was grown by LIGNUM(Sievanen et al., 2008) in southern Finland under medium site fertility conditions;stand density decreased from 15 000 1/ha to 7000 1/h during the 8-year simula-tion. Height of the tree is 2.37 m, diameter at base 3.4 cm, heartwood diameter1.1 cm and number of internodes 295. Needle area of internodes decreaseswith age; 1-, 2- and 3-year-old internodes have 98, 86 and 16 % of initialneedle area, respectively; all needles are lost at age 4; 36 internodes have noneedles. The total needle area (all sides) is 7.1 m2. The parts of the tree for

which results are shown (stem, branch, side branch) are shown with arrows.

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xylem sapwood (Px) and phloem (Pp) and the amount of sucrosesolute in phloem (Np) for each internode as follows:

dPx

dt= ax(Qx,in − Qx,out + Qxp − QT)

dPp

dt= ap(Qp,in − Qp,out − Qxp)

dNp

dt= (Qp,in − Qp,out)c(Np) + QP − QU

where Qx,in, Qx,out and Qp,in, Qp,out are the axial inflow andoutflow rates of water in xylem and phloem, Qxp is the flowrate of water from phloem to xylem in the internode, QT is thesink of water depending on transpiration rate of the internode,QP is the rate of loading of sucrose caused by photosynthesis inthe internode, QU is the rate of unloading of sucrose in the inter-node, c(Np) is the sucrose concentration of phloem sap, and ap

and ax are coefficients dependent on the volume and elasticmodulus of the internodes. The internodes are coupled to otherinternodes through the axial flows. The model of xylem andphloem transport in LIGNUM is thus a system of coupledordinary differential equations with 3 × (number of internodes)variables. The ordinary differential equations for Px and Pp arebased on Darcy’s law (Siau, 1984) and that for Np is accordingto the Munch hypothesis (as formulated in Holtta et al., 2006).We solved the set of ordinary differential equations with theRosenbrock stiffly stable method using automatic step size ad-justment (numerical recipes for C++). A more detailed math-ematical definition of the model and a list of parameter valuesare given in the Appendix.

We performed the calculations with a simulated (grown byLIGNUM, Sievanen et al., 2008), 2.37-m tall, 8-year-old Scotspine tree (Fig. 1B) whose structural (permeability) and function-al (sink strength) properties were consequently changed to studytheir influence on flow rates and pressure gradients. The incom-ing radiation for shoots (i.e. internodes) was calculated using thebackward ray tracing method in LIGNUM (Sievanen et al., 2008)considering only self-shading, the radiation distribution of thesky being that of an overcast day. The tree was thus assumed bein a rather sparse sapling stand.

We first ran the model to steady state with mean values ofenvironmental variables for normal summer day conditions forsouthern Finland (Fig. 2; base case simulations are shown inFigs 3–7). The rates of photosynthesis and transpiration andtheir responses to environmental conditions (e.g. light and tem-perature) were from field observations at our research forest(Kolari et al., 2009). Environmental drivers in the steady-stateanalyses were as follows (Fig. 2): radiation equalled 600mmol(PAR) m– 2 s– 1 on a horizontal surface, temperature was20 8C and saturation deficit of water vapour was 0.6 mol(H2O)m– 3. Xylem sapwood thicknesses were according to theLIGNUM parameters for Scots pine (Sievanen et al., 2008)and phloem thickness was assumed to be a constant 2 mm(Holtta et al., 2013). The axial permeabilities of xylem andphloem increased basipetally in proportion to the square rootof distance from the apex. This pattern stems from conduit taper-ing as described in West et al. (1999) and conduit packing densityin Savage et al. (2010) (see Appendix). We also tried scaling

relative to branch and stem diameters (Savage et al., 2010), butthese yielded very similar results (not shown) since stem andbranch diameters scaled with distance from the leaf apex in oursimulated tree. The reference permeability at the tree base forxylem and phloem was taken from the literature. In these simula-tions we assumed that the sinkof sugars occurs only in internodesthat do not have needles, to approximate a situation after shootgrowth has terminated and the main carbohydrate sink is in thelower part of the tree. At the base we assumed a zero-flow bound-ary condition. The carbon sink of the lowest stem internode was10-fold higher than that of the rest of the internodes to mimic thecarbohydrate sink of the below ground parts.

10

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(°C

)

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C

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PA

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defic

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wat

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apou

r (m

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FI G. 2. Daily courses of (A) radiation, (B) temperature and (C) saturation deficitof water vapour used in the calculations. Data are the from SMEARII stationat theHyytiala Forestry Field Station of UniversityofHelsinki (Hari andKulmala, 2005)(sunny early summer conditions). Daily mean values (used in the steady-state ana-lyses) are 600 mmol(PAR) m–2 s–1, 20 8C and 0.6 mol(H2O) m–3 for radiation,

temperature and saturation deficit of water vapour, respectively.

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In the simulations, soil water pressure was an input to themodel; the xylem of the base internode was connected to soilwith the same resistance as to the internode above. Soil waterpressure was set at –0.35 MPa in all simulations.

The incoming radiation (from all directions) to shoots withneedles varied between 23 and 907 mmol(PAR) m–2 s–1 insteady-state conditions. The photosynthetic and transpirationrates per unit needle area varied within the ranges 0.26–3.7mmol(CO2)m

–2 s–1 and72–462mmol(H2O)m–2 s–1 respectively.The model sensitivity was tested with respect to xylem and

phloem permeabilities (Figs 8–10), the axial change in perme-ability according to West et al. (1999) versus keeping it constant(Fig. 11) and against different formulations of sink strength inthe stem segments (Fig. 12). As the literature values, particularlyfor phloem permeabilities, are quite uncertain, large variation

around the base case values were used by increasing the perme-abilities of xylem and phloem, first alone and then both at thesame time by 10 % and by 10-fold, and decreasing the permeabil-ities in a similar fashion by 10–90 %. To simulate the effect ofdifferent patterns of canopy sink strength on the modeloutcome, we assumed in the base case that the sink of sugarsoccurred only in internodes that did not have needles, to approxi-mate the situation after shoot growth has terminated and the maincarbohydrate sink is in the lower part of the tree. To analysemodel behaviour during the early growing season, when shootsare actively growing, we assumed two different sink formula-tions: (1) when the sink strength would be directly proportionalto carbohydrate concentration; and (2) when sink strengthwould be subject to rate limitation, such that, if sugar concentra-tion was in the range 0.0–0.85 mol L– 1, unloading was propor-tional to sugar concentration and concentration values.0.85 mol L– 1 did not further increase unloading.

−1·5

−1·0

−0·5

0StemBranchSide branch

Xyl

em w

ater

pre

ssur

e (M

Pa)

A

1·7

1·8

1·9

2·0

2·1

Phl

oem

wat

er p

ress

ure

(MP

a)

B

0 0·5 1·0 1·5 2·0 2·50·5

1·0

1·5

Distance from base (m)

Suc

rose

con

c. in

phl

oem

(m

ol L

–1) C

FI G. 3. Steady-state profiles of water pressure in xylem (A) and phloem (B) andsucrose concentration in phloem (C) in different parts of the tree: stem, branch and

side branch (as indicated in the key).

0

0·04

0·08

0·12

Wat

er fl

ow r

ate

(L h

–1)

A

0 0·5 1·0 1·5 2·0 2·5

0

0·2

0·4

0·6

0·8


Sug

ar fl

ow r

ate

(g h

–1)

B

FI G. 4. Steady-state vertical distribution of flow rate of (A) water (upwards) inxylem and (B) sugars (downwards) in phloem. The flows are displayed at both thetop and the bottom of the internodes. Step changes are due to the contributions of

branches to flow.




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RESULTS

The steady-state xylem water pressure was evenly � –1.4 MPaat the tips of the stem, branches and side branches in the basecase (Fig. 3A). There was an abrupt change in the main axis pres-sure at �1.4 m height where the needle-bearing internodesstopped. The water pressure gradient was somewhat steeper inbranches and side branches in comparison with the main stem,in accordance with the steeper permeability change there. Theturgor pressure at the leader shoot was slightly greater than2 MPa and the pressure gradient between the top and the basewas only �0.1 MPa m– 1, which was about one-fifth of that inxylem but in the opposite direction (note that we did not considerthe influence of gravitation because it had only a minor effect onthe results for the small tree simulated here; Fig. 3B). There was aclear difference between the phloem water pressure between the

leader and the side branches, consequently with a steeper pres-sure gradient in the leader. This also means a considerablylower assimilate flow per leaf area downwards from the sidebranches, as theyall had approximately the same sucrose concen-tration (Fig. 3C). Xylem pressure and phloem sucrose concentra-tion behaved like mirror images in these simulations, as theosmotic strength of the phloem sap balanced the xylem waterpotential to maintain phloem turgor pressure, and the onlyosmotic substance included in the model was sucrose.Consequently, in the steady-state situation assimilates partici-pate both in maintaining the phloem water potential balancewith the xylem by osmosis and phloem transport.

Vertically, the largest drop in the stem sap flow rate occurred inthe upper middle part of the crown, where the bulk of the foliageis (Fig. 4A). The main axis sugar flow increased in the upper half

−1·5

−1·2

−0·9

−0·6

−0·3

0X

ylem

wat

er p

ress

ure

(MP

a)A

0 0·5 1·0 1·5 2·0 2·5

1·7

1·8

1·9

2·0

2·1


Phl

oem

wat

er p

ress

ure

(MP

a)

B

FI G. 5. Vertical distribution of (A) xylem and (B) phloem pressures at branchapices in the steady state. Apices belonging to different branch whorls are indi-cated with colours. The colours are, from the lowest branch whorl to the topone (also includes the leader shoot), in the order red, blue, magenta, green,

brown, cyan and black.

−1·5

−1·0

−0·5

0

Xyl

em w

ater

pre

ssur

e (M

Pa)

A

1·0

1·4

1·8

Phl

oem

wat

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ress

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(MP

a)

A

B

C

0 5 10 15

0·6

1·0

1·4

Time (d)

Suc

rose

con

c. in

phl

oem

(m

ol L

–1)

StemApex shootsBase shoots

BranchSide branch

FI G. 6. Dynamics of water pressure in xylem (A) and phloem (B) and of sucroseconcentration in phloem (C) in the base and apex shoots of the stem, branch and

side branch (see key in B) from the initial state to the steady state.




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of the crowns, along the needle-bearing main axis internodes andwith steps as the branches joined the main stem (Fig. 4B). Belowthe needle-bearing part, the flow rate started to decrease due toour assumption of a carbon sink in the non-needle bearinginternodes only. In this part also, the lower whorls of branchesslowed down transport in the main axis, which means that theywere consuming carbon from the main stem.

The xylem water pressure varied roughly between 1.2 and1.4 MPa in the branch tips attached to the samewhorl of branchesin the upper section of the crown (Fig. 5A). Phloem turgorseemed to be much more constant in the same whorl of branchesor even in branches at adjacent whorls but there were abrupt step-

like changes in the pressure that were caused by discretization ofthe tree to finite internodes in the numerical solution.

The model reached steady state quickly for xylem pressure andwater transport (Fig. 6A) but almost 5 d elapsed before asteady-state phloem pressure was attained (Fig. 6B). The longtime taken to reach steady state depended partially on the largedifference between the initial state of the tree and the steady-statecondition. The slow development of phloem transport was due tothe need to reach a steady-state sugar concentration in each inter-node, which required that sugars were transported there fromtheir sources. The overall sugar content of phloem is large incomparison with the assimilation rate. In addition, phloemtissue is more elastic than xylem, which also contributes toslower time constants.

−2·0

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Xyl

em w

ater

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e (M

Pa)

1·4

1·8

2·2

2·6

Phl

oem

wat

er p

ress

ure

(MP

a)

0 0·5 1·0 1·5 2·0 2·5 3·00·8

1·0

1·2

1·4

Time (d)

Suc

rose

con

c. in

phl

oem

(m

ol L

–1) C

B

A

StemBase of main stemTop of main stem

BranchSide branch

FI G. 7. Daily pattern of (A) xylem and (B) phloem pressure and (C) sugarconcentrations at the top and the base of main stem, branch and side branch

(see key in C).

−100

−50

0

50

Px

rela

tive

chan

ge (

%)

A

−100

−50

0

50

Pp

rela

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%)

B

0 0·5 1·0 1·5 2·0 2·5−100

−50

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50


Suc

rose

rel

ativ

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ange

(%

) C

StemBranchSide branch

FI G. 8. Effect of halving xylem and phloem permeabilities. Shown are relativechanges in comparison with the base case of Fig. 3 [(value – base case)/base case]in steady-state values of water pressure in xylem (A) and phloem (B) and sucroseconcentration in phloem (B) in different parts of the tree (stem, branch and side

branch; see key) versus distance from tree base.




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When simulated over several days using the measured vari-ation in light, vapour pressure deficit and temperature (Fig. 2) aclear daily pattern was established (Fig. 7). Xylem pressuredropped rapidly in the morning in the top internodes as transpir-ation increased. The lower internodes in branches and the treebase followed almost simultaneously with the pressure drop,but not as steeply. The recovery of pressure in the evening wasnot quite as symmetrical in comparison with the morning. Thepressure gradually increased throughout the night until itstarted decreasing again the next morning (Fig. 7A). Phloemturgor pressure followed a quite symmetrical daily pattern atthe tree base (Fig. 7B). Turgor pressure at the top quickly

reached its maximum in the evening as transpiration and photo-synthesis stopped. However, it started to decline slowly asphloem transported sugars towards the base, down the steepnight-time turgor pressure gradient. It is interesting that duringthe day the turgor pressure dropped so much at the top that itfell below the turgor pressure at the base, meaning that phloemtransport reversed (Fig. 7B). The pressures in the simulationsbehaved similarly in all of the branch internodes, regardless ofwhether they were at the tip or at the base of the branch, andthey were distinct from the values for the tree base internode.However, the sugar concentrations at the base of a side branchshowed a pattern similar to that at the base of the tree. Thesugar concentration in the top branches followed the dailypattern of photosynthesis but with a delay of �6 h, so that the

−50

50

100

Px

rela

tive

chan

ge (

%)

A

−50

0

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50

100

Pp

rela

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%)

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0 0·5 1·0 1·5 2·0 2·5−50

0

50

100


Suc

rose

rel

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ange

(%

) C


FI G. 10. Effect of increasing xylem and phloem permeabilities 10-fold. Shownare relative changes in comparison with the base case of Fig. 3 [(value – basecase)/base case] of steady-state values of water pressure in xylem (A) andphloem (B) and sucrose concentration in phloem (B) in different parts of the

tree: stem, branch and side branch (see key) versus distance from tree base.

0

1

2

3

4

5

6P

hloe

m w

ater

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e (M

Pa)

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Phl

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wat

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(MP

a)

C

FI G. 9. Steady-state distribution of phloem pressure in the branch apices when(A) xylem and phloem permeabilities were 10 % of the base case, (B) xylem per-meability was 10 % and the phloem was at the base case value, and (C) phloempermeability was 10 % and the xylem was at the base case value (cf. Fig. 3).Apices belonging to different branch whorls are indicated with colours. Thecolours are, from the lowest branch whorl to the top one (includes also the

leader shoot), in the order red, blue, magenta, green, brown, cyan and black.




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minimum value occurred at about sunrise and the maximal valuein the late afternoon to early evening (Fig. 7C). The pattern in thelower internodes was almost opposite, such that the maximumconcentration was early in the morning and the minimum con-centration in the late afternoon to evening.

The results thus show that the daily dynamics of phloem sugartransport were influenced much more strongly by the dynamicsof transpiration [which followed the daily pattern of vapour pres-sure deficit (Fig. 2)] rather than by the dynamics of photosyn-thesis [which followed the daily pattern of incident radiation(Fig. 2)]; the phloem transport rate decreased during the dayand was highest during the night. Increasing transpirationincreased the xylem water potential gradient, which in turndecreased, and even temporarily reversed, the phloem turgor

pressure gradient. This trend in the dynamics of phloem transportratewas also responsible for the simultaneous increase in phloemsugar concentration in the branch and decrease in the tree baseduring the day (Fig. 7C).

We further simulated the model’s sensitivity to xylem andphloem permeabilities and sink activity. When xylem andphloem permeabilities were decreased by 50 %, the xylem pres-sure at the top internodes decreased almost proportionally, as thetranspiration rate was not assumed to be influenced by the treewater potential in the model of Makela et al. (2006) (Fig. 8A).Also, the gradient of phloem pressure increased in proportionto permeability change (approximately doubled with a 50 % de-crease in permeability), although the change in absolute pressurewas small (Fig. 8B). The change in the phloem pressure gradientwas mainly due to the decline in base pressure while the sugarconcentration gradient increased, mainly due to the increase insucrose concentration at the top (Fig. 8C). Larger permeabilitychanges [up to one-tenth of the base case value (not shown)]caused disproportionally large phloem pressure changes, indi-cating the influence of increasing sap viscosity with high sugarconcentrations.

The relative xylem pressure between the shoots with a 10 %change in permeability of both xylem and phloem behaved asin the base case, but the sugar concentration showed more vari-ation between shoots at different positions. In particular, thephloem pressure was very variable between shoots, indicatinga new type of distribution of phloem transport (Fig. 9A). Thechange in this pattern was mainly caused by the change inxylem permeability, as similar variation between shoots,although at very different absolute values, was observed whenonly the xylem permeability was decreased to 10 % of the basecase value (Fig. 9B). The pattern was closer to the base case,although at very different pressure values, when only phloempermeability was lowered (Fig. 9C). When phloem permeabilitywas decreased alone to one-tenth of the base case value (resultsnot shown), it did not change xylem pressure from the base casevalue. However, phloem pressure became almost 10-fold higherwhen it was only 2.5-fold higher relative to the base case when

0 0·5 1·0 1·5 2·0 2·50·6

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FI G. 12. Steady-state profiles of water pressure in phloem in different parts ofthe tree: stem, branch and side branch (see key) with different unloading formula-tions. Solid line, base case; dashed line, unloading in all internodes; dotted line,unloading in all internodes but maximum rate was limited. Rate limitation: ifsugar concentration was in the range 0.8–0.85 mol L– 1 the rate of unloadingwas proportional to sugar concentration; concentration values .0.85 mol L– 1

did not increase unloading.

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FI G. 11. Effect of having constant xylem and phloem permeabilities throughoutthe tree. Shown are relative changes in comparison with the base cases of Figs 3and 5 [(value – base case)/base case] in steady-state values of water pressure inxylem versus distance from tree base. (A) Changes in different parts of the tree:stem, branch and side branch (see key). (B) Changes at branch apices, from thelowest branch whorl to the top one (includes also the leader shoot), are in the

order red, blue, magenta, green, brown, cyan and black.




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both permeabilities were dropped to one-tenth of the base casevalue. This would mean that phloem transport was effectivelyblocked and high xylem pressures allowed pressure build-up inphloem. When xylem permeability was decreased alone,xylem pressures dropped to the same values as those seenwhen both permeabilities were dropped, but phloem pressureattained lower values in comparison with the base case and thesugar concentration remained at somewhat lower values thanwhen both permeabilities were dropped simultaneously.

The 10-fold increase in xylem and phloem permeabilitiescaused proportional changes in gradients of xylem and phloempressure and phloem sugar concentration. There was a clearchange in absolute pressure in xylem but rather small changes inphloem absolute pressure and sucrose concentration, the largestchanges in xylem pressure and sugar concentration occurring atthe tree top and in phloem pressure at the tree base (Fig. 10).

We also tested how the pressure and sugar gradients would beinfluenced if there was no change in axial permeability from thebase to the tip, i.e. contrary to the formulation of West et al.(1999) (Fig. 11). Xylem pressure increased towards the tree topin comparison with the base case, making the gradient frombase to top smaller and the difference between the main axisand side branches larger, emphasizing the influence of transportdistance.

The model behaviour was very sensitive to the formulation ofsink activity in the internodes. The base case results were calcu-lated assuming that the sink was only in non-needle bearinginternodes and was proportional to internode sugarconcentration(see Appendix). If a sink was also assumed in the needle-bearinginternodes, the phloem turgor gradient could temporarily invertfrom top-down to base-up, depending on the assumed unloadingspeed (Fig. 12).

DISCUSSION

This is the first time that xylem and phloem transport have beenmodelled explicitly, driven by the observed level of foliar gasexchange within a realistic 3-D branching architecture. Themodel was parameterized to represent a Scots pine tree inboreal conditions, and gas exchange rate parameters and valuesfor meteorological conditions were both derived from measure-ments taken at our field station. The structural parameters werethose used for the LIGNUM model and are based on field mea-surements in the same region (Sievanen et al., 2008). The onlynew structural parameter value was the thickness of phloemtissue, which was also measured in the same region (Holttaet al., 2013). We did not conduct any new measurements ofaxial and radial permeabilities but used a scaling exponentform the metabolic scaling theory (West et al., 1999) andchose the multipliers to give values within the published range(e.g. Genard et al., 2001; Zwieniecki et al., 2001; De Schepperand Steppe, 2010). Therewas considerable uncertainty regardingaxial phloem permeability and the radial permeability betweenxylem and phloem. Despite this uncertainty, the model gaveresults consistent with observations. The steady-state xylempressure was � –1.4 MPa at the branch tips in the upper crownunder overcast conditions (when the daily pattern was simu-lated), and the lowest values were slightly below –2 MPa.Scots pine follows the isohydric pattern of water use andminimum water pressures drop to � –2.5 MPa (Holtta et al.,

2005). In these measurements, the needle water potentials inthe light crowns of Scots pine varied mostly between –1 and–2 MPa. There is relatively little information on variation inturgor pressure, but it has been found to vary around 1.5 MPa(Sovonick-Dunford et al., 1981; Turgeon, 2010), which wasalso observed in our simulations. The sugar concentrationsvaried between 0.9 and 1.5 mol L– 1, which are also realisticbut somewhat high values (Sovonick-Dunford et al., 1981).These values were also reached in steady-state simulationsusing observed gas exchange rates, which implies that realisticoverall transport rates in xylem and phloem were obtained withrealistic pressure and sugar gradients within the tree. As thesecomparisons show, combining models of structure, transportand gas exchange allow concrete case studies of whole-treephysiology that have not been possible previously.

The simulations indicated that shoot tip water potentialswere similar in the light crown (varying between –1.2 and–1.4 MPa), independently of their position within the crown,which is in agreement with earlier observations of the hydraulicarchitecture of tree crowns: leaf water potentials tend to besimilar in the branch tips in different parts of the crown(Zimmermann, 1983). The water potential at the transpiring siteresults from the soil water potential, transpiration rate and xylempathway conductivity to that spot. Studies have shown that thereare large differences in leaf specific conductance between thethicker and more vigorous main stem and the more slender sidebranches (Zimmermann, 1983). This type of trend arises fromthe regular pattern of cell size as a function of distance from thestem apices (West et al., 1999; Anfodillo et al., 2006), and itseems that the trend as a function of axis thickness is not toodifferent between the main stem and side branches. Here weassumed that each branch tip had similar sapwood conductivity,which increased along the axis relative to the square root of the dis-tance from the tip, according to West et al. (1999) (see Appendix).The change was steeper the shorter the branch, as the base value atthe forking point gave the same value as the main stem. Thisseemed to result in realistic behaviour, as in the simulation inwhich we assumed constant permeability the longer main axishad lower water potential than the side branches and the shootwater pressure seemed to be influenced only by transport distance.

We assumed a constant leaf area:sapwood area relationshipthroughout the tree, as suggested by the pipe model (Shinozakiet al., 1964). In real trees there are differences between branchingorders and along the crown (Berninger and Nikinmaa, 1994) thatcould influence the results. Over the whole crown there were largedifferences in shoot water potentials, particularly between thelower and upper crown. Here, the photosynthesis and transpirationrates were determined by the environmental conditions of theshoots. In tree crowns, however, they also depend on the waterstatus of other parts of the crown (Whitehead et al., 1996). Werecently showed that stomatal conductance also depends on thefactors influencing xylem and phloem transport to and from ashoot in addition to external shoot conditions (Nikinmaa et al.,2013), which was not considered here. In principle, such feedbackcould further balance the pressure differences between the shoots.The main new result of this modelling exercise is that now we areable to study the interaction between tree hydraulic architectureand its environment with respect to tree behaviour. In these simu-lations we used only one standard environment, but this could beeasily varied according to measurements or, for example, by




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simulating a forest consisting of many individual trees (Sievanenet al., 2008).

The daily pattern of xylem pressure was similar to thatobserved previously (Peramaki et al., 2005). In the presentstudy, the asymmetrical pattern in which pressure decreasesquickly in the morning in response to the rise in transpirationbut increases more gradually in the evening was observed in avery small tree. In an earlier study by Peramaki et al. (2005)this type of behaviour was seen even more clearly as it was con-ducted on larger trees (.25 m tall) and was assumed to resultfrom the capacitive influence of gradual pressure release andassociated diameter change that propagates through a largestem volume and differences between axial and radial permeabil-ities. Here, this type of delay effect was smaller due to the smallertree size, and asymmetry was partly due to asymmetry in dailytemperature versus the radiation pattern, but we also consideredexplicitly the influence of phloem transport, which probably con-tributed to the asymmetry.

There is considerably less information on the daily variation inphloem turgor pressure and sap sugar concentration. Mencucciniet al. (2013) reported estimates that were derived from observedvariation in the above- and below-bark diameters of treesgrowing in the same stand as that from which we obtained ourtree parameters and climate values. No exact match with thetree-top values could be expected as our simulated tree was a2.5 m tall sapling while the measured trees in Mencucciniet al. (2013) were 15 m tall and the upper measurement wasseveral metres below the tree top. Overall, the trends weresimilar in that the minimum phloem pressure took place at thebase some time after the xylem minimum and also in that thedaily amplitude of pressure change was much lower at the basethan higher up in the stem. The sugar concentration behaved ina similar manner in that it showed almost an opposite daily vari-ation between the top, where the maximum concentration oc-curred during the daytime, and the base, where there was aminimum at that time. However, the maximum sugar concentra-tion occurred in the late afternoon in our simulations but a shorttime before noon in the observations. Differences of this kind ofare not unexpected as our formulation for sink activity was onlyfor needleless internodes and was quite crude. In this study themaximum sugar concentration occurred in the late afternoonfor two reasons. Phloem transport was at its minimum (it actuallyreversed) in the middle of the day due to a strong pressure gradi-ent in the opposite direction in xylem created by transpiration,and photosynthesis continuously produced new sugars that accu-mulated in the upper crown. In reality, there is most likely also asink component that is presumably correlated with temperatureand would thus consume more sugars in the afternoon than inthe morning. Secondly, there is also sugar–starch conversion,which tends to transform accumulated sugars to starch and de-crease the afternoon peak. Thirdly, stomatal regulation andsugar accumulation might slow down the gas exchange in theafternoon (Nikinmaa et al., 2013), which was not considered inour simulations. However, it is clear from the simulations thattranspiration causes strong demands on phloem transport and itmay well be that it is the collapse of phloem turgor pressureunder drought that may eventually lead to tree mortality, asSevanto et al. (2014) suggested.

Much of the variation in the bark sugar concentration was dueto the need to maintain a balance of water potential between

xylem and phloem. In the simulations we assumed that onlysugars were responsible for the osmotic strength of the phloemsap. This was particularly evident in the sensitivity runs, inwhich the sugar concentration behaved as a mirror image for thexylem pressure. In real trees there are a number of other solutes,particularly potassium, which also have a significant role inturgor regulation in living cells (Thompson and Zwieniecki,2005). Therefore, the simulated sugar concentrations were prob-ably high in comparison with real plants. However, if thesources are located at the tree top and main sinks are at the baseor below ground, as is the case, for example, in late summer(Iivonen et al., 2001), the fact remains that photosynthesizedsugars need to be transported from the leaves, which requires themaintenance of a phloem pressure gradient and a fairly largesugar concentration difference between the top and the base.Also, potassium participates in creating the turgor gradient. Theflow of water down the gradient also flushes the potassium downand, unlike sugars, which are consumed in sinks, potassium islikely retranslocated to the xylem and transported back to top,for example in the xylem sap (Zwieniecki et al., 2004).

The time constant of the whole-tree system was of the order ofdays. Xylem transport reached steadystate quite rapidly, in�1 h.However, it took several days before a steady-state condition inphloem transport was reached. This can be expected since theosmotic balance was only reached by transporting sugarsthrough the phloem and the phloem sugar content was severaltimes larger than the daily sugar assimilation. It has been shownthat sugars have an important role in the daily adjustment ofosmotic strength (Turner et al., 1978). In real trees the responsecould be faster as other ions also contribute to osmotic balanceand trees have substantial carbohydrate stores. Sugar–starch con-version could be a means of rapidly balancing osmotic gradients atthe whole-tree level. Geigenberger et al. (1997) showed that waterstress favoured sugar–starch conversion. The rapidly movingpressurewaves within tree stems convey information about gas ex-change (Thompson and Holbrook, 2004) and can create localwater stress in the bark tissue that can result in a balancingsugar–starch conversion, much more rapidly than would be pos-sible by transport only. If such a pressure-propagation-relatedmechanism does not exist in trees, our result suggests thatphloem transport is never in a steady-state condition.

Sensitivity runs with permeability produced the expectedresult that when the transpiration and photosynthetic ratesremained constant the pressure gradients varied proportionallywith permeability. For both xylem and phloem the variation inthe gradient was mainly due to changes in the minimum pressure,next to transpiring leaves in the xylem and at the tree base in thephloem. Also, the sugar concentration in the phloem closely fol-lowed the xylem pressure changes as sugars were used to balancethe water potential between xylem and phloem at each height.The permeability changes also revealed that the permeabilityrange used (0.5- to 2-fold the base case) probably containedthe true values for xylem and phloem permeabilities, as itproduced reasonable pressure gradients and maximum andminimum values when realistic transpiration and photosynthesisrates were used. When permeability was dropped drastically (to10 % of the base case) the phloem results were influenced by theviscosity build-up due to increasing sugar concentrations. Thesesensitivity runs assumed that the gas exchange rate was not influ-enced by the internal status of the tree. If feedback between gas




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exchange and pressures was considered, then all the observedvariables (xylem and phloem pressures and xylem sugar concen-tration) would have been contained within narrower marginswhile the transpiration and photosynthetic rates would havevaried. Stomatal conductance has been linked to the conductanceof the soil–tree–atmosphere continuum (Hubbard et al., 2001;Buckley, 2005; Duursma et al., 2008) and the hypothesis thatleaf gas exchange would be in balance with the rate of assimilatetransport from the leaves seems to produce realistic gas exchangebehaviour (Nikinmaa et al., 2013). The latter approach combinesboth phloem and xylem transport with gas exchange.

Low permeability values exaggerated the differences in pres-sure and concentration between shoots, particularly in turgor pres-sure and in the upper crown. Such differences would also translateinto shoot growth (e.g. Pantin et al., 2012), which would bringabout differences in crown architecture. When the permeabilitieswere within a normal range, the turgor differences were mainlyalong tree height. However, as the transport capacity of the con-ducting tissue was altered, large differences developed betweendifferent branches, even at the same height. This illustrates thetight linkage between the existing structure, gas exchange rateand the potential for new growth. The sensitivity analysis for per-meability demonstrates a potential mechanism by which changesin water availability could also influence crown architecture assuggested (Nikinmaa et al., 2003), in contrast to the currentlymost often used model of crown development (i.e. light-driven)(e.g. Sievanen et al., 2000). Linkage of both gas exchange (thesource) and growth (the sink) with transport capacity suggests anew very versatile mechanism to study whole tree growth and de-velopment dynamically with increasing tree size and in varyingabove- and below-ground environmental conditions.

The formulation used for sink activity influenced phloemtransport considerably. In the base case simulations weassumed that the sink was at the tree base, which produced theexpected downward pressure gradient in the phloem. However,when concentration-dependent unloading of sugars to the sinkwas also assumed at the tree top, the gradient disappeared oreven turned upside down. How we simulated the unloading activ-ity is not necessarily very realistic as carbohydrate consumptionat each height along the tree axis depends on the respiration rateand sugar–starch conversion and consumption for growth. Atleast the last two terms are influenced by turgor pressure(Lalonde et al., 2003; Holtta et al., 2010) and therefore driversof the unloading rate are more complex than just the concentra-tion of sugars. This is clearly a point to improve in the future asour sensitivity runs showed a very high response of transport tothe unloading rate. Diurnal sugar–starch conversion could, forexample, smooth out the daily variations in sugar concentrationsthat could reflect the daily variation in the phloem gradient.Nikinmaa et al. (2013) showed that daily storing of sugars asstarch in the crown could boost total canopy production, depend-ing of the tree’s sugar status, as it would prevent accumulation ofassimilates to levels that are causing downregulation of photo-synthesis. As the unloading rate is most likely responsive tolocal conditions, its strong linkage with transport introducesvery interesting whole-tree dynamics of resource distribution,which can be mechanistically modelled with the approachpresented here.

The approach presented here offers a promising new tool toanalyse processes at whole-tree level, but has limitations.

There are 295 internodes in the tree (Fig. 1) we used in the simu-lations. This means 3 × 295 ¼ 885 coupled ordinary differentialequations. The set of ordinary differential equations is stiff; thereare different time scales on which water pressure in xylem andwater pressure and sugar concentration in phloem change(Fig. 6). This difference is further exaggerated by the large vari-ability in the sizes of the internodes: the equations for rates ofchange contain the multiplier 1/(xylem volume) or 1/(phloemvolume) (see Appendix). In our tree, xylem volumes, and thusalso the multipliers, can differ by a factor of 2000. We useda method for stiff ordinary differential equations, as the speedof computation was �100-fold faster than with an ordinarymethod (the Runge–Kutta method) Nevertheless, the progresswas 8- to 120-fold faster in the simulations than in real time de-pending on how fast environmental variables were changing.This means that simulation of trees that are an order of magnitudelarger (several thousands of internodes) will not be feasible withthe current configuration. The speed of computation can prob-ably be somewhat increased by optimizing the program codebut more sophisticated methods of numerical mathematics willbe needed if considerable improvements are to be made.

CONCLUSIONS

Realistic simulation of linked water and assimilate transport, gasexchange and sink activity at whole-tree level is feasible in3-D. This opens a new avenue for plant studies. Different for-mulations of local processes and their responses to local and ex-ternal conditions can be tested and their implications forwhole-tree-level behaviour can be evaluated. The results canrapidly show whether the approach used has implications thatare not within realistic physical boundaries. Also, very versatilemechanistic linkages between tree development and differentprocesses can be tested, making the approach a good test-benchfor studies of tree physiology. The main problem of the approachis that, with the current formulation, large trees cannot be studiedwithout simplifying the structural description, developing nestedapproaches or parallel computing.

ACKNOWLEDGEMENTS

We thank Pasi Kolari for advice regarding the SPP model, theAcademy of Finland Centre of Excellence (grant no. 1118615),Top-level Research Initiative (NCoE CRAICC) and the Academyof Finland projects DECADE and MultiTree.

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APPENDIX

Mathematical formulation of the model

The equations of the model given at the general level in the maintext are specified here in detail. The flows of water and sucrose ina branched tree architecture are according to Fig. A1. Apart fromthe application to a branched architecture, the equations are ba-sically the same as in Holtta et al. (2006). We consider in an inter-node i water pressure in xylem sapwood Pxi and in phloem Ppi,and amount of sucrose solute in phloem internode Npi. In the fol-lowing, subscripts x and p refer to xylem and phloem and sub-scripts i and j to internode number. The variables are defined inTable A1 and the values of constants and parameters are givenin Table A2. The xylem and phloem transport model is a set ofinterconnected ordinary differential equations. The equationsfor each internode i are

dPxi

dt= Erx

Vxi

1

h0

∑j[Bi

kx(i,j)Pxj + Phj − Pxi − Phi

Li + Lj

2

Ax(i,j)

+ Erx

Vxi

lr Ppi − Pxi −Npi

Vpi

RT

[ ]Ari −

Erx

Vxi

QTi + Dxi

dPpi

dt= Erp

V pi

∑j[Bi

k p(i,j)h p(i,j)

P pj + Phj − P pi − Phi

Li + Lj

2

A p(i,j)

− Erp

V pi

lr P pi − Pxi −N pi

V pi

RT

[ ]Ari

dN pi

dt=

∑j[Bi

k p(i,j)h p(i,j)

P pj + Phj − P pi − Phi

Li + Lj

2

A p(i,j)n p(i,j) +QPi

12+ Ui

where the factor 1/12 in the equation for Npi is derived by convert-ing the molar amounts of CO2 assimilated into molar amounts ofsucrose loaded into the phloem. The permeabilities are

ks(i,j) = 1/2 ksi + ksj

( ), s = x, p

Note that permeability (m2) is hydraulic conductivity (m4 MPa– 1

s– 1) divided by cross-sectional area (m2) and dynamic viscosity(MPa × s).

The cross-sectional areas are

As(i,j) =min( Asi, Asj), if i, j are in the same axis

1

2min( Asi, Asj), if i or j is lateral s = x, p

⎧⎨⎩

where the multiplier 12

in the cross-sectional areas means that thelateral internodes are connected with half of the cross-sectionalarea to the internode above the branching point and with halfof the cross-sectional area to the internode below the branchingpoint.

The dynamic viscosity of water in phloem is

hp(i,j) = 1/2(hpi + hpj)

TABLE A1. Definition of variables

Variable Definition Unit

Ar Area between xylem and phloem m2

Ap, Ax Cross-sectional area of phloem, xylem m2

B Set of neighbouring internodesDbase Boundary condition of xylem water flow at baseL Length of internode mNp Molar amount of sucrose molnp Sucrose concentration in phloem ( ¼ Np/Vp) mol m– 3

Ph Hydrostatic pressure induced by gravitation MPaPp, Px Water pressure, phloem, xylem MPaQT Transpiration rate of internode mol(H2O) s– 1

QP Photosynthesis rate of internode mol(CO2) s– 1

SA Surface area of internode m2

T Absolute temperature KU Rate of unloading of sucrose mol s– 1

Vp, Vx Volume of phloem, xylem sapwood m3

hp Dynamic viscosity of water in phloem MPa × s

TABLE A2. Parameters and constants of the model

Parameter Meaning Value

c0 Reference sucrose concentration forunloading

800 mol m– 3

Erp Phloem elastic modulus in radialdirection

30 MPa

Erx Xylem elastic modulus in radialdirection

1000 MPa

kp Phloem axial permeability at the baseof the tree

1.2 × 10– 13 m2

kx Xylem axial permeability at the baseof the tree

1.5 × 10– 13 m2

lr Radial hydraulic conductivity 1.0 × 10– 7 m(MPa × s)– 1

R Universal gas constant 8.314 J mol– 1 K–1

h0 Dynamic viscosity of pure waterat 20 8C

1.0 × 10– 9 MPa × s

us Slope parameter of unloading function 3.5 × 10– 7 m s– 1

The values are mainly from Holtta et al. (2006).

2

3

1

FI G. A1. The principle of treating flows in a branched tree architecture: theinternodes exchange material with the neighboring internodes in the same axis(grey shading) and the lateral internodes (internode #3). The set of neighborsof internode #1 is thus B1 ¼ {2,3}. The lateral internodes are connected to inter-nodes both below and above the branching point, the set of neighbors of internode#3 is B3 ¼ {1,2} The cross-sectional areas (A), permeabilities (k) and sucroseconcentrations (n) appearing in equations for exchange of material between inter-

nodes are determined separately for each pair of internodes.




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and the sucrose concentration in phloem is

cp(i,j) =1

2(cpi + cpj) =

1

2

Npi

Vpi

+ Npj

Vpj

( )

The boundary condition for xylem water pressure assumes con-stant water pressure of roots, hence

Di =Exbase

Vxbase

kxbase

h0

−0·25 − Pxbase

Lbase

, i = base

0, otherwise

⎧⎨⎩

The unloading takes place only in internodes that have no needles(unless otherwise indicated) and its rate is proportional to sucroseconcentration and the surface area of the internode, if it exceedsthe threshold concentration c0:

Ui = SA× max 0, us

N pi

V pi

− c0

( )[ ]

To mimic the sinkof sugars in roots, the value of the unloadingslope parameter us is multiplied by 10 in the base internode in thestem.

The axial permeabilities of xylem and phloem change as afunction of distance from the branch tip as:

ks = ks,top

√d s = x,p

where d isdistance fromthe branch tip and ks,top is the permeabilityof the uppermost internode. This means that for the longest dis-tance in the tree (in the stem) ks,top/ks ¼ 0.648. The above relationis derived by combining the Hagen–Poiseuille equation, WBEtheory and conduit density per cross-sectional area, i.e. thepacking ratio of conduits (Savage et al., 2010). The Hagen–Poisseuille equation states that k / nr4, where r is conduitradius and n is the number of conduits (xylem tracheids/phloemsieve tubes). WBE theory states that r / x1/4, where x is the dis-tance from the apex (West et al., 1999; Anfodillo, 2006).Conduit density, i.e. the number of conduits per cross-sectionalarea, is n/ 1/r2 ¼ r –2. This is derived from a purely geometricalconsideration assuming constant packing of conduits, and is veri-fied by experimental data (Savage et al., 2010). Combining theabove relations yields k / r –2r4 / r2 / x1/2.




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Dynamics of leaf gas exchange, xylem and phloem transport ...

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