Annals of Botany Page 1 of 14
doi:10.1093/aob/mcu068, available online at www.aob.oxfordjournals.org
PART OF A SPECIAL ISSUE ON FUNCTIONAL –STRUCTURAL PLANT MODELLING
Dynamics of leaf gas exchange, xylem and phloem transport, water potential
and carbohydrate concentration in a realistic 3-D model tree crown
Eero Nikinmaa1,*, Risto Sievänen2 and Teemu Hölttä1
1
Department of Forest Sciences, University of Helsinki, PO Box 27, Helsinki, 00014, Finland
and 2Finnish Forest Research Institute, Vantaa Research Unit, PO Box 18, Vantaa 01301, Finland
* For correspondence. E-mail [email protected]
† Background and Aims Tree models simulate productivity using general gas exchange responses and structural
relationships, but they rarely check whether leaf gas exchange and resulting water and assimilate transport and
driving pressure gradients remain within acceptable physical boundaries. This study presents an implementation
of the cohesion –tension theory of xylem transport and the Münch hypothesis of phloem transport in a realistic
3-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 and
transpiration model in a realistic tree architecture to simulate leaf gas exchange and water and carbohydrate transport
within an 8-year-old Scots pine tree. The model solved the dynamics of the amounts of water and sucrose solute in the
xylem, 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 sugar concentration. Diurnal
variation 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 structural
and 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 new
avenue for plant studies, as features that are difficult to measure can be studied dynamically with the model. Tree-level
responses to local and external conditions can be tested, thus making the approach described here a good test-bench
for 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, tree
crown, canopy gas exchange.
IN T RO DU C T IO N
The development of a tree crown is one of the principal means by
which trees respond to their growing environment (Nikinmaa,
1992). Plasticity of crown development determines the efficiency of light capture (Posada et al., 2012), nitrogen use
(Mooney and Gulmon, 1979; Field, 1983) and tree-level water
relations (Zimmermann, 1983). The connection between tree
stems and branches and their water transporting role has intrigued scientists for hundreds of years [see Leonardo’s notes
(MacCurdy, 2002)]. The study of regularities between woody
axes and their functional role has a long history in botany [see
Pressler (1865) and Jaccard (1913), cited by Zimmermann,
1983; Huber, 1928], and this line of thought has led to general
theories about the build-up of trees (Shinozaki et al., 1964;
West et al., 1999). The hydraulic limitation of tree size (Ryan
and Yoder, 1997; Koch et al., 2004) has been considered as the
fundamental boundary condition for the development and
scaling of water-conducting tissue within a tree (West et al.,
1999). Also, hydraulic architecture, which determines how transpiration translates into changes in water potential, has been considered to be a fundamental design property of crown patterns
(Zimmermann, 1983; Nikinmaa et al., 2003).
Tree development results from a complex interaction between
carbohydrate sources and sinks that is mediated by the conductive properties of tissue responsible for long-distance transport.
Growth of new tissue requires assimilated carbohydrates for
cell wall synthesis, but these compounds also have a role in
creating sufficient turgor pressure for cell wall extension
(Hölttä et al., 2010; Pantin et al., 2012). Transpiration of tree
crowns is directly connected to hydraulic conductivity from
soil to transpiring leaves. Assimilate transport in phloem is
closely connected to xylem transport (Ferrier et al., 1975;
Boersma et al., 1991; Hölttä et al., 2006; Hall and Minchin,
2013). Low water potential in leaves will slow down phloem
transport or cause even a transient reversal of flow (Hölttä
et al., 2006). Together, the magnitude of negative pressure in
xylem and the osmotic strength that assimilated sugars and
other osmolites maintain in growing tissue determine the
extent to which new tissue can expand (Hölttä et al., 2010;
Pantin et al., 2012). Because of these linkages, growth of tree
crowns follows from an interesting interaction between existing
woody structure, its transport properties and the gas exchange
behaviour of leaves within the crown.
The environmental variables that drive photosynthesis and
transpiration vary a lot within tree crowns and canopies,
# The Author 2014. Published by Oxford University Press on behalf of the Annals of Botany Company. All rights reserved.
For Permissions, please email: [email protected]
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Received: 9 December 2013 Returned for revision: 12 February 2014 Accepted: 12 March 2014
Page 2 of 14
Nikinmaa et al. — Xylem and phloem transport in a 3-D model tree crown
discuss the significance of the results for functional – structural
tree modelling.
M AT E R I A L S A N D M E T H O D S
We combined the xylem and phloem transport model of Hölttä
et al. (2006) and the assimilation and transpiration model SPP
of Mäkelä et al. (2006) with the 3-D tree model for Scots pine
(Pinus sylvestris) trees (Sievänen et al., 2008) in the LIGNUM
modelling framework (Sievänen et al., 2010). The xylem and
phloem transport model deals with the dynamics of water and
sucrose transport and associated axial and radial concentration
and pressure gradients in and between xylem and phloem. The
SPP model evaluates the rates of photosynthesis of a Scots
pine shoot as a function of radiation [ photosynthetically active
radiation (PAR)], temperature and saturation deficit of water
vapour. LIGNUM provides the tree hydraulic architecture and
evaluates the environmental conditions of individual shoots. In
LIGNUM, Scots pine trees consist of internodes that contain
heartwood, sapwood and possibly needles. For this application
we also consider the phloem layer [constant thickness 2 mm
(Hölttä et al., 2013)] (Fig. 1A). The transport model (Hölttä
et al., 2006) considers the amounts of water and sucrose solute
in xylem, cambium and phloem using a fine-grained (in comparison with the size of the tree) mesh and solves their dynamics as a
system of coupled ordinary differential equations. We use this
approach to suit tree level analyses and formulate the transport
model as ordinary differential equations of water pressure in
Sapwood
Heartwood
Phloem
F I G . 1. (A) An internode in LIGNUM consisting of heartwood (brown),
sapwood (yellow), phloem (green) and needles. The internode and sizes of heartwood 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 is
2 mm. (B) The sapling tree used in the calculations was grown by LIGNUM
(Sievänen 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 simulation. Height of the tree is 2.37 m, diameter at base 3.4 cm, heartwood diameter
1.1 cm and number of internodes 295. Needle area of internodes decreases
with age; 1-, 2- and 3-year-old internodes have 98, 86 and 16 % of initial
needle area, respectively; all needles are lost at age 4; 36 internodes have no
needles. 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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influencing the attainable photosynthetic and transpiration rates.
Leaf stomata, which control gas exchange with the exterior, have
well known responses to the main environmental variables: light,
vapour pressure deficit and temperature (Landsberg, 1986).
Stomata are also responsive to the water-conducting pathways
of trees and soil water status (Buckley, 2005; Duursma et al.,
2008). The optimal water use scheme describes well the environmental responses of leaf gas exchange (e.g. Hari and Mäkelä,
2003) and is compatible with the currently most widely used
gas exchange models (Medlyn et al., 2011). Nikinmaa et al.
(2013) recently showed that such behaviour arises if trees
attempt to control their leaf gas exchange to maximize assimilate export from leaves. These findings thus suggest that gas
exchange is linked to carbohydrate sink activity, enhancing
the links between growth, structure, transport and canopy gas
exchange.
Tree crown architecture is responsive to even the species
identity of the tree’s neighbours (Lintunen and Kaitaniemi,
2010), although the exact mechanisms behind such behaviour
are not known. Functional – structural plant models have been
developed for describing the plasticity of crown shape in response to variation in the growing environment. In these
models the development of the tree crown results from initiation,
extension and thickening of individual apical meristems and later
shoots (Sievänen et al., 2000). These developmental processes
depend on the local environment, the position within the tree
topology and often also on the availability of resources, such
as carbon and nitrogen, for example at an annual level
(Prusinkiewicz and Lindermayer, 1990). The success of these
models depends on how well they predict resource capture and
its dependency on environmental conditions and how well
growth rules, which are often determined empirically, capture
the growth process. Although connections seem to exist
between gas exchange, structure, transport and growth, very
few models consider the interactions in resource capture and
solve growth as a local process in which growth results from
the local conditions and resource transport from sources (Allen
et al., 2005).
The linkages between leaf gas exchange, tree architecture and
vascular structure and transport set strong boundary conditions
for possible process rates within a given structure that have not
been fully exploited in plant studies. For example, cavitation in
xylem and viscosity build-up in phloem are feedforward-type
processes that reduce the hydraulic conductivity of the xylem
and phloem, setting limits to maximal transport and gas exchange rates for a given structure (Hölttä and Nikinmaa, 2013).
Although modelled separately in a number of papers, here we
combine for the first time the implementation of the cohesion –
tension theory of xylem transport and the Münch hypothesis of
phloem transport and a tested leaf gas exchange model in a
realistic 3-D tree structure and study the resulting water and assimilate transport. This makes it possible to study the distribution
of water tension, turgor pressure and osmotic concentration
within the different crown axes in more detail than with previous
models (e.g. Nikinmaa et al., 2013), because we can now make
assumptions about structural properties and driving gas exchange rates that are as realistic as possible and can also test
these rates against observed patterns. We further study how the
different aspects of within-crown transport are influenced by
variation in xylem and phloem properties and sink strength and
Nikinmaa et al. — Xylem and phloem transport in a 3-D model tree crown
xylem sapwood (Px) and phloem (Pp) and the amount of sucrose
solute in phloem (Np) for each internode as follows:
Page 3 of 14
A
dPp
= ap (Qp,in − Qp,out − Qxp )
dt
dNp
= (Qp,in − Qp,out )c(Np ) + QP − QU
dt
Temperature (°C)
25
dPx
= ax (Qx,in − Qx,out + Qxp − QT )
dt
20
15
10
Incident radiation
(mmol PAR m–2 s–1)
1500
B
1000
500
0
Saturation deficit of
water vapour (mol m–3)
1·0
C
0·8
0·6
0·4
0·2
0
4
8
12
16
Time (h)
20
24
F I G . 2. Daily courses of (A) radiation, (B) temperature and (C) saturation deficit
of water vapour used in the calculations. Data are the from SMEAR II station at the
Hyytiälä Forestry Field Station of University of Helsinki (Hari and Kulmala, 2005)
(sunny early summer conditions). Daily mean values (used in the steady-state analyses) 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.
relative to branch and stem diameters (Savage et al., 2010), but
these yielded very similar results (not shown) since stem and
branch diameters scaled with distance from the leaf apex in our
simulated tree. The reference permeability at the tree base for
xylem and phloem was taken from the literature. In these simulations we assumed that the sink of sugars occurs only in internodes
that do not have needles, to approximate a situation after shoot
growth has terminated and the main carbohydrate sink is in the
lower part of the tree. At the base we assumed a zero-flow boundary condition. The carbon sink of the lowest stem internode was
10-fold higher than that of the rest of the internodes to mimic the
carbohydrate sink of the below ground parts.
Downloaded from http://aob.oxfordjournals.org/ at Finnish Forest Research Institute / Library on June 18, 2014
where Qx,in, Qx,out and Qp,in, Qp,out are the axial inflow and
outflow rates of water in xylem and phloem, Qxp is the flow
rate of water from phloem to xylem in the internode, QT is the
sink of water depending on transpiration rate of the internode,
QP is the rate of loading of sucrose caused by photosynthesis in
the internode, QU is the rate of unloading of sucrose in the internode, c(Np) is the sucrose concentration of phloem sap, and ap
and ax are coefficients dependent on the volume and elastic
modulus of the internodes. The internodes are coupled to other
internodes through the axial flows. The model of xylem and
phloem transport in LIGNUM is thus a system of coupled
ordinary differential equations with 3 × (number of internodes)
variables. The ordinary differential equations for Px and Pp are
based on Darcy’s law (Siau, 1984) and that for Np is according
to the Münch hypothesis (as formulated in Hölttä et al., 2006).
We solved the set of ordinary differential equations with the
Rosenbrock stiffly stable method using automatic step size adjustment (numerical recipes for C+ +). A more detailed mathematical definition of the model and a list of parameter values
are given in the Appendix.
We performed the calculations with a simulated (grown by
LIGNUM, Sievänen et al., 2008), 2.37-m tall, 8-year-old Scots
pine tree (Fig. 1B) whose structural ( permeability) and functional (sink strength) properties were consequently changed to study
their influence on flow rates and pressure gradients. The incoming radiation for shoots (i.e. internodes) was calculated using the
backward ray tracing method in LIGNUM (Sievänen et al., 2008)
considering only self-shading, the radiation distribution of the
sky being that of an overcast day. The tree was thus assumed be
in a rather sparse sapling stand.
We first ran the model to steady state with mean values of
environmental variables for normal summer day conditions for
southern Finland (Fig. 2; base case simulations are shown in
Figs 3 –7). The rates of photosynthesis and transpiration and
their responses to environmental conditions (e.g. light and temperature) were from field observations at our research forest
(Kolari et al., 2009). Environmental drivers in the steady-state
analyses were as follows (Fig. 2): radiation equalled 600
mmol(PAR) m – 2 s – 1 on a horizontal surface, temperature was
20 8C and saturation deficit of water vapour was 0.6 mol(H2O)
m – 3. Xylem sapwood thicknesses were according to the
LIGNUM parameters for Scots pine (Sievänen et al., 2008)
and phloem thickness was assumed to be a constant 2 mm
(Hölttä et al., 2013). The axial permeabilities of xylem and
phloem increased basipetally in proportion to the square root
of distance from the apex. This pattern stems from conduit tapering as described in West et al. (1999) and conduit packing density
in Savage et al. (2010) (see Appendix). We also tried scaling
Page 4 of 14
Nikinmaa et al. — Xylem and phloem transport in a 3-D model tree crown
A
−1·0
0·08
0·04
B
2·0
0
1·9
B
0·8
Sugar flow rate (g h–1)
1·8
1·7
1·5
C
0·6
0·4
0·2
1·0
0
0
0·5
0
0·5
1·0
1·5
2·0
Distance from base (m)
2·5
F I G . 3. Steady-state profiles of water pressure in xylem (A) and phloem (B) and
sucrose concentration in phloem (C) in different parts of the tree: stem, branch and
side branch (as indicated in the key).
In the simulations, soil water pressure was an input to the
model; the xylem of the base internode was connected to soil
with the same resistance as to the internode above. Soil water
pressure was set at – 0.35 MPa in all simulations.
The incoming radiation (from all directions) to shoots with
needles varied between 23 and 907 mmol(PAR) m – 2 s – 1 in
steady-state conditions. The photosynthetic and transpiration
rates per unit needle area varied within the ranges 0.26–3.7
mmol(CO2)m – 2 s – 1 and 72–462 mmol(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 permeability according to West et al. (1999) versus keeping it constant
(Fig. 11) and against different formulations of sink strength in
the stem segments (Fig. 12). As the literature values, particularly
for phloem permeabilities, are quite uncertain, large variation
0·5
1·0
1·5
2·0
Distance from base (m)
2·5
F I G . 4. Steady-state vertical distribution of flow rate of (A) water (upwards) in
xylem and (B) sugars (downwards) in phloem. The flows are displayed at both the
top and the bottom of the internodes. Step changes are due to the contributions of
branches to flow.
around the base case values were used by increasing the permeabilities of xylem and phloem, first alone and then both at the
same time by 10 % and by 10-fold, and decreasing the permeabilities in a similar fashion by 10 –90 %. To simulate the effect of
different patterns of canopy sink strength on the model
outcome, we assumed in the base case that the sink of sugars
occurred only in internodes that did not have needles, to approximate the situation after shoot growth has terminated and the main
carbohydrate sink is in the lower part of the tree. To analyse
model behaviour during the early growing season, when shoots
are actively growing, we assumed two different sink formulations: (1) when the sink strength would be directly proportional
to carbohydrate concentration; and (2) when sink strength
would be subject to rate limitation, such that, if sugar concentration was in the range 0.0– 0.85 mol L – 1, unloading was proportional to sugar concentration and concentration values
.0.85 mol L – 1 did not further increase unloading.
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2·1
Phloem water pressure (MPa)
0·12
−0·5
−1·5
Sucrose conc. in phloem (mol L–1)
A
Stem
Branch
Side branch
Water flow rate (L h–1)
Xylem water pressure (MPa)
0
Nikinmaa et al. — Xylem and phloem transport in a 3-D model tree crown
0
Xylem water pressure (MPa)
A
−0·3
−0·6
−0·9
−0·5
−1·0
−1·5
Phloem water pressure (MPa)
−1·2
−1·5
2·1
B
2·0
B
1·8
Base shoots
Apex shoots
Stem
Branch
Side branch
1·4
1·0
1·9
1·8
1·7
0
0·5
1·0
1·5
2·0
Distance from base (m)
2·5
Sucrose conc. in phloem (mol L–1)
Phloem water pressure (MPa)
A
C
1·4
1·0
0·6
0
F I G . 5. Vertical distribution of (A) xylem and (B) phloem pressures at branch
apices in the steady state. Apices belonging to different branch whorls are indicated with colours. The colours are, from the lowest branch whorl to the top
one (also includes the leader shoot), in the order red, blue, magenta, green,
brown, cyan and black.
R E S U LT S
The steady-state xylem water pressure was evenly – 1.4 MPa
at the tips of the stem, branches and side branches in the base
case (Fig. 3A). There was an abrupt change in the main axis pressure at 1.4 m height where the needle-bearing internodes
stopped. The water pressure gradient was somewhat steeper in
branches and side branches in comparison with the main stem,
in accordance with the steeper permeability change there. The
turgor pressure at the leader shoot was slightly greater than
2 MPa and the pressure gradient between the top and the base
was only 0.1 MPa m – 1, which was about one-fifth of that in
xylem but in the opposite direction (note that we did not consider
the influence of gravitation because it had only a minor effect on
the results for the small tree simulated here; Fig. 3B). There was a
clear difference between the phloem water pressure between the
5
10
15
Time (d)
F I G . 6. Dynamics of water pressure in xylem (A) and phloem (B) and of sucrose
concentration 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.
leader and the side branches, consequently with a steeper pressure gradient in the leader. This also means a considerably
lower assimilate flow per leaf area downwards from the side
branches, as they all had approximately the same sucrose concentration (Fig. 3C). Xylem pressure and phloem sucrose concentration behaved like mirror images in these simulations, as the
osmotic strength of the phloem sap balanced the xylem water
potential to maintain phloem turgor pressure, and the only
osmotic substance included in the model was sucrose.
Consequently, in the steady-state situation assimilates participate both in maintaining the phloem water potential balance
with the xylem by osmosis and phloem transport.
Vertically, the largest drop in the stem sap flow rate occurred in
the upper middle part of the crown, where the bulk of the foliage
is (Fig. 4A). The main axis sugar flow increased in the upper half
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Xylem water pressure (MPa)
0
Page 5 of 14
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Nikinmaa et al. — Xylem and phloem transport in a 3-D model tree crown
50
A
Px relative change (%)
Xylem water pressure (MPa)
0
−1·0
A
Stem
Branch
Side branch
0
−50
−100
−2·0
Pp relative change (%)
Phloem water pressure (MPa)
2·6
2·2
1·8
0
−50
−100
1·4
50
1·4
C
1·2
Top of main stem
Base of main stem
Stem
Branch
Side branch
1·0
Sucrose relative change (%)
Sucrose conc. in phloem (mol L–1)
B
C
0
−50
−100
0
0·5
1·0
1·5
2·0
Distance from base (m)
2·5
0·8
0
0·5
1·0
1·5
2·0
2·5
3·0
Time (d)
F I G . 7. Daily pattern of (A) xylem and (B) phloem pressure and (C) sugar
concentrations at the top and the base of main stem, branch and side branch
(see key in C).
of the crowns, along the needle-bearing main axis internodes and
with steps as the branches joined the main stem (Fig. 4B). Below
the needle-bearing part, the flow rate started to decrease due to
our assumption of a carbon sink in the non-needle bearing
internodes only. In this part also, the lower whorls of branches
slowed down transport in the main axis, which means that they
were consuming carbon from the main stem.
The xylem water pressure varied roughly between 1.2 and
1.4 MPa in the branch tips attached to the same whorl of branches
in the upper section of the crown (Fig. 5A). Phloem turgor
seemed to be much more constant in the same whorl of branches
or even in branches at adjacent whorls but there were abrupt step-
F I G . 8. Effect of halving xylem and phloem permeabilities. Shown are relative
changes 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 sucrose
concentration in phloem (B) in different parts of the tree (stem, branch and side
branch; see key) versus distance from tree base.
like changes in the pressure that were caused by discretization of
the tree to finite internodes in the numerical solution.
The model reached steady state quickly for xylem pressure and
water transport (Fig. 6A) but almost 5 d elapsed before a
steady-state phloem pressure was attained (Fig. 6B). The long
time taken to reach steady state depended partially on the large
difference between the initial state of the tree and the steady-state
condition. The slow development of phloem transport was due to
the need to reach a steady-state sugar concentration in each internode, which required that sugars were transported there from
their sources. The overall sugar content of phloem is large in
comparison with the assimilation rate. In addition, phloem
tissue is more elastic than xylem, which also contributes to
slower time constants.
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50
B
Nikinmaa et al. — Xylem and phloem transport in a 3-D model tree crown
100
A
5
Px relative change (%)
Phloem water pressure (MPa)
6
4
3
2
1
0
Stem
Branch
Side branch
100
Pp relative change (%)
0·6
0·2
B
50
0
−50
−0·2
100
C
Sucrose relative change (%)
Phloem water pressure (MPa)
50
−50
B
25
A
20
15
10
5
C
50
0
−50
0
0
0·5
1·0
1·5
2·0
Distance from base (m)
2·5
F I 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 permeability was 10 % and the phloem was at the base case value, and (C) phloem
permeability was 10 % and the xylem was at the base case value (cf. Fig. 3).
Apices belonging to different branch whorls are indicated with colours. The
colours 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.
When simulated over several days using the measured variation in light, vapour pressure deficit and temperature (Fig. 2) a
clear daily pattern was established (Fig. 7). Xylem pressure
dropped rapidly in the morning in the top internodes as transpiration increased. The lower internodes in branches and the tree
base followed almost simultaneously with the pressure drop,
but not as steeply. The recovery of pressure in the evening was
not quite as symmetrical in comparison with the morning. The
pressure gradually increased throughout the night until it
started decreasing again the next morning (Fig. 7A). Phloem
turgor pressure followed a quite symmetrical daily pattern at
the tree base (Fig. 7B). Turgor pressure at the top quickly
0
0·5
1·0
1·5
2·0
Distance from base (m)
2·5
F I G . 10. Effect of increasing xylem and phloem permeabilities 10-fold. Shown
are relative changes in comparison with the base case of Fig. 3 [(value – base
case)/base case] of steady-state values of water pressure in xylem (A) and
phloem (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.
reached its maximum in the evening as transpiration and photosynthesis stopped. However, it started to decline slowly as
phloem transported sugars towards the base, down the steep
night-time turgor pressure gradient. It is interesting that during
the day the turgor pressure dropped so much at the top that it
fell below the turgor pressure at the base, meaning that phloem
transport reversed (Fig. 7B). The pressures in the simulations
behaved similarly in all of the branch internodes, regardless of
whether they were at the tip or at the base of the branch, and
they were distinct from the values for the tree base internode.
However, the sugar concentrations at the base of a side branch
showed a pattern similar to that at the base of the tree. The
sugar concentration in the top branches followed the daily
pattern of photosynthesis but with a delay of 6 h, so that the
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Phloem water pressure (MPa)
0
Page 7 of 14
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50
Nikinmaa et al. — Xylem and phloem transport in a 3-D model tree crown
A
1·8
Stem
Branch
Side branch
Pp (MPa)
Px relative change (%)
40
30
Stem
Branch
Side branch
20
1·2
0·6
10
0
2·5
0
F I G . 12. Steady-state profiles of water pressure in phloem in different parts of
the tree: stem, branch and side branch (see key) with different unloading formulations. Solid line, base case; dashed line, unloading in all internodes; dotted line,
unloading in all internodes but maximum rate was limited. Rate limitation: if
sugar concentration was in the range 0.8– 0.85 mol L – 1 the rate of unloading
was proportional to sugar concentration; concentration values .0.85 mol L – 1
did not increase unloading.
−10
50
B
Px relative change (%)
40
30
20
10
0
−10
0
0·5
1·0
1·5
2·0
Distance from base (m)
2·5
F I G . 11. Effect of having constant xylem and phloem permeabilities throughout
the tree. Shown are relative changes in comparison with the base cases of Figs 3
and 5 [(value – base case)/base case] in steady-state values of water pressure in
xylem 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 the
lowest branch whorl to the top one (includes also the leader shoot), are in the
order red, blue, magenta, green, brown, cyan and black.
minimum value occurred at about sunrise and the maximal value
in the late afternoon to early evening (Fig. 7C). The pattern in the
lower internodes was almost opposite, such that the maximum
concentration was early in the morning and the minimum concentration in the late afternoon to evening.
The results thus show that the daily dynamics of phloem sugar
transport were influenced much more strongly by the dynamics
of transpiration [which followed the daily pattern of vapour pressure deficit (Fig. 2)] rather than by the dynamics of photosynthesis [which followed the daily pattern of incident radiation
(Fig. 2)]; the phloem transport rate decreased during the day
and was highest during the night. Increasing transpiration
increased the xylem water potential gradient, which in turn
decreased, and even temporarily reversed, the phloem turgor
pressure gradient. This trend in the dynamics of phloem transport
rate was also responsible for the simultaneous increase in phloem
sugar concentration in the branch and decrease in the tree base
during the day (Fig. 7C).
We further simulated the model’s sensitivity to xylem and
phloem permeabilities and sink activity. When xylem and
phloem permeabilities were decreased by 50 %, the xylem pressure at the top internodes decreased almost proportionally, as the
transpiration rate was not assumed to be influenced by the tree
water potential in the model of Mäkelä et al. (2006) (Fig. 8A).
Also, the gradient of phloem pressure increased in proportion
to permeability change (approximately doubled with a 50 % decrease in permeability), although the change in absolute pressure
was small (Fig. 8B). The change in the phloem pressure gradient
was mainly due to the decline in base pressure while the sugar
concentration gradient increased, mainly due to the increase in
sucrose concentration at the top (Fig. 8C). Larger permeability
changes [up to one-tenth of the base case value (not shown)]
caused disproportionally large phloem pressure changes, indicating the influence of increasing sap viscosity with high sugar
concentrations.
The relative xylem pressure between the shoots with a 10 %
change in permeability of both xylem and phloem behaved as
in the base case, but the sugar concentration showed more variation between shoots at different positions. In particular, the
phloem pressure was very variable between shoots, indicating
a new type of distribution of phloem transport (Fig. 9A). The
change in this pattern was mainly caused by the change in
xylem permeability, as similar variation between shoots,
although at very different absolute values, was observed when
only the xylem permeability was decreased to 10 % of the base
case value (Fig. 9B). The pattern was closer to the base case,
although at very different pressure values, when only phloem
permeability was lowered (Fig. 9C). When phloem permeability
was decreased alone to one-tenth of the base case value (results
not shown), it did not change xylem pressure from the base case
value. However, phloem pressure became almost 10-fold higher
when it was only 2.5-fold higher relative to the base case when
Downloaded from http://aob.oxfordjournals.org/ at Finnish Forest Research Institute / Library on June 18, 2014
0·5
1·0
1·5
2·0
Distance from base (m)
Nikinmaa et al. — Xylem and phloem transport in a 3-D model tree crown
D IS C US S IO N
This is the first time that xylem and phloem transport have been
modelled explicitly, driven by the observed level of foliar gas
exchange within a realistic 3-D branching architecture. The
model was parameterized to represent a Scots pine tree in
boreal conditions, and gas exchange rate parameters and values
for meteorological conditions were both derived from measurements taken at our field station. The structural parameters were
those used for the LIGNUM model and are based on field measurements in the same region (Sievänen et al., 2008). The only
new structural parameter value was the thickness of phloem
tissue, which was also measured in the same region (Hölttä
et al., 2013). We did not conduct any new measurements of
axial and radial permeabilities but used a scaling exponent
form the metabolic scaling theory (West et al., 1999) and
chose the multipliers to give values within the published range
(e.g. Génard et al., 2001; Zwieniecki et al., 2001; De Schepper
and Steppe, 2010). There was considerable uncertainty regarding
axial phloem permeability and the radial permeability between
xylem and phloem. Despite this uncertainty, the model gave
results consistent with observations. The steady-state xylem
pressure was – 1.4 MPa at the branch tips in the upper crown
under overcast conditions (when the daily pattern was simulated), and the lowest values were slightly below – 2 MPa.
Scots pine follows the isohydric pattern of water use and
minimum water pressures drop to – 2.5 MPa (Hölttä et al.,
2005). In these measurements, the needle water potentials in
the light crowns of Scots pine varied mostly between – 1 and
– 2 MPa. There is relatively little information on variation in
turgor pressure, but it has been found to vary around 1.5 MPa
(Sovonick-Dunford et al., 1981; Turgeon, 2010), which was
also observed in our simulations. The sugar concentrations
varied between 0.9 and 1.5 mol L – 1, which are also realistic
but somewhat high values (Sovonick-Dunford et al., 1981).
These values were also reached in steady-state simulations
using observed gas exchange rates, which implies that realistic
overall transport rates in xylem and phloem were obtained with
realistic pressure and sugar gradients within the tree. As these
comparisons show, combining models of structure, transport
and gas exchange allow concrete case studies of whole-tree
physiology that have not been possible previously.
The simulations indicated that shoot tip water potentials
were 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 hydraulic
architecture of tree crowns: leaf water potentials tend to be
similar in the branch tips in different parts of the crown
(Zimmermann, 1983). The water potential at the transpiring site
results from the soil water potential, transpiration rate and xylem
pathway conductivity to that spot. Studies have shown that there
are large differences in leaf specific conductance between the
thicker and more vigorous main stem and the more slender side
branches (Zimmermann, 1983). This type of trend arises from
the regular pattern of cell size as a function of distance from the
stem apices (West et al., 1999; Anfodillo et al., 2006), and it
seems that the trend as a function of axis thickness is not too
different between the main stem and side branches. Here we
assumed that each branch tip had similar sapwood conductivity,
which increased along the axis relative to the square root of the distance from the tip, according to West et al. (1999) (see Appendix).
The change was steeper the shorter the branch, as the base value at
the forking point gave the same value as the main stem. This
seemed to result in realistic behaviour, as in the simulation in
which we assumed constant permeability the longer main axis
had lower water potential than the side branches and the shoot
water pressure seemed to be influenced only by transport distance.
We assumed a constant leaf area:sapwood area relationship
throughout the tree, as suggested by the pipe model (Shinozaki
et al., 1964). In real trees there are differences between branching
orders and along the crown (Berninger and Nikinmaa, 1994) that
could influence the results. Over the whole crown there were large
differences in shoot water potentials, particularly between the
lower and upper crown. Here, the photosynthesis and transpiration
rates were determined by the environmental conditions of the
shoots. In tree crowns, however, they also depend on the water
status of other parts of the crown (Whitehead et al., 1996). We
recently showed that stomatal conductance also depends on the
factors influencing xylem and phloem transport to and from a
shoot in addition to external shoot conditions (Nikinmaa et al.,
2013), which was not considered here. In principle, such feedback
could further balance the pressure differences between the shoots.
The main new result of this modelling exercise is that now we are
able to study the interaction between tree hydraulic architecture
and its environment with respect to tree behaviour. In these simulations we used only one standard environment, but this could be
easily varied according to measurements or, for example, by
Downloaded from http://aob.oxfordjournals.org/ at Finnish Forest Research Institute / Library on June 18, 2014
both permeabilities were dropped to one-tenth of the base case
value. This would mean that phloem transport was effectively
blocked and high xylem pressures allowed pressure build-up in
phloem. When xylem permeability was decreased alone,
xylem pressures dropped to the same values as those seen
when both permeabilities were dropped, but phloem pressure
attained lower values in comparison with the base case and the
sugar concentration remained at somewhat lower values than
when both permeabilities were dropped simultaneously.
The 10-fold increase in xylem and phloem permeabilities
caused proportional changes in gradients of xylem and phloem
pressure and phloem sugar concentration. There was a clear
change in absolute pressure in xylem but rather small changes in
phloem absolute pressure and sucrose concentration, the largest
changes in xylem pressure and sugar concentration occurring at
the tree top and in phloem pressure at the tree base (Fig. 10).
We also tested how the pressure and sugar gradients would be
influenced if there was no change in axial permeability from the
base to the tip, i.e. contrary to the formulation of West et al.
(1999) (Fig. 11). Xylem pressure increased towards the tree top
in comparison with the base case, making the gradient from
base to top smaller and the difference between the main axis
and side branches larger, emphasizing the influence of transport
distance.
The model behaviour was very sensitive to the formulation of
sink activity in the internodes. The base case results were calculated assuming that the sink was only in non-needle bearing
internodes and was proportional to internode sugar concentration
(see Appendix). If a sink was also assumed in the needle-bearing
internodes, the phloem turgor gradient could temporarily invert
from top-down to base-up, depending on the assumed unloading
speed (Fig. 12).
Page 9 of 14
Page 10 of 14
Nikinmaa et al. — Xylem and phloem transport in a 3-D model tree crown
xylem and phloem. In the simulations we assumed that only
sugars were responsible for the osmotic strength of the phloem
sap. This was particularly evident in the sensitivity runs, in
which the sugar concentration behaved as a mirror image for the
xylem pressure. In real trees there are a number of other solutes,
particularly potassium, which also have a significant role in
turgor regulation in living cells (Thompson and Zwieniecki,
2005). Therefore, the simulated sugar concentrations were probably high in comparison with real plants. However, if the
sources are located at the tree top and main sinks are at the base
or below ground, as is the case, for example, in late summer
(Iivonen et al., 2001), the fact remains that photosynthesized
sugars need to be transported from the leaves, which requires the
maintenance of a phloem pressure gradient and a fairly large
sugar concentration difference between the top and the base.
Also, potassium participates in creating the turgor gradient. The
flow of water down the gradient also flushes the potassium down
and, unlike sugars, which are consumed in sinks, potassium is
likely 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 of
days. Xylem transport reached steady state quite rapidly, in 1 h.
However, it took several days before a steady-state condition in
phloem transport was reached. This can be expected since the
osmotic balance was only reached by transporting sugars
through the phloem and the phloem sugar content was several
times larger than the daily sugar assimilation. It has been shown
that sugars have an important role in the daily adjustment of
osmotic strength (Turner et al., 1978). In real trees the response
could be faster as other ions also contribute to osmotic balance
and trees have substantial carbohydrate stores. Sugar–starch conversion could be a means of rapidly balancing osmotic gradients at
the whole-tree level. Geigenberger et al. (1997) showed that water
stress favoured sugar–starch conversion. The rapidly moving
pressure waves within tree stems convey information about gas exchange (Thompson and Holbrook, 2004) and can create local
water stress in the bark tissue that can result in a balancing
sugar–starch conversion, much more rapidly than would be possible by transport only. If such a pressure-propagation-related
mechanism does not exist in trees, our result suggests that
phloem transport is never in a steady-state condition.
Sensitivity runs with permeability produced the expected
result that when the transpiration and photosynthetic rates
remained constant the pressure gradients varied proportionally
with permeability. For both xylem and phloem the variation in
the gradient was mainly due to changes in the minimum pressure,
next to transpiring leaves in the xylem and at the tree base in the
phloem. Also, the sugar concentration in the phloem closely followed the xylem pressure changes as sugars were used to balance
the water potential between xylem and phloem at each height.
The permeability changes also revealed that the permeability
range used (0.5- to 2-fold the base case) probably contained
the true values for xylem and phloem permeabilities, as it
produced reasonable pressure gradients and maximum and
minimum values when realistic transpiration and photosynthesis
rates were used. When permeability was dropped drastically (to
10 % of the base case) the phloem results were influenced by the
viscosity build-up due to increasing sugar concentrations. These
sensitivity runs assumed that the gas exchange rate was not influenced by the internal status of the tree. If feedback between gas
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simulating a forest consisting of many individual trees (Sievänen
et al., 2008).
The daily pattern of xylem pressure was similar to that
observed previously (Perämäki et al., 2005). In the present
study, the asymmetrical pattern in which pressure decreases
quickly in the morning in response to the rise in transpiration
but increases more gradually in the evening was observed in a
very small tree. In an earlier study by Perämäki et al. (2005)
this type of behaviour was seen even more clearly as it was conducted on larger trees (.25 m tall) and was assumed to result
from the capacitive influence of gradual pressure release and
associated diameter change that propagates through a large
stem volume and differences between axial and radial permeabilities. Here, this type of delay effect was smaller due to the smaller
tree size, and asymmetry was partly due to asymmetry in daily
temperature versus the radiation pattern, but we also considered
explicitly the influence of phloem transport, which probably contributed to the asymmetry.
There is considerably less information on the daily variation in
phloem turgor pressure and sap sugar concentration. Mencuccini
et al. (2013) reported estimates that were derived from observed
variation in the above- and below-bark diameters of trees
growing in the same stand as that from which we obtained our
tree parameters and climate values. No exact match with the
tree-top values could be expected as our simulated tree was a
2.5 m tall sapling while the measured trees in Mencuccini
et al. (2013) were 15 m tall and the upper measurement was
several metres below the tree top. Overall, the trends were
similar in that the minimum phloem pressure took place at the
base some time after the xylem minimum and also in that the
daily amplitude of pressure change was much lower at the base
than higher up in the stem. The sugar concentration behaved in
a similar manner in that it showed almost an opposite daily variation between the top, where the maximum concentration occurred during the daytime, and the base, where there was a
minimum at that time. However, the maximum sugar concentration occurred in the late afternoon in our simulations but a short
time before noon in the observations. Differences of this kind of
are not unexpected as our formulation for sink activity was only
for needleless internodes and was quite crude. In this study the
maximum sugar concentration occurred in the late afternoon
for two reasons. Phloem transport was at its minimum (it actually
reversed) in the middle of the day due to a strong pressure gradient in the opposite direction in xylem created by transpiration,
and photosynthesis continuously produced new sugars that accumulated in the upper crown. In reality, there is most likely also a
sink component that is presumably correlated with temperature
and would thus consume more sugars in the afternoon than in
the morning. Secondly, there is also sugar – starch conversion,
which tends to transform accumulated sugars to starch and decrease the afternoon peak. Thirdly, stomatal regulation and
sugar accumulation might slow down the gas exchange in the
afternoon (Nikinmaa et al., 2013), which was not considered in
our simulations. However, it is clear from the simulations that
transpiration causes strong demands on phloem transport and it
may well be that it is the collapse of phloem turgor pressure
under drought that may eventually lead to tree mortality, as
Sevanto et al. (2014) suggested.
Much of the variation in the bark sugar concentration was due
to the need to maintain a balance of water potential between
Nikinmaa et al. — Xylem and phloem transport in a 3-D model tree crown
There are 295 internodes in the tree (Fig. 1) we used in the simulations. This means 3 × 295 ¼ 885 coupled ordinary differential
equations. The set of ordinary differential equations is stiff; there
are different time scales on which water pressure in xylem and
water pressure and sugar concentration in phloem change
(Fig. 6). This difference is further exaggerated by the large variability in the sizes of the internodes: the equations for rates of
change contain the multiplier 1/(xylem volume) or 1/( phloem
volume) (see Appendix). In our tree, xylem volumes, and thus
also the multipliers, can differ by a factor of 2000. We used
a method for stiff ordinary differential equations, as the speed
of computation was 100-fold faster than with an ordinary
method (the Runge –Kutta method) Nevertheless, the progress
was 8- to 120-fold faster in the simulations than in real time depending on how fast environmental variables were changing.
This means that simulation of trees that are an order of magnitude
larger (several thousands of internodes) will not be feasible with
the current configuration. The speed of computation can probably be somewhat increased by optimizing the program code
but more sophisticated methods of numerical mathematics will
be needed if considerable improvements are to be made.
CO NC L US IO NS
Realistic simulation of linked water and assimilate transport, gas
exchange and sink activity at whole-tree level is feasible in
3-D. This opens a new avenue for plant studies. Different formulations of local processes and their responses to local and external conditions can be tested and their implications for
whole-tree-level behaviour can be evaluated. The results can
rapidly show whether the approach used has implications that
are not within realistic physical boundaries. Also, very versatile
mechanistic linkages between tree development and different
processes can be tested, making the approach a good test-bench
for studies of tree physiology. The main problem of the approach
is that, with the current formulation, large trees cannot be studied
without simplifying the structural description, developing nested
approaches or parallel computing.
ACK NOW LED GE MENTS
We thank Pasi Kolari for advice regarding the SPP model, the
Academy of Finland Centre of Excellence (grant no. 1118615),
Top-level Research Initiative (NCoE CRAICC) and the Academy
of Finland projects DECADE and MultiTree.
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presented here.
The approach presented here offers a promising new tool to
analyse processes at whole-tree level, but has limitations.
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Nikinmaa et al. — Xylem and phloem transport in a 3-D model tree crown
Page 13 of 14
APP E ND IX
Mathematical formulation of the model
2
dPxi Erx 1 Pxj + Phj − Pxi − Phi
kx(i,j)
=
Ax(i,j)
Li + Lj
dt
Vxi h0 j[Bi
2
Npi
Erx
Erx
+
lr Ppi − Pxi −
RT Ari −
QTi + Dxi
Vxi
Vpi
Vxi
dN pi k p(i,j) P pj + Phj − P pi − Phi
QPi
=
+ Ui
A p(i,j) n p(i,j) +
L
+
L
i
j
dt
h
12
j[Bi p(i,j)
2
where the factor 1/12 in the equation for Npi is derived by converting the molar amounts of CO2 assimilated into molar amounts of
sucrose 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)
if i, j are in the same axis
if i or j is lateral
1
F I G . A1. The principle of treating flows in a branched tree architecture: the
internodes exchange material with the neighboring internodes in the same axis
(grey shading) and the lateral internodes (internode #3). The set of neighbors
of internode #1 is thus B1 ¼ {2,3}. The lateral internodes are connected to internodes 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 sucrose
concentrations (n) appearing in equations for exchange of material between internodes are determined separately for each pair of internodes.
TA B L E A1. Definition of variables
Variable
dPpi Erp k p(i,j) P pj + Phj − P pi − Phi
=
A p(i,j)
Li + Lj
dt
V pi j[Bi h p(i,j)
2
Erp
N pi
−
lr P pi − Pxi −
RT Ari
V pi
V pi
⎧
⎨ min( Asi , Asj ),
= 1
⎩ min( Asi , Asj ),
2
3
s = x, p
where the multiplier 12 in the cross-sectional areas means that the
lateral internodes are connected with half of the cross-sectional
area to the internode above the branching point and with half
of the cross-sectional area to the internode below the branching
point.
The dynamic viscosity of water in phloem is
Ar
Ap, Ax
B
Dbase
L
Np
np
Ph
Pp, Px
QT
QP
SA
T
U
Vp, Vx
hp
Unit
Area between xylem and phloem
Cross-sectional area of phloem, xylem
Set of neighbouring internodes
Boundary condition of xylem water flow at base
Length of internode
Molar amount of sucrose
Sucrose concentration in phloem ( ¼ Np/Vp)
Hydrostatic pressure induced by gravitation
Water pressure, phloem, xylem
Transpiration rate of internode
Photosynthesis rate of internode
Surface area of internode
Absolute temperature
Rate of unloading of sucrose
Volume of phloem, xylem sapwood
Dynamic viscosity of water in phloem
m2
m2
m
mol
mol m – 3
MPa
MPa
mol(H2O) s – 1
mol(CO2) s – 1
m2
K
mol s – 1
m3
MPa × s
TA B L E A2. Parameters and constants of the model
Parameter
c0
Erp
Erx
kp
kx
lr
R
h0
us
hp(i,j) = 1/2(hpi + hpj )
Definition
Meaning
Reference sucrose concentration for
unloading
Phloem elastic modulus in radial
direction
Xylem elastic modulus in radial
direction
Phloem axial permeability at the base
of the tree
Xylem axial permeability at the base
of the tree
Radial hydraulic conductivity
Universal gas constant
Dynamic viscosity of pure water
at 20 8C
Slope parameter of unloading function
The values are mainly from Hölttä et al. (2006).
Value
800 mol m – 3
30 MPa
1000 MPa
1.2 × 10 – 13 m2
1.5 × 10 – 13 m2
1.0 × 10 – 7 m
(MPa × s) – 1
8.314 J mol – 1 K – 1
1.0 × 10 – 9 MPa × s
3.5 × 10 – 7 m s – 1
Downloaded from http://aob.oxfordjournals.org/ at Finnish Forest Research Institute / Library on June 18, 2014
The equations of the model given at the general level in the main
text are specified here in detail. The flows of water and sucrose in
a branched tree architecture are according to Fig. A1. Apart from
the application to a branched architecture, the equations are basically the same as in Hölttä et al. (2006). We consider in an internode i water pressure in xylem sapwood Pxi and in phloem Ppi,
and amount of sucrose solute in phloem internode Npi. In the following, subscripts x and p refer to xylem and phloem and subscripts i and j to internode number. The variables are defined in
Table A1 and the values of constants and parameters are given
in Table A2. The xylem and phloem transport model is a set of
interconnected ordinary differential equations. The equations
for each internode i are
Page 14 of 14
Nikinmaa et al. — Xylem and phloem transport in a 3-D model tree crown
and the sucrose concentration in phloem is
cp(i,j)
1
1 Npi Npj
= (cpi + cpj ) =
+
2
2 Vpi Vpj
The boundary condition for xylem water pressure assumes constant water pressure of roots, hence
i = base
otherwise
The unloading takes place only in internodes that have no needles
(unless otherwise indicated) and its rate is proportional to sucrose
concentration and the surface area of the internode, if it exceeds
the threshold concentration c0:
N pi
− c0
Ui = SA × max 0, us
V pi
√
ks = ks,top d s = x,p
where d is distance from the branch tip and ks,top isthe permeability
of the uppermost internode. This means that for the longest distance in the tree (in the stem) ks,top/ks ¼ 0.648. The above relation
is derived by combining the Hagen–Poiseuille equation, WBE
theory and conduit density per cross-sectional area, i.e. the
packing ratio of conduits (Savage et al., 2010). The Hagen–
Poisseuille equation states that k / nr 4, where r is conduit
radius and n is the number of conduits (xylem tracheids/phloem
sieve tubes). WBE theory states that r / x 1/4, where x is the distance from the apex (West et al., 1999; Anfodillo, 2006).
Conduit density, i.e. the number of conduits per cross-sectional
area, is n / 1/r 2 ¼ r – 2. This is derived from a purely geometrical
consideration assuming constant packing of conduits, and is verified by experimental data (Savage et al., 2010). Combining the
above relations yields k / r – 2r 4 / r 2 / x 1/2.
Downloaded from http://aob.oxfordjournals.org/ at Finnish Forest Research Institute / Library on June 18, 2014
⎧
⎨ Exbase kxbase −0·25 − Pxbase ,
Di = Vxbase h0
Lbase
⎩
0,
To mimic the sink of sugars in roots, the value of the unloading
slope parameter us is multiplied by 10 in the base internode in the
stem.
The axial permeabilities of xylem and phloem change as a
function of distance from the branch tip as: