Plant, Cell and Environment (2011) 34, 1819–1834
doi: 10.1111/j.1365-3040.2011.02377.x
A carbon cost–gain model explains the observed patterns
of xylem safety and efficiency
pce_2377
1819..1834
TEEMU HÖLTTÄ1, MAURIZIO MENCUCCINI2 & EERO NIKINMAA1
1
Department of Forest Sciences, 00014 University of Helsinki, Helsinki, Finland and 2School of GeoSciences, University of
Edinburgh, Crew Building, West Mains 13 Road, EH9 3JN Edinburgh, UK
ABSTRACT
Efficient water transport from the soil to the leaves is essential for plant function, while building and maintaining the
water transport structure in the xylem require a major proportion of the assimilated carbon of the tree. Xylem transport also faces additional challenges as water in the xylem is
under tension and therefore cavitation cannot be completely avoided. We constructed a model that calculates the
xylem structure that maximizes carbon-use efficiency while
simultaneously taking into account pit structure in increasing the resistance to water transport and constricting the
spreading of embolisms. The optimal xylem structure predicted by the model was found to correspond well to the
generally observed trends: xylem conduits grew in size from
the apex towards the base while simultaneously decreasing
in number, and vulnerability to cavitation increased with
conduit size. These trends were caused primarily by the
axial water potential gradient in the xylem. The pits have to
be less porous near the apex where water potential is lower
to restrict the spreading of embolisms, while whole-plant
carbon-use efficiency demands that conduit size decreases
and conduit number increases simultaneously. The model
predictions remained qualitatively the same regardless of
the exact optimality criterion used for defining carbon-use
efficiency.
Key-words: cavitation; conduit tapering; hydraulic architecture; Murray’s law; optimality; pipe model; pit membrane;
WBE model.
INTRODUCTION
Water transport from the soil to the leaves is crucial for tree
performance as tree carbon assimilation and photosynthetic production are largely dependent on the supply of
water to the leaves (Tyree & Sperry 1988; Bond &
Kavanagh 1997). In unison, the construction and maintenance of the water transport system, the xylem, require a
large proportion of the carbon assimilated by the tree. As a
tree grows in height, the transport distance from the soil to
the leaves increases and the supply of water to the leaves
grows increasingly limiting for tree performance and
Correspondence: T. Hölttä. E-mail: [email protected]
© 2011 Blackwell Publishing Ltd
growth, while the construction and maintenance of the
xylem consume an increasing proportion of the carbon
assimilated by the tree (Koch et al. 2004). The efficiency of
the xylem transport system can therefore be considered a
crucial factor for tree competence and survival.
Several theories have been put forth to describe the
structural properties of the xylem. The pipe model theory
states that there is a linear relationship between leaf and
sapwood area (Shinozaki et al. 1964). Although the pipe
model was originally put forward without interpretation of
tree hydraulic properties, it has been later interpreted to
signify that conduit size remains constant and conduit
number increases in relation to leaf area as a tree grows
(e.g. West, Brown & Enquist 1999; McCulloh, Sperry &
Adler 2003). This type of hydraulic architecture will lead to
a linear decrease in xylem conductance per leaf area as the
transport distance increases and is therefore unsustainable
at larger tree heights (West et al. 1999). West et al. (1999)
(from now onwards, WBE) proposed from the context of
general biological scaling laws that xylem conduits should
increase in size from the apex downwards according to a
power law to prevent the loss of water transport capacity
caused by increased transport length. Anatomical measurements of the axial distribution of conduit size generally tend
to support the theory in that conduits have been found to
increase from the apex downwards according to a power
law, but the power law exponent has shown variation
between studies (e.g. Anfodillo et al. 2006; Weitz, Ogle &
Horn 2006; Mencuccini et al. 2007; Petit, Anfodillo &
Mencuccini 2008). However, many argue that the theory is
too general and inaccurate to be applicable to real trees, as
in practice the whole-tree water transport capacity is not
maintained with increasing transport distance at realistic
tree heights with the WBE power law (Mencuccini 2002;
Mäkelä & Valentine 2006). Other theories for optimal
xylem structure include the maximization of hydraulic conductance per units of carbon invested in its construction,
which leads to Murray’s law (McCulloh et al. 2003), and the
hypothesis that plants should maximize their net carbon
gain (i.e. the amount of carbon that a plant can assimilate
with a given xylem structure minus the construction costs
of the system) (Mencuccini et al. 2007). The generally
observed trend of increasing vulnerability to cavitation with
increasing conduit diameter has also received much attention in the literature (e.g. Kavanagh, Bond & Knowe 1999;
1819
1820 T. Hölttä et al.
Martinez-Vilalta et al. 2002) because a direct causal relationship cannot be established between these two properties, except in the case of freezing- and thawing-induced
embolism (e.g. Davis, Sperry & Hacke 1999).
None of the previously presented theories for optimal
xylem structure have taken into account that the plant
water transport tissue is a unique system in nature because
of its reliance on meta-stable water, where water pressure is
negative and therefore embolism formation and its spreading to the conduit system is a frequent event (Tyree 2003;
Cochard 2006). Xylem embolism formation causes a
decrease in plant hydraulic conductance (Tyree & Sperry
1989), which would eventually cut off all water supply to the
leaves if the tree transpiration was not restrained by stomatal control (Tyree & Sperry 1988). Because embolism
formation cannot be completely avoided, and it is in fact a
common occurrence in all plants, the xylem cannot be just a
collection of ‘uninterrupted pipes’. Otherwise, initially small
emboli would spread to fill the whole xylem eliminating its
capacity to transport water. To reduce the spread of embolisms, xylem conduits need to have a finite size with relatively non-porous pits joining adjacent conduits. Small
pores in the pit membranes adjoining conduits are able, to a
certain extent, to prevent gas emboli from spreading from
one conduit to another by surface tension forces (Sperry &
Hacke 2004).
The pits also constitute a major proportion, roughly half
or even much more, of the hydraulic resistance of the xylem
(Lancashire & Ennos 2002; Sperry, Hacke & Wheeler 2005;
Choat et al. 2006). Therefore, they can be argued to be as
crucial a component in the xylem hydraulic architecture as
the conduit lumens are. It could even be argued that the
xylem is a very inefficient conduit system due its small
conduit size if the requirement for constricting emboli is not
taken into account, because water transport capacity is proportional to the fourth power of the conduit lumen diameter (Zimmermann 1983) while the construction cost of
conduits is proportional to the second power of it (Hacke
et al. 2001). Without the pits, each increase in the order of
magnitude of conduit diameter would therefore bring about
an increase in two orders of magnitude in the ratio of
hydraulic conductance to invested carbon. Larger conduits
would always be much more efficient in transporting water,
but because the pits adjoining the conduits will necessarily
have a relatively high resistance, very large conduit lumens
would add very little extra to the hydraulic conductance
(Sperry & Hacke 2004). On the other hand, the structure of
the pits must be dependent on the water potential that the
xylem experiences. Lower water potentials demand less
porous and therefore less conductive pit membranes in
order to restrain the spreading of embolisms by air seeding.
Therefore, the xylem hydraulic structure has to be ‘tuned’ to
satisfy the needs of the local water potential environment.
Should a model of xylem hydraulic structure be built
using strict optimality criteria and, if so, according to which
criteria? Models built using optimality criteria are useful to
help identify the main biophysical and ecological constraints that limit plant performance (see e.g. Mencuccini,
Hölttä & Martinez-Vilalta 2011, for a review of such models
in hydraulics). One can only speculate what the exact optimization criteria would be for a tree (Niklas 1994). The
WBE model ‘minimizes’ transport resistance while being
constrained by biomechanics. Murray’s law (McCulloh et al.
2003) maximizes hydraulic conductance per unit of vascular
investment, and the model of Mencuccini et al. (2007) maximizes the plant’s net carbon gain (i.e. the gross carbon
assimilation rate minus the carbon costs of building the
hydraulic system of a tree). Furthermore, carbon-use efficiency (i.e. the carbon assimilation rate divided by the
carbon costs of building the hydraulic system) might be
optimized. It would be sensible to consider them in terms of
evolutionary fitness. Why would a plant maximize hydraulic
or carbon-use efficiency instead of maximizing the net
carbon surplus, which may support its other functions (e.g.
nutrient scavenging and uptake, carbon storage and, especially, reproduction)? In addition to this uncertainty related
to the choice of the optimality criterion, these previous
models of plant hydraulics incorporated a variable level of
detail in the description of the plant hydraulic system. Are
some of their limitations caused by the lack of a fundamental process in their representation of the structure of the
vascular structure? We argue that this is the case and that
such process relates to pit functioning.
A model is presented that predicts xylem conduit structure and the operating xylem potential that maximizes the
net carbon gain of the tree, while taking into account the
structural necessities to prevent the spreading of emboli.
We also compare the behaviour of this model under different optimization schemes, to test whether our results are
sensitive to the choice of the actual criterion employed. A
summary of the characteristics of this model in relation to
previous models of xylem hydraulic architecture is presented in Table 1. Unlike other previous models describing
optimal xylem structure, no restrictions (except on conduit
length) are given a priori to conduit size or the number of
conduits at the apex, pit structure or the water potential
that the plant operates at. The objective of this study was
not the testing of the WBE theory or of any other hydraulic
structure theory. Rather, we aimed to go deeper into the
theory of xylem water transport to identify the connections
between xylem structural design and its carbon requirements, and the connections between xylem water potential,
pit structure and conduit lumen size. We emphasize the
special characteristic of plant water transport, the metastability of water, as it sets large limitations to the design of
the xylem hydraulic architecture. Pure degassed water can
tolerate pressures (~ water potentials) down to -35 MPa at
room temperatures when not in contact inpure surfaces
(e.g. Briggs 1950), but the xylem structure contains inpurities and the sap is not pure degassed water. The specific aim
was to explain simultaneously from a ‘closed form’ theoretical construction: (1) why conduit size and number are what
they are; (2) why conduits taper and furcate; (3) why plants
operate at the water potentials that they do; (4) why there is
a correlation between conduit size and vulnerability to cavitation; and (5) why tree relative productivity declines
© 2011 Blackwell Publishing Ltd, Plant, Cell and Environment, 34, 1819–1834
Carbon net gain
Criteria
Models
This model
Conduit number
Conduit size
Pits
© 2011 Blackwell Publishing Ltd, Plant, Cell and Environment, 34, 1819–1834
Optimized, starting
from fixed size
at the top
Not consideredd
Optimized
Not considered
Fixed
Considered
Not considered Considered
Not considered
Violates
requirements
Yes
No
No
Moderate
Moderate
Low
Very high
Dependency on
environmental Level of
conditions
detail
Not considered,
Yes
but can be added
Not considered Not considered Considered
Considered
Mechanical
Mechanical
support against support against
xylem tension
outside forces
We used the term ‘efficiency’ to indicate the ratio of input to outputs of a system (i.e. ratio of carbon gains to carbon requirements to build a system; we used ‘efficacy’ to indicate instead an unbounded estimate of efficiency
because costs are not accounted for.
b
This is true mathematically when tree height approaches infinity.
c
Bounded by biomechanical constraints. Computes conductance of each tapered pipe.
d
But extension by Becker et al. (2003) shows results do not change when pit conductance scales with lumen conductance.
Efficiency; computes Fixed
costs of conduit
wall construction
Carbon net gain
Fixed
Mencuccini et al.
net C gain
Efficacyc
Hydraulic conductance Efficiency; computes Optimized together Optimized together Not considered
for a given carbon
costs of conduit
with tapering
with furcation
investment
wall construction
a
Water
potential
Efficiency; computes Optimized together Optimized together Optimized to balance Optimized
costs of conduit
with tapering
with furcation
conductance and
wall construction
prevention
of cavitation
Efficiency versus
efficacya
Murray’s law
WBE (West et al. Maintain hydraulic
1999)
conductanceb
Optimization
criteria
Table 1. Summary of previous xylem hydraulic architecture models
A carbon gain-cost model for xylem structure 1821
1822 T. Hölttä et al.
because of hydraulic limitations as it grows taller. Our
results are obtained on the basis of only four assumptions:
(1) that Hagen–Poiseuille applies to describe water flow in
capillaries; (2) that Laplace law applies to describe the pressures required to drain a capillary; (3) that conduit lumen
and pit resistances can be approximated to be connected in
series; and (4) that conduit wall thickness is proportional to
conduit lumen diameter.
THEORY
Hydraulic conductance
The resistance to water flow in the xylem consists of two
compartments: the conduit lumens and the pits connecting
the lumens. Lumen conductance (kl) per conduit length
(m3 Pa-1 s-1) can be calculated from the Hagen–Poiseuille
law (Zimmermann 1983)
π r( x)
,
8 μ lc
4
kl ( x ) =
(1)
where r is conduit radius, lc is conduit length and m is the
dynamic viscosity. The variables which can change with
axial position (x) in our model are so marked. Conduit
length can also vary with height, but this was not considered
in our model.
Pit conductance (kp) per conduit length (m3 Pa-1 s-1) for
angiosperm pits can be estimated by multiplying the conductance of one pit (Sperry & Hacke 2004) times the
number of pits in the whole conduit
npores 8r ( x )p,ave
3
kp ( x ) = npits ( x ) ∗ 0.5
24 μ
(2)
,
where npits is the number of pits in a conduit, npores is the
number of pores in a pit and rp,ave is the average pore radius.
The coefficient 0.5 is included to account for the effect of
the pit aperture (Sperry & Hacke 2004) on pit conductance.
Hydraulic conductance per conduit length (kc) is calculated
from lumen (kl) and pit (kp) conductances per conduit connected in series (Lancashire & Ennos 2002)
−1
−1
−1
k ( x )c = k ( x ) l + k ( x ) p ,
rp,max ( x ) =
(5)
Transpiration and carbon assimilation rates
The transpiration rate from the leaves is (e.g. Jarvis &
McNaughton 1986)
T = ags ΔwS,
(6)
where gs is stomatal conductance, a (= 1.6) is the relation
between stomatal conductance for water and CO2 (as gs is
expressed for CO2), Dw is the vapour pressure deficit and S
is leaf area. This equation assumes infinitely high boundary
layer and aerodynamic conductances in the canopy, conditions which are roughly satisfied in the case of trees.
In steady state, the water flow rate through the xylem
must be equal to the transpiration rate, so by combining
Eqns 4 and 6, we obtain the maximum stomatal conductance which can be sustained for a given leaf water potential
and a given water supply to the leaves
gs = −
The water flow rate through the xylem (J) is the same
everywhere at each height in the tree at steady state (Zimmermann 1983)
2γ
,
Ψ ( x)
where g is the surface tension of water. While the conductance of the pits will depend on an average pit pore size,
constriction of embolism spreading (i.e. air seeding) will
depend on the maximum pore size in the conduit. Following
the approach of Sperry & Hacke (2004), the average pore
size was made to scale with the largest pore size in the
conduit so that the former is 0.63 times the size of the latter.
Furthermore, experimental evidence supports the assumption that the cavitation pressures in individual conduits are
correlated with the average pore diameter of the pit membranes (Jansen, Choat & Pletsers 2009). In reality, the pit
pore size is likely to stretch under tension (Sperry & Hacke
2004), but this is not taken into account here.
(3)
Water flow, water potential and confining the
spreading of embolisms
J = (Ψ s −Ψ l − ρ gh ) ktot ,
In order to prevent the spreading of embolisms by
surface tension forces, the maximum pit pore size rp,max must
be restricted in relation to the water potential y at each
height according to Laplace’s law so its value is at most
(Zimmermann 1983)
(Ψ l −Ψ s + ρ gh ) ktot
,
aΔwS
(7)
The carbon assimilation rate used for xylem production (G)
which can be achieved in a year with the given stomatal
conductance is
G=
gsCa f
St,
gs + f
(8)
(4)
where ys is soil water potential, yl is leaf water potential,
r is density of water, g is the gravitational constant
(9.81 ms-2), h is plant height and ktot is the whole xylem
hydraulic conductance, which can be integrated provided
the axial distribution of the conduit properties are known.
where t is the time in the year during which photosynthetic
production takes place, Ca is the ambient carbon dioxide
concentration and f is a function of the photosynthetically
active radiation (I) (an average value over the year when
photosynthesis takes place) and photosynthetic parameters
(a) and (G) (Mäkelä, Berninger & Hari 1996), where
© 2011 Blackwell Publishing Ltd, Plant, Cell and Environment, 34, 1819–1834
A carbon gain-cost model for xylem structure 1823
f (t ) =
αI
,
I +Γ
(9)
Carbon costs of the xylem
The carbon cost (Cc) associated with each xylem conduit
equates to the amount of carbon deposited in the xylem
conduit walls. Conduit wall thickness (tw) was modelled to
be dependent on xylem water potential and lumen radius
(Hacke et al. 2001)
tw ( x ) = r ( x ) − A1Ψ ( x ) + A2 ,
(10)
and
Cc ( x ) = Fc
θ
2
2
n ( x ) lπ ⎡⎣( r ( x ) + tw ( x ) ) − r ( x ) ⎤⎦ ,
Y
(11)
where A1 and A2 parameters related to the mechanical reinforcement of the conduit walls to withstand water tension,
and they have been empirically estimated from (Hacke
et al. 2001). q is a constant determined by the relationship
between conduit wall volume and carbon content, and Y is
conduit functional longetivity in years. The carbon costs
were multiplied by a factor of Fc to account for the growth
and maintenance respiration of the xylem.
NUMERICAL MODEL
For numerical calculations, a model tree with a single vertical axis supporting a leaf area at the top was divided into
N numerical elements of equal length l (l = h/N where h is
tree height). There was no branching, and transpiration
occurred from only the end point element i = N, so that the
model tree followed essentially the WBE structure. The
distance from the apex (x) was now substituted by i, succession of the numerical element, so that i = 1 is the numerical
element at the tree base and N is the numerical element at
the apex. The hydraulic conductance k(i) per numerical
element i is then
−1
−1
4
n ( i ) npits ( i ) npores 8r ( i )p,ave ⎞
⎛
⎛ n ( i )π r ( i ) ⎞
−1
k (i ) = ⎜
⎟ ,
⎟ + ⎜ 0.5 ∗
ns 24 μ
⎝ ns 8 μ lc ⎠
⎝
⎠
(12)
3
where n(i) is the number of conduits in parallel in an
element (i.e. the number of conduits per cross section at
that height) and ns (element length divided by conduit
length) is the number of conduits in series per element. The
variation in conduit length with height was not considered,
but it was given a constant value. The water potential at
each numerical element was then calculated according to
Ψ ( i ) = Ψ ( i − 1) −
and
ags ΔwS
k (i )
for i > 1,
(13a)
Ψ ( 1) = Ψ s −
ags ΔwS
kr
for i = 1,
(13b)
where kr is the root and soil hydraulic conductance. For
simplicity, kr was made to scale with xylem hydraulic conductance so that it was always twice the whole xylem conductance (i.e. it accounted for one-third of the total
hydraulic resistance between the soil and the top of the
xylem).
The optimization task was to choose conduit radius
r(i), the number of conduits n(i) and leaf water potential
y(N) in such a way that the net gain of carbon (G - Cc)
or, alternatively, some other criterion, such as the carbonuse efficiency (G/Cc) or the hydraulic conductance per
unit of carbon use (ktot/Cc) was maximized. Maximization
carbon gain or hydraulic conductance per se (i.e. without
giving consideration to costs) would be unreasonable
as it always leads to infinitely large and many conduits.
Altogether, five independent parameters (A, B, C, D
and E) were varied simultaneously without constriction
whereby:
r ( i ) = Ax B ,
(14a)
n ( i ) = Cx D ,
(14b)
Ψ ( N ) = E,
(14c)
where x is the distance from the apex. Conduit pit pore size
[which affects k(i) and the xylem water potential profile]
and xylem water potential were solved repeating the following iterative process until the water potential and pore
size were in equilibrium according to Eqn 5 at each numerical element:
1 The axial water potential profile was calculated using
Eqn 13. The pore size distribution was given an axial
distribution assuming a linear change in xylem water
potential with tree position in the first iteration round.
Note that the choice for the initial pore size distribution
does not affect the model results. It affects only the rate
of convergence of the solution.
2 The pore size distribution was calculated to match the
local water potential using Eqn 5 so that pore size at each
element was made to be as large as possible, but still small
enough to be able to constrict the spreading of embolisms
from one conduit to another. The condition for equilibrium was fulfilled when the change in pit pore radius
between two consecutive iteration rounds in all numerical elements was less than 0.1 nm.
Finding the maximum net gain (or maximum carbon-use
efficiency) by optimizing the independent parameters (A to
E) together was done using a Markov chain Monte Carlo
simulation. The details of the method, the demonstration of
convergence of the solution, discussion on the possibility of
finding a global (instead of a local) maximum and the
© 2011 Blackwell Publishing Ltd, Plant, Cell and Environment, 34, 1819–1834
1824 T. Hölttä et al.
program code used are given in the Supporting Information
Appendix S1.
Parameterization for the model above is shown in
Table 2. Furthermore, sensitivity of the model to the
choice of the parameters is demonstrated in the model
calculations that follow. The sensitivity analysis was
carried out as a Monte Carlo analysis so that all of the
model parameters which can have variation in reality were
varied simulataneously. Altogether, 1500 repetition runs,
where the parameter values varied, were conducted. At
each repetition run, the parameters were chosen randomly
from a distribution. We used a logarithmic distribution for
the parameter values with mean value representing the
value presented in Table 2. The logarithmic scale was used
to be able to include also extreme but rare occurrences in
the sensitivity analysis.
Conduit pit number was always made to scale linearly
with conduit wall area so that conduit wall pit density
remained constant. Conduit length was given a constant
value. The power law form (in Eqn 14) was chosen to
model the change in conduit size and number as a function of distance from the apex, as this gave larger xylem
net carbon gain compared to the linear form for either
conduit size or number. Furthermore, the power law
form has been found to describe conduit tapering best
in anatomical measurements (e.g. Mencuccini et al. 2007).
Total leaf area was made to scale with tree height raised
to the second power. In the simulations, the soil water
potential should be thought to represent a minimum value
which the tree is likely to encounter. The carbon cost of
the xylem was assumed to be double that of the carbon
allocated to the cell walls (i.e. Fc in Eqn 11 was set to 2
because of respiratory losses within the xylem).
In gymnosperms, the pit structure and cavitation mechanisms are more complicated than what is described above,
and the relationships between pit conductivity and vulnerability to air seeding described above do not directly apply
to them. Nevertheless, a similar trade-off between pit vulnerability and pit conductance can also be expected for
gymnosperms from both theoretical (Hacke, Sperry & Pittermann 2004) and empirical (Domec, Lachenbruch &
Meinzer 2006; Domec et al. 2008; Ambrose, Sillett &
Dawson 2009) grounds. In gymnosperms, the vulnerability
to cavitation appears to be primarily linked to the pit
aperture to torus diameter ratio. Pit aperture and torus
diameter also affect pit conductance, and hence the tradeoff between the two. Pit membrane pore resistance is very
small and unimportant for the trade-off in gymnosperms.
However, a recent study by Cochard et al. (2009) showed
that surface tension forces also play a key role in gymnosperm cavitation. In addition, the structure and function of
the pits in temperate conifer species are also required to
tolerate freeze-thaw events with the associated winter
embolism formation, and also to allow needles to have
access to water on warmer winter days (Hammel 1967;
Pittermann & Sperry 2003). In summary, the model is
directly applicable to angiosperms only, and application to
conifers is possible but not directly attempted here.
RESULTS
Optimal xylem structure, based on the criterion
of maximum net carbon gains
Maximizing net carbon gain led to similar patterns in xylem
architecture as have been empirically found: increasing
conduit radius (Fig. 1a) and decreasing number of conduits
from top to bottom (Fig. 1a), and approximately equal distribution of resistances between the lumens and pits
(Fig. 1b). The tapering exponent (B in Eqn 14a) was 0.17,
with 0.25 being the case for the WBE model (see Anfodillo
et al. 2006 for a comparison of the tapering exponent
expressed per unit length versus WBE per unit segment).
The water potential gradient steepened towards the apex
(Fig. 1b) as the hydraulic conductivity decreased because of
decreasing conduit size and pit conductance. The axial
profile of the conducting xylem cross-sectional area
(Fig. 1c) (i.e. conduit radius raised to the second power
multiplied by conduit number) remained close to constant
(as is predicted by the pipe model theory), while the axial
profile of conduit number multiplied by conduit radius
raised to the third power, which is predicted to remain
constant according to Murray’s law, increased sharply from
tree top to bottom.
Monte Carlo sensitivity analysis
The absolute values for leaf water potential, conduit size
and tapering, conduit number and furcation, etc., were
affected by the choice of parameterization. Especially soil
water potential, soil hydraulic conductance (i.e. the proportion of soil conductance to xylem conductance) and conduit
length had major influence on the predicted hydraulic structure (Table 3). The hydraulic structure was most sensitive to
soil water potential, soil hydraulic conductance and tree
height (Table 3). Nevertheless, the most important qualitative trends always remained the same between all of the
parameterizations: conduits always increased in size from
top to bottom (average tapering 0.22 ⫾ 0.19) and decreased
in number from tree top to bottom (average furcation
0.44 ⫾ 0.36). Out of the 1500 Monte Carlo simulations, 330
parameter combinations lead to the result that no xylem
structure was able to produce a positive net gain. These
combinations were excluded from the statistical analysis as
they represent ‘impossible’ conditions (otherwise referred
to as ‘non-behavioural’ runs, e.g. Beven & Binley 1992).
When all model results from the Monte Carlo simulations were put together, a clear relationship emerged
between conduit radius and water potential (which is the
same as cavitation water potential) at the tree top (Fig. 2a).
The data points showed a power law relationship (with y a
x-0.81, R2 = 0.50). The same was true when xylem water
potential and conduit size across all heights in all of the
simulations were compared (not shown) with y a x-0.85,
R2 = 0.48. A clear relationship was also found between
conduit furcation and tapering. Larger furcation was associated with larger tapering and vice versa (with y a 2.37x,
R2 = 0.97, Fig. 2b). A smaller number of conduits was
© 2011 Blackwell Publishing Ltd, Plant, Cell and Environment, 34, 1819–1834
Soil water potential
Vapour pressure deficit (VPD)
Photosynthetically active radiation (PAR)
Tree height
Conduit length
Whole-tree leaf area
Photosynthetic parameter
Photosynthetic parameter
Conduit functional longetivity
Hours of photosynthesis in a year
Number of pits in a conduit
Number of pores in a pit
Hydraulic conductance from soil to tree in
relation to total xylem conductance
Empirical parameter relating cell wall thickness
to conduit size and water potential
Empirical parameter relating cell wall thickness
to conduit size and water potential
Ambient CO2 concentration
Factor by which xylem carbon costs are
multiplied by to account for carbon costs of
xylem respiration
Carbon required per cell wall volume
Surface tension of water
Dynamic viscosity of water
ys
Dw
I
H
L
S
A
G
Y
T
npits
npores
kr/ktot
© 2011 Blackwell Publishing Ltd, Plant, Cell and Environment, 34, 1819–1834
Fc
G
m
Estimateda
400 ppm
0.001 Pa s-1
0.073 Nm
(Siau 1984), assuming carbon
is 50% of dry matter
CRC Handbook of Chemistry
and Physics
CRC Handbook of Chemistry
and Physics
Fitted from Hacke et al. (2001)
0.005 mm
46 875 mol C m-3
Fitted from Hacke et al. (2001)
6.0a10-9 Pa-1
Estimateda
Not included in
sensitivity analysis
1
Estimateda
Hypothetical value
2
0.00167 mm
Mäkelä et al. (1996)
Mäkelä et al. (1996)
Estimateda
Estimateda
Wheeler et al. (2005), constant
pit density is assumedb
0.04 mmol m-2 s-1
400 mmol m-2 s-1
3 years
3 888 000 s (180 d, 6 h d-1)
5% of conduit wall area occupied by pits.
1000 for a 20 mm in radius and 1 cm in length
conduit, scales linearly with conduit wall area
100
2.0
Not included in
sensitivity analysis
Not included in
sensitivity analysis
Not included in
sensitivity analysis
2
33
0.6
-5.0 MPa
1.3 kPa
167 mmol m-2 s-1
6.7 m
3 mm
26.6 m
Scales with tree
height S a h2
0.0133 mmol m-2 s-1
133 mmol m-2 s-1
1 year
1 295 870 s
1.67%
Estimateda
Estimateda
Estimateda
Estimateda
Estimateda
Estimateda
-1.0 MPa
4 kPa
500 mmol m-2 s-1
20 m
3 cm
80 m
Scales with tree height, S a h2
Lower bound in
sensitivity analysis
Reference
Value
Not included in
sensitivity analysis
Not included in
sensitivity analysis
Not included in
sensitivity analysis
Not included in
sensitivity analysis
4
0.0167 mm
20
333
6
-0.05 MPa
13 kPa
1670 mmol m-2 s-1
67 m
30 cm
266 m
Scales with tree
height S a h2
0.133 mmol m-2 s-1
1333 mmol m-2 s-1
10 years
12 958 704 s
16.7%
Upper bound in
sensitivity analysis
b
Estimated to be a typical value for a tree in typical conditions in temperate forest.
Constant pit density is assumed. Assuming that pore area is 50% pit area, or 7% is pore area is 33% pit area. Actually, the number of pits times the number of pores in the variable matters,
not the values of those per se.
a
Q
Ca
A2
A1
Paremeter description
Parameter
symbol
Table 2. Parameterization of the model
A carbon gain-cost model for xylem structure 1825
1826 T. Hölttä et al.
Sensitivity to soil water potential, tree height
and conduit length
Next, only one input parameter was varied at a time, while
the other parameters were maintained at the values given
in Table 2. Low soil water potential (Fig. 3a), low tree
heights (Fig. 3c) and long conduits (Fig. 3e) lead to larger
and fewer conduits, and to larger tapering and furcation,
and vice versa (see also Table 3). Leaf water potential
was most affected by soil water potential with low leaf
water potential corresponding to low soil water potential
(Fig. 3a). Greater tree heights and conduit lengths were
also associated with slightly lower leaf water potential
(Fig. 3c,e). The fraction of the total hydraulic resistance
allocated to the pits (not shown) was found to be quite
insensitive to soil water potential and tree height, but it
was found to decrease with increasing conduit length.
Carbon gains, net carbon gains and xylem hydraulic conductance per leaf area decreased, and carbon costs
increased when soil water potential decreased (Fig. 3b),
tree height increased (Fig. 3d) and conduit length
decreased (Fig. 3f).
Results for other optimization criteria
Figure 1. Xylem properties when net carbon gain is maximized:
the axial profile of conduit radius and conduit number (a),
proportion of total hydraulic resistance of the xylem allocated to
pits and xylem water potential (b) and the sum of the conduit
number multiplied by conduit radius raised to the second power
(equivalent to the total cross-sectional area of the xylem conduit
lumens) and the sum of the conduit number multiplied by
conduit radius raised to the third power (c).
associated with larger conduits at the top (Fig. 2c, y a x-2.46,
R2 = 0.70). In addition, larger conduits at the top were associated with more tapering (Fig. 2d, y a x0.20, R2 = 0.17), and
fewer conduit number with more furcation (Fig. 2e, y a
x-0.09, R2 = 0.27).
Allocation of resistance between the pit and lumen, and
the axial profiles of the conducting xylem cross-sectional
area were relatively unaffected by changes in the parameterization. Both ratios remained quite constant with
a fraction of 0.66 ⫾ 0.060 (standard deviation) of the
hydraulic resistance allocated to the pits, and a xylem
cross-sectional area of 1.03 ⫾ 0.13 (standard deviation) at
other heights in the tree in relation to the apex across all
of the parameterizations.
Maximizing carbon-use efficiency (i.e. the assimilation
rate divided by the carbon costs of building the hydraulic
system) or the hydraulic conductance per unit of carbon
costs gave qualitatively the same patterns for all properties as maximizing net carbon gain (Fig. 4): conduit size
still increased from apex downwards with a simultaneous
decrease in conduit number; larger conduits were more
vulnerable to cavitation. In both of these alternative
optimization criteria, larger furcation was associated with
larger values of tapering, and smaller conduit radius was
associated with larger conduit number (not shown) similarly to the case where the maximum net gain was
maximized.
However, maximizing carbon-use efficiency gave smaller
conduits at the apex with larger tapering downwards
(Fig. 4a), a larger conduit number at the apex with more
furcation (Fig. 4a), a lower leaf water potential (Fig. 4b) and
much larger axial variation in the fraction of total hydraulic
resistance allocated to pits and the cross-sectional conducting area of the xylem. In this case, the absolute number of
conduits (parameter C in Eqn 14b) was not allowed to vary
freely, but it was fixed to be the same as in the optimal
solution for the net carbon gain, because the smallest
number of conduits at the apex would always give the
highest efficiency.
Maximizing hydraulic conductance per unit of carbon
costs (i.e. our third optimality criterion) led to an almost
identical pattern in all of the variables as maximizing
net carbon gain (Fig. 4a–c). Furthermore, in this case, the
absolute number of conduits at the apex had to be fixed to
be the same as in the optimal solution for the net gain,
as also now the solution was independent of the
absolute number of conduits. Additionally, the xylem
© 2011 Blackwell Publishing Ltd, Plant, Cell and Environment, 34, 1819–1834
A carbon gain-cost model for xylem structure 1827
Table 3. Linear regression correlation coefficients between input and output variables (a) and sensitivity (linear) of model output
variables to input variables (b)
(a)
Variable
Tree height (h)
Conduit functional longetivity (Y)
Photosynthetically active radiation (PAR, I)
Vapour pressure deficit (VPD, Dw)
Whole-tree leaf area (S)
Soil water potential (ys)
Conduit length (l)
Number of pits in a conduit (npits)
Number of pores in a pit (npores)
Hydraulic conductance from soil to tree in relation to total xylem
conductance (kr/ktot)
Empirical parameter relating cell wall thickness to conduit size and water
potential (A1)
Empirical parameter relating cell wall thickness to conduit size and water
potential (A2)
Photosynthetic parameter (a)
Photosynthetic parameter (G)
Hours of photosynthesis in a year (t)
Factor by which xylem carbon costs are multiplied by to account for
carbon costs of xylem respiration (Fc)
Leaf water
potential (yl)
Conduit
number
Conduit
furcation
Conduit
radius
Conduit
tapering
-0.11
-0.06
0.04
0.08
-0.11
0.99
-0.07
-0.06
-0.06
-0.13
0.30
0.16
0.05
-0.07
0.35
-0.23
-0.20
-0.07
-0.07
0.09
-0.35
-0.08
0.01
0.03
-0.25
0.48
0.18
0.04
0.03
0.19
-0.13
-0.06
0.06
0.09
-0.09
0.51
0.50
0.09
0.14
-0.12
-0.40
-0.07
0.00
0.02
-0.28
0.42
0.18
0.04
0.03
0.30
0.03
-0.06
-0.02
-0.03
0.01
-0.07
0.00
-0.01
-0.03
-0.02
0.05
0.01
-0.09
0.01
0.15
0.00
0.13
-0.08
-0.01
0.01
-0.06
0.00
0.04
-0.02
0.02
0.04
-0.01
0.00
-0.05
0.00
Leaf water
potential (yl)
Conduit
number
Conduit
furcation
Conduit
radius
Conduit
tapering
-0.10
-0.05
0.03
0.07
0.00
0.29
-0.02
-0.05
-0.05
-0.11
0.22
0.11
0.03
-0.05
0.00
-0.05
-0.05
-0.05
-0.05
0.06
-0.35
-0.07
0.01
0.02
0.00
0.15
0.05
0.04
0.02
0.18
-0.11
-0.05
0.05
0.07
0.00
0.14
0.13
0.07
0.11
-0.10
-0.36
-0.06
0.00
0.02
0.00
0.12
0.05
0.04
0.02
0.25
0.02
-0.04
-0.01
-0.02
0.01
-0.06
0.00
-0.01
-0.03
-0.02
0.04
0.01
-0.07
0.01
0.10
0.00
0.08
-0.11
-0.01
0.00
-0.05
0.00
0.03
-0.01
0.01
0.06
-0.01
0.00
-0.04
0.01
(b)
Variable
Tree height (h)
Conduit functional longetivity (Y)
Photosynthetically active radiation (PAR, I)
Vapour pressure deficit (VPD, Dw)
Whole-tree leaf area (S)
Soil water potential (ys)
Conduit length (l)
Number of pits in a conduit (npits)
Number of pores in a pit (npores)
Hydraulic conductance from soil to tree in relation to total xylem
conductance (kr/ktot)
Empirical parameter relating cell wall thickness to conduit size and water
potential (A1)
Empirical parameter relating cell wall thickness to conduit size and water
potential (A2)
Photosynthetic parameter (a)
Photosynthetic parameter (G)
Hours of photosynthesis in a year (t)
Factor by which xylem carbon costs are multiplied by to account for
carbon costs of xylem respiration (Fc)
The results are based on 1500 Monte Carlo simulations where the parameter values were varied as shown in Table 2. The sensitivity values
are scaled so that the sum of each model prediction (conduit size, number . . .) is equal to unity. The correlation is statistically significant
(P < 0.05) when the correlation coefficient is at least 0.06.
water potential at the apex had to be fixed (it was fixed to
equal the leaf water potential in the optimal solution for
the net gain), as the highest xylem water potential always
gave the highest hydraulic conductance per unit carbon
cost. Note that this optimizing criterion is not equivalent
to Murray’s law’s optimizing criterion, and thus it also
leads to a different prediction. The optimization criterion
for Murray’s law does not take into account the relation
between xylem water potential and pit conductance
(Laplace’s law). In addition, unlike in the case of Murray’s
law, the amount of carbon used for construction is not
fixed here.
© 2011 Blackwell Publishing Ltd, Plant, Cell and Environment, 34, 1819–1834
1828 T. Hölttä et al.
Figure 2. Structural relationships from combining all model results from the 1500 Monte Carlo simulations: leaf water potential
(equivalent to air seeding pressure at the leaf) versus conduit radius at tree top (a), conduit tapering versus furcation (b), conduit radius
at tree top versus conduit number at tree top (c), conduit radius at tree top versus conduit tapering (d) and conduit number at tree top
versus conduit furcation (e). Conduit furcation and tapering are expressed with respect to numerical element (i) and not with respect to
actual distance (i*le) as in Eqn 14a,b, so that variation in tree height is comparable with variation in the other parameters.
The sensitivity of carbon gains and costs in
response to changes in xylem structure
We also examined how much a departure from the
optimal xylem structure would influence carbon gains and
costs. Firstly, we considered a case without tapering and
furcation. The maximum net carbon gain was achieved
with a constant conduit diameter of 46 mm and a conduit
number of 0.6 million conduits (Fig. 5a, where the contour
has the highest value), when xylem water potential at the
apex was given its optimal value of -2.5 MPa. The optimal
conduit number and conduit size were of the same value
as approximately at the middle of the tree when tapering
and furcation were allowed (see Fig. 1a). Next, conduit
radius and number at the apex were kept at their optimal
value (as in the results in Fig. 1). Figure 5b demonstrates
the sensitivity of the net carbon gains to changes in
tapering and furcation. The net carbon gains showed only
a small deviation from the optimal value when tapering
and furcation changed simultaneously in the same direction, but a large deviation when tapering and furcation
were changed in opposite directions.
DISCUSSION
As restrictions such as conduit properties at the apex or
water potentials were not imposed on the possible xylem
structure, the xylem water potential profile and pit porosity were central in determining the xylem conduit size and
number, and their axial distribution along the xylem. The
optimal xylem structure was predicted to be qualitatively
similar under all environmental conditions and according
to all of the optimizing criteria that were considered: conduits grew in size from the apex towards the base while
© 2011 Blackwell Publishing Ltd, Plant, Cell and Environment, 34, 1819–1834
A carbon gain-cost model for xylem structure 1829
Figure 3. Conduit radius, conduit number, conduit tapering, conduit furcation, xylem water potential, carbon gain (G in Eqn 9), carbon
cost (Cc in Eqn 11), net carbon gain (G - Cc) and total xylem hydraulic conductance per unit leaf area when soil water potential (a and
b), tree height (c and d) and conduit length (e and f) are varied.
simultaneously decreasing in number. However, maximizing the carbon-use efficiency (instead of maximizing the
carbon net gain or hydraulic conductance per unit of
carbon costs) gave unrealistically large (in comparison to
values found in the literature, e.g. Mencuccini et al. 2007)
values for conduit tapering and furcation, and unrealistically large variation in the axial profile of cross-sectional
conducting area of the xylem and the proportion of total
hydraulic resistance in pits. This indicates that maximizing
carbon-use efficiency is not very close to the design criteria for plant xylem hydraulic architecture. As water potential decreases towards the apex, the pits adjoining the
conduits have to become less porous and conductive so
conduit diameter will also decrease towards the apex to
fulfil any optimization criteria which include the carbon
construction cost of the xylem (i.e. the WBE model is
inherently unable to make such a prediction). The
model simultaneously provides an explanation as to why
conduit vulnerability increases with conduit size. Large
conduits are efficient only when pit conductance is high,
and large pit conductance leads to increased vulnerability
to cavitation.
The trends predicted by our model are well supported
by empirical evidence: conduits diameter increases (e.g.
Anfodillo et al. 2006; Mencuccini et al. 2007) and their
number decreases (Sperry, Meinzer & McCulloh 2008)
from tree top to bottom. The hydraulic resistance of the
xylem is divided approximately equally among the conduit
lumens and pits (Sperry et al. 2005) with larger and more
conductive conduits always being more vulnerable to cavitation in intra-specific comparisons (e.g. Tyree & Sperry
1989; Domec & Gartner 2001; Domec et al. 2008). The
© 2011 Blackwell Publishing Ltd, Plant, Cell and Environment, 34, 1819–1834
1830 T. Hölttä et al.
comparisons (represented by the results in Fig. 2a) provided that other structural features, namely conduit length
and the proportion of conduit wall occupied by pits in a
conduit, are similar among the species compared. The
nearly inverse relationship between conduit size and water
potential (the same as vulnerability to cavitation) arises
from the Laplace equation (Eqn 5), as conduit size must
scale approximately with pit pore size (pit conductance
must scale with lumen conductance) if carbon use is efficient. The power law exponent of slightly less than -1 (and
not -1) predicted by the model results from the different
Figure 4. Xylem properties for alternative optimization criteria:
maximizing carbon-use efficiency (i.e. the gross carbon gain
divided by the carbon costs of building the xylem) (G/Cc), or the
total hydraulic conductance per carbon cost (ktot/Cc): the axial
profile of conduit radius and number (a), proportion of total
resistance in pits and xylem water potential (b), and the sum of
the conduit number multiplied by conduit radius raised to the
second power (equivalent to the total cross-sectional area of the
xylem conduit lumens) and the sum of the conduit number
multiplied by conduit radius raised to the third power (c).
relationship between conduit size and vulnerability has
also been found in many inter-specific comparisons (e.g.
Kavanagh et al. 1999; Martinez-Vilalta et al. 2002; Choat
et al. 2005), although in some other cases this comparison
has not held (e.g. Cochard 1992). The nearly inverse relationship between conduit size and vulnerability is predicted by the model to hold also for inter-specific
Figure 5. Net carbon net gain as a function of conduit radius
and conduit number at its optimal xylem water potential without
tapering or furcation (a) and with varying tapering (B in Eqn
14a) and furcation (D in Eqn 14b) with the optimal conduit
diameter and number at the apex (b). The location of the highest
net carbon gain has been marked with a symbol (x).
© 2011 Blackwell Publishing Ltd, Plant, Cell and Environment, 34, 1819–1834
A carbon gain-cost model for xylem structure 1831
scaling between pit and lumen radius with their respective
condutances (i.e. pit conductance is proportional to radius
raised to the third power, whereas lumen conductance is
proportional to radius raised to the fourth power). We
also confirmed that the optimal xylem structure predicted
by any of the optimization criteria would not change,
except the absolute conduit number in the case of maximizing net carbon gain, when carbon usage outside the
xylem (e.g. leaves, bark and transport to soil) was explicitly taken into consideration (see Supporting Information
Appendix S1).
According to the ‘rare pit hypothesis’, larger conduits are
more vulnerable to cavitation because they have a larger pit
area, and hence there is a greater probability of finding a
large-enough pit to seed embolisms (e.g.Wheeler et al. 2005;
Christman, Sperry & Adler 2009). Our approach provides
an alternative explanation which states that the lumen and
pit properties, and hence cavitation resistance, must vary in
concert within a tree and also roughly among trees because
of the optimization of plant carbon use. This explanation
follows on the lines put forward by Sperry & Hacke (2004)
and Hacke et al. (2004), who calculated that increasing
conduit size beyond a certain limit does not increase total
hydraulic conductance while increasing the carbon costs. In
addition, plants in dry environments exhibit smaller conduits (e.g. Sterck et al. 2008) and operate at low xylem water
potentials. Note that this does not mean that the model
predicts anisohydric behaviour during a drought, as the
model does not say anything about the dynamic behaviour
under varying environmental conditions. Our model predicts that efficient conduits are produced when water
potentials are high and safe conduits when water potentials
are low.
Our model predicted that total conduit cross-sectional
area remains relatively constant, in agreement with the pipe
model theory. It is important to note that this constancy was
obtained while the number of conduits and conduit size
varied independently, and conduit furcation occurred, and it
is therefore an emerging property of our model. Furthermore, this partially vindicates the pipe model theory and
shows that current analyses of this theory (i.e. WBE, Murray’s law) have misinterpreted the constancy of conducting
area per unit of leaf area to mean necessarily a fixed
number of constant diameter pipes that never furcate or
taper (West et al. 1999; McCulloh et al. 2003). This was never
explicitly specified in the original theory (Shinozaki et al.
1964).
In addition, the model predicted smaller conduits to been
associated with more conduits (Table 3), which has also
been found in the literature (Sperry et al. 2008; Zanne et al.
2010). The decline in conduit tapering with tree height predicted by the model has been reported in the literature
(Mencuccini et al. 2007). Tapering has been found to be the
smallest in conifers, intermediate in diffuse porous trees and
largest in ring-porous trees (Fan, Cao & Becker 2009),
which is consistent with our predictions of increasing tapering with increasing soil water potential and conduit length
(Fig. 3a,e). The prediction of larger tapering with larger
conduit size has not been widely tested. Petit et al. (2008)
found the opposite to be true for sycamore (Acer pseudoplatanus) trees. However, in that study, larger conduits were
associated with taller trees. Petit et al. (2008) also found that
shorter trees of the same species tapered more than the
taller ones, which is consistent with our model predictions
(Fig. 3c).
Tall trees were predicted to display a decline in wholeplant hydraulic conductance per unit of leaf area, and therefore also to have reduced photosynthetic production per
unit of leaf area compared to smaller trees (Mencuccini
2002). Maintaining hydraulic conductance per leaf area
would be too costly in terms of carbon use as height
increases. Conduit radius and water potential at the apex
were predicted to decrease slightly with tree height because
of gravity (both remained constant at the apex as a function
of tree height when gravity was switched off from the
model, not shown). These same trends, with approximately
the same magnitude, have also been reported in the literature and have been hypothesized to occur because of
decreasing turgor pressure needed for cell expansion
(Woodruff, Bond & Meinzer 2004; Woodruff, Meinzer &
Lachenbruch 2008).
We considered only the average value of conduit diameter and xylem water potential for air seeding. In reality,
xylem conduit properties and vulnerability to air seeding
will have a wide distribution even at a given height and
growth ring (Melcher, Zwieniecki & Holbrook 2003;
Choat et al. 2005). We also made the assumption that cavitation was completely avoided although in reality cavitation is a common occurrence (Tyree & Sperry 1989) and it
can be even beneficial for the tree to maintain the stomata
open at the expense of some cavitation (Jones & Sutherland 1991; Meinzer, Clearwater & Goldstein 2001; Hölttä
et al. 2009) especially as many, but not all, species can refill
embolised conduits in favourable conditions (Zwieniecki
& Holbrook 2009). As the optimal xylem structure
depends on environmental conditions, it seems logical that
a tree will in reality have varying conduit sizes and vulnerability to cavitation, as the environmental conditions to
which it is subjected to will also vary. Furthermore, we did
not optimize conduit length as it was given a constant
value. If conduit length was allowed to vary freely, the
optimization solution would always predict conduits with
a length equal to the length of the plant. We hypothesize
that something else, presently not considered in our
model, restricts conduit length in reality. One candidate
could be that shorter conduits should be more efficient in
constricting the spreading of embolism a network of
xylem conduits (Loepfe et al. 2007). Nevertheless, longer
conduits always resulted in larger conduit diameter, which
is a generally observed trend in the literature (Sperry,
Hacke & Pittermann 2006). We also did not consider the
mechanical aspects of xylem structure, other than those
caused by the water tension itself, which will also impose
constraints on the xylem structure, as the whole stem has
to maintain a form which is able to cope with, for
example, wind and snow loads. In the future, it will be
© 2011 Blackwell Publishing Ltd, Plant, Cell and Environment, 34, 1819–1834
1832 T. Hölttä et al.
possible to develop the model further to simulate the
cumulative xylem formation on a yearly basis to take into
account the effect of previous rings’ conduits on the water
transport capacity and mechanical stability of subsequent
rings. In the future, the modelling approach presented
might also allow the prediction of the upper limit to tree
height based on the interconnections between the xylem
water transport capacity and its carbon costs. The net
carbon budget of the tree will necessarily turn negative at
some tree height (i.e. when the building costs of the xylem
become larger than the carbon assimilation rate which can
be maintained with the hydraulic structure). For this
purpose, carbon usage outside the xylem has to be considered more carefully.
The model presented here is very general, in the sense
that xylem structure is predicted without imposing boundary conditions on xylem water potential, conduit radius and
conduit number at any point in the structure. Our only
assumptions are based on widely accepted physical principles to describe flow and drainage of capillaries, and the
known proportionality between conduit lumen diameters
and conduit wall thickness. Previous models of xylem
hydraulic structure are based on stricter optimization criteria and stricter a priori restrictions given to the structure.
The WBE theory predicts that conduits increase in size
from tree top to bottom to minimize the loss of conductance
as the tree grows, while assuming that the terminal elements
at the apex are constrained and remain fixed. In a later
development by Becker, Gribben & Schulte (2003), the
predictions of WBE were still shown to hold when pore
conductance was assumed to scale with lumen conductance.
Our model predicts, not assumes, based on optimization of
carbon use, the scaling between conduit lumen and pit conductance. It is also worth noting that while the WBE theory
predicts a single optimal scaling of conduit sizes, our theory
demonstrates that a range of values are possible. Indeed,
the curve for the net carbon gains showed rather a long
crest as a function of tapering (Fig. 5b), as opposed to a
single peak, demonstrating that a range of tapering values
can give nearly optimal solutions, provided that furcation
varies accordingly.
Murray’s law predictions, on the other hand, stem
from the maximization of hydraulic conductance for
a given carbon investment. Our model also predicts
decreasing conduit number from tree top to bottom
(although our model predicts a substantially smaller
decrease in conduit number for all parameterizations) for
qualitatively the same reasons as Murray’s law, to ensure
that carbon is distributed efficiently along the axis so that
there is not a large ‘bottleneck’ anywhere along the xylem.
However, contrary to our approach involving the relation
between water potential and pit size, Murray’s law leads
to decreasing number of conduits from tree top to bottom,
because the conduit size and number are restricted at the
apex. Clearly, the most efficient conduit system would
always be one in which the conduits would be unrealistically large and few in number if pit structure and xylem
vulnerability to cavitation are not taken into account.
Similarly, neither WBE nor Murray’s law can make predictions about the central feature of our model, namely
the modelling of the vertical profile in xylem water potential, because the behaviour of pits is not incorporated in
those models.
ACKNOWLEDGMENT
The research was funded by the Academy of Finland
(#1132561).
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Received 19 February 2011; received in revised form 27 May 2011;
accepted for publication 8 June 2011
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1834 T. Hölttä et al.
SUPPORTING INFORMATION
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Appendix S1. Numerical model.
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