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- NUMERICAL MODEL
- Supporting Information
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.
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- NUMERICAL MODEL
- Supporting Information
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 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; 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 meta-stability 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 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.
Table 1. Summary of previous xylem hydraulic architecture models
|Criteria||Optimization criteria||Efficiency versus efficacya||Conduit number||Conduit size||Pits||Water potential||Mechanical support against xylem tension||Mechanical support against outside forces||Dependency on environmental conditions||Level of detail|
|Models|| || || || || || || || || || |
|This model||Carbon net gain||Efficiency; computes costs of conduit wall construction||Optimized together with tapering||Optimized together with furcation||Optimized to balance conductance and prevention of cavitation||Optimized||Considered||Not considered, but can be added||Yes||Very high|
|WBE (West et al. 1999)||Maintain hydraulic conductanceb||Efficacyc||Fixed||Optimized, starting from fixed size at the top||Not consideredd||Not considered||Not considered||Considered||No||Low|
|Murray's law||Hydraulic conductance for a given carbon investment||Efficiency; computes costs of conduit wall construction||Optimized together with tapering||Optimized together with furcation||Not considered||Not considered||Considered||Violates requirements||No||Moderate|
|Mencuccini et al. net C gain||Carbon net gain||Efficiency; computes costs of conduit wall construction||Fixed||Optimized||Not considered||Fixed||Considered||Not considered||Yes||Moderate|
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- NUMERICAL MODEL
- Supporting Information
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
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
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 ψ(N) in such a way that the net gain of carbon (G − Cc) or, alternatively, some other criterion, such as the carbon-use 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:
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:
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.
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 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.
Table 2. Parameterization of the model
|Parameter symbol||Paremeter description||Value||Reference||Lower bound in sensitivity analysis||Upper bound in sensitivity analysis|
|ψs||Soil water potential||−1.0 MPa||Estimateda||−5.0 MPa||−0.05 MPa|
|Δw||Vapour pressure deficit (VPD)||4 kPa||Estimateda||1.3 kPa||13 kPa|
|I||Photosynthetically active radiation (PAR)||500 µmol m−2 s−1||Estimateda||167 µmol m−2 s−1||1670 µmol m−2 s−1|
|H||Tree height||20 m||Estimateda||6.7 m||67 m|
|L||Conduit length||3 cm||Estimateda||3 mm||30 cm|
|S||Whole-tree leaf area||80 m||Estimateda||26.6 m||266 m|
|Scales with tree height, S α h2||Scales with tree height S α h2||Scales with tree height S α h2|
|A||Photosynthetic parameter||0.04 µmol m−2 s−1||Mäkeläet al. (1996)||0.0133 µmol m−2 s−1||0.133 µmol m−2 s−1|
|Γ||Photosynthetic parameter||400 µmol m−2 s−1||Mäkeläet al. (1996)||133 µmol m−2 s−1||1333 µmol m−2 s−1|
|Y||Conduit functional longetivity||3 years||Estimateda||1 year||10 years|
|T||Hours of photosynthesis in a year||3 888 000 s (180 d, 6 h d−1)||Estimateda||1 295 870 s||12 958 704 s|
|npits||Number of pits in a conduit||5% of conduit wall area occupied by pits.||Wheeler et al. (2005), constant pit density is assumedb||1.67%||16.7%|
|1000 for a 20 µm in radius and 1 cm in length conduit, scales linearly with conduit wall area|
|npores||Number of pores in a pit||100||Estimateda||33||333|
|kr/ktot||Hydraulic conductance from soil to tree in relation to total xylem conductance||2.0||Hypothetical value||0.6||6|
|A1||Empirical parameter relating cell wall thickness to conduit size and water potential||6.0a10−9 Pa−1||Fitted from Hacke et al. (2001)||2||20|
|A2||Empirical parameter relating cell wall thickness to conduit size and water potential||0.005 µm||Fitted from Hacke et al. (2001)||0.00167 µm||0.0167 µm|
|Ca||Ambient CO2 concentration||400 ppm||Estimateda||Not included in sensitivity analysis||Not included in sensitivity analysis|
|Fc||Factor by which xylem carbon costs are multiplied by to account for carbon costs of xylem respiration||2||Estimateda||1||4|
|Θ||Carbon required per cell wall volume||46 875 mol C m−3||(Siau 1984), assuming carbon is 50% of dry matter||Not included in sensitivity analysis||Not included in sensitivity analysis|
|Γ||Surface tension of water||0.073 Nm||CRC Handbook of Chemistry and Physics||Not included in sensitivity analysis||Not included in sensitivity analysis|
|µ||Dynamic viscosity of water||0.001 Pa s−1||CRC Handbook of Chemistry and Physics||Not included in sensitivity analysis||Not included in 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 trade-off 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.
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- NUMERICAL MODEL
- Supporting Information
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 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 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 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 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 whole-plant 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; Hölttäet 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 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.