In the case of the needle, water loss is diffusional, driven by the concentration gradient in water vapour between the intercellular air spaces of the leaf and the atmosphere. Because leaf water potential has only a minor effect on the equilibrium vapour pressure (Nobel 1983), the rates of evaporation along the needle will be essentially independent of the pressure within the xylem and instead depend on the driving gradient for vapour diffusion and the conductance of the needle surface to water vapour. Nevertheless, the liquid phase pathway for water movement through leaves does influence the ability of a leaf to support high rates of evaporation through the impact of leaf water status on stomatal aperture. Thus, for a given investment in xylem permeability there is a limit on needle length before the pressure in the xylem reaches the threshold for stomatal closure at the tip. A similar role for the venation in limiting leaf size was proposed as the basis for differences in leaf size and shape within the crown of Quercus rubra trees (Zwieniecki, Boyce & Holbrook 2004a). However, leaf vasculature also contributes to the mechanical properties of leaves, and functions in the export of photosynthate. Thus, it is possible that axial permeability exceeds what is needed to efficiently deliver water to evaporation sites (Martre, Cochard & Durand 2001) and that needle size is not limited by its water distribution system.
To explore if the hydraulic system of pine needles evolved to optimize delivery of water per fixed investment in axial permeability, we asked how the hydraulic design of pine needles compares with the distribution of hydraulic permeability needed to maximize needle length or minimize the pressure drop along the vein. The presence of such optimization would suggest that xylem hydraulic properties play an important role in leaf performance, and thus, are a potential constraint on leaf size. The simplicity of the geometry of the pine needle allows for detailed analysis using a formal mathematical approach. Here, we construct a model of the hydraulic system of single-vein leaves and present a mathematical procedure that calculates the optimum distribution of axial hydraulic permeability. We compare theoretical findings with measurements conducted on three species with a wide range of needle length, but exhibiting the same general morphology (i.e. all species have needles in fascicles of three such that the cross-sectional shapes of both the needle and the vascular bundle are similar): Pinus palustris (Engl.) Miller in its juvenile stage has needles ranging from 45 to 50 cm in length, Pinus ponderosa Lawson & Lawson has needles ranging from 20 to 25 cm in length and Pinus rigida Miller produces needles 10–11 cm in length. Our results suggest that xylem hydraulics plays an important role in constraining needle length, especially in species with long needles, and that species optimize their xylem permeability to efficiently use the materials required for constructing xylem conduits.
Theoretical analysis: a model for pressure distribution in the needle
Using cylindrical coordinates, let z = 0 denote the base of the needle, and z = l denote the tip, where we assume there is no loss of water. Let A be the cross-sectional area of the tracheid lumens, which we assume to be constant in this initial development (Fig. 1 & Table 1). Then we can examine a section between z and z + Δz and write the mass balance as
Figure 1. Model representation of a pine needle as a one-dimensional distribution system. (a) Control model of the needle with xylem cross-sectional area independent of length. (b) Model of the needle with xylem cross-sectional area functionally related to distance from the needle base. pbase, pressure at the needle base; A, cross-sectional area of the tracheid lumens; u(z), fluid velocity; q, evaporation rate per needle length; pl, pressure at the needle tip; l, needle length; A(z), tracheid lumen cross-sectional area.
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Table 1. Units and descriptions of symbols used in the text
|A(z)||(m2)||Tracheid lumen cross-sectional area|
|N(z)|| ||Number of tracheids|
|K(z)||(m4)||Grand permeability of xylem|
|Q||(m3 s−1)||Total evaporation from the needle|
|q||(m3 m−1 s−1)||Evaporation rate per needle length|
|u(z)||(m s−1)||Fluid velocity|
|z||(m)||Distance from the needle base|
- A[u(z) − u(z + Δz)] − qΔz = 0,(1)
where u denotes the average water velocity, and qΔz denotes the evaporation rate out of the leaf section of length Δz. After dividing the mass balance formula by Δz and taking the limit of Δz 0, we arrive at
Next, we can use Darcy's law for the relationship between u and pressure p in the following form:
where k is the xylem permeability and µ is the viscosity of the liquid. Now, combining Eqns 2 and 3, and assuming that q, A, µ and k are constant, we have
This equation can be integrated twice to give
We next apply boundary conditions. At z = 0, we prescribe the pressure at the needle base p = pbase. At z = l, we assume that there is no fluid flux in the axial direction, so u = dp/dz = 0 (i.e. all water flowing into the leaf has evaporated by the distance l). We can now solve Eqn 5 for the two constants c1 and c2, and so we arrive at
The pressure distribution in the vein is parabolic and exhibits a monotonic decay starting with negative pressure (i.e. pbase < 0, at z = 0 (at the needle base) and decays to p = pbase − µql2/2kA at z = l (the needle tip) where it has zero slope.
Any change in needle length (l) will alter the final pressure distribution in the needle near the tip but not the general shape of the curve. This simple model shows that the pressure distribution in the needle is a result of flow due to the evaporation (q) along the needle.
Because the cross-sectional area of the xylem tissue within the vein was assumed in this first model as constant, we can immediately see that near the tip very low axial flow results in a low pressure drop, while near the base of the needle the much higher water flow is associated with a more pronounced loss in pressure. Thus, we can pose an optimization problem to find the distribution of permeability along the leaf that minimizes the pressure drop for a given investment in conductive tissue.
We can use the aforementioned model, but generalize it to allow for axial variation in permeability and cross-sectional area of the xylem. Let us keep the above notation, but now allow A(z) (the cross-sectional area) and k(z) (the permeability) to vary with distance from the base of the needle. We also assume that q (evaporation rate per needle length) is constant along the length of the needle. For the one-dimensional mass balance, we now have
Because q is constant, we can integrate this equation to find
If we impose the boundary condition of no-fluid flux out of the tip of the needle, u = 0 at z = l, then we find
We use Darcy's law, Eqn 3, and so using the form of velocity distribution just obtained, we have
which can be integrated to yield the pressure drop along the needle Δp = pbase − ptip,
Now, let us introduce a ‘grand permeability’ (K) relating the volume flow rate Q = A(z)u(z) to the pressure gradient, which corresponds to a permeability function K(z) =k(z)A(z), that is,
Thus, we have a pressure drop in the form
If we assume that tracheid dimensions are uniform along the length of the needle [k(z) constant], K(z) will be directly proportional to A(z). Thus, we can introduce a function N(z) for the number of tracheids at position z along the needle, and ask the question: What is the distribution of N(z) that minimizes the pressure drop along the leaf length l subject to the constraint that the total number of tracheids remains constant? In other words, what is the optimal investment in conductivity such that
It is simplest to address this optimization question by rewriting Eqns 13 and 14 in terms of the relative distance from the tip s = z/l. Therefore, we have an optimization question
This optimization problem may be considered using the calculus of variations. The Euler–Lagrange equation corresponding to the solution of the problem statement in Eqn 15 is
where λ is a constant to be determined (a Lagrange multiplier). Equation 16 can be integrated and rearranged to obtain
Equation 17 represents the optimized grand permeability, equivalent to the distribution of equal-sized tracheids, along the needle. Now, we can contrast the pressure drop in needles with uniform and optimized permeabilities. Firstly, we need to calculate the pressure drop assuming the distribution of tracheids is uniform (i.e. with Nuniforml = constant). From Eqn 15 we have
Similarly, for the optimized distribution we have
Therefore, taking the ratio of Eqns 18 and 19 and recognizing that the requirement to conserve the total number of tracheids is such that
This result shows that by optimally redistributing tracheids along the length of the needle according to hydraulic principles, there might be a reduction in pressure drop by about 10%.
An alternative optimization question is to find the maximum needle length for a given pressure drop given a fixed vascular investment. We can address this question using Eqn 15 because the maximum leaf length corresponds to the minimum value of cΔp/µql2. Hence, for fixed Δp, we can determine the value of l corresponding to both Nuniform and Noptimized. In this case we use Eqn 18 which gives
while for the optimized distribution (Eqn 19) we have
Equations 22 and 23 subject to the requirement that the total number of tracheids is conserved lead to
This result gives an optimized length increase of about 4% resulting from the redistribution of tracheids.