## Introduction

Interest in the link between carbon uptake and water loss has yielded numerous efforts to quantify the efficiency of the water transport network of plants (Banavar *et al*., 1999; McCulloh *et al*., 2003, 2004; McCulloh & Sperry, 2005a,b; Anfodillo *et al*., 2006; Weitz *et al*., 2006; Atala & Lusk, 2008). Much of this work has focused on the xylem tissue of woody stems, where the xylem conduits (vessels or tracheids) may be providing structural support to the plant in addition to transporting water. The additional support task compromises the potential hydraulic efficiency of water transport (McCulloh *et al*., 2004). Within leaves, though, substantial structural support is provided by hydrostatic pressure and specialized nonvascular tissues such as collenchyma and sclerenchyma. To the extent that this nonxylary structural support holds up the leaf, this would theoretically leave the xylem conduits free to achieve their maximum transport efficiency. Assessing transport efficiency within the lamina is complicated by the typically reticulate networks that are losing water to transpiration (Canny, 1993; McCulloh & Sperry, 2005b). By contrast, it is relatively easy to quantify network efficiency in compound leaves because the vascular tissue can be compared at discrete branching levels, the petiole and petiolule. Using these distinct levels in 13 species, we compared compliance with two theoretical optima that maximize the hydraulic conductance of this simple two-ranked vascular network.

The first optimum, Murray's law (ML), predicts how the conduit diameters should change across branching points of a vascular network so as to minimize the power (work/time) driving a given flow rate through a network of fixed total volume (Murray, 1926). As Sherman (1981) demonstrated, a Murray law network also maximizes the hydraulic conductance (volume flow rate per pressure drop). Maximizing conductance (rather than minimizing power at one flow rate) is undoubtedly the real adaptive significance of Murray's law, because a maximum-conductance network is optimal at all flow rates (Sherman, 1981). Murray's law was derived for the specific case of the cardiovascular system which delivers blood through a single branching tube. The law states that conductance is maximized when the sum of the conduit radii cubed (Σ*r*^{3}) is conserved at all points along the flow path from aorta to the beginning of the capillaries. Conservation of Σ*r*^{3} requires an increase in the cross-sectional area (Σ*r*^{2}) from aorta to capillaries.

Unlike the cardiovascular system for which it was developed, Murray's law does not define the sole hydraulic optimum for the plant vascular system. In the cardiovascular system the branching of the vascular network equals the branching of a single tube. The ratio of the number of daughter-to-mother tubes (*N*_{R}) cannot be varied independently of the network branching structure. Murray's law gives the ‘best’ daughter-to-mother tube taper ratio (*D*_{R}) for the *N*_{R} dictated by branching architecture. The evolution of *N*_{R} is constrained by the need to deliver fluids spatially, and Murray's law *D*_{R} evolved to maximize the conductance of this mammalian vascular topography.

Xylem networks are more complex because they are composed of many conduits in parallel and in series at every branch level, including the trunk. That means that *N*_{R} is potentially independent of the branching network. When two daughter arteries join at the aorta, *N*_{R} = 2. When two petiolules each containing hundreds of conduits join at the petiole, *N*_{R} does not have to be 2. *N*_{R} can be any number greater than zero or less than the theoretical maximum for a petiole with just one conduit. Optimization of the network conductance is not just a matter of *D*_{R}, because *N*_{R} can also vary independently of the network branching topography.

The fact that there are at least two variables that are potentially free to vary across the branch junctions of plants means that there are at least two theoretical optima for maximizing conductance. The distinction between the two optima is shown graphically in Fig. 1. The contour lines on this figure are hydraulic conductances of a simple network with one petiole and two petiolules. The tubes within each branch rank are of constant diameter, and the total number of petiolule tubes is constant. All other parameters, including the volume of the network, are constant except for *N*_{R} and *D*_{R}.

The contours in Fig. 1 describe a diagonal ridge that ascends to greater conductance as *N*_{R} increases and *D*_{R} decreases. The ascent of the ridge corresponds to fewer, larger tubes in the petiole. The ridge cannot be ascended indefinitely because of limits on permissible values of *D*_{R} and *N*_{R}. The *N*_{R} reaches its maximum theoretical value when the petiole has just a single tube (*N*_{R} max on ‘*y*’ axis; McCulloh *et al*., 2003). At *N*_{R}=*N*_{R }max, the greatest conductance is achieved at the horizontal tangent of the conductance contour where *D*_{R} conforms to Murray's law (asterisk on ML diagonal in Fig. 1). This global optimum is represented by the cardiovascular system which is a single branched tube that approximates Murray law taper (Sherman, 1981).

Plants are never plumbed with a single branched tube and so are never at their *N*_{R} max. Negative sap pressures make a single-tube network vulnerable to complete failure in the event of a single air leak or cavitation event (Ewers *et al*., 2007). Greater *N*_{R} also results in a greater departure of the xylem network from area preservation (Fig. 1; AP diagonal) towards area-increasing branching at Murray's law. Extreme increases in the cross-sectional area of conduits would result in top-heavy networks that could be mechanically unstable. Hydraulic and mechanical safety considerations combine to keep the *N*_{R} of plant xylem well below its theoretical maximum and potentially free to vary independently of the plant's branching system.

The alternative scenario to Murray's law for plant xylem is that *D*_{R} is more constrained than *N*_{R} for developmental or functional reasons. Such a constraint could result from a developmental link between vessel diameter and branch rank, or a limit to maximum vessel diameter. In this case, the ‘best’ network would have the minimum permissible *D*_{R}, and the greatest conductance would be achieved at the vertical tangent of the conductance contour where *N*_{R} conforms to the ‘YO’ diagonal. Because this optimum is the opposite of Murray's law, we call it the ‘Yarrum’ optimum (YO; Sperry *et al*., 2008).

Here, we derive both Murray's law and the Yarrum optimum for a simple network composed of the conduits within a petiolule and two daughter petiolules of equal length. The derivations make explicit the distinction between the two optima. Following Sherman's example (1981), the optima are derived from a maximum conductance criterion because this is more biologically relevant than the equivalent minimum power derivation. The series conductance of the network is represented as the reciprocal resistances (*R*), because they are additive in series. The Hagen–Poiseuille equation predicts:

where* l* is the axis length, *n* is the number of conduits in parallel, and *r* is the conduit radius, which is assumed constant within a rank. The subscript ‘0’ denotes petiole values, and subscript ‘1’ denotes the totals for the daughter petiolules. The *D*_{R} is *r*_{1}/*r*_{0}. The mean radiuses of petiole or petiolule vessels are assumed to substitute for *r*_{0} or *r*_{1}, and mean petiolule vessel diameters are not expected to differ within a compound leaf when leaflets are of approximately similar size. Constant η is the fluid viscosity and *k* is the factor by which the resistivity exceeds the Hagen–Poiseuille prediction because of end-walls and other obstructions (*k* ≥ 1). The volume, *V*, of this network is

Network volume is proportional to the conduit wall volume if wall thickness is proportional to conduit diameter (McCulloh & Sperry, 2006); the latter is approximately true for xylem of a given cavitation resistance (Hacke *et al*., 2001). In both derivations, we hold the vascular investment, *V*, constant and solve for the minimum *R* under either Murray's law or Yarrum conditions. To derive Murray's law, we hold the numbers of conduits constant (*n*_{0} and *n*_{1}) and vary the conduit taper (*D*_{R}) to find the value that minimizes *R*. Because volume (*V*) is constant, we must also allow *r*_{0} to vary with *D*_{R}. We use the Lagrange multiplier method to find where *R *(*D*_{R}, *r*_{0}) (Eqn 1) is minimized subject to the constraint that *V*(*D*_{R}, *r*_{0}) = 0, where we have subtracted a constant from Eqn 2 to set *V* = 0. At the minimum *R*, the Lagrange multiplier, λ, defines the two equalities (Edwards & Penney, 1998, p. 864):

From Eqns 1 and 2 the partial derivatives in Eqn 3 are:

Substituting 4a and 4b into 3a yields

Substituting λ, 4c and 4d into 3b, and simplifying, yields

*D*

_{R}= (

*n*

_{0}/

*n*

_{1})

^{1/3}

or

*D*

_{R}=

*N*

_{R}

^{−1/3}(Eqn 6)

for *N*_{R}=*n*_{1}/*n*_{0}. Equation 6 is Murray's law for the special case where tubes within a rank are of equal diameter. However, we have previously shown that conservation of Σ*r*^{3} across branch points is within 3% whether computed from actual vessel distributions or their mean values (McCulloh & Sperry, 2006). Note that the lengths cancel out, indicating that the optimal taper is independent of branch lengths.

To find the Yarrum optimum, we hold the taper constant (*D*_{R}) and vary the relative numbers of conduits across ranks (varying *n*_{0} for a constant *n*_{1}) to find the value that minimizes *R*. Because volume (*V*) is constant, we must allow *r*_{0} to vary with *n*_{0}. As before, we use the Lagrange multiplier method to find where *R*(*n*_{0}, *r*_{0}) (Eqn 1) is minimized subject to the constraint that *V*(*n*_{0}, *r*_{0}) = 0. At the minimum *R*, the Lagrange multiplier, λ, defines the two equalities:

From Eqns 1 and 2 the partial derivatives in 7a are:

The derivatives for 7b are 4c and 4d. Substituting 8a and 8b into 7a yields:

Substituting λ, 4c and 4d into 7b, and simplifying, gives:

or

where the branch length ratio *L*_{R} = *l*_{1}/*l*_{0}. Using the quadratic formula to solve for *N*_{R} yields the following relevant root (the other root gives negative *N*_{R}; signs in the valid root have been simplified):

which is the Yarrum optimum. Note that the branch lengths do not cancel out, indicating that the optimal NR depends on the relative lengths of the petiole and petiolule. The shorter the petiolules relative to the petiole (smaller the *L*_{R}), the fewer conduits they need in parallel (relative to the petiole) to minimize the total resistance (the smaller the optimal *N*_{R}). As *L*_{R} approaches zero, the lowest optimal *N*_{R} converges on 1.41 . For typical *D*_{R} < 1, Yarrum's *N*_{R} will always exceed 1.4, meaning that petiolule conduits will always be more numerous than petiole conduits at the Yarrum optimum.

The existence of two hydraulic optima raises the question of what variable is most constrained during evolution and development. If the evolution of *N*_{R} is more constrained (as in the single-tube mammalian cardiovascular system where it is at *N*_{R} max), then the vascular conductance through the branching system can only be optimized by evolving the value of *D*_{R} predicted by Murray law. Alternatively, if the *D*_{R} and *L*_{R} are more constrained, then the vascular conductance through the branch system would be maximized by evolving the Yarrum optimum *N*_{R}. To date, the Yarrum optimum has not been recognized, or evaluated, in any system. To address the question of whether leaf vasculature follows Murray's law, or the Yarrum optimum, or conduit area preservation, or none of these patterns, we measured *D*_{R}, *L*_{R}, and *N*_{R} across the petiole vs petiolule ranks of compound leaves and compared them with the predictions. Leaves were studied in six tropical and five temperate angiosperm species, and in two temperate fern species. In addition to evaluating these two optima, we examined the evidence for size- and species-independent scaling patterns in the hydraulic architecture of leaves.