Scaling the paths of resistance


The height of self-supporting stems ranges over six orders of magnitude, and an individual seedling may span this entire size range as it grows to old age. One obvious result is a dramatic increase in the root-to-leaf hydraulic path-length. Another is a concomitant increase in the resistance to water flow through this pathway – a feature that is illustrated, albeit too simplistically, by means of the Hagen–Poiseuille equation for capillary tubes with length l and radius r (Fig. 1a).

Figure 1.

Hydraulic path-lengths, resistances and velocity-area products. (a) Tubular conduit with length l and radius r has a parabolic transverse velocity profile (see arrows) and a resistance to flow that is proportional to l/r4. (b) With increasing growth in height (increasing hydraulic path-length lT), the resistance to flow increases linearly with respect to conduit length. (c) Topology of a simple bifurcating hydraulic network (n = 2) with three levels of branching (k = 0, 1 and 2) and distally tapered conduits. (d) Fluid velocity–area products (VA) through the topology shown in (c). Local velocities are proportional to arrow lengths and widths of cross-hatched areas. The sum of the velocity–area products at any level of branching must be equal to one another (e.g. V0A0 = Σ V2A2).

Assuming a constant pressure gradient, this equation predicts that the total resistance to water transport RT is proportional to l/r4. Therefore, if r is constant, the relationship between hydraulic resistance and path-length (a rough gauge of plant height) is linear, as predicted by the pipe-model for tree vasculature (Fig. 1b). Naturally, the biological situation is far more complex because the l/r4 relationship neglects other resistances (e.g. flow through bordered pits or vessel end-wall perforation plates). Nevertheless, it shows that one strategy to reduce RT is to increase r.

Curiously, this is not generally observed when tracheary cell dimensions are measured along the lengths of branches or along the lengths of whole trees. For example, according to ‘Sanio's Laws’ (Sanio, 1872), tracheid length tends to increase radially (from the inside to the outside of successive annual growth rings) and distally up to some maximum height (from the base to the top of stems). Likewise, Carlquist (1975) reports that tracheary cell diameters in the most recent growth ring tend to decrease from the bottom to the top of trees. These trends likely reflect adaptive compromises to competing developmental and functional demands. For example, tracheary cell length is largely defined by the length of fusiform initals, whereas widening cell diameters to reduce hydraulic resistance can weaken mechanical tissues and increase the chances of embolism formation, particularly in branches sustaining high transpirational demands. Trade-offs between reducing hydraulic resistance and protecting against physical damage have been partially resolved by adaptive differences in the cell dimensions of early vs late wood. But the unavoidable ontogenetic extension of the hydraulic path-length, coupled with dramatic phyletic increases in plant height, undoubtedly required complex evolutionary innovations.

‘This is one of the few attempts to test some of the basic assumptions of the WBE theory.’

The WBE theory: a conceptual advance

A major conceptual advance to unraveling evolutionary responses to increasing size is the ‘global’ theory advanced by Geoffey B. West, James H. Brown, and Brian J. Enquist (henceforth, the WBE theory) for fractal-like resource distribution systems (West et al., 1999a,b). This theory assumes that evolution by natural selection has resulted in the maximization of the effective surface areas over which mass and energy are exchanged with the environment and the minimization of internal transport path-lengths (or the total energy required to transport resources). This optimization process is posited to have resulted in size-invariant terminal resource delivery conduits (e.g. capillaries, bronchioles and distalmost tracheary elements).

This last proposition is particularly interesting in the context of the principal of hydraulic continuity, because no matter how a network branches, contracts or changes cross-sectional shape, the product of local fluid velocity V and conduit cross-sectional area A (the velocity–area product) must remain constant everywhere in the entire network (Fig. 1c,d). For example, if the sum of the cross-sectional areas of all terminal delivery units exceeds the cross-sectional area of the network's base, local fluid velocities in these units must be less than at the base, and the time a fluid spends in contact with the walls of these conduits is increased, which is particularly important for passive diffusion.

This is important because, by modeling total hydraulic resistance RT in a hierarchically branched network consisting of equal length conduits varying in radius, the WBE theory predicts that conduits should narrow rather than expand distally. Specifically, this theory yields the formula

image( Eqn 1)

where n is the average number of branches at each branch juncture, lT is the total (trunk-to-petiole) path-length, lp is the petiole length, rp is the petiole hydraulic resistance and α is the scaling exponent governing the tapering of conduits (West et al., 1999b; see also Enquist, 2003). Note that, regardless of the numerical values of all other variables, RT = 0 when α = 1/6. Also, when α = 1/6, the total hydraulic resistance is predicted to be independent of total path-length (and thus plant height) as size increases beyond mathematically well-defined limits (Fig. 2).

Figure 2.

Total hydraulic resistance RT vs total path-length lT relationships predicted by Eqn 1 for conduits with different taperings α, assuming n = 2, l= 0.02 and rp = 0.5. With the exception of untapered conduits (α = 0, the ‘pipe model’; see Fig. 1b), total hydraulic resistance decreases with increasing path-lengths as conduits become more tapered distally.

Putting theory to the test

These and other claims of the WBE theory have been subjected to intense debate and criticism, in part because few attempts have been made to test its underlying assumptions empirically. That this has changed recently is attested to by the work of Anfodillo et al. (this issue, pp. 279–290), who set out to determine whether trees differing in height (path-length) manifest similar tracheary taperings and whether the exponent of taper complies with α = 1/6, as predicted by theory. Anfodillo et al. sampled 50 dicot and conifer trees ranging between 0.5 and 44.4 m in height and measured the variation in the mean hydraulic-weighted diameters of tracheids or vessel members. Their analyses show that the interspecific exponent of taper for actively growing trees statistically complies with α = 1/6, but that trees approaching their maximum heights or growing in environments conducive to embolism formation manifest suboptimal taperings.

This study is significant in three ways. First, it weds ‘Sanio's Laws’ directly to the hypothesis that hydraulic resistance limits plant height (see Meinzer et al., 2001; Koch et al., 2004; Koch & Fredeen, 2005). Second, it provides detailed information about longitudinal changes in tracheary cell dimensions for hitherto unstudied conifer and dicot species. And, third, it is one of the few attempts to test some of the basic assumptions of the WBE theory.

But does it really provide a test of the WBE theory? Arguably, the answer depends on our statistical predilections and philosophical perspective on what distinguishes a ‘test’ from a ‘quiz’.

As Anfodillo et al. note, the ‘… sharp mathematical limit of [the] optimized vascular network (α = 1/6) cannot be easily compared’ for ‘real trees because the empirical degree of tapering always has a certain confidence interval, and, more importantly, the variation of xylem tapering when plants grow taller occurs continuously.’ Statistical tests of allometric theories typically rest on whether the 95% confidence intervals for observed scaling exponents include or exclude those of the fractions posited by a particular theory. Given the considerable natural variation observed in most biological phenomena, these intervals tend to be very broad (Niklas, 2004), which makes the assessment of any theory extremely difficult.

Consider the hypothetical case where an observed scaling exponent equals 0.1429 and has 95% confidence intervals of 0.1179 and 0.1679. The upper limit agrees with the theoretical prediction that α = 1/6, but the intervals also agree with alternative predictions – for example, α = 1/7 or 1/8. This ‘ambiguity’ is exacerbated by the fact that the prediction α = 1/6 sets no discrete upper limit for the confidence intervals of α. This is particularly troublesome because, as noted, some of the trees examined by Anfodillo et al. have suboptimal tracheary taperings (a disputation of the WBE theory that can be either rationalized away by ecological reasoning, or used as evidence that the WBE theory is too vague, or worse, simply wrong).

A first approximation

It is fair to say that the slowly mounting body of evidence is, on the whole, more in alignment with the WBE theory than not. For example, the lower limits of empirically observed tapering exponents are remarkably close to 1/6, particularly since Eqn 1 represents a first approximation. Likewise, the WBE theory's prediction that tracheary elements must widen progressively at the base of trunks to maintain an optimal or nearly optimal hydraulic transport system is consistent with the now classical observations of Sanio (1872) and Carlquist (1975). Finally, no other allometric theory serves to explain the phenomenology of hydraulics and mechanics as completely as does the WBE theory.

Any ‘global theory’ has to scale various levels of resistance and well reasoned debate until it is accepted or rejected. Along the way, a really good theory evokes a scramble for evidence to either reject or accept it. In this sense (and many others), the WBE theory has had a profoundly positive affect on the science of allometry, in general, and on the plant sciences, in particular.