West, Brown & Enquist's (1997, 1999) allometric scaling theory (hereafter, WBE) is based on a biophysical model of resource transport in branched networks in an ‘ideal’ organism. WBE derives equations that predict how biological attributes (B, e.g. metabolic rate, rates of resource use, even morphological features) scale with body mass (M), i.e. BMα, where α is a scaling exponent. Many such exponents are predicted by WBE to be simple multiples of 1/4. These predictions have largely been confirmed by statistical comparisons with extensive data sets.

WBE is explanatory in the sense of ‘makes comprehensible’ as well as ‘accounts for most of the variation in the data’. This is remarkable, given the huge morphological and physiological diversity that exists among organisms. WBE deliberately ignores specific details and concentrates on general principles. It is clearly not intended to explain the detailed form or function of particular species. Just as the Gas Laws of classical physics account for the average behaviour of large populations of ‘ideal’ molecules but not of individual molecules, WBE deals with the gross patterns seen when many taxa, habitats or body sizes are compared. Choose a particular species at random and the probability is that its features will deviate from the WBE ‘ideal’. Nevertheless, if WBE is largely correct, most individuals presumably conform, on average, to the principles on which it is based.

That presumption has yet to be verified. Until it is, doubts will remain about whether WBE provides unique insights into how nature works or is just one of many theories that can make similar predictions but from different starting points (e.g. Makarieva, Gorshkov & Li 2003). Other challenges to WBE's theoretical or statistical bases have been made (e.g. Dodd, Rothman & Weitz 2001; Kozlowski, Konarzewski & Gawelczyk 2003; White & Seymour 2003). It is simply worth noting here that other scaling exponents (e.g. 2/3) that explain the data statistically do not yet have the theoretical justification of WBE's 1/4-power exponents.

WBE's parameters usually have clear structural or functional identities. This contrasts with the parameters of the polynomial equations used in standard growth analysis. Comparisons between data and WBE can be discussed on the basis of general biophysical, physiological or morphological principles, and not just in terms of statistical goodness-of-fit (West, Brown & Enquist 2001). WBE also offers a way to predict general ecological patterns using some fairly basic and widespread information – stem diameter, plant height, species number, etc. (Enquist & Niklas 2001). For functional ecologists this must increase the rigour with which we can interrogate the natural world, even if WBE eventually comes to be regarded as one of several (and not necessarily the most fundamental) models that fulfil this need.

One test of whether WBE is useful is if it can shed new light on important ecological questions. Here I explore how mass allocation patterns in vegetation can be analysed using WBE. Enquist & Niklas (2002) developed the original WBE model to explain general allocation patterns in plants. My analysis shows that despite its strengths, the current WBE model is insufficient to explain these patterns across all size scales; the alternative conclusion is that WBE suggests that the below-ground carbon (C) stocks of vegetation could be much larger than the current estimates based on biomass inventories.

Enquist & Niklas (2002) derived an equation relating the above-ground mass of vegetation (MA, comprising leaves and stems) to that below-ground (MR). The equation is (using Enquist & Niklas’ notation):

  • MA = MR1213) + (MR13)3/4(eqn 1)

where β12 and β13 are allometric constants, estimated empirically from a dataset that included many entries for forest trees (Cannell 1982) to be 8·33 and 2·44, respectively. The scaling exponents in eqn 1, 1 and 3/4, emerge directly from WBE. As Enquist & Niklas (2002) acknowledged, eqn 1 deviates systematically from the best-fit regression for MA on MR for small (or young) plants. The reason suggested for this was that access to seed reserves (particularly carbohydrate) favours the production of relatively more root material than predicted by WBE. Another reason for this deviation may be because the constants in eqn 1 were estimated from inaccurate root data.

There is a practical problem with root data for large trees. MR is inevitably under-estimated by unknown amounts because it is impossible to sample all parts of the root system. Enquist & Niklas (2002) noted this problem, but assumed any errors to be negligible. However, the extent of this under-estimation is potentially serious. For example, Le Goff & Ottorini (2001) could account for as little as 65% of the root mass of 30-year-old beech (Fagus sylvatica) trees after excavation compared with the masses recovered after meticulous sampling and statistical estimation of unsampled root fragments. Only 60% of the root mass of 2-year-old hybrid poplar trees (Populus trichocarpa × P. deltoides) was recovered by direct excavation and 24% was recovered as fine roots extracted from soil from which roots had already been coarsely sieved (Friend et al. 1991). Crude excavations – usual in large-scale biomass inventories of forest trees – will presumably recover even smaller fractions of the true MR, especially from among fine roots. The spatial extent of root sampling can be limited (e.g. Schulze et al. 1996), roots below 1 m deep are rarely sampled although many roots grow much deeper than this (Schenk & Jackson 2002), and seasonal variations in peak MR can easily be missed (Lauenroth 2000). For these reasons MR will be under-estimated and the error likely to increase with plant size.

A related issue is that ‘root systems’ are usually symbioses between plants and mycorrhizal fungi. This symbiosis is functionally important for the capture of nutrients and water from soil, and is a significant C sink. WBE considers ‘the root system’ as a single functional unit of mass MR that will (presumably) include any hyphal mass associated with resource capture. Yet, measurements of MR almost never deliberately include external hyphae because these are even more elusive than the roots to which they are attached.

Under-estimating MR is much less of a problem with small plants grown hydroponically or in pots of soil, so these offer the means to parameterise eqn 1 with greater confidence. If eqn 1 is applicable to all size scales it should not matter if the parameterisation is done on small plants rather than large; what is important is that the parameterisation is based on reliable data. Figure 1 shows the MA vs. MR relation for a large (n = 271) diverse group of hydroponically or container-grown plants ranging in total mass (MA + MR) from 0·3 mg to 35·6 g. The reduced major axis (RMA) regression of ln MA on ln MR revealed that:


Figure 1. Above- (MA) and below-ground (MR) masses of plants grown hydroponically or in soil containers. Data comprise one C4 grass (Robinson & Conroy 1999), six C3 grasses (Jarvis 1987; Robinson & Rorison 1988; Harmens et al. 2000; D. Robinson, unpublished data), 20 species of woody tree (Huante & Rincon 1998; Atkin et al. 1999), and 68 wetland angiosperm species (Shipley & Peters 1990). Curve A: eqn 1 with β12 and β13 = 8·33 and 2·44, respectively (Enquist & Niklas 2002). Curve B: eqn 1 with β12 and β13 = 4·86 and 3·01, respectively. Curve C: RMA regression (Niklas 1994), inline image where α = 0·983 (95% confidence interval, 0·963–1·003) and β = 2·609 (95% CI 2·393–2·845), n = 271, r2 = 0·987, P < 0·001, eqn 2). Unlike curve A, curve B intersects the geometric means of MR and MA (indicated by the cross-hairs).

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  • image(eqn 2)

As Enquist & Niklas (2002) found, MA scales isometrically with MR, i.e. MA ∝ MR. But eqn 1 deviates systematically and significantly from this relation and in a direction consistent with an under-estimation by eqn 1 of MR for a given MA (compare curves A and C in Fig. 1). A better fit, such that eqns 1 and 2 are statistically indistinguishable, was obtained by estimating new values for β12 and β13. This was done by iteratively fitting simulated values of MA derived using eqn 1 for > 100 different values of MR such that the scaling exponent (α) and allometric constant (β) for the resulting regression converged on 0·983 (± 0·001) and 2·609 (± 0·001), respectively, as in eqn 2. The corresponding value of β12 was 4·86, and for β13, 3·01 (curve B, Fig. 1).

If these β12 and β13 values are more accurate than the originals and are constant, the re-parameterised eqn 1 should also describe biomass allocation in larger plants such as forest trees, but it does not (curve B, Fig. 2). From the RMA regression of ln MA on ln MR for the forest data (curve C, Fig. 2) we find that


Figure 2. Above- (MA) and below-ground (MR) masses of forest trees. Data compiled by Cannell (1982), including only cases where root and above-ground masses, and plant density, were reported, to allow masses per plant to be calculated. Curves A and B: as in Fig. 1. Curve C: RMA regression, inline image where α = 1·048 (95% CI 1·024–1·071) and β = 2·723 (95% CI 2·163–3·430), n = 471, r2 = 0·938, P < 0·001, eqn 3).

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  • image(eqn 3)

(Curve C in Fig. 2 corresponds closely to the original version of eqn 1, i.e. curve A, because many of the data that Enquist & Niklas (2002) used to estimate β12 and β13 are included in the figure.)

At this point we can conclude either that eqn 1 is not universally applicable or that it reveals a large discrepancy between measured and predicted MR for forests. The relatively few data for small trees (MR < 300 g) in Fig. 2 seem to track curve B more closely than they do curve A, but those for the largest trees track curve A; this would be expected if MR for small tress is estimated more accurately than for large. To explore further the possibility of a general under-estimate in forest MR, the re-parameterised eqn 1 (curve B in Fig. 2) can also be described by a regression analogous to eqn 3:

  • image(eqn 4)

Assuming a relatively small error in estimating above-ground mass, and because MAMR, we can derive the average extent to which the measured forest tree masses (described by eqn 3) deviate from those predicted by the re-parameterised eqn 1 (and which is described by eqn 4). The simple result is that, on average, we may have been measuring only about 1·650/2·723, i.e. 60%, of the true below-ground mass of forest trees.

If correct, this has serious implications for estimates of how much C is stored in vegetation and for closing the global C balance (Houghton 2001). Instead of the 160 Pg C currently estimated to be stored in root systems globally, the true amount could be about 267 Pg C, i.e. about half of the 492 Pg C estimated for the planet's above-ground vegetation (figures based on data compiled by Saugier, Roy & Mooney (2001) and assuming that C is half of MR). Taking the more conservative view that the under-estimation of MR applies only to forests (tropical, temperate and boreal) and that data for other biomes are correct gives a total of 239 Pg C in root systems. This would mean that the current 160 Pg estimate of global root C was still only two-thirds of the true figure.

What are the implications for C dynamics if below-ground stocks have genuinely been under-estimated by this extraordinarily large margin? Assuming that estimates of ecosystem C fluxes (i.e. photosynthesis and respiration) obtained from micrometeorological and ecophysiological techniques are broadly accurate, a larger below-ground C pool can be accommodated within existing budgets only if the mean residence time (MRT) of C in various pools is longer than we think. For the forest C budgets reported by Malhi, Baldocchi & Jarvis (1999), the required increases in MRT vary among biomes and C pools, and depend on how C is partitioned between fine roots and detritus (including fine root turnover, exudation and fluxes to mycorrhizal fungi), but they are generally modest. For example, in temperate forests the MRT of C in plant biomass (Table 1) would have to increase by only 17% from 10 to 12 years assuming equal partitioning of C between roots and detritus to accommodate the ‘extra’ C implied by Fig. 2. This longer MRT would still be consistent with the 3–18 years mean ages of root C estimated for temperate trees (Gaudinski et al. 2001; Matamala et al. 2003). Longer MRTs of root C would extend the duration of the effective forest sink for atmospheric CO2 (Malhi et al. 1999), but would not increase the annual C flux below-ground. Longer MRTs would also reduce ‘the contribution of roots to the global annual NPP of terrestrial ecosystems’ (Matamala et al. 2003). However, the effects of this reduction on long-term C sequestration might be offset by the larger accumulation of C in roots suggested by Fig. 2. Current C cycle models would be influenced significantly by these revised estimates of C storage – if they are correct.

Table 1.  Mean residence time (MRT) of carbon (C) in living plant biomass (including that below-ground, MR), soil and litter, and in whole ecosystems, and their calculated percentage increases assuming that the measurements of MR from which MRTs were originally derived were only 60% of the true values. Increases in MRT are shown for three ways of partitioning C between root detritus (which includes mycorrhizal symbionts) and fine roots: all as detritus; all as fine root; or 50% as detritus plus 50% as fine root
Forest biomeC poolMean MRT of C (years)*Calculated increase in MRT of C (%)
All as root detritusAll as fine root50% as root detritus, 50% as fine root
  • *

    Data from Table 6 in Malhi et al. (1999).

  • Calculated by: (1) dividing the C stocks of fine plus coarse roots reported in Table 5 of Malhi et al. (1999) by 0·6; (2) partitioning this C between root detritus and fine roots and to total and biomass C pools; (3) adjusting the C increments in roots detritus according to the partitioning assumption applied; (4) ensuring that NPP estimates obtained by gas exchange match those obtained by summing biomass and detrital increments (see Table 6 of Malhi et al. 1999); (5) dividing the adjusted C stocks by the appropriate rates of C input, and expressing the resulting MRTs as a percentage of those reported by Malhi et al. (1999).

TropicalBiomass 16 018 6
Soil & litter 1552 929
Ecosystem 29241010
TemperateBiomass 10 01417
Soil & litter 10311112
Ecosystem 1814 8 8
BorealBiomass 12 01210
Soil & litter106 417 6
Ecosystem 89 4 2 2

But how likely is it that they are correct? Could we really have underestimated global MR by so wide a margin simply by not recovering all the roots from soil samples? The studies by Le Goff & Ottorini (2001) and Friend et al. (1991) suggest that recoveries of root mass as small as 60% are plausible for large trees. But at the global scale, the difference between 160 and 267 Pg C is equivalent to about 1·4 kg root dry weight m−3 soil averaged over the planet's entire land area to a depth of 1 m. This would mean that more than 40 km root m−3 soil (i.e. 4 cm cm−3) must routinely remain unsampled (assuming a mean specific live-root length of 30 m g−1: Jackson, Mooney & Schulze 1997). This seems an excessive oversight even for the least assiduous root ecologist.

However, large mass losses can occur after roots have been recovered. Between 20 and 40% of the sample's original mass can be lost during subsequent handling, washing and storage (Oliveira et al. 2000). This relatively mundane source of error in estimating MR is often ignored, but it will be important when many samples must be processed, as during forest biomass inventories. Additionally, some of the unsampled but functional ‘root’ material will actually be fungal. Mycorrhizal fungi often form extensive mycelia external to the root: Smith, Smith & Timonen (2003) quoted lengths for the external hyphae of ectomycorrhizal fungi of 103−104 m m−1 root. In terms of C, though, these hyphal lengths are probably equivalent to about only 15 Pg globally (assuming that 1 m hypha contains 1 µg C: see Metadata.jsp). A conservative 10% post-harvest mass loss (27 Pg C), unsampled external hyphae of mycorrhizal fungi (15 Pg C), plus the 160 Pg root C estimated from existing biomass inventories (Saugier et al. 2001), accounts for about 200 Pg of the global 267 Pg C implied by Fig. 2. Some of the remaining 67 Pg will be as recently functional ‘root’ material unrecognisable as such because it is so finely fragmented, and which would probably be classed as soil organic matter, not MR. But even if it were all simply left behind in unrecovered roots, that amount of C would be equivalent to root lengths ‘missed’ of only 1 or 2 cm cm−3, a much more believable magnitude of operator error. So, errors arising from root sampling and processing (which would increase the estimated size of the root C pool) and longer MRTs of root C (to balance current ecosystem C fluxes) could combine to make the under-estimates of global MR predicted by WBE rather more plausible than they might initially seem.

Of course, part of the apparent under-estimation of MR could simply be because eqn 1 is wrong. Eqn 1, however parameterised, might not be universally applicable and more work needed to derive a genuinely universal scaling relationship for mass allocation. β12 and β13 in eqn 1 could be size-dependent, and the large apparent under-estimation of MR in forests an artefact of assuming that these parameters are constant. This will remain an open question until the biophysical basis of β12 and β13 is understood. (Unlike most other developments of WBE, Enquist & Niklas's (2002) model has a complex allometry in which the values of the constants – i.e. β12 and β13– are as crucial to its predictions as are those of the scaling exponents.) Until this uncertainty is resolved, the large under-estimates of MR identified here should be regarded as pessimistic. However, the possibility raised by this use of WBE that global MR is genuinely and significantly under-estimated should not be dismissed entirely and the implications considered when compiling C budgets from biomass inventories. WBE can offer a rational way to estimate, or at least put limits on, poorly constrained data (global MR and below-ground C in this case), sharpening our confidence in their subsequent use to provide answers to big ecological questions.

Further questions about the WBE model, as applied mainly to plants, to which answers would be valuable, include the following:

  • • 
    One of the assumptions of WBE is that the diameters of ‘terminal capillaries’ are size-invariant, but to what extent are they genuinely size-invariant in real plants of diverse morphology and phenology?
  • • 
    Does it matter how a ‘terminal capillary’ is defined – xylem vessels in the finest lateral roots and petioles, or the micropathways between cellulose fibres in cell walls?
  • • 
    How do the internal transport systems in plants maintain themselves sufficiently close to the WBE ideal as they develop and grow?
  • • 
    How does the turnover – i.e. development, growth and senescence – of and damage to plant organs affect the functioning of the whole plant in terms of its capacity to operate like a WBE system?
  • • 
    How do developmental responses to environmental heterogeneities or transient conditions influence a plant's convergence onto the WBE ideal?
  • • 
    How do symbionts (e.g. mycorrhizal fungi), parasites (e.g. root hemi-parasites) and parasite-like organisms (e.g. lianas) fit into the WBE scheme?
  • • 
    Should the external hyphae of mycorrhizal fungi be included as part of a plant's ‘functional’ allometry since most plants form such symbioses and these structures are involved in resource capture and transport?
  • • 
    Does WBE also apply to the scaling of transport processes even within the non-living fractal-like structures of the soils and sediments in which plants grow?

The WBE model proposes a generic design for life. The tolerances of such a design are sufficiently wide that natural selection can produce apparently limitless variations within the constraints set by the basic theme (Enquist 2003). Traditional studies of mass allocation in plants have focused on this variation, i.e. detailed inter- and intraspecific comparisons of similarly sized plants, and on their responses to environmental factors. Such studies have failed to produce a unified, mechanistic understanding of allocation patterns and much confusion has reigned (Enquist & Niklas 2002; Reich 2002). This was because, pre-WBE, we had numerous datasets from which it was difficult to draw generalisations that provided satisfying functional insights. Crucially, we had no idea that we might not be looking along a main axis of variation in allocation patterns, but only at variation around that axis: we were literally unable to see the wood for the trees. But, as highlighted here, we still need to keep looking at the trees.


  1. Top of page
  2. Acknowledgements
  3. References

Thanks to Owen Atkin for providing Acacia data, Ruth Hasty for help with data processing, and Pete Smith, Dominic Standing, David Genney and two reviewers for their helpful comments.


  1. Top of page
  2. Acknowledgements
  3. References
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