871
Biol. Rev. (2004), 79, pp. 871–889. f Cambridge Philosophical Society
DOI : 10.1017/S1464793104006499 Printed in the United Kingdom
Plant allometry: is there a grand
unifying theory?
Karl J. Niklas*
Department of Plant Biology, Cornell University, Ithaca, New York, 14853, USA (E-mail : kjn2@cornell.edu)
(Received 10 June 2003 ; revised 16 April 2004; accepted 19 April 2004)
ABSTRACT
The study of size and its biological consequences – called allometry – has fascinated biologists for centuries.
Recent advances in this area of study have stimulated a renewed interest in these scaling phenomena, especially
in terms of the search for mechanistic explanations that transcend mere descriptive phenomenology. These
advances are reviewed in the context of plant biology. Allometric derivations are presented that predict how
annual growth in total body biomass is partitioned to construct new leaf, stem, and root tissues at the level of
an individual plant. Derivations are also presented to predict annual reproductive effort and to predict how the
biomass of body parts changes as a function of the number of plants per unit area in communities. The predictions emerging from these derivations are then examined empirically by comparing predicted and observed
scaling exponents for each relationship using a world-wide data compendium gathered from the primary literature. These comparisons provide strong statistical support for each of the allometric predictions. This support is
taken as evidence that a general unifying allometric theory for plant biology is near at hand. Nevertheless, the
validation of this theory requires much additional work and raises a number of procedural and conceptual
concerns that must be resolved before a single ‘ global ’ theory is accepted.
Key words : annual plant growth rates, biomass allocation, body mass, community structure, density effects, organ
partitioning, plant reproductive effort, size-correlated changes.
CONTENTS
I.
II
III.
IV.
V.
VI.
VII.
VIII.
IX.
X.
Introduction .................................................................................................................................................
The allometric formula : uses and abuses .................................................................................................
Derivations for organ biomass partitioning .............................................................................................
Derivations for annual growth rates of vegetative organs .....................................................................
Derivations for reproductive biomass .......................................................................................................
Derivations for plant community properties ...........................................................................................
Assessing organ biomass partitioning and annual growth .....................................................................
Assessing reproductive effort ......................................................................................................................
Assessing plant community relations ........................................................................................................
Is there a single unifying theory? ..............................................................................................................
(1) Size-dependent biases and ad hoc hypotheses ....................................................................................
(2) Outliers: do exceptions prove a rule ? ................................................................................................
(3) Proportionality ‘ constants’ are not constant .....................................................................................
(4) Scaling exponents : fractions or numbers ? ........................................................................................
(5) Different ‘first principles ’ can yield the same predictions ...............................................................
XI. Conclusions ..................................................................................................................................................
872
873
874
877
877
878
878
881
883
884
885
886
886
887
887
887
* Address for correspondence : Karl J. Niklas, Department of Plant Biology, Cornell University, Ithaca, New York, 14853, USA. Tel. :
+01 607 255 8727 ; fax : +01 607 255 5407.
Karl J. Niklas
872
XII. List of symbols ..............................................................................................................................................
XIII. Acknowledgements ......................................................................................................................................
XIV. References ....................................................................................................................................................
I. INTRODUCTION
Size is one of the most important features of any organism.
Well before the time of Darwin, natural historians observed
and recorded manifold changes in plant and animal shape
attending ontogenetic growth in size at the level of the individual. By the 1920s, these changes were mathematically
explored by Julian S. Huxley (1924, 1932) who showed that
the relative growth rates of different body parts can be deduced from the slopes of log–log plots of organ versus total
body size. During the decades that followed, physiologists
explored how metabolic rates varied as a function of body
size across taxa from mice to elephants. The expectation was
that metabolic rate would remain proportional to the 2/3
power of body size, reflecting the simple scaling relationship
between body surface areas with respect to body volumes.
However, the work of Samuel Brody and Max Kleiber
revealed surprisingly that the slope of the log–log plots of
metabolic rate versus body size was closer to 3/4 rather than
2/3 (Brody, 1945; Kleiber, 1947). The emergent 3/4 scaling
‘rule ’ was subsequently found to govern many other phenomena,includingsomerelatingtoplants(Hemmingsen,1960).
Since the seminal works of Huxley, Kleiber, and
Hemmingsen, the study of size and its consequences – called
allometry – has burgeoned in scope and approach. Numerous physiological, morphogenetic, ecological, and evolutionary size-correlated trends for plants and animals have
been observed, archived, and discussed conceptually (see
McMahon & Bonner, 1983 ; Peters, 1983 ; Calder, 1984;
Schmidt-Nielson, 1984; Niklas, 1994b ; Brown & West,
2000). Many of these trends appear to hold true across organisms as phyletically and ecologically diverse as microbes
and trees or mosses and whales. These broad interspecific
size-correlated trends are undeniably useful. Regardless
of the mechanisms underlying them, they permit the description (and thus potential prediction) of many important
biological relationships, at least within the boundary conditions proscribed by their statistical properties. Accordingly, the study of allometry is justifiable on the grounds
of strict empirical enquiry.
However, the importance of the study of allometry
extends beyond description or prediction. If certain trends
are size-dependent and ‘ invariant ’ with regard to phyletic
affinity or habitat, they draw sharp attention to the existence
of properties that are deeply rooted in all, or at least most
living things. Identifying these properties using a first principles approach, therefore, has become something of a Holy
Grail in biological allometry because any successful theory
would unify as many diverse phenomena in biology as
Einstein’s general theory of relativity has for physics. It is
understandable, therefore, that numerous attempts have
been made to provide an all-inclusive, unifying theory for
broad interspecific allometric trends. However, most have
888
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not held up against well-reasoned criticism or withstood
empirical tests.
For example, McMahon (1973, 1980) argued that metabolic rates increase as the 3/4 power of body mass because
the cross-sectional areas of structural members must scale
as the 3/4 power of the weight they support. Unfortunately,
this explanation has little or no bearing on unicellular
organisms or aquatic life forms for which the 3/4 power
‘ rule ’ also appears to hold true. Alternatively, Gray (1981)
proposed that local variations in body temperature affect
metabolism such that metabolic rates remain proportional
to the 3/4 power of body mass. Yet, it is hard to imagine
that ‘ local ’ variations in the body temperature of bacteria
and unicellular algae are substantial. Equally provocative,
Blum (1977) suggested that metabolic rates depend on the
functional surface areas of organisms, which vary in time as
well as in three dimensions. Noting that the surface area of
a hypervolume increases as a function of volume raised
to the 1xnx1 (where n is the number of dimensions), the
3/4 ‘ rule ’ immediately follows mathematically provided
that n=4. Unfortunately, Blum (1977) never fully explored
the ramifications of this idea (nor did he note that the variance for the numerical values of the slopes of metabolism
versus body mass includes 0.80 such that organisms may exist
in a five-dimensional space-time continuum).
However, no attempt at a unifying allometric theory is
more comprehensive or far-reaching than the theory proposed by Geoffrey West, James Brown, and Brian Enquist
(1997, 1999, 2000), which purports to explain everything
from Kleiber’s mouse-to-elephant 3/4 ‘ law ’ to the packaging of species in communities. Although similar to Blum’s
(1977) hypervolume explanation, this theory argues that
allometric relationships are governed by 1/4 power rules
(or multiples of 1/4) because all organisms have internalized
fractal delivery networks for energy and mass transfer,
which have evolved to minimize the energy and time required to absorb, distribute and deliver resources internally.
Although criticized on empirical and theoretical grounds
and challenged by alternative conceptual approaches (see
Banavar, Maritan & Rinaldo, 1999; Dodds, Rothman &
Wertz, 2001 ; Darveau et al., 2002 ; Weibel, 2002), the WestBrown-Enquist theory currently remains the most comprehensive, so much so that it may be called a ‘ theory for
everything. ’
The goal and scope of this article are far more circumspect. Here, my objective is to explore plant-size-correlated
trends that are explicable in terms of two empirically undeniable trends. The first of these is that, across vastly different
species and habitats, annual growth in biomass at the level of
the individual plant scales as the 3/4 power of body mass.
The second is that annual growth remains proportional
to the capacity of the individual to harvest sunlight as judged
by either cell pigment content or leaf biomass (Niklas &
Plant allometry
Enquist, 2001). These two size-correlated trends in conjunction with a comparatively small number of assumptions,
each of which is predicated on simple biophysical principles,
are used to predict the scaling exponents for four important
aspects of plant allometry: (a) the partitioning of total body
biomass among leaves, stems, and roots (Enquist & Niklas,
2002), (b) the annual growth rates for new leaf, stem, and
root tissues (Niklas & Enquist, 2002 a, b), (c) the biomass
annually invested in reproductive effort at the level of individual plants (Niklas & Enquist, 2003), and (d) a number of
ecologically important properties of mixed or monotypic
plant communities (Enquist & Niklas, 2001 ; Niklas, Midgley
& Enquist, 2003 a ; Niklas, Midgley & Enquist, 2003b). The
predictions of these derivations are then tested by examining
a large database for plant organ biomass relations gathered
from the primary literature.
This foray into plant allometry is preceded by a brief
treatment of the statistical protocols used in allometric
analyses. This treatment is necessary because the interpretation of any size-dependent relationship relies on regression
analysis to obtain the proportional (scaling) relationship between biological variables (which is given by the numerical
value of the slope of the regression curve) and the proportionality factor governing the relationship (which is given
by the Y-intercept of the regression curve). Thus, every
empirical study of allometry and every theory for allometric
relationships rests ultimately on the edifice of statistical inference. As we shall see, this edifice is remarkably robust and
sophisticated, but it is nevertheless not as firm as one would
like, largely because alternative regression models exist that
can often obtain different slopes and constants depending
on the nature of the data at hand (or the conceptual biases of
the researcher).
II. THE ALLOMETRIC FORMULA : USES
AND ABUSES
The term ‘ allometry’ was first coined by Julian S. Huxley
(1924, 1932) who evaluated a large number of size-correlated
trends and proposed that each could be approximated by
the power function Y=bXa, where Y denotes the size of a
body part (as gauged by its length or mass), X is a comparable measure of the size of the organism minus the size of the
body part of interest, b is the allometric constant, and a is the
allometric (scaling) exponent. Huxley’s formula is typically
expressed logarithmically as log Y=log b+a log X (Gould,
1966 ; Smith, 1980 ; Sokal & Rohlf, 1981 ; LaBarbera, 1986,
1989 ; Niklas, 1994a). In this form, we see that log b is the
Y-intercept (the numerical value of log Y when log X=1) and
a is the slope of log Y versus log X (Sokal & Rohlf, 1981).
Provided that the values of Y and X are measured for an
individual organism over its ontogeny, for any unit time
interval t, the change in the logarithm of Y equals a times
the change in the logarithm of X, i.e. d (log Y)/dt=ad
(log X)/dt such that dY/dX=aY/X (Huxley, 1932). The
scaling exponent a is thus the ratio of the relative growth
rate in the size of Y with respect to the relative growth rate
in the size of X.
873
Huxley (1932) presented this formula as a true biological
‘law ’ – one that provides an a priori ‘ theory ’ for understanding the effects of size on morphometry. However, several workers have argued for a more pluralistic approach to
the study of relative size, one that does not rest exclusively
on the power formula log Y=log b+a log X (e.g. Smith,
1980 ; Harvey, 1982; Chappell, 1989). Indeed, Huxley’s first
detractor was the individual to whom he dedicated his 1942
book, D’Arcy Thompson. Reevaluating the data sets used
by Huxley, Thompson (1942) quickly showed that simple
linear equations often fit some of Huxley’s untransformed
data sets as well as or better than Y=bXa power functions
and that linearly correlated pair-wise data remain linear
when log-transformed. Indeed, many authors fail to provide
a sound theoretical basis for accepting the power function
relationship, and reevaluations of their data indicate that
statistical analysis of untransformed data can give equally
good results as that of the transformed data (e.g. Smith,
1980).
Another problem in allometric analyses is the predictability of log–log regression curves, especially in the case of
interspecific comparisons, which obtain high coefficients
of correlation suggesting that Y values can be accurately
predicted based on their corresponding X values. This
problem arises when the variation in Y is large, as is often
the case. Under these circumstances, the predictive capacity
of log–log regression curves is remarkably low and analyses
of residuals and per cent prediction errors are required.
Another problem with using log-transformed data is that
regression techniques fit a line to the mean values of the Y
variable, but the mean of log-transformed variables is the
median of the log-normal distribution (Gould, 1966 ; Sokal
& Rohlf, 1981). Therefore, without correction, values
reported for the antilog of the Y-intercept of the regression
line are consistently biased (Prothero, 1986; Niklas, 1994 b).
The correction factor, however, is simple ; it can be estimated from the standard error of log b. Unfortunately,
many workers fail to report this important statistical parameter.
Nevertheless, there are at least three good reasons for
using log-transformed data. First, transformation of many
forms of data typically reduces the problem of working with
outliers ; second, log-transformed data typically comply with
the statistical assumptions of normality and homoscedacity
(Kermack & Haldane, 1950 ; Sokal & Rohlf, 1981) ; and,
third, it provides a convenient means of examining proportionality that is unaffected by the unit of measurement,
since the slope of the log–log regression line (log y2xlog y1)/
(log x2xlog x1) becomes ( y2/y1)/(x2/x1) when converted to a
linear scale. For these reasons, power functions taking the
form Y=bXa have become the traditional ‘ theory ’ in contemporary allometric analyses (see Gould, 1966 ; Peters,
1983 ; Calder, 1984 ; Schmidt-Nielson, 1984).
A far more controversial issue is the type of regression
analysis used in allometric analyses, because the type of
analysis can profoundly influence the numerical values
of scaling exponents and thus the extent to which observed
exponents accord statistically with those predicted by
a particular allometric theory (Sokal & Rohlf, 1981;
Seim, 1983 ; Niklas, 1994 b). Based on standard statistical
Karl J. Niklas
874
Table 1. Estimates of the allometric constant b and scaling exponent a for the formula Y2=b+aY1, where Y and b are
log-transformed
Regression method
OLS
Expression to be minimized
n
X
ð y2i xbxay1i Þ2
Estimate of b
Estimate of a
y2 x
ay1
sy2 y1
sy21
y2 x
ay1
1
2sy2 y1
i=1
MA
n
X
( y2 xbxay1 )2
i
1+a2
i=1
RMA
i
n
X
( y2 xbxay1 )2
i
i=1
i
a
y2 x
ay1
sy2
sy1
sy
2
sy1
(
1=2 )
2
sy22 sy21 + sy22 sy21 +4sy2 y
2 1
if sy2 y1 > 0
if sy2 y1 < 0
sy21 , sample variance of Y1 ; sy22 , sample variance of Y2 ; sy22 y1 , sample covariance of Y2 and Y1 ; OLS, ordinary least-squares regression analysis ;
MA, major axis regression analysis ; RMA, reduced major axis regression analysis ; y1 and y2, numerical values of interdependent variables Y1
and Y2, respectively.
inference, ordinary least-squares (Model Type I) regression
analysis (denoted here as OLS) can be used provided that
(a) the error term (log e) is normally distributed with a mean
of zero and constant variance, (b) the distribution of log Y
is normal at each value of log X, (c) the variance of log Y is
constant across the range of log X, and (d) log X is an independent variable whose values are known without error
(Sokal & Rohlf, 1981). Unfortunately, these four assumptions rarely if ever hold true for many data sets subjected
to allometric analyses. Certainly, there is no ‘ independent
variable’ when the size of the body parts of the same organism are compared.
For these reasons, many allometricists have turned to
Model Type II regression analyses, e.g. major axis (MA) or
reduced major axis (RMA) regression analyses (see Harvey
& Mace, 1982 ; Rayner, 1985; LaBarbara, 1986 ; McArdle,
1988; Niklas, 1994b). Model Type II analyses typically
identify the variables of interest as Y2 and Y1 rather than Y
and X to indicate their biological and functional interdependence. The choice of which kind of Model Type II
regression analysis to use nevertheless remains. Each Model
Type II regression protocol involves different assumptions
about the error structure and variance relations between Y2
and Y1 to estimate the allometric constant b and the scaling
exponent a (Table 1).
In terms of selecting which Model Type II protocol to
use, it is instructive to compare the assumptions underlying MA or RMA. The former is sensitive to the absolute
measurement scales for the variables of interest and it is not
especially robust to rotating the coordinate axes, whereas
RMA is insensitive to both of these concerns. RMA is also
less sensitive to assumptions about the error structure in a
data set and it is less biased in terms of estimating the
functional (allometric) relationship between two dependent
variables. For these reasons, RMA has become one of the
standard regression techniques for allometric analyses,
although arguments can be made for MA (see Harvey &
Mace, 1982 ; Rayner, 1985 ; LaBarbara, 1986 ; McArdle,
1988; Niklas, 1994 b).
Importantly, the numerical value of the scaling exponent
for the RMA regression line (aRMA) can be computed for the
exponent of the corresponding OLS regression line (aOLS)
using the formula aRMA=aOLS/r, where r is the OLS
correlation coefficient. Likewise, the numerical value of the
Y-intercept of the RMA regression curve (log bRMA) can
be computed using the formula log bRMA=log Ŷ 2xaRMA
log Ŷ1, where Ŷ denotes the mean value of Y. Methods for
calculating the 95% confidence intervals of aRMA and bRMA
are available ( Jolicoeur & Mosimann, 1968 ; Sokal & Rohlf,
1981; Rayner, 1985; McArdle, 1988), although each tends
to give a highly conservative estimate of the intervals and
thus inclines a worker to commit Type I error (the rejection
of a valid allometric hypothesis).
Provided that the coefficient of correlation r2 equals
or exceeds 0.95, the choice of regression model is largely
moot, since OLS, MA, and RMA obtain nearly equivalent
numerical values for a. However, when r2<0.95, either
Model Type II regression protocol (i.e. MA or RMA) gives
higher numerical values for a than OLS. Throughout this
article, RMA regression analyses are used to compare
predicted with observed allometric trends (see Sections
VII–IX). These comparisons emphasize the numerical
values of scaling exponents, i.e. the slopes of log–log linear
RMA regression curves, because this parameter emerges
directly from allometric derivations for plant-size-correlated
trends (see Sections III–VI). By contrast, the derivations
to be reviewed do not predict the numerical values of allometric constants, which can be taxon-specific and can vary
as a function of habitat.
III. DERIVATIONS FOR ORGAN BIOMASS
PARTITIONING
In this and the following two sections, I present mathematical derivations that predict the scaling exponents,
denoted by a, for a number of size-correlated trends.
Plant allometry
875
A
3
2
1
0
−1
Log G T
−2
−3
−4
−5
−6
−7
−8
−9
−10
−11
−12
−13
−16 −15 −14 −13 −12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1
0
1
2
3
4
2
3
Log M T
B
3
2
1
0
−1
Log G T
−2
−3
−4
−5
−6
−7
−8
z algal species
−9
herbaceous species and
juveniles of woody species
−10
−11
−12
−13
−16 −15 −14 −13 −12 −11 −10 −9
x woody species
−8 −7 −6
−5 −4 −3 −2
−1
0
1
Log H
Fig. 1. Log–log bivariate plots of total annual growth rate in body mass GT versus total body mass MT(A) and the capacity to
harvest sunlight H(B). Lines denote reduced major axis regression curves for entire data set (A) and for portions of the data set
(B). Scaling exponents and allometric constants for these relationships provided in Table 2. Data gathered from the primary
literature (see text).
The corresponding allometric constants for each relationship are denoted by b. Since the derivations in each section
are extensive, successive allometric constants are denoted by
sequentially numerical subscripts, i.e. b1, b2, etc. All of these
derivations are predicated on two empirically well-established allometric trends (Fig. 1). Specifically, across a broad
spectrum of aquatic and terrestrial nonvascular and vascular
plant species, annual growth in plant body biomass (net
annual gain in dry mass per individual, GT) remains
proportional to the 3/4 power of total body mass (total dry
mass per individual, MT) and directly proportional (scales
isometrically with respect) to the capacity of an individual
plant to harvest sunlight H as measured by the concentration of photosynthetic pigments per algal cell or the
standing leaf biomass of a vascular plant with woody, nonphotosynthetic stems (see Niklas & Enquist, 2001).
The evidence for these two trends comes from an analysis
of three data sets, one for algal species (Niklas, 1994 b), one
for tree-sized species (Cannell, 1982), and another recently
compiled data set for small nonwoody species and juvenile
Karl J. Niklas
876
Table 2. Statistical comparisons between predicted and observed relations among annual total body growth rates GT, total
body mass MT, light-harvesting capacity H and standing leaf biomass ML. Scaling exponents and allometric constants are
based on reduced major axis regression (aRMA and bRMA) of log10-transformed data (original units for growth rates are kg dry
mass plantx1 ; original units for H and ML are kg dry mass plantx1). In all cases, P<0.001 or less. ++, prediction accepted ;
+, prediction accepted because scaling exponent reflects differences in bRMA among data sets ; x, reject prediction ; CI,
confidence intervals
aRMA¡S.E.
Y1 versus Y2
Predicted
Antilog bRMA¡S.E.
Observed
95 % CI
Observed
r2
n
Within the size range of the entire data base : 5.65r10x16 kg f MT f 3.18r103 kg
GT versus MT
0.75
0.76¡0.004
0.75–0.77
0.59¡0.017
0.97
1173
GT versus H
1.00
0.82¡0.003
0.81–0.70
2.05¡0.020
0.99
453
x16
x10
Within the size range of the algal data base : 5.65r10
kg f MT f 1.88r10
kg
GT versus MT
0.75
0.75¡0.007
0.73–0.76
0.12¡0.100
0.99
66
GT versus H
1.00
1.05¡0.045
0.95–1.14
637¡0.647
0.89
66
Within the size range of the entire vascular plant data base : 5.93r10x6 kg f MT f 3.18r103 kg
GT versus MT
0.75
0.66¡0.006
0.65–0.68
0.38¡0.017
0.90
1107
GT versus ML
1.00
0.83¡0.008
0.81–0.85
2.05¡0.021
0.96
387
Within the size range of the Cannell (1982) vascular plant data base : 4.16r10x1 kg f MT f 3.18r103 kg
GT versus MT
0.75
0.77¡0.030
0.71–0.84
0.26¡0.066
0.80
132
GT versus ML
1.00
1.03¡0.039
0.95–1.12
1.59¡0.036
0.76
175
Within the size range of the new vascular plant data set : 5.93r10x6 kg f MT f 6.41r10x2 kg
GT versus MT
0.75
0.72¡0.011
0.69–0.75
0.39¡0.028
0.79
975
GT versus ML
1.00
0.98¡0.021
0.95–1.03
5.51¡0.070
0.90
212
trees that have not accumulated substantial quantities
of secondary tissues in stems and roots, which has been
gathered from the primary literature published between
1992 and 2002 (Table 2). When the three data sets are
pooled together in various ways, the scaling exponents
are observed to depart, often significantly, from either 3/4
or unity but only as a consequence of group-specific differences in the allometric constant. For example, annual
growth is observed to scale as the 0.82 – power of lightharvesting capacity across the entire pooled data base,
but the one-to-one (isometric) relationship holds true
when each of the three components of the data base is
examined separately (Table 2). Thus, across all species,
GT=b1MT3/4=b2H, where the numerical values of b1 and
b2 are group-specific and may vary as a function of habitat.
This relationship provides a basis for predicting the allocation of total body mass to leaf, stem, and root biomass
for vascular plants with a stereotypical body plan (i.e. those
consisting of leaves, stems, and roots). For these organisms,
total vegetative body mass equals the sum of standing leaf,
stem, and root biomass (ML, MS, and MR, respectively) such
that GT=b1MT3/4=b1(ML+MS+MR)3/4. Likewise, since
light harvesting capacity is a function of standing leaf biomass, the relationship GT=b2 ML holds true across species.
Thus, M L =(b 1 /b2 )(M L +M S +M R ) 3/4=b 3 (M L +M S +
MR)3/4. This last relation can be used to derive the individual
scaling relationships among leaf, stem, and root biomass,
provided a few assumptions are made (Enquist & Niklas,
2002; Niklas & Enquist, 2002 a ; Niklas, 2003).
This derivation begins by noting that standing stem and
root biomass must scale as some function of organ bulk
F
Prediction
35 624
62 737
++
+
12 186
496
+
+
9829
9495
+
+
533
545
+
+
3684
1993
+
+
tissue density r, organ diameter D, and organ length L.
Denoting these stem and root features by the subscripts S
and R, respectively, standing stem and root biomass can
be expressed as MS=b4rSDS2LS and MR=b5rRDR2LR, respectively. Provided that stem and root bulk tissue densities
are relatively constant for any species (i.e. b4rS=b6 and
b5rR=b7), such that MS=b6DS2LS and MR=b7DR2LR, we
see that ML=b3[ML+b6DS2LS+b7DR2LR]3/4.
A critical assumption is required at this point in the derivation. Standing leaf biomass is assumed to scale as average
cross-sectional stem and root area, i.e. ML=b8DS2=b9DR2.
This assumption is biologically and physically reasonable.
The mass of water flowing through stems and roots must
be conserved and it must depend on the number and
average cross section of water-conducting cell types (i.e.
tracheids and vessel members), which are dependent on
stem cross section (Carlquist, 1975 ; Zimmermann, 1983 ;
Niklas, 1994b), particularly sapwood in woody species
(Bond-Lamberty, Wang & Gower, 2001). Provided that
ML=b8DS2=b9DR2 holds for any particular species, it
then follows that ML=b3[1+(b6/b8)LS+(b7/b9)LR]3/4
(ML)3/4=b34[1+(b6/b8)LS+(b7/b9)LR]3.
This last scaling relationship can be simplified further
if it is assumed that average root and stem lengths scale
isometrically with respect to one another (i.e. LR=b10LS).
This assumption is based on the observation that, across
diverse vascular species, stem and root meristematic growth
and extension in organ length (and diameter) is coordinated,
presumably through hormone-mediated developmental processes (e.g. Genard et al., 2001). If this assumption is true, then
it follows that ML=b34[(1/LS)+(b6/b8)+(b7b10/b9)]3LS3.
Plant allometry
Noting that, for any species characterized by indeterminate
growth in body size, 1/LSp0 as LS increases, if also follows
that MLyb34[(b6/b8)+(b7b10/b9)]3LS3=b11LS3.
Collectively, the foregoing scaling relationships indicate that MS=(b6/b111/3b8)MLML1/3=b12ML4/3, that MR=
(b7b10/b9b111/3)MLML1/3=b13ML4/3, and that MS=(b12/b13)
MR. It is also seen that the standing biomass of the shoot
(the sum of all leaves and stems per plant) scales with respect
to standing root biomass as ML+MS=(b12/b13)MR+
(MR/b13)3/4 (Enquist & Niklas, 2002).
The allometric (taxon-specific) constants for these relationships may vary widely between different groups of plants as
a consequence of phyletic or local environmental differences. However, in terms of their scaling exponents, the
foregoing derivations indicate that leaf biomass should scale
as the 3/4 power of stem (or root) biomass such that stem
and root biomass will scale isometrically with respect to one
another, i.e. ML ! MS3/4, ML !MR3/4, and MS ! MR.
IV. DERIVATIONS FOR ANNUAL GROWTH
RATES OF VEGETATIVE ORGANS
The relations empirically observed among total annual
growth GT, body mass MT, and standing leaf biomass ML
(Niklas & Enquist, 2001) also permit the derivation of the
relationships among the annual growth rates of leaves,
stems, and roots, i.e. organ biomass production per plant per
growth season (denoted as GL, GS, and GR, respectively)
(Niklas & Enquist, 2002 a, b).
From first principles, total annual growth in vegetative
body mass must equal the sum of the annual growth of
leaves, stems, and roots such that GT=GL+GS+GR. As
noted, across vascular plants with woody, nonphotosynthetic stems, GT=b2ML and, for deciduous species, it is
reasonable to assume that annual leaf growth scales as an
isometric function of standing leaf biomass (i.e. GL=b14ML)
such that GT=(b2/b14)GL. It therefore follows that GL=
[b14/(b2xb14)](GS+GR)=b15(GS+GR). However, for nondeciduous species, annual leaf growth is likely to scale as an
isometric function of the difference between standing leaf
biomass and the leaf biomass produced in the previous
growth season (Ml), which in turn is may be assumed to
be some function of ML. That is, GL=b16(ML x Ml) and
Ml=b17ML, respectively. If these assumptions are correct,
then we see that, for nondeciduous species, GL=
b16(1 x b17)ML=b18ML and GT=(b2/b18)GL=b19GL, from
which it follows that GL=(GS+GR)/(b19x1)=b8(GS+GR).
Noting that GL=b15(GS+GR) and GL=b20(GS+GR)
are mathematically (but not numerically) equivalent, the
general expression for the relationship between leaf growth
and stem plus root growth across all vascular plants is predicted to be GL=b15 or b20(GS+GR)=b21(GS+GR). This general formula can be used to derive the scaling relations among
the growth rates of all three vegetative organs as follows.
It is reasonable to assume that stem and root growth
rates scale as some function of the average diameter and
length of stems and roots, i.e. GS=b22rSDS2LS and GR=
b23rRDR2LR, respectively (see Section III). This assumption
877
is predicated on the dependency of organ growth rates
on standing organ biomass. It is also reasonable to assume
that the average bulk tissue densities of stems and roots
are relatively constant for any particular taxon, i.e.
b22rS=b24 and b23rR=b25, such that GS=b24DS2LS and
GR=b25DR2LR. Under these circumstances, GL=b21
(b24DS2LS+b25DR2LR). Assuming, once again, that average
root cross-sectional areas scale isometrically with respect to
average stem cross-sectional areas (i.e. DR2=b26DS2) and
that root length scales isometrically with respect to stem
length (i.e. LR=b27LS), we see that GL=b21(b24+b25b26b27)
DS2LS=b 28DS2LS. Therefore, GL=(b 28/b24)GS=b129GS,
GL=[b29/(b29xb21)]GR=b30GR, and GS=(b30/b29)GR=
b131GR. Likewise, total shoot growth (leaf and stem growth
per year) is predicted to scale with respect to root growth as
GL+GS=b30GR+b31GR=b32GR.
The numerical values of the allometric ‘ constants’
featured in these derivations are expected to vary among
taxa and as a result of local environmental conditions.
However, in terms of proportional relationships, annual
leaf growth rate is predicted to scale isometrically with
respect to stem or root annual growth rates, i.e. GL !
GS !GR, whereas the annual growth rate of the shoot in toto
will scale isometrically with respect to the annual growth
rate of roots, i.e. GL+GS ! GR.
V. DERIVATIONS FOR REPRODUCTIVE
BIOMASS
As noted, for most seed plants, annual growth in body mass
scales as the 3/4 power of total body mass, which equals
the sum of standing leaf, stem, and root biomass. That
is, GT=b33MT3/4=b33(ML+MS+MR)3/4, where b33 is a
taxon-specific (allometric) constant. With relatively few assumptions, it is possible to derive a relationship between
the annual biomass invested in reproduction MP and the
partitioning pattern of vegetative organ biomass, provided
that plants are large such that MP is not a substantial fraction
of total body mass (Niklas & Enquist, 2003).
This derivation begins by stressing that, with the exception of very small or annual monocarpic species, MP does
not contribute significantly to MT, because reproductive
body parts, even for many slow-growing conifer species, are
typically shed in less than one year. By contrast, when a
plant enters its reproductive phase of growth, GT is the sum
of GL, GS, GR, and GP, where GP is annual reproductive
growth, which requires an annual expenditure in metabolic
production.
Therefore, across large and perennial seed plant species,
GL+GS+GR+GP=b33(ML+MS+MR)3/4. As noted, MS
and MR each scale as the 4/3 power of ML, whereas GL,
GS, and GR scale isometrically with respect to each other.
In other words, MS=b34ML4/3, MR=b35ML4/3, GS=b36GL,
and GR=b37GL (Enquist & Niklas, 2002 ; Niklas & Enquist,
2002 b). We also know that, on average, GL=b38 ML, where
b6 includes units of yearx1 (Niklas & Enquist, 2002 a).
Accordingly, GP=b33(ML+MS+MR)3/4x(1+b36+b37)b38
ML. Assuming that the relation between annual reproductive
878
growth and biomass scales isometrically as GP=b39MP,
where b39 includes units of yearx1, the scaling relation between MP and ML, MS, or MR is MP=b40(ML+b41ML4/3)3/4
xb42ML, MP=b40[(MS/b34)3/4+(b41/b34)MS]3/4xb42(MS/
b34)3/4, and MP=b40[(MR/b35)3/4+(b41/b35)MR]3/4xb42
(MR/b35)3/4, where b40=b33/b39, b41=b34+b35, and b42=
(1+b36+b37)(b38/b39). Each of these equations describes a
slightly nonlinear log–log (concave) relation for MP versus
ML, MS, or MR. However, as will be seen, each predicted
trend is approximated reasonably well by a linear log–log
relationship.
Importantly, none of the scaling relations used to derive
the scaling relationships between reproductive biomass and
leaf, stem, or root biomass directly or indirectly relates MP
to ML, MS, or MR. Therefore, no mathematical ‘circularity ’
exists if MP is predicted based on the values of ML, MS, or
MR reported in the literature for plants (regardless of their
reproductive status). The scaling exponent and the allometric constant for the relationship MP versus ML, MS, or MR
depend exclusively on the numerical values of b40–42. In
turn, these ‘ constants’ depend on the vegetative biomass
partitioning pattern of a particular species or higher taxon.
Although all of the foregoing derivations cannot predict
the numerical values of the allometric constants a priori, they
can be tested directly by evaluating whether predicted
values for the scaling exponents and the allometric constants
agreed with those observed for inter – and intraspecific
reproductive trends (see Niklas & Enquist, 2003).
VI. DERIVATIONS FOR PLANT COMMUNITY
PROPERTIES
Here, my focus shifts from the level of the individual to that
of an entire plant community. Once again, the whole-plant
annual growth rate GT is observed to scale as the 3/4 power
of MT and isometrically with respect to the capacity to
intercept sunlight H, i.e. GT=b43MT3/4 and GT=b44H1,
where b43 and b44 are group-specific constants. For vascular
plants, H is proportional to ML (Niklas & Enquist, 2001).
Therefore, GT=b43MT3/4=b44H=b45ML and ML=
b46MT3/4, where b46=b43/b45 (allometric constants, which
may or may not vary across taxa). Based on hydraulic and
other biophysical considerations, ML is assumed to scale
isometrically with respect to stem cross-sectional area, which
is proportional to the square of basal stem diameter DS2.
Therefore, ML=b46 MT3/4=b47DS2.
Assuming that the three-dimensional space ai occupied
by an individual plant is proportional to basal stem crosssectional area DS2, i.e. ai=b48DS2, the maximum number of
individual plants possible in a community per unit area
sampled N (‘ plant density ’) equals the quotient of the total
unit area occupied by all individuals AT and the area occupied by an average individual in any given community.
Therefore, it mathematically follows that N=AT/ai=
AT/b48DS2=(b47/b46 b48) AT/MT3/4=b49 AT/MT3/4 (Niklas
et al., 2003 a).
Because AT is a constant whenever comparisons across
different communities employ equivalent sample areas,
Karl J. Niklas
i.e., AT equals some constant k, the preceding scaling relations take the form of N=k/b48DS2=(b47/b46b48)k/MT3/4
=b49k/MT3/4 such that, at the level of an individual plant,
four proportional relations emerge, i.e. MT ! Nx4/3,
GT ! Nx1, ML ! Nx1, and DS2 ! Nx1 (Enquist & Niklas,
2001; Niklas et al., 2003 a). As in all cases, these relationships
are influenced by potential taxon-specific variation in biomass allocation, a variety of allometric constants, and by
rates of limiting resource supply for a given environment.
However, the scaling exponents for these relationships are
expected to be insensitive to these factors.
Finally, because the numerical value for each variable Y
measured at the level of an individual plant equals the
total community value for the variable Ŷ divided by N, i.e.
Y=Ŷ/N, the scaling relationship for an entire community
becomes Ŷ ! N1+a. Thus, from the foregoing proportional relationships, we see that total plant biomass M̂T is expected to scale as N 1x4/3 or N x1/3, whereas total community
growth in biomass per year ĜT, total community standing
leaf biomass M̂L, and total basal stem area D̂S2 are each
predicted to scale as N 1x1 or N 0. Accordingly, at the level
of entire plant communities, total standing biomass will
scale as the x1/3 power of plant density, whereas total
community growth, standing leaf biomass, and total basal
stem area are expected to be invariant provided that communities have reached an equilibrium with their available
resources (Niklas et al., 2003 a).
VII. ASSESSING ORGAN BIOMASS
PARTITIONING AND ANNUAL GROWTH
To assess the predictions for standing leaf, stem, and root
biomass and for annual leaf, stem, and root growth in biomass (see Sections III–IV), data for these parameters were
gathered from the primary literature. For the majority of
tree species, the bulk of these data comes from Cannell
(1982) who compiled data sets for tree-sized dicot, monocot,
and conifer species as well as a limited number of bamboo
species (see Enquist & Niklas, 2002; Niklas and Enquist,
2002a, b).
Each of the Cannell (1982) data sets is standardized to
1.0 ha and represents approximately 600 sites world-wide,
published in a standardized tabular format that provides the
primary citation and, when supplied by authors, longitude,
elevation, the age of the dominant species (or conspecific
in the case of monotypic managed stands), the number of
plants per 1.0 ha (‘ plant density ’), height, total basal stem
cross-sectional area, and the standing biomass and net biomass production of stem wood, bark, branches, fruits, foliage, and roots (in units of metric tons dry matter per year).
The values for annual stem wood, bark, foliage, etc. production used here reflect as much as possible annual losses
of dry matter due to mortality, litter-fall, decay, and consumption (see Cannell, 1982).
Organ biomass and productivity were determined by
authors from direct measurements of fully dissected representative plants (typically f5 individuals) for the majority
of the Cannell (1982) sites. Authors regressed these data to
Plant allometry
879
Table 3. Statistical comparisons between predicted and observed relations among standing leaf, stem, root, and shoot biomass per
plant (ML, MS, MR, and MSH, respectively). Scaling exponents and allometric constants are based on reduced major axis regression
(aRMA¡S.E. and bRMA¡S.E.) of log10-transformed data (original units for biomass are kg dry mass plantx1). In all cases, P<0.0001
or less. +, prediction accepted ; x, prediction rejected ; MT, total individual body mass ; CI, confidence intervals
aRMA¡S.E.
Y1 versus Y2
Predicted
Antilog bRMA¡S.E.
Observed
95 % CI
Observed
x6
r2
n
Within the size range of the entire vascular plant data base : 5.93r10 kg f MT f 3.18r10 kg
ML versus MS
0.75
0.75¡0.004
0.74–0.76
0.20¡0.011
0.97
1117
ML versus MR
0.75
0.84¡0.006
0.82–0.85
0.40¡0.015
0.96
678
1.00
1.12¡0.006
1.11–1.13
2.66¡0.012
0.98
736
MS versus MR
MSH versus MR
y1.00
1.07¡0.005
1.06–1.08
3.28¡0.014
0.97
1244
Within the size range of the Cannell (1982) vascular plant data base : 4.16r10x1 kg f MT f 3.18r103 kg
ML versus MS
0.75
0.75¡0.008
0.73–0.76
0.12¡0.012
0.910
661
ML versus MR
0.75
0.79¡0.016
0.76–0.82
0.41¡0.016
0.861
338
MS versus MR
1.00
1.09¡0.009
1.05–1.13
2.59¡0.012
0.971
366
MSH versus MR
y1.00
1.08¡0.014
1.05–1.11
3.54¡0.020
0.94
345
Within the size range of the new vascular plant data set : 5.93r10x6 kg f MT f 6.41r10x2 kg
ML versus MS
0.75
0.87¡0.010
0.85–0.89
0.43¡0.035
0.94
456
ML versus MR
0.75
0.96¡0.016
0.92–0.99
0.74¡0.016
0.91
340
MS versus MR
1.00
1.03¡0.020
0.99–1.07
1.01¡0.066
0.86
370
MSH versus MR
y1.00
1.07¡0.012
1.06–1.08
2.32¡0.041
0.89
899
estimate total organ biomass per 1.0 ha community sample.
Data based on estimated regression variables were rejected
when entering the Cannell (1982) data sets for the purposes
of the analyses presented here and elsewhere (Enquist &
Niklas, 2002; Niklas & Enquist, 2002a, b, 2003). Importantly, most of these data sets are for even-aged conspecific
stands (n=600 out of 880 usable data sets) and biomass
production values are typically averaged values for two or
more years. Therefore, for each site used in the following
analyses, the variance in standing organ biomass and biomass production was assumed to be comparatively small
and annual production rates were considered representative
of ‘ normal ’ rather than idiosyncratic growth seasons.
Standing leaf, stem and root biomass per ‘average ’ plant
was computed for each of the Cannell (1982) sites using the
quotient of total community standing organ biomass and
plant density. Annual leaf, stem, and root production rates
were similarly calculated using the quotient of annual organ
type production per community sample and plant density.
However, it must be noted that most of the Cannell (1982)
data sets probably underestimate standing root biomass
and biomass production, particularly those of fine and small
roots, because these are more difficult to excavate completely for increasingly larger root systems and because these
fine and small roots are reported to increase disproportionately with increasing plant size (see Niklas &
Enquist, 2002 a). Thus, numerically higher scaling exponents than those predicted were anticipated for any regression analysis using root biomass or biomass production
as the variable plotted on the abscissa.
Since the Cannell (1982) data sets emphasize mature
and large plant body sizes, additional data were gathered
from the primary literature published between 1990 and
2002 for species with comparatively small mature body sizes
F
Prediction
33 893
16 444
29 568
49 165
+
x
x
+
8425
2439
13 621
5786
+
+
x
+
7441
3371
2181
7297
x
x
x
+
3
(e.g. species of Arabidopsis, Bromus, Lactuca, Lycopersicum, Plantago, Spartina), or for seedlings and saplings of tree species
(e.g., Betula, Quercus, and Thuja) (see Table 2 for size range).
These additional data, which come from 57 species not
represented in the Cannell (1982) data sets, are from
laboratory or field studies of plants grown under normal
field or experimental conditions (e.g. elevated CO2, UV-B
radiation, salinity, or soil micronutrient levels). The only
criterion used to select data was that the variance per treatment was small as gauged by the standard errors reported
for standing biomass or biomass production (Enquist &
Niklas, 2002 ; Niklas & Enquist, 2002a, b ; Niklas, 2003).
For these additional species, standing organ biomass and
annual organ biomass production were calculated as for the
Cannell (1982) data sets.
Table 3 provides comparisons among the predicted and
observed scaling exponents for the relationships among
standing leaf, stem, root, and shoot (leaf and stem) biomass,
i.e. ML, MS, MR, and MSH, respectively. The 95 % confidence intervals for the scaling exponent aRMA (the slope of
the log–log linear reduced major axis regression curve)
for each relationship provide an assessment of whether an
observed scaling exponent complies statistically with that
predicted. Provided that the 95% confidence intervals of the
observed exponent include the predicted numerical value of
the scaling exponent, the relevant allometric derivation is
judged to be valid. In turn, the 95 % confidence intervals for
the allometric constant bRMA for each relationship are used
to assess whether there are statistically significant differences
in the Y-intercepts of the log–log linear reduced major axis
regression curves for different portions of the entire data set.
If the 95 % confidence intervals of the allometric constant
for a particular scaling relationship differ among the different portions of the entire data set, then comparisons of the
Karl J. Niklas
880
A 2
1
0
log ML
−1
−2
−3
−4
herbaceous species and
juveniles of woody species
−5
−6
−7
x woody species
−6
−5
−4
−3
−2
−1
0
1
2
3
4
log MS
B
2
1
log ML
0
−1
−2
−3
−4
−5
−6
−7
−6
−5
−4
−3
−2
−1
0
1
2
3
2
3
log MR
log MS
C
D
log MSH
slopes of the regression curves for the different portions of
the entire data set are used to assess the validity of the relevant allometric derivations.
Inspection of Table 3 indicates that, across the entire data
set, the scaling relations predicted for leaf biomass with respect to stem biomass and for shoot biomass with respect to
root biomass agree with those predicted by the allometric
derivations, whereas the scaling relations between leaf
or stem biomass with respect to root biomass are not in
agreement. This discrepancy appears to be the result of a
statistically significant difference in the allometric constants
of the regression curves for the Cannell (1982) data sets
(predominantly tree-sized individuals) and the data from
smaller sized plants (herbaceous species and juveniles of tree
species) (Fig. 2). Regression analyses across the plant size
range represented in the Cannell (1982) data sets (i.e.,
4.16r10x1 kg f MT f 3.18r103 kg) reveals that the allometric derivations and observation are compatible in terms
of the relationship between standing leaf and root biomass,
whereas the relation predicted for standing stem versus root
biomass remains incompatible with the trend in the data.
Likewise, for smaller sized plants (i.e. 5.93r10x6 kg
f MT f 6.41r10x2 kg), the predicted and observed
scaling exponents for the relationships among standing leaf,
stem and root biomass are incompatible. Nevertheless, isometric scaling exponents are observed for the relationship
between shoot and root biomass (Table 3).
The aforementioned discrepancies are reconciled in part
by turning attention to the relations among annual organ
growth in biomass, because standing organ biomass and
annual organ growth in biomass are equivalent for very
small herbaceous plants and for juveniles of tree-sized
species that have yet to accumulate or produce secondary
tissues (wood and periderm). During the first year of growth,
standing stem biomass and annual stem growth are the
same quantities, i.e. MS=GS. Likewise, standing root biomass and annual root growth in biomass are equivalent
quantities, i.e. MR=GR. In successive years, however, secondary tissues (specifically, wood) accumulate in the stems
and roots of woody species (see Franco & Kelly, 1998).
Thus, over time, the biomass of standing stems and roots
increases with age such that MS>GS and MR>GR. This
feature of plant growth is relevant to the preceding comparisons of standing organ biomass, because the scaling
relations for ML, MS, and MR are nearly isometric for the
data drawn from small herbaceous plants and for juveniles
in the size range of 5.93r10x6 kg f MT f 6.41r10x2 kg
(Table 3).
Indeed, the predictions of the allometric derivations for
standing organ biomass emerge as statistically strong when
they are placed in the context of annual leaf, stem, and root
growth, i.e. GL, GS, and GR, respectively (Table 4). Specifically, comparisons of annual organ growth rates indicate
that isometric scaling relations exist across the entire data set
and within each of the two portions of the data set (tree-sized
plants, and herbaceous plants and juveniles of tree-sized
species). The single exception is the relationship between
annual leaf and stem growth for tree-sized plants (i.e.
4.16r10x1 kg f MT f 3.18r103 kg), which may be the result of sampling biases or neglecting the effects of herbivory
4
3
2
1
0
−1
−2
−3
−4
−5
−6
−7
−7
−6
−5
−4
−3
4
3
2
1
0
−1
−2
−3
−4
−5
−6
−7
−7
−6
−5
−4
−3
−2
−1
0
1
−2
−1
0
1
log MR
log MR
2
3
Fig. 2. Log–log bivariate plots for the relationships among
standing leaf, stem, root, and shoot biomass (ML, MS, MR, and
MSH, respectively). Lines denote reduced major axis regression
curves. Scaling exponents and allometric constants for these
relationships provided in Table 3. Data gathered from the primary literature (see text).
(which would reduce standing leaf biomass). Since standing
organ biomass and annual organ growth rates are equivalent for herbaceous and juvenile tree individuals, the absence of the 3/4 power relations predicted for leaf biomass
with respect to stem or root biomass for small herbaceous
species and juveniles of tree-sized species (see Table 3) is
Plant allometry
881
Table 4. Statistical comparisons between predicted and observed relations among annual leaf, stem, and root growth rates (GL, GS,
and GR, respectively). Scaling exponents and allometric constants are based on reduced major axis regression (aRMA¡S.E. and
bRMA¡S.E.) of log10-transformed data (original units for growth rates are kg dry wgt./plt./yr.). In all cases, P<0.0001 or less. ++,
prediction accepted ; +, prediction accepted because scaling exponent reflects differences in bRMA among data sets ; x, prediction
rejected ; CI, confidence intervals
aRMA¡S.E.
Y1 versus Y2
Predicted
Antilog bRMA¡S.E.
Observed
95 % CI
r2
Observed
n
Within the size range of the entire vascular plant data base : 5.93r10x6 kg f MT f 3.18r103 kg
GL versus GS
1.00
0.98¡0.006
0.97–1.01
0.62¡0.012
0.97
709
GL versus GR
1.00
1.05¡0.010
1.03–1.06
1.82¡0.022
0.96
422
GS versus GR
1.00
1.17¡0.012
1.15–1.20
3.09¡0.025
0.96
402
x1
3
Within the size range of the Cannell (1982) vascular plant data base : 4.16r10 kg f MT f 3.18r10 kg
GL versus GS
1.00
0.93¡0.015
0.90–0.96
0.62¡0.013
0.88
431
GL versus GR
1.00
1.00¡0.034
0.92–1.07
2.02¡0.023
0.77
205
Gi versus GR
1.00
1.02¡0.019
1.00–1.07
3.69¡0.013
0.93
205
Within the size range of the new vascular plant data set : 5.93r10x6 kg f MT f 6.41r10x2 kg
GL versus GS
1.00
0.98¡0.018
0.94–1.01
0.71¡0.051
0.91
278
GL versus GR
1.00
1.02¡0.022
0.97–1.06
1.17¡0.065
0.90
217
GS versus GR
1.00
1.08¡0.031
1.00–1.13
0.88¡0.094
0.84
197
Analyses of the Cannell (1982) world-wide data base for seed
plant biomass relations supplemented with data collected
from the primary literature (see Section VII) provides robust
statistical support for the allometric derivations for plant
reproductive effort (Table 5). Across all species, the allometric constants emerging from these derivations (see Section V) are b40=0.027, b41=4.22, and b42=0.018. Inserting
these values into the derivations for reproductive biomass
indicates that MP should scale as the 0.861 power of standing leaf biomass ML and have an allometric constant (Y
intercept ; bRMA) of 0.067 (Fig. 4A). This scaling relationship
is statistically indistinguishable from that observed : aRMA=
0.841 and bRMA=0.064 (Table 5). Likewise, the scaling
relations predicted for MP based on observed values of MS
or MR were statistically indistinguishable from those observed statistically (Table 5 ; Fig. 4 B, C).
In passing, the accuracy of the derivations for reproductive effort is comparable to that of direct regression
analysis of the raw (non-transformed) data. The smallest
difference between predicted and observed MP is observed
when ML is used as the predictive variable for both methods
of estimating reproductive biomass. Likewise, both methods
underestimate MP for some of the largest tree species,
perhaps because published values of MP for these species
were measured by some authors after leaf-fall or herbivory.
Nonetheless, the derivations are strikingly accurate for the
majority of the plants in the database, even within the size
range of small annual species (Fig. 4).
log GL
B
log GL
VIII. ASSESSING REPRODUCTIVE EFFORT
A
C
log GS
entirely consistent with the allometric derivations, especially
in light of the differences in the numerical values of the
allometric constants governing these relations (Table 4;
Fig. 3).
2
1
0
−1
−2
−3
−4
−5
−6
−6
−5
−4
−3
2
1
0
−1
−2
−3
−4
−5
−6
−6
−5
−4
−3
2
1
0
−1
−2
−3
−4
−5
−6
−6
−5
−4
−3
F
Prediction
21 559
10 184
9983
++
+
+
3238
671
2771
x
++
++
2692
1974
1024
++
++
++
herbaceous species and
juveniles of woody species
x woody species
−2
−1
0
1
2
−2
−1
0
1
2
−2
−1
0
1
2
log GS
log GR
log GR
Fig. 3. Log–log bivariate plots for the relationships among
annual growth rates of leaf, stem, and root biomass (GL, GS, and
GR, respectively). Lines denote reduced major axis regression
curves. Scaling exponents and allometric constants for these
relationships provided in Table 4. Data gathered from the primary literature (see text).
Karl J. Niklas
882
Table 5. Representative statistical comparisons between predicted and observed scaling exponents (aRMA) and taxon-specific
(allometric) constants (bRMA) for inter- and intraspecific relations of reproductive, leaf, stem, and root biomass (MP, ML, MS, MR,
respectively) based on reduced major axis regression of log10-transformed data (original units in kg of dry mass plantx1). In all cases,
P < 0.0001. F- and r-values for predicted relations o85 000 and o0.998, respectively. CI, 95 % confidence intervals. See Section
V for definitions of b40–42
aRMA¡S.E.
95 % CI
Antilog bRMA¡S.E.
Across all species (b40=0.027, b41=4.22, b42=0.018)
MP versus ML
Predicted
0.861¡0.002
0.856–0.865
0.067¡0.004
Observed
0.841¡0.025
0.784–0.898
0.064¡0.046
MP versus MS
Predicted
0.657¡0.001
0.655–0.659
0.059¡0.003
Observed
0.674¡0.016
0.637–0.709
0.051¡0.039
MP versus MR
Predicted
0.654¡0.001
0.652–0.656
0.049¡0.003
Observed
0.700¡0.020
0.656–0.745
0.044¡0.046
MP versus ML
Across angiosperms (b40=0.023, b41=7.77, b42=0.015)
Predicted
0.918¡0.003
0.912–0.923
0.101¡0.006
Observed
0.924¡0.035
0.858–0.990
0.115¡0.065
MP versus ML
Across conifers (b40=0.036, b41=9.87, b42=0.056)
Predicted
0.961¡0.001
0.958–0.963
0.161¡0.001
Observed
0.778¡0.072
0.515–1.042
0.167¡0.066
MP versus MS
Within species
Pinus rigida (conifer) (b40=0.142, b41=25.6, b42=0.001)
Predicted
0.909¡0.002
0.905–0.909
1.66¡0.002
Observed
0.909¡0.015
0.876–0.942
1.66¡0.021
Dolicus lablab (dicot) (b40=0.090, b41=10.2, b42=0.002)
Predicted
1.24¡0.002
1.23–1.24
21.9¡0.002
Observed
1.25¡0.081
1.06–1.42
22.1¡0.291
Pennisetum glaucum (monocot) (b40=0.024, b41=5.61, b42=0.017)
Predicted
0.775¡0.002
0.774–0.776
0.235¡0.002
Observed
0.776¡0.017
0.739–0.810
0.232¡0.008
Significant numerical differences in b40 –42 are nevertheless
evident between angiosperms and conifers as well as among
individual species (Niklas & Enquist, 2003). These differences account for most of the ‘data scatter ’ observed in
bivariant plots because, in each case, allometric derivations
accurately predict all observed inter- and intraspecific MP
trends (Table 5). For example, across all angiosperms,
statistical analyses reveal that b40=0.023, b41=7.77,
and b42=0.015 such that MP is expected to scale as the
0.918-power of ML with bRMA=0.101. These values are
statistically indistinguishable from those observed. Similarly
accurate results are obtained when the data from conifer
species are examined (Table 5).
The reproductive trends of phyletically and ecologically
disparate species for which b40–42 values can be determined
are also accurately predicted by the allometric derivations
presented here (Fig. 5). For example, in the case of Pinus
rigida, MP is predicted to scale as the 0.909 power of MS with
bRMA=1.66, whereas aRMA=0.909¡0.015 and bRMA=
1.66¡0.021 are observed (Table 5). Likewise, for the large
annual monocot species Pennisetum glaucum, MP is predicted
to scale as the 0.775 power of MS with bRMA=0.235,
whereas aRMA=0.776¡0.017 and bRMA=0.232¡0.008
are observed.
n
r2
F
0.068–0.069
0.057–0.072
279
279
—
0.754
—
851.1
0.048–0.049
0.047–0.055
418
418
—
0.754
—
1331
0.048–0.050
0.040–0.048
204
204
—
0.827
—
967.0
0.098–0.104
0.092–0.143
195
195
—
0.799
—
768.2
0.159–0.160
0.167–0.259
84
84
—
0.296
—
34.5
1.58–21.60
1.49–1.84
16
16
—
0.822
—
193.7
44
44
—
0.822
—
193.7
50
50
—
0.976
—
1920
95 % CI
21.7–22.1
5.71–85.5
0.233–0.237
0.227–0.237
These analyses suggest that tradeoffs are involved when a
finite amount of total body mass is partitioned between or
among different organ types (Enquist & Niklas, 2002 ; Niklas
& Enquist, 2003). Clearly, the expenditures required to
construct new leaf, stem, or root tissues annually (as gauged
by their annual growth in biomass or by the difference in
standing biomass between successive years) are removed
from a pool of resources available for the construction
of reproductive organs. That these tradeoffs have been reconciled differently by different species is evident from the
numerical differences observed among the allometric constants b40–42 for different plant lineages (angiosperms versus
conifers) or different individual species within each lineage
of seed plant. These differences indicate that taxa with dissimilar biomass partitioning patterns with respect to their
vegetative organs will have different allometric trends in
their reproductive effort, whereas those with the same or
very similar vegetative partitioning patterns will share similar b40 –42 values and thus share similar MP scaling relations.
An important caveat is that reproductive effort can and
has been measured by different authors using very different
‘ currencies, ’ e.g. seed number or biomass, and flower or fruit
biomass. Likewise, it is not immediately obvious whether
some portions of the plant body should be included when
Plant allometry
log MP
B
2
1
0
−1
−2
−3
−4
−5
−6
−7
−6
A
1
x
−5
−4
−3
−2
log ML
−1
0
1
log MP
2
1
0
−1
−2
−3
−4
−5
−6
−7
−5
−1
−2
−3
−4
angiosperms
conifers
−5
−6
−6
2
B
−5
−4
−3
−2
log M L
−1
0
1
2
0
−1
−2
−5
−4
−3
−2
−1
0
1
2
3
4
P. glaucum
P. rigida
−3
−4
log MS
C
2
0
log MP
2
1
0
−1
−2
−3
−4
−5
−6
−7
−6
log MP
log MP
A
883
−5
−5
D. lablab
−4
−3
−2
−1
0
1
log M S
−4
−3
−2
−1
0
1
2
3
log MR
Fig. 4. Log–log bivariate plots for the relationships among reproductive biomass and standing leaf, stem, and root biomass
(MP, ML, MS, and MR, respectively) for angiosperm and conifer
species. Solid lines denote reduced major axis regression curves
of data ; dashed lines denote predicted relationships based on
allometric derivations for predicting MP. Scaling exponents and
allometric constants for predicted and observed relationships
are provided in Table 5. Data gathered from the primary
literature (see text).
estimating reproductive effort, e.g. should the scale-bract
complexes of conifer megasporangiate cones be assigned a
‘ vegetative ’ or ‘ reproductive ’ organ status; are the pedicles
of flowers ‘vegetative ’ or ‘ reproductive ’ organs? The data
used here to measure reproductive effort undoubtedly reflect a broad range of ‘currencies ’ depending on the protocols used by different authors (see Körner, 1994, for a
discussion regarding vegetative biomass fractionation protocols). An additional concern is that none of the data
examined here incorporate the biomass of pollen grains,
which are clearly metabolically costly (Niklas, 1994 b).
Therefore, although the derivations pertaining to reproductive effort appear to be remarkably robust in terms of
predicting observed trends at the level of individual plants,
Fig. 5. Log–log bivariate plots for the relationships among reproductive biomass and standing leaf and stem biomass (MP,
ML, and MS, respectively) for angiosperm species (A) and three
individual species (Dolicus lablab, a legume vine ; Pinus rigida, a
conifer tree ; and Pennisetum glaucum, a herbaceous monocot) (B).
Solid lines denote reduced major axis regression curves of data ;
dashed lines denote predicted relationships based on allometric
derivations for predicting MP. Scaling exponents and allometric constants for predicted and observed relationships are
provided in Table 5. Data gathered from the primary literature
(see text).
the concordance between predicted and observed scaling
exponents must be viewed cautiously and with some degree
of skepticism.
IX. ASSESSING PLANT COMMUNITY
RELATIONS
Statistical analysis of the broad database discussed in
Section V provides strong support for the scaling relationships predicted for the biomass partitioning and annual
growth rates of plants at the individual level and for the
relationships between these parameters and the number of
plants per unit area sampled (‘ plant density ’) (Table 6). As
predicted, total plant biomass MT scales as the x1.27 power
of plant density N (Fig. 6 A) ; the 95 % confidence intervals
including x4/3 and excluding x3/2 (Table 6). Also meeting predictions, annual plant growth rate GT and standing
plant leaf biomass ML scale isometrically with respect to N
(a=x0.98 and x1.00, respectively), whereas basal stem
diameter DS scales as the x0.53 power of N (Fig. 6B–D).
Karl J. Niklas
884
Table 6. Predicted and observed scaling exponents aRMA and allometric constants bRMA for log10-transformed data from the
Cannell (1982) compendium. In each case, P < 0.001 or less. ML, MR, MSH, standing leaf, root, and shoot biomass ; MT, total plant
biomass ; GT, annual total body growth in biomass ; DS, basal stem diameter ; N, plant density
Observed
Log Y versus log X
Predicted
aRMA¡S.E.
Across all communities
Log MT versus log N
x4/3
x1.27¡0.03
Log GT versus log N
x1.0
x0.98¡0.03
Log ML versus log N
x1.0
x1.00¡0.03
Log DS versus log N
x1/2
x0.53¡0.01
Log MSH versus log N
x4/3
x1.31¡0.03
Log MR versus log N
x4/3
x1.17¡0.03
Across angiosperm-dominated communities
Log MT versus log N
x4/3
x1.30¡0.05
Log GT versus log N
x1.0
x1.00¡0.05
Log ML versus log N
x1.0
x1.00¡0.03
x1/2
x0.51¡0.02
Log DS versus log N
Log MSH versus log N
x4/3
x1.25¡0.04
Log MR versus log N
x4/3
x1.08¡0.05
Across conifers-dominated communities
Log MT versus log N
x4/3
x1.34¡0.04
Log GT versus log N
x1.0
x1.04¡0.03
Log ML versus log N
x1.0
x1.12¡0.04
x1/2
x1.28¡0.05
Log DS versus log N
Log MSH versus log N
x4/3
x1.31¡0.03
Log MR versus log N
x4/3
x1.39¡0.04
95 % CI
log10 bRMA¡S.E.
r2
n
F
x1.33 to x1.16
x1.03 to x0.92
x1.05 to x0.95
x0.56 to x0.51
x1.36 to x1.25
x1.23 to x1.10
5.96¡0.11
3.99¡0.09
3.78¡0.09
0.86¡0.04
5.99¡0.10
4.97¡0.11
0.801
0.861
0.669
0.673
0.753
0.773
342
205
670
792
668
347
1368
1257
1348
1624
2029
1172
x1.36 to x1.23
x1.11 to x0.90
x1.05 to x0.95
x0.55 to x0.48
x1.33 to x1.17
x1.18 to x0.98
5.74¡0.16
3.76¡0.15
3.37¡0.10
0.75¡0.06
5.81¡0.14
4.72¡0.16
0.759
0.833
0.751
0.685
0.733
0.729
174
74
331
342
325
178
542.2
359.9
993.1
740.1
888.3
465.9
x1.42 to x1.25
x1.11 to x0.98
x1.19 to x1.05
x1.37 to x1.19
x1.36 to x1.25
x1.46 to x1.31
5.96¡0.11
3.99¡0.09
4.41¡0.10
5.33¡0.16
5.99¡0.10
6.26¡0.13
0.846
0.880
0.746
0.827
0.753
0.782
168
131
339
169
668
343
909.3
949.8
992.1
800.8
2029
1220
The predicted scaling relationship for shoot biomass MSH
versus N at the level of an individual plant is also supported.
Specifically, MSH scales as the x1.31 power of N with 95%
confidence intervals that include x4/3 and exclude x3/2
(i.e., x1.36 to x1.25) (Table 6). However, MR scales as the
x1.17 power of N and in this case the 95% confidence
preclude the predicted x4/3 value (Table 6). This discrepancy can be attributed to a systematic, size-dependent
underestimation of fine and small root biomass, i.e. progressively larger plants have disproportionately more fine
and small root biomass, which are systematically more difficult to excavate with increasing plant size (see Niklas et al.,
2002). This bias is expected to elevate the numerical value
of the scaling exponent for MR versus N. This conjecture is
supported by the exponent for conifer MR versus N. Conifer
MR, which tends to be more shallowly buried and thus more
easily excavated than angiosperm MR, scales as the x1.28
power of N with 95% CI that include the predicted value of
x4/3 (Table 6).
The derivations presented for plant density relationships
are also supported based on empirically observed total
community mass-growth-density relationships (Table 7). For
example, individual MT and GT are predicted to scale as the
x4/3 and the x1 power of N, respectively, but total community biomass M̂T is predicted to scale as the as the x1/3
power of N, whereas total community growth ĜT is predicted to be independent of N. Regression of M̂T versus N and
ĜT versus N gives scaling exponents of x0.266¡0.03 and
0.007¡0.03, which are statistically compatible with the
predictions of the model (Table 7). Total community leaf
biomass M̂L is also predicted to be independent of N, and the
best fit regression curve for the observed data of M̂L versus N
gives a scaling exponent statistically indistinguishable from
zero. Similar comparisons show that the predicted and
observed scaling exponents for other total community
biomass-plant density relationships are statistically indistinguishable, although it is evident that the coefficients of
correlation are low (Table 7).
X. IS THERE A SINGLE UNIFYING THEORY ?
This article outlines a set of previously published allometric
derivations interrelating individual plant growth, vegetative
and reproductive organ biomass partitioning, and a variety
of density-correlated community properties. All of these
derivations are predicated on two statistically robust observed trends, each of which is shown to span many orders
of magnitude and a broad spectrum of taxonomically
and ecologically divergent species. These trends are the 3/4
scaling relationship between total annual plant growth in
biomass and total body biomass and the one-to-one (isometric) relationship between the capacity to harvest sunlight
and body biomass (see Niklas & Enquist, 2001).
The predictions emerging from these derivations are
shown here and elsewhere to receive strong statistical
support, as judged by comparing the numerical values of
predicted scaling exponents and those observed for an
extensive database accumulated from the published literature. However, this concordance must be viewed with a
Plant allometry
885
log MT
A 4
log GT
B
3
2
1
0
−1
−2
−3
3
2
1
0
−1
−2
−3
−4
angiosperms
conifers
2
3
2
3
4
log N
4
5
6
7
5
6
7
5
6
7
5
6
7
log N
C 3
log ML
2
1
0
−1
−2
−3
log DS
D
2
3
4
2
3
4
log N
0
−1
−2
log N
Fig. 6. Log–log bivariate plots for the relationships among
total body mass MT, total annual growth in biomass GT,
standing leaf biomass ML, basal stem diameter DS, and plant
density N (number of plants per unit sampled area) for mixedspecies and monospecific angiosperm and conifer-dominated
communities. Lines denote reduced major axis regression
curves of data. Scaling exponents and allometric constants for
predicted and observed relationships are provided in Table 6
(see Table 7 for entire community relationships). Data taken
from Cannell (1982) (see text).
critical eye. Indeed, the study of allometry is intrinsically
limited (if not outright defined) by the availability and the
statistical structure of data just as it is rife with methodological and philosophical pitfalls. Space does not permit a
systematic in-depth treatment of each of the problems facing
the allometricist. However, some problems require special
attention.
(1 ) Size-dependent biases and ad hoc hypotheses
Size-dependent biases are manifold within and across biological data sets. That these biases can have a profound
effect on suppositions regarding allometric trends in terms of
statistical techniques is obvious. What is less obvious is the
temptation to erect ad hoc hypotheses in light of a theoretical
framework to account for biases that may not actually exist.
For example, the scaling exponent for the relationship
between standing leaf and root biomass is predicted to be
3/4, whereas the lower 95% confidence interval for the
observed scaling exponent exceeds 0.75 (see Table 3, Fig. 2).
This discrepancy between observation and theory is explicable provided that standing root biomass is systematically
underestimated with increasing body size. Larger root systems are arguably increasingly more difficult to excavate,
whereas the biomass of small and fine roots is reported to
increase disproportionately as total root biomass increases
(see Makkonen & Helmissar, 2001). Likewise, feeder roots,
which increase in number as a function of overall root system size, typically decompose annually (e.g. Niklas et al.,
2002). Importantly, systematic size-dependent biases such as
these will progressively ‘deflate’ reported values of standing
root biomass for larger plants and thus artificially ‘inflate ’
the numerical values of the scaling exponents of bivariate
regression curves whenever root biomass is plotted along
the abscissa.
However, it cannot escape attention that an alternative
explanation is that the observed relationship between leaf
and root biomass is in fact accurate. Indeed, no concrete
evidence has been provided demonstrating that standing
root biomass is systematically underestimated. Regardless
of how ‘reasonable ’ an ad hoc hypothesis may appear, its
plausibility does not equate with proof of existence. Unfortunately, some workers are so beguiled by a particular
theory that they neglect the possibility that it may be wrong.
A related concern is the ease with which mathematical
post hoc hypotheses can be erected and the difficulties of
testing them empirically. In this context, consider the many
biophysically plausible assumptions that were required to
simplify the otherwise intractable mathematical expressions
for certain morphometric relationships. These assumptions
give the appearance of being robust theoretically but they
are nevertheless untested.
For example, when treating the scaling relationships
among standing leaf, stem, and root biomass, it was assumed
that average stem diameter (and thus cross-sectional area)
scales isometrically with respect to average root diameter
(and cross section). The biophysical basis for this assumption
is that the volume of water flowing per unit time from roots
to stems must be conserved. Although the amount of water
flowing from roots and through stems must be equivalent
at the level of an individual plant, differences in the
volume fractions and average cross-sectional areas of waterconducting cells in roots and stems undeniably exist across
species. Thus, for some species, average root and stem diameters may not be equivalent for all species at the level
of representative individuals (see Zimmermann, 1983;
Gartner, 1995) especially for succulents and arid-adapted
species (Holbrook, 1995 ; Niklas et al., 2002).
Likewise, all of the derivations presented here assume that
foliar leaves are the principal or sole light-harvesting organs,
whereas it is clear that many species rely on photosynthetic
stem tissues for capturing radiant energy (see Niklas, 2002;
Pfanz et al., 2002). Taxon-specific differences in body plan
Karl J. Niklas
886
Table 7. Predicted scaling exponents at the levels of the individual plant (see Table 6) and entire communities and those observed
for total community mass–growth–density relationships
Predicted
Observed
Individual
Community
a¡S.E.
95 % CI
r2
n
F
MT ! Nx4/3
GT ! Nx1.0
ML ! Nx1.0
DS ! Nx1/2
MSH ! Nx4/3
MR ! Nx4/3
M̂T ! N x1/3
ĜT ! N 0
M̂L ! N 0
D̂S ! N 1/2
M̂SH ! N x1/3
M̂R ! N x1/3
x0.266¡0.03
0.007¡0.03
0.000¡0.03
0.467¡0.01
x0.310¡0.03
x0.165¡0.03
x0.33 to x0.16
x1.03 to 0.92
x0.05 to 0.05
0.42 to 0.51
x0.36 to x0.25
x0.23 to x0.10
0.135
0.000
0.000
0.610
0.143
0.034
342
495
675
792
668
347
53.28***
0.002
0.000
1237***
111.8***
23.44**
*** Significant at the 0.001 level.
** Significant at the 0.01 level.
ML, MR, MSH, standing leaf, root, and shoot biomass ; MT, total plant biomass ; GT, annual total body growth in biomass ; DS, basal stem
diameter ; N, plant density.
architecture or physiological features thus clearly exist and
may account for numerical differences observed in allometric constants.
(2 ) Outliers : do exceptions prove a rule ?
Here we turn to the question of statistical ‘outliers ’ in allometric trends. Many exist that are not immediately accounted for by the obvious differences in phylogenetic
affinity that translate into body plan differences. This raises
a number of related issues.
One is the relative frequency of outliers, is it high or low
with respect to species that appear to comply with a posited
interspecific trend ? The analyses presented here indicate
that comparatively few outliers exist for the majority of
allometric trends explored. Nevertheless, if broad interspecific allometric trends are indicative of deeply rooted and
fundamental biological phenomena, then data that deviate
substantially from otherwise well-confined trends point to
circumstances where general ‘ rules ’ are obviated in one
manner or another.
Clearly, outliers may be the result of unusual environmental circumstances or unusual taxon-specific features.
In this regard, optimization theory indicates that organisms
are likely to reflect form-function compromises within the
context of their particular environmental circumstances
(see Niklas, 1994 a). Every organism must perform manifold
tasks to grow, survive, and reproduce. Many of these tasks
have contradictory ‘best solutions’ and it is the reconciliation of these tasks that often leads to evolutionary diversification and the biodiversity we see today. For example,
vertically oriented stems minimize the mechanical bending
moments induced by self-loading, but they also minimize
the amount of sunlight that can be captured daily. In a
desert environment, columnar unbranched stems may be
adaptive, whereas in other habitats the reverse may be true.
The trilogy of form-function-environment is often neglected
as is the fact that seemingly very different morphologies
may be equally effective at performing all their functional
obligations simultaneously in the same habitat. In this sense,
allometric outliers are extremely instructive, even if they
give the theorist momentary grief. Under any circumstances, if a ‘global ’ theory for size-dependent trends is
advocated, understanding why ‘ outliers ’ exist (and what
they tell us about biology) can only foster a deeper understanding of biodiversity in light of theory.
( 3) Proportionality ‘ constants ’ are not constant
Yet another important concern is that allometric theory
and practice are currently incapable of predicting a priori
the numerical values of the allometric constants governing
size-dependent trends. That these ‘ constants’ can numerically differ significantly for the same size-correlated relationship depending on the taxonomic group studied is selfevident from the analyses discussed here. The historical
emphasis on scaling exponents in allometric theory and
practice is understandable, since these exponents shed light
on the proportional changes among correlated variables such
as leaf and stem biomass or annual growth rates. But, from a
biological perspective, the manner in which the absolute
quantities of these variables change as a function of body
size is equally important.
Consider the hypothetical case of two plant species A and
B for which leaf biomass ML scales as the 3/4 power of stem
biomass MS but for which the allometric constants are
b=1.5 and b=10, i.e. ML=1.5MS3/4 and ML=10MS3/4,
respectively. Both of these hypothetical species evince the
same proportional relationship, since a=3/4. However, for
any value of MS, species B will have an order of magnitude
more standing leaf biomass than species A.
Indeed, differences in the values of allometric ‘constants ’
often account for much of the ‘data scatter ’ observed for
broad interspecific allometric trends, because many of these
trends are ‘composites ’ of numerous parallel intraspecific
allometries sharing the same proportional relationships differing in their Y intercepts. Such differences in the numerical
values of allometric constants probably reflect fundamental
phylogenetic differences in how the representatives of
different lineages are architecturally constructed. This
aspect of allometric study awaits detailed empirical exploration and conceptual explanation.
Plant allometry
(4 ) Scaling exponents : fractions or numbers ?
A serious concern is the inclination of allometricists to convert the numerical values of scaling exponents into fractions
as they explore a favoured theoretical framework.
Consider a hypothetical database that obtains a log–log
regression curve with a slope of 0.79 and 95 % confidence
intervals of 0.74 and 0.84. Is this slope indicative of a 3/4
scaling relationship, or a 4/5 or 5/6 relationship ? A strictly
empirical approach to allometry can rely on statistical
inference employing the numerical value of 0.79 within the
boundary conditions of 0.74 and 0.84. However, the temptation to seek a mechanistic explanation may lead a particular worker to claim 0.79 as evidence for a 3/4, 4/5, or 5/
6 ‘ rule ’ depending on the theoretical bias.
Which, if any among these contending ‘ rules ’ is real must
remain problematic unless the numerical values (and posited
fractions) of the slopes of many different theoretically interrelated allometric trends can be used to eliminate some
alternatives (e.g. Niklas, 2003). Yet, even this approach is
not immune to the foibles of ‘ fractions versus numbers.’
For example, suppose that a certain allometric theory
predicts that variable Y1 scales as the a1 power of variable Y2
and that variable Y2 scales as a2 power of variable Y3, i.e.
Y1 ! Y2 a1 and Y2 !Y3a2 . It then follows that this theory
makes three predictions that can be subjected collectively
(rather than individually) to empirical enquiry, i.e. Y1 ! Y2 a1 ,
Y2 ! Y3 a2 , and Y1 ! Y3a1 a2 . But let us cast this hypothetical
case in terms of regression analyses involving numbers
rather than fractions. Suppose our theory predicts that
a1=3/4 and a2=2 such that a1a2 is predicted to equal 3/2.
Suppose in turn that regression of data for Y1 versus Y3a1 a2
gives a slope equal to 1.385 with lower and upper 95%
confidence intervals equal to 1.22 and 1.55. Does this ‘ test ’
confirm our theory, or is it possible that an alternative theory positing a1=2/3 and a2=2 such that a1a2 is predicted
to equal 4/3 is correct ? Clearly, 4/3=1.333 falls well within
the empirically determined 95 % confidence intervals of the
data for Y1 versus Y3 a1 a2 , whereas both theories predict that
a2=2. Thus, the only remaining test to determine which of
the two theories is correct is whether the slope of Y1 versus
Y2 a1 is numerically more in accord with a1=3/4 or 2/3.
Accordingly, the multiple predictions of the two theories are
reduced to one in terms of statistical testing.
(5 ) Different ‘ first principles’ can yield the
same predictions
Currently, the most comprehensive allometric theory in
biology is that of West, Brown & Enquist (1997, 1999, 2000 ;
see also Brown, West & Enquist, 2000; Enquist, 2002). The
WBE theory attempts to explain an extraordinarily broad
spectrum of scaling relationships, ranging from physiological processes to community-level and macroecological
phenomena. Importantly, it rests on three assumptions relating to the exchange, distribution and delivery networks
for energy and mass transfer in biological systems. These
networks are argued to (1) branch three-dimensionally
throughout biological structures (within cells or entire organisms), (2) have terminal units that do not vary in size
887
within organisms (e.g. capillaries and peripheral xylem
elements), and (3) minimize the total resistance (and thus
the energy and time) involved in the distribution and delivery of resources.
In simplistic terms, the WBE theory states that organisms,
whether unicellular or multicellular, have internal networks
with fractal-like architectures that have evolved through
natural selection to minimize the energy and time required
to absorb, distribute, and deliver resources internally. These
assertions mathematically obtain the prediction that biological surface areas scale as the 3/4 power of body volumes
and that many other size-correlated biological phenomena
obey 1/4 (or its multiples) power ‘ rules. ’ In this way, the
historically classic 3/4 scaling relationship observed by
Kleiber and many others receives a formal (mathematical
and conceptual) explanation.
Support for the WBE theory, however, is currently circumstantial. True, many allometric trends seem to have
scaling exponents of 1/4 or multiples thereof. However,
from a purely logical perspective, the verification of any
theory requires the validation of its basic assumptions and
not its predictions if for no other reason than that alternative
theories may obtain the same predictions starting with different ‘first principle ’ assumptions.
The allometric derivations presented here are a case in
point. Each set of derivations is based in part or whole
on the empirical observation that annual plant growth rates
appear to scale as the 3/4 power of body mass. From this
basic scaling ‘ rule, ’ a variety of other allometric relationships
emerge, each of which is governed by quarter-power scaling
exponents. It is nevertheless illogical to interpret this as evidence supporting the WBE theory because any set of derivations that employs the 3/4 scaling relationship between
growth and body mass (for whatever reason) may predict
many of the same scaling relations as those presented here.
The argument that no ‘ first principles ’ theory currently
exists other than the WBE theory is both logically irrelevant
and untrue (see Weibel, 2000; Darveau et al., 2002).
Because its basic assumptions have yet to be subjected to
fastidious and extensive tests and because ad hoc explanations
can always be advanced to explain away discrepancies
between observed and predicted scaling exponents, it is
premature to pass judgment on the WBE theory. Certainly,
it has reinvigorated the study of biological allometry in
general and the study of plant allometry in particular. What
can be said with more certainty is that there are size-correlated trends that span many orders of magnitude in body
size across virtually every plant lineage and habitat. That
these trends exist at all suggests that they reflect very fundamental properties of living things that demand a mechanistic explanation.
XI. CONCLUSIONS
(1) Two allometric and empirical trends receive strong
statistical support based on broad interspecific comparisons :
annual growth in biomass at the level of an individual
plant GT scales as the 3/4 power of body mass MT and
Karl J. Niklas
888
isometrically with respect to the ability of an individual
plant (unicellular or multicellular) to harvest light H, i.e.
GT ! M3T/4 ! H.
(2) With the aid of only a few assumptions concerning
stem and root hydraulics (e.g. standing leaf biomass ML
scales as the square of basal stem diameter DS), these two
trends provide a basis to analytically derive a spectrum of
hypotheses concerning size-correlated phenomena, each
of which is seen to receive strong statistical support, e.g.,
standing leaf biomass scales as the 3/4 power of standing
stem (or root) biomass ; annual growth in leaf biomass scales
isometrically with respect to annual growth in stem (or root)
biomass ; annual reproductive biomass scales as a speciesspecific complex function of how annual growth is partitioned among leaf, stem, and root biomass ; total body
biomass scales as the x4/3 power of plant density.
(3) Although the general allometric theory emerging
from these derivations is far-reaching and statistically
robust, the theoretical basis for the allometric trends
GT ! MT3/4 ! H remains problematic because any theory
that starts with the assumption that body surface area scales
as the 3/4 power of body volume (mass) will obtain the same
predictions regarding such features as standing leaf, stem,
and root biomass allocation patterns.
(4) Additionally, a number of statistical and mathematical issues regarding allometric analyses (e.g., the choice of
regression analysis, and converting the numerical values
of scaling exponents into fractions) must be carefully explored before rival allometric theories can be resolved using
large data sets.
XII. LIST OF SYMBOLS
a, scaling exponent (slope of log–log linear regression curve).
aRMA, scaling exponent (slope of log–log linear reduced
major axis regression curve).
b, allometric constant (Y-intercept of log–log linear
regression curve).
bRMA, allometric constant (Y-intercept of log–log linear
reduced major axis regression curve).
ai, unit area occupied by an individual in a community.
AT, total area occupied by all individuals in a community.
DS, DR, average diameter of stem and root, respectively.
GL, GS, GR, GP, annual growth in leaf, stem, root, and
reproductive biomass plantx1, respectively.
GT, annual growth in biomass plantx1.
k, a numerical constant.
LS, LR, average length of stem and root, respectively.
MT, total biomass plantx1.
ML, MS, MR, MSH, standing leaf, stem, root, and shoot
(leaf and stem) biomass plantx1, respectively.
MP, standing reproductive biomass plantx1.
M̂L, M̂S, M̂R, M̂SH, standing leaf, stem, root, and shoot
(leaf and stem) biomass communityx1, respectively.
N, plant density (number of plants occupying AT,).
r, correlation coefficient.
rS, rR, bulk tissue density of stem and root, respectively.
t, time.
X, Y, independent and dependent variables, respectively.
y1, y2, numerical values of interdependent variables Y1, Y2.
Y1, Y2, interdependent variables.
XIII. ACKNOWLEDGEMENTS
Many of the concepts and analyses presented here have been
published together with Brian Enquist, Jeremy Midgley, and Richard Rand, whom I gratefully acknowledge. I am also grateful to
two anonymous reviewers who provided valuable comments on
how to improve the organization of this article. This research has
been supported by funds provided by the College of Agricultural
and Life Sciences, Cornell University.
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