Invariant scaling relationships for interspecific plant
biomass production rates and body size
Karl J. Niklas*
and Brian J. Enquist§
*Department of Plant Biology, Cornell University, Ithaca, NY 18453; National Center for Ecological Analysis and Synthesis, University of California,
Santa Barbara, CA 93110; §The Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501; and Department of Ecology and
Evolutionary Biology, University of Arizona, Tucson, AZ 85721
Communicated by Simon A. Levin, Princeton University, Princeton, NJ, December 13, 2000 (received for review October 31, 2000)
The allometric relationships for plant annualized biomass production
(``growth'') rates, different measures of body size (dry weight
and length), and photosynthetic biomass (or pigment concentration)
per plant (or cell) are reported for multicellular and unicellular
plants representing three algal phyla; aquatic ferns; aquatic and
terrestrial herbaceous dicots; and arborescent monocots, dicots,
and conifers. Annualized rates of growth G scale as the 3 4-power
of body mass M over 20 orders of magnitude of M (i.e., G M3/4);
plant body length L (i.e., cell length or plant height) scales, on
average, as the 1 4-power of M over 22 orders of magnitude of M
(i.e., L M1/4); and photosynthetic biomass Mp scales as the
3 4-power of nonphotosynthetic biomass Mn (i.e., Mp Mn
3/4).
Because these scaling relationships are indifferent to phylogenetic
affiliation and habitat, they have far-reaching ecological and
evolutionary implications (e.g., net primary productivity is predicted
to be largely insensitive to community species composition
or geological age).
It has long been noted that metabolic rates B scale nonlinearly
with respect to adult body mass M (1) and that this allometric
relationship is best described by a 3 4 scaling exponent (i.e., B
M3/4
) (2, 3). Broad interspecific comparisons for mature
unicellular and multicellular animals by numerous researchers
have experimentally verified this exponent (4­11). Many other
physiological, anatomical, and life-history characteristics also
scale as ``quarter-power'' exponents of animal body mass or
length, e.g., annualized rates of biomass production G scale to
the 3 4-power (9­11). Nevertheless, the origin of the 3 4 and
other quarter-power exponents has long been the subject of
considerable theoretical speculation (7­17).
Allometric relationships have played a central role in the
development of zoology. Unfortunately, with few exceptions,
botanical studies have not traditionally considered the role of
body size (11­13), and thus the extent to which the scaling
relationships observed for animals hold true for photoautotrophs
is unclear. Recently, however, there has been both
empirical and theoretical justification to expect the scaling
relationships observed for animals to apply equally well for
plants (12­18). Theoretical models based on the evolution of
fractal-like networks predict that many attributes of organic
structure and function should scale with quarter-powers of body
mass (14, 17, 18). These models specifically predict numerous
allometric scaling relationships, including that B M3/4
, that L
M1/4
, and that the total effective tissue exchange area (e.g., leaf
or root surface area) with the environment will scale as M3/4
.
Because rates of biomass production are intimately dictated by
metabolic rates B, annualized rates of gross metabolic or biomass
production G are expected to be directly proportional to B (G
B) such that G M3/4
(2­11, 17, 18), although this may not hold
true over ontogenetic time scales.
This theoretical expectation has been verified to a limited
extent based on interspecific comparisons among unicellular
algal species that show rates of biomass production scale as the
3 4-power of cell mass over eight orders of magnitude of body
size (19, 20). Likewise, rates of biomass production in tropical
trees appear to scale as M3/4
(13). However, 11 orders of
magnitude in body size separate the largest known unicellular
algae and the largest multicellular terrestrial plants. It is thus
unclear whether a single or multiple scaling relationships best
describe the relationship between G and M across all or most
major plant clades and grades. Likewise, the relationship between
plant body length and mass is not well established,
although a scaling exponent statistically similar to 1 4 has been
reported for a limited number of unicellular and multicellular
plant species (21).
Here, we report the relationship between annualized biomass
production rates and body size for representative species of
unicellular algae, aquatic ferns, and a broad spectrum of dicot,
monocot, and conifer species. Our analyses of these data, which
span 20 orders of magnitude of body mass, unequivocally
demonstrate that plant allometric relationships (among rates of
biomass production, photosynthetic mass, and body length and
mass) conform both to those previously described for animals
and with the predictions made by recent allometric and biomechanical
theoretical models (12­18).
Materials and Methods
Unicellular Algae. All of the data used in our analyses were
collected from the primary literature (refs. 21­31, compiled in
ref. 12). Values for cell (body) mass M (pg of carbon per cell),
length L ( m), biomass production (``growth'') rates G (pg of
carbon per cell per h), and total cell pigment content (pg of
carbon) C reported in the primary literature were converted to
their corresponding units reported for metaphytes (i.e., kg of
carbon per cell, m, kg of carbon per cell per year, and kg of
carbon per cell, respectively; see below). In the majority of cases,
the values published for G were based on monocultures grown
under optimal or near optimal conditions (12, 19). Three phyla
are represented by these data.
Aquatic Metaphytes. Two fern and one dicot species are represented
in this data set (Azolla caroliniana, Azollaceae; Salvina
natans, Salviniaceae; and Lemna minor, Lemnaceae; respectively).
Stock cultures of these species (from Carolina Biological
Supply) were grown in shallow glass dishes (containing sterilized
pond water) placed in a growth chamber at a temperature of 25°
F (with a day night photoperiod of 15 9 h and an average light
intensity of 354 mol photons m2
s measured at the water-air
interface). Growth in M and L were monitored every 48 h by
culling 80% of each population and counting the number of
culled plants, which were then air-dried and weighed. Body mass
M was computed from the quotient of total kg of dry matter and
Abbreviations: G, annualized biomass production (``growth'') rate; M, body mass; L, body
length; C, total (algal) cell pigment content; Mp, photosynthetic (foliage) biomass; Mn,
nonphotosynthetic (stem and root) biomass; RMA, scaling exponent (slope) of reduced
major axis regression curve.
See commentary on page 2113.
To whom reprint requests should be addressed. E-mail: kjn2@cornell.edu.
The publication costs of this article were defrayed in part by page charge payment. This
article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C.
§1734 solely to indicate this fact.
2922­2927 PNAS February 27, 2001 vol. 98 no. 5 www.pnas.org cgi doi 10.1073 pnas.041590298
culled plant number; annualized G was computed based on total
kg of dry matter per day and the number of culled plants.
Terrestrial Metaphytes. These species include herbaceous and
tree-size monocots (grass and palm species), and herbaceous and
arborescent dicots and conifers. With the exception of the data
reported for Pearl Millet (Pennisetum glaucum, Poaceae) provided
by Steven Mason (Department of Agronomy, University of
Nebraska), the data are primarily those compiled by Cannell (32)
based on the primary literature published up to mid-1981 for
1,200 forest stands from 46 countries worldwide. In most cases,
these data are for even-age monospecific stands grown under
horticulturally controlled field conditions (n 600 of 880 data
sets for which authors reported the data required to compute G
and M) such that the variance in tree G and M is likely low and
thus roughly comparable to that of the unicellular algae and
aquatic metaphyte data (see above).
The Cannell compendium provides the number of plants per
hectare (``plant density''), average plant height, total basal stem
cross-sectional area, and biomass and net production (in units of
metric tons of dry matter subsequently converted into units of kg
of dry matter) of stem wood, bark, branches, fruits, foliage, and
roots whenever reported by an author. Body mass M (kg of dry
matter per plant) was computed from the quotient of total
standing biomass and plant density per 1.0 ha of forest sample.
Annualized biomass production rates (kg of dry matter per plant
per year, accounting for annual losses of dry matter attributable
to mortality, litter-fall, decay, or consumption) were computed
as the quotient of total net production (of stem wood, bark,
foliage, roots, and reproductive organs) and plant density.
Values of foliage (photosynthetic) Mp and stem and root (nonphotosynthetic)
Mn biomass (kg of dry matter per plant) were
computed from the quotients of total biomass allocated to each
body part and plant density. Body length L is plant height, and
thus neglects root length (which was not reported in the Cannell
data sets). Values for L were unavailable for Pearl Millet.
Additional data for metaphyte M and L (taken from ref. 21) were
pooled with the Cannell data in our analyses. These data include
those from the General Sherman tree (Sequoia) and the Giant
Pacific Kelp (Macrocystis).
Statistical Analyses. All computations and analyses used the
software package JMP Version 3 (SAS Institute, Cary, NC).
Model Type II (reduced major axis, denoted as RMA) regression
analysis was used to compute scaling exponents (slopes of curves
designed as RMA) because the error variance resulting from
measurement error and real biological variation was equivalent
among all variables (12). The 95% confidence intervals of RMA
were used to assess whether an empirically determined scaling
exponent complied with that predicted by theory. The 95%
confidence intervals of Y-intercepts were also used to determine
whether data sets sharing the same RMA could be pooled for
additional regression analyses (12).
Results
A 3 4 scaling exponent was statistically verified for the relationship
between annualized rates of biomass production G and body
mass M for the species representing individual higher taxa or
evolutionary grades (e.g., dicots and unicellular algae, respectively;
Fig. 1A). Specifically, among all metaphyte species, RMA
0.749 0.007 and Y-intercept 0.214 0.016 (n 334, r2
0.972, F 11,695, P 0.0001); across tree species, RMA
0.791 0.03 and Y-intercept 0.301 0.16 (n 178, r2
0.705,
F 421, P 0.0001). Similarly, unicellular algae G scaled as the
3 4-power of body mass ( RMA 0.749 0.008 and Y-intercept
0.119 0.107; n 66, r2
0.995, F 9745, P 0.0001).
When the data were pooled and regressed for all plant species,
a single allometric scaling (regression) formula with a 3 4
exponent described the relationship between G and M across 20
orders of magnitude of M ( RMA 0.763 0.003 and Y-intercept
0.208 0.016; n 387, r2
0.996, F 84933, P
0.00001). However, one decade of body size was unoccupied by
these data (i.e., 10 9
M 10 8
kg of dry matter).
A 1 4 scaling exponent was statistically verified for the
relationship between L (maximum cell linear dimension; metaphyte
body length or height) and M across all species. Specifically,
RMA 0.264 0.019 and Y-intercept 2.58 0.012 (n
398, r2
0.995,, F 345.1, P 0.0001; Fig. 1B). Likewise, algal
cell length scaled as the 1 4-power of M (n 66, r2
0.879,
RMA 0.264 0.025, Y-intercept 2.16 0.026, F 489.3,
P 0.0001). However, for tree species, L scaled approximately
as the 1 3-power of M (n 271, r2
0.663, RMA 0.345
0.012, Y-intercept 3.71 0.026, F 529.5, P 0.0001). This
deviation from the 1 4 exponent was not attributable to neglecting
root length; the scaling exponent for the relationship
between plant (stem) height and above-ground body mass
( stem and leaf mass) was higher than that predicted by a 1 3
scaling relationship (i.e., RMA 0.367 0.009). Rather, we
attribute the deviation in the exponent to the log­log nonlinear
relationship between L and M for tree species (i.e., the slope for
the regression of L against M is statistically ambiguous). This
nonlinearity is likely attributable to variation in the ability of
above-ground branching structures to efficiently occupy
space (16).
The ``light-harvesting capability'' of plants H (measured here
as either total algal cell pigment content or total foliage biomass)
scaled as the 3 4-power of M and as the 1.0-power of G. These
scaling relationships are allometrically internally consistent;
because H M3/4
and G M3/4
, it follows that G H. Thus,
because total cell pigment content C scaled as the 3 4-power of
cell mass M ( RMA 0.718 0.035 and Y-intercept 1.12
10 5
0.497; n 56, r2
0.871, F 363.1, P 0.0001; Fig. 2A),
it follows that algal G will scale isometrically with respect to C
( RMA 1.04 0.049 and Y-intercept 438 0.709; n 56,
r2
0.880, F 396.8, P 0.0001; Fig. 2B). Likewise, total foliage
biomass Mp scaled as the 3 4-power of total nonphotosynthetic
biomass Mn ( RMA 0.762 0.023 and Y-intercept 0.192
0.047; n 292, r2
0.747, F 856.2, P 0.0001; Fig. 3A). Thus,
G scaled isometrically with respect to Mp ( RMA 1.02 0.049
and Y-intercept 1.81 0.024; n 337, r2
0.720, F 861.5,
P 0.0001; Fig. 3B). Noting that total metaphyte body mass M
and Mp are autocorrelated, Mp nonetheless scaled approximately
as the 3 4-power of M ( RMA 0.779 0.022 and Y-intercept
0.166 0.047; n 292, r2
0.765, F 856.2, P 0.0001), such
that the isometric relationship between G and Mp holds across
species.
We note in passing that, because G Mp Mn
3/4, short-term
(ontogenetic) growth rates for woody species can vary as a
function of shifts in the relative amounts of photosynthetic and
nonphotosynthetic biomass. Juveniles will have higher growth
rates in comparison to more mature conspecifics provided they
produce disproportionately large foliage biomass. However, for
tree species, analyses indicate that Mn Mp scales nearly isometrically
with respect to plant age and that, on average, G M3/4
within as well as across species.
Discussion
Regardless of clade, grade, or habitat, annualized rates of plant
biomass production, G, on average, scale with respect to body
mass or length in the same manner previously described for
unicellular and multicellular animals (i.e., Kleiber's 3 4-power
``rule''). However, unlike animal clades, which have similar
allometric exponents but different intercepts (i.e., normalization
constants), all plants comply with a single allometric formula
that spans 20 orders of magnitude in body mass. We attribute this
invariance to the fact that, unlike heterotrophs, all photoauNiklas
and Enquist PNAS February 27, 2001 vol. 98 no. 5 2923
PLANTBIOLOGY
totrophs share the same basic physiological machinery and
requirements. It is also evident that the relationship between
plant G and M is intimately related to light-harvesting capacity
H (pigment content per algal cell; foliage biomass per metaphyte),
because H also scales as the 3 4-power of M such that G
H. These scaling relationships reflect the ability of unicellular
and multicellular plants and animals to use available energy for
growth. Thus, it is likely that a common explanation for quarterpower
allometric relationships rests on general principles applicable
to all manner of living things.
One global explanation, advanced on theoretical grounds (13,
16­18), argues that all unicellular and multicellular organisms
have either virtual or real fractal-like distribution networks for
the internal transport of metabolites, thereby endowing them
with a ``fourth spatial dimension'' (17). These networks are
purported to maximize metabolic capacity and efficiency by
maximizing the available exchange surface area for absorption of
limiting resources from the environment but yet minimize
transport distance and time (17). This ``zeroth order'' model
predicts that scaling (allometric) exponents for G, M, and L (as
well as other biological properties) must be either 1 4 or
multiples of 1 4. Accordingly, the fractal body length, which
accounts for the internal spatial dimension of an organism, is
predicted to increase as the 1 4-power of body mass (L M1/4
),
whereas growth (and metabolic) rates are predicted to scale as
the 3 4-power of body mass (G B M3/4
), regardless of an
organism's phyletic affiliation or habitat.
Our data are consistent with this model in virtually all
respects. In addition to showing that G M3/4
and that L M1/4
hold true across 20 decades of plant body mass, we demonstrate
that the scaling relationship between photosynthetic to nonphotosynthetic
biomass for metaphytes and the relationship between
total cell pigment content and cell size for unicellular algae abide
by a 3 4-power ``rule'' such that G H. We interpret this
isometric relationship to indicate that an approximate constant
fraction of short-term biomass production is allocated to the
construction of photosynthetic and nonphotosynthetic body
parts. Such invariance suggests that, over the evolutionary
Fig. 1. Annualized biomass production (growth) rates G (A) and body length L (B) of unicellular and multicellular plants plotted against body mass M. Solid
lines are reduced major axis (Model Type II) regression curves (hair lines indicate 95% confidence intervals); scaling exponents (upper left) based on
log10-transformed data.
2924 www.pnas.org cgi doi 10.1073 pnas.041590298 Niklas and Enquist
diversification of plant clades, there has been no major change
in the basic physiological basis for biomass production and in the
allocation pattern of biomass to energy-harvesting relative to the
biomass supporting other body functions. This proportionality
helps to explain phenomenologically why plant annualized
growth rates scale in the same manner regardless of clade or
grade. More significantly, it is precisely what is expected if
natural selection maximized metabolic capacity by maximizing
effective surface area while simultaneously minimizing metabolite
transport distance and time across or within cells (14­17).
The implications of this theory, which are supported here for
plants, are ecologically and evolutionarily far reaching. For
example, allometric scaling critically influences many attributes
of ecological organization. In particular, extension of allometric
theory predicts that the number of individuals N within any large
sample of a plant community will be proportional to the 3 4power
of body mass (i.e., N M 3/4
), whereas annualized
production rates will be proportional to the 3 4-power of body
mass (i.e., G M3/4
) (33). Assuming that G is a reasonable
surrogate measure of the rate of resource use per individual Q,
we note that the rate of total (community) metabolic production
QTot (which is proportional to the rate of community biomass
production GTot) is the product of the number of individuals N
and Q. Thus, total (community) resource use RTot is predicted to
Fig. 2. Total cell pigment content C plotted against body mass M (A) and annualized biomass production (growth) rates G plotted against total cell pigment
content for unicellular plants (B). Solid lines are reduced major axis (Model Type II) regression curves (hair lines indicate their 95% confidence intervals); scaling
exponents (upper left) based on log10-transformed data.
Niklas and Enquist PNAS February 27, 2001 vol. 98 no. 5 2925
PLANTBIOLOGY
be proportional to NTotQ and, because Q G M3/4
, it follows
that RTot GTot QN M3/4
M 3/4
M0
(33). This relationship
implies that, within communities, the intrinsic capacities for
annualized rates of resource use (as well as biomass production)
will be invariant with respect plant size or species composition.
Allometric theory in tandem with the data presented here
generates similarly important hypotheses when juxtaposed with
biomechanical theory. For example, engineering first principles
indicate that the critical load that can be supported by a vertical
columnar support member Pcr is proportional to the quotient of
flexural rigidity EI and the square of column L (i.e., Pcr EI/L2
)
(34). This proportionality indicates that stem biomass will scale
as the 8 3-power of stem diameter (Ms D8/3
) and that
additional lateral loads, such as photosynthetic (foliage) biomass,
will scale as the 2-power of stem diameter (Mp D2
) (34).
Noting that our data for metaphytes show that photosynthetic
(foliage) biomass scales as the 3 4-power of stem biomass (Mp
Ms
3/4) and that annualized metaphyte growth rates scale as the
3 4-power of foliage biomass (G Mp
3/4), it follows that
annualized growth rates are predicted to scale as the 3 2-power
of stem diameter (G D3/2
). If this hypothesis is true, then
size-frequency distributions of stem area could be used as
surrogate measures for the annualized growth rates of plant
communities (33).
Fig. 3. Photosynthetic (foliage) biomass Mp plotted against nonphotosynthetic (stem and root) body mass Mn (A) and annualized biomass production (growth)
rates G plotted against photosynthetic body mass Mp for metaphytes (data taken from ref. 32) (B). Solid lines are reduced major axis (Model Type II) regression
curves (hair lines indicate their 95% confidence intervals); scaling exponents (upper left) based on log10-transformed data.
2926 www.pnas.org cgi doi 10.1073 pnas.041590298 Niklas and Enquist
These and other arguably provocative allometric hypotheses
must be tested based on additional broad interspecific and
intercommunity comparisons. However, all of the available data
for unicellular and multicellular plants demonstrate scaling
relationships that thus far comply in every respect with allometric
and biomechanical theory (33, 34). Because present-day
plants and animals appear to abide by the same or very similar
scaling ``rules,'' there is good reason to expect these rules extend
into deep geological (evolutionary) time. As such, allometric and
biomechanical theory provides a potentially powerful tool for
predicting many important properties for past as well as presentday
organisms and the communities in which they live.
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