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First published online April 26, 2005
Journal of Experimental Biology 208, 1575-1592 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01589
Review article: Basal metabolic rate and cellular energetics |
The origin of allometric scaling laws in biology from genomes to ecosystems: towards a quantitative unifying theory of biological structure and organization
1 The Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501,
USA
2 Los Alamos National Laboratory, Los Alamos, NM 87545, USA
3 Department of Biology, University of New Mexico, Albuquerque, NM 87131,
USA
* Author for correspondence (e-mail: gbw{at}santafe.edu)
Accepted 14 March 2005
Summary
Life is the most complex physical phenomenon in the Universe, manifesting an extraordinary diversity of form and function over an enormous scale from the largest animals and plants to the smallest microbes and subcellular units. Despite this many of its most fundamental and complex phenomena scale with size in a surprisingly simple fashion. For example, metabolic rate scales as the 3/4-power of mass over 27 orders of magnitude, from molecular and intracellular levels up to the largest organisms. Similarly, time-scales (such as lifespans and growth rates) and sizes (such as bacterial genome lengths, tree heights and mitochondrial densities) scale with exponents that are typically simple powers of 1/4. The universality and simplicity of these relationships suggest that fundamental universal principles underly much of the coarse-grained generic structure and organisation of living systems. We have proposed a set of principles based on the observation that almost all life is sustained by hierarchical branching networks, which we assume have invariant terminal units, are space-filling and are optimised by the process of natural selection. We show how these general constraints explain quarter power scaling and lead to a quantitative, predictive theory that captures many of the essential features of diverse biological systems. Examples considered include animal circulatory systems, plant vascular systems, growth, mitochondrial densities, and the concept of a universal molecular clock. Temperature considerations, dimensionality and the role of invariants are discussed. Criticisms and controversies associated with this approach are also addressed.
Key words: allometry, quarter-power scaling, laws of life, circulatory system, ontogenetic growth
Introduction
Life is almost certainly the most complex and diverse physical system in the universe, covering more than 27 orders of magnitude in mass, from the molecules of the genetic code and metabolic process up to whales and sequoias. Organisms themselves span a mass range of over 21 orders of magnitude, ranging from the smallest microbes (10-13 g) to the largest mammals and plants (108 g). This vast range exceeds that of the Earth's mass relative to that of the galaxy (which is `only' 18 orders of magnitude) and is comparable to the mass of an electron relative to that of a cat. Similarly, the metabolic power required to support life over this immense range spans more than 21 orders of magnitude. Despite this amazing diversity and complexity, many of the most fundamental biological processes manifest an extraordinary simplicity when viewed as a function of size, regardless of the class or taxonomic group being considered. Indeed, we shall argue that mass, and to a lesser extent temperature, is the prime determinant of variation in physiological behaviour when different organisms are compared over many orders of magnitude.
Scaling with size typically follows a simple power law behaviour of the
form:
 | (1) |
where Y is some observable biological quantity,
Y0 is a normalization constant, and Mb
is the mass of the organism (Calder,
1984
; McMahon and Bonner,
1983; Niklas,
1994; Peters,
1986; Schmidt-Nielsen,
1984). An additional simplification is that the exponent,
b, takes on a limited set of values, which are typically simple
multiples of 1/4. Among the many variables that obey these simple
quarter-power allometric scaling laws are nearly all biological rates, times,
and dimensions; they include metabolic rate (b 
3/4), lifespan
(b 1/4), growth rate (b -1/4), heart rate
(b -1/4), DNA nucleotide substitution rate (b -1/4),
lengths of aortas and heights of trees (b 1/4), radii of aortas
and tree trunks (b 3/8), cerebral gray matter (b
5/4), densities of mitochondria, chloroplasts and ribosomes
(b=-1/4), and concentrations of ribosomal RNA and metabolic enzymes
(b -1/4); for examples, see Figs
1,
2,
3,
4. The best-known of these
scaling laws is for basal metabolic rate, which was first shown by Kleiber
(Brody, 1945; Kleiber,
1932,
1975) to scale approximately
as Mb3/4 for mammals and birds
(Fig. 1).Subsequent researchers
showed that whole-organism metabolic rates also scale as
Mb3/4 in nearly all organisms, including
animals (endotherms and ectotherms, vertebrates and invertebrates;
Peters, 1986), plants
(Niklas, 1994), and
unicellular microbes (see also Fig.
7). This simple 3/4 power scaling has now been observed at
intracellular levels from isolated mammalian cells down through mitochondria
to the oxidase molecules of the respiratory complex, thereby covering fully 27
orders of magnitude (Fig. 2;
West et al., 2002b). In the
early 1980s, several independent investigators
(Calder, 1984;
McMahon and Bonner, 1983;
Peters, 1986;
Schmidt-Nielsen, 1984)
compiled, analyzed and synthesized the extensive literature on allometry, and
unanimously concluded that quarter-power exponents were a pervasive feature of
biological scaling across nearly all biological variables and life-forms.
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Another simple characteristic of these scaling laws is the emergence of
invariant quantities (Charnov,
1993). For example, mammalian lifespan increases approximately as
Mb1/4, whereas heart-rate decreases as
Mb-1/4, so the number of heart-beats per
lifetime is approximately invariant (
1.5 x109),
independent of size. A related, and perhaps more fundamental, invariance
occurs at the molecular level, where the number of turnovers of the
respiratory complex in the lifetime of a mammal is also essentially constant
(1016). Understanding the origin of these dimensionless
numbers should eventually lead to important fundamental insights into the
processes of aging and mortality. Still another invariance occurs in ecology,
where population density decreases with individual body size as
Mb-3/4 whereas individual power use increases
as Mb3/4, so the energy used by all individuals
in any size class is an invariant (Enquist
and Niklas, 2001).
It seems impossible that these `universal' quarter-power scaling laws and the invariant quantities associated with them could be coincidental, independent phenomena, each a `special' case reflecting its own unique independent dynamics and organisation. Of course every individual organism, biological species and ecological assemblage is unique, reflecting differences in genetic make-up, ontogenetic pathways, environmental conditions and evolutionary history. So, in the absence of any additional physical constraints, one might have expected that different organisms, or at least each groups of related organisms inhabiting similar environments, might exhibit different size-related patterns of variation in structure and function. The fact that they do not - that the data almost always closely approximate a power law, emblematic of self-similarity, across a broad range of size and diversity - raises challenging questions. The fact that the exponents of these power laws are nearly always simple multiples of 1/4 poses an even greater challenge. It suggests the operation of general underlying mechanisms that are independent of the specific nature of individual organisms.
We argue that the very existence of such ubiquitous power laws implies the existence of powerful constraints at every level of biological organization. The self-similar power law scaling implies the existence of average, idealized biological systems, which represent a `0th order' baseline or point of departure for understanding the variation among real biological systems. Real organisms can be viewed as variations on, or perturbations from, these idealized norms due to influences of stochastic factors, environmental conditions or evolutionary histories. Comparing organisms over large ranges of body size effectively averages over environments and phylogenetic histories. Sweeping comparisons, incorporating organisms of different taxonomic and functional groups and spanning many orders of magnitude in body mass, reveal the more universal features of life, lead to coarse-grained descriptions, and motivate the search for general, quantitative, predictive theories of biological structures and dynamics.
Such an approach has been very successful in other branches of science. For example, classic kinetic theory is based on the idea that generic features of gases, such as the ideal gas law, can be understood by assuming atoms to be structureless `billiard balls' undergoing elastic collisions. Despite these simplifications, the theory captures many essential features of gases and spectacularly predicts many of their coarse-grained properties. The original theory acted as a starting point for more sophisticated treatments incorporating detailed structure, inelasticity, quantum mechanical effects, etc, which allow more detailed calculations. Other examples include the quark model of elementary particles and the theories describing the evolution of the universe from the big bang. This approach has also been successful in biology, perhaps most notably in genetics. Again, the original Mendelian theory made simplifying assumptions, portraying each phenotypic trait as the expression of pairs of particles, each derived from a different parent, which assorted and combined at random in offspring. Nevertheless, this theory captured enough of the coarse-grained essence of the phenomena so that it not only provided the basis for the applied sciences of human genetics and plant and animal breeding, but also guided the successful search for the molecular genetic code and supplied the mechanistic underpinnings for the modern evolutionary synthesis. Although the shortcomings of these theories are well-recognized, they quantitatively explain an extraordinary body of data because they do indeed capture much of the essential behavior.
Scaling as a manifestation of underlying dynamics has been instrumental in gaining deeper insights into problems across the entire spectrum of science and technology, because scaling laws typically reflect underlying general features and principles that are independent of detailed structure, dynamics or other specific characteristics of the system, or of the particular models used to describe it. So, a challenge in biology is to understand the ubiquity of quarter-powers - to explain them in terms of unifying principles that determine how life is organized and the constraints under which it has evolved. Over the immense spectrum of life the same chemical constituents and reactions generate an enormous variety of forms, functions, and dynamical behaviors. All life functions by transforming energy from physical or chemical sources into organic molecules that are metabolized to build, maintain and reproduce complex, highly organized systems. We conjecture that metabolism and the consequent distribution of energy and resources play a central, universal role in constraining the structure and organization of all life at all scales, and that the principles governing this are manifested in the pervasive quarter-power scaling laws.
Within this paradigm, the precise value of the exponent, whether it is
exactly 3/4, for example, is less important than the fact that it approximates
such an ideal value over a substantial range of mass, despite variation due to
secondary factors. Indeed, a quantitative theory for the dominant behaviour
(the 3/4 exponent, for example) provides information about the residual
variation that it cannot explain. If a general theory with well-defined
assumptions predicts 3/4 for average idealized organisms, then it is possible
to erect and test hypotheses about other factors, not included in the theory,
which may cause real organisms to deviate from this value. On the other hand,
without such a theory it is not possible to give a specific meaning to any
scaling exponent, but only to describe the relationship statistically. This
latter strategy has usually been employed in analyzing allometric data and has
fueled controversy ever since Kleiber's original study (Kleiber,
1932,
1975). Kleiber's contemporary
Brody independently measured basal metabolic rates of birds and mammals,
obtained a statistically fitted exponent of 0.73, and simply took this as the
`true' value (Brody, 1945).
Subsequently a great deal of ink has been spilled debating whether the
exponent is `exactly' 3/4. Although this controversy appeared to be settled
more than 20 years ago (Calder,
1984; McMahon and Bonner,
1983; Peters,
1986; Schmidt-Nielsen,
1984), it was recently resurrected by several researchers
(Dodds et al., 2001;
Savage et al., 2004b;
White and Seymour, 2003).
A deep understanding of quarter-power scaling based on a set of underlying principles can provide, in principle, a general framework for making quantitative dynamical calculations of many more detailed quantities beyond just the allometric exponents of the phenomena under study. It can raise and address many additional questions, such as: How many oxidase molecules and mitochondria are there in an average cell and in an entire organism? How many ribosomal RNA molecules? Why do we stop growing and what adult weight do we attain? Why do we live on the order of 100 years - and not a million or a few weeks - and how is this related to molecular scales? What are the flow rate, pulse rate, pressure and dimensions in any vessel in the circulatory system of any mammal? Why do we sleep eight hours a day, a mouse eighteen and an elephant three? How many trees of a given size are there in a forest, how far apart are they, how many leaves does each have and how much energy flows in each or their branches? What are the limits on the sizes of organisms with different body plans?
Basic principles
All organisms, from the smallest, simplest bacterium to the largest plants
and animals, depend for their maintenance and reproduction on the close
integration of numerous subunits: molecules, organelles and cells. These
components need to be serviced in a relatively `democratic' and efficient
fashion to supply metabolic substrates, remove waste products and regulate
activity. We conjecture that natural selection solved this problem by evolving
hierarchical fractal-like branching networks, which distribute energy and
materials between macroscopic reservoirs and microscopic sites
(West et al., 1997). Examples
include animal circulatory, respiratory, renal, and neural systems, plant
vascular systems, intracellular networks, and the systems that supply food,
water, power and information to human societies. We have proposed that the
quarter-power allometric scaling laws and other features of the dynamical
behaviour of biological systems reflect the constraints inherent in the
generic properties of these networks. These were postulated to be: (i)
networks are space-filling in order to service all local biologically active
subunits; (ii) the terminal units of the network are invariants; and (iii)
performance of the network is maximized by minimizing the energy and other
quantities required for resource distribution.
These properties of the `average idealised organism' are presumed to be consequences of natural selection. Thus, the terminal units of the network where energy and resources are exchanged (e.g. leaves, capillaries, cells, mitochondria or chloroplasts), are not reconfigured or rescaled as individuals grow from newborn to adult or as new species evolve. In an analogous fashion, buildings are supplied by branching networks that terminate in invariant terminal units, such as electrical outlets or water faucets. The third postulate assumes that the continuous feedback and fine-tuning implicit in natural selection led to `optimized' systems. For example, of the infinitude of space-filling circulatory systems with invariant terminal units that could have evolved, those that have survived the process of natural selection, minimize cardiac output. Such minimization principles are very powerful, because they lead to `equations of motion' for network dynamics.
Using these basic postulates, which are quite general and independent of
the details of any particular system, we have derived analytic models for
mammalian circulatory and respiratory systems
(West et al., 1997) and plant
vascular systems (West et al.,
1999b). The theory predicts scaling relations for many structural
and functional components of these systems. These scaling laws have the
characteristic quarter-power exponents, even though the anatomy and physiology
of the pumps and plumbing are very different. Furthermore, our models derive
scaling laws that account for observed variation between organisms
(individuals and species of varying size), within individual organisms (e.g.
from aorta to capillaries of a mammal or from trunk to leaves of a tree), and
during ontogeny (e.g. from a seedling to a giant sequoia). The models can be
used to understand the values not only for allometric exponents, but also for
normalization constants and certain invariant quantities. The theory makes
quantitative predictions that are generally supported when relevant data are
available, and - when they are not - that stand as a priori
hypotheses to be tested by collection and analysis of new data
(Enquist et al., 1999;
Savage et al., 2004a; West et
al., 1997,
1999a,b,
2001,
2002a,b).
Metabolic rate and the vascular network
Metabolic rate, the rate of transformation of energy and materials within
an organism, literally sets the pace of life. Consequently it is central in
determining the scale of biological phenomena, including the sizes and
dimensions of structures and the rates and times of activities, at levels of
organization from molecules to ecosystems. Aerobic metabolism in mammals is
fueled by oxygen whose concentration in blood is invariant, so cardiac output
or blood volume flow rate through the cardiovascular system is a proxy for
metabolic rate. Thus, characteristics of the circulatory network constrain the
scaling of metabolic rate. We shall show how the body-size dependence for
basal and field metabolic rates, B
Mb3/4, where B is total
metabolic rate, can be derived by modeling the hemodynamics of the
cardiovascular system based on the above general assumptions. In addition, and
just as importantly, this whole-system model also leads to analytic solutions
for many other features of the blood supply network. These results are derived
by solving the hydrodynamic and elasticity equations for blood flow and vessel
dynamics subject to space-filling and the minimization of cardiac output
(West et al., 1997). We make
certain simplifying assumptions, such as cylindrical vessels, a symmetric
network, and the absence of significant turbulence. Here, we present a
condensed version of the model that contains the important features pertinent
to the scaling problem.
In order to describe the network we need to determine how the radii, rk, and lengths, lk, of vessels change throughout the network; k denotes the level of the branching, beginning with the aorta at k=0 and terminating at the capillaries where k=N. The average number of branches per node (the branching ratio), n, is assumed to be constant throughout the network.
Space-filling (Mandelbrot,
1982) ensures that every local volume of tissue is serviced by the
network on all spatial scales, including during growth from embryo to adult.
The capillaries are taken to be invariant terminal units, but each capillary
supplies a group of cells, referred to as a `service volume',
vN, which can scale with body mass. The total volume to be
serviced, or filled, is therefore given by
VS=NNvN, where
NN is the total number of capillaries. For a network with
many levels, N, space-filling at all scales requires that this same
volume, VS, be serviced by an aggregate of the volumes,
vk, at each level k. Since
rk<<lk, vk
lk3, so VS
Nkvk
Nklk3 for every
level, k. Thus lk+1/lk
n-1/3, so space-filling constrains only branch
lengths, lk.
The equation of motion governing fluid flow in any single tube is the
Navier-Stokes equation (Landau and
Lifshitz, 1978). If non-linear terms responsible for turbulence
are neglected, this reads:
 | (2) |
where v is the local fluid velocity at some time t,
p the local pressure, 
, blood density and µ, blood viscosity.
Assuming blood is incompressible, then local conservation of fluid requires
. When combined with Eq. 2, this gives:
 | (3) |
The beating heart generates a pulse wave that propagates down the arterial
system causing expansion and contraction of vessels as described by the Navier
equation, which governs the elastic motion of the tube. This is given by:
 | (4) |
where 
is the local displacement of the tube wall, w
its density, and E its modulus of elasticity. These three coupled
equations, Eq. 2-4, must be solved subject to boundary conditions that require
the continuity of velocity and force at the tube wall interfaces. In the
approximation where the vessel wall thickness, h, is small compared
to the static equilibrium value of the vessel radius, r, i.e.
h<<r, the problem can be solved analytically, as first
shown by Womersley (Caro et al.,
1978; Fung, 1984),
to give:
 | (5) |
where Jn denotes the Bessel function of order n. Here,
is the angular frequency of the wave, 
(/µ)1/2r is a dimensionless
parameter known as the Womersley number, c0
(Eh/2r)1/2 is the classic
Korteweg-Moens velocity, and Z is the vessel impedance. Both
Z and the wave velocity, c, are complex functions of
so, in general, the wave is attenuated and dispersed as it propagates along
the tubes. The character of the wave depends critically on whether
|| is less than or greater than 1. This can be seen
explicitly in Eq. 5, where the behavior of the Bessel functions changes from a
power-series expansion for small || to an expansion with
oscillatory behavior when || is large. In humans,
|| has a value of around 15 in the aorta, 5 in the
arteries, 0.04 in the arterioles, and 0.005 in the capillaries. When
|| is large (>>1), Eq. 5 gives
cc0, which is a purely real quantity, so the
wave suffers neither attenuation nor dispersion. Consequently, in these large
vessels viscosity plays almost no role and virtually no energy is dissipated.
In an arbitrary unconstrained network, however, energy must be expended to
overcome possible reflections at branch junctions, which would require
increased cardiac power output. Minimization of energy expenditure is
therefore achieved by eliminating such reflections, a phenomenon known as
impedance matching. From Eq. 5,
Zc0/
r2 for
large vessels, and impedance matching leads to area-preserving branching:
rk2=nrk+12,
so that
rk+1/rk=n-1/2. For
small vessels, however, where ||<<1, the role of
viscosity and the subsequent dissipation of energy become increasingly
important until they eventually dominate the flow. Eq. 5 then gives
c(1/4)i1/2c0
0,
in quantitative agreement with observation
(Caro et al., 1978;
Fung, 1984). Because
c now has a dominant imaginary part, the traveling wave is heavily
damped, leaving an almost steady oscillatory flow whose impedance is, from Eq.
5, just the classic Poiseuille formula,
Zk=8µlk/rk4.
Unlike energy loss due to reflections at branch points, energy loss due to
viscous dissipative forces cannot be entirely eliminated. It can be minimized,
however, using the classic method of Lagrange multipliers to enforce the
appropriate constraints (Marion and
Thornton, 1988; West et al.,
1997). To sustain a given metabolic rate in an organism of fixed
mass with a given volume of blood, Vb, the cardiac output
must be minimized subject to a space-filling geometry. The calculation shows
that area-preserving branching is thereby replaced by
area-increasing branching in small vessels, so blood slows down
allowing efficient diffusion of oxygen from the capillaries to the cells.
Branching, therefore, changes continuously down through the network, so that
the ratio rk+1/rk is not independent
of k but changes continuously from n-1/2 at the
aorta to n-1/3 at the capillaries. Consequently, the
network is not strictly self-similar, but within each region (pulsatile in
large vessels and Poiseuille in small ones), self-similarity is a reasonable
approximation that is well supported by empirical data
(Caro et al., 1978;
Fung, 1984;
Zamir, 1999).
In order to derive allometric relations between animals of different sizes
we need to relate the scaling of vessel dimensions within an organism to its
body mass, Mb. A natural vehicle for this is the total
volume of blood in the network, Vb, which can be shown to
depend linearly on Mb if cardiac output is minimized, i.e.
Vb Mb, in agreement with data
(Caro et al., 1978;
Fung, 1984). This is
straightforwardly given by
Vb=
NkVk=nkrk2lk,
where Nk=nk is the number of vessels
at level k. Provided there are sufficiently large vessels in the
network with ||>1 so that pulsatile flow dominates,
the leading-order behavior for the blood volume is Vb
n4N/3VN. Conservation
of blood requires that the flow rate in the aorta, Q0=
NNQN, where QN is
the flow rate in a capillary and NN
nN, the total number of capillaries. But
Q0 B, the total metabolic rate, so putting
these together we obtain B
(Vb/VN)3/4QN.
However, capillaries are invariant units, so VN and
QN are both independent of Mb, whereas
from minimization of energy loss, Vb
Mb, so we immediately obtain the seminal result
B Mb3/4.
The allometric scaling of radii, lengths and many other physiological
characteristics, such as the flow, pulse and dimensions in any branch of a
mammal of any size, can be derived from this whole-system model and shown to
have quarter-power exponents. Quantitative predictions for all these
characteristics of the cardiovascular system are in good agreement with data
(West et al., 1997). For
example, even the residual pulse wave component in capillaries is determined:
it is predicted to be attenuated to 0.1% with its velocity being 10 cm
s-1, compared to 580 cm s-1 for the unattenuated
wave in the aorta, both numbers being invariant with respect to body size.
To summarise: there are two independent contributions to energy expenditure: viscous energy dissipation, which is important only in smaller vessels, and energy reflected at branch points, which is important only in larger vessels and is eliminated by impedance matchings In large vessels (arteries), pulse-waves suffer little attenuation or dissipation, and impedance matching leads to area-preserving branching, such that the cross-sectional area of daughter branches equals that of the parent; so radii scale as rk+1/rk=n-1/2 with the blood velocity remaining constant. In small vessels (capillaries and arterioles) the pulse is strongly damped since Poiseuille flow dominates and substantial energy is dissipated. Here minimization of energy dissipation leads to area-increasing branching with rk+1/rk=n-1/3, so blood slows down, almost ceasing to flow in the capillaries. Consequently, the ratio of vessel radii between adjacent levels, rk+1/rk, changes continuously from n-1/2 to n-1/3 down through the network, which is, therefore, not strictly self-similar. Nevertheless, since the length ratio lk+1/lk remains constant throughout the network because of space-filling, branch-lengths are self-similar and the network has some fractal-like properties. Quarter-power allometric relations then follow from the invariance of capillaries and the prediction from energy optimization that total blood volume scales linearly with body mass.
The dominance of pulsatile flow, and consequently of area-preserving
branching, is crucial for deriving power laws, including the 3/4 exponent for
metabolic rate, B. However, as body size decreases, narrow tubes
predominate and viscosity plays an ever-increasing role. Eventually even the
major arteries would become too constricted to support wave propagation, blood
flow would become steady and branching exclusively area-increasing, leading to
a linear dependence on mass. Since energy would be dissipated in all
branches of the network, the system is now highly inefficient; such an
impossibly small mammal would have a beating heart (with a resting heart-rate
in excess of approximately 1000 beats min-1) but no pulse! This
provides a framework to estimate the size of the smallest mammal in terms of
fundamental cardiovascular parameters. This gives a minimum mass
Mmin1 g, close to that of a shrew, which is
indeed the smallest mammal (Fig.
3; West et al.,
2002b). Furthermore, the predicted linear extrapolation of
B below this mass to the mass of a single cell should, and does, give
the correct value for the metabolic rate of mammalian cells growing in culture
isolated from the vascular network (Fig.
2).
The derivation that gives the 3/4 exponent is only an approximation,
because of the changing roles of pulsatile and Poiseuille flow with body size.
Strictly speaking, the theory predicts that the exponent for B is
exactly 3/4 only where pulsatile flow completely dominates. In general, the
exponent is predicted to depend weakly on M, manifesting significant
deviations from 3/4 only in small mammals, where only the first few branches
of the arterial system can support a pulse wave (West et al.,
1997,
2002b). Since small mammals
dissipate relatively more energy in their networks, they require elevated
metabolic rates to generate the increased energy expenditure to circulate the
blood. This leads to the prediction that the allometric exponent for
B should decrease below 3/4 as Mb decreases to
the smallest mammal, as observed (Dodds et
al., 2001; Savage et al.,
2004b).
If the total number of cells, Nc, increases linearly
with Mb, then both cellular metabolic rate,
Bc(B/Nc), and specific
metabolic rate, B/Mb, decrease as
Mb-1/4. In this sense, therefore, larger
animals are more efficient than smaller ones, because they require less power
to support unit body mass and their cells do less work than smaller animals.
In terms of our theory this is because the total hydrodynamic resistance of
the network decreases with size as Mb-3/4. This
has a further interesting consequence that, since the `current' or volume rate
of flow of blood in the network, Q0, increases as
Mb3/4, whereas the resistance decreases as
Mb-3/4, the analog to Ohm's law
(pressure=current xresistance) predicts that blood pressure must be an
invariant, as observed (Caro et al.,
1978; Fung, 1984).
This may seem counterintuitive, since the radius of the aorta varies from
approximately 0.2 mm in a shrew up to approximately 30 cm in a whale!
Scaling up the hierarchy: from molecules to mammals
At each organisational level within an organism, beginning with molecules
and continuing up through organelles, cells, tissues and organs, new
structures emerge, each with different physical characteristics, functional
parameters, and resource and energy network systems, thereby constituting a
hierarchy of hierarchies. Metabolic energy is conserved as it flows through
this hierarchy of sequential networks. We assume that each network operates
subject to the same general principles and therefore exhibits quarter-power
scaling (West et al., 2002b).
From the molecules of the respiratory complex up to intact cells, metabolic
rate obeys 3/4-power scaling. Continuity of flow imposes boundary conditions
between adjacent levels, leading to constraints on the densities of invariant
terminal units, such as respiratory complexes and mitochondria, and on the
networks of flows that connect them (West
et al., 2002b). The total mitochondrial mass relative to body mass
is correctly predicted to be
(Mminmm/mcMb)1/4
0.06Mb-1/4, where mm is
the mass of a mitochondrion, Mmin is minimum mass,
mc is average cell mass and Mb is
expressed in g. Since mitochondria are assumed to be approximately invariant,
the total number in the body is determined in a similar fashion. This shows
why there are typically only a few hundred per human cell, whereas there are
several thousand in a shrew cell of the same type.
As already stressed, a central premise of the theory is that general
properties of supply networks constrain the coarse-grained, and therefore the
scaling properties, of biological systems. An immediate qualitative
consequence of this idea is that, if cells are liberated from the network
hegemony by culturing them in vitro, they are likely to behave
differently from cells in vivo. An alternative possibility is that
cellular metabolic rates are relatively inflexible. This, however, would be a
poor design, because it would prevent facultative adjustment in response to
variation in body size over ontogeny and in response to the varying metabolic
demands of tissues. If the metabolic rate and number of mitochondria per cell
are indeed tuned facultatively in response to variations in supply and demand,
the theory makes an explicit quantitative prediction: cells isolated from
different mammals and cultured in vitro under conditions of unlimited
resource supply should converge toward the same metabolic rate
(predicted to be 6 x10-11 W), rather than scaling as
M-1/4 as they do in vivo
(Fig. 5); cells in
vitro should also converge toward identical numbers of mitochondria,
losing the M-1/4 scaling that they exhibit in
vivo. The in vivo and in vitro values are predicted to
coincide at Mmin, which we estimated above to be
approximately 1 g. So cells in shrews work at almost maximal output, which, no
doubt, is reflected in their high levels of activity and the shortness of
their lives. By contrast cells in larger mammals are constrained by the
properties of vascular supply networks and normally work at lower rates.
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All of these results depend only on generic network properties, independent of details of anatomy and physiology, including differences in the size, shape and number of mitochondria among different tissues within a mammal. Since quarter-power scaling is observed at intracellular, as well as whole-organism and cellular levels, this suggests that metabolic processes at subcellular levels are also constrained by optimized space-filling, hierarchical networks, which have similar properties to the more macroscopic ones. A major challenge, both theoretically and experimentally, is to understand quantitatively the mechanisms of intracellular transport, about which surprisingly little is known.
Extensions
Ontogenetic growth
The theory developed above naturally leads to a general growth equation
applicable to all multicellular animals (West et al.,
2001,
2002a). Metabolic energy
transported through the network fuels cells where it is used either for
maintenance, including the replacement of cells, or for the production of
additional biomass and new cells. This can be expressed as:
 | (6) |
where Nc is the total number of cells in the organism
at time t after birth and Ec the energy needed to
create a new cell. Since
Nc=m/mc, where m is
the ontogenetic mass and mc the average cell mass, this
leads to an equation for the growth rate of an organism:
 | (7) |
where B0 is the taxon-dependent normalization constant
for the scaling of metabolic rate: B
B0m3/4. The parameters of the
resulting growth equation are therefore determined solely by fundamental
properties of cells, such as their metabolic rates and the energy required to
produce new ones, which can be measured independently of growth. The model
gives a natural explanation for why animals stop growing: the number of cells
supplied (Nc m) scales faster than the
number of supply units (since B NN
m3/4), and leads to an expression for the asymptotic
mass of the organism:
Mb=(B0mc/Bc)4.
Eq. 7 can be solved analytically to determine m as a function of
t, leading to a classic sigmoidal growth curve. By appropriately
rescaling m and t as prescribed by the theory, the solution
can be recast as a universal scaling curve for growth. When rescaled in this
way, growth data from many different animals (including endotherms and
ectotherms, vertebrates and invertebrates) all closely fit a single universal
curve (Fig. 6). Ontogenetic
growth is therefore a universal phenomenon determined by the interaction of
basic metabolic properties at cellular and whole-organism levels. Furthermore,
this model leads to scaling laws for other growth characteristics, such as
doubling times for body mass and cell number, and the relative energy devoted
to production vs maintenance. Recently, Guiot et al.
(2003) applied this model to
growth of solid tumors in rats and humans. They showed that the growth curve
derived from Eq. 7 gave very good fits, even though the parameters they used
were derived from statistical fitting rather than determined from first
principles, as in ontogenetic growth. This is just one example of the exciting
potential applications of metabolic scaling theory to important biomedical
problems.
|
Temperature and universal biological clocks
Temperature has a powerful effect on all biological systems because of the
exponential sensitivity of the Boltzmann factor, e-E/kT,
which controls the temperature dependence of biochemical reaction rates; here,
E is a chemical activation energy, T absolute temperature,
and k Boltzmann's constant. Combined with network constraints that
govern the fluxes of energy and materials, this predicts a joint universal
mass and temperature scaling law for all rates and times connected
with metabolism, including growth, embryonic development, longevity and DNA
nucleotide substitution in genomes. All such rates are predicted to scale as:
 | (8) |
and all times as:
 | (9) |
The critical points here are the separable multiplicative nature of the
mass and temperature dependences and the relatively invariant value of
E, reflecting the average activation energy for the rate-limiting
biochemical reactions (Gillooly et al.,
2001). Data covering a broad range of organisms (fish, amphibians,
aquatic insects and zooplankton) confirm these predictions with Ê0.65 eV
(Fig. 7). These results suggest
a general definition of biological time that is approximately invariant and
common to all organisms: when adjusted for size and temperature,
determined by just two numbers (1/4 and E0.65 eV), all
organisms to a good approximation run by the same universal clock with similar
metabolic, growth, and evolutionary rates!
(Gillooly et al., 2005).
Metabolic scaling in plants: independent evolution of M3/4
One of the most challenging facts about quarter-power scaling relations is
that they are observed in both animals and plants. Our theory offers an
explanation: both use fractal-like branching structures to distribute
resources, so both obey the same basic principles despite the large
differences in anatomy and physiology. For example, in marked contrast to the
mammalian circulatory system, plant vascular systems are effectively fiber
bundles of long micro-capillary tubes (xylem and phloem), which transport
resources from trunk to leaves, driven by a non-pulsatile pump
(Niklas, 1994). If the
microcapillary vessels were of uniform radius, as is often assumed in models
of plants, a serious paradox results: the supply to the tallest branches,
where most light is collected, suffers the greatest resistance. This problem
had to be circumvented in order for the vertical architecture of higher plants
to have evolved. Furthermore, the branches that distribute resources also
contain substantial quantities of dead wood which provide biomechanical
support, so the model must integrate classic bending moment equations with the
hydrodynamics of fluid flow in the active tubes.
Assuming a space-filling branching network geometry with invariant terminal units (petioles or leaves) and minimization of energy use as in the cardiovascular system, the theory predicts that tubes must have just enough taper so that the hydrodynamic resistance of each tube is independent of path length. This therefore `democratises' all tubes in all branches, thereby allowing a vertical architecture. As in the mammalian system, many scaling relations can be derived both within and between plants, and these make quantitative testable predictions for metabolic rate (the 3/4 exponent), area-preserving branching, xylem vessel tapering and conductivity, pressure gradients, fluid velocity, and the relative amount of non-conducting wood to provide biomechanical support (Figs 8, 9).
|
|
Of particular relevance here is the fact that these two systems, in plants
and mammals, which have evolved independently by natural selection to solve
the problem of efficient distribution of resources from a central `trunk' to
terminal `capillaries' and which have such fundamentally different anatomical
and physiological features, nevertheless show identical
Mb3/4 scaling of whole-organism metabolic rate,
and comparable quarter-power scaling of many structures and functions. Our
model accurately predicts scaling exponents for 17 parameters of trees
(West et al., 1999b). These
sets of comparable quarter-power scaling laws reflect convergent solutions of
trees and mammals to the common problems of vascular network design satisfying
the same set of basic principles.
The fourth dimension
We have shown that power law scaling reflects generic properties of energy
and resource distribution networks: space-filling, invariant terminal units
and minimization of energy expenditure are sufficient to determine scaling
properties, regardless of the detailed architecture of the network. For
example, area-preserving branching and the linearity of the network volume
with body mass both follow from optimising the solution to the dynamical
equations for network flow and are sufficient to derive quarter-powers in both
mammals and plants. Nevertheless, one can ask why it is that exponents are
destined to be quarter-powers in all cases, rather than some other power, and
why should this be a universal behaviour extending even to unicellular
organisms with no obvious branching network. Is there a more general argument,
independent of dynamics and hierarchical branching that determines the `magic'
number 4? This question was addressed by Banavar et al.
(1999) who, following our work,
also assumed that allometric relations reflect network constraints. They
proposed that quarter-powers arise from a more general class of directed
networks that do not necessarily have fractal-like hierarchical branching.
They assumed that the network volume scales linearly with body mass
Mb and claimed on general grounds that a lower bound on
the overall network flow rate scales as Mb3/4.
Although intriguing, the biological significance of this result is unclear,
not only because it is a lower bound rather than an optimization, but more
importantly, because it was derived assuming that the flow is
sequential between the invariant terminal units being supplied (e.g.
from cell to cell, or from leaf to leaf) rather than hierarchically
terminating on such units, as in most real biological networks.
Whether their result can be generalized to general networks of more relevance
in biology is unclear.
A general argument (West et al.,
1999a) can be motivated from the observation that in d
dimensions our derivations of the metabolic exponent obtained from solving the
dynamical equations leads to d/(d+1), which in three
dimensions reduces to the canonical 3/4. Thus, the ubiquitous `4' is actually
the dimensionality of space ('3') plus `1'. In our derivations this can be
traced partially to the space-filling constraint, which typically leads to an
increase in effective scaling dimensionality
(Mandelbrot, 1982). For
example, the total area of two-dimensional sheets filling three-dimensional
washing machines clearly scales like a volume rather than an area. In this
scaling sense, organisms effectively function as if in four spatial
dimensions. Natural selection has taken advantage of the generalised
fractality of space-filling networks to maximise the effective network surface
area, A, of the terminal units interfacing with their resource
environments. This can be expressed heuristically in the following way: if
terminal units are invariant and the network space-filling, then metabolic
rate, B A, which scales like a volume, rather than an
area; that is, B A L3
(rather than L2), where L is some
characteristic length of the network, such as the length of the aorta in the
circulatory system of mammals or the length of the stem in plants. However,
the volume of the network, Vnet AL
L4. So, if we assume that Vnet
M (proven from energy minimisation in our theory), we then
obtain Vnet M L4.
Thus, L M1/4 leading to B
A L3 M3/4.
This therefore provides a geometrical interpretation of the quarter-powers,
and, in particular, a geometrical `derivation' of the 3/4 exponent for basal
metabolic rate (West et al.,
1999a).
Criticisms and controversies
Since our original paper was published
(West et al., 1997), there
have been several criticisms (Darveau et
al., 2002; Dodds et al.,
2001; White and Seymour,
2003). Some of these revolve around matters of fact and
interpretation that still need to be resolved - such as the scaling of maximal
metabolic rates in mammals or the precise value of the exponent. Others claim
to provide empirical information or theoretical calculations that refute our
models. We have not found any of these latter criticisms convincing for two
reasons. First, most of them rest on single technical issues, for which there
are at least equally supportable alternative explanations, and some that are
simply incorrect. Furthermore, most of these have been concerned solely with
mammalian metabolic rate, so they fail to appreciate that our theory offers a
single parsimonious explanation, rooted in basic principles of biology,
physics and geometry, for an enormous variety of empirical scaling relations.
None of the criticisms offer alternative models for the complete design of
vascular networks or for the Mb3/4 scaling of
whole-organism metabolic rate. Here, we address some of the more salient
issues.
Scaling of metabolic rate: is it 3/4, 2/3 or some other number?
Some of the recent criticisms have centered around whether whole-organism
metabolic rate really does scale as Mb3/4
(Dodds et al., 2001;
White and Seymour, 2003).
Indeed, Kleiber himself (Kleiber,
1932,
1975) and many others
(Brody, 1945;
Calder, 1984;
McMahon and Bonner, 1983;
Niklas, 1994;
Peters, 1986;
Schmidt-Nielsen, 1984) had
expected that basal mammalian metabolic rates (BMR) should scale as
Mb2/3, reflecting the role of body surface area
in heat dissipation. Heusner
(1982) presented a statistical
analysis focusing on intra-specific comparisons and suggested that the
exponent was indeed 2/3 rather than 3/4, indicative of a simple Euclidean
surface rule. The statistical argument was strongly countered by Feldman and
McMahon (1983), and by Bartels
(1982), after which the debate
subsided and the ubiquity of quarter powers was widely accepted
(Calder, 1984;
McMahon and Bonner, 1983;
Peters, 1986;
Schmidt-Nielsen, 1984). In his
synthetic book on biological scaling, Schmidt-Nielsen seemed to have settled
the argument when he declared that `the slope of the metabolic regression
line for mammals is 0.75 or very close to it, and most definitely not
0.67' (Schmidt-Nielsen,
1984).
Arguments that the scaling of whole-organism metabolic rate is effectively
a Euclidean surface phenomenon were overshadowed by two lines of opposing
evidence. First, metabolic rates of many groups of ectothermic organisms,
whose body temperatures fluctuate closely with environmental temperatures, so
that control of heat dissipation is not an issue, were also shown to scale as
Mb3/4. Second, extensive work on temperature
regulation in endotherms elucidated powerful mechanisms for heat dissipation,
in which body surface area per se played an insignificant role
(Schmidt-Nielsen, 1984).
Recently, however, this controversy was resurrected by Dodds (2001) and by
White and Seymour (2003), who
concluded that a reanalysis of data supported 2/3, especially for smaller
mammals (<10 kg). These and earlier authors
(Heusner, 1982) argued for an
empirical exponent of 2/3 based on their reanalyses of large data sets, using
various criteria for excluding certain taxa and data, and employing
non-standard statistical procedures. For every such example, however, it is
possible to generate a counter-example using at least equally valid data and
statistical methods (Bartels,
1982; Feldman and McMahon,
1983; Savage et al.,
2004b). Observed deviations from perfect
Mb3/4 seems attributable to some combination of
elevated rates for the smallest mammals, as first observed empirically by
Calder (1984) and predicted
theoretically by our model (see above;
West et al., 1997), and
statistical artefacts due to small errors in precisely estimating the
characteristic body masses of species. Ironically, one might argue that
deviations from the 3/4 exponent for small mammals is another of the
successful predictions of our theory.
More telling than repeated reanalyses of the largely overlapping data on
basal metabolic rates of mammals would be a reanalysis of all of the multiple
studies of scaling of whole-organism metabolic rate in different groups of
organisms. Peters (1986)
published such a meta-analysis of the large number of studies available at the
time of his synthetic monograph. He obtained an approximately normal-shaped
frequency distribution of exponents, with a sharp peak at almost exactly 3/4.
Savage et al. (2004b) recently
performed a similar analysis, incorporating data from additional studies, and
obtained virtually identical results: the mean value of the exponent is
0.749±0.007, so the 95% confidence intervals include 3/4 but exclude
2/3 (Fig. 10). This analysis
was extended to a variety of other rates and times leading to similar results:
the mean value of the exponent for mass-specific rates was found to be
-0.247±0.011 and for times, 0.250±0.011, so both of these
clearly exclude a 1/3 exponent. It is worth emphasizing that these
meta-analyses include studies of a wide variety of processes in a broad range
of taxa (including ectotherms, invertebrates, and plants, as well as birds and
mammals) measured in a very large number of independent studies by many
different investigators working over more than 50 years. This kind of evidence
had earlier led Calder (1984)
to conclude that `Despite shortcomings and criticisms [including the lack
of a theoretical model], empirically most of the scaling does seem to fit
M1/4 scaling......', and Peters
(1986) to remark that
'......one cannot but wonder why the power formula, in general, and the
mass exponents of 3/4, 1/4, and -1/4, in particular, are so effective in
describing biological phenomena.' We see no reason to change this
assessment in the light of the very few recent studies that have once again
argued for Mb2/3 scaling.
|
Ours are whole-system models: how do other parameters scale?
Many critics lose sight of the fact that our theory generates a complete,
whole-system model for the structure and function of the mammalian arterial
system as well as quantitatively predicting many other unrelated biological
phenomena. The model quantifies the flow of blood from the heart to the
capillaries. It predicts the scaling exponents for 16 variables, including
blood volume, heart rate, stroke volume, blood pressure, radius of the aorta,
volume of tissue served by a capillary, number and density of capillaries,
dimensions of capillaries and oxygen affinity of hemoglobin
(Fig. 3;
West et al., 1997). No
alternative whole-system model has been developed that makes different
predictions.
The most serious theoretical criticism, by Dodds et al.
(2001), took issue with our
derivation of the 3/4 exponent for mammals based on an analysis of the
pulsatile circulatory systems Their calculation, however, naively minimized
the total complex impedance of the network, which includes analogs to
capacitance and inductance effects not directly associated with energy
dissipation. This is a subtle, but important, point. The meaningful, physical
quantity associated with energy dissipation due to viscous forces is the real
part of the impedance, and it is only this that must be minimized in order to
minimize the total energy dissipated. In addition, and as already emphasized
above, the total energy expended in a pulsatile system is the sum of two
contributions: the viscous energy dissipated (determined from the real part of
the impedance) and the loss due to reflections at branch points. Dodds et al.
neglected, however, to consider this critical latter effect and so failed to
impose impedance matching, thereby allowing arbitrary reflections at all
branch points, so the total energy expended is no longer minimised.
Consequently, they did not obtain area-preserving branching in large vessels
nor, therefore, a 3/4 exponent nor a realistic description of the flow. Put
simply, their criticism is invalid because they failed to correctly minimize
the total energy expended in the network.
For those who would have mammalian BMR scale as M2/3,
the onus is on them to explain the scaling of other components of the
metabolic resource supply systems In particular, they need to explain why
cardiac output and pulmonary exchange also scale as M3/4
in mammals (Schmidt-Nielsen,
1984). Heart rate (fH), stroke volume
(VS), respiration rate (Rl) and tidal
volume (VT) can all be measured with at least as much
precision and standardization as metabolic rates It is well established that
fH Rl
M-1/4, and VS
VT M, so the cardiac output,
fH xVS, scales as
M-1/4M=M3/4, and similarly
for the rate of respiratory ventilation, Rl
VT, again scaling as M3/4. Of course
this is not surprising if metabolic rate scales as M3/4,
because the rate of respiratory gas exchange in the lungs and the rate of
blood flow from the heart with its contained oxygen and fuel must match the
rate of metabolism of the tissues. There is, however, a serious unexplained
inconsistency in the quarter-power scalings of these components of the
circulatory and respiratory systems if metabolic rate scales as
M2/3.
Supply and demand at the cellular level: why do the cells care about the size of the body?
Darveau et al. (2002) and
Suarez et al. (2004)
criticized our theory as being `flawed' for implying that `there is a
single rate-limiting step or process that accounts for the b value in
equation (1)' (i.e. the allometric exponent for metabolic rate). As an
alternative, they suggested a much more complicated, multiple-causes,
allometric cascade model, in which metabolic rate is the sum of all
`ATP-utilising processes',
Bi:B=Bi. This sum simply
represents overall conservation of energy, so it must be correct if it is
carried out consistently. Therefore, it cannot be in conflict with our theory.
Darveau et al. incorporated many details of metabolic processes both at
whole-organism and at cellular-molecular levels in their sum: from pulmonary
capacity, alveolar ventilation and cardiac output to
Na+,K+-ATPase activity, protein synthesis and
capillary-mitochondria diffusion. All were added as if they were independent
and in parallel. However, many of these processes are primarily in series,
thereby leading to multiple-counting and therefore to a violation of energy
conservation. Each Bi was assumed to scale allometrically
as
so
,
where the ci(=ai/a) were
identified with conventional control coefficients defined as `the
fractional change in organismal flux divided by the fractional change in
capacity' of the ith contributing process. As such, the
ci are dimensionless. Unfortunately, however, it is
obvious that the ci as used by Darveau et al. in the above
equations cannot be dimensionless since the bi that were
used all have different values. Consequently, their results are inconsistent
and incorrect (West et al.,
2003).
Even without this fatal flaw, their model makes no a priori predictions about the scaling of metabolic rate, since no explanation is offered for the origin or values of the scaling exponents for the contributing processes, bi. If the sum is carried out correctly, it simply verifies the conservation of metabolic energy. From the Darveau et al. point of view the bi are simply inputs; from ours, they are potentially outputs determined from network constraints. It is surely no accident that almost all of the bi cluster around 0.75.
Although we take issue with the characterization of our theory as `single-cause' - and point out that it predicts the scaling of 16 variables for the mammalian cardiovascular system in addition to metabolic rate and, in addition, makes similar predictions for plants - we regard its relative simplicity as a strength rather than a weakness. We contend that for a given metabolic state, scaling of metabolic rate between different-sized organisms (that is, its relative value: M varying, but with a and ai fixed) is indeed constrained by the network and this is the origin of quarter-powers. However, the absolute rate of resource flow and power output (given by a and ai) within an individual organism (that is, with fixed M) is not rate-limited by the network: as in any transport system, changes in supply and demand cause the network flow to change accordingly. The concept of an absolute `single-cause' as used by Darveau et al. simply does not arise. Because of this, our model deliberately leaves out much of what is known about the biochemistry and physiology of metabolism at cellular-molecular levels.
More generally, analytic models are typically deliberate oversimplifications of more complex realities. They are intended to ignore many details, to capture just the most fundamental essence of a phenomenon, to provide a useful conceptual framework, and to make robust testable predictions. To make an analogy, the basic theory of Mendelian genetics still provides the conceptual foundation for most of modern population and evolutionary genetics, even though it does not incorporate much of what is known about genetics at the cellular (chromosomal) and molecular level. Indeed, for at least a century Mendel's laws have helped to guide the research into the structure and function of the genetic material. Mendel's laws ignore linkage, epistasis and crossing over, not to mention such features of genomic architecture as regulatory regions, introns and transposons. They can now be amended or extended to account for these phenomena if and when it is important to incorporate such details into a conceptual framework or an empirical analysis.
We therefore find it surprising that certain features of metabolic
processes at molecular and cellular levels
(Darveau et al., 2002) are
viewed as irreconcilable alternatives to our model. We view them as generally
consistent and complementary. So, for example, cellular-molecular processes
related to BMR generally scale close to M3/4 (but with
higher exponents for some processes linked more directly to maximal metabolic
rate (Weibel et al., 2004).
This is used as input to the `allometric cascade model' of Darveau et al.
(2002), who claim that the
M3/4 scaling of metabolic rate is determined by
demand-generated processes at the cellular-molecular level rather than by
supply-generated processes at the whole-organism level. We fail to see the
logic of this argument, which makes an explicit distinction between supply-
and demand-driven processes. We conjecture that metabolic systems at the
molecular, organelle and cellular levels, and the circulatory, respiratory,
and other network systems that supply metabolic requirements at the
whole-organism level, are co-adjusted and co-evolved so as to match supply to
demand and vice versa. More importantly, if whole-organism metabolic
rate is determined entirely by cellular and molecular processes, why should it
scale at all and why should it scale as M
