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Unformatted text preview: Life’s Universal Scaling Laws
The life process covers more than 27
orders of magnitude in mass—from
molecules of the genetic code and
metabolic machinery to whales and sequoias—and the metabolic power required to support life across that range
spans over 21 orders of magnitude.
Throughout those immense
ranges, life uses basically the same
chemical constituents and reactions to create an amazing
variety of forms, processes, 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. Understanding the origins, structures,
and dynamics of living systems from molecules to the biosphere is one of the grand challenges of modern science.
Finding the universal principles that govern life’s enormous diversity is central to understanding the nature of
life and to managing biological systems in such diverse
contexts as medicine, agriculture, and the environment. Biological systems have evolved branching networks that
transport a variety of resources. We argue that common
properties of those networks allow for a quantitative theory of
the structure, organization, and dynamics of living systems.
Geoffrey B. West and James H. Brown
years ago, the eminent biologist D’Arcy
Nearly 100 began his wonderful by quotingGrowth and
Thompson
book On
Form (Cambridge U. Press, 1917)
Immanuel
Kant. The philosopher had observed that “chemistry . . . was
a science but not Science . . . for that the criterion of true
Science lay in its relation to mathematics.” Thompson then
declared that, since a “mathematical chemistry” now existed, chemistry was thereby elevated to Science; whereas
biology had remained qualitative, without mathematical
foundations or principles, and so it was not yet Science.
Although few today would articulate Thompson’s position so provocatively, the spirit of his characterization remains to a large extent valid, despite the extraordinary
progress during the intervening century. The basic question implicit in his discussion remains unanswered: Do biological phenomena obey underlying universal laws of life
that can be mathematized so that biology can be formulated as a predictive, quantitative science? Most would regard it as unlikely that scientists will ever discover “Newton’s laws of biology” that could lead to precise calculations
of detailed biological phenomena. Indeed, one could convincingly argue that the extraordinary complexity of most
biological systems precludes such a possibility.
Nevertheless, it is reasonable to conjecture that the
coarsegrained behavior of living systems might obey
quantifiable universal laws that capture the systems’ essential features. This more modest view presumes that, at
every organizational level, one can construct idealized biological systems whose average properties are calculable.
Such ideal constructs would provide a zerothorder point
of departure for quantitatively understanding real biological systems, which can be viewed as manifesting “higherorder corrections” due to local environmental conditions or
historical evolutionary divergence.
The search for universal quantitative laws of biology
that supplement or complement the Mendelian laws of inheritance and the principle of natural selection might
seem to be a daunting task. After all, life is the most complex and diverse physical system in the universe, and a
systematic science of complexity has yet to be developed.
Geoffrey West is a senior fellow at the Los Alamos National
Laboratory and a distinguished research professor at the Santa
Fe Institute, both in New Mexico. Jim Brown is a distinguished
professor of biology at the University of New Mexico in
Albuquerque. 36 September 2004 Physics Today Allometric scaling laws
In marked contrast to the amazing diversity and complexity of living organisms is the remarkable simplicity of
the scaling behavior of key biological processes over a
broad spectrum of phenomena and an immense range of
energy and mass. Scaling as a manifestation of underlying dynamics and geometry is familiar throughout physics.
It has been instrumental in helping scientists gain deeper
insights into problems ranging across the entire spectrum
of science and technology, because scaling laws typically
reflect underlying generic features and physical principles
that are independent of detailed dynamics or specific characteristics of particular models. Phase transitions, chaos,
the unification of the fundamental forces of nature, and
the discovery of quarks are a few of the more significant
examples in which scaling has illuminated important universal principles or structure.
In biology, the observed scaling is typically a simple
power law: Y ⊂ Y0 M b, where Y is some observable, Y0 a constant, and M the mass of the organism.1⊗3 Perhaps of even
greater significance, the exponent b almost invariably approximates a simple multiple of 1/4. Among the many fundamental variables that obey such scaling laws—termed
“allometric” by Julian Huxley4—are metabolic rate, life
span, growth rate, heart rate, DNA nucleotide substitution
rate, lengths of aortas and genomes, tree height, mass of
cerebral grey matter, density of mitochondria, and concentration of RNA.
The most studied of those variables is basal metabolic
rate, first shown by Max Kleiber to scale as M3/4 for mammals and birds.5 Figure 1 illustrates Kleiber’s now 70yearold data, which extend over about four orders of magnitude in mass. Kleiber’s work was generalized by
subsequent researchers to ectotherms (organisms whose
© 2004 American Institute of Physics, S0031922804090106 Figure 1. The basal metabolic rate of mammals and birds was originally plotted by Max
Kleiber in 1932. In this reconstruction, the
slope of the best straightline fit is 0.74, illustrating the scaling of metabolic rate with the
3/4 power of mass. The diameters of the
circles represent estimated data errors of
10%. Presentday plots based on many
hundreds of data points support the 3/4
exponent, although evidence exists of a
deviation to a smaller value for the smallest
mammals. (Adapted from ref. 5.) LOG METABOLIC RATE (kcal/day) 4 3 –1 0 1
LOG MASS (kg) 5 LOG METABOLIC RATE (W) Shrew
Elephant
Mammals Steer Steer 2 body temperature is determined by their surroundings),
unicellular organisms, and even plants. It was then further extended to intracellular levels, terminating at the
mitochondrial oxidase molecules (the respiratory machinery of aerobic metabolism). The metabolic exponent b 3/4
is found over 27 orders of magnitude;6 figure 2 shows data
spanning most of that range. Other examples of allometric scaling include heart rate (b ⊗1/4, figure 3a), life span
(b 1/4), the radii of aortas and tree trunks (b 3/8), unicellular genome lengths (b 1/4, figure 3b), and RNA concentration (b ⊗1/4).
An intriguing consequence of these “quarterpower”
scaling laws is the emergence of invariant quantities,7
which physicists recognize as usually reflecting fundamental underlying constraints. For example, mammalian
life span increases as approximately M1/4, whereas heart
rate decreases as M⊗1/4, so the number of heartbeats per
lifetime is approximately invariant (about 1.5 × 109), in 0 dependent of size. Hearts are not fundamental, but the molecular machinery of aerobic metabolism is, and it has an analogous
invariant: the number of ATP (adenosine
triphosphate) molecules synthesized in a
3
lifetime (of order 1016). Another example
arises in forest communities where population density decreases with individual body
size as M⊗3/4, whereas individual power use
increases as M3/4; thus the power used by all individuals in
any size class is invariant.8
The enormous amount of allometric scaling data accumulated by the early 1980s was synthesized in four
books that convincingly showed the predominance of quarter powers across all scales and life forms.1,2 Although several mechanistic models were proposed, they focused
mostly on very specific features of a particular taxonomic
group. For example, in his explanation of mammalian
metabolic rates, Thomas McMahon assumed the elastic
similarity of limbs and the invariance of muscle speed,1
whereas Mark Patterson addressed aquatic organisms
based on the diffusion of respiratory gases.9 The broader
challenge is to understand the ubiquity of quarter powers
and to explain them in terms of unifying principles that
determine how life is organized and the constraints under
which it has evolved.
Cow Sheep
Woman
Man Dog Dog Hen Dove 1 Rat
Rat
Pigeon 2 Origins of scaling
A general theory should provide a scheme for making quantitative dynamical calculations in addition to explaining
the predominance of quarter powers. The kinds of problems that a theory might address include, How many oxidase molecules and mitochondria are there in a cell? Why
do we live approximately 100 years, not a million years or
a few weeks, and how is life span related to molecular
scales? What are the flow rate, pulse rate, pressure, and –5 –10
Average mammalian
cell, in culture –15
Respiratory
–20 complex –20 –15 Mitochondrion
(mammalian
myocyte) –10 –5
0
LOG MASS (g) http://www.physicstoday.org 5 10 Figure 2. The 3/4power law for the metabolic rate as a
function of mass is observed over 27 orders of magnitude.
The masses covered in this plot range from those of individual mammals (blue), to unicellular organisms (green), to
uncoupled mammalian cells, mitochondria, and terminal
oxidase molecules of the respiratory complex (red). The
blue and red lines indicate 3/4power scaling. The dashed
line is a linear extrapolation that extends to masses below
that of the shrew, the lightest mammal. In reference 6, it
was predicted that the extrapolation would intersect the
datum for an isolated cell in vitro, where the 3/4power
reemerges and extends to the cellular and intracellular
levels. (Adapted from ref. 6.)
September 2004 Physics Today 37 LOG HEART RATE (beats/min)
LOG GENOME LENGTH 3.0 Figure 3. Simple scaling laws are not limited to metabolic rates. (a) A log–log plot of heart rate as a function
of body mass for a variety of mammals. The best straightline fit has a slope very close to ⊗1/4. (Adapted from
ref. 14.) (b) A log–log plot of genome length (number of
base pairs) as a function of cell mass for a variety of
unicellular organisms. The best straightline fit has a
slope very close to 1/4. a 2.6
2.2
1.8
1.4
1.0 8.0 0 2 4
LOG MASS (g) 6 8 b 7.5 Nonphotosynthetic prokaryotes
Cyanophyta 7.0
6.5
6.0
5.5
5.0
–15 –14 –13
–12
–11
LOG MASS (g) –10 –9 dimensions in any vessel of any circulatory system? How
many trees of a given size are in a forest, how far apart
are they, and how much energy flows in each branch? Why
does an elephant sleep only 3 hours and a mouse 18?
Beginning in the late 1990s, we attempted to address
such questions, first with Brian Enquist and later with
others.6,10,11 The starting point was to recognize that highly
complex, selfsustaining, reproducing, living structures require close integration of enormous numbers of localized
microscopic units that need to be serviced in an approximately “democratic” and efficient fashion. To solve that
challenge, natural selection evolved hierarchical fractallike branching networks that distribute energy, metabolites, and information from macroscopic reservoirs to
microscopic sites. Examples include animal circulatory
systems, plant vascular systems, and ecosystem and intracellular networks. We proposed that scaling laws and
the generic coarsegrained dynamical behavior of biological systems reflect the constraints inherent in universal
properties of such networks. These constraints were postulated as follows:
Networks service all local biologically active regions in
both mature and growing biological systems. Such networks are called spacefilling.
The networks’ terminal units are invariant within a
class or taxon.
Organisms evolve toward an optimal state in which the
energy required for resource distribution is minimized.
These properties, which characterize an idealized biological organism, are presumed to be consequences of natural selection. Thus, terminal units—the basic building
blocks of the network in which energy and resources are
exchanged—are not reconfigured as individuals grow from
newborn to adult nor reinvented as new species evolve. Examples of such units include capillaries, mitochondria,
leaves, and chloroplasts. Analogous architectural terminal
units, such as electrical outlets or water faucets, are also
38 September 2004 Physics Today approximate invariants, independent of building size or
location. The third postulate assumes that the continuous
feedback and finetuning implicit in natural selection lead
to nearoptimized systems. For example, of the infinitude
of spacefilling circulatory systems with invariant terminal
units that could have evolved, the ones that did evolve minimize cardiac output. Minimization principles are potentially very powerful because they can be expressed mathematically as equations that describe network dynamics.
Guided by the three postulates, we and our colleagues
built on earlier work to derive analytic models of the mammalian circulatory and respiratory systems and of plant
vascular systems. The theory enables one to address the
types of questions we raised at the beginning of this section and predicts quarterpower scaling of diverse biological phenomena even though the networks and associated
pumps are very different. It allowed us to derive many
scaling laws not only between organisms of varying size
but also within an individual organism—for example, laws
that relate the aorta to capillaries and growth laws that
connect, say, a seedling to a giant sequoia. Where data
exist, one generally finds excellent agreement, and where
they do not, the theory provides testable predictions. Metabolic rate
Aerobic metabolism is fueled by oxygen, whose concentration in hemoglobin is fixed. Consequently, the rate at
which blood flows through the cardiovascular system is a
proxy for metabolic rate so that the properties of the circulatory network partially control metabolism. The requirement that the network be spacefilling constrains the
branch lengths lk to scale as lk⊕1/lk ⊂ n⊗1/3 within networks,
where n is the branching ratio, k is the branching level,
and the lowestlevel branch is the aorta. The 3 in the
branchingratio exponent reflects the dimensionality of
space.
In 1997, we and Enquist derived an analytic solution
for the entire network from the hydrodynamic and elasticity equations for blood flow and vessel dynamics.10 We
assumed, for simplicity, that the network was symmetric
and composed of cylindrical vessels and that the blood flow
was not turbulent. We also imposed the requirements that
the network be spacefilling and that dissipated energy be
minimized.
Two factors independently contribute to energy loss:
viscous energy dissipation, which is only important in
smaller vessels, and energy reflected at branch points,
which is eliminated by impedance matching. In large vessels such as arteries, viscous forces are negligible and the
resulting pulsatile flow suffers little attenuation or dissipation. In that case, impedance matching leads to areapreserving branching. That is, the crosssectional area of
the daughter branches equals that of the parent, so radii
scale as rk⊕1/rk ⊂ n⊗1/2 and blood velocity remains constant.
In small vessels such as capillaries and arterioles, the
pulse is strongly damped by viscous forces, socalled
Poiseuille flow dominates, and significant energy is dissihttp://www.physicstoday.org pated. For such flow, minimization of dissipated energy
leads to areaincreasing branching with rk⊕1/rk ⊂ n⊗1/3, so
blood slows down and almost ceases to flow in the capillaries. Because rk⊕1/rk changes continuously from n⊗1/2 to
n⊗1/3 as branching increases, the network is not strictly
selfsimilar. Nevertheless, the length ratio lk⊕1/lk does remain constant throughout the network and the network
has some fractallike properties.
Allometric relations follow from the invariance of capillaries and the prediction from energy optimization that
the total blood volume is proportional to the body mass, as
observations confirm. We derived the scaling of radii,
lengths, and many other physiological characteristics and
showed them to have quarterpower exponents. Quantitative predictions for those and other characteristics of the
cardiovascular system, such as the flow, pulse, and dimensions in any branch of a mammal, are in good agreement with data.
The dominance of pulsatile flow, and consequently of
areapreserving branching, is crucial for deriving quarter
powers and, in particular, the 3/4 power describing metabolic rate B. However, as body size decreases, tubes narrow and viscosity plays an increasing role. Eventually,
even major arteries become too constricted to support wave
propagation, and steady Poiseuille flow dominates. As a result, branching becomes predominantly area increasing
and metabolic rate becomes proportional to M, rather than
M3/4. Networks with constricted arteries are highly inefficient because energy is dissipated in all branches; a limitingcase animal whose network supported only steady flow
would have a beating heart but no pulse and would not
have evolved. The limitingcase idea allows one to estimate, in terms of fundamental parameters, the size of the
smallest animal. For mammals, theory predicts Mmin of
about 1 g. That’s close to the mass of a shrew, which is indeed the smallest mammal. Although no mammals exist
with masses smaller than the shrew, a linear extrapolation of B to lower masses is meaningful: As figure 2 shows,
the extrapolation intersects metabolicpower data at the
location of an isolated mammalian cell, a tiny “mammal
without a network.”
Because of the changing roles of pulsatile and
Poiseuille flow with body size, as mass decreases, the exponent for B should depend weakly on M, exhibiting calculable deviations from 3/4 as observed. From molecules to forests
Metabolism is organized at a number of levels, and at each
level new structures emerge. The result is a hierarchy of
networks, each with different physical characteristics and
effective degrees of freedom. Yet metabolic rate continues
to obey 3/4power scaling. That invariance is in contrast to
the analogous situation in physics. Scaling, as manifested
http://www.physicstoday.org –9 –10 LOG METABOLIC RATE (W) Figure 4. Cells in living organisms and cells cultured in
vitro have different metabolic rates. The plot shows the
metabolic rates of mammalian cells in vivo (blue) and in
vitro (red) as a function of organism mass M. While still
in the body and constrained by vascular supply networks, cellular metabolic rates scale as M⊗1/4 (blue line).
Cells removed from the body and cultured in vitro generally take on a constant metabolic rate (red line) predicted
by theory. Consistent with theory, the in vivo and in vitro
lines meet at the mass Mmin of a theoretical smallest
mammal, which is close to that of a shrew.
(Adapted from ref. 6.) M = Mmin –11 –12 –13
0 2 4
LOG MASS (g) 6 8 in structure functions or phase transitions, for example,
persists from quarks through hadrons, atoms, and ultimately to matter. Yet no continuous universal behavior
emerges: Each level manifests different scaling laws.
Metabolic energy is conserved as it flows through the
hierarchy of sequential networks, each presumed to satisfy
the same general principles and, therefore, the same quarterpower scaling. The continuity of flow imposes boundary
conditions between adjacent levels. Those conditions, in
turn, lead to constraints on densities of the invariant terminal units, such as mitochondria and respiratory molecules, that interact between levels. So, for example, the
total mitochondrial mass relative to body mass is correctly
predicted to be (Mmin mm /mc M )1/4 0.06 M⊗1/4, where mm is
the mitochondrial mass, mc is the average cell mass, and M
is in grams.
The control exercised by networks is further exemplified by culturing cells in vitro and so liberating them from
network hegemony. Cells in vivo adjust their number of
mitochondria appropriately to the size of the host mammal
as dictated by the resource supply networks. In vivo cellular metabolic rate thereby scales as M⊗1/4, as seen in figure
4. In vitro cultured cells from different mammals, however,
are predicted to develop the same metabolic rate, about
3 × 10⊗11 watts. The figure shows that the in vivo and in
vitro values coincide at Mmin, so cells in shrews work at almost maximal output. No wonder shrews live short lives!
The calculations that yield quarterpower scaling depend only on generic network properties. The observation
of such scaling at intracellular levels therefore suggests
that subcellular structure and dynamics are constrained
by optimized spacefilling, hierarchical networks. A major
challenge, both theoretically and experimentally, is to understand quantitatively the nature and structure of intracellular pathways, about which surprisingly little is
known.
Energy transported through the network fuels the
metabolic machinery that maintains biological systems.
September 2004 Physics Today 39 Figure 5. The universality of growth is illustrated by
plotting a dimensionless mass variable against a dimensionless time variable. Data for mammals, birds, fish,
and crustacea all lie on a single universal curve. The
quantity M is the mass of the organism at age t, m0 its
birth mass, m its mature mass, and a is a parameter
determined by theory in terms of basic cellular properties that can be measured independently of growth data.
(Adapted from ref. 11.) DIMENSIONLESS MASS (M/m)1/4 1.25 1.00
Swine
Shrew
Rabbit
Cod
Rat
Guinea pig
Shrimp
Salmon
Guppy
Chicken
Robin
Heron
Cow 0.75 0.50 0.25 0.00
0 2 4
6
(at/4m1/4) – [1 – (m0 /m)1/4]
DIMENSIONLESS TIME 8 10 In addition, that energy is used to grow new cells for
added tissue. Thus metabolic rate has two components,
maintenance and growth, and can be expressed as
B ⊂ Nc Bc ⊕ Ecd Nc /d t, where Nc is the number of cells, Bc
is the metabolic rate per cell in mature individuals, Ec is
the energy required to grow a cell, and t is time. The equation gives a natural explanation for why we all stop growing: The number of cells to be supported (Nc } M) increases faster than the rate at which they are supplied
with energy (B } M3/4 } Nc3/4), which allows a determination of the mass at maturity. Moreover, the parameters in
the growth equation are determined by fundamental
properties of cells. As a consequence, one can derive a universal scaling curve valid for the growth of any organism.
As figure 5 shows, the curve fits the data well for a variety of organisms, including mammals, birds, fish, and
crustacea. The idea behind the universal growth curve
has recently been extended by Caterina Guiot and colleagues to parameterize tumor growth.12 Thus, growth
and lifehistory events are, in general, universal phenomena governed primarily by basic cellular properties
and quarterpower scaling.
Temperature has a powerful effect on those basic properties—indeed, on all of life—because of its exponential effect on biochemical reaction rates. The Boltzmann factor
e⊗E/kT, where E is an activation energy, k is Boltzmann’s
constant, and T is the temperature, describes the effect
quantitatively. Combined with network constraints, the
Boltzmann factor predicts a joint universal mass and temperature scaling law for times and rates connected with
metabolism, including longevity and rates of growth, embryonic development, and DNA nucleotide substitution in
genomes. All times associated with metabolism should
scale as M1/4 eE/kT and all rates as M⊗1/4 e⊗E/kT, with approximately the same value for E. Data covering fish, amphibians, aquatic insects, and zooplankton confirm the prediction. The bestfit value for E, about 0.65 eV, may be
interpreted as an average activation energy for the ratelimiting biochemical reactions.
Size and temperature considerations suggest a general definition of biological time determined by just two
universal numbers, the scaling exponent 1/4 and the energy
40 September 2004 Physics Today E. When adjusted for size and temperature, all organisms,
to a good approximation, run by the same universal clock
with similar metabolic, growth, and even evolutionary
rates.
The basic principles that yield allometric scaling in
animals may also be applied to plants, whose vascular systems are effectively bundles of long microcapillary tubes
driven by a nonpulsatile pump. One can derive many scaling relationships within and between plants, including
those for conductivity, fluid velocity, and, as first observed
by Leonardo da Vinci, areapreserving branching. Metabolic rate scales as M3/4 and trunk diameter (like aorta diameter) scales as M3/8. Thus B scales as the square of trunk
diameter.
Steadystate forest ecosystems, too, can be treated as
integrated networks satisfying appropriate constraints.
The network elements are not connected physically, but
rather by the resources they use. Scaling in the forest as
a whole mimics that in individual trees. So, for example,
the number of trees as a function of trunk diameter scales
just like the number of branches in an individual tree as
a function of branch diameter. As figure 6 shows, both scalings are described by predicted inversesquare laws. Criticisms and controversies
The theoretical framework reviewed here has now been
published for long enough to have attracted a number of critical responses. Broadly speaking, they fall into three categories: data, technical issues, and conceptual questions.
In 1982, Alfred Heusner analyzed data on mammalian
metabolic rates and concluded that the powerlaw exponent was 2/3 rather than 3/4, indicative of a simple surfacetovolume rule. His suggestion met with strong opposition,
and after the statistical debate subsided, the ubiquity of
quarter powers was widely accepted.2 Recently, however,
the controversy was resurrected by Peter Dodds and
coworkers and by Craig White and Roger Seymour, all of
whom concluded that a reanalysis of data indeed supports
the 2/3 exponent, especially for the smaller mammals with
masses less than 10 kg that dominate the dataset.13 More
recently we, Van Savage, and others, assembled and analyzed the largest compilation of such data to date.14 We
gave equal weight to all sizes and found a best singlepower fit of 0.74 (⊕0.02, ⊗0.03), although we did confirm
for small mammals a trend towards smaller exponents
that is in qualitative agreement with earlier theoretical
predictions.10
Although we authors disagree with critiques of the
3/4scaling exponents, we recognize that such criticisms
have raised important empirical and statistical issues
about data interpretation.
Dodds and colleagues also criticized our derivation of
the 3/4 exponent for mammals. In their reanalysis, they
minimized the total impedance of the pulsatile circulatory
network rather than the total energy loss, which is the sum
of viscous energy dissipated (related to the real part of the
http://www.physicstoday.org a b d 3.0 LOG NUMBER OF TREES c 100 NUMBER 50 10
5 2.5 2.0 1.5 1.0 0.5
0.5
0.1 0.5
1
DIAMETER (cm) 5 10 1.0 1.5 2.0 2.5 3.0 LOG TRUNK DIAMETER (cm) Figure 6. Trees and forests exhibit similar scaling behavior. (a) A typical tree exhibits multiple branching levels. (b) A page
from Leonardo da Vinci’s notebooks illustrates his discovery of areapreserving branching. (c) The number of branches
(closed circles) and roots (open circles) in a tree varies roughly as the predicted inverse square of the diameter, indicated by
the straight lines. The data are from a Japanese forest. (Adapted from ref. 17.) (d) The number of trees of a given size as a
function of trunk diameter also follows a predicted inversesquare law (to within the error of the bestfit lines). The data,
from a forest in Malaysia, were collected in 1947 (open circles) and 1981 (filled circles) and illustrate the robustness of the
result—the composition of the forest changed over the 34 years separating the data, but the inversesquare behavior
persisted. (Adapted from ref. 18.) impedance) and the loss due to reflections at branch points
(related to the imaginary part). They did not, however, impose impedance matching, so their analysis allowed reflections at branch points and therefore did not minimize
total energy loss. Consequently, they failed to obtain a 3/4
exponent.
Space filling, invariant terminal units, areapreserving branching, and the linearity of network volume with
M are sufficient to derive quarter powers. The last two
properties follow from network dynamics by way of minimizing energy loss. Is there a more general argument, independent of dynamics and hierarchical branching, that
determines the special number 4? Jayanth Banavar and
coworkers assumed, like us, that allometric relations reflect network constraints.15 But they proposed that quarter powers arise from a more general class of directed networks that need not have fractallike hierarchical
branching. They showed that if the flow is sequential behttp://www.physicstoday.org tween the invariant units being supplied—cells or leaves,
for example—rather than hierarchically terminating on
such units, then a scaling exponent of 3/4 is obtained. Their
result follows from minimizing flow rate rather than minimizing energy loss and assumes, in agreement with observation, that network volume scales linearly with body
mass. A further consequence of their model (and also ours)
is that in d dimensions, the metabolic exponent is d/(d⊕1):
The special number 4 thus reflects the threedimensionality of space.
The cascade model of CharlesAntoine Darveau and
colleagues provides another alternative.16 In that model,
the total metabolic rate is expressed as the sum of fundamental, mostly intracellular contributions, such as ATP
synthesis. Darveau and coworkers assume each contribution Bi obeys a power law; conservation of energy requires
the metabolic rate to be B ⊂ S Bi with each Bi equal to a
coefficient ci multiplying the mass raised to a power ai.
September 2004 Physics Today 41 Using data for the ci and ai, they obtained a good fit for
B, consistent with the 3/4power law, but rejected the idea
that transport networks constrain cellular behavior.
However, they did not offer any firstprinciples explanation for why the Bi scale or why most of the exponents ai
cluster around 3/4.
The alternate models summarized in this article do
not provide a general dynamical scheme or set of principles for calculating detailed properties of specific systems
or phenomena, even at a coarsegrained level. They were
designed almost exclusively to understand only the scaling of mammalian metabolic rates and do not address the
extraordinarily diverse, interconnected, integrated body of
scaling phenomena across different species and within individuals. They do, however, raise significant conceptual
questions about universality classes of biological networks
and the minimal set of assumptions needed to construct a
general quantitative theory of biological phenomena.
Scaling is a potent tool for revealing universal behavior and its corresponding underlying principles in any
physical system. Ubiquitous quarterpower scaling is
surely telling us something fundamental about biological
systems. Major technical and conceptual challenges remain, including extensions to neural systems, intracellular transport, evolutionary dynamics, and genomics. One
of the big questions is, Why does the theory work so well?
Does some fixed point or deep basin of attraction in the dynamics of natural selection ensure that all life is organized
by a few fundamental principles and that energy is a prime
determinant of biological structure and dynamics among
all possible variables? References
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The Ecological Implications of Body Size, Cambridge U.
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(1997); J. F. Gillooly et al., Nature 417, 70 (2002). See also R.
Lewin, New Sci. 162, 34 (1999).
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This note was uploaded on 10/25/2010 for the course BIOPHYS 2090 taught by Professor R.r.l during the Fall '10 term at Maple Springs Baptist Bible College.
 Fall '10
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