Life's Universal Scaling LawsBiological systems have evolved branching networks thattransport a variety of resources. We argue that commonproperties of those networks allow for a quantitative theory ofthe structure, organization, and dynamics of living systems.

Geoffrey B. West and James H. Brown

Nearly 100 years ago, the eminent biologist D'ArcyThompson began his wonderful book On Growth and

Form iCamhridge U. Press, 1917) by quoting Immanue!Kant. The philosopher had observed that "chemistry. .. wasa science but not Science . . . for that the criterion of trueScience lay in its relation to mathematics." Thompson thendeclared that, since a "mathematical chemistry" now ex-isted, chemistry was thereby elevated to Science; whereasbiology had remained qualitative, without mathematicalfoundations or principles, and so it was not yet Science.

Although few today would articulate Thompson's po-sition so provocatively, the spirit of his characterization re-mains to a large extent valid, despite the extraordinaryprogress during the intervening century. The basic ques-tion implicit in his discussion remains unanswered: Do bi-ological phenomena obey underlying universal laws of lifethat can be mathematized so that biology can be formu-lated as a predictive, quantitative science? Most would re-gard it as unlikely that scientists will ever discover "New-ton's laws of biology" that could lead to precise calculationsof detailed biological phenomena. Indeed, one could con-vincingly argue that the extraordinary complexity of mostbiological systems precludes such a possibility.

Nevertheless, it is reasonable to conjecture that thecoarse-grained behavior of living systems might obeyquantifiable universal laws that capture the systems' es-sential features. This more modest view presumes that, atevery organizational level, one can construct idealized bi-ological systems whose average properties are calculable.Such ideal constructs would provide a zeroth-order pointof departure for quantitatively understanding real biolog-ical systems, which can be viewed as manifesting "higher-order corrections" due to local environmental conditions orhistorical evolutionary divergence.

The search for universal quantitative laws of biologythat supplement or complement the Mendelian laws of in-heritance and the principle of natural selection mightseem to be a daunting task. After all, life is the most com-plex and diverse physical system in the universe, and asystematic science of complexity has yet to be developed.

Geoffrey West is a senior fellow at the Los Alamos NationalLaboratory and a distinguished research professor at the SantaFe Institute, both in New Mexico. Jim Brown is a distinguishedprofessor of biology at the University of New Mexico inAlbuquerque.

The life process covers more than 27orders of magnitude in mass—frommolecules of the genetic code andmetabolic machinery to whales and se-quoias—and the metabolic power re-quired to support life across that rangespans over 21 orders of magnitude.

Throughout those immenseranges, life uses basically the same

chemical constituents and reactions to create an amazingvariety of forms, processes, and dynamical behaviors. Alllife functions by transforming energy from physical orchemical sources into organic molecules that are metabo-lized to build, maintain, and reproduce complex, highly or-ganized systems. Understanding the origins, structures,and dynamics of living systems from molecules to the bios-phere is one of the grand challenges of modern science.Finding the universal principles that govern life's enor-mous diversity is central to understanding the nature oflife and to managing biological systems in such diversecontexts as medicine, agriculture, and the environment.

Allometric scaling lawsIn marked contrast to the amazing diversity and com-plexity of living organisms is the remarkable simplicity ofthe scaling behavior of key biological processes over abroad spectrum of phenomena and an immense range ofenergy and mass. Scaling as a manifestation of underly-ing dynamics and geometry is familiar throughout physics.It has been instrumental in helping scientists gain deeperinsights into problems ranging across the entire spectrumof science and technology, because scaling laws typicallyreflect underlying generic features and physical principlesthat are independent of detailed dynamics or specific char-acteristics of particular models. Phase transitions, chaos,the unification of the fundamental forces of nature, andthe discovery of quarks are a few of the more significantexamples in which scaling has illuminated important uni-versal principles or structure.

In biology, the observed scaling is typically a simplepower law: Y - Y^M^, where y is some observable, Y,, a con-stant, and M the mass of the organism.' •' Perhaps of evengreater significance, the exponent 6 almost invariably ap-proximates a simple multiple of 'A. Among the many fun-damental variables that obey such scaling laws^termed"allometric" by Julian Huxley''—are metabolic rate, lifespan, growth rate, heart rate, DNAnucleotide substitutionrate, lengths of aortas and genomes, tree height, mass ofcerebral grey matter, density of mitochondria, and con-centration of RNA.

The most studied of those variables is basal metabolicrate, first shown by Max Kleiber to scale as Af" for mam-mals and birds."' Figure 1 illustrates Kleiber's now 70-year-old data, which extend over ahout four orders of magni-tude in mass. Kleiber's work was generalized bysubsequent researchers to ectotherms (organisms whose

36 September 2004 Physics Today © 2004 American Institute ol Ptiysics, S-0031-9228-0409-010-6

Figure 1. The basal metabolic rale oE mam-mals and birds was originally plolted by MaxKleiber in 1932. In this reconstruction, theslope of the best straight-line tit is 0.74, illus-trating the scaling of metabolic rate with the'A power of mass. The diameters of thecircles represent estimated data errors of±10%. Present-day plots based on manyhundreds of data points support the ^kexponent, although evidence exists of adeviation to a smaller value for the smallestmammals. [Adapted from ref. 5.)

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body temperature is determined by their surroundings),unicellular organisms, and even plants. It was then fur-ther extended to intracellular levels, terminating at themitochondria] oxidase molecules (the respiratory machin-ery of aerobic metaholism). The metabolic exponent h ^ %is found over 27 orders of magnitude;*^ figure 2 shows dataspanning most of that range. Other examples of allomet-ric scaling include heart rate (6 - - V4, figure 3a), life span(6 - 'A), the radii of aortas and tree trunks Kb ^ %), uni-cellular genome lengths {h = V4, figure 3b), and RNA con-centration (6 - -'A).

An intriguing consequence of these "quarter-power"scaling laws is the emergence of invariant quantities,^which physicists recognize as usually reflecting funda-mental underlying constraints. For example, mammalianlife span increases as approximately J W \ whereas heartrate decreases as M ''', so the number of heartbeats perlifetime is approximately invariant (about 1.5 x 10^), in-

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dependent of size. Hearts are not funda-mental, hut the molecular machinery of aer-ohic metabolism is, and it has an analogousinvariant: the number of ATP (adenosinetriphosphate) molecules synthesized in alifetime (of order 10"*). Another examplearises in forest communities where popula-tion density decreases with individual bodysize as Af ''', whereas individual power use

increases as W^"-, thus the power used hy all individuals inany size class is invariant.^

The enormous amount of allometric scaling data ac-cumulated by the early 1980s was synthesized in fourbooks that convincingly showed the predominance of quar-ter powers across all scales and life forms. ' Although sev-eral mechanistic models were proposed, they focusedmostly on very specific features of a particular taxonomicgroup. For example, in his explanation of mammalianmetahohc rates, Thomas McMahon assumed the elasticsimilarity of limbs and the invariance of muscle speed,'whereas Mark Patterson addressed aquatic organismshased on the diffusion of respiratory gases.'' The broaderchallenge is to understand the ubiquity of quarter powersand to explain them in terms of unifying principles thatdetermine how life is organized and the constraints underwhich it has evolved.

Origins of scalingA general theory should provide a scheme for making quan-titative dynamical calculations in addition to explainingthe predominance of quarter powers. The kinds of prob-lems that a theory might address include. How many oxi-dase molecules and mitochondria are there in a cell? Whydo we live approximately 100 years, not a million years ora few weeks, and how is life span related to molecularscales? What are the flow rate, pulse rate, pressure, and

Figure 2. The V4-power law for the metabolic rate as afunction of mass is observed over 27 orders of magnitude.The masses covered in this plot range from those of indi-vidual mammals (blue), to unicellular organisms (green), touncoupled mammalian cells, mitochondria, and terminaloxidase molecules of the respiratory complex (red). Theblue and red lines indicate 'A-power scaling. The dashedline is a linear extrapolation that extends to masses belowthat of the shrew, the lightest mammal. In reference 6, itwas predicted that the extrapolation would intersect thedatum for an isolated cell in vitro, where the 'A-powerreemerges and extends to the cellular and intracellularlevels. (Adapted from ref. 6.)

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dimensions in any vessel of any circulatory system? Howmany trees of a given size are in a forest, how far apartare they, and how much energy flows in each branch? Whydoes an elephant sleep only 3 hours and a mouse 18?

Beginning in the late 1990s, we attempted to addresssuch questions, first with Brian Enquist and later withothers.'''"" The starting point was to recognize that highlycomplex, self-sustaining, reproducing, living structures re-quire close integration of enormous numbers of localizedmicroscopic units that need to be serviced in an approxi-mately "democratic" and efficient fashion. To solve thatchallenge, natural selection evolved hierarchical fractal-like branching networks that distribute energy, metabo-lites, and information from macroscopic reservoirs tomicroscopic sites. Examples include animal circulatorysystems, plant vascular systems, and ecosystem and in-tracellular networks. We proposed that scaling laws andthe generic coarse-grained dynamical behavior of biologi-cal systems reflect the constraints inherent in universalproperties of such networks. These constraints were pos-tulated as follows;• Networks service all local hiologically active regions inboth mature and growing biological systems. Such net-works are called space-filling.• The networks' terminal units are invariant within aclass or taxon.• Organisms evolve toward an optimal state in which theenergy required for resource distribution is minimized.

These properties, which characterize an idealized bio-logical organism, are presumed to he consequences of nat-ural selection. Thus, terminal units—the hasic huildinghlocks of the network in which energy and resources areexchanged—are not reconfigured as individuals grow fromnewborn to adult nor reinvented as new species evolve. Ex-amples of such units include capillaries, mitochondria,leaves, and chloroplasts. Analogous architectural terminalunits, such as electrical outlets or water faucets, are also

Figure 3. Simple scaling laws are not limited to meta-bolic rates, (a) A log-lofi plot of heart rate as a functionof body mass for a variety of mammals. The best straight-line fit has a slope very close to - ' / i . (Adapted fromref. 14.) (b) A log-log plot of genome length (number ofbase pairs) as a function of cell mass for a variety ofunicellular organisms. The best straight-line fit has aslope very close to 'A.

approximate invariants, independent of building size orlocation. The third postulate assumes that the continuousfeedback and fine-tuning implicit in natural selection leadto near-optimized systems. For example, of the infinitudeof space-filling circulatory systems with invariant terminalunits that could have evolved, the ones that did evolve min-imize cardiac output. Minimization principles are poten-tially very powerful because they can he expressed mathe-matically as equations that describe network dynamics.

Guided by the three postulates, we and our colleaguesbuilt on earlier work to derive analytic models of the mam-malian circulatory and respiratory systems and of plantvascular systems. The theory enables one to address thetypes of questions we raised at the beginning of this sec-tion and predicts quarter-power scaling of diverse biologi-cal phenomena even though the networks and associatedpumps are very different. It allowed us to derive manyscaling laws not only between organisms of varying sizebut also within an individual organism—for example, lawsthat relate the aorta to capillaries and growth laws thatconnect, say, a seedling to a giant sequoia. Where dataexist, one generally finds excellent agreement, and wherethey do not, the theory provides testable predictions.

Metabolic rateAerobic metabolism is fueled by oxygen, whose concentra-tion in hemoglobin is fixed. Consequently, the rate atwhich hlood flows through the cardiovascular system is aproxy for metabolic rate so that the properties of the cir-culatory network partially control metabolism. The re-quirement that the network be space-filling constrains thebranch lengths l^ to scale as ik^Jh ~ '^ '' within networks,where n is the branching ratio, k is the branching level,and the lowest-level branch is the aorta. The 3 in thebranching-ratio exponent reflects the dimensionality ofspace.

In 1997, we and Enquist derived an analytic solutionfor the entire network from the hydrodynamic and elas-ticity equations for hlood flow and vessel dynamics.'" Weassumed, for simplicity, that the network was symmetricand composed of cylindrical vessels and that the blood flowwas not turbulent. We also imposed the requirements thatthe network be space-filling and that dissipated energy heminimized.

Two factors independently contribute to energy loss:viscous energy dissipation, which is only important insmaller vessels, and energy reflected at branch points,which is eliminated by impedance matching. In large ves-sels such as arteries, viscous forces are negligible and theresulting pulsatile flow suffers little attenuation or dissi-pation. In that case, impedance matching leads to area-preserving branching. That is, the cross-sectional area ofthe daughter branches equals that of the parent, so radiiscale as r , Jr^ = n~"'^ and blood velocity remains constant.In small vessels such as capillaries and arterioles, thepulse is strongly damped by viscous forces, so-calledPoiseuille flow dominates, and significant energy is dissi-

38 September 2004 Physics Today http://www.physicstoday.org

Figure 4. Ceils in living organisms and cells cullured invitro have different metabolic rates. The plot shows the

metabolic rates of mammalian cells in vivo (blue) and invitro (red) as a function of organism mass M. While slill

in the body and constrained by vascular supply net-works, cellular metabolic rates scale as M '•' (blue line).

Cells removed from the body and cultured in vitro gener-ally take on a constani metabolic rale (red line) predictedby theory. Consistent with theory, the in vivo and in vitro

lines meet at the mass M,, ,, of a theoretical smallestmammal, which is close to that of a shrew.

(Adapted from rel. 6.)

pated. For such flow, minimization of dissipated energyleads to area-increasing branching with r +i/r - n "^, soblood slows down and almost ceases to flow in the capil-laries. Because r^^^lr^ changes continuously from n '' ton ' ^ as branching increases, the network is not strictlyself-similar. Nevertheless, the length ratio Z ., Jl^ does re-main constant throughout the network and the networkhas some fractal-like properties.

Allometric relations follow from the invariance of cap-illaries and the prediction from energy optimization thatthe total blood volume is proportional to the body mass, asobservations confirm. We derived the scaling of radii,lengths, and many other physiological characteristics andshowed them to have quarter-power exponents. Quantita-tive predictions for those and other characteristics of thecardiovascular system, such as the flow, pulse, and di-mensions in any branch of a mammal, are in good agree-ment with data.

The dominance of pulsatile flow, and consequently ofarea-preserving branching, is crucial for deriving quarterpowers and, in particular, the % power describing meta-bolic rate B. However, as body size decreases, tubes nar-row and viscosity plays an increasing role. Eventually,even major arteries become too constricted to support wavepropagation, and steady Poiseuille flow dominates. As a re-sult, branching becomes predominantly area increasingand metabolic rate becomes proportional to M, rather thanilf' . Networks with constricted arteries are highly ineffi-cient because energy is dissipated in all branches; a limit-ing-case animal whose network supported only steady flowwould have a beating heart but no pulse and would nothave evolved. The limiting-case idea allows one to esti-mate, in terms of fundamental parameters, the size of thesmallest animal. For mammals, theory predicts M,,,,,, ofabout 1 g. That's close to the mass of a shrew, which is in-deed the smallest mammal. Although no mammals existwith masses smaller than the shrew, a linear extrapola-tion of B to lower masses is meaningful: As figure 2 shows,the extrapolation intersects metabolic-power data at thelocation of an isolated mammalian cell, a tiny "mammalwithout a network."

Because of the changing roles of pulsatile andPoiseuille flow with body size, as mass decreases, the ex-ponent for B should depend weakly on M, exhibiting cal-culable deviations from 3/4 as observed.

From molecules to forestsMetabolism is organized at a number of levels, and at eachlevel new structures emerge. The result is a hierarchy ofnetworks, each with different physical characteristics andeffective degrees of freedom. Yet metabolic rate continuesto obey •V4-power scaling. That invariance is in contrast tothe analogous situation in physics. Scaling, as manifested

LOG MASS (g)

in structure functions or phase transitions, for example,persists from quarks through hadrons, atoms, and ulti-mately to matter. Yet no continuous universal behavioremerges: Each level manifests different scaling laws.

Metabolic energy is conserved as it flows through thehierarchy of sequential networks, each presumed to satisfythe same general principles and, therefore, the same quar-ter-power scaling. The continuity of flow imposes boundaryconditions between adjacent levels. Those conditions, inturn, lead to constraints on densities of the invariant ter-minal units, such as mitochondria and respiratory mole-cules, that interact between levels. So, for example, thetotal mitochondria] mass relative to body mass is correctlypredicted to be iM^-^^mJm^Mf'^ = 0.06 M ''*, where m,,, isthe mitochondrial mass, m^ is the average cell mass, and Mis in grams.

The control exercised by networks is further exempli-fied by culturing cells in vitro and so liberating them fromnetwork hegemony. Cells in vivo adjust their number ofmitochondria appropriately to the size of the host mammalas dictated by the resource supply networks. In vivo cellu-lar metabolic rate thereby scales as M'' •*, as seen in figure4. In vitro cultured cells from different mammals, however,are predicted to develop the same metabolic rate, about3 X 10 " watts. The figure shows that the in vivo and invitro values coincide at M ^ ,,, so cells in shrews work at al-most maximal output. No wonder shrews live short lives!

The calculations that yield quarter-power scaling de-pend only on generic network properties. The observationof such scaling at intracellular levels therefore suggeststhat subcellular structure and dynamics are constrainedby optimized space-filling, hierarchical networks. A majorchallenge, both theoretically and experimentally, is to un-derstand quantitatively the nature and structure of intra-cellular pathways, about which surprisingly little isknown.

Energy transported through the network fuels themetabolic machinery that maintains biological systems.

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In addition, that energy is used to grow new cells foradded tissue. Thus metabolic rate has two components,maintenance and growth, and can be expressed asB ^ N^B^ + E^dNJdt, where N^ is the number of cells, .6is the metabolic rate per cell in mature individuals, E^ isthe energy required to grow a cell, and t is time. The equa-tion gives a natural explanation for why we all stop grow-ing: The number of cells to be supported (A^ " M) in-creases faster than the rate at which they are suppliedwith energy (B ^ Af^"* ^ A , '*), which allows a determina-tion of the mass at maturity. Moreover, the parameters inthe growth equation are determined by fundamentalproperties of cells. As a consequence, one can derive a uni-versal scaling curve valid for the growth of any organism.As figure 5 shows, the curve fits the data well for a vari-ety of organisms, including mammals, birds, fish, andCrustacea. The idea behind the universal growth curvehas recently been extended by Caterina Guiot and col-leagues to parameterize tumor growth.'^ Thus, growthand life-history events are, in general, universal phe-nomena governed primarily by basic cellular propertiesand quarter-power scaling.

Temperature has a powerful effect on those basic prop-erties—indeed, on all of life—because of its exponential ef-fect on biochemical reaction rates. The Boltzmann factore -'*-' , where E is an activation energy, k is Boltzmann'sconstant, and T is the temperature, describes the effectquantitatively. Combined with network constraints, theBoltzmann factor predicts a joint universal mass and tem-perature scaling law for times and rates connected withmetabolism, including longevity and rates of growth, em-bryonic development, and DNA nucleotide substitution ingenomes. All times associated with metabolism shouldscale as Af'* e™^ and all rates as M"' ' e" '*'', with approx-imately the same value for E. Data covering fish, amphib-ians, aquatic insects, and zooplankton confirm the predic-tion. The best-fit value for E, about 0.65 eV, may beinterpreted as an average activation energy for the rate-limiting biochemical reactions.

Size and temperature considerations suggest a gen-eral definition of biological time determined by just twouniversal numbers, the scaling exponent V4 and the energy

Figure 5. The universality of growth is illustrated byplotting a dimensionless mass variable against a dimen-sionless time variable. Data for mammals, birds, fish,and Crustacea all lie on a single universal curve. Thequantity M is the mass of the organism at age t, m^ itsbirth mass, m its mature mass, and a is a parameterdetermined by theory in terms of basic cellular proper-ties that can be measured independently of growth data.(Adapted from ref. 11.)

E. When adjusted for size and temperature, all organisms,to a good approximation, run by the same universal clockwith similar metabolic, growth, and even evolutionaryrates.

The basic principles that yield allometric scaling inanimals may also be applied to plants, whose vascular sys-tems are effectively bundles of long microcapillary tubesdriven by a nonpulsatile pump. One can derive many scal-ing relationships within and between plants, includingthose for conductivity, fluid velocity, and, as first observedby Leonardo da Vinci, area-preserving branching. Meta-bolic rate scales as Af " and trunk diameter (like aorta di-ameter) scales as Af'**. Thus B scales as the square of trunkdiameter.

Steady-state forest ecosystems, too, can be treated asintegrated networks satisfying appropriate constraints.The network elements are not connected physically, butrather by the resources they use. Scaling in the forest asa whole mimics that in individual trees. So, for example,the number of trees as a function of trunk diameter scalesjust like the number of branches in an individual tree asa function of branch diameter. As figure 6 shows, both scal-ings are described by predicted inverse-square laws.

Criticisms and controversiesThe theoretical framework reviewed here has now beenpublished for long enough to have attracted a number of crit-ical responses. Broadly speaking, they fall into three cate-gories: data, technical issues, and conceptual questions.

In 1982, Alfred Heusner analyzed data on mammalianmetabolic rates and concluded that the power-law expo-nent was % rather tban %, indicative of a simple surface-to-volume rule. His suggestion met with strong opposition,and after the statistical debate subsided, the ubiquity ofquarter powers was widely accepted.^' Recently, however,the controversy was resurrected by Peter Dodds andcoworkers and by Craig White and Roger Seymour, all ofwhom concluded that a reanalysis of data indeed supportsthe % exponent, especially for the smaller mammals withmasses less than 10 kg that dominate the dataset.'-^ Morerecently we. Van Savage, and others, assembled and ana-lyzed the largest compilation of such data to date.''' Wegave equal weight to all sizes and found a best single-power fit of 0.74 (+0.02, -0.03), although we did confirmfor small mammals a trend towards smaller exponentsthat is in qualitative agreement with earlier theoreticalpredictions.'"

Although we authors disagree with critiques of the%-scaling exponents, we recognize that such criticismshave raised important empirical and statistical issuesabout data interpretation.

Dodds and colleagues also criticized our derivation ofthe 'Vi exponent for mammals. In their reanalysis, theyminimized the total impedance of the pulsatile circulatorynetwork rather than the total energy loss, which is the sumof viscous energy dissipated (related to the real part of the

40 September 2004 Physics Today http://www.physicstoday.org

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Figure 6. Trees and forests exhibit similar scaling behavior, (a) A typical tree exhibits multiple branching levels, (b) A pagefrom Leonardo da Vinci's notebooks illustrates his discovery of area-preserving 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 bythe straight lines. The data are from a Japanese forest. [Adapted from ref. 1 7.) (d) The number of trees of a given size as afunction of trunk diameter also follows a predicted inverse-square law (to within the error of the best-fit lines). The data,from a forest in Malaysia, were collected in 1947 (open circles) and 1981 (filled circles) and illustrate the robustness of theresult—the composition of the forest changed over the 34 years separating the data, but the inverse-square behaviorpersisted. (Adapted from ref. 18.)

impedance) and the loss due to reflections at branch points(related to the imaginary parti. They did not, however, im-pose impedance matching, so their analysis allowed re-flections at branch points and therefore did not minimizetotal energy loss. Consequently, they failed to obtain a %exponent.

Space filling, invariant terminal units, area-preserv-ing branching, and the linearity of network volume withM are sufficient to derive quarter powers. The last twoproperties follow from network dynamics by way of mini-mizing energy loss. Is there a more general argument, in-dependent of dynamics and hierarchical branching, thatdetermines the special number 4? Jayanth Banavar andcoworkers assumed, like us, that allometric relations re-flect network constraints.'"' But they proposed that quar-ter powers arise from a more general class of directed net-works that need not have fractal-like hierarchicalbranching. They showed that if the flow is sequential be-

tween the invariant units being supplied—cells or leaves,for example—rather than hierarchically terminating onsuch units, then a scaling exponent of % is obtained. Theirresult follows from minimizing flow rate rather than min-imizing energy loss and assumes, in agreement with ob-servation, that network volume scales linearly with bodymass. Afurther consequence of their model (and also ours)is that in d dimensions, the metabolic exponent is d/(d+ly.The special number 4 thus reflects the three-dimensional-ity of space.

The cascade model of Charles-Antoine Darveau andcolleagues provides another alternative.'^ In that model,the total metabolic rate is expressed as the sum of fun-damental, mostly intracellular contributions, such as ATPsynthesis. Darveau and coworkers assume each contribu-tion B^ obeys a power law; conservation of energy requiresthe metabolic rate to be B ^ S S, with each B^ equal to acoefficient c, multiplying the mass raised to a power a,.

http://www.physicstoday.org September 2004 Physics Today 41

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Using data for the c, and «„ they obtained a good fit forB, consistent with the %-power law, but rejected the ideathat transport networks constrain cellular behavior.However, they did not offer any first-principles explana-tion for why the B^ scale or why most of the exponents a,cluster around %.

The alternate models summarized in this article donot provide a general dynamical scheme or set of princi-ples for calculating detailed properties of specific systemsor phenomena, even at a coarse-grained level. They weredesigned almost exclusively to understand only the scal-ing of mammalian metabolic rates and do not address theextraordinarily diverse, interconnected, integrated body ofscaling phenomena across different species and within in-dividuals. They do, however, raise significant conceptualquestions about universality classes of biological networksand the minimal set of assumptions needed to construct ageneral quantitative theory of biological phenomena.

Scaling is a potent tool for revealing universal behav-ior and its corresponding underlying principles in anyphysical system. Ubiquitous quarter-power scaling issurely telling us something fundamental about biologicalsystems. Major technical and conceptual challenges re-main, including extensions to neural systems, intracellu-lar transport, evolutionary dynamics, and genomics. Oneof the big questions is. Why does the theory work so well?Does some fixed point or deep basin of attraction in the dy-namics of natural selection ensure that all life is organizedby a few fundamental principles and that energy is a primedeterminant of hiological structure and dynamics amongall possible variables?

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American Library, New York (1983).2. K. Schmidt-Nielsen, Scaling: Why Is Animal Size So Impor-

tant?, Cambridge U. Press, New York (1984); R. H. Peters,The Ecological Implications of Body Size, Cambridge U,Press, New York (1983); W. A. Calder'lII, Size. Function andLife History, Harvard U. Press, Cambridge, MA (1984).

3. K. J. Niklas, Plant Allometry: The Scaling of Form andProcess, U. of Chicago Press, Chicago (1994); J. H. Brown,G. B. West, eds.. Scaling in Biology, Oxford U. Press, NewYork (2000).

4. J. S. Huxley, Problems of Relative Growth, Dial Press, NewYork (1932).

5. M. Kleiber, The Fire of Life: An Introduction to Animal Ener-getics, Robert E. Krieger, Huntington, NY (1975),

6. G. B, West, W. H. Woodruff, J, H, Brown, Proc. Nail. Acad.Sci. f/SA 99, 2473(2002),

7. E. L, Charnov, Life History Invariants: Some Explorations ofSymmetry in Evolutionary Ecology, Oxford U. Press, NewYork (1993); M. Ya, Azbel, Proc. Natl. Acad. Sci. USA 91,12453(1994),

8. B. J. Enquist, K. J. Niklas, Nature 410, 655 (2001).9. M. R. Patterson, Science 255, 1421 (1992).

10. G. B. West, J. H. Brown, B. J. Enquist, Science 276, 122(1997); J. F. Gillooly et al., Nature 417, 70 (2002). See also R.Lewin, Neiv Sci. 162, 34 (1999).

11. G. B. West, J. H, Brown, B. J, Enquist, Nature 413, 628 (2001).12. C. Guiot et al., J. Theor. Biol. 225, 147 (2003).13. P. S. Dodds, D. H. Rotbman, J. S. Weitz, J. Theor. Bio. 209, 9

(2001); C, R. White, R, S, Seymour, Proc. Natl. Acad. Sci.USA 100, 4046 (2003),

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42 September 2004 Physics Today