p888, which contains a Bam HI to Not I fragmentencoding a full-length profilin cDNA (16); p989,which encodes a mutant form of profilin, Pfy1p-3,lacking the last three amino acids (18); p890, whichcontains the Bgl II to Stu I fragment from p182 (26),encoding Bni1p(1227–1397); p813, which con-tains the Bgl II to Not I fragment from p182, encod-ing Bni1p(1414–1953); and p951, which containsthe Hpa I to Not I fragment from p182, encodingBni1p(1647–1953). The pJG4-5–derived plasmidswere p561, which contains the Bam HI to Not Ifragment from p532 (26), encoding Bni1p(1–1953);p717, which contains the Bam HI to Eco47 III frag-ment from p532, encoding Bni1p(1–1214); p558,which contains the Eco 47III to Not I fragment fromp182, encoding Bni1p(1215–1953); p913, whichcontains the Bgl II to Stu I fragment from p182,encoding Bni1p(1227–1397); p929, which con-tains the Bgl II to Not I fragment from p182, encod-ing Bni1p(1414–1953); p952, which contains theHpa I to Not I fragment from p182, encodingBni1p(1647–1953); and p887, which contains theBam HI to Not I fragment encoding a full-lengthprofilin cDNA (16). The pACT-derived plasmid wasp1124, encoding full-length Act1p as isolated in acatch and release screen (22). The pGAD-C–de-rived plasmid was p688, encoding the COOH-ter-minal 311 amino acids (478–788) of Bud6p, asisolated in a catch and release screen (22).

29. For localization of Bni1p, SY2625 (11) cells carrying amulticopy plasmid encoding either HA-tagged Bni1p[pY39tet1 (9)] or nontagged Bni1p were induced toform mating projections (12). HA-Bni1p was localizedby immunofluorescence with monoclonal antibodyHA.11(Berkeley Antibody Company) as described [J.R. Pringle, A. E.M. Adams, D. G. Drubin, B. K. Haarer,Methods Enzymol. 194, 565 (1991)]. For localizationof Bud6p, SY2625 cells expressing GFP-Bud6p (23)or containing the control plasmid pRS316 (26) wereinduced to form mating projections (12), then ob-served by fluorescence microscopy with the use of afluorescein isothiocyanate filter set.

30. Yeast cells of strain B5459 (MATa pep4::HIS3prb1D1-6R ura3 trp1 lys2 leu2 his3D200 can1) car-rying p1025 (26) were grown to mid-log phase inraffinose medium, and galactose was added toinduce the production of HA-tagged Bni1p(1215–1953). After 1 hour, extracts were prepared bygrinding cells with glass beads in lysis buffer [0.6 Msorbitol, bovine serum albumin (1%), 140 mMNaCl, 5 mM EDTA, 50 mM tris-HCl (pH 7.6), 0.06%Triton X-100, 2 mM phenylmethylsulfonyl fluoride,aprotinin (10 mg/ml)] as described (2). Escherichiacoli strain BL 21 (Novagen) was transformed withpGEX-3X (Pharmacia) or p907 (26) and induced forexpression of GST or GST-profilin, respectively.GST proteins were purified on glutathione-Sepha-rose (Pharmacia) and washed twice with phos-phate-buffered saline (PBS) [140 mM NaCl, 2.7mM KCl, 10 mM Na2HPO4, 1.8 mM KH2PO4 (pH7.3)]. Glutathione-Sepharose beads with GST orGST-profilin bound were then added to the yeastextract containing HA-Bni1p(1215–1953) and in-cubated on ice. After 45 min, the beads were col-lected and washed twice with PBS. The GST pro-teins and associated proteins were eluted with glu-tathione [10 mM glutathione, 50 mM tris-HCl (pH8.0)] and subjected to immunoblot analysis withantibodies to GST (Pharmacia) or the HA epitope(29) as described (27).

31. We thank D. Amberg, B. Andrews, R. Brent, R.Dorer, S. J. Elledge, S. Givan, B. K. Haarer, J.Horecka, P. James, I. Sadowski, M. Tyers, and J.Zahner for plasmids and yeast strains; B. Brand-horst, J. Brown, N. Davis, S. Kim, B. Nelson, and I.Pot for comments on the manuscript; and G. Pojeand I. Pot for assistance with experiments. Support-ed by grants to C.B. from the Natural Sciences andEngineering Research Council of Canada and theNational Cancer Institute of Canada; by a grant fromthe Swiss National Science Foundation to M.P.; andby NIH grant GM31006 to J.R.P.

2 December 1996; accepted 10 February 1997

A General Model for the Origin of AllometricScaling Laws in Biology

Geoffrey B. West, James H. Brown,* Brian J. EnquistAllometric scaling relations, including the 3/4 power law for metabolic rates, are char-acteristic of all organisms and are here derived from a general model that describes howessential materials are transported through space-filling fractal networks of branchingtubes. The model assumes that the energy dissipated is minimized and that the terminaltubes do not vary with body size. It provides a complete analysis of scaling relations formammalian circulatory systems that are in agreement with data. More generally, themodel predicts structural and functional properties of vertebrate cardiovascular andrespiratory systems, plant vascular systems, insect tracheal tubes, and other distributionnetworks.

Biological diversity is largely a matter ofbody size, which varies over 21 orders ofmagnitude (1). Size affects rates of all bio-logical structures and processes from cellu-lar metabolism to population dynamics (2,3). The dependence of a biological variableY on body mass M is typically characterizedby an allometric scaling law of the form

Y 5 Y0Mb (1)

where b is the scaling exponent and Y0 aconstant that is characteristic of the kindof organism. If, as originally thought, theserelations reflect geometric constraints,then b should be a simple multiple ofone-third. However, most biological phe-nomena scale as quarter rather than thirdpowers of body mass (2–4): For example,metabolic rates B of entire organisms scaleas M3/4; rates of cellular metabolism,heartbeat, and maximal populationgrowth scale as M21/4; and times of bloodcirculation, embryonic growth and devel-opment, and life-span scale as M1/4. Sizesof biological structures scale similarly: Forexample, the cross-sectional areas of mam-malian aortas and of tree trunks scale asM3/4. No general theory explains the ori-gin of these laws. Current hypotheses,such as resistance to elastic buckling interrestrial organisms (5) or diffusion ofmaterials across hydrodynamic boundarylayers in aquatic organisms (6), cannotexplain why so many biological processesin nearly all kinds of animals (2, 3), plants(7), and microbes (8) exhibit quarter-pow-er scaling.

We propose that a common mechanism

underlies these laws: Living things are sus-tained by the transport of materialsthrough linear networks that branch tosupply all parts of the organism. We de-velop a quantitative model that explainsthe origin and ubiquity of quarter-powerscaling; it predicts the essential features oftransport systems, such as mammalianblood vessels and bronchial trees, plantvascular systems, and insect trachealtubes. It is based on three unifying princi-ples or assumptions: First, in order for thenetwork to supply the entire volume ofthe organism, a space-filling fractal-likebranching pattern (9) is required. Second,the final branch of the network (such asthe capillary in the circulatory system) is asize-invariant unit (2). And third, the en-ergy required to distribute resources isminimized (10); this final restriction isbasically equivalent to minimizing the to-tal hydrodynamic resistance of the system.Scaling laws arise from the interplay be-tween physical and geometric constraintsimplicit in these three principles. Themodel presented here should be viewed asan idealized representation in that we ig-nore complications such as tapering ofvessels, turbulence, and nonlinear effects.These play only a minor role in determin-ing the dynamics of the entire networkand could be incorporated in more de-tailed analyses of specific systems.

Most distribution systems can be de-scribed by a branching network in whichthe sizes of tubes regularly decrease (Fig.1). One version is exhibited by vertebratecirculatory and respiratory systems, anoth-er by the “vessel-bundle” structure of mul-tiple parallel tubes, characteristic of plantvascular systems (11). Biological networksvary in the properties of the tube (elasticto rigid), the fluid transported (liquid togas), and the nature of the pump (a pul-satile compression pump in the cardiovas-cular system, a pulsatile bellows pump inthe respiratory system, diffusion in insect

G. B. West, Theoretical Division, T-8, Mail Stop B285,Los Alamos National Laboratory, Los Alamos, NM87545, and The Santa Fe Institute, 1399 Hyde ParkRoad, Santa Fe, NM 87501, USA.J. H. Brown and B. J. Enquist, Department of Biology,University of New Mexico, Albuquerque, NM 87131, andThe Santa Fe Institute, 1399 Hyde Park Road, Santa Fe,NM 87501, USA.

*To whom correspondence should be addressed. E-mail:jhbrown@unm.edu

SCIENCE z VOL. 276 z 4 APRIL 1997 z http://www.sciencemag.org122

tracheae, and osmotic and vapor pressurein the plant vascular system). In spite ofthese differences, these networks exhibitessentially the same scaling laws.

For convenience we shall use the lan-guage of the cardiovascular system, name-ly, aorta, arteries, arterioles, and capillar-ies; the correspondence to other systems isstraightforward. In the general case, thenetwork is composed of N branchings fromthe aorta (level 0) to the capillaries (levelN, denoted here by a subscript c) (Fig.1C). A typical branch at some intermedi-ate level k has length lk, radius rk, andpressure drop Dpk (Fig. 1D). The volumerate of flow is Qk 5 prk

2uk where uk is theflow velocity averaged over the cross sec-tion and, if necessary, over time. Eachtube branches into nk smaller ones (12), sothe total number of branches at level k isNk 5 n0n1 . . . nk. Because fluid is con-served as it flows through the system

Q0 5 NkQk 5 Nkprk2uk 5 Ncprc

2uc (2)

which holds for any level k. We next intro-duce the important assumption, the secondabove, that the terminal units (capillaries)are invariant, so rc, lc, uc, and, consequently,Dpc are independent of body size. Becausethe fluid transports oxygen and nutrients formetabolism, Q0 } B; thus, if B } Ma (wherea will later be determined to be 3/4), thenQ0 } Ma. Equation 2 therefore predicts thatthe total number of capillaries must scale asB, that is, Nc } Ma.

To characterize the branching, we in-troduce scale factors bk [ rk11/rk and gk [lk11/lk. We shall prove that in order tominimize the energy dissipated in the sys-tem in the sense of the third principleabove, the network must be a convention-al self-similar fractal in that bk 5 b, gk 5g, and nk 5 n, all independent of k (animportant exception is bk in pulsatile sys-tems). For a self-similar fractal, the num-ber of branches increases in geometric pro-portion (Nk 5 nk) as their size geometri-cally decreases from level 0 to level N.Before proving self-similarity, we first ex-amine some of its consequences.

Because Nc 5 nN, the number of gener-ations of branches scales only logarithmi-cally with size

N 5a ln~M/Mo)

ln n(3)

where M0 is a normalization scale for M(13). Thus, a whale is 107 times heavierthan a mouse but has only about 70% morebranchings from aorta to capillary. The to-tal volume of fluid in the network (“blood”volume Vb) is

Vb 5 Ok 5 0

N

NkVk 5 Ok 5 0

N

prk2lknk

5~ngb2!2~N 1 1! 2 1

~ngb2!21 2 1nNVc (4)

where the last expression reflects the fractalnature of the system. As shown below, onecan also prove from the energy minimiza-tion principle that Vb } M. Because ngb2 ,1 and N .. 1, a good approximation to Eq.4 is Vb 5 V0/(1 2 ngb2) 5 Vc(gb2)2N/(1 2 ngb2). From our assumption that cap-illaries are invariant units, it therefore fol-lows that (gb2)2N } M. Using this relationin Eq. 3 then gives

a 5 2ln n

ln~gb2!(5)

To make further progress requires knowl-edge of g and b. We shall show how theformer follows from the space-filling fractalrequirement, and the latter, from the energyminimization principle.

A space-filling fractal is a natural struc-ture for ensuring that all cells are servicedby capillaries. The network must branch sothat a group of cells, referred to here as a“service volume,” is supplied by each capil-lary. Because rk ,, lk and the total numberof branchings N is large, the volume sup-plied by the total network can be approxi-mated by the sum of spheres whose diame-ters are that of a typical kth-level vessel,namely 4/3p(lk/2)

3Nk. For large N, this es-timate does not depend significantly on thespecific level, although it is most accuratefor large k. This condition, that the fractalbe volume-preserving from one generationto the next, can therefore be expressed as4/3p(lk/2)

3Nk ' 4/3p(lk11/2)3Nk11. This

relation gives g3k [ (lk11/lk)3 ' Nk/Nk11 5

1/n, showing that gk ' n21/3 ' g must beindependent of k. This result for gk is ageneral property of all space-filling fractalsystems that we consider.

The 3/4 power law arises in the simplecase of the classic rigid-pipe model, wherethe branching is assumed to be area-pre-

serving, that is, the sum of the cross-sec-tional areas of the daughter branches equalsthat of the parent, so prk

2 5 nprk211. Thus,

bk [ rk11/rk 5 n21/2 5 b, independent of k.When the area-preserving branching rela-tion, b 5 n21/2, is combined with thespace-filling result for g, Eq. 5 yields a 53/4, so B } M3/4. Many other scaling lawsfollow. For example, for the aorta, r0 5b2Nrc 5 Nc

1/2rc and l0 5 g2Nrc 5 Nc1/3lc,

yielding r0 } M3/8 and l0 } M1/4. Thisderivation of the a 5 3/4 law is essentially ageometric one, strictly applying only to sys-tems that exhibit area-preserving branch-ing. This property has the further conse-quence, which follows from Eq. 2, that thefluid velocity must remain constantthroughout the network and be indepen-dent of size. These features are a naturalconsequence of the idealized vessel-bundlestructure of plant vascular systems (Fig. 1B),in which area-preserving branching arisesautomatically because each branch is as-sumed to be a bundle of nN2k elementaryvessels of the same radius (11). Pulsatilemammalian vascular systems, on the otherhand, do not conform to this structure, sofor them, we must look elsewhere for theorigin of quarter-power scaling laws.

Some features of the simple pipe modelremain valid for all networks: (i) The quan-tities g and b play a dual scaling role: theydetermine not only how quantities scalefrom level 0 (aorta) to N (capillary) withina single organism of fixed size, but also howa given quantity scales when organisms ofdifferent masses are compared. (ii) The frac-tal nature of the entire system as expressed,for example, in the summation in Eq. 4leads to a scaling different from that for asingle tube, given by an individual term inthe series. These network systems musttherefore be treated as a complete integrat-ed unit; they cannot realistically be mod-eled by a single or a few representativevessels. (iii) The scaling with M does not

Fig. 1.Diagrammatic examples ofsegments of biological distribu-tion networks: (A) mammalian cir-culatory and respiratory systemscomposed of branching tubes;(B) plant vessel-bundle vascularsystem composed of divergingvessel elements; (C) topologicalrepresentation of such networks,where k specifies the order of thelevel, beginning with the aorta(k 5 0) and ending with the capil-lary (k5N ); and (D) parameters ofa typical tube at the kth level.

REPORTS

http://www.sciencemag.org z SCIENCE z VOL. 276 z 4 APRIL 1997 123

depend on the branching ratio n.We next consider the dynamics of the

network and examine the consequences ofthe energy minimization principle, which isparticularly relevant to mammalian vascu-lar systems. Pulsatile flow, which dominatesthe larger vessels (aorta and major arteries),must have area-preserving branching, sothat b 5 n21/2, leading to quarter-powerscaling. The smaller vessels, on the otherhand, have the classic “cubic-law” branch-ing (10), where b 5 n21/3, and play arelatively minor role in allometric scaling.

First consider the simpler problem ofnonpulsatile flow. For steady laminar flowof a Newtonian fluid, the viscous resistanceof a single tube is given by the well-knownPoiseuille formula Rk 5 8mlk/prk

4, where m isthe viscosity of the fluid. Ignoring smalleffects such as turbulence and nonlineari-ties at junctions, the resistance of the entirenetwork is given by (14)

Z 5 Ok 5 0

N RkNk

5 Ok 5 0

N 8mlkprk

4nk

5@1 2 ~nb4/g!N 1 1#Rc

~1 2 nb4/g!nN (6)

Now, nb4/g , 1 and N .. 1, so a goodapproximation is Z 5 Rc/(1 2 nb4/g)Nc.Because Rc is invariant, Z } Nc

21 } M2a,which leads to two important scaling laws:blood pressure Dp 5 Q0Z must be indepen-dent of body size and the power dissipatedin the system (cardiac output)W 5 Q0Dp }Ma, so that the power expended by theheart in overcoming viscous forces is a size-independent fraction of the metabolic rate.Neither of these results depends on detailedknowledge of n, b, or g, in contrast toresults based on Vb } M, such as Eq. 5, a 53/4, and r0 } M3/8. From Eq. 2, Q0 5 pr0

2u0,which correctly predicts that the velocity ofblood in the aorta u0 } M0 (2). However, anarea-preserving scaling relation b 5 n21/2

also implies by means of Eq. 2 that uk 5 u0for all k. This relation is valid for fluid flowin plant vessels (because of the vascular bun-dle structure) (11, 15) and insect tracheae(because gas is driven by diffusion) (16);both therefore exhibit area-preservingbranching, which leads to 3/4 power scalingof metabolic rate. Branching cannot be en-tirely area-preserving in mammalian circula-tory systems because blood must slow downto allow materials to diffuse across capillarywalls. However, the pulsatile nature of themammalian cardiovascular system solves theproblem.

Energy minimization constrains the net-work for the simpler nonpulsatile systems.Consider cardiac output as a function of allrelevant variables: W(rk, lk, nk, M). To sus-tain a given metabolic rate in an organismof fixed mass M with a given volume of

blood Vb(rk, lk, nk, M), the minimizationprinciple requires that the cardiac output beminimized subject to a space-filling geome-try. To enforce such a constraint, we use thestandard method of Lagrange multipliers (l,lk, and lM) and so need to minimize theauxiliary function

F(rk, lk, n) 5 W(rk, lk, nk, M)

1 lVb(rk, lk, nk, M) 1 Ok 5 0

N

lkNklk3 1 lMM

(7)

Because B } Q0 and W 5 Q02Z, this prob-

lem is tantamount to minimizing the im-pedance Z, which can therefore be used inEq. 7 in place of W. First, consider the casewhere nk 5 n, so that we can use Eqs. 4 and6 for Vb and Z, respectively. For a fixed massM, the auxiliary Lagrange function F,which incorporates the constraints, must beminimized with respect to all variables forthe entire system (rk, lk, and n). This re-quires ]F/]lk 5 ]F/]rk 5 ]F/]n 5 0, whichstraightforwardly leads to bk 5 n21/3. Moregenerally, by considering variations withrespect to nk, one can show that nk 5 n,independent of k. The result, bk 5 n21/3, isa generalization of Murray’s finding (17),derived for a single branching, to the com-plete network. Now varying M and mini-mizing F in Eq. 7 (]F/]M 5 0) leads to Vb} M, which is just the relation needed toderive Eq. 5. Although the result bk 5 n21/3

is independent of k, it is not area-preservingand therefore does not give a 5 3/4 whenused in Eq. 5; instead, it gives a 5 1. It does,however, solve the problem of slowingblood in the capillaries: Eq. 2 gives uc/ u0 5(nb2)2N 5 Nc

21/3. For humans, Nc ' 1010,so uc/ u0 ' 1023, in reasonable agreementwith data (18). On the other hand, it leadsto an incorrect scaling law for this ratio:uc/ u0 } M21/4. Incorporating pulsatile flownot only solves these problems, giving thecorrect scaling relations (a 5 3/4 and uc/ u0} M0), but also gives the correct value foruc/ u0.

A complete treatment of pulsatile flowis complicated; here, we present a simpli-fied version that contains the essentialfeatures needed for the scaling problem.When an oscillatory pressure p of angularfrequency v is applied to an elastic (char-acterized by modulus E) vessel with wallthickness h, a damped traveling wave iscreated: p 5 p

0ei(vt 2 2pz/l). Here, t is time,

z is the distance along the tube, l is thewavelength, and p0 is the amplitude aver-aged over the radius; the wave velocityc 5 2pvl. Both the impedance Z and thedispersion relation that determines c arederived by solving the Navier-Stokesequation for the fluid coupled to theNavier equations for the vessel wall (19).

In the linearized incompressible-fluid,thin-wall approximation, this problem canbe solved analytically to give

S cc0D2

' 2J2~i3/2a!

J0~i3/2a!and Z '

c02r

pr2c(8)

Here a [ (vr/m)1/2r is the dimensionlessWomersley number (13), and c0 [ (Eh/2rr)1/2 is the Korteweg-Moens velocity. Ingeneral, both c and Z are complex functionsof v, so the wave is attenuated and dispersesas it propagates. Consider the consequencesof these formulas as the blood flows throughprogressively smaller tubes: For large tubes,a is large (in a typical human artery, a '5), and viscosity plays almost no role. Equa-tion 8 then gives c 5 c0 and Z 5 rc0/pr

2;because both of these are real quantities,the wave is neither attenuated nor dis-persed. The r dependence of Z has changedfrom the nonpulsatile r24 behavior to r22.Minimizing energy loss now gives hk/rk (and,therefore, ck) independent of k and, mostimportantly, an area-preserving law at thejunctions, so bk 5 n21/2. This relation en-sures that energy-carrying waves are notreflected back up the tubes at branch pointsand is the exact analog of impedancematching at the junctions of electricaltransmission lines (18). As k increases, thesizes of tubes decrease, so a3 0 (in humanarterioles, for example, a ' 0.05), and therole of viscosity increases, eventually dom-inating the flow. Equation 8 then gives c 'i1/2ac0/4 3 0, in agreement with observa-tion (18). Because c and, consequently, lnow have imaginary parts, the travelingwave is heavily damped, leaving an almoststeady oscillatory flow whose impedance is,from Eq. 8, given by the Poiseuille formula;that is, the r24 behavior is restored. Thus,for large k, corresponding to small vessels,bk 5 n21/3. We conclude that for pulsatileflow, bk is not independent of k but ratherhas a steplike behavior (Fig. 2). This picture

Fig. 2. Schematic variation of theWomersley num-ber ak and the scaling parameters bk and gk withlevel number (k) for pulsatile systems. Note thesteplike change in bk at k5 k from area-preservingpulse-wave flow inmajor vessels to area-increasingPoiseuille-type flow in small vessels.

SCIENCE z VOL. 276 z 4 APRIL 1997 z http://www.sciencemag.org124

is well supported by empirical data (18, 20,21). The crossover from one behavior to theother occurs over the region where the waveand Poiseuille impedances are comparable insize. The approximate value of k where thisoccurs (say, k) is given by r k

2 /l k ' 8m/rc0,leading toN2 k[N' ln(8mlc/rc0rc

2)/ln n,independent of M. Thus, the number ofgenerations where Poiseuille flow dominatesshould be independent of body size. On theother hand, the crossover point itselfgrows logarithmically: k } N } ln M. Forhumans, with n 5 3 (21), N ' 15 andN ' 22 (assuming Nc ' 2 3 1010), where-as with n 5 2, N ' 24 and N ' 34. Thesevalues mean that in humans Poiseuilleflow begins to compete with the pulsewave after just a few branchings, dominat-ing after about seven. In a 3-g shrew,Poiseuille flow begins to dominate shortlybeyond the aorta.

The derivation of scaling laws based on bkderived from Eqs. 7 and 8 (Fig. 2) leads to thesame results as before. For simplicity, assumethat the crossover is sharp; using a gradualtransition does not change the resulting scal-ing laws. So, for k . k, define bk [ b. 5n21/3 and, for k , k, bk [ b, 5 n21/2. Thispredicts that area preservation only persistsin the pulsatile region from the aortathrough the large arteries, at most until k 'k. First consider the radius of the aorta r0: itsscaling behavior is now given by r0 5rcb.

k 2Nb,k 5 rcn

1/3N11/6k 5 rcn1/2N21/6N,

which gives r0 } M3/8 and, for humans, r0/rc' 104, in agreement with data (2) . UsingEq. 3 we obtain, for the ratio of fluidvelocity in the aorta to that in the capil-lary, u0 / uc 5 Nc(rc/r0)

2 5 nN/3u0 / uc '250, independent of M, again in agree-ment with data. Because g reflects the

space-filling geometry, it remains un-changed, so we still have l0 } M1/4. Bloodvolume Vb, however, is more complicated

Vb 5Vc

~b.2 g!N HSb.

b,D2k 1 2 ~nb,

2 g!k

1 2 ~nb,2 g!

1 F1 2 ~nb.2 g!N

1 2 ~nb.2 g!

21 2 ~nb.

2 g!k

1 2 ~nb.2 g! GJ (9)

This formula is a generalization of Eq. 4and is dominated by the first term, whichrepresents the contribution of the largetubes (aorta and arteries). Thus, Vb }nN11/3k } n4/3N, which, because it mustscale as M, leads, as before, to a 5 3/4. Assize decreases, the second term, represent-ing the cubic branching of small vessels,becomes increasingly important. This be-havior predicts small deviations fromquarter-power scaling (a * 3/4), observedin the smallest mammals (2). An expres-sion analogous to Eq. 9 can be derived forthe total impedance of the system Z. It isdominated by the small vessels (arteriolesand capillaries) and, as before, gives Dpand u0 } M0.

In order to understand allometric scal-ing, it is necessary to formulate an integrat-ed model for the entire system. The presentmodel should be viewed as an idealizedzeroth-order approximation: it accounts formany of the features of distribution net-works and can be used as a point of depar-ture for more detailed analyses and models.In addition, because it is quantitative, thecoefficients, Y0 of Eq. 1, can also, in prin-ciple, be derived. It accurately predicts theknown scaling relations of the mammalian

cardiovascular system (Table 1); data areneeded to test other predictions. For exam-ple, the invariance of capillary parametersimplies Nc } M3/4 rather than the naıveexpectationNc } M, so the volume servicedby each capillary must scale as M1/4, andcapillary density per cross-sectional area oftissue, as M21/12.

A minor variant of the model describesthe mammalian respiratory system. Al-though pulse waves are irrelevant becausethe tubes are not elastic, the formula for Z isquite similar to Eq. 8. The fractal bronchialtree terminates in NA } M3/4 alveoli. Thenetwork is space-filling, and the alveoli playthe role of the service volume accountingfor most of the total volume of the lung,which scales as M. Thus, the volume of analveolus VA } M1/4, its radius rA } M1/12,and its surface area AA } rA

2 } M1/6, so thetotal surface area of the lung AL 5 NAAA }M11/12. This explains the paradox (22) thatAA scales with an exponent closer to 1 thanthe 3/4 seemingly needed to supply oxygen.The rate of oxygen diffusion across an alve-olus, which must be independent of M, isproportional to DpO2AA/rA. Thus, DpO2 }M21/12, which must be compensated for by asimilar scaling of the oxygen affinity of he-moglobin. Available data support these pre-dictions (Table 1).

Our model provides a theoretical, mech-anistic basis for understanding the centralrole of body size in all aspects of biology.Considering the many functionally inter-connected parts of the organism that mustobey the constraints, it is not surprising thatthe diversity of living and fossil organisms isbased on the elaboration of a few successfuldesigns. Given the need to redesign theentire system whenever body size changes,

Table 1. Values of allometric exponents for variables of the mammaliancardiovascular and respiratory systems predicted by the model compared

with empirical observations. Observed values of exponents are taken from (2,3); ND denotes that no data are available.

Cardiovascular Respiratory

VariableExponent

VariableExponent

Predicted Observed Predicted Observed

Aorta radius r0 3/8 5 0.375 0.36 Tracheal radius 3/8 5 0.375 0.39Aorta pressure Dp0 0 5 0.00 0.032 Interpleural pressure 0 5 0.00 0.004Aorta blood velocity u0 0 5 0.00 0.07 Air velocity in trachea 0 5 0.00 0.02Blood volume Vb 1 5 1.00 1.00 Lung volume 1 5 1.00 1.05Circulation time 1/4 5 0.25 0.25 Volume flow to lung 3/4 5 0.75 0.80Circulation distance l 1/4 5 0.25 ND Volume of alveolus VA 1/4 5 0.25 NDCardiac stroke volume 1 5 1.00 1.03 Tidal volume 1 5 1.00 1.041Cardiac frequency v 21/4 5 20.25 20.25 Respiratory frequency 21/4 5 20.25 20.26Cardiac output E 3/4 5 0.75 0.74 Power dissipated 3/4 5 0.75 0.78Number of capillaries Nc 3/4 5 0.75 ND Number of alveoli NA 3/4 5 0.75 NDService volume radius 1/12 5 0.083 ND Radius of alveolus rA 1/12 5 0.083 0.13Womersley number a 1/4 5 0.25 0.25 Area of alveolus AA 1/6 5 0.083 NDDensity of capillaries 21/12 5 20.083 20.095 Area of lung AL 11/12 5 0.92 0.95O2 affinity of blood P50 21/12 5 20.083 20.089 O2 diffusing capacity 1 5 1.00 0.99Total resistance Z 23/4 5 20.75 20.76 Total resistance 23/4 5 20.75 20.70Metabolic rate B 3/4 5 0.75 0.75 O2 consumption rate 3/4 5 0.75 0.76

REPORTS

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either during ontogeny or phylogenetic di-versification, small deviations from quarter-power scaling sometimes occur (3, 23).However, when body sizes vary over manyorders of magnitude, these scaling laws areobeyed with remarkable precision. More-over, the predicted scaling properties do notdepend on most details of system design,including the exact branching pattern, pro-vided it has a fractal structure (24). Signif-icantly, nonfractal systems, such as combus-tion engines and electric motors, exhibitgeometric (third-power) rather than quar-ter-power scaling (1). Because the fractalnetwork must still fill the entire D-dimen-sional volume, our result generalizes to a 5D/(D 1 1). Organisms are three-dimension-al, which explains the 3 in the numerator ofthe 3/4 power law, but it would be instruc-tive to examine nearly two-dimensional or-ganisms such as bryozoans and flatworms.The model can potentially explain howfundamental constraints at the level of in-dividual organisms lead to correspondingquarter-power allometries at other levels.The constraints of body size on the rates atwhich resources can be taken up from theenvironment and transported and trans-formed within the body ramify to causequarter-power scaling in such diverse phe-nomena as rate and duration of embryonicand postembryonic growth and develop-ment, interval between clutches, age of firstreproduction, life span, home range andterritory size, population density, and max-imal population growth rate (1–3). Becauseorganisms of different body sizes have dif-ferent requirements for resources and oper-ate on different spatial and temporal scales,quarter-power allometric scaling is perhapsthe single most pervasive theme underlyingall biological diversity.

REFERENCES AND NOTES___________________________

1. T. A. McMahon and J. T. Bonner, On Size and Life(Scientific American Library, New York, 1983); J. T.Bonner, The Evolution of Complexity by Means ofNatural Selection (Princeton Univ. Press, Princeton,NJ, 1983); J. H. Brown,Macroecology (Univ. of Chi-cago Press, Chicago, 1995).

2. K. Schmidt-Nielsen, Scaling: Why Is Animal Size soImportant? (Cambridge Univ. Press, Cambridge,1984); W. A. Calder III,Size, Function and Life History(Harvard Univ. Press, Cambridge, MA, 1984).

3. R. H. Peters, The Ecological Implications of BodySize (Cambridge Univ. Press, Cambridge, 1983).

4. H. A. Feldman and T. A. McMahon, Respir. Physiol.52, 149 (1983).

5. T. A. McMahon, Science 179, 1201 (1973).6. M. R. Patterson, ibid. 255, 1421 (1992).7. K. J. Niklas, Plant Allometry: The Scaling of Form and

Process (Univ. of Chicago Press, Chicago, 1994);Am. J. Bot. 81, 134 (1994).

8. A. M. Hemmingsen, Rep. Steno Mem. Hosp.(Copenhagen) 4, 1 (1950); ibid. 9, 1 (1960).

9. B. B. Mandelbrot, The Fractal Geometry of Nature(Freeman, New York, 1977).

10. D’A. W. Thompson, On Growth and Form (Cam-bridge Univ. Press, Cambridge, 1942).

11. K. Shinozaki et al., Jpn. J. Ecol. 14, 97 (1964); ibid.,

p. 133; see also M. H. Zimmerman, Xylem Structureand the Ascent of Sap (Springer-Verlag, Berlin,1983); M. T. Tyree and F. W. Ewers, New Phytol.119, 345 (1991).

12. The branching of a vessel at level k into nk smallervessels (Fig. 1) is assumed to occur over some small,but finite, distance that is much smaller than either lkor lk11. This relation is similar to that assumed in theStrahler method [A. N. Strahler, Trans. Am. Geo-phys. Union 34, 345 (1953); (11, 21)]. A generaliza-tion to nonuniform branching, where the radii andlengths at a given level may vary, is straightforward.

13. Normalization factors, such as M0, will generally besuppressed, as in Eq. 1. In general, all quantitiesshould be expressed in dimensionless form; note,however, that this does not guarantee that they aresize independent and scale asM 0. For example, theWomersley number, a of Eq. 8, although dimension-less, scales as M 1/4.

14. This formula is not valid for plant vessel bundlesbecause plants are composed of multiple parallelvessel elements. Their resistance is given by Z 58ml /Ncprc

4, where l is the length of a single vesselelement, rc is its radius, and Nc is their total number.

15. This relation holds for plant vessels from the roots tothe leaves, but not within leaves [M. J. Canney, Phi-los. Trans. R. Soc. London Ser. B 341, 87 (1993)].

16. A. Krogh, Pfluegers Arch. Gesamte Physiol. Men-schen Tiere 179, 95 (1920).

17. C. D. Murray, Proc. Natl. Acad. Sci. U.S.A. 12, 207(1926).

18. C. G. Caro et al., The Mechanics of Circulation (Ox-ford Univ. Press, Oxford, 1978).

19. J. R. Womersley, Philos. Mag. 46, 199 (1955);J. Physiol. (London) 127, 553 (1955).

20. See, for example, A. S. Iberall, Math. Biosci. 1, 375(1967) and T. F. Sherman, J. Gen. Physiol. 78, 431(1981), which contain summaries of earlier data; alsoM. Zamir et al., J. Biomech. 25, 1303 (1992) and J.K.-J. Li, Comparative Cardiovascular Dynamics of

Mammals (CRC Press, Boca Raton, FL, 1996). Caremust be taken in comparing measurements withprediction, particularly if averages over many suc-cessive levels are used. For example, if Ak [ (kprk

2 isthe total cross-sectional area at level k, then for theaorta and major arteries, where k , k and thebranching is area-preserving, we predict A0 5 Ak.Suppose, however, that the firstK levels are groupedtogether. Then, if the resulting measurement givesAK, area-preserving predicts AK 5 KA0 (but not AK 5A0). It also predicts r0

3 ' n 1/2(Nkrk3. Using results

from M. LaBarbera [Science 249, 992 (1990)], whoused data averaged over the first 160 vessels (ap-proximately the first 4 levels), gives, for human be-ings, A0 ' 4.90 cm2, AK ' 19.98 cm2, r0

3 ' 1.95cm3, and(Nkrk

3 ' 1.27 cm3, in agreement with areapreservation. LaBarbera, unfortunately, took the factthat AK Þ A0 and r0

3 ' (Nkrk3 as evidence for cubic

rather than area-preserving branching. For smallvessels, where k . k, convincing evidence for thecubic law can be found in the analysis of the arteriolarsystem by M. L. Ellsworth et al.,Microvasc. Res. 34,168 (1987).

21. Y. C. Fung, Biodynamics (Springer-Verlag, NewYork, 1984).

22. P. Gehr et al., Respir. Physiol. 44, 61 (1981).23. P. M. Bennett and P. H. Harvey, J. Zool. 213, 327

(1987); A. F. Bennett, Am. Zool. 28, 699 (1988); P. H.Harvey andM. D. Pagel, The Comparative Method inEvolutionary Biology (Oxford Univ. Press, Oxford,1991).

24. This is reminiscent of the invariance of scaling expo-nents to details of the model that follow from renor-malization group analyses, which can be viewed as ageneralization of classical dimensional analysis.

25. J.H.B. is supported by NSF grant DEB-9318096,B.J.E. by NSF grant GER-9553623 and a FulbrightFellowship, and G.B.W. by the Department of Energy.

13 September 1996; accepted 12 February 1997

Flexibility in DNA Recombination: Structure ofthe Lambda Integrase Catalytic Core

Hyock Joo Kwon, Radhakrishna Tirumalai, Arthur Landy,*Tom Ellenberger*

Lambda integrase is archetypic of site-specific recombinases that catalyze intermolec-ular DNA rearrangements without energetic input. DNA cleavage, strand exchange, andreligation steps are linked by a covalent phosphotyrosine intermediate in which Tyr342

is attached to the 39-phosphate of the DNA cut site. The 1.9 angstrom crystal structureof the integrase catalytic domain reveals a protein fold that is conserved in organismsranging fromarchaebacteria to yeast and that suggests amodel for interactionwith targetDNA. The attacking Tyr342 nucleophile is located on a flexible loop about 20 angstromsfrom a basic groove that contains all the other catalytically essential residues. Thisbipartite active site can account for several apparently paradoxical features of integrasefamily recombinases, including the capacity for both cis and trans cleavage of DNA.

The integrase protein (Int) of Escherichiacoli phage lambda (l) belongs to a largefamily of site-specific DNA recombinasesfrom archaebacteria, eubacteria, and yeast(1–3) that catalyze rearrangements be-

tween DNA sequences with little or nosequence homology to each other (4–8).Like l Int, many of these recombinasesfunction in the integration and excision ofviral genomes into and out of the chromo-somes of their respective hosts. Othersfunction in the decatenation or segrega-tion of newly replicated chromosomes,conjugative transposition, regulation ofplasmid copy number, or expression of cellsurface proteins. Integrase family membershave the distinctive ability to carry out acomplete site-specific recombination reac-

H. J. Kwon and T. Ellenberger, Department of BiologicalChemistry and Molecular Pharmacology, Harvard Medi-cal School, Boston MA 02115, and the Graduate Pro-gram in Biophysics, Harvard University, Cambridge, MA02138, USA.R. Tirumalai and A. Landy, Division of Biology and Med-icine, Brown University, Providence, RI 02912, USA.

*Corresponding authors.

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