A General Model for the Origin of Allometric Scaling Laws in Biology PDF

p888, which contains a Bam HI to Not I fragment encoding a full-length profilin cDNA (16); p989, A General Model for the Origin of Allometric which encodes a mutant form of profilin, Pfy1p-3, lacking the last three amino acids (18); p890, which Sca...

p888, which contains a Bam HI to Not I fragment encoding a full-length profilin cDNA (16); p989, A General Model for the Origin of Allometric which encodes a mutant form of profilin, Pfy1p-3, lacking the last three amino acids (18); p890, which Scaling Laws in Biology contains 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- Geoffrey B. West, James H. Brown,* Brian J. Enquist ing Bni1p(1414 –1953); and p951, which contains Allometric scaling relations, including the 3/4 power law for metabolic rates, are char- the Hpa I to Not I fragment from p182, encoding Bni1p(1647–1953). The pJG4-5 – derived plasmids acteristic of all organisms and are here derived from a general model that describes how were p561, which contains the Bam HI to Not I essential materials are transported through space-filling fractal networks of branching fragment from p532 (26), encoding Bni1p(1–1953); tubes. The model assumes that the energy dissipated is minimized and that the terminal p717, which contains the Bam HI to Eco47 III frag- tubes do not vary with body size. It provides a complete analysis of scaling relations for ment from p532, encoding Bni1p(1–1214); p558, which contains the Eco 47III to Not I fragment from mammalian circulatory systems that are in agreement with data. More generally, the p182, encoding Bni1p(1215 –1953); p913, which model predicts structural and functional properties of vertebrate cardiovascular and contains the Bgl II to Stu I fragment from p182, respiratory systems, plant vascular systems, insect tracheal tubes, and other distribution encoding Bni1p(1227–1397); p929, which con- tains the Bgl II to Not I fragment from p182, encod- networks. ing Bni1p(1414 –1953); p952, which contains the Hpa I to Not I fragment from p182, encoding Bni1p(1647–1953); and p887, which contains the Bam HI to Not I fragment encoding a full-length profilin cDNA (16). The pACT-derived plasmid was Biological diversity is largely a matter of underlies these laws: Living things are sus- Downloaded from https://www.science.org at CASA Institution Identity on August 31, 2023 p1124, encoding full-length Act1p as isolated in a catch and release screen (22). The pGAD-C– de- body size, which varies over 21 orders of tained by the transport of materials rived plasmid was p688, encoding the COOH-ter- minal 311 amino acids (478 –788) of Bud6p, as magnitude (1). Size affects rates of all bio- through linear networks that branch to isolated in a catch and release screen (22). logical structures and processes from cellu- supply all parts of the organism. We de- 29. For localization of Bni1p, SY2625 (11) cells carrying a lar metabolism to population dynamics (2, velop a quantitative model that explains multicopy plasmid encoding either HA-tagged Bni1p 3). The dependence of a biological variable the origin and ubiquity of quarter-power [pY39tet1 (9)] or nontagged Bni1p were induced to form mating projections (12). HA-Bni1p was localized Y on body mass M is typically characterized scaling; it predicts the essential features of by immunofluorescence with monoclonal antibody by an allometric scaling law of the form transport systems, such as mammalian HA.11(Berkeley Antibody Company) as described [J. blood vessels and bronchial trees, plant R. Pringle, A. E. M. Adams, D. G. Drubin, B. K. Haarer, Y 5 Y 0M b (1) vascular systems, and insect tracheal Methods Enzymol. 194, 565 (1991)]. For localization where b is the scaling exponent and Y0 a tubes. It is based on three unifying princi- of Bud6p, SY2625 cells expressing GFP-Bud6p (23) or containing the control plasmid pRS316 (26) were constant that is characteristic of the kind ples or assumptions: First, in order for the induced to form mating projections (12), then ob- of organism. If, as originally thought, these network to supply the entire volume of served by fluorescence microscopy with the use of a relations reflect geometric constraints, the organism, a space-filling fractal-like fluorescein isothiocyanate filter set. 30. Yeast cells of strain B5459 (MATa pep4::HIS3 then b should be a simple multiple of branching pattern (9) is required. Second, prb1D1-6R ura3 trp1 lys2 leu2 his3D200 can1) car- one-third. However, most biological phe- the final branch of the network (such as rying p1025 (26) were grown to mid-log phase in nomena scale as quarter rather than third the capillary in the circulatory system) is a raffinose medium, and galactose was added to induce the production of HA-tagged Bni1p(1215 – powers of body mass (2–4): For example, size-invariant unit (2). And third, the en- 1953). After 1 hour, extracts were prepared by metabolic rates B of entire organisms scale ergy required to distribute resources is grinding cells with glass beads in lysis buffer [0.6 M as M3/4; rates of cellular metabolism, minimized (10); this final restriction is sorbitol, bovine serum albumin (1%), 140 mM heartbeat, and maximal population basically equivalent to minimizing the to- NaCl, 5 mM EDTA, 50 mM tris-HCl (pH 7.6), 0.06% Triton X-100, 2 mM phenylmethylsulfonyl fluoride, growth scale as M21/4; and times of blood tal hydrodynamic resistance of the system. aprotinin (10 mg/ml)] as described (2). Escherichia circulation, embryonic growth and devel- Scaling laws arise from the interplay be- coli strain BL 21 (Novagen) was transformed with opment, and life-span scale as M1/4. Sizes tween physical and geometric constraints pGEX-3X (Pharmacia) or p907 (26) and induced for expression of GST or GST-profilin, respectively. of biological structures scale similarly: For implicit in these three principles. The GST proteins were purified on glutathione-Sepha- example, the cross-sectional areas of mam- model presented here should be viewed as rose (Pharmacia) and washed twice with phos- malian aortas and of tree trunks scale as an idealized representation in that we ig- phate-buffered saline (PBS) [140 mM NaCl, 2.7 mM KCl, 10 mM Na2HPO4, 1.8 mM KH2PO4 (pH M3/4. No general theory explains the ori- nore complications such as tapering of 7.3)]. Glutathione-Sepharose beads with GST or gin of these laws. Current hypotheses, vessels, turbulence, and nonlinear effects. GST-profilin bound were then added to the yeast such as resistance to elastic buckling in These play only a minor role in determin- extract containing HA-Bni1p(1215 –1953) and in- cubated on ice. After 45 min, the beads were col- terrestrial organisms (5) or diffusion of ing the dynamics of the entire network lected and washed twice with PBS. The GST pro- materials across hydrodynamic boundary and could be incorporated in more de- teins and associated proteins were eluted with glu- layers in aquatic organisms (6), cannot tailed analyses of specific systems. tathione [10 mM glutathione, 50 mM tris-HCl (pH 8.0)] and subjected to immunoblot analysis with explain why so many biological processes Most distribution systems can be de- antibodies to GST (Pharmacia) or the HA epitope in nearly all kinds of animals (2, 3), plants scribed by a branching network in which (29) as described (27). (7), and microbes (8) exhibit quarter-pow- the sizes of tubes regularly decrease (Fig. 31. We thank D. Amberg, B. Andrews, R. Brent, R. er scaling. 1). One version is exhibited by vertebrate Dorer, S. J. Elledge, S. Givan, B. K. Haarer, J. Horecka, P. James, I. Sadowski, M. Tyers, and J. We propose that a common mechanism circulatory and respiratory systems, anoth- Zahner for plasmids and yeast strains; B. Brand- er by the “vessel-bundle” structure of mul- horst, J. Brown, N. Davis, S. Kim, B. Nelson, and I. G. B. West, Theoretical Division, T-8, Mail Stop B285, tiple parallel tubes, characteristic of plant Pot for comments on the manuscript; and G. Poje Los Alamos National Laboratory, Los Alamos, NM and I. Pot for assistance with experiments. Support- 87545, and The Santa Fe Institute, 1399 Hyde Park vascular systems (11). Biological networks ed by grants to C.B. from the Natural Sciences and Road, Santa Fe, NM 87501, USA. vary in the properties of the tube (elastic Engineering Research Council of Canada and the J. H. Brown and B. J. Enquist, Department of Biology, to rigid), the fluid transported (liquid to National Cancer Institute of Canada; by a grant from University of New Mexico, Albuquerque, NM 87131, and the Swiss National Science Foundation to M.P.; and The Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, gas), and the nature of the pump (a pul- by NIH grant GM31006 to J.R.P. NM 87501, USA. satile compression pump in the cardiovas- * To whom correspondence should be addressed. E-mail: cular system, a pulsatile bellows pump in 2 December 1996; accepted 10 February 1997 [email protected] the respiratory system, diffusion in insect 122 SCIENCE z VOL. 276 z 4 APRIL 1997 z http://www.sciencemag.org  REPORTS tracheae, and osmotic and vapor pressure Fig. 1. Diagrammatic examples of in the plant vascular system). In spite of segments of biological distribu- these differences, these networks exhibit tion networks: (A) mammalian cir- essentially the same scaling laws. culatory and respiratory systems composed of branching tubes; For convenience we shall use the lan- (B) plant vessel-bundle vascular guage of the cardiovascular system, name- system composed of diverging ly, aorta, arteries, arterioles, and capillar- vessel elements; (C) topological ies; the correspondence to other systems is representation of such networks, straightforward. In the general case, the where k specifies the order of the network is composed of N branchings from level, beginning with the aorta the aorta (level 0) to the capillaries (level (k 5 0) and ending with the capil- N, denoted here by a subscript c) (Fig. lary (k 5 N ); and (D) parameters of 1C). A typical branch at some intermedi- a typical tube at the kth level. ate level k has length lk, radius rk, and pressure drop Dpk (Fig. 1D). The volume rate of flow is Q̇k 5 pr2k uk where uk is the flow velocity averaged over the cross sec- tion and, if necessary, over time. Each tube branches into nk smaller ones (12), so Downloaded from https://www.science.org at CASA Institution Identity on August 31, 2023 the total number of branches at level k is ~ngb2!2~N 1 1! 2 1 N serving, that is, the sum of the cross-sec- Nk 5 n0n1... nk. Because fluid is con- 5 n Vc (4) tional areas of the daughter branches equals ~ngb2!21 2 1 served as it flows through the system that of the parent, so pr2k 5 npr2k 11. Thus, where the last expression reflects the fractal bk [ rk11/rk 5 n21/2 5 b, independent of k. Q̇0 5 NkQ̇k 5 Nkpr2k uk 5 Ncpr2c uc (2) nature of the system. As shown below, one When the area-preserving branching rela- which holds for any level k. We next intro- can also prove from the energy minimiza- tion, b 5 n21/2, is combined with the duce the important assumption, the second tion principle that Vb } M. Because ngb2 , space-filling result for g, Eq. 5 yields a 5 above, that the terminal units (capillaries) 1 and N.. 1, a good approximation to Eq. 3/4, so B } M3/4. Many other scaling laws are invariant, so rc, lc, uc, and, consequently, 4 is Vb 5 V0/(1 2 ngb2) 5 Vc(gb2)2N/ follow. For example, for the aorta, r0 5 Dpc are independent of body size. Because (1 2 ngb2). From our assumption that cap- b2Nrc 5 Nc1/2rc and l0 5 g2Nrc 5 Nc1/3lc, the fluid transports oxygen and nutrients for illaries are invariant units, it therefore fol- yielding r0 } M3/8 and l0 } M1/4. This metabolism, Q̇0 } B; thus, if B } Ma (where lows that (gb2)2N } M. Using this relation derivation of the a 5 3/4 law is essentially a a will later be determined to be 3/4), then in Eq. 3 then gives geometric one, strictly applying only to sys- Q̇0 } Ma. Equation 2 therefore predicts that ln n tems that exhibit area-preserving branch- the total number of capillaries must scale as a52 (5) ing. This property has the further conse- ln~gb2! B, that is, Nc } Ma. quence, which follows from Eq. 2, that the To characterize the branching, we in- To make further progress requires knowl- fluid velocity must remain constant troduce scale factors bk [ rk11/rk and gk [ edge of g and b. We shall show how the throughout the network and be indepen- lk11/lk. We shall prove that in order to former follows from the space-filling fractal dent of size. These features are a natural minimize the energy dissipated in the sys- requirement, and the latter, from the energy consequence of the idealized vessel-bundle tem in the sense of the third principle minimization principle. structure of plant vascular systems (Fig. 1B), above, the network must be a convention- A space-filling fractal is a natural struc- in which area-preserving branching arises al self-similar fractal in that bk 5 b, gk 5 ture for ensuring that all cells are serviced automatically because each branch is as- g, and nk 5 n, all independent of k (an by capillaries. The network must branch so sumed to be a bundle of nN2k elementary important exception is bk in pulsatile sys- that a group of cells, referred to here as a vessels of the same radius (11). Pulsatile tems). For a self-similar fractal, the num- “service volume,” is supplied by each capil- mammalian vascular systems, on the other ber of branches increases in geometric pro- lary. Because rk ,, lk and the total number hand, do not conform to this structure, so portion (Nk 5 nk) as their size geometri- of branchings N is large, the volume sup- for them, we must look elsewhere for the cally decreases from level 0 to level N. plied by the total network can be approxi- origin of quarter-power scaling laws. Before proving self-similarity, we first ex- mated by the sum of spheres whose diame- Some features of the simple pipe model amine some of its consequences. ters are that of a typical kth-level vessel, remain valid for all networks: (i) The quan- Because Nc 5 nN, the number of gener- namely 4/3p(lk/2)3Nk. For large N, this es- tities g and b play a dual scaling role: they ations of branches scales only logarithmi- timate does not depend significantly on the determine not only how quantities scale cally with size specific level, although it is most accurate from level 0 (aorta) to N (capillary) within a ln~M/Mo) for large k. This condition, that the fractal a single organism of fixed size, but also how N5 (3) be volume-preserving from one generation a given quantity scales when organisms of ln n to the next, can therefore be expressed as different masses are compared. (ii) The frac- where M0 is a normalization scale for M 4/3p(lk/2)3Nk ' 4/3p(lk11/2)3Nk11. This tal nature of the entire system as expressed, (13). Thus, a whale is 107 times heavier relation gives g3k [ (lk11/lk)3 ' Nk/Nk11 5 for example, in the summation in Eq. 4 than a mouse but has only about 70% more 1/n, showing that gk ' n21/3 ' g must be leads to a scaling different from that for a branchings from aorta to capillary. The to- independent of k. This result for gk is a single tube, given by an individual term in tal volume of fluid in the network (“blood” general property of all space-filling fractal the series. These network systems must volume Vb) is systems that we consider. therefore be treated as a complete integrat- The 3/4 power law arises in the simple ed unit; they cannot realistically be mod- Vb 5 O N V 5 O pr l n N k50 k k N k50 2 k k k case of the classic rigid-pipe model, where the branching is assumed to be area-pre- eled by a single or a few representative vessels. (iii) The scaling with M does not http://www.sciencemag.org z SCIENCE z VOL. 276 z 4 APRIL 1997 123 depend on the branching ratio n. blood Vb(rk, lk, nk, M), the minimization In the linearized incompressible-fluid, We next consider the dynamics of the principle requires that the cardiac output be thin-wall approximation, this problem can network and examine the consequences of minimized subject to a space-filling geome- be solved analytically to give the energy minimization principle, which is try. To enforce such a constraint, we use the particularly relevant to mammalian vascu- lar systems. Pulsatile flow, which dominates the larger vessels (aorta and major arteries), standard method of Lagrange multipliers (l, lk, and lM) and so need to minimize the auxiliary function SD c c0 2 '2 J2~i3/2a! J0~i3/2a! and Z' c20r pr2c (8) must have area-preserving branching, so F(rk, lk, n) 5 W(rk, lk, nk, M) Here a [ (vr/m)1/2r is the dimensionless that b 5 n21/2, leading to quarter-power Womersley number (13), and c0 [ (Eh/ scaling. The smaller vessels, on the other hand, have the classic “cubic-law” branch- 1 lVb(rk, lk, nk, M) 1 O lNl 1l M N k50 k 3 kk M 2rr)1/2 is the Korteweg-Moens velocity. In general, both c and Z are complex functions ing (10), where b 5 n21/3, and play a (7) of v, so the wave is attenuated and disperses relatively minor role in allometric scaling. as it propagates. Consider the consequences First consider the simpler problem of Because B } Q0 and W 5 Q̇02Z, this prob- of these formulas as the blood flows through nonpulsatile flow. For steady laminar flow lem is tantamount to minimizing the im- progressively smaller tubes: For large tubes, of a Newtonian fluid, the viscous resistance pedance Z, which can therefore be used in a is large (in a typical human artery, a ' of a single tube is given by the well-known Eq. 7 in place of W. First, consider the case 5), and viscosity plays almost no role. Equa- Poiseuille formula Rk 5 8mlk/pr4k , where m is where nk 5 n, so that we can use Eqs. 4 and tion 8 then gives c 5 c0 and Z 5 rc0/pr2; the viscosity of the fluid. Ignoring small 6 for Vb and Z, respectively. For a fixed mass because both of these are real quantities, Downloaded from https://www.science.org at CASA Institution Identity on August 31, 2023 effects such as turbulence and nonlineari- M, the auxiliary Lagrange function F, the wave is neither attenuated nor dis- ties at junctions, the resistance of the entire which incorporates the constraints, must be persed. The r dependence of Z has changed network is given by (14) minimized with respect to all variables for from the nonpulsatile r24 behavior to r22. O NR 5 O pr8mln N N the entire system (rk, lk, and n). This re- Minimizing energy loss now gives hk/rk (and, k k Z5 4 k quires ]F/]lk 5 ]F/]rk 5 ]F/]n 5 0, which therefore, ck) independent of k and, most k50 k k50 k straightforwardly leads to bk 5 n21/3. More importantly, an area-preserving law at the @1 2 ~nb4/g!N 1 1#Rc generally, by considering variations with junctions, so bk 5 n21/2. This relation en- 5 respect to nk, one can show that nk 5 n, sures that energy-carrying waves are not ~1 2 nb4/g!nN (6) independent of k. The result, bk 5 n21/3, is reflected back up the tubes at branch points Now, nb /g , 1 and N.. 1, so a good 4 a generalization of Murray’s finding (17), and is the exact analog of impedance approximation is Z 5 Rc/(1 2 nb4/g)Nc. derived for a single branching, to the com- matching at the junctions of electrical Because Rc is invariant, Z } Nc21 } M2a, plete network. Now varying M and mini- transmission lines (18). As k increases, the which leads to two important scaling laws: mizing F in Eq. 7 (]F/]M 5 0) leads to Vb sizes of tubes decrease, so a 3 0 (in human blood pressure Dp 5 Q̇0Z must be indepen- } M, which is just the relation needed to arterioles, for example, a ' 0.05), and the dent of body size and the power dissipated derive Eq. 5. Although the result bk 5 n21/3 role of viscosity increases, eventually dom- in the system (cardiac output) W 5 Q̇0Dp } is independent of k, it is not area-preserving inating the flow. Equation 8 then gives c ' Ma, so that the power expended by the and therefore does not give a 5 3/4 when i1/2ac0/4 3 0, in agreement with observa- heart in overcoming viscous forces is a size- used in Eq. 5; instead, it gives a 5 1. It does, tion (18). Because c and, consequently, l independent fraction of the metabolic rate. however, solve the problem of slowing now have imaginary parts, the traveling Neither of these results depends on detailed blood in the capillaries: Eq. 2 gives uc/ u0 5 wave is heavily damped, leaving an almost knowledge of n, b, or g, in contrast to (nb2)2N 5 Nc21/3. For humans, Nc ' 1010, steady oscillatory flow whose impedance is, results based on Vb } M, such as Eq. 5, a 5 so uc/ u0 ' 1023, in reasonable agreement from Eq. 8, given by the Poiseuille formula; 3/4, and r0 } M3/8. From Eq. 2, Q̇0 5 pr20u0, with data (18). On the other hand, it leads that is, the r24 behavior is restored. Thus, which correctly predicts that the velocity of to an incorrect scaling law for this ratio: for large k, corresponding to small vessels, blood in the aorta u0 } M0 (2). However, an uc/ u0 } M21/4. Incorporating pulsatile flow bk 5 n21/3. We conclude that for pulsatile area-preserving scaling relation b 5 n21/2 not only solves these problems, giving the flow, bk is not independent of k but rather also implies by means of Eq. 2 that uk 5 u0 correct scaling relations (a 5 3/4 and uc/ u0 has a steplike behavior (Fig. 2). This picture for all k. This relation is valid for fluid flow } M0), but also gives the correct value for in plant vessels (because of the vascular bun- uc / u0. dle structure) (11, 15) and insect tracheae A complete treatment of pulsatile flow (because gas is driven by diffusion) (16); is complicated; here, we present a simpli- both therefore exhibit area-preserving fied version that contains the essential branching, which leads to 3/4 power scaling features needed for the scaling problem. of metabolic rate. Branching cannot be en- When an oscillatory pressure p of angular tirely area-preserving in mammalian circula- frequency v is applied to an elastic (char- tory systems because blood must slow down acterized by modulus E) vessel with wall to allow materials to diffuse across capillary thickness h, a damped traveling wave is walls. However, the pulsatile nature of the created: p 5 p0ei(vt 2 2pz/l). Here, t is time, mammalian cardiovascular system solves the z is the distance along the tube, l is the problem. wavelength, and p0 is the amplitude aver- Energy minimization constrains the net- aged over the radius; the wave velocity Fig. 2. Schematic variation of the Womersley num- work for the simpler nonpulsatile systems. c 5 2pvl. Both the impedance Z and the ber ak and the scaling parameters bk and gk with Consider cardiac output as a function of all dispersion relation that determines c are level number (k) for pulsatile systems. Note the relevant variables: W(rk, lk, nk, M). To sus- derived by solving the Navier-Stokes steplike change in bk at k 5 k from area-preserving tain a given metabolic rate in an organism equation for the fluid coupled to the pulse-wave flow in major vessels to area-increasing of fixed mass M with a given volume of Navier equations for the vessel wall (19). Poiseuille-type flow in small vessels. 124 SCIENCE z VOL. 276 z 4 APRIL 1997 z http://www.sciencemag.org  REPORTS is well supported by empirical data (18, 20, space-filling geometry, it remains un- cardiovascular system (Table 1); data are 21). The crossover from one behavior to the changed, so we still have l0 } M1/4. Blood needed to test other predictions. For exam- other occurs over the region where the wave volume Vb, however, is more complicated ple, the invariance of capillary parameters HS D and Poiseuille impedances are comparable in 2¯k ¯ implies Nc } M3/4 rather than the naı̈ve size. The approximate value of k where this Vc b. 1 2 ~nb2,g!k expectation Nc } M, so the volume serviced occurs (say, k) is given by r 2k /l k ' 8m/rc0, Vb 5 2 N by each capillary must scale as M1/4, and leading to N 2 k [ N ' ln(8mlc/rc0rc2)/ln n, ~b.g! b, 1 2 ~nb2,g! capillary density per cross-sectional area of tissue, as M21/12. F GJ independent of M. Thus, the number of ¯ generations where Poiseuille flow dominates 1 2 ~nb2.g!N 1 2 ~nb2.g!k A minor variant of the model describes should be independent of body size. On the 1 2 (9) the mammalian respiratory system. Al- other hand, the crossover point itself 1 2 ~nb2.g! 1 2 ~nb2.g! though pulse waves are irrelevant because grows logarithmically: k } N } ln M. For the tubes are not elastic, the formula for Z is humans, with n 5 3 (21), N ' 15 and This formula is a generalization of Eq. 4 quite similar to Eq. 8. The fractal bronchial N ' 22 (assuming Nc ' 2 3 1010), where- and is dominated by the first term, which tree terminates in NA } M3/4 alveoli. The as with n 5 2, N ' 24 and N ' 34. These represents the contribution of the large network is space-filling, and the alveoli play values mean that in humans Poiseuille tubes (aorta and arteries). Thus, Vb } the role of the service volume accounting flow begins to compete with the pulse nN11/3k } n4/3N, which, because it must for most of the total volume of the lung, wave after just a few branchings, dominat- scale as M, leads, as before, to a 5 3/4. As which scales as M. Thus, the volume of an ing after about seven. In a 3-g shrew, size decreases, the second term, represent- alveolus VA } M1/4, its radius rA } M1/12, Downloaded from https://www.science.org at CASA Institution Identity on August 31, 2023 Poiseuille flow begins to dominate shortly ing the cubic branching of small vessels, and its surface area AA } rA2 } M1/6, so the beyond the aorta. becomes increasingly important. This be- total surface area of the lung AL 5 NAAA } The derivation of scaling laws based on bk havior predicts small deviations from M11/12. This explains the paradox (22) that derived from Eqs. 7 and 8 (Fig. 2) leads to the quarter-power scaling (a * 3/4), observed AA scales with an exponent closer to 1 than same results as before. For simplicity, assume in the smallest mammals (2). An expres- the 3/4 seemingly needed to supply oxygen. that the crossover is sharp; using a gradual sion analogous to Eq. 9 can be derived for The rate of oxygen diffusion across an alve- transition does not change the resulting scal- the total impedance of the system Z. It is olus, which must be independent of M, is ing laws. So, for k. k, define bk [ b. 5 dominated by the small vessels (arterioles proportional to DpO2AA/rA. Thus, DpO2 } n21/3 and, for k , k, bk [ b, 5 n21/2. This and capillaries) and, as before, gives Dp M21/12, which must be compensated for by a predicts that area preservation only persists and u0 } M0. similar scaling of the oxygen affinity of he- in the pulsatile region from the aorta In order to understand allometric scal- moglobin. Available data support these pre- through the large arteries, at most until k ' ing, it is necessary to formulate an integrat- dictions (Table 1). k. First consider the radius of the aorta r0: its ed model for the entire system. The present Our model provides a theoretical, mech- scaling behavior is now given by r0 5 model should be viewed as an idealized anistic basis for understanding the central rcbk.2Nb,k 5 rcn1/3N11/6k 5 rcn1/2N21/6N, zeroth-order approximation: it accounts for role of body size in all aspects of biology. which gives r0 } M3/8 and, for humans, r0/rc many of the features of distribution net- Considering the many functionally inter- ' 104, in agreement with data (2). Using works and can be used as a point of depar- connected parts of the organism that must Eq. 3 we obtain, for the ratio of fluid ture for more detailed analyses and models. obey the constraints, it is not surprising that velocity in the aorta to that in the capil- In addition, because it is quantitative, the the diversity of living and fossil organisms is lary, u0 / uc 5 Nc(rc/r0)2 5 nN/3u0 / uc ' coefficients, Y0 of Eq. 1, can also, in prin- based on the elaboration of a few successful 250, independent of M, again in agree- ciple, be derived. It accurately predicts the designs. Given the need to redesign the ment with data. Because g reflects the known scaling relations of the mammalian entire system whenever body size changes, Table 1. Values of allometric exponents for variables of the mammalian with empirical observations. Observed values of exponents are taken from (2, cardiovascular and respiratory systems predicted by the model compared 3); ND denotes that no data are available. Cardiovascular Respiratory Exponent Exponent Variable Variable Predicted Observed Predicted Observed Aorta radius r0 3/8 5 0.375 0.36 Tracheal radius 3/8 5 0.375 0.39 Aorta pressure Dp0 0 5 0.00 0.032 Interpleural pressure 0 5 0.00 0.004 Aorta blood velocity u0 0 5 0.00 0.07 Air velocity in trachea 0 5 0.00 0.02 Blood volume Vb 1 5 1.00 1.00 Lung volume 1 5 1.00 1.05 Circulation time 1/4 5 0.25 0.25 Volume flow to lung 3/4 5 0.75 0.80 Circulation distance l 1/4 5 0.25 ND Volume of alveolus VA 1/4 5 0.25 ND Cardiac stroke volume 1 5 1.00 1.03 Tidal volume 1 5 1.00 1.041 Cardiac frequency v 21/4 5 20.25 20.25 Respiratory frequency 21/4 5 20.25 20.26 Cardiac output Ė 3/4 5 0.75 0.74 Power dissipated 3/4 5 0.75 0.78 Number of capillaries Nc 3/4 5 0.75 ND Number of alveoli NA 3/4 5 0.75 ND Service volume radius 1/12 5 0.083 ND Radius of alveolus rA 1/12 5 0.083 0.13 Womersley number a 1/4 5 0.25 0.25 Area of alveolus AA 1/6 5 0.083 ND Density of capillaries 21/12 5 20.083 20.095 Area of lung AL 11/12 5 0.92 0.95 O2 affinity of blood P50 21/12 5 20.083 20.089 O2 diffusing capacity 1 5 1.00 0.99 Total resistance Z 23/4 5 20.75 20.76 Total resistance 23/4 5 20.75 20.70 Metabolic rate B 3/4 5 0.75 0.75 O2 consumption rate 3/4 5 0.75 0.76 http://www.sciencemag.org z SCIENCE z VOL. 276 z 4 APRIL 1997 125 either during ontogeny or phylogenetic di- p. 133; see also M. H. Zimmerman, Xylem Structure Mammals (CRC Press, Boca Raton, FL, 1996). Care and the Ascent of Sap (Springer-Verlag, Berlin, must be taken in comparing measurements with versification, small deviations from quarter- 1983); M. T. Tyree and F. W. Ewers, New Phytol. prediction, particularly if averages over many suc- power scaling sometimes occur (3, 23). 119, 345 (1991). cessive levels are used. For example, if Ak [ (kpr2k is However, when body sizes vary over many 12. The branching of a vessel at level k into nk smaller the total cross-sectional area at level k, then for the vessels (Fig. 1) is assumed to occur over some small, aorta and major arteries, where k , k and the orders of magnitude, these scaling laws are but finite, distance that is much smaller than either lk branching is area-preserving, we predict A0 5 Ak. obeyed with remarkable precision. More- or lk11. This relation is similar to that assumed in the Suppose, however, that the first K levels are grouped over, the predicted scaling properties do not Strahler method [A. N. Strahler, Trans. Am. Geo- together. Then, if the resulting measurement gives depend on most details of system design, phys. Union 34, 345 (1953); (11, 21)]. A generaliza- AK, area-preserving predicts AK 5 KA0 (but not AK 5 tion to nonuniform branching, where the radii and A0). It also predicts r03 ' n 1/2(Nkrk3. Using results including the exact branching pattern, pro- lengths at a given level may vary, is straightforward. from M. LaBarbera [Science 249, 992 (1990)], who vided it has a fractal structure (24). Signif- 13. Normalization factors, such as M0, will generally be used data averaged over the first 160 vessels (ap- icantly, nonfractal systems, such as combus- suppressed, as in Eq. 1. In general, all quantities proximately the first 4 levels), gives, for human be- should be expressed in dimensionless form; note, ings, A0 ' 4.90 cm2, AK ' 19.98 cm2, r03 ' 1.95 tion engines and electric motors, exhibit however, that this does not guarantee that they are cm3, and (Nkrk3 ' 1.27 cm3, in agreement with area geometric (third-power) rather than quar- size independent and scale as M 0. For example, the preservation. LaBarbera, unfortunately, took the fact ter-power scaling (1). Because the fractal Womersley number, a of Eq. 8, although dimension- that AK Þ A0 and r03 ' (Nkrk3 as evidence for cubic less, scales as M 1/4. rather than area-preserving branching. For small network must still fill the entire D-dimen- 14. This formula is not valid for plant vessel bundles vessels, where k. k, convincing evidence for the sional volume, our result generalizes to a 5 because plants are composed of multiple parallel cubic law can be found in the analysis of the arteriolar D/(D 1 1). Organisms are three-dimension- vessel elements. Their resistance is given by Z 5 system by M. L. Ellsworth et al., Microvasc. Res. 34, 8ml /Ncprc4, where l is the length of a single vessel 168 (1987). al, which explains the 3 in the numerator of element, rc is its radius, and Nc is their total number. 21. Y. C. Fung, Biodynamics (Springer-Verlag, New the 3/4 power law, but it would be instruc- Downloaded from https://www.science.org at CASA Institution Identity on August 31, 2023 15. This relation holds for plant vessels from the roots to York, 1984). tive to examine nearly two-dimensional or- the leaves, but not within leaves [M. J. Canney, Phi- 22. P. Gehr et al., Respir. Physiol. 44, 61 (1981). los. Trans. R. Soc. London Ser. B 341, 87 (1993)]. 23. P. M. Bennett and P. H. Harvey, J. Zool. 213, 327 ganisms such as bryozoans and flatworms. 16. A. Krogh, Pfluegers Arch. Gesamte Physiol. Men- (1987); A. F. Bennett, Am. Zool. 28, 699 (1988); P. H. The model can potentially explain how schen Tiere 179, 95 (1920). Harvey and M. D. Pagel, The Comparative Method in fundamental constraints at the level of in- 17. C. D. Murray, Proc. Natl. Acad. Sci. U.S.A. 12, 207 Evolutionary Biology (Oxford Univ. Press, Oxford, dividual organisms lead to corresponding (1926). 1991). 18. C. G. Caro et al., The Mechanics of Circulation (Ox- 24. This is reminiscent of the invariance of scaling expo- quarter-power allometries at other levels. ford Univ. Press, Oxford, 1978). nents to details of the model that follow from renor- The constraints of body size on the rates at 19. J. R. Womersley, Philos. Mag. 46, 199 (1955); malization group analyses, which can be viewed as a which resources can be taken up from the J. Physiol. (London) 127, 553 (1955). generalization of classical dimensional analysis. 20. See, for example, A. S. Iberall, Math. Biosci. 1, 375 25. J.H.B. is supported by NSF grant DEB-9318096, environment and transported and trans- (1967) and T. F. Sherman, J. Gen. Physiol. 78, 431 B.J.E. by NSF grant GER-9553623 and a Fulbright formed within the body ramify to cause (1981), which contain summaries of earlier data; also Fellowship, and G.B.W. by the Department of Energy. quarter-power scaling in such diverse phe- M. Zamir et al., J. Biomech. 25, 1303 (1992) and J. K.-J. Li, Comparative Cardiovascular Dynamics of 13 September 1996; accepted 12 February 1997 nomena as rate and duration of embryonic and postembryonic growth and develop- ment, interval between clutches, age of first reproduction, life span, home range and Flexibility in DNA Recombination: Structure of territory size, population density, and max- the Lambda Integrase Catalytic Core imal population growth rate (1–3). Because organisms of different body sizes have dif- Hyock Joo Kwon, Radhakrishna Tirumalai, Arthur Landy,* ferent requirements for resources and oper- Tom Ellenberger* ate on different spatial and temporal scales, quarter-power allometric scaling is perhaps Lambda integrase is archetypic of site-specific recombinases that catalyze intermolec- the single most pervasive theme underlying ular DNA rearrangements without energetic input. DNA cleavage, strand exchange, and all biological diversity. religation 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 structure REFERENCES AND NOTES of the integrase catalytic domain reveals a protein fold that is conserved in organisms ___________________________ ranging from archaebacteria to yeast and that suggests a model for interaction with target 1. T. A. McMahon and J. T. 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Brown, and Brian J. Enquist Science, 276 (5309),. DOI: 10.1126/science.276.5309.122 Downloaded from https://www.science.org at CASA Institution Identity on August 31, 2023 View the article online https://www.science.org/doi/10.1126/science.276.5309.122 Permissions https://www.science.org/help/reprints-and-permissions Use of this article is subject to the Terms of service Science (ISSN 1095-9203) is published by the American Association for the Advancement of Science. 1200 New York Avenue NW, Washington, DC 20005. The title Science is a registered trademark of AAAS.