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

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Science  04 Apr 1997:
Vol. 276, Issue 5309, pp. 122-126
DOI: 10.1126/science.276.5309.122

Abstract

Allometric scaling relations, including the 3/4 power law for metabolic rates, are characteristic of all organisms and are here derived from a general model that describes how essential materials are transported through space-filling fractal networks of branching tubes. The model assumes that the energy dissipated is minimized and that the terminal tubes do not vary with body size. It provides a complete analysis of scaling relations for mammalian circulatory systems that are in agreement with data. More generally, the model predicts structural and functional properties of vertebrate cardiovascular and respiratory systems, plant vascular systems, insect tracheal tubes, and other distribution networks.

Biological diversity is largely a matter of body size, which varies over 21 orders of magnitude (1). Size affects rates of all biological structures and processes from cellular metabolism to population dynamics (2, 3). The dependence of a biological variable Y on body mass M is typically characterized by an allometric scaling law of the form Embedded Image(1) where b is the scaling exponent and Y0 a constant that is characteristic of the kind of organism. If, as originally thought, these relations reflect geometric constraints, then b should be a simple multiple of one-third. However, most biological phenomena scale as quarter rather than third powers of body mass (2-4): For example, metabolic rates B of entire organisms scale as M3/4; rates of cellular metabolism, heartbeat, and maximal population growth scale as M-1/4; and times of blood circulation, embryonic growth and development, and life-span scale as M1/4. Sizes of biological structures scale similarly: For example, the cross-sectional areas of mammalian aortas and of tree trunks scale as M3/4. No general theory explains the origin of these laws. Current hypotheses, such as resistance to elastic buckling in terrestrial organisms (5) or diffusion of materials across hydrodynamic boundary layers in aquatic organisms (6), cannot explain why so many biological processes in nearly all kinds of animals (2, 3), plants (7), and microbes (8) exhibit quarter-power scaling.

We propose that a common mechanism underlies these laws: Living things are sustained by the transport of materials through linear networks that branch to supply all parts of the organism. We develop a quantitative model that explains the origin and ubiquity of quarter-power scaling; it predicts the essential features of transport systems, such as mammalian blood vessels and bronchial trees, plant vascular systems, and insect tracheal tubes. It is based on three unifying principles or assumptions: First, in order for the network to supply the entire volume of the organism, a space-filling fractal-like branching pattern (9) is required. Second, the final branch of the network (such as the capillary in the circulatory system) is a size-invariant unit (2). And third, the energy required to distribute resources is minimized (10); this final restriction is basically equivalent to minimizing the total hydrodynamic resistance of the system. Scaling laws arise from the interplay between physical and geometric constraints implicit in these three principles. The model presented here should be viewed as an idealized representation in that we ignore complications such as tapering of vessels, turbulence, and nonlinear effects. These play only a minor role in determining the dynamics of the entire network and could be incorporated in more detailed analyses of specific systems.

Most distribution systems can be described by a branching network in which the sizes of tubes regularly decrease (Fig. 1). One version is exhibited by vertebrate circulatory and respiratory systems, another by the "vessel-bundle" structure of multiple parallel tubes, characteristic of plant vascular systems (11). Biological networks vary in the properties of the tube (elastic to rigid), the fluid transported (liquid to gas), and the nature of the pump (a pulsatile compression pump in the cardiovascular system, a pulsatile bellows pump in the respiratory system, diffusion in insect tracheae, and osmotic and vapor pressure in the plant vascular system). In spite of these differences, these networks exhibit essentially the same scaling laws.

Fig. 1.

Diagrammatic examples of segments of biological distribution networks: (A) mammalian circulatory and respiratory systems composed of branching tubes; (B) plant vessel-bundle vascular system composed of diverging vessel elements; (C) topological representation of such networks, where k specifies the order of the level, beginning with the aorta (k = 0) and ending with the capillary (k = N); and (D) parameters of a typical tube at the kth level.

For convenience we shall use the language of the cardiovascular system, namely, aorta, arteries, arterioles, and capillaries; the correspondence to other systems is straightforward. In the general case, the network is composed of N branchings from the aorta (level 0) to the capillaries (level N, denoted here by a subscript c) (Fig. 1C). A typical branch at some intermediate level k has length lk, radius rk, and pressure drop δpk (Fig. 1D). The volume rate of flow is Embedded Imagek=πrk2ūk where ūk is the flow velocity averaged over the cross section and, if necessary, over time. Each tube branches into nk smaller ones (12), so the total number of branches at level k is Nk = n0n1 . . . nk. Because fluid is conserved as it flows through the system Embedded Image(2) which holds for any level k. We next introduce the important assumption, the second above, that the terminal units (capillaries) are invariant, so rc, lc, ūc, and, consequently, Δpc are independent of body size. Because the fluid transports oxygen and nutrients for metabolism, Embedded Image0B; thus, if BMa (where a will later be determined to be 3/4), then Embedded Image0Ma. Equation 2 therefore predicts that the total number of capillaries must scale as B, that is, Nc Embedded Image Ma.

To characterize the branching, we introduce scale factors βkrk+1/rk and γklk+1/lk. We shall prove that in order to minimize the energy dissipated in the system in the sense of the third principle above, the network must be a conventional self-similar fractal in that βk = β, γk = γ, and nk = n, all independent of k (an important exception is βk in pulsatile systems). For a self-similar fractal, the number of branches increases in geometric proportion (Nk = nk) as their size geometrically decreases from level 0 to level N. Before proving self-similarity, we first examine some of its consequences.

Because Nc = nN, the number of generations of branches scales only logarithmically with size Embedded Image(3) where M0 is a normalization scale for M (13). Thus, a whale is 107 times heavier than a mouse but has only about 70% more branchings from aorta to capillary. The total volume of fluid in the network ("blood" volume Vb) is Embedded ImageEmbedded Image (4) where the last expression reflects the fractal nature of the system. As shown below, one can also prove from the energy minimization principle that Vb Embedded Image M. Because nγβ2 < 1 and N >> 1, a good approximation to Eq. 4 is Vb = V0/(1 - nγβ2) = Vc(γβ2)-N/(1 - nγβ2). From our assumption that capillaries are invariant units, it therefore follows that (γβ2)-N Embedded Image M. Using this relation in Eq. 3 then gives Embedded Image(5)

To make further progress requires knowledge of γ and β. We shall show how the former follows from the space-filling fractal requirement, and the latter, from the energy minimization principle.

A space-filling fractal is a natural structure for ensuring that all cells are serviced by capillaries. The network must branch so that a group of cells, referred to here as a "service volume," is supplied by each capillary. Because rk << lk and the total number of branchings N is large, the volume supplied by the total network can be approximated by the sum of spheres whose diameters are that of a typical kth-level vessel, namely 4/3π(lk/2)3Nk. For large N, this estimate does not depend significantly on the specific level, although it is most accurate for large k. This condition, that the fractal be volume-preserving from one generation to the next, can therefore be expressed as 4/3π(lk/2)3Nk ≈ 4/3π(lk+1/2)3Nk+1. This relation gives γ3k ≡ (lk+1/lk)3Nk/Nk+1 = 1/n, showing that γkn-1/3 ≈ γ must be independent of k. This result for γk is a general property of all space-filling fractal systems that we consider.

The 3/4 power law arises in the simple case of the classic rigid-pipe model, where the branching is assumed to be area-preserving, that is, the sum of the cross-sectional areas of the daughter branches equals that of the parent, so πrk2 = nπrk2+1. Thus, βkrk+1/rk = n-1/2 = β, independent of k. When the area-preserving branching relation, β = n-1/2, is combined with the space-filling result for γ, Eq. 5 yields a = 3/4, so BM3/4. Many other scaling laws follow. For example, for the aorta, r0 = β-Nrc = Nc1/2rc and l0 = γ-Nrc = Nc1/3lc, yielding r0M3/8 and l0M1/4. This derivation of the a = 3/4 law is essentially a geometric one, strictly applying only to systems that exhibit area-preserving branching. This property has the further consequence, which follows from Eq. 2, that the fluid velocity must remain constant throughout the network and be independent of size. These features are a natural consequence of the idealized vessel-bundle structure of plant vascular systems (Fig. 1B), in which area-preserving branching arises automatically because each branch is assumed to be a bundle of nN-k elementary vessels of the same radius (11). Pulsatile mammalian vascular systems, on the other hand, do not conform to this structure, so for them, we must look elsewhere for the origin of quarter-power scaling laws.

Some features of the simple pipe model remain valid for all networks: (i) The quantities γ and β play a dual scaling role: they determine not only how quantities scale from level 0 (aorta) to N (capillary) within a single organism of fixed size, but also how a given quantity scales when organisms of different masses are compared. (ii) The fractal nature of the entire system as expressed, for example, in the summation in Eq. 4 leads to a scaling different from that for a single tube, given by an individual term in the series. These network systems must therefore be treated as a complete integrated unit; they cannot realistically be modeled by a single or a few representative vessels. (iii) The scaling with M does not depend on the branching ratio n.

We next consider the dynamics of the network and examine the consequences of the energy minimization principle, which is particularly relevant to mammalian vascular systems. Pulsatile flow, which dominates the larger vessels (aorta and major arteries), must have area-preserving branching, so that β = n-1/2, leading to quarter-power scaling. The smaller vessels, on the other hand, have the classic "cubic-law" branching (10), where β = n-1/3, and play a relatively minor role in allometric scaling.

First consider the simpler problem of nonpulsatile flow. For steady laminar flow of a Newtonian fluid, the viscous resistance of a single tube is given by the well-known Poiseuille formula Rk = 8µlkrk4, where µ is the viscosity of the fluid. Ignoring small effects such as turbulence and nonlinearities at junctions, the resistance of the entire network is given by (14) Embedded ImageEmbedded Image (6)

Now, nβ4/γ < 1 and N >> 1, so a good approximation is Z = Rc/(1 - nβ4/γ)Nc. Because Rc is invariant, ZNc-1M-a, which leads to two important scaling laws: blood pressure Δp = Embedded Image0Z must be independent of body size and the power dissipated in the system (cardiac output) W = Embedded Image0ΔpMa, so that the power expended by the heart in overcoming viscous forces is a size-independent fraction of the metabolic rate. Neither of these results depends on detailed knowledge of n, β, or γ, in contrast to results based on VbM, such as Eq. 5, a = 3/4, and r0M3/8. From Eq. 2, Embedded Image0 = πr02ū0, which correctly predicts that the velocity of blood in the aorta ū0M0 (2). However, an area-preserving scaling relation β = n-1/2 also implies by means of Eq. 2 that ūk = ū0 for all k. This relation is valid for fluid flow in plant vessels (because of the vascular bundle structure) (11, 15) and insect tracheae (because gas is driven by diffusion) (16); both therefore exhibit area-preserving branching, which leads to 3/4 power scaling of metabolic rate. Branching cannot be entirely area-preserving in mammalian circulatory systems because blood must slow down to allow materials to diffuse across capillary walls. However, the pulsatile nature of the mammalian cardiovascular system solves the problem.

Energy minimization constrains the network for the simpler nonpulsatile systems. Consider cardiac output as a function of all relevant variables: W(rk, lk, nk, M). To sustain a given metabolic rate in an organism of fixed mass M with a given volume of blood Vb(rk, lk, nk, M), the minimization principle requires that the cardiac output be minimized subject to a space-filling geometry. To enforce such a constraint, we use the standard method of Lagrange multipliers (λ, λk, and λM) and so need to minimize the auxiliary function Embedded ImageEmbedded Image (7)

Because BQ0 and W = Embedded Image02Z, this problem is tantamount to minimizing the impedance Z, which can therefore be used in Eq. 7 in place of W. First, consider the case where nk = n, so that we can use Eqs. 4 and 6 for Vb and Z, respectively. For a fixed mass M, the auxiliary Lagrange function F, which incorporates the constraints, must be minimized with respect to all variables for the entire system (rk, lk, and n). This requires ∂F/∂lk = ∂F/∂rk = ∂F/∂n = 0, which straightforwardly leads to βk = n-1/3. More generally, by considering variations with respect to nk, one can show that nk = n, independent of k. The result, βk = n-1/3, is a generalization of Murray's finding (17), derived for a single branching, to the complete network. Now varying M and minimizing F in Eq. 7 (∂F/∂M = 0) leads to VbM, which is just the relation needed to derive Eq. 5. Although the result βk = n-1/3 is independent of k, it is not area-preserving and therefore does not give a = 3/4 when used in Eq. 5; instead, it gives a = 1. It does, however, solve the problem of slowing blood in the capillaries: Eq. 2 gives ūc/ū0 = (nβ2)-N = Nc-1/3. For humans, Nc ≈ 1010, so ūc/ū0 ≈ 10-3, in reasonable agreement with data (18). On the other hand, it leads to an incorrect scaling law for this ratio: ūc/ū0M-1/4. Incorporating pulsatile flow not only solves these problems, giving the correct scaling relations (a = 3/4 and ūc/ū0M0), but also gives the correct value forūc/ū0.

A complete treatment of pulsatile flow is complicated; here, we present a simplified version that contains the essential features needed for the scaling problem. When an oscillatory pressure p of angular frequency Ω is applied to an elastic (characterized by modulus E) vessel with wall thickness h, a damped traveling wave is created: p = p0ei(ωt - 2πz/λ). Here, t is time, z is the distance along the tube, λ is the wavelength, and p0 is the amplitude averaged over the radius; the wave velocity c = 2πΩλ. Both the impedance Z and the dispersion relation that determines c are derived by solving the Navier-Stokes equation for the fluid coupled to the Navier equations for the vessel wall (19). In the linearized incompressible-fluid, thin-wall approximation, this problem can be solved analytically to give Embedded Image(8)

Here ω ≡ (Ωρ/µ)1/2r is the dimensionless Womersley number (13), and c0 ≡ (Eh/2ρr)1/2 is the Korteweg-Moens velocity. In general, both c and Z are complex functions of Ω, so the wave is attenuated and disperses as it propagates. Consider the consequences of these formulas as the blood flows through progressively smaller tubes: For large tubes, ω is large (in a typical human artery, ω ≈ 5), and viscosity plays almost no role. Equation 8 then gives c = c0 and Z = ρc0r2; because both of these are real quantities, the wave is neither attenuated nor dispersed. The r dependence of Z has changed from the nonpulsatile r-4 behavior to r-2. Minimizing energy loss now gives hk/rk (and, therefore, ck) independent of k and, most importantly, an area-preserving law at the junctions, so βk = n-1/2. This relation ensures that energy-carrying waves are not reflected back up the tubes at branch points and is the exact analog of impedance matching at the junctions of electrical transmission lines (18). As k increases, the sizes of tubes decrease, so ω → 0 (in human arterioles, for example, ω ≈ 0.05), and the role of viscosity increases, eventually dominating the flow. Equation 8 then gives ci1/2ωc0/4 → 0, in agreement with observation (18). Because c and, consequently, λ now have imaginary parts, the traveling wave is heavily damped, leaving an almost steady oscillatory flow whose impedance is, from Eq. 8, given by the Poiseuille formula; that is, the r-4 behavior is restored. Thus, for large k, corresponding to small vessels, βk = n-1/3. We conclude that for pulsatile flow, βk is not independent of k but rather has a steplike behavior (Fig. 2). This picture is well supported by empirical data (18, 20, 21). The crossover from one behavior to the other occurs over the region where the wave and Poiseuille impedances are comparable in size. The approximate value of k where this occurs (say, ) is given by r2/l ≈ 8μ/ρc0, leading to N -; ≈ ln(8μlcc0rc2)/ln n, independent of M. Thus, the number of generations where Poiseuille flow dominates should be independent of body size. On the other hand, the crossover point itself grows logarithmically: N ∝ ln M. For humans, with n = 3 (21), ≈ 15 and N ≈ 22 (assuming Nc ≈ 2 × 1010), whereas with n = 2, ≈ 24 and N ≈ 34. These values mean that in humans Poiseuille flow begins to compete with the pulse wave after just a few branchings, dominating after about seven. In a 3-g shrew, Poiseuille flow begins to dominate shortly beyond the aorta.

Fig. 2.

Schematic variation of the Womersley number ωk and the scaling parameters βk and γk with level number (k) for pulsatile systems. Note the steplike change in βk at k = from area-preserving pulse-wave flow in major vessels to area-increasing Poiseuille-type flow in small vessels.

The derivation of scaling laws based on βk derived from Eqs. 7 and 8 (Fig. 2) leads to the same results as before. For simplicity, assume that the crossover is sharp; using a gradual transition does not change the resulting scaling laws. So, for k > , define βk ≡ β> = n-1/3 and, for k < , βk ≡ β< = n-1/2. This predicts that area preservation only persists in the pulsatile region from the aorta through the large arteries, at most until k. First consider the radius of the aorta r0: its scaling behavior is now given by r0 = rcβ>-Nβ< = rcn1/3N+1/6 = rcn1/2N-1/6, which gives r0M3/8 and, for humans, r0/rc ≈ 104, in agreement with data (2) . Using Eq. 3 we obtain, for the ratio of fluid velocity in the aorta to that in the capillary, ū0/ūc = Nc(rc/r0)2 = n/3ū0/ūc ≈ 250, independent of M, again in agreement with data. Because γ reflects the space-filling geometry, it remains unchanged, so we still have l0M1/4. Blood volume Vb, however, is more complicated Embedded ImageEmbedded Image (9)

This formula is a generalization of Eq. 4 and is dominated by the first term, which represents the contribution of the large tubes (aorta and arteries). Thus, VbnN+1/3n4/3N, which, because it must scale as M, leads, as before, to a = 3/4. As size decreases, the second term, representing the cubic branching of small vessels, becomes increasingly important. This behavior predicts small deviations from quarter-power scaling (a ≳ 3/4), observed in the smallest mammals (2). An expression analogous to Eq. 9 can be derived for the total impedance of the system Z. It is dominated by the small vessels (arterioles and capillaries) and, as before, gives Δp and ū0M0.

In order to understand allometric scaling, it is necessary to formulate an integrated model for the entire system. The present model should be viewed as an idealized zeroth-order approximation: it accounts for many of the features of distribution networks and can be used as a point of departure for more detailed analyses and models. In addition, because it is quantitative, the coefficients, Y0 of Eq. 1, can also, in principle, be derived. It accurately predicts the known scaling relations of the mammalian cardiovascular system (Table 1); data are needed to test other predictions. For example, the invariance of capillary parameters implies NcM3/4 rather than the naïve expectation NcM, so the volume serviced by each capillary must scale as M1/4, and capillary density per cross-sectional area of tissue, as M-1/12.

Table 1.

Values of allometric exponents for variables of the mammalian cardiovascular 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.

View this table:

A minor variant of the model describes the mammalian respiratory system. Although pulse waves are irrelevant because the tubes are not elastic, the formula for Z is quite similar to Eq. 8. The fractal bronchial tree terminates in NAM3/4 alveoli. The network is space-filling, and the alveoli play the role of the service volume accounting for most of the total volume of the lung, which scales as M. Thus, the volume of an alveolus VAM1/4, its radius rAM1/12, and its surface area AArA2M1/6, so the total surface area of the lung AL = NAAAM11/12. This explains the paradox (22) that AA scales with an exponent closer to 1 than the 3/4 seemingly needed to supply oxygen. The rate of oxygen diffusion across an alveolus, which must be independent of M, is proportional to ΔpO2AA/rA. Thus, ΔpO2M-1/12, which must be compensated for by a similar scaling of the oxygen affinity of hemoglobin. Available data support these predictions (Table 1).

Our model provides a theoretical, mechanistic basis for understanding the central role of body size in all aspects of biology. Considering the many functionally interconnected parts of the organism that must obey the constraints, it is not surprising that the diversity of living and fossil organisms is based on the elaboration of a few successful designs. Given the need to redesign the entire system whenever body size changes, either during ontogeny or phylogenetic diversification, small deviations from quarter-power scaling sometimes occur (3, 23). However, when body sizes vary over many orders of magnitude, these scaling laws are obeyed with remarkable precision. Moreover, the predicted scaling properties do not depend on most details of system design, including the exact branching pattern, provided it has a fractal structure (24). Significantly, nonfractal systems, such as combustion engines and electric motors, exhibit geometric (third-power) rather than quarter-power scaling (1). Because the fractal network must still fill the entire D-dimensional volume, our result generalizes to a = D/(D + 1). Organisms are three-dimensional, which explains the 3 in the numerator of the 3/4 power law, but it would be instructive to examine nearly two-dimensional organisms such as bryozoans and flatworms. The model can potentially explain how fundamental constraints at the level of individual organisms lead to corresponding quarter-power allometries at other levels. The constraints of body size on the rates at which resources can be taken up from the environment and transported and transformed within the body ramify to cause quarter-power scaling in such diverse phenomena as rate and duration of embryonic and postembryonic growth and development, interval between clutches, age of first reproduction, life span, home range and territory size, population density, and maximal population growth rate (1-3). Because organisms of different body sizes have different requirements for resources and operate on different spatial and temporal scales, quarter-power allometric scaling is perhaps the single most pervasive theme underlying all biological diversity.

REFERENCES AND NOTES

  1. The branching of a vessel at level k into nk smaller vessels (Fig. 1) is assumed to occur over some small, but finite, distance that is much smaller than either lk or lk+1. This relation is similar to that assumed in the Strahler method [A. N. Strahler, Trans. Am. Geophys. Union 34, 345 (1953); (11, 21)]. A generalization to nonuniform branching, where the radii and lengths at a given level may vary, is straightforward.
  2. Normalization factors, such as M0, will generally be suppressed, as in Eq. 1. In general, all quantities should be expressed in dimensionless form; note, however, that this does not guarantee that they are size independent and scale as M0. For example, the Womersley number, ω of Eq. 8, although dimensionless, scales as M1/4.
  3. This formula is not valid for plant vessel bundles because plants are composed of multiple parallel vessel elements. Their resistance is given by Z = 8µl/Ncπrc4, where l is the length of a single vessel element, rc is its radius, and Nc is their total number.
  4. This relation holds for plant vessels from the roots to the leaves, but not within leaves [ M. J. Canney, Philos. Trans. R. Soc. London Ser. B 341, 87 (1993)].
  5. 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; also M. Zamir, et al., J. Biomech. 25, 1303 (1992) and J. K.-J. Li, Comparative Cardiovascular Dynamics of Mammals (CRC Press, Boca Raton, FL, 1996). Care must be taken in comparing measurements with prediction, particularly if averages over many successive levels are used. For example, if Ak = +kπrk2 is the total cross-sectional area at level k, then for the aorta and major arteries, where k < and the branching is area-preserving, we predict A0 = Ak. Suppose, however, that the first K levels are grouped together. Then, if the resulting measurement gives ĀK, area-preserving predicts ĀK = KA0 (but not ĀK = A0). It also predicts r03n1/2ΣNkrk3. Using results from M. LaBarbera [Science 249, 992 (1990)], who used data averaged over the first 160 vessels (approximately the first 4 levels), gives, for human beings, A0 ≈ 4.90 cm2, ĀK ≈ 19.98 cm2, r03 ≈ 1.95 cm3, and ΣNkrk3 ≈ 1.27 cm3, in agreement with area preservation. LaBarbera, unfortunately, took the fact that ĀKA0 and r03 ≈ ΣNkrk3 as evidence for cubic rather than area-preserving branching. For small vessels, where k > , convincing evidence for the cubic law can be found in the analysis of the arteriolar system by M. L. Ellsworth, et al., Microvasc. Res. 34, 168 (1987).
  6. This is reminiscent of the invariance of scaling exponents to details of the model that follow from renormalization group analyses, which can be viewed as a generalization of classical dimensional analysis.
  7. J.H.B. is supported by NSF grant DEB-9318096, B.J.E. by NSF grant GER-9553623 and a Fulbright Fellowship, and G.B.W. by the Department of Energy.
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