Allometry: revealing evolution's engineering principles

Three years after the founding of Journal of Experimental Biology, the ‘philosopher/geneticist’ J. B. S. Haldane (1892–1964) wrote an intriguing essay entitled, ‘On Being the Right Size’. He began by describing body size as the most obvious, but least studied, of the features that distinguish animals from one another (Haldane, 1926). Over the past century, many of those differences have not only been identified, but in the process, many quantitative patterns have emerged. It is now apparent that an animal's body size imposes ‘design limitations’ that apparently regulate many aspects of animal structure and function. As articulated by the comparative physiologist, C. R. Taylor (1939–1995): ‘the million-fold range in body weight found in terrestrial vertebrates provides us with a powerful natural experiment for understanding the design constraints under which vertebrates have been built’ (Taylor, 1977).

There are at least two reasons why this approach is insightful. First, comparing a shrew to an elephant provides an enormous ratio of signal-to-noise, making otherwise invisible underlying principles of design apparent. However, the robust signal-to-noise ratio (see Glossary) and the biological insights provided only become apparent when the magnitude of the variables measured span a broad body size range. For this reason, W. A. Calder III (1934–2002) cautioned that this comparative tool should only be used for animals spanning several orders of magnitude in body size (Calder, 1984). Furthermore, as the foundations of anatomy and physiology are based on first principles of chemistry and physics, making comparisons using body size as a discriminator may suggest that critical physical principles drive evolution.

In this Review, we provide a brief history of body size scaling in biology and identify apparent principles and insights that have emerged from these investigations. We revisit the ‘pathway for oxygen’ to contrast maximal and basal (resting) scaling within the respiratory system. We conclude by examining alternative avenues of exploration and how simple scaling conclusions can provide both insights and possible confusion.

For body size-dependent design constraints to be recognized, a mathematical tool is required. Expressing variables in the form of a simple power law equation, Y=aM^b, provides a quantitative and statistically testable means to identify body size-dependent patterns in biology. Thus, when any variable of interest (Y) is expressed as a function of body mass (M), the exponent b is the slope of the regression line when plotted on logarithmic coordinates and the intercept a, is the value of Y when M=1.

Perhaps the first description of biological scaling or ‘allometry’ (see Glossary) was the iconic image of bone shapes drawn by Galileo in the 16th century, showing the robust, thick shape of a large bone compared with a small one if drawn to the same scale. The bioengineer Thomas McMahon (1943–1999) analysed the scaling of bone dimensions from an engineering perspective and, as the strength of any column is proportional to its cross-sectional area, he reasoned that bones must not scale linearly with bone length, but rather that they must scale with diameter more so than with length (diameters as M^3/8 and bone lengths as M^1/4; McMahon, 1975). Because of this allometric scaling, the mass of the skeleton must scale relative to body mass with an exponent greater than one (e.g. M^1.1). Hence, the mass of the skeleton is about 5% of a shrew's body mass and 25% of an elephant's. If we begin with the assumption that all structures within the body scale linearly, i.e. contribute equally to body mass, because the skeleton is proportionately much heavier in large animals, what is compensating? Owing to a higher surface to volume ratio in small animals, the scaling of integument mass is complementary to that of skeletal mass, with a slope of b=0.9. Together, skin and skeleton constitute ∼25–30% of a mammal's body mass, independently of body size.

In contrast to the skeleton, most ‘volumes’ scale linearly with body mass (i.e. b=1). For example, in mammals, heart mass is about 0.5% of body mass, the volume of blood 7% and the largest contributor to body mass is skeletal muscle, contributing about 40% of the total body mass (Lindstedt and Schaeffer, 2002). We have found these volumes to be handy references when working with animals of diverse body sizes.

In addition to anatomical features, body size impacts almost all aspects of an animal's physiology and life history. Indeed, the per gram rate of metabolic fuel consumption in a mouse would be sufficient to cause the body fluids to boil in a cow. Max Rubner (1854–1932) may have been one of the first to attempt to understand and explain this scaling. As all objects exchange heat with their environment through their surface areas, so warm-blooded animals must do as well. Rubner's concept later became known as the ‘surface hypothesis’. Assuming all animals are of similar shape (an oversimplification), then their surface area is proportional to a characteristic length, symbolized as L, squared as L² and volume as L³. Thus surface area is proportional to volume (or mass) raised to the 2/3 power, M^2/3 in all similarly shaped objects. As heat production must equal heat loss, Rubner postulated that it should vary as M^2/3 (Rubner, 1883). However, although metabolic heat production must be lost to the environment, that does not mean that surface area is the trigger that sets metabolism (see Seymour and White, 2011 for discussion).

Experimental data on resting or basal metabolism was collected by Max Kleiber (1893–1976), professor of animal husbandry at the agriculture campus of the University of California, Davis. His interests in meat production led him to ask how animals of different sizes metabolize food. He compiled metabolic data for multiple ‘warm-blooded’ species resulting in the ‘Mouse to Steer’ curve which demonstrated that metabolism varies not as M^2/3 as postulated by the surface hypothesis but rather as M^3/4 (Kleiber, 1932).

It must be pointed out that attempting to find the definitive values of allometric exponents (if they exist) is a fool's errand. An excellent examination of the perils of attempting to do so is provided by White and colleagues who concluded that both exponents (2/3 and 3/4) are weakly, if at all, supported by the data (White et al., 2009). Acknowledging that there is sufficient uncertainty regarding the exact scaling exponent of metabolic rate, what must be true is that heat production (metabolic rate) must equal heat loss for body temperature to remain constant.

One consequence of the linear scaling of structures supplying metabolism (e.g. heart, lung), and the allometric scaling of the metabolic demand for oxygen, is a disproportionate metabolic body size-dependent ‘load’ on those structures. For example, in the equation cardiac output (CO, see Glossary) equals heart rate (HR, see Glossary) multiplied by stroke volume, CO=HR×SV, as stroke volume (SV, see Glossary) scales linearly with mass but cardiac output scales with oxygen demand as ∼M^3/4, the only way to increase cardiac output in small animals is by increasing heart rate. Indeed, resting heart rate scales as M^–1/4 resulting in resting cardiac output scaling in parallel with metabolism (CO=HR×SV: M^3/4=M^–1/4×M^1.0). However, heart rate is but one of many biological rates that scale close to M^–1/4. As rate (r) is the inverse of time (t) (r=1/t) biological cycles such as heart or respiratory rates can be expressed in terms of their durations or biological times, scaling inversely with size as M^1/4. These cycle times range in duration from milliseconds (muscle contraction times) to years or decades (gestation period or even lifespan itself). If these biological times are universal in nature, we can even speculate on how they may have varied among the largest now extinct land vertebrates (Fig. 1).

In his 1945 book, Samuel Brody (1890–1956) introduced a body size dependency he called ‘Physiological time and the equivalence of age’ (Brody, 1945). Although Brody did not provide support for this largely theoretical concept, A. V. Hill (1886–1977) did, suggesting that so many cycle times vary with body size that animals essentially have two clocks: the chronological one and a physiological one that is a function of body size (Hill, 1950). To reinforce this concept, W. R. Stahl (1929–1967) combined allometric equations to form dimensionless ratios (see Glossary). For example, as both heart rate and respiratory rate vary with nearly identical body scaling (b≈¼), their dimensionless ratio is ∼4 and independent of body size. Thus, at rest, mammals experience about 4 heart beats per breath – although this occurs in 250 ms in a shrew and 10 s in an elephant (Stahl, 1962). The suggestion of a body size-dependent ‘physiological time’ clock was further bolstered when the range of biological times was expanded and further quantified, spanning muscle fibre twitch contraction times lasting milliseconds to lifespan itself, lasting years. Biological rates as diverse as glomerular filtration rate to growth rate and gestation period to the life span of an erythrocyte, all vary with nearly identical body size dependency (Lindstedt and Calder, 1981; Calder, 1984). Employing Stahl's ratios, under basal conditions, the number of heartbeats or even joules of energy burned per gram of tissue per lifetime is roughly constant in all mammals, regardless of the chronological time over which these occur (Fig. 1).

Animals spend most of their lives living under basal (parasympathetic nervous system-dominant) conditions with the energy flux through the system minimized, which is of obvious evolutionary advantage. Indeed, the discussion up to this point has focused on animals in resting or basal conditions. One could argue however, that the strongest selection pressures may occur when fleeing a predator or capturing prey; conditions that require maximal effort and thus ‘enhanced’ anatomical and physiological support (sympathetic nervous system-dominant). Just as there are insights provided by allometric comparisons under basal conditions, the question arises, ‘what unique insights can be gained by studying these comparisons under maximal conditions?’.

As mentioned previously, most structures (i.e. anatomy) scale isometrically; however, their uses (i.e. physiology) scale allometrically. One mathematical consequence is that large animals have much greater ‘reserve capacity’ (see Glossary) relative to small ones (Lindstedt and Schaeffer, 2002; Schaeffer and Lindstedt, 2013). For example, in the smallest mammals, the pulmonary transit time under basal conditions is so short (∼400 ms) that the erythrocytes require nearly the full length of the pulmonary capillaries to reach oxygen saturation. In contrast, saturation occurs in the first fraction of the pulmonary capillary transit duration in the largest mammals (Lindstedt, 1984, 2021). Hence, in large animals, a significant increase in cardiac output (and consequent reduction in pulmonary transit time) is possible with no sacrifice in red blood cell oxygen saturation, which is simply not possible in the smallest animals. This would seem to confine maximum oxygen consumption to a lower multiple of rest in the smallest animals. Indeed, it is now apparent that maximum metabolic rate in mammals scales with a steeper slope than basal metabolism. Exercise-induced V̇_O2,max (see Glossary) scales with 0.872 power of body mass in 34 eutherian mammals covering a body mass range from 7 g to 500 kg (nearly five orders of magnitude in body mass; Weibel and Hoppeler, 2005). The scaling of V̇_O2,max is thus significantly different from the scaling of basal metabolic rate (BMR, see Glossary; 0.66<b<0.75); the slope is significantly larger for athletic than for non-athletic species and the metabolic scope is greater in large animals than in small ones (Weibel et al., 2004). The same study reports data for a subset of 11 species with congruent measurements of V̇_O2,max and the relevant structural variables on all levels of the respiratory cascade. For these 11 species, V̇_O2,max was found to scale with a body mass exponent of 0.96. These sets of data are further explored below.

E. R. Weibel and C. R. Taylor coupled allometry with compartmental analysis (see Glossary) as experimental tools to investigate the design of the mammalian respiratory system. In using this approach, they made the first quantitative analysis of the entire respiratory system, ‘the pathway for oxygen,’ from atmospheric source to mitochondrial sink (Taylor and Weibel, 1981). For the system analysis, they choose the respiratory system because it serves an overall dominant function, the transfer of oxygen from the environment to the organs. The respiratory system consists of a sequence of linked obligatory structures such that, at least in mammals, all oxygen is transported by the respiratory system. The overall flow of oxygen has a measurable and experimentally reproducible upper limit (V̇_O2,max). V̇_O2,max is induced by activation of a single organ system, the locomotor musculature, which in exercise consumes over 90% of the oxygen taken up (Weibel et al., 1991). For the allometric analysis they divided the respiratory system into four critical compartments, each analysed separately: diffusive flow in the lung across the air-blood barrier into the blood, convective flow in the circulatory system, diffusive flow into the cells from the capillaries, and finally the translation of oxygen into ATP within the mitochondrial sink. Their analysis also included separating functional and structural variables in all compartments of the respiratory cascade allowing for a quantitative assessment of the relevant structural variables with morphometric technique. As the demand for ATP sets the demand for oxygen flow, it is appropriate to start this summary of the results with the mitochondrial sink.

What is the magnitude of the mitochondrial sink? Answering this question required estimating the total volume of mitochondria in all skeletal muscles with an appropriate sampling scheme (Hoppeler et al., 1984). Indeed, the scaling of mitochondrial volume and of V̇_O2,max share nearly identical slopes (Weibel et al., 2004). It should come as no surprise that mitochondrial volume alone sets the demand for oxygen, hence V̇_O2,max, in mammalian species. Therefore, maximal mitochondrial oxygen consumption is invariant, it is a simple function of mitochondrial volume (5 ml O₂ ml^-1 mito. min⁻¹) for mammalian species (Hoppeler and Lindstedt, 1985). Indeed, when expressed further as a function of inner mitochondrial membrane, maximal mitochondrial oxygen consumption seems identical in all homeotherms (Schaeffer and Lindstedt, 2013).

Total capillary volume in muscles must support the maximal flow of oxygen. The scaling exponent of capillary length, volume or surface area is 0.98, not significantly different from the scaling of V̇_O2,max or mitochondrial volume (Weibel et al., 2004). The oxygen flow into skeletal muscles is invariant, at 15 ml O₂ ml⁻¹ capillary volume (Hoppeler and Lindstedt, 1985). As both the mitochondria and capillaries are phenotypically plastic, it is understandable that they are tuned to oxygen demand. Likewise, when comparing athletic and non-athletic mammals of the same body mass, athletic animals have a systematically larger haemoglobin (Hb) concentration (Conley et al., 1987). This indicates a shared adaptive effort of two structural variables: a larger capillary volume containing a larger erythrocyte volume that contributes about equally to a larger capacity for oxygen delivery (Weibel and Hoppeler, 2005). Poole et al. (2022) provide important updates to the concept of oxygen diffusion from capillaries to muscle mitochondria in their recent review.

Circulatory oxygen transport is given by stroke volume×heart rate×haemoglobin concentration of the blood. We can use heart size as proxy for stroke volume and find that both heart size (0.58% of body mass) and the haemoglobin concentration in the blood (13 g Hb 100 ml⁻¹ blood) do not vary with body mass. From this it follows that higher (maximal) heart rates must be responsible for the larger circulatory oxygen transport in small animals. We find maximal heart rate to scale to body mass as −0.15 in 34 eutherian species ranging in body mass from 7 g to 500 kg (Weibel and Hoppeler, 2005). This is close to the body mass-specific scaling of V̇_O2,max of 0.13 of the same species. Critically, as noted above, basal metabolic rate (BMR) and maximal metabolic rate (MMR) scaling exponents differ significantly. This difference is exaggerated in athletic species, which have larger hearts, and hence stroke volumes, as well as larger haemoglobin concentrations in their blood than sedentary species (Lindstedt et al., 1991).

With all downstream steps of the respiratory cascade tuned to maximal oxygen flow, it was expected that the relevant structural variable, pulmonary diffusing capacity (D_LO2, see Glossary), would also be proportional to V̇_O2,max in both allometric and adaptive variation (see Glossary). With allometric variation (see Glossary), the scaling exponent of D_LO2 was found to be significantly larger (1.08) than that of V̇_O2,max, indicating that small animals operate their air-blood barrier with a larger partial pressure gradient for oxygen (P_O2 gradient) than large animals (Weibel et al., 1991). Because the structure of the lung appears to have limited phenotypic plasticity, it seems to be built in ‘excess’ in sedentary animals, including humans. Thus, sedentary animals need a shorter stretch of the pulmonary capillary bed to reach oxygen saturation of the blood (58% in steers versus 75% in horses; Constantinopol et al., 1989).

The great tragedy of science - the slaying of a beautiful hypothesis by an ugly fact.

Thomas Huxley (1825–1895)

When patterns are identified in nature, they are begging for explanations. Nowhere has that been more evident than in the past century of allometric discovery. The difficulty is that, as appealing as overarching principles may be, exceptions and ‘ugly facts’ often result in their reconsideration. The difficulties and misdirections inherent in this exercise were nicely outlined by Kozłowski and Weiner (1997). We provide one example here. In an influential article in Science, West et al. (1997) proposed that the ¾ power law for metabolic rates in all organisms can be explained by a general model. They reasoned that animals and plants need to be supplied by linear, branching networks such as a bronchial trees, blood vessels, tracheoles or plant vasculature. For the mathematical treatment of distribution networks, they made three assumptions: (1) the supplying network is of a space-filling, fractal-like nature; (2) the terminal branch of the network is size-invariant; (3) the energy dissipated in the system is minimized. In the rigid-pipe case (plant vasculature and insect tracheoles), the ¾ power law for metabolic rates arises geometrically through the area-preserving (see Glossary) branching of bundles of size invariant elementary vessels (see Glossary), such as capillaries. As a consequence of this design, the fluid velocity is also constant in the plant vasculature, whereas oxygen delivery is driven by diffusion in insect tracheoles. In the mammalian vascular system, pulsatile flow (see Glossary) prevails in the aorta and the major arteries. In these vessels, area-preserving branching is observed, which leads to the ¾ power law scaling. However, in order to allow for gas exchange in the periphery, blood flow must reduce in the smaller branches. Energy minimization constraints require that the cardiac output be minimized within a space-filling geometry, and be treated for pulsatile flow, to obtain the ¾ scaling exponent. As vessels get smaller, the flow is dominated by viscosity (Poiseuille flow, see Glossary). The mathematical model of West et al. (1997) shows that this occurs in humans after a small number of branches, whereas in 3 g shrews, Poiseuille flow prevails immediately after the aorta.

West et al. (1997) propose that ‘…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’. To suggest a universal explanation for scaling laws, spelled out mathematically and rooted in established physical principles governing distribution networks, was an attractive proposition indeed. As indicated above, the general model for allometric scaling proposed by West et al. (1997) did support the iconic ‘Kleiber’ ¾ power relationship of body mass and basal metabolism. Unfortunately, experimental data does not support the sweeping contention of a universal ¾ scaling law in biology. The critical test for a theoretical model of allometric scaling of the components of the circulatory is not basal metabolism but limiting conditions of V̇_O2,max when the muscle circulatory system must accommodate over 90% of the cardiac output. The lack of the quantitative unifying theory of biological structure and organization (West and Brown, 2005) to explain the scaling of structure at V̇_O2,max became apparent at the Journal of Experimental Biology Symposium in 2004 at the Mountain Verita in Ascona (Weibel and Hoppeler, 2005).

In a landmark paper, Felsenstein (1985) pointed out, that in comparative numerical analyses, species should be regarded in their hierarchically organized phylogenetic context and not as independent samples drawn from a single distribution. He noted that shared ancestry could produce phenotypic similarity, thus violating statistical independence. Felsenstein (1985) also developed a method – phylogenetic ‘independent contrasts’ – which was based on the phylogenetic topology, taking evolutionary branch length into account. He modelled trait evolution by assuming ‘Brownian motion’ (see Glossary) along the branch length to deal with genetic drift, mutations and variable selection of quantitative traits on a phylogenetic time scale, which is over millions of years. As pointed out in a review by Huey et al. (2019), Felsenstein's method of independent contrasts avoids the non-independence of species data, but at the cost of a detailed knowledge of the phylogenetic relationship among the species under consideration and a particular mode of trait evolution along the branch length.

Felsenstein's (1985) paper marked the beginning of a paradigm shift. The initial major obstacle to using phylogenetic relationship in scaling was the difficulty in obtaining a detailed phylogenetic tree for the species under analysis. However, owing to the recent exponential growth of genomic data and computer power, this problem has essentially disappeared. It has been more difficult to deal with the rate of trait evolution. The assumption of ‘Brownian motion’ along the branch length accounts for molecular evolution but does not hold for continuous traits, which may evolve at different rates in different clades in a micro- or macroevolutionary context. Huey et al. (2019) points out several options to deal with or check for the complex problem of trait evolution. While we acknowledge the critical importance of phylogeny, we also stress that allometric analysis is one way to identify critical aspects of animal evolution that apparently transcend phylogeny. We believe that both approaches can provide unique insights.

It is critical to include in this Review the limitations to this approach. We believe the utility of this approach is a ‘lumpers’ dream and a ‘sorters’ nightmare. That is, lumping all mammals together, despite their enormous differences, is not an attempt to minimize the significance or vital importance of phylogeny. Likewise, we make no attempt to speculate on causation, though that is a very worthy endeavor. However, it is also a minefield that is out of the scope of this Review (see Kozłowski and Weiner, 1997; White et al., 2022 for insightful discussions of this topic). On the contrary, this purely empirical approach is intended to identify patterns with the caution that any scaling explanations must be mindful of these empirical patterns.

In 1929, August Krogh (1874–1949) introduced what has become known as the ‘Krogh Principle’, namely that there are ideal model systems for deciphering physiological principles (Krogh, 1929; Lindstedt, 2014). We view body size as one of those model systems; borrowing the words of George Somero: ‘allometry provides ‘unity in diversity’ (Somero, 2000). In other words, patterns emerge only when body size comparisons are made that would be undetectable when studying a single species. Understandably, once apparent patterns of design and function are identified, their causation is sought; however, the quest for deeper understanding should not obfuscate the emergent principles of design. In other words, our lack of clearly understanding causation does not invalidate the empirical and repeated patterns that emerge from allometric analysis. Until we agree on the underlying causation of body size dependency in biology, the patterns themselves do provide insights. That allometric relationships themselves are often better understood than their sometimes elusive underpinnings does not diminish the utility of this approach. For example, Calder used these relations to identify those outliers that did not conform to the patterns as evidence of what he termed ‘adaptive deviation’ (Calder, 1984), inferring selective pressure that resulted in a divergence from the expected/observed allometric pattern. For example, pronghorns are among the most aerobic species and the mass of their hearts is about three times the 0.5% allometric prediction (Lindstedt et al., 1991). Calder's point is that selection is most apparent relative to the allometric ‘default’ pattern. Similarly, Weibel et al. (1987) used the term ‘adaptive variation’ when exploring the difference in V̇_O2,max between same-sized athletic and sedentary mammalian species. Allometry thus demonstrates both how physics constrains form and function of animals and how evolution has pushed the limits of these constraints. As the Journal of Experimental Biology embarks on its next century of discovery, we are confident that body size will continue to provide valuable insights in design and function.