How scaling approaches can reveal fundamental principles in physiology and biomechanics

Small and large animals are fundamentally different. Their variation in size is linked to changes in body shape, physiology and locomotor biomechanics. For example, small mammals are crouched, their hearts beat faster and they seem capable of fantastic locomotor feats. A small mammal might run straight up the vertical trunk of a tree, which a larger animal would struggle to ascend. Even as far back as 1638, Galileo noticed these differences, writing ‘I believe that a little dog might carry on his back two or three dogs of the same size, whereas I doubt if a horse could carry even one horse of his own size’ (Galileo, 1638).

There have been many attempts to find a universal law or theory that predicts the relationships between an organism's size and physiological or biomechanical functions. Over the past 100 years, Journal of Experimental Biology has served as a platform for scientists to discuss these concepts and to communicate novel ideas (both empirical and theoretical) regarding how we can use scaling principles to understand the natural world. Thus, it is very appropriate to revisit this subject during the journal's centenary year.

In this Review, we provide key examples of how scaling has revealed fundamental principles in physiology and biomechanics. We compare how empirical measurements hold up against scaling theories and emphasise the importance of considering phylogenetic relationships in scaling analyses. We highlight how phylogenetic constraints can distort these relationships, but also how this distortion can be used to better understand the pathways evolution has travelled along. Finally, we provide forward-looking perspectives on how computational modelling, when combined with scaling theories, can advance our understanding of comparative physiology and biomechanics.

Most biological systems do not change linearly with size, but rather follow a power curve when plotted on an x–y plot. To make understanding these changes easier, we can plot values on a log–log scale. This converts the power curve relationships seen in nature into linear relationships, and reveals some fundamental patterns, which can be explained by a few general scaling ‘laws’. Geometric scaling is the simplest among these laws, and it predicts many (but not all) biological phenomena. In this case, animals are assumed to be scaled up or ‘geometrically similar’ (see Glossary) versions of one another, whereby shape remains unchanged as size increases (Fig. 1A,B). This is the same way we might expect a cube to scale with size. Lengths, when plotted against mass on this log–log plot, are expected to increase with a slope of 0.33; thus, when converted back to the power law formula, lengths would scale as M^0.33 (Fig. 1C; see also Box 1 for a discussion of regression models). Surface areas are expected to scale as M^0.66, whereas volumes scale as M^1.0, as density is assumed to be constant. One consequence of this scaling law is that musculoskeletal stresses (α=F/A, where F is force and A is cross-sectional area) are expected to scale with M^0.33 (see ‘Alterations in musculoskeletal design’, below). This is a result of the mismatch between the scaling of force, which an animal experiences due to gravity (F∝M¹), and the scaling of cross-sectional area (A∝M^0.66) of the muscles and bones that are available to resist this force.

Given this constraint, McMahon published two influential papers that provided alternative scaling laws: elastic similarity (McMahon, 1973) and static stress similarity (McMahon, 1975). Two animals are elastically similar if their support structures are similarly threatened by elastic failure (see Glossary) under their own body weight. Of note, McMahon's analysis of elastic similarity was based on compressive loading of different sized columns under their own weight, which is distinct from the elastic failure that occurs during bending. Even though the columns are turned on their sides, the bending is based on column end loading approaching a buckling failure, not a bending failure. An alternative law, static stress similarity, theorises that maximum stress (typically bending stress) remains the same under equivalent conditions and materials, at different sizes. Although these models (geometric similarity, elastic similarity and static stress similarity) are able to explain how several biological phenomena vary with animal size (Alexander, 1977; McMahon, 1975), there are numerous examples where scaling trends deviate from predictions based on these three models (e.g. Alexander et al., 1979; Biewener, 1982) (Fig. 1D), suggesting that diverse physiological and biomechanical constraints may apply between different taxonomic groups and size scales (Biewener, 2005).

One of the most intriguing uses of scaling has been exploring how metabolic energy use changes with body size. This topic has aroused interest for over 140 years, perhaps because of the counterintuitive relationship between these factors. As animals increase in size, they are likely to be composed of more cells, which will each require energy. Thus, a simple hypothesis might be that metabolic rate should scale with M^1.0. Yet, as the cells use energy they also produce heat, which must be lost from the body to avoid overheating; thus, an alternative hypothesis is that metabolic rate might be linked to the surface area available to lose this heat, thus scaling as M^0.66.

To explore this, scientists measured how metabolic rate varied among animals by comparing the basal metabolic rate (see Glossary), a measure of the maintenance rate of energy use. This was thought to be independent of food acquisition, exercise or stress. One of the earliest scientists to do so was a German physiologist, Rubner, who measured the metabolic rate of dogs, and found that it scaled with M^0.675, close to the surface area estimate (Rubner, 1883). This suggested that surface area determined the rate of energy consumption among animals. It wasn't until 50 years later that this rate was challenged by a Swiss physiologist, Kleiber, who compared the metabolic rates in a much larger group of mammals, from rats to steers, and observed a higher rate of M^0.74 (Kleiber, 1932). This was supported 2 years later by Brody et al. (1934), who looked at animals ranging from mice to elephants (M^0.73), and again by Benedict (1938) 4 years later, who included both birds and mammals. Kleiber finally settled on the exponent of 0.75, as this made calculation easier when using a slide rule or log table (Kleiber, 1961; Schmidt-Nielsen, 1984).

The trouble with metabolic rate scaling with the M^0.75 rule is that it doesn't match the M^0.66 scaling predicted from surface area, nor the M^1.0 predicted from cell number. McMahon (1975) attempted to explain the 0.75 scaling exponent using his concept of elastic similarity. He argued that metabolically related variables should scale with the cross-sectional area of the body or the muscle, rather than the surface area. Because under elastic similarity, smaller animals are expected to be more slender than larger animals, with cross-sectional area being proportional to diameter² (where d∝M^3/8), this would predict metabolic rate to be proportional to M^0.75, with metabolic rate scaling to supply these systems (e.g. the muscular system) with nutrients and oxygen. An alternative mechanism was proposed by West and Brown (2005), who suggested that the fractal nature of the energy-distributing vascular networks (e.g. blood vessels) could explain the scaling of metabolic rate with M^0.75. A similar approach was also taken by Bejan (2005), who used mathematical modelling of fluid flow, which maximised access and minimised heat loss in living systems, and also supported the M^0.75 scaling rule.

Yet, the ‘universality’ of the 0.75 scaling rule has often been questioned, with some studies noting how phylogenetic effects may have driven this result. White and Seymour (2003, 2005) argue that the 0.75 scaling of basal metabolic rate is an artefact of the inclusion of large herbivores (e.g. artiodactyls) in earlier studies. The inclusion of artiodactyls in scaling studies is problematic, as these animals – being ruminants – have very long digestive processes; thus, they are unlikely to enter the postabsorptive state required for the measurement of basal metabolic rate (White and Seymour, 2005). Consequently, their metabolic rate might be higher than expected, and given their large body size, the inclusion of these species might be forcing a higher-than-expected scaling exponent. Indeed, removal of these species from analyses reduces the exponent to 0.686, closer to that expected from consideration of surface area (White and Seymour, 2003, 2005). Other studies have also shown how phylogenetic effects can alter the observed scaling of metabolic rate. Work by Uyeda et al. (2017) suggests that there is not one universal scaling law, but instead this rate has changed throughout vertebrate evolution. By fitting metabolic rates over the vertebrate tree of life, Uyeda and colleagues (2017) showed that, throughout evolutionary history, there have been a number of shifts in both the slope and the intercept relating metabolic rate to body size (Fig. 2A). For example, mammals, salamanders and squamates have slopes well above the rate shared by fish and amphibians at the base of the tree. Understanding the underlying mechanism determining relative changes in metabolic rate among different phylogenetic groups is a fruitful area for future research.

It is perhaps not surprising that there is no single universal scaling law relating metabolic rate to organismal size, as organisms can change their metabolic rate not only over evolutionary time but also over much shorter time periods to meet the demands of digestion, reproduction or movement. Nagy (2005) suggested that an animal's field metabolic rate – the metabolic rate while it is active in its natural environment – may be a more important measure of its energy demands. The field metabolic rate slope can vary from <0.6 to >0.9 in species of mammals, birds and reptiles (Nagy, 2005). Further, the maximal aerobic metabolic rate, measured when an animal is pushed close to its locomotor capacity, has been shown to scale as 0.872, closer to the expected scaling for cell number, M^1.0 (Weibel and Hoppeler, 2005). In fact, the scaling of maximal metabolic rate closely correlates with the volume of mitochondria and capillaries in motor muscle, suggesting that maximal metabolic rate is determined by the energy needs of the cells active during maximal work (Weibel and Hoppeler, 2005).

Why basal metabolic rate scales with a somewhat lower exponent than maximal metabolic rate remains a mystery. White et al. (2022) suggest that it relates to the relative contribution of three fundamental aspects of life: metabolic rate, growth and reproduction. It is the combination of these factors that defines the fitness of an organism in its environment and, by extension, the scaling of metabolic rate. However, irrespective of the detailed mechanism that drives basal metabolic rate, the observation that its empirical scaling is close to 0.75 results in some interesting consequences. The mass-specific metabolic rate, which is simply the metabolic rate divided by the animal's mass, will decrease as an animal gets bigger as ∼M^−0.25. Consequently, biological processes occur at a relatively slower rate for larger-bodied animals. For example, rates of growth, ingestion and defecation all tend to scale close to M^−0.25 (Savage et al., 2004). Similarly, heart-beat frequency decreases with ∼M^−0.25 (Fig. 2B), whereas lifespan increases with ∼M^0.25, with the interesting consequence that the number of heart beats per lifetime for all animals is similar, at around ∼1.5×10⁹ (Levine, 1997). However, much like the scaling of metabolic rate, the scaling of both heart-beat frequency and longevity is linked with phylogeny, with both humans and birds being obvious outliers (Healy et al., 2014; Zhang and Zhang, 2009) (Fig. 2B,C).

Just as the patterns of energy use vary with size, scaling also affects the way in which animals locomote and the musculoskeletal structures that support and permit movement. Locomotion is one of the core themes that has structured the evolutionary form and function of organisms from the earliest Cambrian jellyfish to modern-day elephants. In this section, we discuss musculoskeletal and biomechanical adaptations that animals use to mitigate the mechanical consequences of size, and we provide insights into the scaling of mechanical and energetic demands of animal locomotion.

The millionfold size differences in terrestrial animals impose effects of scale on the structural elements that enable support and propulsion. As size increases, there is a mismatch whereby a muscle's ability to produce force or a bone's ability to resist force increases at a slower rate than the gravitational loads it must bear. One potential solution to deal with the size-predicted increases in stress is to evolve relatively stronger muscles or bones. Yet this does not appear to be a widely used mechanism, as the mechanical properties of bones (Biewener, 1982; Currey, 2006) and muscles (Medler, 2002; Seow and Ford, 1991) remain largely unchanged across mammals ranging in size from <0.05 to 700 kg. Do larger animals simply operate with lower safety factors (see Glossary) and therefore risk the failure of their musculoskeletal tissues? Apparently not. Researchers have determined the peak stresses in long bones of animals during locomotion based on direct measurements of in vivo surface bone strains or estimates from force platform and kinematic analyses of the limb (Biewener, 1990, 1991; Blob and Biewener, 1999; Rubin and Lanyon, 1984). The peak compressive bone stresses in animals ranging in mass from 0.04 to 2500 kg vary between 40 and 80 MPa during strenuous activity, with observed differences owing, in part, to differences in the experimental technique (Biewener, 1990). Thus, it appears that terrestrial animals maintain similar relative stresses during locomotion – operating with safety factors between 2 and 10. Thus, in order to mitigate the effects of increasing body size, there must be alterations in musculoskeletal design, regular changes in limb posture or a decline in locomotor performance.

The biomechanical consequences of scaling in the limbs of terrestrial animals have been historically studied by comparing the macroscopic design requirements for three locomotor elements: bones, muscles and tendons. Do larger animals have a more robust skeletal system to deal with potential increases in stress? As discussed above, McMahon (1973, 1975) proposed that animals scale with ‘elastic similarity’ to maintain similar elastic deformations under equivalent loading conditions. His model was supported by limb-bone scaling in ungulates (Alexander, 1977; McMahon, 1975) and some forelimb bones in reptiles and mammals (Campione and Evans, 2012), but within a wider size range of mammals, bone dimensions scale closer to geometric similarity (Alexander et al., 1979; Biewener, 1983; Christiansen, 1999; Maloiy et al., 1979; Selker and Carter, 1989). Giant animals (>1000 kg) defy both of these scaling laws and tend to have more robust (i.e. shorter, thicker) limb bones, which is likely to have implications for locomotor function in giant extinct species (Hutchinson, 2021).

A muscle's ability to generate force depends on its architecture, which is known to vary as animals increase in size. If animals scaled geometrically, muscle mass would scale with M^1.0, fascicle length would scale with M^0.33 and physiological cross-sectional area (PCSA) would scale proportional to M^0.66. The scaling of muscle design has been studied across a variety of mammals, birds and lizards (Alexander, 1977; Alexander et al., 1981; Bennett, 1996; Bennett and Taylor, 1995; Cieri et al., 2020; Cuff et al., 2016a,b; Dick and Clemente, 2016; Eng et al., 2008; Maloiy et al., 1979; McGowan et al., 2008; Pollock and Shadwick, 1994 b). Together, these studies demonstrate that, across most species, the size and architecture of individual muscles scales in a broadly similar fashion: muscle mass scales with weakly lower to modestly higher slopes when compared with geometric similarity (M^0.9–1.15); PCSA scales with weak to modestly higher slopes (M^0.69–0.91); fascicle length scales with weakly lower to modestly higher slopes (M^0.14–0.5). Yet, the scaling of muscle architecture can vary between different muscle groups; for example, between forelimb and hindlimb muscles or extensors and flexors (e.g. Cieri et al., 2020; Dick and Clemente, 2016). The functional implications of such variation provide an interesting area for future research.

In contrast to bones and muscles, tendon cross-sectional area scales from negative allometry (see Glossary) to geometric similarity (M^0.56–0.68) in animals ranging in size from 0.4 to 545 kg (Bennett, 2000; Pollock and Shadwick, 1994b). Because of the relatively thinner tendons, the ratio of muscle force to tendon area increases with size, such that larger mammals and birds favour elastic energy savings (Pollock and Shadwick, 1994b). Given that tendon mechanical properties are generally similar across species (Bennett et al., 1986; Pollock and Shadwick, 1994a), this scaling relationship suggests that larger animals may operate at lower tendon safety factors. This is certainly the case for large bipedal hopping animals, such a macropodids, whose tendons undergo higher stresses with increased size during locomotion (scaling with M^0.20–0.5; Bennett and Taylor, 1995; McGowan et al., 2008; Thornton et al., 2022). These higher stresses may, in part, underlie the remarkable hopping efficiency seen in macropodids via strain energy return capacity (see Glossary), especially at high speeds (Dawson and Taylor, 1973).

Along with changes in the design of musculoskeletal tissues, animals of different sizes can also adapt their limb posture to reduce size-related increases in stress. Small mammals move with crouched postures whereby their limbs are flexed, whereas larger animals move with an upright posture with extended joints. This transition to ‘uprightness’ offers mechanical benefits, allowing larger mammals to balance rotational moments at their limb joints, with lower muscle forces than expected for their body size. Mammals appear to use these posture changes to their advantage. In a highly influential scaling study, Biewener (1989) demonstrated that the effective mechanical advantage (EMA; see Glossary) scales with M^0.25, nearly matching the predicted increase in stress that would occur if posture remained unchanged with size. However, postural changes reach an upper limit at ∼300 kg, as above this body mass the joints are fully extended. Further, this stress reduction strategy may not be ubiquitously employed across all taxa. Kinematics measurements in felids standing at rest (4–200 kg; Day and Jayne, 2007) or in varanid lizards during mid-stance of locomotion (0.04–8 kg; Clemente et al., 2011) demonstrate that these groups maintain crouched postures across their full size range, offering unique opportunities to understand how the problem of size is solved without modifications in limb posture. If these groups fail to employ biomechanical strategies to reduce stress with increasing body size, the only option left is to decrease locomotor performance.

As we have seen above, scaling laws have often been used as null hypotheses to test mechanisms for how we predict animals might change with size. The top speed of animal locomotion has attracted considerable research interest, yet our scaling laws are poor at predicting how speed changes as animals get bigger. Geometric scaling, which appears to explain changes in shape well, predicts that maximum running speed should be independent of mass (M⁰) (Hill, 1950); however, Hill's analysis may be oversimplified for running speed, as it best argues for similar jump height across animals of different size. The elastic similarity criterion predicts that the speed of running animals should scale as M^0.25, owing to stride frequency (see Glossary) and stride length (see Glossary) scaling as M^−0.125 and M^0.375, respectively (McMahon, 1975). Static stress similarity predicts speed should scale as M^0.40 (McMahon, 1975), whereas the dynamic similarity model (see Glossary; Box 2) predicts a theoretical reduced exponent for velocity of M^0.16 (Gunther, 1975). The problem with these scaling estimates is the observation that speed does not increase monotonically with size. Instead, the fastest animals are not the largest nor the smallest, but rather of an intermediate size (Garland, 1983). This is observed both inter-specifically among mammals (Garland, 1983) and lizards (Clemente et al., 2009; Van Damme and Vanhooydonck, 2001), and intra-specifically in varanids (Clemente et al., 2012) and frogs (Clemente and Richards, 2013).

It is perhaps not too surprising that speed does not scale linearly with size given what we know of the scaling of posture and the differential taxon-specific scaling of the musculoskeletal system, explored above. The speed at which an animal moves is modulated by changes in both stride length and stride frequency. Yet, understanding how stride length and stride frequency change with size is difficult, as animals also change these features as they move at different speeds. The earliest studies to explore these stride parameters did so in mammals at the transition between different gaits. These transitions were thought to be equivalent speeds in different animals, meaning the scaling of stride length and frequency could be determined. The studies revealed that large animals take longer (M^0.38) but less frequent strides (M^{−0.14–−0.17}; Heglund et al., 1974) compared with smaller animals. Similar results have been observed for animals moving in the wild (Pennycuick, 1975) or during laboratory treadmill measurements across different gaits (Heglund and Taylor, 1988).

Exploring stride length and stride frequency changes using a much larger dataset of 103 species (Granatosky and McElroy, 2022) reveals that, like speed, these features probably do not change consistently with size (Fig. 3). Iriarte-Díaz (2002) used change point regression (see Glossary) on relative speeds to suggest that a transition point between two linear scaling patterns occurs at ∼30 kg. Using this transition to subset the data from Granatosky and McElroy (2022) shows that mean stride length initially increases with M^0.32 (slightly less than the estimate given by Heglund et al., 1974), but above the transition this rate slows to M^0.13 (much less than that given by Heglund et al., 1974; Fig. 3A). Stride frequency showed a less dramatic shift, decreasing as M^−0.22 below 30 kg, and scaling as M^−0.19 above (Fig. 3B). Together, this suggests that, for animals less than 30 kg, changes in stride length closely resemble changes in limb length, but as stride frequency decreases by a smaller amount, the overall result is a general increase in speed. Above the 30 kg transition point, however, changes in stride length reduce to slightly less than the scaling of stride frequency, likely causing the slight drop in speed above this body mass. Yet, as we have observed for other factors (e.g. musculoskeletal properties) above, there appears to be some variation between phylogenetic groups. These differences probably represent changes in the mechanisms available to reduce size-related stress, and will probably result in different groups changing speed in appreciably different ways (see Clemente et al., 2009; Dick and Clemente, 2017). Understanding the causes and mechanisms underlying these differences would be an exciting area to explore in future studies. See also Box 2 for a discussion of dynamic similarity.

Although the scaling of stride length might be explained by geometric changes in body shape, the changes in stride frequency are likely to require further exploration. More, Donelan and colleagues have explored two possible limitations to stride frequency: neural delays and inertial properties of the limbs. Of the former, it was noted that larger animals need to convey information over a longer distance. As conduction velocity is linked to nerve diameter, larger animals could circumvent this problem by growing relatively thicker neurons, but this would limit the number of neurons which could be packed into the limbs, resulting in a trade-off between resolution and responsiveness. When More et al. (2010) explored the scaling of conduction velocity in animals ranging from shrews to elephants, they found that it scales very low, as M^0.04. This low exponent means that larger animals are burdened with relatively longer physiological delays, as time lags for both neural and motor signals scale with positive allometry (More et al., 2010). Thus, large animals may partially offset the consequences of increased sensorimotor delays by reducing their frequency of movement (More and Donelan, 2018).

Compared with small animals, larger animals have heavier and longer limbs, partially owing to larger muscles, which together increase inertia. As such, the inertial properties of whole limbs and muscle tissue itself may also limit stride frequency as animals increase in size. Thangal and Donelan (2020) developed a biomechanical model to estimate the scaling of limb inertial delays – defined as the time between the onset and completion of a corrective movement. They found that inertial delays scaled with strong positive allometry (M^0.28–0.35), meaning that larger animals require more absolute time to perform the same movement as small animals. Ross and colleagues (2018) have explored how tissue mass influences an individual muscle's mechanical output. Using models, they demonstrated that the inertial effects from a muscle's mass lead to slower muscle shortening speeds and less work when compared with smaller or massless muscles (Ross et al., 2018). These modelling results were experimentally confirmed by adding external mass to contracting muscles in situ (Ross et al., 2020). Slower muscle shortening speeds would probably act to decrease stride frequency with increasing body size. These studies suggest that during low-amplitude movements, sensorimotor delays are likely to dictate the response time for smaller animals, whereas delays due to inertial properties dominate the response time for larger animals. Further, this highlights that muscle properties measured on single fibres, bundles of fibres or small muscles, which are then used to predict performance, may need to account for mass and inertial effects. Yet, with few experimental studies, our understanding of the multi-scale nature of neural control and muscle mechanics remains incomplete; this is certainly an exciting topic for future research.

Current scaling analysis in terrestrial animals has focused on the well-studied and most common forms of steady-state locomotion; for example walking, running and hopping. It would be exciting to apply scaling principles to understand locomotor performance during non-steady movements; for example, while navigating uneven terrain (Birn-Jeffery et al., 2014) or during turning manoeuvres (Haagensen et al., 2022). These are perhaps more relevant to the diversity of behaviours that animals perform in their natural environments.

Maximum speed is not the only important performance characteristic that underpins locomotion. Animals move for a variety of reasons, such as to find food or a suitable mate, or to escape predators. Although speed might determine the outcome of predator–prey interactions, some animals travel vast distances – for example, during migrations – and would benefit from locomotion that costs very little energy per unit distance travelled.

The energetic demands of locomotion and their relationship with body size have fascinated biologists for more than 50 years. Taylor and colleagues (1970) were the first to demonstrate that the cost of moving (expressed as the oxygen needed to transport 1 kg of body weight over 1 unit of distance) decreased in a regular way with increased body size (scaling with M^−0.40). Twelve years later, Taylor and his colleagues published a series of four highly influential papers in Journal of Experimental Biology, on animals ranging from the 0.124 kg tree shrew to the 254 kg zebu cattle, confirming that larger animals simply use less energy to move compared with smaller animals (scaling with a slightly shallower slope of M^−0.316; Fedak et al., 1982; Heglund et al., 1982a,b; Taylor et al., 1982). In the same series of papers, they demonstrated that the mechanical work that muscles perform to swing the limbs and raise and lower the body's centre of mass does not vary with size (Heglund et al., 1982a).

Heglund and colleagues (1982a,b) suggested that elastic energy storage, muscle shortening velocity and rates of muscle activation and/or deactivation are potential mechanisms that could help to explain the scaling of energetics. For example, stride frequency scales with a negative slope, meaning that smaller animals turn on and off their muscles to develop force at much higher rates than larger animals, even when moving at equivalent speeds. Kram and Taylor (1990) further proposed the ‘cost of generating force’ hypothesis to explain the energetics of locomotion across scales. Their hypothesis posits that the primary determinants of the energy cost required for locomotion are the cost of generating muscle force to support body weight and the time course of applying force to the ground. Using one equation that predicts the rate of metabolic energy expenditure as the product of the animal's body weight, the inverse of ground contact time, and a size-invariant metabolic cost coefficient, they showed that the cost of transport scales with M^−0.25 in animals ranging in size from a 30 g kangaroo rat to a 140 kg horse. Reilly et al. (2007) developed these ideas further, suggesting that the scaling of the cost of locomotion is non-linear, being much steeper for smaller crouched animals (M^−0.383) compared with larger erect species (M^−0.164). Yet, these ideas remain untested across different taxonomic groups.

We have already seen how phylogenetic effects can distort the scaling patterns among diverse groups of animals, for metabolic rates, longevity and speed. The reason for this is that most organisms tend to show a type of phylogenetic inertia, a tendency for closely related species to resemble each other (Harvey and Purvis, 1991). Yet, rather than this limiting our ability to understand how biological processes change with size, by including phylogenetic patterns in our analysis we can understand how different groups have responded to the challenges of evolving larger size.

For example, Cieri et al. (2022) were interested in how muscles increase in size as animals get bigger. Two possible mechanisms are available: (1) they can increase the number of muscle fibres and maintain these fibres at a similar size, or (2) they might increase the size of the individual fibres to maintain the number of fibres within a muscle. Surprisingly, the answer depends on the taxonomic group explored (Fig. 4A). Among birds and mammals, fibre number increases with size, yet among insects and fish, it is fibre diameter that increases (Cieri et al., 2022). Understanding why these differences exist between groups might reveal important functional constraints between taxa. For example, increasing fibre diameter as seen in fish and insects might have the advantage of simplifying control systems, as fewer innervations might be required, whereas increasing fibre number but retaining fibre diameter might facilitate the oxygen diffusion necessary for aerobic metabolism (Kinsey et al., 2011). The result of these effects might have consequences for the scaling of other biological processes, such as metabolic rate, which is linked with muscle performance and locomotory demands (Weibel and Hoppeler, 2005).

In other cases, the simple expectations from scaling laws such as geometric similarity can provide a null hypothesis to understand how problems of size are solved at different taxonomic levels. For example, a diverse group of animals, including insects, spiders, frogs and lizards, are able to climb up and down smooth surfaces. The adhesive force required to grip onto these surfaces is produced from adhesive pads, with the magnitude of this force being dependent on their area. Yet, here we see a similar problem to the force mismatch highlighted for bones and muscles. As animals increase in size, their mass increases faster than their surface area, meaning that larger (geometrically scaled) animals will be faced with having relatively smaller adhesive pads. There are two solutions to this problem, either increase the relative size of adhesive pads (i.e. pad area ∼M^1.0) or increase the ‘stickiness’ of the adhesive pads (i.e. F/A>M⁰). Labonte et al. (2016) investigated this among 225 species ranging from 20 μg to 200 g. When pad area is explored over this size range, it scales with M^0.95, suggesting that these animals are choosing the former solution, growing relatively larger adhesive pads (Fig. 4B). This solution is not without problems, as now relatively more of the body surface area must be devoted to adhesive pads, increasing from 0.02% in the smallest mites up to 4.3% in tokay geckos, the heaviest species capable of sticking to smooth surfaces. This might also set the upper limits for adhesion with size, as an animal the size of a human would require 40% of its body surface area to be a sticky adhesive pad, an impractically large amount.

Yet, among closely related species, pad area scales closer to the expectations for geometric similarity, with phylogenetically corrected statistics bringing the slope to M^0.70 (Labonte et al., 2016). When researchers explored how the force per unit pad area changed within these groups, they showed that it increased with M^0.19, suggesting that at least within some groups, the second option of getting stickier pads is possible. Thus, there are two potential solutions, and the answer depends not only on the taxonomic group explored but also at which taxonomic level it is investigated.

Studies that have used scaling as a tool have highlighted important insights into the evolution of species, and the constraints imposed by the physical laws of nature. Broadly sampling from the phylogenetic tree can further illustrate evolutionary innovations into how different groups have solved the problems of getting larger. Yet, there are relatively few examples where a wide phylogenetic sampling is used. This might be because it is difficult to measure diverse species, given both practical aspects and the biodiversity crisis which threatens many of our wild species. There might also be a limit to the phenotypic variation possible; for example, certain groups may not reach large body sizes (i.e. there is evolutionary constraint) or because these species (especially the largest species) have gone extinct. Finally, the phenotype of every species alive today is the result of enumerable selective pressures, from biophysical and developmental factors to predator–prey interactions and sexual conflict, meaning that no single phenotype has responded solely to the changes imposed by size, but rather all phenotypes are a compromise between competing pressures.

These effects will blur our ability to understand how any one factor, such as size, may have shaped an organism, and even the best statistical tools will be limited to the diversity currently present. In these cases, the only way to explore changes in size is to use a modelling or simulation approach. Using models and simulation tools, we can recreate the physiological systems of animals in silico, enabling us to understand extant or extinct phenotypes further. Doing so allows us to circumvent the potential pitfalls of phylogenetic constraints, to understand the consequences of morphological change or to explore phenotypes that have never (at least yet) existed in nature.

For example, Richards and Clemente (2012, 2013) built a ‘muscle-in-the-loop’ robotic system based on a swimming frog. This approach meant they could independently alter fin morphology, skeletal morphology, muscle morphology and motor coordination, observing each feature's influence on performance, measured as the limbs' propulsive thrust. Further, they used a computational model, based on geometric scaling, to demonstrate how the muscle force–velocity relationship limits swimming speed among frogs as they get larger (Clemente and Richards, 2013). As muscle power is maximum at one-third of maximum shortening velocity, the scaling of muscle force combined with drag forces in water suggests that the optimal shortening velocity for power cannot be reached for all body sizes, but instead follows a curvi-linear pattern, similar to that seen for maximal running speed.

Even more complex musculoskeletal models have been created, for example using open-source tools (Delp et al., 2007) whereby digital representations of bones, muscles and tendons are capable of recreating the kinematics and dynamics of human and animal locomotion. These models allow us to explore complex interactions between nervous, muscular and skeletal systems, and to ask ‘what if’ questions facilitating assessment of the relevance of musculoskeletal form on animal performance. Most commonly, human models have been used in rehabilitation, sports science and clinical settings, but more recently, other animal models have become a powerful tool in comparative and evolutionary biomechanics (Fig. 5). For example, models have been used to understand how the unique musculoskeletal structures in ostriches enable elastic energy storage so they can achieve fast running speeds with high metabolic economy (Rankin et al., 2016). Elsewhere, models have even been used to estimate the locomotor capabilities of extinct giant dinosaur species using fossil records (Hutchinson and Garcia, 2002).

Recent advances in optimisation approaches have allowed us to go one step further and use musculoskeletal models in a simulation framework whereby we can predict movements, without the need for experimental data, de novo (Falisse et al., 2019). This approach has been used to explore links between anatomy and jumping in birds (Bishop et al., 2021a) and to illuminate the crucial function of the tail during running in extinct bipedal dinosaurs (Bishop et al., 2021b). Predictive musculoskeletal simulations provide an exciting technique to explore how locomotor function varies with body size, but have not yet been used in a scaling context. Within models, the dimensions and properties of bones, muscles and tendons can be varied – scaled to represent different exponents observed across animals of different sizes. Optimisation criteria can then be prescribed to represent an ecologically or physiologically relevant task; for example, to maximise running speed or minimise cost of transport. These simulation tools have impressive potential to be used in a scaling context, providing exciting opportunities to advance our understanding of comparative physiology and biomechanics more broadly.

Body size determines many aspects of an animal's energy use, shape, movement patterns and ecology. In this Review, we have illustrated how scaling principles can be used to unveil the extent to which physical laws and evolutionary history can govern form and function. We hope our perspective highlights interesting and open questions; inspires further scaling studies that integrate empirical, theoretical or modelling approaches; and emphasises the exciting potential for comparative physiology and biomechanics to provide a framework for exploring the intimate links between form and function. Finally, the application of scaling laws goes beyond animal physiology and biomechanics, with potential to provide bio-inspiration for the design, control and application of wearable assistive technologies to augment or restore human movement or for legged robotic devices to achieve animal-like motions across a range of body shapes and sizes.

We thank our colleagues and students for insightful discussions on the topic. Three anonymous reviewers each gave excellent, constructive feedback which greatly improved the manuscript.