ABSTRACT
To reconcile the scaling of the mechanics and energetics of locomotion to recent data on the scaling of the mechanics of muscle fibres, I have extended the theory of Taylor and colleagues that the energetic cost of locomotion is determined by the cost of generating force by the fibres. By assuming (1) that the cost of generating force in a fibre is proportional to Vmax (maximum velocity of shortening) and (2) that, at physiologically equivalent speeds, animals of different body sizes recruit the same fibre types, this extension quantitatively predicts the scaling of the energetics of locomotion, as well as other observations, from the scaling of Vmax of the muscle fibres. First, the energetic cost of locomotion at physiologically equivalent speeds scales with Mb−0 16, where Mb is body mass, as does Vmax of a given fibre type. However, the energetic cost at absolute speeds (cost of transport) scales with Mb−0.30, because small animals must compress their recruitment order into a narrower speed range and, hence, recruit faster muscle fibre types at a given running speed. Thus, it costs more for small animals to move 1kg of their body mass 1 km not only because a given muscle fibre type from a small animal costs more to generate force than from a large one, but also because small animals recruit faster fibre types at a given absolute running speed.
In addition, this analysis provides evidence that Vmax scales similarly to 1/tc (where tc is foot contact time) and muscle shortening velocity (V), in agreement with recent models. Finally, this extension predicts that, at physiologically equivalent speeds, the weight-specific energetic cost per step is independent of body size, as has been found empirically.
INTRODUCTION
Over the past 20 years, Taylor and colleagues have examined how the energetics and mechanics of locomotion scale with body size in mammals. They found that although the weight-specific mechanical power used for running is independent of body size (Heglund et al. 1982), the energetic cost to move a gram of tissue a given distance scales with Mb−0.30 (Taylor et al. 1982). Load-carrying experiments led Taylor et al. (1980) to postulate that the energetics of locomotion depended on force generation by the muscles. They further proposed that the increased energetic cost of locomotion in small animals is due to the increased cost of generating force in their fibres, which in turn is associated with their higher maximum velocity of shortening (Vmax).
If this theory is correct, then we should be able to predict the scaling of whole-animal energetics from that of isolated muscle fibres. The paucity of data on the scaling of muscle properties with body size prevented quantitative testing of this theory in the past. With recent information on the scaling of muscle properties with body size (Rome et al. 1990; Altringham and Young, 1991), we can begin to test this hypothesis.
THEORY
is average muscle force (newtons), cost of generating force is given per unit length (watts per newton metre) and average fibre length is given in metres. According to Biewener (1989a), because small animals have a more crouched posture (and thus lower effective mechanical advantage) than large ones, they must generate greater weight-specific forces to support their body weight. Hence, 
Formally, the cost of generating force is a function of where on the force-velocity curve the muscles are operating (i.e. V/Vmax; where V is shortening velocity during locomotion). As V increases, force decreases but the rate of ATP splitting increases, so that when V equals Vmax, the cost of generating force is infinite. For this treatment, however, we assume that the cost of generating force is that during isometric contractions (V=0). This assumption is made for a number of reasons. First, it is probably a good approximation. During locomotion, muscle performs both shortening contractions (in which the cost of generating force is higher than during isometric contractions) and lengthening contractions (in which the cost of generating force is lower than during isometric contractions). Second, with the information available we cannot calculate the energetics of muscles during locomotion any more accurately. There is no information on V/Vmax in mammals during lengthening and shortening (as there is in fish, Rome et al. 1988) and, even if there were, there is very little information on the energetics of lengthening muscle. Third, Alexander’s (1991) analysis (based on a number of assumptions) concludes that the energetic cost of the muscle performing the contractions during locomotion is likely to be a constant multiple of the cost of generating force during isometric contractions. Hence, if energetic cost during isometric contractions underestimates the cost during locomotory contractions, it would do so by a constant which is scale-independent. Finally, the cost of generating isometric force can be approximated by using the Huxley (1957) model (see below) and can be measured in skinned fibres over a wide range of animal size (as suggested in Rome et al. 1990).
Although, there is no information on how the cost of generating force in isolated fibres scales with Mb, Rome et al. (1990) have recently found that Vmax of the slow oxidative (SO, type I) fibres scales with Wb−0.18 and that of fast glycolytic fibres (FG, type lib) scales with an even smaller exponent (Mb−0.07) (N.B. a given fibre type is qualitatively similar in small and large animals in relative Vmax, order of recruitment and aerobic capacity, but differs quantitatively in such parameters as Vmax). Thus, Vmax of a single fibre type scales with a smaller exponent than does the cost of generating force during locomotion (Rome et al. 1990). This quantitative discrepancy in the scaling exponents seems puzzling because from a theoretical viewpoint (see below) one might anticipate that the cost of generating force in isolated fibres is proportional to Vmax. Explaining this discrepancy thus presents a further challenge for Taylor’s hypothesis (equation 1).
Do Vmax and the cost of generating force in isolated fibres scale differently?
The simplest explanation of the apparent discrepancy noted above is that the cost of generating force simply increases more rapidly than Vmax in isolated fibres. The empirical evidence is equivocal on this point. There is some empirical evidence supporting the disproportionate increase in the cost of generating force compared to Vmax (frog vs tortoise muscle, Woledge, 1968; Q10 values of Vmax and the cost of generating force in frog fibres, Rome and Kushmerick, 1983), but there is also empirical evidence suggesting that the cost of generating force and Vmax scale similarly (mouse soleus vs EDL, Crow and Kushmerick, 1982, 1983).
From a theoretical analysis based on the Huxley (1957) model of crossbridge dynamics, it is likely that the cost of generating force scales similarly to Vmax. In this analysis, the rate of crossbridge attachment is f(l-n) and the rate of crossbridge detachment is gn, where f is the attachment rate constant, g is the detachment rate constant, and n is the proportion of attached crossbridges.
How can the cost of generating force during locomotion scale differently from Vmax of a single fibre type if they scale similarly in isolated fibres?
The difference between the exponents for Vmax of a single fibre type and the cost of generating force during locomotion seems paradoxical only because of the implicit assumption that the same muscle fibre types are recruited at a given absolute speed in both small and large animals. Only in this case would there be the expectation that the scaling exponent from a particular muscle fibre type should agree with the scaling exponent for the cost of generating force (or cost of transport) in locomoting animals. This assumption, however, seems unlikely. Fibres are recruited in a fixed order (Henneman et al. 1965) and, as speed increases, animals recruit faster fibre types (Rome et al. 1984). Because large animals can run faster than small ones (at least over much of the size range), it seems inevitable that the small animals will go through their recruitment order at a slower running speed.
Although muscle recruitment in mammals has not been measured as a function of running speed, it is reasonable to expect that animals cannot sustain exercise intensities where there is a significant recruitment of anaerobic fibres. Maximum sustainable galloping speed (where presumably all aerobic fibres are recruited) scales with Mb0 17 (e.g. 3.1ms−1 in rats and 11ms−1 in horses as in Fig. 1A; calculated from Heglund and Taylor, 1988). Thus, small animals must recruit their aerobic fibre types (SO and fast oxidative glycolytic, FOG) over a smaller running speed range and, thus, their recruitment order will be effectively ‘compressed’ into a narrower range of locomotion speeds (as has been previously reported in cold fish vs warm ones, Rome et al. 1984). Hence, at a given running speed, the small animal will have to recruit faster fibre types than the large animal, as illustrated in Fig. 1A. This means that, when measuring Ė at a given absolute running speed (e.g. 3.1 ms−1), we might be comparing the cost of generating force (and Vmax) of FOG fibres in the small animal (rat) with the cost of generating force (and Vmax) of SO fibres in the large animal (horse). This would result in a larger difference in the cost of generating force (and Vmax) between the large and small animal than would be given by the scaling exponent for a single fibre type (Fig. 1B), and hence potentially explain the discrepancy in scaling exponents between the cost of generating force during locomotion and Vmax of a single fibre type.
How compression of recruitment order in small animals influences scaling of the energetics of locomotion. (A) A schematic representation of recruitment of different fibre types (open symbols=SO, closed symbols=FOG) as a function of running speed in the rat (diamonds) and the horse (triangles). The Vmax values for each fibre type (data from Eddinger et al. 1986; Rome et al. 1990) have been measured from skinned fibres at 15°C except for the rat FOG, which is calculated from the Vmax of rat FG data. This figure demonstrates five points. First, at physiologically equivalent speeds, the animals use the same fibre types. Thus, at the maximum sustainable galloping speed (3.1 ms−1 in rats, 11ms−1 in horses), all aerobic (SO and FOG) fibres are recruited. Second, the physiologically equivalent speed occurs at a much higher running speed in large animals than in small ones (e.g. maximum sustainable galloping speeds are 11ms−1 and 3.1ms−1 for the horse and rat, respectively; calculated from Heglund and Taylor, 1988). Third, the recruitment order is therefore compressed into a slower range of running speeds in the small animal. Fourth, at an absolute running speed, the small animal will be recruiting faster fibre types. For instance at 3.1ms−1, the horse is probably using only slow fibres (SO) as this is close to its minimal trotting speed (Heglund and Taylor, 1988). Fifth, Vmax increases more rapidly with running speed in the small animal than in the large one, in a similar fashion to the way l/fc does. B shows how Vmax of SO and FOG fibres scale with Mb. If one measures Ė at physiologically equivalent speeds (i.e. maximum sustainable galloping speed), then the animals would be using the same fibre types. Hence the Vmax and the cost of generating force in the fibres being utilized would scale the same as Vmax of the FOG fibres (Mb≈−0 16). This agrees with how E of animals running at physiologically equivalent speeds scale. If Ė is measured at an absolute speed, such as 3.1ms−1, then the rat would be using both its FOG and SO fibres, but the horse at this speed will be using only its SO fibres. Thus, we would be comparing the mechanics and energetics of the FOG fibres in the rat to that of SO fibres in the large animals. This would give an equivalent scaling exponent of Mt≈−0.34 for the cost of generating force and Vmax of the recruited fibres, which is close to the scaling exponent for Ė at absolute running speeds.
How compression of recruitment order in small animals influences scaling of the energetics of locomotion. (A) A schematic representation of recruitment of different fibre types (open symbols=SO, closed symbols=FOG) as a function of running speed in the rat (diamonds) and the horse (triangles). The Vmax values for each fibre type (data from Eddinger et al. 1986; Rome et al. 1990) have been measured from skinned fibres at 15°C except for the rat FOG, which is calculated from the Vmax of rat FG data. This figure demonstrates five points. First, at physiologically equivalent speeds, the animals use the same fibre types. Thus, at the maximum sustainable galloping speed (3.1 ms−1 in rats, 11ms−1 in horses), all aerobic (SO and FOG) fibres are recruited. Second, the physiologically equivalent speed occurs at a much higher running speed in large animals than in small ones (e.g. maximum sustainable galloping speeds are 11ms−1 and 3.1ms−1 for the horse and rat, respectively; calculated from Heglund and Taylor, 1988). Third, the recruitment order is therefore compressed into a slower range of running speeds in the small animal. Fourth, at an absolute running speed, the small animal will be recruiting faster fibre types. For instance at 3.1ms−1, the horse is probably using only slow fibres (SO) as this is close to its minimal trotting speed (Heglund and Taylor, 1988). Fifth, Vmax increases more rapidly with running speed in the small animal than in the large one, in a similar fashion to the way l/fc does. B shows how Vmax of SO and FOG fibres scale with Mb. If one measures Ė at physiologically equivalent speeds (i.e. maximum sustainable galloping speed), then the animals would be using the same fibre types. Hence the Vmax and the cost of generating force in the fibres being utilized would scale the same as Vmax of the FOG fibres (Mb≈−0 16). This agrees with how E of animals running at physiologically equivalent speeds scale. If Ė is measured at an absolute speed, such as 3.1ms−1, then the rat would be using both its FOG and SO fibres, but the horse at this speed will be using only its SO fibres. Thus, we would be comparing the mechanics and energetics of the FOG fibres in the rat to that of SO fibres in the large animals. This would give an equivalent scaling exponent of Mt≈−0.34 for the cost of generating force and Vmax of the recruited fibres, which is close to the scaling exponent for Ė at absolute running speeds.
Alexander (1991) suggests that it might be more important for Vmax to scale as V rather than to scale as l/tc, to maintain a constant V/Vmax for maximum efficiency as Rome et al. (1988) have shown to occur in fish. However, it seems possible that both V and l/tc scale similarly, and that Vrnax scales appropriately for both. Lindstedt et al. (1985), by a combination of anatomical and physiological analyses, showed that the absolute length (Δl) by which knee extensors shorten scales with Mb026. Because the muscle fibres in small animals are shorter than in large animals (equation 3), the strain (Δl/fibre length) should be approximately scale-dent (depending on exact fibre architecture), and hence V, at both physiologically equivalent and absolute running speeds, should scale as stride frequency (as does l/tc and Vmax).
DISCUSSION
This extension of Taylor’s hypothesis explains from first principles why the cost of transport has a larger scaling exponent than for Vmax in a single fibre type, while Ė at physiologically equivalent speeds and Vmax in a single fibre type have similar scaling exponents. Thus, it provides strong support for Taylor et al.’s (1980) original hypothesis that the energetic cost of locomotion is determined by the cost of generating force in the fibres. Further, this treatment recognizes that the recruitment order in small animals is probably compressed into a narrower speed range than in large ones. Thus, it costs more for small animals to move 1 kg of their body mass 1 km not only because a given muscle fibre type from a small animal costs more to generate force than from a large one, but also because small animals recruit faster fibre types at a given absolute running speed (Fig. 1).
Finally, this extension of Taylor’s hypothesis, unlike previous papers (Heglund and Taylor, 1988), affords no special significance to the constancy of the cost per step at physiologically equivalent speeds, because it can explain the scaling of the energetics of locomotion without invoking it. It has been suggested that the constant cost per step is a basic property of locomotion and it costs small animals more to run at a given speed because they must take more steps and incur more cost associated with turning their muscles on and off (Heglund and Taylor, 1988).
To test this extension of Taylor’s hypothesis further, it is necessary to test the new assumptions on which it is based. One must determine scaling of the cost of generating force and Vmax with body size and determine the running speed of recruitment of different fibre types in mammals. As suggested by Alexander (1991), it would also be useful to compare the energetics of muscle undergoing lengthening and shortening contractions to the isometric case.
ACKNOWLEDGEMENTS
The author thanks Professors R. McN. Alexander, A. A. Biewener and C. R. Taylor for making helpful comments on the manuscript. This work was supported by NIH grant AR38404.
