Introduction

It is possible for animals of very different sizes to use similar patterns of locomotion. For example, kangaroo rats and kangaroos hop bipedally, quails and ostriches walk and run bipedally, and quadrupeds of a wide range of different sizes walk, trot and gallop. A more detailed comparison of locomotion in animals of different sizes can be made by comparing dimensionless parameters, such as stride length divided by leg length ('relative stride length', RSL), stance time divided by stride time ('duty factor', DF), peak ground reaction force (GRF) divided by body weight ('relative peak force', RPF) and the phase relationships of the limbs. Alexander and Jayes (1983) showed that mammalian species of very different sizes use the same values of these dimensionless parameters when moving at speeds corresponding to equal values of another dimensionless parameter, the Froude number (u²/gh, where u is forward speed, g is the acceleration due to gravity and h is hip height).

All of the dimensionless parameters considered above are `mechanical' dimensionless parameters, because the parameters from which they are constructed can be defined in terms of forces, lengths and times. Systems that have equal values of mechanical dimensionless parameters are said to be `dynamically similar' (Isaacson and Isaacson, 1975), so Alexander and Jayes (1983) described animals that have equal values of the above dimensionless parameters as moving in a `dynamically similar fashion'.

Dynamically similar locomotion is not, however, inevitable in animals moving at equal Froude number. For example, Alexander and Jayes (1983) found that cursorial and non-cursorial mammals do not move in a dynamically similar fashion at equal Froude number and Donelan and Kram (2000) found that humans do not move in a dynamically similar fashion when running at equal Froude number in different levels of simulated reduced gravity. Engineering theory tells us that, for dynamic similarity to occur in some aspect of a system, all relevant system parameters must scale appropriately with size (Isaacson and Isaacson, 1975). Previously, we have applied this to understanding dynamic similarity in animal locomotion. We argued that, because tendon elastic modulus does not increase with size as required for dynamic similarity, compensatory changes in other parameters with size are required for dynamically similar locomotion to be possible (Bullimore and Burn, 2004). We referred to these changes as `compensatory distortions', after a term that has been used in engineering to describe the changes that must be made in physical models in order to make them dynamically similar to the systems that they represent (Baker et al., 1973). We showed that the changes in limb posture that occur with size in mammals (Biewener, 1989) should compensate for the size-independence of tendon elastic modulus sufficiently for dynamically similar locomotion to be possible in species of different sizes.

This is an example of a general principle that is frequently encountered in studies of size effects in biology: where organisms of different sizes are made of the same materials, they cannot remain functionally similar unless systematic changes in form compensate for size effects. For example, for animal bones and tree trunks to maintain the same resistance to elastic buckling, their thickness must increase disproportionately with length (McMahon, 1973). Conversely, where form remains the same, differently sized organisms differ functionally. For example, the functional characteristics of the hairy appendages of crustaceans are size-dependent (Koehl, 2004). In the case of mammalian locomotion, changes in limb posture with size allow functional similarity in locomotor mechanics to be maintained despite the size-independence of tendon material properties. However, if no factor compensates for size effects, changes in locomotor patterns with size can be expected, so locomotion will not be dynamically similar.

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Fig. 1.

Consequences of the nonlinear stress-strain relationship of tendon for the scaling of tendon strain. Data of Riemersma and Schamhardt (1985) for equine superficial digital flexor tendon. Because the tendon is stiffer at higher stresses, a given increase in stress causes a smaller increase in strain when it occurs at a higher initial stress (compare shaded regions). A consequence of this is that the scaling exponent for strain is lower than the scaling exponent for stress. For example, over the part of the curve within and between the shaded regions, a stress proportional to causes a strain proportional to . Therefore, a nonlinear tendon stress-strain relationship can reduce the effects of size differences on tendon strain.

Size not only varies between species, but also within species. This occurs under two circumstances: (a) between adult individuals, due to genetic and environmental factors, and (b) within an individual during growth. The same principle will apply within species as between species: if tendon elastic modulus is size-independent, and a compensatory distortion does not occur, then individuals of different sizes would not be expected to move in a dynamically similar fashion. This raises two questions. (1) Over the smaller size range present within a species, could the resulting deviations from dynamically similar locomotion be large enough to be detectable and physiologically significant? (2) If so, do systematic deviations from dynamically similar locomotion occur with size within species, or does some factor (a compensatory distortion or a change in tendon elastic modulus with size) prevent this? Answering these questions is important in order to understand the biomechanical consequences of growth and the factors that affect the optimal size for the adults of a species.

Here we address both of these questions for the first circumstance under which intraspecific size differences occur: variation between adult individuals. We chose to study the domestic horse (Equus caballus) because it occurs in a wide range of sizes, allowing small deviations from dynamic similarity to be detected, and because there is a large volume of published anatomical and biomechanical literature on this species. To address the question of whether detectable deviations from dynamic similarity could occur (question 1 above), we modelled `idealised' horses that were identical except for size differences, so that there were no compensatory distortions, and predicted by how much locomotion would deviate from dynamic similarity in these animals. To address the question of whether real horses of different sizes move in a dynamically similar fashion (question 2 above), we measure RSL and DF in trotting horses of body mass 86-714 kg. The theoretical predictions are presented in Part 1 of this paper and the experimental results in Part 2.

A factor that could potentially decrease the effects of size on locomotion is the nonlinear stress-strain relationship of tendon. Because tendons are stiffer at higher stresses, a given increase in stress causes a smaller increase in strain when it occurs at higher stresses. A consequence of this is that tendon strain will scale less markedly with size than tendon stress (Fig. 1). If tendon elastic modulus does not increase with size as required for dynamic similarity, and there are no compensatory distortions, then it would be expected that larger animals would experience greater tendon strains, causing deviations from dynamically similar locomotion (Bullimore and Burn, 2004). The capacity of the nonlinear tendon stress-strain relationship to decrease the scaling exponent for strain would reduce this effect. This is not a compensatory distortion, because it does not depend upon something changing with animal size. It also could not compensate completely for a size-independent tendon elastic modulus, because it would never give the same tendon strain for different stresses. However, it could reduce deviations from dynamically similar locomotion to some extent. In order to estimate the magnitude of this effect, we made the predictions in Part 1 both with and without taking these nonlinear tendon properties into account.

(Part 1) Predicted scaling exponents

Two possibilities exist as to the effects of size in horses: (i) there may be no systematic changes with size, or (ii) there may be systematic changes in some parameters with size, for example, in anatomy or muscle activation patterns. In either case, there will also be individual variation due to other factors. Our approach to determining whether size differences could cause detectable deviations from dynamically similar locomotion in the absence of compensatory distortions was to model `idealised' horses which did not differ in any way except for in size. They therefore had identical muscle activation patterns and tendon properties and were geometrically similar (so all lengths scaled in proportion to and all areas scaled in proportion to , where M<sub>b</sub> is body mass). This represents situation (i) above, except that variation due to factors other than size is ignored. The relationships of RSL and DF to body mass predicted for these `idealised' horses were then compared to the relationships measured in real horses in Part 2. A difference between the predicted and measured relationships was taken to indicate that systematic changes with size do occur, i.e. that possibility (ii) above is correct.

The predictions were made in four stages. We first predicted the scaling of tendon strain with size, and used this to estimate the scaling of joint angular excursion. From this, we predicted the scaling of limb stiffness, which we defined as the ratio of peak GRF to limb shortening during the stance phase. Lastly, we used the planar spring-mass model of locomotion to predict how this scaling of limb stiffness would affect RSL, DF and RPF. We made the predictions for both a linear tendon stress-strain relationship and a realistic nonlinear relationship, in order to determine how much this would affect deviations from dynamic similarity.

In order to predict limb stiffness, i.e. in stages (i) to (iii) of the predictions below, we took peak GRF to be proportional to M_b in horses moving at equal Froude number. In horses that are not moving in a dynamically similar fashion, peak GRF will not be exactly proportional to M_b. However, because the force-length relationships of the limbs of animals tend to be approximately linear (Farley et al., 1991; McGuigan and Wilson, 2003), limb stiffness is not sensitive to the GRF for which it is calculated. The scaling exponents predicted below for RPF are small (Table 1), justifying this approximation.

We also assumed that peak tendon force would be proportional to peak GRF, and therefore also to M_b. Again, this assumption will only be exact for dynamically similar horses. If larger horses experience greater joint angular excursions, as expected if tendon elastic modulus is size-independent and there are no compensatory distortions, the moment arms of the GRF about the limb joints will be relatively greater. This will increase the tendon forces required to counteract a given GRF and so could potentially increase deviations from dynamic similarity. From this point of view, the predictions made here are conservative.

(Part 2) Measured scaling exponents

Results

The allometric equations calculated for the anatomical measurements are given in Table 3. The four smallest horses (horses 18-21) had relatively short legs for their mass so that, when all horses were included in the analysis, the scaling exponents for wither height, greater trochanter height and radius and metacarpus length were significantly higher than would occur in geometrically similar animals. Only the exponent for girth was not significantly different from 0.33. When these horses were excluded from the analysis, the scaling exponents for wither height, radius and metacarpus length and girth were not significantly different from 0.33, indicating that the other horses were geometrically similar in these parameters. The scaling exponent for greater trochanter height was still significantly greater than 0.33 (P=0.045), however. To determine the effect of the proportionally shorter legs of horses 18 to 21, the relationships of RSL and DF to body mass were determined both with and without the data for these horses.

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Fig. 2.

Relationships of relative stride length (RSL) and duty factor (DF) to body mass (M_b) in horses trotting at three different Froude numbers (Fr). Allometric equations describing the data are shown (scaling exponents given ± 95% confidence intervals) and are indicated by the solid lines. The scaling exponents were not significantly different from 0, indicating that RSL and DF are independent of body mass in horses trotting at equal Froude number.

The number of trials and range of speeds obtained for each horse are shown in Table 2. The Froude numbers at which comparisons were made were well within the range of speeds achieved by all but one of the horses (horse 10), which only reached a Froude number greater than 1.0 in one trial, while all the other trials were at Froude numbers of 0.80 or slower. This horse was excluded from the analysis at a Froude number of 1.0, but was included in the rest of the analysis. The quadratic equations were a good fit to the relationships of RSL to Froude number, with R² values between 0.95 and 1.00. The data for DF showed more scatter around the fitted lines, with R² values between 0.60 and 0.98.

At all three of the Froude numbers considered, the scaling exponents for RSL and DF against body mass were not significantly different from zero (Fig. 2, Table 4). This conclusion was not altered by including horses 18-21 in the analysis. Including data from these horses increased the body mass range without greatly increasing the scatter of the data, so resulted in narrower confidence intervals (Table 4). The results support the hypothesis that RSL and DF are independent of size in horses and therefore that these parameters are dynamically similar in horses trotting at equal Froude number.

Discussion

The first question that we addressed was whether detectable and physiologically significant deviations from dynamic similarity could occur over the size range present within a species. When the nonlinearity of the tendon stress-strain relationship was taken into account in the theoretical predictions, RSL was predicted to scale in proportion to approximately (Table 1). This illustrates the argument that we have made previously that dynamically similar locomotion would not be expected in geometrically similar animals with identical tendon properties (Bullimore and Burn, 2004). The scaling exponent of 0.10 is well outside the 95% confidence intervals that we obtained experimentally when all horses were included in the analysis (Table 4), so would have been detectable in our experimental measurements (Fig. 3). This suggests that detectable deviations from dynamically similar locomotion are possible over the size range present in adult horses.

Horses have been bred by humans to occur in a wider range of sizes than the adults of most wild species. Over a narrower size range, the scaling exponent of 0.10 predicted for RSL would not be detectable over inter-individual variation. For example, if RSL was 1.50 in an animal of 400 kg, a scaling exponent of 0.10 would give an RSL of 1.55 in a 550 kg individual. This difference is smaller than the differences that would occur between individuals of the same size given a MAPD of 3.5% as measured here. This suggests that, with a nonlinear tendon stress-strain relationship, locomotion would not deviate sufficiently from dynamic similarity to be detectable, or physiologically significant, among the adult members of many species.

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Fig. 3.

Measured and predicted relationships of relative stride length (RSL) to body mass (M_b) in horses trotting at a Froude number of 0.5. Crosses are measured values, solid line is allometric equation fitted to measured values , broken line is relationship predicted with linear tendon properties , circles are relationship predicted with realistic nonlinear tendon properties . The predicted effect of size is substantially reduced by taking into account the nonlinear properties of tendon, but is still greater than measured experimentally. This suggests that additional factors compensate for size effects in horses.

In this study, we have only considered the consequences of size differences between adult individuals. However, the size changes that occur during growth are often much greater. Therefore, it seems likely that detectable alterations in locomotion could arise within an individual during growth. It would be of interest to determine whether this occurs and to identify the biomechanical consequences. In addition to the effects of size, locomotion will also be influenced by changes in anatomical proportions and tendon properties during growth. For example, the increase in tendon elastic modulus that occurs with age in some species (Gillis et al., 1995; Yamamoto et al., 2004) may allow dynamically similar locomotion to be maintained despite large size differences.

The question of the physiological significance of the predicted deviations from dynamic similarity is more difficult to answer. A useful first approach is to compare the magnitude of the predicted size effects with the amount of interindividual variability that arises from other sources. The values of RSL that we measured at a Froude number of 0.75 ranged between 1.50 and 1.69, although some of this variability probably arose from experimental error. With a scaling exponent of 0.10, mean RSL would increase from 1.50 to 1.85 over the size range of horses used in this study. This effect is larger than the measured interindividual variability, so could potentially be physiologically significant.

The scaling exponents predicted for DF and RPF, both with and without taking nonlinear tendon properties into account, were very low. This indicates that these parameters are unlikely to vary much with size, even without compensatory distortions, and that they are insensitive to the scaling of leg stiffness. This may explain why Alexander and Jayes (1983) found greater differences in RSL than in DF when comparing mammalian species of different sizes. Because the exponents predicted for these parameters were within, or close to, our measured confidence intervals, we could not distinguish them statistically from zero.

The second question that we addressed was whether systematic deviations from dynamically similar locomotion occur with size in adult horses. We found that the scaling exponents for RSL and DF in horses trotting at equal Froude number were not significantly different from zero. The 95% confidence intervals for the scaling exponents were narrow (Table 4), indicating that any deviations from dynamic similarity that occur in RSL and DF are likely to be very small. We did not measure the phase relationships of the limbs or the RPF. However, all of the horses were trotting, so their diagonal limb pairs must have had a phase difference of half a stride. Griffin et al. (2004) have also shown that the abrupt change in limb phase at the walk-trot transition occurs at similar Froude numbers in horses of different sizes, as would occur in dynamically similar locomotion. It can be deduced that RPF must have been close to independent of size because, if vertical GRF rises and falls as a half sinusoid during the stance phase, animals moving with equal DF must also have equal RPF (Alexander et al., 1979). Vertical GRF is approximately sinusoidal in horses (Witte et al., 2004), and we found that DF was independent of size, so RPF must have been independent, or close to independent, of size.

Our results are therefore consistent with the hypothesis that horses of different sizes move in a dynamically similar manner when trotting at equal Froude number. This does not necessarily imply that the same would be found in an undomesticated species that is subject to natural selection. However, it does show that dynamic similarity in RSL and DF is possible within a species over a more than eightfold range of body mass. Because we only considered trotting, further work would be needed to establish whether horses of different sizes move in a dynamically similar manner at other gaits. It is also important to note that dynamic similarity in some mechanical parameters cannot be taken to imply that other mechanical parameters are dynamically similar (Bullimore and Burn, 2004).

We predicted theoretically that, in `idealised' horses which were identical except for size differences, RSL at equal Froude number would increase with size. However, we did not find this experimentally; the measured scaling exponents were significantly different from the theoretically predicted values and were not significantly different from zero. This discrepancy suggests that some systematic change occurs with size in horses that compensates for the effects of size differences on locomotion. It would be of interest to identify this factor. One possibility is that, while tendon elastic modulus does not increase with size across species (Pollock and Shadwick, 1994), it does increase with size in horses. Alternatively, a compensatory distortion may occur. For example, larger horses may have relatively thicker tendons. We have discussed possible sites of compensatory distortions previously (Bullimore and Burn, 2004) and concluded that an important compensatory distortion between species is the scaling of the effective mechanical advantage (EMA) of the limb (Biewener, 1989). Large animals tend to have a more upright limb posture than smaller species, and Biewener (1989) demonstrated that this results in a systematic increase the ratio of muscle moment arms to GRF moment arms (the EMA) with animal size. This compensates for the size-independence of tendon elastic modulus in three ways: (i) by decreasing the muscle and tendon forces required to counteract a given GRF, (ii) by decreasing the joint angular excursion that arises from a given tendon elongation, and (iii) by decreasing the limb shortening that arises from a given joint angular excursion. Griffin et al. (2004) have shown that no gross changes in limb posture occur with size in horses. However the changes in EMA that would be required to maintain dynamically similar locomotion in horses would be small and therefore difficult to detect. To illustrate this, the metacarpophalangeal joint can be taken as an example. In a 450 kg horse, the digital flexors have a moment arm about this joint of approximately 30 mm (Brown et al., 2003) and the GRF moment arm at midstance is approximately 100 mm (S.R.B. and J.F.B., unpublished data). If EMA is independent of size, and muscle moment arms scale in proportion to , then the corresponding GRF moment arms would be 57 mm in an 80 kg horse and 116 mm in a 700 kg horse, while the muscle moment arms would be 17 mm and 35 mm, respectively. However, a small change in muscle moment arms, to 11 mm and 39 mm respectively, would give an EMA that was proportional to , as has been found between species (Biewener, 1989). This is a difference of only a few millimetres, suggesting that careful measurements from a large number of individuals would be required to detect it.

An important question is how such a compensation for the effects of size could occur in a species where the size range has developed through artificial selection by humans. Horses are often used for physically demanding activities and are selectively bred for a wide range of characteristics including speed, endurance, agility and resistance to injury. The fact that this has resulted in similar patterns of locomotion being maintained across a wide range of sizes suggests that this way of moving confers desirable qualities. The lack of even a small amount of locomotor scaling with size suggests that relatively slight alterations in movement would be detrimental. Similarly, the (different) selection pressures acting on wild species could potentially also result in dynamically similar locomotion by selecting for optimal locomotor patterns.

Our third aim was to estimate the extent to which nonlinear tendon properties could compensate for the effects of size. We found that taking nonlinear tendon properties into account substantially reduced the scaling exponents predicted for tendon strain, joint angular excursion and RSL, so that predicted deviations from dynamic similarity were smaller than for linear tendon properties. These predictions were made for horses, but tendon properties are similar among species (Pollock and Shadwick, 1994), and the tendons of many species are loaded predominantly in the nonlinear region of their stress-strain relationships (Ker et al., 1988). Therefore these conclusions are likely to be widely applicable. The effect of tendon nonlinearity will be greatest when tendons are used in the low stress region, where the stress-strain relationship is at its most nonlinear. Therefore we expect the effect of tendon nonlinearity to decrease as speed increases during locomotion. Because the digital flexor tendons of horses are loaded at higher stresses than many other tendons (Ker et al., 1988), we also expect the effect of nonlinear tendon properties to be greater in many other species and tendons than was found here. Where tendons are used at low stresses, their nonlinear properties will effectively help to `buffer' the effects of size differences on locomotion. In addition to reducing locomotor differences between individuals, this could also reduce the effects of body mass fluctuations within an individual due to, for example, growth, pregnancy or seasonal variation.

Appendix