## SUMMARY

It is possible for animals of very different sizes to use the same patterns of locomotion, i.e. to move in a `dynamically similar fashion'. This will only occur, however, if relevant biomechanical parameters scale with size in such a way that they compensate for the effects of size differences. Here we apply this principle to understanding the effects of size on locomotion within a species: the domestic horse. We predict that, without any factor to compensate for size differences, detectable deviations from dynamically similar locomotion would occur over the size range present in adult horses. We measured relative stride length (RSL) and duty factor (DF) in 21 trotting horses (body mass: 86-714 kg), and interpolated the data to predict RSL and DF at equivalent speeds (Froude numbers: 0.5, 0.75, 1.0). RSL and DF at equal Froude number were not significantly related to body mass. This is consistent with the hypothesis that horses trot in a dynamically similar fashion at equal Froude number. We show that the nonlinear stress-strain relationship of tendon can contribute to reducing deviations from dynamic similarity, `buffering' the effects of variation in body mass, but conclude that this effect is unlikely to explain fully our results. This suggests that a `compensatory distortion' may occur in horses, counteracting the effects of size differences. The approach used here is also applicable to understanding the consequences of size changes within an individual during growth.

## 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
(*u*^{2}/*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.

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*_{b} 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.

### (i) Predicted scaling of tendon strain

For a peak tendon force proportional to *M*_{b}, and a
tendon cross-sectional area proportional to
, tendon stress will increase with
size in proportion to . For
tendons with a linear stress-strain relationship, strain will scale in
proportion to stress and therefore also to
. In order to predict the scaling
exponent for strain in tendons with a nonlinear stress-strain relationship, we
used the data of Riemersma and Schamhardt
(1985), digitised from their
fig. 4. This gives stress-strain relationships for a superficial digital
flexor tendon (SDFT), a deep digital flexor tendon (DDFT) and a suspensory
ligament (SL) from an equine hindlimb. These structures cross the
metacarpophalangeal joint in the forelimb and the metatarsophalangeal joint in
the hindlimb. Hyperextension of these joints is responsible for a large
proportion of the shortening of the limb during the stance phase in trotting
horses (McGuigan and Wilson,
2003) and, because the muscle fibres associated with these
structures are relatively short (Ker et
al., 1988), it is likely that most of this hyperextension occurs
through tendon elongation. Therefore, the properties of these tendons have a
substantial influence on overall limb compliance. To determine whether the
data of Riemersma and Schamhardt
(1985) were typical we fitted
straight lines to the stress-strain relationships between strains of 3.5% and
6.5% (*R*^{2}>0.99). This gave elastic moduli of 1.15 GPa,
1.44 GPa and 0.56 GPa and intercepts with the strain axis of 1.6%, 2.1% and
2.1%, for the SDFT, DDFT and SL, respectively. Comparison with published
values indicated that these are typical properties for mammalian tendons: mean
elastic modulus for mammalian limb tendons is 1.24±0.23 GPa
(Pollock and Shadwick, 1994)
and mean intercept strain for the equine SDFT is 1.5%
(Wilson, 1991). In order to
make predictions for horses with identical tendon material properties, we used
the stress-strain relationships measured by Riemersma and Schamhardt
(1985) for horses of all
sizes.

The relationship between the scaling exponents for stress and strain depends upon which part of the tendon stress-strain relationship is used. Biewener (1998) calculated peak tendon stresses of approximately 16, 20 and 13 MPa in the SDFT, DDFT and SL, respectively, in trotting horses of approximately 275 kg. We used these values as a starting point for calculating the scaling exponent for strain, which corresponds to a stress proportional to . Tendon stresses for horses between 80 and 800 kg were predicted using the equation: (1)

where the scaling constant, *k*, was calculated from the above
stresses for a 275 kg horse, giving values of 2.46, 3.08 and 2.00 for the
SDFT, DDFT and SL, respectively. The tendon stress-strain data of Riemersma
and Schamhardt (1985) were
fitted with third order polynomials (*R*^{2}>0.99) and these
were used to predict tendon strains corresponding to the stresses predicted by
Eqn 1. Allometric equations relating these predicted strains to body mass were
obtained by log-transforming the data and fitting linear least-squares
regression equations (*R*^{2}>0.99), as described by
Schmidt-Nielsen (1984). This
gave scaling exponents of 0.19, 0.16 and 0.18 for strain in the SDFT, DDFT and
SL, respectively. To determine how robust these predictions were, we varied
the stresses used for calculating *k*. Large variations (±5 MPa)
altered the predicted exponents by less than 0.01. The predicted exponents are
considerably lower than the exponent of 0.33 for a tendon with linear
properties, indicating that nonlinear tendon properties could substantially
reduce the effects of size on tendon strain.

### (ii) Predicted scaling of joint angular excursion

If the joint is modelled as having a circular profile, the joint angular excursion arising from a given tendon strain can be calculated as: (2)

For geometrically similar horses, joint radii and tendon lengths will be proportional to , so that joint angular excursions will be directly proportional to tendon strain. Therefore, we take joint angular excursion to be proportional to for a tendon with linear properties and to (the mean of the scaling exponents calculated above) for a tendon with nonlinear properties. In both cases this represents an increase in joint angular excursion with size, but the predicted increase is substantially smaller for a tendon with nonlinear properties.

The above calculation ignores the contribution of muscle strain to joint angular excursion. This seems reasonable in trotting horses because most of the length change in the limb occurs distally (McGuigan and Wilson, 2003) where the muscle-tendon units have relatively short fibres and long tendons (Ker et al., 1988). However, muscle strains would also be expected to increase with size, by the following argument. Muscle stress would increase with size in the same way as tendon stress, i.e. as . The horses we are modelling have identical muscle activation patterns. Therefore, due to the nature of the force-velocity relationship of muscle, these higher stresses would tend to decrease the ability of the muscle to shorten and to increase the possibility of it lengthening, resulting in greater muscle strains in larger animals.

### (iii) Predicted scaling of limb stiffness

In order to predict the scaling of limb stiffness, we needed to be able to
predict limb shortening from joint angular excursion, where limb shortening is
defined as the change in the distance between the proximal and distal ends of
the limb between ground contact and midstance. This cannot be done without
knowledge of limb morphology because the relationship depends upon limb
segment lengths and initial joint angles. For this reason, we used
experimental data from three horses (horses 7 and 8 from
Table 2 and a third horse of
630 kg) as a basis for predicting the scaling exponent for limb shortening.
Reflective markers were placed over the joint centres of the right fore- and
hindlimbs and their positions were recorded by optical motion capture (240
frames s^{-1}; Proreflex, Qualisys, Sweden) during trot. The marker
positions were used to calculate the limb segment lengths, joint angles at
ground contact and joint angular excursions between ground contact and
midstance. Each leg of each horse was then used separately as the basis for
calculating scaling exponents for limb shortening. Limb segment lengths for
horses of 80, 200, 400, 600 and 800 kg were predicted by scaling the measured
lengths in proportion to (so that
they were geometrically similar), and joint angular excursions were predicted
by scaling measured values in proportion to
to model a tendon with linear
properties, or to to model a
tendon with nonlinear properties, as predicted above. Limb shortening was then
calculated trigonometrically, assuming initial joint angles were equal to
measured values. This produced a total of twelve sets of values for predicted
limb shortening against *M*_{b}: six corresponding to the fore-
and hindlimbs of each of the three horses for linear tendon properties, and
another six for nonlinear tendon properties. For each of these twelve
datasets, an allometric equation relating predicted limb shortening to body
mass was calculated by log-transforming the data and fitting a linear
least-squares regression equation (*R*^{2}>0.99). Because
the calculated scaling exponents were very similar for the data based on the
three different horses, and on the fore- and hindlimbs, the means of these
scaling exponents were used. The mean exponent for limb shortening with linear
tendon properties was 0.77±0.01 (± s.e.m.) which, in combination
with a GRF proportional to *M*_{b}, gives a limb stiffness
proportional to . The mean
exponent for limb shortening with nonlinear tendon properties was
0.58±0.01 (± s.e.m.), which gives a limb stiffness proportional
to . Therefore, the predicted
scaling exponent for limb stiffness is higher with nonlinear tendon
properties, but is still substantially lower than the exponent of 0.67 that
would be required for dynamic similarity
(Bullimore and Burn, 2004).

### (iv) Predicted scaling of dimensionless locomotor parameters

The effect of the above scaling of limb stiffness on the dynamics of locomotion was predicted using the planar spring-mass model (Blickhan, 1989; McMahon and Cheng, 1990). This is a simple model of running gaits, such as trot, in which the animal is represented by a point mass bouncing on a spring. It describes the mechanical relationships between basic locomotor parameters such as limb stiffness, stride length, stance time and GRF, and has been shown to be able to model the mechanics of trotting in mammalian species of a wide range of different sizes remarkably well (Farley et al., 1993).

GRF data (1000 Hz; model 9287 force plate, Kistler Instruments,
Switzerland) and kinematic data (240 frames s^{-1}; Proreflex) from
one horse of intermediate size (horse 8;
Table 2) were used to determine
model parameter values representative of a horse trotting at Froude numbers of
0.5, 0.75 and 1.0. With the calculated parameter values, the model predicted
RSL, DF and RPF to within 5% of the values measured in this horse. Parameter
values representing horses of 80, 200, 400, 600 and 800 kg were generated by
scaling limb stiffness as calculated above (i.e. in proportion to
for linear tendon properties and
to for nonlinear properties) and
keeping the other model parameters dynamically similar to the values
calculated for horse 8. The parameter values that were used are listed in the
Appendix. Predictions of RSL, DF and RPF for the horses of different sizes
were then obtained by numerical integration of the equations of motion for the
model using a function written in Matlab (version 6.5, The MathWorks, Inc.,
MA, USA). The method used has been described in detail previously
(Bullimore and Burn, 2006).
Allometric equations relating the predicted values of RSL, DF and RPF to body
mass were obtained by log-transforming the data and fitting linear
least-squares regression equations (*R*^{2}>0.98). The
scaling exponents predicted for both linear and nonlinear tendon properties
are shown in Table 1. The
exponents for RSL and DF are positive, indicating that these dimensionless
parameters are predicted to increase with size, while the exponents for RPF
are negative, indicating that it is predicted to decrease with size.
Incorporating the nonlinear characteristics of tendon into the predictions
substantially reduced the exponents predicted for RSL, while the exponents
predicted for DF and RPF were small for both linear and nonlinear tendon
properties.

In summary, the results obtained in Part 1 predict that: (a) in horses that do not exhibit any systematic changes with size, RSL at equal Froude number will increase with size in proportion to approximately , while DF and RPF will be close to independent of size, and (b) relative to a tendon with a linear stress-strain relationship, a tendon with a realistic nonlinear relationship could reduce the effects of size on tendon strain, joint angular excursion and RSL.

## (Part 2) Measured scaling exponents

### Materials and methods

#### Overview

RSL and DF were measured in 21 adult horses *Equus callabus* (86-714
kg, Table 2) trotting at a
range of speeds. The data for each horse were interpolated to predict the RSL
and DF that the horse would use at Froude numbers of 0.5, 0.75 and 1.0.
Dynamic similarity of RSL and DF was defined as these parameters being
independent of body mass in horses trotting at equal Froude number.

#### Anatomical measurements

Girth was defined as the circumference of the trunk just behind the forelegs. Wither height was defined as the height above the ground of the highest point at the base of the neck. Body mass was measured using a weighbridge, girth was measured using a tape measure and the height of the withers and the greater trochanter above the ground were determined using a measuring stick.

To calculate RSL and Froude number, a measure of leg length is required. Previous studies (e.g. Alexander and Jayes, 1983; Donelan and Kram, 2000) used greater trochanter height in the standing animal. However, the greater trochanter is difficult to palpate in horses and its height above the ground varies with standing posture. Preliminary calculations based on a pilot study indicated that error in this measurement was likely to be a significant source of interindividual variability. Therefore, instead we measured radius and metacarpus length between the most distal prominence on the lateral side of the elbow joint and the prominence on the lateral, proximopalmar aspect of the proximal phalanx. This measurement proved to be substantially more repeatable than greater trochanter height. For convenience, we multiplied the measured radius and metacarpus lengths by a scale factor of 2.05 to obtain a `calculated leg length', which was approximately equal to greater trochanter height. This was done to make our results comparable with previous studies. The scale factor of 2.05 was the mean value of the ratios of greater trochanter height to radius and metacarpus length for all the horses. This ratio was independent of body mass (Table 3). Because the same scale factor was used for all horses it did not contribute to interindividual variability and so did not affect the capacity to detect deviations from dynamic similarity. The calculated leg lengths are shown in Table 2.

#### Experimental procedure

All experimental procedures were approved by the University ethics
committee. Reflective markers were attached to the medial side of the left
fore hoof, to the lateral side of the right fore hoof and overlying the dorsal
spinous process of the fifth thoracic vertebra ('t5 marker'). The horses were
trotted along a track by an experienced handler. The first section of the
track was used for acceleration, the central section for data collection and
the final section to slow the horses to a stop. The positions of the
reflective markers were recorded at 240 frames s^{-1} using a 3D
optical motion capture system (Proreflex) as the horses passed through the
data collection region. The field of view was sufficient for at least one
stride to be recorded from all horses at all speeds. For all except horses 7
and 8, one stance phase of the left or right fore leg was recorded at 250
frames s^{-1} using a high speed digital video camera (MotionCorder
SR-1000, Kodak, Herts, UK). Trials were obtained over as wide a range of
trotting speeds as possible while maintaining an approximately constant speed
through the data collection region.

#### Data analysis

Allometric equations relating girth, wither height, greater trochanter height and radius and metacarpus length to body mass were calculated by log-transforming the data and fitting linear regression lines. Reduced major axis (RMA) regression was used because this method is appropriate for data in which error occurs in both the dependent and independent variables (Rayner, 1985). The fit of the allometric equations was assessed by calculating the mean absolute percent deviation (MAPD) of the data from the values predicted by the equations (Prothero, 1986). 95% confidence intervals for the scaling exponent were calculated using the t-distribution. If these confidence intervals included 0.33, the horses were considered to be geometrically similar in the measurement in question (Schmidt-Nielsen, 1984). The ratio of greater trochanter height to radius and metacarpus length was calculated and an allometric equation relating this parameter to body mass was calculated as above, except that least-squares regression was used because the relationship obtained using RMA regression did not fit the data well, probably because the range of the dependent variable was small.

Forward speed was calculated as the mean velocity of the t5 marker over one
complete stride. Stride length was calculated as the distance travelled by the
left or right fore-hoof marker between consecutive stance phases of that hoof.
Stance time was determined from the high speed video. Stride time was
calculated as stride length divided by forward speed. RSL was calculated as
stride length divided by calculated leg length, DF was calculated as stance
time divided by stride time and Froude number was calculated as
*u*^{2}/*gh*, where *u* is forward speed,
*g* is the acceleration due to gravity (9.81 m s^{-2}) and
*h* is calculated leg length.

For each horse, the RSL and DF that would be used at Froude numbers of 0.5,
0.75 and 1.0 were predicted by fitting quadratic equations to the data for RSL
and DF against Froude number. Quadratics were chosen on the basis of a pilot
study in six horses in which linear, quadratic, power and logarithmic
functions were tested and quadratics gave the highest *R*^{2}
values. Higher order polynomials were not used, despite sometimes giving
higher *R*^{2} values, because the resulting curves often
exhibited multiple extrema, which were thought unlikely to be representative
of the true relationship.

For each of the three Froude numbers considered, allometric equations relating RSL and DF to body mass were calculated as described above. Least-squares regression was used because RMA regression gave a poor fit to the data. MAPD and 95% confidence intervals were calculated as described above. If the confidence intervals included zero, the parameter in question was considered to be independent of body mass and therefore to be dynamically similar.

## 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.

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 *R*^{2} values between 0.95 and 1.00. The data for
DF showed more scatter around the fitted lines, with *R*^{2}
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.

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

### Spring-mass model parameters used

The planar spring-mass model is specified by four dimensionless parameters:
dimensionless spring stiffness (*K*), dimensionless horizontal landing
velocity (*U*_{0}), dimensionless vertical landing velocity
(*V*_{0}) and initial spring angle (θ_{0}), where
*K*=*kl*_{0}/*M*_{b}*g*,
(the square root of Froude number),
and *k* is spring stiffness, *l*_{0} is initial spring
length, *M*_{b} is mass, *g* is the magnitude of the
acceleration due to gravity, *u*_{0} is horizontal landing
velocity and *v*_{0} is vertical landing velocity
(McMahon and Cheng, 1990). The
values of *K, U*_{0} and *V*_{0} that were used
to simulate trotting horses of different sizes in Part 1(iv) are listed in
Table A1. θ_{0}
was chosen so that the model bounced symmetrically, by the method described
previously (Bullimore and Burn,
2006).

- List of abbreviations
- DDFT
- deep digital flexor tendon
- DF
- duty factor
- EMA
- effective mechanical advantage
- GRF
- ground reaction force
- MAPD
- mean absolute percent deviation
*M*_{b}- body mass
- RMA
- reduced major axis
- RPF
- relative peak force
- RSL
- relative stride length
- SDFT
- superficial digital flexor tendon
- SL
- suspensory ligament

## ACKNOWLEDGEMENTS

We thank Rachel Williams of Hartpury College, Tikki Adorian of Toyhorse International and Tracey Wilson and family for providing horses, Kelly Heath, Fiona Steel, Gina Cobb, Julie Ford and the first year Equine Science MSc students at Hartpury College (2002-2003) for technical assistance and Anne Gutmann and Kirk Holloway for helpful comments on the manuscript.

- © The Company of Biologists Limited 2006