## Abstract

This laboratory exercise demonstrates fundamental principles of mammalian locomotion. It provides opportunities to interrogate aspects of locomotion from biomechanics to energetics to body size scaling. It has the added benefit of having results with robust signal to noise so that students will have success even if not “meticulous” in attention to detail. First, using respirometry, students measure the energetic cost of hopping at a “preferred” hop frequency. This is followed by hopping at an imposed frequency half of the preferred. By measuring the O_{2} uptake and work done with each hop, students calculate mechanical efficiency. Lessons learned from this laboratory include *1*) that the metabolic cost per hop at half of the preferred frequency is nearly double the cost at the preferred frequency; *2*) that when a person is forced to hop at half of their preferred frequency, the mechanical efficiency is nearly that predicted for muscle but is much higher at the preferred frequency; *3*) that the preferred hop frequency is strongly body size dependent; and *4*) that the hop frequency of a human is nearly identical to the galloping frequency predicted for a quadruped of our size. Together, these exercises demonstrate that humans store and recover elastic recoil potential energy when hopping but that energetic savings are highly frequency dependent. This stride frequency is dependent on body size such that frequency is likely chosen to maximize this function. Finally, by requiring students to make quantitative solutions using appropriate units and dimensions of the physical variables, these exercises sharpen analytic and quantitative skills.

- eccentric contractions
- locomotion
- frequency
- metabolism
- respirometry
- allometry
- scaling

for most animals, moving about is one of the fundamental, and most energetically costly, of physiological processes. Have selective pressures resulted in predictable and quantifiable relationships between patterns of activity and energy costs of locomotion? In this laboratory, students explore the relationship between stride frequency and energetic cost. The laboratory exercises link biomechanics of locomotion to metabolic rate measurements and power generation during locomotion, using students as subjects. By varying stride frequency during standardized hopping (two-legged, vertical movement), the role of elastic recoil energy in locomotion can be quantified (see Ref. 12 and the references therein). As stride frequency in galloping mammals is one of very few variables that is purely size dependent (4, 12), these exercises allow students to quantify the energetic consequences of stride frequency selection and to compare their performance with other mammals.

Exercises will consider both metabolic power and mechanical power, and, therefore, students should be familiar with the basic principles of muscle use in locomotion (biomechanics) as well as gas exchange and metabolism before this laboratory. Measurement of metabolic power relies on the determination of O_{2} consumption used during activity. Simultaneous determination of mechanical power output via determination of work rate during hopping permits students to explore the relationship between the mechanics and energetics of locomotion, a relationship that is strongly body size dependent. Demand for metabolic power increases linearly with exercise intensity (e.g., running speed) during submaximal exercise (15).

During each locomotor cycle, the nature and timing of muscle force production are driven by nervous input. Muscles often cycle between shortening (concentric) contractions and lengthening (eccentric) contractions (12). For example, when a person runs, the hip extensors (quadriceps) are activated but stretch (eccentric contraction) during the first half of a stride and actively shorten (concentric contraction) during the second half (12). Furthermore, when a muscle produces force and lifts a load, the muscle performs work, calculated as the product of the force produced by the muscle and the distance shortened.

If we consider a person hopping in terms of mechanics, the calf muscle shortens in order for the foot to produce force against the ground, and the ground produces force on the foot. The force produced on the person's foot by the ground overcomes the force of gravity and displaces the subject, resulting in the person being propelled into the air, producing work. Because we trust the first law of thermodynamics, we account for the difference between energy in (metabolism) and out (mechanics) through heat production, thereby calculating muscle efficiency (work out/energy in). An internal combustion engine is at best ∼25% efficient (that's why cars have radiators), and muscle, whether insect, frog or mammalian, also operates at a maximum efficiency of ∼25%. Upon landing, due to the elastic properties of tendons and muscle, potential energy is stored during lengthening (eccentric) contractions, which can be recovered to help power the shortening (concentric) phase of muscle contraction (8). When an individual hops in place, stored potential energy (due to gravity) is transferred to kinetic energy as the person falls back to the ground. When the person hits the ground, some of this kinetic energy is transferred to the elastic elements of muscle as potential energy. In this way, the muscles are able to act as springs, transferring the stored potential energy from the previous hop into the kinetic energy of the next hop. This phenomenon can result in considerable energy savings. For instance, the efficiency of a human running is ∼50%. If the mechanical efficiency of muscle is only 25%, then up to half of the work in running must be due to the recovery of elastic recoil potential energy (14).

The energy cost of moving a unit body mass a given distance can be standardized across species as the cost of transport (CoT). CoT is body size dependent. As body mass increases, CoT decreases (13). CoT also varies predictably with the mode of locomotion (e.g., running, flying, or swimming) such that CoT is highest for runners and lowest for swimmers (13, 18). Curiously, CoT is similar for different clades of animals (e.g., mammals, amphibians, and reptiles) of the same body mass and mode of locomotion (13, 18), further emphasizing the body size dependence of CoT. Within a given body size, CoT is the lowest at the stride frequency that maximizes this recovery of elastic recoil potential energy. This stride frequency decreases with increase in body size.

This laboratory is designed to illustrate two fundamental principles of mammalian locomotion. First, mammals select a stride frequency that minimizes the energy cost of locomotion by maximizing the recovery of elastic recoil potential energy (8), and, second, body size alone determines this frequency (4). In exploring these two core questions, students will measure metabolic rates using respirometry and calculate mechanical and metabolic properties of the locomotor machinery. The final exercise introduces the importance of body size in establishing physiological constraints. We present each exercise in the laboratory with detailed instructions, equations for calculations of each variable, and suggested discussion questions. We also include an appendix presenting selected physical principles and conversion factors with discussion points for students (see Tables 3 and 4). The activities described below are thus designed to reinforce understanding of the physical principles of metabolism, muscle physiology, and locomotion. Finally, an additional goal of these exercises is to develop students' ability to make quantitative solutions using appropriate units and dimensions of the physical variables.

## MATERIALS AND METHODS

### Overview

This exercise is designed for upper-division undergraduate physiology students and can be completed in one laboratory period meeting for 2.5–3 h. The procedures may be limited by available equipment (in our case, one O_{2} and CO_{2} analyzer), but several stations permit the engagement of 12–16 students in any given laboratory period. The timeframe assumes that students are familiar with basic data analysis and can make graphs and perform statistical analyses independently outside of the class period. It may be beneficial for students to have already been exposed to concepts such as metabolic rate measurement, concentric versus eccentric muscle contractions, and work/power calculations. The core of the laboratory is the hopping exercises presented as *activities 1–3*. An option to include cycling as a comparison is presented as *activity 2b*, creating the possibility of running this laboratory over two instructional periods. Elimination of cycling exercises permits easy completion in one period.

The activities are designed to reinforce understanding of metabolism (gas exchange) as well as skeletal muscle mechanics and body size constraints. To facilitate making quantitative solutions to problems, and with attention to units and dimensions, we recommend including Tables 3 and 4 in a worksheet for students (see the appendix). Students will alternate roles as subjects and investigators, quantifying mechanical and metabolic measures of locomotion. Students will determine the energy costs and efficiency of hopping, in which elastic recoil energy stored during eccentric contractions can contribute to performance. Finally, hopping frequency data are compared with values derived from an equation derived from literature values spanning a large size range of mammals. This comparison permits determination of the extent to which human hopping is constrained by body size, thus representing “typical” mammalian locomotion.

### Equipment

The equipment required for the exercise is as follows:

Indirect calorimetry system

Douglas bags

Metronome

Bathroom scale

Hop height guide (see Fig. 1)

Timers

Tape measure or other device to determine height

Optional: stationary bicycle ergometer [one that displays power output (in W)]

### Safety Precautions

Some of the students will act as the experimental subjects for each of these exercises (we suggest that no subject undertake more than one active session of hopping or cycling). Willing students should be instructed to wear the appropriate footwear and clothing for the exercises. Although neither of the exercises described below is particularly strenuous, the volunteers should have no apparent health problems or contraindications. Furthermore, the hopping exercise often results in mild delayed onset muscle soreness due to the novel levels of eccentric contractions engaged. This exercise may require Institutional Review Board approval, which would then be obtained before the laboratory. Students could be required to fill out a physical activity readiness questionnaire to get them prepared to better understand the challenges of conducting human subject work.

### Experimental Details: Background, Activities, Practical Tips, and Suggestions

While each group of students can rotate through each of the activities in any order, it is recommended that standing metabolic rate (*activity 1*) be determined before participation in either the hopping activity (*activity 2*) or the optional cycling activity (*activity 2b*) to ensure a more valid standing rate. Standing metabolic rate (*activity 1*) should be determined for every student who will participate in either exercise activity. Should you chose to include cycling (*activity 2b*), we do not recommend that any students participate as subjects in both hopping and cycling activities, due to time constraints. The primary bottleneck for this exercise is the collection of expired gases. Once gases are collected, students are easily able to determine the O_{2} and CO_{2} content of the collected expired gas and to begin to make calculations. However, proper guidance is important to keep them moving through each station. Typically, only a subset of students is interested in participating as subjects, and we find it sufficient to collect data on only three to four subjects in the hopping exercise (as well as for the optional cycling exercise). While not engaged in other activities, time is available for all students to complete *activity 3* as this requires very little time. Before beginning these activities, students should be familiarized with the concepts shown in Tables 3 and 4.

### Activity 1: Overview of the Measurement of Metabolic Rate: Inactive Metabolism

#### Learning outcomes.

Learning outcomes are as follows:

*1*. Students will become familiar with the basic concepts of determining the metabolic rate through measurement of expired gas concentrations.

*2*. Students will master the quantitative determination of metabolic rate using standard units and appropriate conversion factors.

The most common method to measure the mammalian metabolic rate is to estimate the rate of O_{2} consumption (V̇o_{2}) or CO_{2} production (V̇co_{2}) through indirect calorimetry. When both are measured, the ratio of V̇co_{2} to V̇o_{2} returns the respiratory quotient (see Table 3), which permits the estimation of the relative contribution of carbohydrate and lipid fuels. With this information, O_{2} consumed can be used to calculate the total energy consumed (in J) or V̇o_{2} can be converted into Watts (Table 4). For example, when at rest, a typical adult (70 kg) human consumes ∼220 ml O_{2}/min (5). Assuming that carbohydrate and lipid are consumed at equal rates, this is equivalent to a metabolic rate of ∼75 W, most of which is ultimately emitted as heat.

#### Activity details.

Activity details are as follows:

*1*. To begin, O_{2} and CO_{2} analyzers should be calibrated with dried^{1} room air following standard protocols.

*2*. It is assumed that the instructors of this laboratory exercise are familiar with measurements of O_{2} consumption and CO_{2} production. We find that it is useful to use the Douglas bag method for this laboratory as it allows students to work in parallel. Multiple samples can be collected and held until ready to pass through the gas analyzers, as that final step requires little time. We have also undertaken these activities using a more standard approach in which individual subjects are measured in series using a flow-through respirometry system. However, whatever method is used, each student subject should sit quietly for a few minutes before measuring expired gas concentrations. Essentially, the metabolic rate is measured while the subject is standing quietly. In that way, this value is equal to the hopping metabolic rate without the cost of hopping.

*3*. Once collected, dried gas samples are passed through the gas analyzers to determine both the fractional content of O_{2} in expired air (Fe_{O2}) and the fractional content of CO_{2} in expired air (Fe_{CO2}). These values are then recorded, noting that most gas analyzers return percent gas content, which must be divided by 100 to express the fractional content.

#### Calculations.

The following are example calculations for the determination of V̇o_{2} and V̇co_{2} using Douglas bags that are simple and accessible while demonstrating the suggested quantitative approach. Similar learning outcomes can be accomplished using a flow-through system by directing student attention to standard temperature and pressure dry (STPD) corrections and considerations of relative humidity. Note that these calculations can be done after the laboratory period has ended, although we find that if students work in parallel that there is sufficient time to at least begin calculations while other subjects are actively collecting subsequent samples. Once students have a template (which can be provided), calculations will progress rapidly.

*1*. Determination of V̇o_{2} and V̇co_{2}.

*A*. Determination of the standard volume of gas expired per unit time (V_{e}).

The volume of expired air contains water vapor exhaled by the subject. The first step in the calculation of V_{e} is to subtract the water vapor volume from the exhaled volume. Water vapor partial pressure is equal to 24 mmHg in the Douglas bag (at 25°C; or it can be calculated from actual room temperature, see http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/watvap.html) and is used to calculate the volume of dry air in the bag as follows:
(1)
where 15 liters refers to an example volume of the Douglas bag after the collection of expired gases, Ph_{2}o is the partial pressure of water at room temperature, and P_{bar} is the barometric pressure in the classroom (in mmHg; use local weather reports if a barometer is not available).

Once the volume of water vapor is removed, the remaining volume, measured under ambient conditions, is converted to standard temperature and pressure using the formula in Table 4. The resultant STPD volume is then divided by the time required to fill the bag to return the flow rate of expired gas (V_{e}; in ml/min).

*B*. Calculation of V̇o_{2} and V̇co_{2}. Practically, V̇o_{2} and V̇co_{2} are calculated using *Eqs. 2* and *3* as follows:
(2)
(3)
where Fi_{O2} and Fi_{CO2} are the fractional content of O_{2} or CO_{2} in inspired air, which is room air and contains 20.95% O_{2} and 0.04% CO_{2} (as determined during calibration). Similarly, Fe_{O2} and Fe_{CO2} vary with activity and are determined from expired gas collected from each student. V̇o_{2} and V̇co_{2} are expressed as the volume of O_{2} consumed or CO_{2} produced per unit time (typically in ml/min, the same units as used in determining V_{e}). Note that these are somewhat simplified equations. For a more detailed and precise examination of the factors that contribute to determination of metabolic rate using indirect calorimetry, see Withers (19), Fedak et al. (2), and Lighton (9).

*2*. Determination of the respiratory quotient (RQ).

Both V̇o_{2} and V̇co_{2} are used to determine RQ using the formula shown in Table 3. Once RQ is determined, students can calculate the ratio of carbohydrate and lipid fuels consumed assuming that an RQ of 1.0 reflects the consumption of only carbohydrate fuels and an RQ of 0.7 is due to the consumption of only lipid fuels. RQ is typically ∼0.80 at rest. Note that ignoring the contribution of protein fuels has a negligible effect under these conditions.

*3*. Conversion of V̇o_{2} to Watts.

After estimating the contribution of carbohydrates and lipids, students will use the proportionate volume of O_{2} consumed with each fuel together with the corresponding conversion factor shown in Table 4 to convert the measured V̇o_{2} to Watts. This value will be used in *activity 2* (and potentially *activity 2b*) below for calculations of efficiency.

### Activity 2: Storage and Recovery of Elastic Potential Energy While Hopping

#### Learning outcomes.

Learning outcomes are as follows:

*1*. Students will become familiar with the basic concepts of muscle force generation during locomotion (i.e., concentric and eccentric contractions).

*2*. Students will master the quantitative determination of metabolic costs of hopping using standard units and the appropriate conversion factors.

In *activity 2*, students will examine their ability to behave like a spring, storing and recovering energy like a pogo stick. They will ask the following question: Is there a running stride frequency that results in a greater running efficiency? One way that animals maximize efficiency is to behave like springs, alternately storing and recovering energy as they walk, run, fly, or swim. Students will test the hypothesis that humans hopping up and down likewise store and recover elastic recoil energy. Furthermore, they will test the additional hypothesis that we select a hopping frequency that “feels comfortable” because it maximizes our ability to recover elastic potential energy when we hop up and down. Students will plot the metabolic power consumed during hopping (V̇o_{2} converted to Watts, as in *activity 1*) against mechanical power produced (calculated from the work of hopping, see below) and describe the resultant relationship. Students will calculate efficiency at each work rate measured while hopping.

#### Activity details.

Activity details are as follows:

*1*. Each student participant should first be weighed on a bathroom scale. Weight will be used to calculate work done per hop and as a predictor of hopping frequency in *activity 3*.

*2*. After the determination of standing metabolism (*activity 1*), O_{2} consumption is measured (as described above) while the subject hops at a preferred frequency that feels “natural” to them. It is important that the student hops at a constant height (guide set to 1.07 × height at the eyes; see Fig. 1 for a description of setting hop height). Each subject should practice hopping to a standard height before beginning. Expired gases should then be collected as in *activity 1*. While the subject is hopping, another student will record the subject's preferred hopping frequency (and record the time needed to fill the Douglas bag if this approach is used). This preferred hopping frequency will also be used as a comparison in *activity 3* (below). We find that the best data for hopping frequency can be collected after a steady state is achieved (after 1–2 min). Expired gas collection typically requires 3–5 min.

*3*. After a brief rest, set a metronome to one-half of the preferred hopping frequency and have the subject hop at that frequency while measuring expired gas concentrations (as above). Again, each subject should practice hopping at this frequency and constant height, as this can be surprisingly difficult (mentally rather than physically). Instruct students to record the actual hopping frequency rather than believing the metronome.

*4*. In both cases, determine Fe_{O2} and Fe_{CO2} as described above in *activity 1*.

#### Calculations.

Calculations are as follows:

*1*. Using a similar approach as that in *activity 1*, determine the metabolic rate (in W, using both V̇o_{2} and V̇co_{2}) of the subjects at each hopping frequency.

*2*. Using the values determined in *activity 1*, subtract the standing metabolic rate from the hopping metabolic rate to determine the energy cost of hopping and then determine the total energy cost of an average minute's hopping (in J rather than in W).

*3*. Divide this value of Joules expended by the number of hops per minute to arrive at the cost in Joules per hop.

*4*. Using the subject's weight and vertical distance moved, calculate the work done per hop using *Eq. 4* (this should be the same at both frequencies for each subject), as follows:
(4)With attention to units, work (in J) (N·m ≡ kg·m^{2}·s^{−2}) is calculated from the subject’s mass times gravity and the height of each hop (eye height × 0.07). This calculation emphasizes the importance of maintaining constant hopping height.

*5*. By dividing mechanical work by metabolic energy consumed per hop, students can estimate the efficiency of the energy conversion during hopping. This should be done at each hopping frequency. This calculation emphasizes the importance of measuring metabolism at steady state.

#### Discussion questions.

The following are discussion questions:

*1*. Does energy cost increase as a linear function of hopping frequency?

*2*. Does the natural frequency minimize the energy cost per hop?

*3*. What impact does storage of elastic energy have on the efficiency of hopping?

*4*. What is the role of concentric versus eccentric muscle contractions in the storage of elastic energy?

*5*. What role does the timing of neuronal activation play in the storage of elastic energy?

### Optional Activity 2b: O_{2} Consumption While Cycling: Work, Power, and Efficiency

#### Learning outcomes.

Learning outcomes are as follows:

*1*. Students will become familiar with the basic concepts of work and power during exercise.

*2*. Students will master the quantitative determination of metabolic costs of cycling using standard units and the appropriate conversion factors.

While the hopping exercise is the central focus of this laboratory, some instructors may wish to include a segment in which the storage of elastic potential energy is not relevant for comparison, e.g., cycling. This exercise agrees with the general perception that as we work harder, we all have the feeling that our metabolism increases. As a reference, elite endurance athletes may increase V̇o_{2} to 5–6 liters O_{2}/min during maximal exercise, an ∼25-fold increase compared with rest. While exercise typically results in the execution of work, the majority of the energy used leads to the generation of heat due to the low efficiency of energy conversion. In *activity 2b*, students will begin by examining the rate of increase in metabolic power with increase in mechanical power. Measurements of the metabolic rate at rest and during activity allow students to estimate the energy demands of activity and the efficiency with which metabolism and work are coupled. Students will test the hypothesis that any increase in metabolic power is a linear function of mechanical power output. Note that the chosen workloads should be sufficiently submaximal that anaerobic contributions will be small. At higher workloads, the slope of the relationship between aerobic power in and mechanical power output will fall due to the contribution of anaerobic metabolism. Students will cycle at a range of fixed power outputs on a bicycle ergometer while V̇o_{2} and V̇co_{2} are measured at each of those workloads. Students will plot metabolic power (converted to W as in *activity 1*) against mechanical power (set on the bicycle ergometer) and describe the resultant relationship. Again, given that we believe the first law of thermodynamics, we account for the difference between power (or energy) in and out through heat production (that's why cars have radiators and animals must be capable of heat dissipation during intense activity). Students will calculate efficiency at each work rate measured while cycling.

#### Activity details.

Activity details are as follows:

*1*. After the determination of standing metabolism (*activity 1*), student subjects should adjust the ergometer and ride for a few minutes at a comfortable cadence until steady (we recommend 80 RPM for someone familiar with cycling, otherwise a bit slower). Set the resistance so that power is set at 50 W.

*2*. Expired gases should be collected while the subject maintains a constant power output on the ergometer throughout.

*3*. After a rest sufficient to return to resting heart rate, repeat the above steps after resetting the ergometer to a power output of 100 W. Time permitting, further trials at 150 W will permit calculation of regressions.

*4*. In both cases, pass a dried sample of the gas through the gas analyzers and record both Fe_{O2} and Fe_{CO2} as described above.

#### Calculations.

Calculations are as follows:

*1*. Determination of the activity cost of cycling. Determine V̇o_{2} and V̇co_{2} at each power output. Calculate RQ at each power output and convert V̇o_{2} to Watts as described above in *activity 1*. Subtract the standing metabolic rate determined in *activity 1* from the cycling metabolic rate at each power output to determine the cost of activity.

*2*. To determine the relationship between metabolic and mechanical power, students should graph the calculated metabolic power as a function of the defined mechanical power. If more points are collected, regression analysis can be used to describe the constancy of the relationship; however, within one 2.5-h laboratory period, we find that we do not have time for more than two activity levels.

*3*. By dividing mechanical power by metabolic power, students can estimate the efficiency of the energy conversion. This should be done at each defined power output.

#### Discussion questions.

The following are discussion question:

*1*. How is metabolic rate (power in) related to mechanical demand (power out) across the range of exertion tested?

*2*. Is efficiency constant across the measured activity range? Why or why not?

*3*. How do the metabolic rate changes while on the bicycle ergometer differ from those during hopping?

### Activity 3: Introduction to Body Size Constraints on Locomotion

#### Learning outcomes.

Learning outcomes are as follows:

*1*. Students will become familiar with the basic concept of the body size dependence of stride frequency.

*2*. Students will master the quantitative determination of functional parameters (in this case, stride frequency) using predictive exponential allometric equations.

In an ingenious pair of articles, C. R. Taylor and N. C. Heglund (15, 16) made the compelling argument that hopping in bipeds is biomechanically identical to galloping in quadrupeds. In fact, while stride frequency is highly constrained by body size (6), animals of a given body mass, including hopping kangaroos (1) and galloping quadrupeds (3), use nearly identical stride frequency over a large range of speeds. Thus, Taylor's article (17) was the inspiration for this activity.

In this exercise, students will hop at their preferred frequency and compare this value with predictions based solely on body size. While a full discussion of the role of body size in physiology is beyond the scope of this laboratory exercise, this activity can serve as an entry point or supplement to other classroom discussion of this important trait.

#### Activity details.

Activity details are as follows:

*1*. Each willing student who did not participate in *activity 2* should weigh themselves on the scale (converting their weight to kg if necessary). Students who participated in *activity 2* should simply proceed to the calculations.

*2*. The subject should then hop to the appropriate height (see Fig. 1), as in *activity 2*, and at their preferred frequency. As in *activity 2*, an observer should count the number of hops with a timer. The resulting frequencies are then recorded as the preferred stride frequency of each subject.

#### Calculations.

Calculations are as follows:

*1*. After recording the observed frequency, students will calculate their stride frequency using *Eq. 5* (4), which predicts stride frequency as a function of body weight in galloping mammals (they will need a calculator that can calculate power functions for this), as follows:
(5)

*2*. Using data collected from each participant, make a table of observed versus predicted stride frequencies. Ask students to consider the best way to graphically present and statistically analyze class data. Consider the fit to a regression spanning several orders of magnitude in body size (e.g., a 30 g. mouse to 500-kg horse).

#### Discussion questions.

The following are discussion questions:

*1*. To what extent are the students “typical galloping mammals?” How can this be best expressed statistically?

*2*. Given the large range of speeds in animals with similar mass [e.g., domestic goats and pronghorn antelope (6)], how do some animals attain higher speed if stride frequency does not change?

## RESULTS AND DISCUSSION

In Table 1, we show data for an individual from a recent class. We measured hopping frequency and metabolic rate at rest and while hopping at both the preferred and half of the preferred frequency. We then calculated the cost per hop and efficiency at both frequencies. The overall metabolic rate while hopping at each frequency was very similar. However, the energy cost per hop nearly doubled with the reduced hopping frequency. Because the mechanical output (work) is constant at both frequencies, these data also demonstrate that the efficiency of hopping is much higher at the preferred frequency.

While allometric equations often faithfully reproduce organismal function, they generally serve best as predictive functions. As such, we compared the predicted hopping frequencies calculated from *Eq. 5* with actual recorded hopping frequencies of student subjects. Table 2 shows the outcome of a recent class demonstrating the close matching of predicted and observed values. Overall, students tend to hop at slightly lower frequencies than predicted for galloping quadrupeds (4), but class values are typically within 10% of those values, strongly suggesting that the same biomechanical constraints must be operating in hopping humans as a similar sized galloping deer.

All of these results provide the instructor with a great deal of material for class discussion and interpretation. The approach presented above permits instructors to improve skills in muscle physiology, respirometry, and energy metabolism while developing understanding of the importance of SI units and physical constants. The interrelationship between mechanical and metabolic properties during locomotion is an excellent avenue in which to reinforce physical principles within a biological context.

In summary, this laboratory exercise provides results that are extremely dependable and highly reproducible. From experience, sources of errors arise from variance in hop height leading to inconsistent work output, inaccurate time keeping, or not hopping long enough to reach steady state. Calculations of cost, efficiency (work out/energy in), as well as allometric relations all provide quantitative reasoning opportunities. Furthermore, discussions can explore the very timely and ongoing investigations regarding the anatomical location and physiological mechanisms of the muscle-tendon spring (10, 11).

## GRANTS

Support for this work was provided by National Institutes of Health Grants 1-R15-DK-085497-01A1 (to P. J. Schaeffer) and R21-AG-18701-01A1 (to S. L. Lindstedt).

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

## AUTHOR CONTRIBUTIONS

Author contributions: S.L.L. and P.J.S. conception and design of research; S.L.L., P.M.M., and P.J.S. analyzed data; S.L.L., P.M.M., and P.J.S. interpreted results of experiments; S.L.L., P.M.M., and P.J.S. drafted manuscript; S.L.L., P.M.M., and P.J.S. edited and revised manuscript; S.L.L. and P.J.S. approved final version of manuscript; P.M.M. and P.J.S. performed experiments; P.M.M. and P.J.S. prepared figures.

## ACKNOWLEDGMENTS

This laboratory was the direct result of discussions with and inspiration from Claire Farley and the late C. R. Taylor; the authors are indebted to both. The authors are also thankful for the many physiology class students that allowed the authors to perfect the techniques presented in this laboratory.

## Footnotes

↵1 To dry the gas sample, pass it through a column containing Drierite (W. A. Hammond Drierite, Xenia, OH) before the O

_{2}and CO_{2}analyzers.

- Copyright © 2013 the American Physiological Society