## Abstract

Students are almost universally interested in animals, and especially endotherms, including mammals and birds. According to Bergmann's rule, endotherms that live in colder climates at higher latitudes are larger than those living in warmer climates. As with most biological principles, hands-on investigation will provide a better understanding of why size is important in endotherm thermal regulation. One easily observable aspect of this principle is that larger organisms have a lower ratio of body surface area to total body volume. This affects how efficiently they can retain or radiate heat, which can be easily tested in the laboratory using commonly available materials. In this activity, simple models of endotherms of different sizes are used to assess the effects of body size on heat loss.

An interesting principle related to physiological ecology and biogeography is expressed in Bergmann's rule. In 1847, Carl Bergmann formalized a concept suggesting that among members of an endotherm species, or among closely related endotherm groups, larger specimens will occur at higher latitudes (i.e., farther from the Equator) where average temperatures are lower (Stiling, 2015). While it is probably not the only principle at work (Ashton et al., 2000), the pattern appears to hold within many species of mammals, with larger species adhering to the principle better than smaller ones (Freckleton et al., 2003).

According to Louw (1993), the most critical parameter affecting an organism's thermoregulation is the ratio of the organism's surface area to body mass. Similarly, one of the presumed foundations of Bergmann's rule is the ratio of an endotherm's surface area to its volume (S/V). For the purpose of this discussion, a reference to “more” or “less” surface area of an organism will refer to the *relative* surface area of an endotherm compared to its volume, in order to provide an appropriate point of comparison among subjects rather than the *absolute* surface area. A key concept here is that an organism's volume increases to the third power as surface area increases to the second, resulting in declining S/V with increasing body size. Thus, a smaller organism has relatively more surface area through which to radiate heat with respect to its volume than a larger organism.

In 1878, J. C. Allen expanded on Bergmann's concept, demonstrating that endotherms in cold climates tend to have shorter appendages (legs, ears, etc.) and more rounded or compact bodies than those living in warm climates (Stiling, 2015). Thus, two endotherms of the same absolute volume may have different S/V ratios owing to differences in body architecture, resulting in different rates of heat loss. The principle of Allen's rule is similar to Bergmann's in that endotherms like the Arctic hare (*Lepus arcticus*) adapted to cold climates have smaller ears, shorter limbs, and rounder bodies, effectively presenting less surface area through which to lose heat in relation to the body volume. The antelope jackrabbit (*L. alleni*), although in the same genus as the Arctic hare, has much larger ears, longer legs, a longer body, and a longer, narrower rostrum, all of which provide greater surface area through which to radiate heat in relation to the animal's volume.

Over the years, I have used diagrams, formulas for spheres and cubes, and wooden building blocks to explain the essentials of the S/V ratio and how this is an important concept at both the cellular and organismal levels. To demonstrate the concept, I developed a simple exercise to clarify the role of the S/V ratio in Bergmann's rule. My goal was to use commonly available, low-cost materials while maintaining biological validity.

## Materials

Plastic drink bottles (2 L, 1 L, and 0.5 L) with holes drilled in the caps to accept a temperature probe

Instant-read kitchen thermometer or laboratory digital temperature probe

Measuring tape or string to measure circumference of bottles

Measuring cup or graduated cylinder for more accurate measurement of volumes

Ruler

Timer

Box fan

## Methods

I used three sizes of plastic soft drink or water bottles (2 L, 1 L, and 0.5 L) with holes drilled in the caps to accept the probe of an instant-read digital thermometer to simulate different endotherm body sizes. I filled each bottle with water at ~37°C and recorded the starting temperature of each bottle as the temperature at “time zero” (Figure 1). I placed a box fan 1 m in front of the bottles, turned it to the “low” setting, and started a timer, recording the temperatures at 5-minute intervals for a total of 20 minutes. Of course, the water in the bottles will cool in still air, but using the fan provides for additional convective cooling, which will speed up the temperature changes to give better results in a shorter amount of time.

In order to minimize cost, I used only one instant-read thermometer and moved it between bottles. Obviously, separate thermometers for each bottle would simplify measurements, but it would also introduce an issue of calibration variance that could bias results. Since the probe takes a few seconds to stabilize between readings, I measured the temperatures in the same order at each interval so that the measurement interval remained roughly equivalent for each volume across the course of the activity.

As always, replication averages-out individual variances. Pooling class data to obtain overall means for each size of model will allow for this. In this example, I used two replicates of each bottle size to calculate run means and repeated the exercise three times. I rearranged the positions of the 0.5 L and 1 L bottles between runs to account for potential differences in airflow. As long as the smaller bottles were not blocked by the larger 2 L bottles, their positions had no effect.

## Calculations & Visualizations

Although they are not perfect cylinders, I used the standard formula for the surface area of a cylinder to approximate the surface area of each bottle. Given that the circumference of the bottle = 2πr, I calculated the radius using the formula r = circumference/2π. I measured the height (h) from the bottom of the bottle to the level of fill. Given these values, I used the formula surface area = area of top and bottom of cylinder + area of side, or

Converting the volumes to milliliters, the approximate S/V values were 0.812 for 500 mL, 0.655 for 1000 mL, and 0.543 for 2000 mL (Figure 2).

If it takes 1 calorie of heat to change the temperature of 1 g of water by 1°C, and 1 mL of water has a mass of 1 g, how many calories were lost when 500 mL of water changed by 6.58°C over the course of the observations, or 1000 mL changed by 4.55°C, or 2000 mL changed by 3.17°C? If we consider *absolute* amounts of heat lost, the larger the volume the more heat is lost.

However, the *rate* of heat loss is more important in this exercise. I plotted temperature changes with a line graph to directly demonstrate a more rapid rate of heat loss for the smaller volume with the higher S/V (Figure 3). Students could also use a bar graph to compare temperatures at each recording time. Comparing these types of figures can open discussions on the value of different types of graphs and allows students to make a judgment as to which is more useful for a given set of data. For time-course measurements like the ones made here, the line graph is more informative.

Finally, I calculated the rate of heat loss for each model by subtracting the temperature at time 20 from the temperature at time zero and dividing by 20 minutes (i.e., [(T_{0} − T_{20})/20]) and compared the rates with a bar graph (Figure 4). The pattern in Figure 4 is very similar to that of Figure 2, reinforcing the close relationship of the S/V and the rate of heat loss. From these calculations and graphs, it is apparent that the model with the smallest volume and the highest S/V loses heat more rapidly.

## Special Considerations

Make sure that the holes drilled in the caps are large enough to allow the thermometer probe to be inserted easily. If the hole is too small and the probe must be forced, you run the risk of breakage or injury.

The only safety concern would be that the probe on some thermometers may have a sharp point, although no sharper than a pencil or pen. Students should be reminded to exercise caution when moving the thermometer from bottle to bottle.

## Further Investigation

The difference in the rate of heat loss based on size and the associated S/V differences open the door to a discussion of metabolic rates. Thought questions might include the following:

*How would the metabolic rate for a smaller endotherm compare to the metabolic rate for a larger endotherm in a cold climate?*In order for a small endotherm (higher S/V) in a cold climate to maintain a constant body temperature, it would be necessary to have a higher metabolic rate. A larger endotherm (lower S/V) would lose heat less rapidly, allowing the metabolic rate to be lower.

*What would be the advantage to having a smaller body size in a hot climate?*larger endotherm (lower S/V) in a hot climate would be at a disadvantage because it holds more heat. A smaller endotherm (higher S/V) in a hot climate dissipates heat more efficiently.

*Based on your observations of heat loss in models of different sizes, how would thermal regulation differ between a newborn mammal and an adult mammal?*Newborn mammals have high S/V ratios and lose heat more rapidly. Despite higher metabolic rates, they may be incapable of maintaining constant body temperatures and must rely on external heat sources to help with thermoregulation.

*Siberian tigers are larger than Sumatran tigers. Polar bears are larger than Tennessee black bears or Asian sun bears. Which of these best demonstrates Bergmann's rule?*Both the tigers and the bears can be used as examples of Bergmann's rule. However, since Siberian tigers and Sumatran tigers are members of the same species, they provide a more focused example.

## Suggestions for Audience & Assessment

Depending on the specific calculations assigned, this exercise can be scaled for upper elementary or middle school through introductory-level college laboratories. Students can prepare graphs by hand to introduce younger students to data visualization concepts, or they could use easily accessible spreadsheet software like Microsoft Excel or LibreOffice Calc. I recommend using a spreadsheet for graph preparation for more advanced students.

You can assess concept mastery by having students complete S/V calculations for each size model, calculate rates of heat loss, accurately produce and interpret relevant graphs, propose experiments to test related concepts (i.e., other ecogeographic rules like Allen's rule), and provide supported responses to thought questions such as those outlined above.

## Conclusion

According to Ricklefs (1990), “Size changes everything in ecology.” I remember having a discussion with a Minnesota native who had relocated to the South. She observed that the trophy white-tailed bucks she saw down south were tiny compared to the bucks up north. I assured her that the southern deer were not inferior to northern deer; they were just following Bergmann's rule.

Bergmann's rule is one of those common-sense principles that practicing biologists often take for granted. Students, and especially those with less of an interest in science, may not see those principles quite so easily. Using readily available materials, students can investigate why body size matters in terms of thermal regulation in endotherms. Being able to see the principle in action should provide a better practical framework on which to build other physiological concepts.

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