Body Weight and the Energetics of Temperature Regulation

The recent rebirth of interest in the 200-year-old field of homoiothermic energetics may be dated from three papers published in 1950 by Scholander, Irving and colleagues. These papers furnished many data and a theoretical analysis of the energetics of temperature regulation in tropical and arctic birds and mammals. A massive body of data has since become available through the efforts of many workers. Unfortunately, there has been little effort to extend Scholander and Irving’s model (which was based on Newton’s law of cooling) to meet the challenge of new data. One of the peculiarities of this model is that it is presented in a weight-independent form, although two of its parameters, basal rate of metabolism and thermal conductance, are well-known to be weight-dependent.

In this paper the influence of weight on the energetics of homoiotherms will be re-examined; special attention will be given to the applicability of Newton’s law of cooling.

Equation (1) indicates that the net heat loss from an object warmer than its surroundings is proportional to the temperature differential between the object and its environment. A proportionality constant C (usually referred to as thermal conductance, but maybe more properly called the coefficient of heat transfer) is defined such that C( T_b– T_a) is equal to the rate of heat loss (not C, as erroneously stated by me, 1966 a, b). If an animal regulates its temperature perfectly, T_b is constant and the rate of metabolism equals C(T_b– T_a) when there is no thermal source for regulation other than metabolism.

Two aspects of this relationship must be kept always in mind: (1) C(T_b– T_a) is an approximation of the summed heat loss by a warm animal to a cool environment, and (2) there are biological limits placed on this approximation. Each of these conditions must be examined briefly to avoid any misunderstandings.

Newton’s law of cooling, as represented in equations (1) and (4), resembles the Fourier equation for heat exchange with two exceptions. (1) A core temperature is used in Newton’s law, which is justified by being easier to measure and by the fact that most of the temperature gradient between an endotherm and its environment exists between the surface and the environment (Porter & Gates, 1969), so that a surface temperature can be replaced by a core temperature with little error, except in animals that are both large and naked. (2) The surface area is not generally used in Newton’s law. The units for thermal conductance in this paper are cal/g-h °C (or equivalent). It has been argued that these units are not appropriate because heat loss occurs per unit of surface area, not per unit of weight. There is no question that the surface area is an important parameter of heat loss, but it is nearly impossible to measure the effective area, since it may be modified by postural changes and by changes in conductivity brought about by local vasoconstriction and vasodilation. Faced with this difficulty some observers advocate that surface area measurements be replaced by (constant) W^0·67. But surely it is better to use the simpler weight-specific units for conductance than to calculate a pseudo surface-specific conductance.

There are biological limits to the application of Newton’s law. It is an appropriate expression only at cooler ambient temperatures, since the evaporative heat loss (which does not depend upon a temperature differential) becomes the predominate means of heat dissipation at higher temperatures. Another complication derives from the behaviour of C. As the ambient temperatures fall from moderate values, C gradually decreases to a minimal value. Any further fall in T_a requires an increase in heat production by the animal to maintain a constant body temperature. The ambient temperature above which changes in C are the major means of thermal adjustment to a varying T_a and below which changes in M predominate is called the lower limit of thermoneutrality (T₁)). If the separation between the temperatures at which changes in C or M are used is sharp, the curve of metabolism on T_a below T_l has a slope equal to C and extrapolates to T_b when M = o. But if an animal does not rigidly distinguish between ‘physical’ and ‘chemical’ regulation, C will decrease and M will increase with a fall in ambient temperature. As a result the slope of the curve of M on T_a below T_l is not equal to C and the curve extrapolates to a temperature greater than T_b when M = o. Contrary to the opinion of some authors, this result does not invalidate the application of Newton’s law to the heat exchange of homoiotherms, since this law is simply a mathematical approximation of the physics of heat exchange and per se does not incorporate any biological features, except for the dependence upon non-evaporative heat loss.

The two major criticisms of Newton’s law, then, are that it is either physically too simple or that it is biologically inappropriate. But as long as the Newtonian simplification is used within its proper limits, e.g. at ambient temperatures below the mid-point of thermoneutrality, its great mathematical convenience can be profitably employed.

Homoiotherms accommodate their energetics to unique environments mainly through a modification of thermal conductance and the basal rate of metabolism, M_b (McNab, 1966b). Since both parameters vary as a function of weight, species that differ greatly in weight may be compared if M_b and C are expressed relative to the values expected by weight from some ‘standard’, such as the curves of Kleiber (1960) for M_b and Herreid and Kessel (1967) for C. This transformation demonstrates that much of the observed variation in the level of T_b is due to relative variation in the basal rate and thermal conductance. The reason for this influence may be seen from rearranging Newton’s law: T_b= M/C+T_a. Thus, T_b increases with an increase in the ratio M/C, which becomes M_b/C, when evaluated at the lower limit of thermoneutrality.

What is the physical basis for the influence of M_b/C on T_b ? This ratio is a measure of the temperature differential between a homoiotherm and the environment at the lower limit of thermoneutrality. Although temperature determines the rate of chemical reaction, the rate of chemical reaction may also determine the temperature of the system. Consider a furnace having a steady-state temperature distribution and consuming coal at a fixed rate. If the amount of coal fed to the furnace is increased, the rate of combustion will increase (assuming no other limiting factors) and the temperature of the furnace will rise to a new steady-state distribution, the level of which is dictated by the temperature differential needed to dissipate the heat produced by the furnace. Endotherms can be compared to small furnaces, their temperatures determined, in part, by the proportionality existing between the rate of heat production and the temperature differential.

The biological significance of variations in the ratio M_b/C may be seen in Fig. 1. Temperate species generally have higher basal rates of metabolism than expected from weight, but they have only a modest increase, if any, in conductance (unless they are very large). The ratio M_b/C in these species tends, then, to be higher than expected from weight, a condition that leads to the high, precisely regulated body temperatures typical of species living in a thermally unstable environment. Temperate species that live in deserts have a low M_b (to minimize water exchange) and in compensation have a low conductance. Fossorial rodents have both low basal rates and high conductances to reduce the threat of overheating in closed burrows. Tropical mammals also have low basal rates and high conductances, but like fossorial mammals, can tolerate the resulting low body temperatures by living in environments characterized by thermal stability.

Recent data on tropical bats (McNab, 1969) have shown that body weight also influences the level of T_b (contrary to the conclusion of Morrison & Ryser, 1952, but in agreement with Rodbard, 1950). If it is recalled that the absolute values of M_b and C are weight-dependent, the possibility arises that the level of body temperature is determined by the weight-dependent relationships of M_b and C, as well as the extent to which actual values conform to these expectations. Thus, the measured ratio M_b/C may be represented by the product (M_b/C)r(M_b/C)_e, where the expected ratio (M_b/C)_e = (3·W^–0·25)/(10·2W^–0·51) = 3·33W^0·28 (McNab, 1966b; Herreid & Kessel, 1967) and the relative ratio (M_b/C)_r is the dimensionless fraction: (measured ratio)/(expected ratio) = (M_b/C)/(M_b/C)_e. The relative ratio is identical to that described in previous analyses (McNab, 1966b, 1969).

whether obtained by extrapolating the curves in Figs. 2B and 3 to a common body temperature (38 °C), or by taking the critical weights from the (M_b/C)_r constant curves in these figures. A similar equation is obtained from equation (5 a) by substituting T_b = 38 °C and T_l = 27 °C (the mean T_l for all mammals in Table 1).

A ‘critical’ weight results from the mathematical form of equation (5) and from the setting of a maximal T_b. Mammalian temperatures seem set near 38 °C, all deviations being due either to a small weight or to deviations from the ‘standard’ curves of metabolism and conductance on weight.

A high, precisely regulated temperature, therefore, can be attained at any weight, provided that an appropriate energy expenditure, relative to conductance and weight, is made. The exact balance differs somewhat among mammals, but it has a consistent pattern: to regulate temperature precisely a species weighing 10 kg must have the ratio (M_b/C)_r equal to or greater than 0·3, one weighing 500 g (in bats about 250 g) must have a ratio equal to 0·7, and one weighing 100 g (in bats 70 g) must have a ratio equal to 1·0.

A very small species must pay a high price for a high T_b. Morrison, Ryser & Dawe (1959) have shown that a common shrew, Sorex cinereus, even though weighing only 3·3 g, maintains a temperature of 39 °C by having (M_b/C)_r equal to about 2·9 (assuming that the mean minimal rate of metabolism measured in Sorex was equal to 1·4 M_b, due to the specific dynamic action of protein metabolism). Small shrews are near the lower limit of weight for continuous homoiothermy. Animals of still smaller weights are theoretically capable of continuous homoiothermy, but apparently it is prohibitively expensive. For example, a mammal weighing 0·25 g requires a ratio (M_b/C)_r equal to about 4·9 for T_b to be independent of weight. The intermittently endothermic sphinx moth (Celerio, thorax weight = 0·25 g) has (M_b/C)_r equal to 0·6, only 12% of the expenditure needed for continuous homoiothermy (Heath & Adams, 1967). As a result this moth can thermoregulate only with the high heat production associated with wing movements; during quiescent periods the moth is poikilothermic.

In the evolution of a small weight, then, an endotherm has two alternatives: (1) to maintain the relative levels of M_b and C, as may well be dictated by its habits and its environment, and suffer a reduction in the level and precision of thermoregulation, or (2) maintain thermoregulation by means of a compensatory increase in the relative rate of metabolism, or a compensatory decrease in the relative conductance. It appears that the second ‘choice’ is utilized whenever energetically (that is, ecologically) feasible (e.g. McNab, 1969).

Data on the lower limit of thermoneutrality in mammals (Table 1) are plotted in Fig. 5 as a function of weight. Clearly the variation in T_l, whether found among species or seasonally within species, conforms to the predictions of equations (7) and (8).

There is another source of variation inherent in K. It will be noticed in Figs. 2 B and 3 that the curves below the ‘critical’ weight, although similar in form in the various groups of mammals, do not extrapolate to a common K; K is higher in bats than in monotremes and edentates (Fig. 6a). And K is even higher in Australian marsupials. It should be re-emphasized that the analysis used above accounts for variations in the size of the temperature differential T_b– K = T_b– T_l and will account for T_b only when K is equal in all mammals. This is shown in Fig. 6B by the fact that much of the variation in T_b among the mammals in Fig. 6 A disappears when the differential T_b–T_l is used. In this case the differential approaches 0·0 as either (M_b/C)_r or (M_b/C)_e approaches o·o. Therefore, the variation of K represents a variation in the placement of the differential along a temperature axis and not in the ability of M_b/C to predict the size of the differential.

If there is significance in the variation of K, it should follow some meaningful pattern. The search for such a pattern is facilitated by substituting T₀+ z for K, where T₀ is the lowest limit of thermoneutrality for any mammal whose T_b depends upon (M_b/C)_e (here equal to 22·2 °C in Tachyglossus), and z is the difference between the actual T_l for a particular species and T₀. The range of z, then, is a measure of the flexibility of the set point of body temperature independent of the influence of W, M_b, and C. Of two mammals having the same weight, basal rate of metabolism, and conductance, the species with the highest set point for temperature regulation, that is with the largest z, will have a higher rate of metabolism at all ambient temperatures below the lower limit of thermoneutrality of the warmer species. One may therefore expect that mammals that live in cold climates or that have unusually narrow food habits might have small values for z, since this may be a means of reducing energy expenditure at a fixed weight without sacrificing the effectiveness of thermoregulation. These conclusions agree with the data on bats (Table 2). Furthermore, Didelphis has the lowest z for known marsupials, and it has moved farther into the temperate zone than any other marsupial; Sorex has a low z, which insures that its rate of metabolism is less than it would be for a mammal of its size with a large z; and z is small in edentates and monotremes, tropical mammals with specialized food habits. Body temperature thus seems to be capable of some variation independent of M_b, C and W in a manner adaptive to the climate. It is not clear what sets the limits of variation in z.

Beyond the ‘critical’ weight, body temperature is influenced by the ratio (M_b/C)_r. One can expect from equation (5b) that the slope of the curve of body temperature on the relative ratio would be equal to 3·33, irrespective of weight, since variation in W^0·28 is compensated for by variation in T_l. The same relationship between T_b and (M_b/C)_r should also exist when T_b depends upon (M_b/C)_e, provided that weight and z are constant. Both of these expectations generally hold (Fig. 7), except at some low values for (M_b/C)_r, where there is variation in z.

Since a large weight can compensate for a low ratio (M_b/C)_r in the maintenance of precise thermoregulation, large mammals tend to have lower ratios (M_b/C)_r (Fig. 1) and therefore lower body temperatures than species of an intermediate size (Figs. 2B and 3). This trend raises the possibility that the largest of mammals, the pachyderms and cetaceans, have even lower relative ratios and temperatures. One can conclude from Fig. 7 that T_b in mammals whose temperature is independent of (M_b/C)_e should approach 35·7° C as (M_b/C)_r approaches 0·0 and weight becomes infinitely large. Unfortunately, there are no adequate measurements of the energetics of large mammals; it is even difficult to obtain reliable measurements of their body temperatures.

The few data indicate a continuing decrease in temperature with an increase in weight, an asymptote in accord with the argument above occurring near 36 ° C (Fig. 8). The decrease in temperature with an increase in weight occurs intraspecifically both in the walrus (Qdobenus rosmaros;Ray & Fay, 1968) and in the northern elephant seal (Mirounga angustirostris;Bartholomew, 1954).

It is clear from this discussion that the parameters T_b, T_l, M_b, C, and W have interrelations that are consistent with the Newtonian model of thermoregulation proposed by Scholander and Irving. It may also be tentatively concluded that the level of temperature regulation in mammals is determined in the following maimer:

Birds usually have higher resting temperatures than mammals; this has been explained as due to the high basal rates of metabolism and low thermal conductances of birds (McNab, 1966a). Two problems with this explanation have since arisen. One is that the higher level of metabolism in birds has been questioned by Lasiewski & Dawson (1967) and the lower conductances disputed by Herreid and Kessel (1967). Secondly, it remains to be determined what effect bringing weight into the analysis of mammalian temperatures has upon the conclusion that bird temperatures are determined in the same manner as those of mammals.

Lasiewski & Dawson (1967) suggest on the basis of energetics that birds can best be broken into two groups, passerines and non-passerines, the curve of metabolism on weight for each having an exponent similar to that of mammals and differing each from the other and from mammals only by a coefficient. Without doubt marked differences exist among birds, but one could as well divide birds into familial or generic groups (e.g. Zar, 1968). How animals are divided into categories depends upon the questions being asked, not simply upon whether one can find statistically significant differences. Thus, one may be interested in the differences existing between homoio-therms and poikilotherms, as was Hemmingsen (1960), who combined mammals and birds into one equation. Or, one may be interested in the differences existing between mammals and birds (King & Farner, 1961; McNab, 1966a). Each of these viewpoints is legitimate.

Herreid & Kessel (1967) measured cooling constants for dead birds from which they calculated thermal conductances; they also summarized the data available in the literature on the conductances of live mammals. They concluded that there is no difference between the conductances of mammals and birds. However, conductances measured on live animals and those calculated from the cooling curves of dead individuals often do not give identical results (e.g. McNab & Morrison, 1963), which may have obscured any inequality in conductance occurring between birds and mammals. This suggestion is borne out by Lasiewski, Weathers & Bernstein (1967), who described an equation for the conductances of live birds as a function of weight that is identical in the exponent to that of Herreid and Kessel for mammals, differing only in the coefficient, which is 17% less than that for mammals. Thus, birds do in fact have lower conductances than mammals of the same weight.

The question remains whether these differences are sufficient to account for the higher temperatures of birds, especially given the effect of weight in mammals. That is, if a mammal of a given weight had the rate of metabolism and conductance of a bird of the same weight, would the mammal have the same temperature as the bird ? Or do mammals operate at lower temperatures than birds for reasons other than those of M_b and C ? Unfortunately, there are very few complete sets of data on birds.

The body temperatures of birds are plotted in Fig. 9 as a function of 8·59W^0·17 (data from the literature, see Table 3): there is an acceptable agreement between the temperatures measured in birds and the parameters of (M_b/C)_r and W^{0· 17}. Furthermore, the interactions that exist between T_b, (M_b/C)_r, and W^{0· 26} in these birds are similar to those found in mammals (Fig. 7), the major difference between these groups being that birds have higher values for the ratio (M_b/C)_r. But an important point must be made here: in spite of the many comparative studies on the metabolism of birds, most of these data have little or no value in studies of the energetics of temperature regulation because body temperatures corresponding to the rates of metabolism were not reported. This deficiency is very important, since conductance cannot be evaluated without T_b, especially when a bird’s response to a change in T_a involves both M and C. To interpret rates of metabolism in terms of temperature regulation, one must have at least the following data: (1) W, (2) T_a, (3) T_b, and (4) the animal’s activity.

Obviously, more complete data on the energetics of birds are required over a large weight range. Nevertheless, one can tentatively conclude that temperature regulation in birds is determined as it is in mammals; the differences in body temperature between these groups are primarily due to differences in M_b and C. The variation of M_b and C among birds depends upon their body weight and the conditions existing in their normal environments (Fig. 1).

I should like to thank the many people with whom I have discussed these ideas; special thanks must go to Drs A. Carr, C. Johnson, D. Johnston, and F. Nordlie and to Mr T. Krakauer, all of whom critically read this manuscript. Mr P. Laessle drew the figures for this paper. Finally, I would like to express my thanks to the National Science Foundation (GB-3477) and the American Philosophical Society (Penrose Fund 4225) for sponsoring the research that led to the ideas found within this essay.