# Effects of Contingency versus Constraints on the Body-Mass Scaling of Metabolic Rate

## Abstract

**:**

_{a}) on metabolic scaling in ectothermic versus endothermic animals. Interspecific comparisons show that increasing T

_{a}results in decreasing metabolic scaling slopes in ectotherms, but increasing slopes in endotherms, a pattern uniquely predicted by the metabolic-level boundaries hypothesis, as amended to include effects of the scaling of thermal conductance in endotherms outside their thermoneutral zone. No other published theoretical model explicitly predicts this striking variation in metabolic scaling, which I explain in terms of contingent effects of T

_{a}and thermoregulatory strategy in the context of physical and geometric constraints related to the scaling of surface area, volume, and heat flow across surfaces. My analysis shows that theoretical models focused on an ideal 3/4-power law, as explained by a single universally applicable mechanism, are clearly inadequate for explaining the diversity and environmental sensitivity of metabolic scaling. An important challenge is to develop a theory of metabolic scaling that recognizes the contingent effects of multiple mechanisms that are modulated by several extrinsic and intrinsic factors within specified constraints.

## 1. Introduction

## 2. Results

## 3. Discussion

#### 3.1. Implications of Results for Theory

_{a}) declines, and thus becomes increasingly different from the relatively high body temperature (T

_{b}) maintained by endotherms. Following Fourier’s Law (which is related to Newton’s law of cooling), rate of heat flow across an organism’s surface is a function of four major factors: surface area, thickness and thermal conductivity of the surface layer of insulation, and the temperature differential (T

_{b}– T

_{a}) between the inside and outside of an organism [25,56]. At thermoneutrality, which is near 30 °C in most endothermic vertebrates, the temperature differential is relatively small, and the cost of thermoregulation is minimal, thus causing the metabolic scaling exponent to approximate 2/3, which follows the classic surface law [11,12], as pointed out by [56]. Within the thermoneutral zone, metabolic heat production exactly balances the heat dissipated, which is chiefly a function of surface area. However, as the cross-surface temperature differential increases, the metabolic scaling slope should approach ~0.45–0.55, as typically observed for the scaling of thermal conductance in birds and mammals [6,79,80,81,82,83,84,85,86]. As predicted, when T

_{a}equals 0 °C (which is greater than 30 °C below T

_{b}), the metabolic scaling exponents for mammals (0.40), passerine birds (0.52) and nonpasserine birds (0.53) ([56,58]; Figure 3 and Figure 4) all approach that observed for the scaling exponent of thermal conductance.

^{0.167})(M

^{0.667})/(W

^{0.333}), which are the hypothesized power relations for the thermal conductivity of the surface layer of insulation (h), surface area (A), and insulation thickness (I), respectively, for endotherms. This dimensional analysis uses a formula for the rate of heat flow across a surface (k) that is based on Fourier’s Law: i.e., k = hA/I(T

_{b}− T

_{a}) (cf. [6,25,28,58]). It also assumes that T

_{b}and body shape are constant and pelage mass increases isometrically with body mass (scaling slope = 1), and ignores the effects of other factors such as radiation and air convection [6,25]. Notably, the hypothesized power relations for h, A and I approximate empirical estimates quite closely [56,83]. Similarly, Schmidt-Nielsen [6] noted that the insulation of mammals scales as M

^{0.17}, and thus thermal conductance should scale as (M

^{0.67})/(W

^{0.17}) or as M

^{0.50}, as approximately observed (cf. [88]). Therefore, the greater insulation of larger mammals causes the scaling slope for heat dissipation at low T

_{a}to approximate 0.5, which in turn requires the scaling slope for metabolic heat production to approximate 0.5, in order to maintain a constant body temperature (cf. [25,89]). In support, maximal non-shivering thermogenesis (induced by noradrenaline injection) scales similarly in mammals (exponent = 0.546 [90]). In addition, if one minimizes the size-related effect of insulation by exposing birds or mammals of different size to low temperatures in a He-O

_{2}atmosphere with high thermal conductivity, surface-area related heat dissipation chiefly influences metabolic scaling, which thus has a scaling slope approximating 2/3 (see [91,92,93]).

_{a}. According to dynamic energy budget theory, as recently modified by [94,95], negative associations between T

_{a}and the metabolic scaling exponent should occur in some, but not all colonial animals. However, the ability of this model to explain T

_{a}effects on the metabolic scaling of unitary organisms is problematic. It invokes effects of T

_{a}on growth rate, which cannot explain negative associations between T

_{a}and the metabolic scaling exponent commonly observed in ectothermic organisms. Decreasing T

_{a}inhibits growth rate, and decreased growth rates are associated with lower, not higher metabolic scaling exponents (see [10,32,44,96]).

#### 3.2. Challenges for Future Research

#### 3.2.1. General Perspective

_{a}and thermoregulatory lifestyle (ectothermy versus endothermy). The metabolic scaling slope varies with T

_{a}, and the nature of this variation differs markedly between ectotherms and endotherms (Figure 1, Figure 2, Figure 3 and Figure 4). One cannot understand this variation in metabolic scaling without recognizing these and other contingencies. In birds and mammals, photoperiod and time of day of metabolic measurements may also affect thermal conductance and its scaling with body mass [6,82], and by association, metabolic rate and its scaling with body mass in the cold [56]. These contingent effects operate within the boundaries of geometric and physical constraints, including surface-area-to-volume relationships, and physical laws of heat flow.

_{a}on metabolic scaling than do birds and mammals, because of differences in ecological habitat (aquatic vs. terrestrial), rather than thermoregulatory strategy (ectothermy vs. endothermy). Increasing temperature not only increases metabolic demand for oxygen both in water and on land, but in water, it also decreases oxygen concentration and thus its availability. The double jeopardy of higher oxygen demand and lower oxygen supply at high T

_{a}in water, especially as body size increases, may therefore cause inverse relationships between T

_{a}and the metabolic scaling exponent to be more prevalent in aquatic vs. terrestrial animals. However, as pointed out in Section 3.1, negative associations between T

_{a}and the metabolic scaling exponent also occur in many terrestrial ectotherms. In addition, the habitat hypothesis cannot explain why associations between T

_{a}and the metabolic scaling exponent are positive in endotherms. Therefore, I believe that it is more likely that the taxonomic differences in how T

_{a}affects metabolic scaling observed in this study relate more to differences in thermoregulatory strategy than to habitat.

#### 3.2.2. Suggestions for Future Research

_{a}affects metabolic scaling are required for comparisons both within and among species. As predicted by the MLBH (as amended in Section 3.1), in the laboratory mouse (Mus musculus domesticus), metabolic level is higher at 21 versus 29 °C, but the metabolic scaling slope is lower [106]. Varied patterns occur for other rodents, but these analyses suffer from very small body-mass ranges [107]. In addition, as predicted by the MLBH, the metabolic rate of marine zooplankton (including diverse taxa) increases in the order of boreal, temperate and tropical species, whereas the interspecific scaling exponent (based on dry masses) decreases (0.830, 0.691, and 0.538, respectively) [108], though this may not be true for planktonic crustaceans [109]. Although inverse relationships between T

_{a}and the exponent (slope) for ontogenetic metabolic scaling are common in ectotherms, other patterns are also possible, including non-significant, positive and nonlinear relationships (reviewed in [16,26]). In this study, I suggest that thermoregulatory lifestyle (ectothermy versus endothermy) can affect the temperature-sensitivity of interspecific metabolic scaling. Other differences in lifestyle may also help explain the varied responses of intraspecific metabolic scaling to T

_{a}observed in ectotherms. One key lifestyle feature may be activity level. According to the MLBH, for resting (inactive) metabolism, increasing metabolic level should be associated with decreasing scaling exponents, whereas for active metabolism, increasing metabolic level should be associated with increasing scaling exponents (because of an increase in the relative influence of volume-related locomotor power production) ([16,23,26]; cf. [92]). Therefore, in sedentery (immobile) organisms, a thermally increased metabolic level should result in a lowered metabolic scaling exponent, whereas, in actively mobile animals, a thermally increased metabolic level may result in a variety of effects on the metabolic scaling exponent, depending on the relative size-specific effects of T

_{a}on activity level (also see [26]). Consistent with this hypothesis, sedentery or mostly stationary organisms (including plants, oysters, mussels, chitons, and ascideans) usually show strong negative associations between T

_{a}and the resting metabolic scaling exponent (e.g., [110,111,112,113,114]), whereas actively mobile animals show a variety of responses (as reviewed in [16,26]; and as shown in an unpublished data set). As further evidence, when the effects of activity are removed in an actively mobile species, such as the fish Coregonus albula, T

_{a}and the resting metabolic scaling exponent are strongly negatively correlated [115], as predicted by the MLBH [16,26]. Further studies of the effects of various abiotic and biotic ecological factors on metabolic scaling would also be worthwhile (see also [10,16,22,26,32,41,44,45,46,48]).

_{a}, but also the mode of regulation of T

_{b}, affect metabolic scaling. My interpretation of these contingent thermal effects contributes to a growing revival of the old, controversial view that thermoregulation is importantly involved in the metabolic scaling of endotherms ([10,11,12,16,23,24,25,26,27,28,32,56,58,87,89,93], but see [9,21,22,33,92,116,117]). This thermoregulatory view is testable. For example, since thermal insulation scales allometrically (slope ~ 0.17) in mammals with a body mass <10 kg, but remains constant in larger mammals [6], the thermoregulatory view predicts that at low T

_{a}below the thermoneutral zone, the scaling exponent for metabolic rate should approach 0.50 in small mammals, but be near 0.67 in large mammals (cf. Section 3.1). The data provided in Figure 4 are only for mammals ≤14 kg, which show metabolic scaling exponents of 0.39–0.56 at T

_{a}below the thermoneutral zone, which encompass the predicted slope of 0.50. As a further test, we now need data for larger mammals exposed to the cold. A thermoregulatory view may also help explain the curvilinearity of the scaling of mammalian basal metabolic rate in the thermoneutral zone, showing a slope near 2/3 in small mammals, and a higher slope (≥3/4) in larger mammals (e.g., [93,118,119]). Is it a coincidence that the breakpoint in the scaling of mammalian basal metabolic rate occurs near a body mass of 10 kg (4.25 or 20 kg, according to the two-segmented linear models of Kozłowski and Konarzewski [120] and Makarieva and colleagues [121], respectively)?

_{b}and metabolic rate during torpor and hibernation [125]: as a result, their metabolic scaling slope shifts markedly from ~0.67–0.75 to ~1, as predicted by the MLBH [10,23,26,93]. This observation also reinforces the view that thermoregulation plays a major role in the metabolic scaling of endotherms. Therefore, I recommend that increased attention should be given to how various regulatory systems at various levels of biological organization (from biochemical signaling pathways to neuroendocrine systems) affect metabolic scaling. Metabolic scaling is not merely the result of physical constraints, but is also mediated by various regulated processes involving resource supply and demand, metabolic waste removal, and heat dissipation [10,27,96].

_{a}on the metabolic scaling exponent (see Section 3.1), it can, if combined with the MLBH. Thermally increased metabolic level may cause decreases in the scaling exponent for resting metabolic rate, not only because of the increased influence of surface-area related metabolic processes at the whole body level, but also at the cellular level, a hypothesis requiring testing [10,26,129]. Other examples of multi-mechanistic models of metabolic scaling are reviewed in [10,32].

## 4. Materials and Methods

_{a}affects the scaling slopes of existence metabolism and oxygen consumption rates in similar ways. As shown for existence metabolism (Figure 3), the interspecific scaling slope for oxygen consumption rate of birds decreases at lower T

_{a}(30 °C: summer and winter scaling slopes = 0.658, 0.688 for passerines, and 0.701, 0.728 for non-passerines; 0 °C: summer and winter scaling slopes = 0.528, 0.531 for passerines, and 0.571, 0.594 for non-passerines [56]). Figure 3 depicts scaling relationships for existence metabolism, because they had larger sample sizes (and thus were more reliable) than those for oxygen consumption.

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Interspecific scaling of metabolic rate in relation to body mass of marine crustaceans at three different habitat temperatures, 29, 25 and 20 °C (data from [53]). The correlation coefficients (r) and sample sizes (n) are 0.963 and 212, 0.727 and 249, and 0.977 and 247, respectively. All probability values are <0.001. The inset graph shows the negative relationship between the metabolic scaling slope (±95% confidence intervals) and habitat temperature. The crab picture is from [54].

**Figure 2.**Ontogenetic scaling of metabolic rate in relation to body mass averaged for multiple species of marine teleost fishes at three different habitat temperature ranges, ≥20, 10 to <20, and <10 °C (data from [48]). The sample sizes (n) are 30, 19 and 40, respectively. The inset graph shows the negative relationship between the mean metabolic scaling slope (±95% confidence intervals) and habitat temperature. The fish picture is from [55].

**Figure 3.**Interspecific scaling of existence metabolism in relation to body mass for multiple species of passerine and non-passerine birds at two different air temperatures, 0 and 30 °C (data from [56]). The sample sizes (n) are 71 for passerine birds, and 40 for non-passerine birds. The inset graph shows the positive relationships between the metabolic scaling slopes and temperature for both passerine and non-passerine birds. The bird picture is from [57].

**Figure 4.**Interspecific scaling of metabolic rate in relation to body mass for multiple species of eutherian mammals at four different air temperatures, 0, 10, 20 and 30 °C (data from [58]). The correlation coefficients (r) and sample sizes (n) are 0.91 and 61, 0.84 and 69, 0.88 and 73, and 0.89 and 81, respectively. All probability values are <0.001. The inset graph shows the positive relationship between the metabolic scaling slope (±95% confidence intervals) and temperature. The mammal picture is from [59].

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Glazier, D.S.
Effects of Contingency versus Constraints on the Body-Mass Scaling of Metabolic Rate. *Challenges* **2018**, *9*, 4.
https://doi.org/10.3390/challe9010004

**AMA Style**

Glazier DS.
Effects of Contingency versus Constraints on the Body-Mass Scaling of Metabolic Rate. *Challenges*. 2018; 9(1):4.
https://doi.org/10.3390/challe9010004

**Chicago/Turabian Style**

Glazier, Douglas S.
2018. "Effects of Contingency versus Constraints on the Body-Mass Scaling of Metabolic Rate" *Challenges* 9, no. 1: 4.
https://doi.org/10.3390/challe9010004