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Metabolic Scaling in Animals: Methods, Empirical Results, and Theoretical Explanations

Craig R. White, Michael R. Kearney

10.1002/cphy.c110049

Source: Volume 4, Issue 1, January 2014

Published online: January 2014

Full Article on Wiley Online Library



Abstract

Life on earth spans a size range of around 21 orders of magnitude across species and can span a range of more than 6 orders of magnitude within species of animal. The effect of size on physiology is, therefore, enormous and is typically expressed by how physiological phenomena scale with massb. When b ≠ 1 a trait does not vary in direct proportion to mass and is said to scale allometrically. The study of allometric scaling goes back to at least the time of Galileo Galilei, and published scaling relationships are now available for hundreds of traits. Here, the methods of scaling analysis are reviewed, using examples for a range of traits with an emphasis on those related to metabolism in animals. Where necessary, new relationships have been generated from published data using modern phylogenetically informed techniques. During recent decades one of the most controversial scaling relationships has been that between metabolic rate and body mass and a number of explanations have been proposed for the scaling of this trait. Examples of these mechanistic explanations for metabolic scaling are reviewed, and suggestions made for comparing between them. Finally, the conceptual links between metabolic scaling and ecological patterns are examined, emphasizing the distinction between (1) the hypothesis that size‐ and temperature‐dependent variation among species and individuals in metabolic rate influences ecological processes at levels of organization from individuals to the biosphere and (2) mechanistic explanations for metabolic rate that may explain the size‐ and temperature‐dependence of this trait. © 2014 American Physiological Society. Compr Physiol 4:231‐256, 2014.

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Figure 1. Figure 1. Scaling of surface area with body mass (M, kg) in mammals (data from 333). The solid line is the phylogenetically informed scaling relationship (surface area = 0.092 M 0.67) (333). The lower dashed line is the relationship between the surface area and volume of a sphere with a density of 1.08 g cm−3, a rough estimate of a mixture of muscle, fat, and bone: muscle has a density of 1.06 g cm−3; bone has a density of 2.00 g cm−3; and fat has a density of 0.93 g cm−3 (7).
Figure 2. Figure 2. (A) Standard deviation of 10,000 scaling exponents (b, where Y = aMb ) calculated for mass ranges of 0.2 to 6 orders of magnitude. (B) Standard deviation of 10,000 scaling exponents (b, where Y = aMb ) calculated for sample sizes of 10 to 500 “species.” For each of the 10,000 simulations in (A) 100 values of log(M) spanning the appropriate range were randomly generated and values of Y were calculated as M 0.75 plus a normal deviate with a mean of 0 and a standard deviation equal to 20% of M 0.75. For each of the 10,000 simulations in (B), values of log(M) spanning two orders of magnitude were randomly generated and values of Y were calculated as M 0.75 plus a normal deviate with a mean of 0 and a standard deviation equal to 20% of M 0.75. Values of b were then calculated as the slope of the relationship between log(Y) and log(M).
Figure 3. Figure 3. The scaling of mean arterial blood pressure (P, mmHg) in mammals (data and analysis from 448). Solid line is the significant three parameter power equation: P = 6.9 M 0.24 + 93. The scaling exponent is significantly greater than 0, and not significantly different from the scaling exponent of 0.33 predicted by geometric scaling of the vertical distance between the heart and head (448).
Figure 4. Figure 4. Scaling of skeletal mass (Ms ) and relative skeletal mass with body mass (M) in mammals. Relative skeletal mass is calculated by dividing skeletal mass by body mass. Data are for 73 species and were compiled from published sources (21,241,319,322,335), and matched to a supertree of mammals (28). Data were analyzed using phylogenetic generalized least squares (PGLS) (128,146,251) in the APE (Analysis of Phylogenetics and Evolution) package (310) within R (189) according to established procedures (98,156,432). In addition to the dated branch lengths associated with the supertree, a range of branch length transformations were compared: star, log e , punctuated, Grafen's (146), Nee's (324), and Pagel's (308). For each of these models, a measure of phylogenetic correlation, λ (119,306), was estimated by fitting PGLS models with different values of λ and finding the value that maximizes the log likelihood. The degree to which trait evolution deviates from Brownian motion (λ = 1) was accommodated by modifying the covariance matrix using the maximum likelihood value of λ, which is a multiplier of the off‐diagonal elements of the covariance matrix (i.e., those quantifying the degree of relatedness between species). All models were compared on the basis of Akaike's Information Criterion (AIC) as a measure of model fit (49). The solid line is a significant second‐order polynomial regression relating log(M s) to log(M) and [log(M) + 4]2: the best model included a phylogeny with all branch lengths equal to 1 (wi = 0.18, λ = 0.32); log(Ms ) = −1.53 + 0.87 log(M) + 0.02 [log(M) + 4]2. Dashed line is the best model for data excluding Loxodonta africana and Elephas maximas and including a phylogeny with all branch lengths equal to 1 (wi = 0.68, λ = 0.37), which is a linear model: log(Ms ) = −1.19 + 1.02 log(M). The 95% confidence interval of the scaling exponent for the linear model includes 1 (95%CI: 0.98‐1.06).
Figure 5. Figure 5. Scaling of field metabolic rate (FMR) with body mass (M) in arid (filled symbols, solid line) and nonarid (unfilled symbols, dashed line) birds (data from 405). Phylogenetically informed relationships: arid birds, FMR = 5.24 (2.86‐9.59) M 0.691 (0.610‐0.772); non‐arid birds, FMR = 9.31 (7.79‐11.12) M 0.676 (0.383‐0.969); values in parentheses are 95% confidence limits. Analyzed using independent contrasts (110), the scaling exponent of FMR does not differ among arid and non‐arid birds (t = −1.57, P = 0.12), whereas environment (desert or non‐desert) does have a significant effect (t = 2.11, P < 0.04) (405).
Figure 6. Figure 6. Scaling of heart mass with body mass for birds (unfilled symbols, dashed line) and mammals (filled symbols, dashed line) (data from 362). The scaling exponents of heart mass are different for birds (b = 0.91) and mammals (b = 1.06). Application of the Johnson‐Neyman technique (196) demonstrates that at P = 0.05, regression elevations are not significantly different at masses above 4.26 kg (427), thus confirming the conclusion that the hearts of flightless birds are not significantly larger than those of similarly sized mammals (362). The vertical dashed line represents the lower limit of the region of nonsignificance. Within the region of nonsignificance there is no significant difference in elevation between the scaling relationships, so in this example the groups differ significantly in elevation to the left of the vertical line. Birds: heart mass (g) = 8.08 M 0.91, mammals: heart mass = 4.04 M 1.06.
Figure 7. Figure 7. Scaling of heart rate (f H, bpm) with body mass (M, kg) for mammals. Data are for 58 species compiled by White and Seymour (444), matched to a supertree of mammals (28), and analyzed according to the PGLS methods described in the legend to Figure 3. Solid line is the scaling relationship fitted by PGLS with all branch lengths equal and equal to 1, with a maximum likelihood λ of 0.71 (f H = 150 M−0.26 [95% CI: −0.30 to −0.22]).
Figure 8. Figure 8. Interspecific relationship between heart rate (f H, bpm) and basal metabolic rate (BMR, mL h−1) assessed using phylogenetic generalized least squares. Data are for 58 species (444) matched to a supertree of mammals (28). The relationship between BMR and f H was then assessed by comparing the fit of statistical models that either included BMR (logf H ∼ logM + logBMR) or did not include BMR (logf H ∼ logM). The best model was chosen on the basis of its Akaike weight (w i, the probability that a model is the best of a candidate set, given the data). The best model included logBMR and logM (w i = 0.19) with a maximum likelihood λ of 0.64 and all branch lengths set equal to one (a punctuated model of evolution). Summed over all evolutionary models (dated, punctuated, Grafen's, Nee's and Pagel's), the probability that the best model for logf H includes logBMR in addition to logM is 0.66 (see legend to Figure 3 for full details of analysis procedure). Residuals are shown to account for the relationships between logM and both logBMR and logf H, but the analysis of the relationship between f H and BMR was not assessed using residuals; the solid line is the parameter estimate for logBMR from the best model (logf H ∼ logM + logBMR) plotted through the bivariate mean of residuals.
Figure 9. Figure 9. (A) scaling of basal metabolic for 148 species of murid rodent. Data adapted, with permission, from White and Seymour (443) synonymized to match the supertree of Bininda‐Emonds et al. (28); see White et al. (432) for details. Solid blue line is the ordinary least squares relationship; dashed blue lines are the 95% prediction interval of the ordinary least squares (OLS) relationship. Solid red line is the relationship for Notomys alexis estimated by independent contrasts (PIC) (110) following Garland and Ives (128); dotted red lines are the 95% prediction interval for the phylogenetically informed regression. (B) BMR measured for 11 individual Notomys alexis (mean mass 33 g) using indirect calorimetry (440) shown ± SEM and compared with predicted BMR for a 33 g murid rodent (shown ± SEE) for the OLS and PIC regressions presented in (A). Error bounds of BMR value predicted by OLS encompass the measured value, but absolute BMR is overestimated by 29%, and the OLS relationship estimates BMR with considerable uncertainty. The PIC estimate of BMR for Notomys alexis is more accurate and overestimates BMR by only 4%, an error similar to the measurement error associated with experimental determination of metabolic rate by indirect calorimetry (e.g., 439), but the error bounds associated with the PIC estimate of BMR are wider than those of OLS. Note that the error bars for predicted BMR are asymmetric because of back‐transformation from log‐transformed data.
Figure 10. Figure 10. Mean coefficient of determination (r 2, filled diamonds and solid line) and standard error of estimate (filled squares and dashed line) for 10,000 scaling exponents (b, where Y = aMb ) calculated for mass ranges of 0.2 to 6 orders of magnitude. For each of the 10,000 simulations, 100 values of log(M) spanning the appropriate range were randomly generated and values of Y were calculated as M 0.75 plus a normal deviate with a mean of 0 and a standard deviation equal to 20% of M 0.75. The value of b was then calculated as the slope of the relationship between log(Y) and log(M). For a given quantity of residual variation, the coefficient of determination increases with mass range but the standard error of estimate does not.
Figure 11. Figure 11. Scaling of (A) maximum metabolic rate ( V · O 2 max, filled diamonds and solid black line), (B) field metabolic rate (FMR, filled and unfilled circles, blue and red solid lines), and (C) basal metabolic rate (BMR, unfilled diamonds, blue and red dashed lines) with body mass (M) for mammals. V · O 2 max = 3300 M 0.87 (see also 93 for a recent analysis including additional species that reported a scaling exponent of 0.85, data adapted, with permission, from 445). Data for FMR are shown for terrestrial (filled circles) and aquatic (unfilled circles) animals (2,289,385); blue and red solid lines in (B) are maximum sustained metabolic rates predicted by the Heat Dissipation Limit theory for endotherms at 10 and 30°C, respectively (385). Blue and red dashed lines in (B) and (C) are predicted BMR for tropical/xeric/desert and widespread species, respectively, and are curvilinear on log‐log axes (429).
Figure 12. Figure 12. Size dependence of (A) factorial aerobic scope ( = maximum aerobic metabolic rate, V · O 2 max, divided by basal metabolic rate, BMR) and (B) factorial activity scope ( = field metabolic rate, FMR, divided by BMR) for bats (unfilled symbols) and other mammals (filled symbols). Both aerobic and activity scope show a triangular pattern, with a wider range of aerobic scopes observed for large mammals than small ones, and a wider range of activity scopes observed for small animals than large ones. Aerobic and activity scopes were calculated using published data for V · O 2 max (198,234,412,445), FMR (385), and BMR (101,267).


Figure 1. Scaling of surface area with body mass (M, kg) in mammals (data from 333). The solid line is the phylogenetically informed scaling relationship (surface area = 0.092 M 0.67) (333). The lower dashed line is the relationship between the surface area and volume of a sphere with a density of 1.08 g cm−3, a rough estimate of a mixture of muscle, fat, and bone: muscle has a density of 1.06 g cm−3; bone has a density of 2.00 g cm−3; and fat has a density of 0.93 g cm−3 (7).


Figure 2. (A) Standard deviation of 10,000 scaling exponents (b, where Y = aMb ) calculated for mass ranges of 0.2 to 6 orders of magnitude. (B) Standard deviation of 10,000 scaling exponents (b, where Y = aMb ) calculated for sample sizes of 10 to 500 “species.” For each of the 10,000 simulations in (A) 100 values of log(M) spanning the appropriate range were randomly generated and values of Y were calculated as M 0.75 plus a normal deviate with a mean of 0 and a standard deviation equal to 20% of M 0.75. For each of the 10,000 simulations in (B), values of log(M) spanning two orders of magnitude were randomly generated and values of Y were calculated as M 0.75 plus a normal deviate with a mean of 0 and a standard deviation equal to 20% of M 0.75. Values of b were then calculated as the slope of the relationship between log(Y) and log(M).


Figure 3. The scaling of mean arterial blood pressure (P, mmHg) in mammals (data and analysis from 448). Solid line is the significant three parameter power equation: P = 6.9 M 0.24 + 93. The scaling exponent is significantly greater than 0, and not significantly different from the scaling exponent of 0.33 predicted by geometric scaling of the vertical distance between the heart and head (448).


Figure 4. Scaling of skeletal mass (Ms ) and relative skeletal mass with body mass (M) in mammals. Relative skeletal mass is calculated by dividing skeletal mass by body mass. Data are for 73 species and were compiled from published sources (21,241,319,322,335), and matched to a supertree of mammals (28). Data were analyzed using phylogenetic generalized least squares (PGLS) (128,146,251) in the APE (Analysis of Phylogenetics and Evolution) package (310) within R (189) according to established procedures (98,156,432). In addition to the dated branch lengths associated with the supertree, a range of branch length transformations were compared: star, log e , punctuated, Grafen's (146), Nee's (324), and Pagel's (308). For each of these models, a measure of phylogenetic correlation, λ (119,306), was estimated by fitting PGLS models with different values of λ and finding the value that maximizes the log likelihood. The degree to which trait evolution deviates from Brownian motion (λ = 1) was accommodated by modifying the covariance matrix using the maximum likelihood value of λ, which is a multiplier of the off‐diagonal elements of the covariance matrix (i.e., those quantifying the degree of relatedness between species). All models were compared on the basis of Akaike's Information Criterion (AIC) as a measure of model fit (49). The solid line is a significant second‐order polynomial regression relating log(M s) to log(M) and [log(M) + 4]2: the best model included a phylogeny with all branch lengths equal to 1 (wi = 0.18, λ = 0.32); log(Ms ) = −1.53 + 0.87 log(M) + 0.02 [log(M) + 4]2. Dashed line is the best model for data excluding Loxodonta africana and Elephas maximas and including a phylogeny with all branch lengths equal to 1 (wi = 0.68, λ = 0.37), which is a linear model: log(Ms ) = −1.19 + 1.02 log(M). The 95% confidence interval of the scaling exponent for the linear model includes 1 (95%CI: 0.98‐1.06).


Figure 5. Scaling of field metabolic rate (FMR) with body mass (M) in arid (filled symbols, solid line) and nonarid (unfilled symbols, dashed line) birds (data from 405). Phylogenetically informed relationships: arid birds, FMR = 5.24 (2.86‐9.59) M 0.691 (0.610‐0.772); non‐arid birds, FMR = 9.31 (7.79‐11.12) M 0.676 (0.383‐0.969); values in parentheses are 95% confidence limits. Analyzed using independent contrasts (110), the scaling exponent of FMR does not differ among arid and non‐arid birds (t = −1.57, P = 0.12), whereas environment (desert or non‐desert) does have a significant effect (t = 2.11, P < 0.04) (405).


Figure 6. Scaling of heart mass with body mass for birds (unfilled symbols, dashed line) and mammals (filled symbols, dashed line) (data from 362). The scaling exponents of heart mass are different for birds (b = 0.91) and mammals (b = 1.06). Application of the Johnson‐Neyman technique (196) demonstrates that at P = 0.05, regression elevations are not significantly different at masses above 4.26 kg (427), thus confirming the conclusion that the hearts of flightless birds are not significantly larger than those of similarly sized mammals (362). The vertical dashed line represents the lower limit of the region of nonsignificance. Within the region of nonsignificance there is no significant difference in elevation between the scaling relationships, so in this example the groups differ significantly in elevation to the left of the vertical line. Birds: heart mass (g) = 8.08 M 0.91, mammals: heart mass = 4.04 M 1.06.


Figure 7. Scaling of heart rate (f H, bpm) with body mass (M, kg) for mammals. Data are for 58 species compiled by White and Seymour (444), matched to a supertree of mammals (28), and analyzed according to the PGLS methods described in the legend to Figure 3. Solid line is the scaling relationship fitted by PGLS with all branch lengths equal and equal to 1, with a maximum likelihood λ of 0.71 (f H = 150 M−0.26 [95% CI: −0.30 to −0.22]).


Figure 8. Interspecific relationship between heart rate (f H, bpm) and basal metabolic rate (BMR, mL h−1) assessed using phylogenetic generalized least squares. Data are for 58 species (444) matched to a supertree of mammals (28). The relationship between BMR and f H was then assessed by comparing the fit of statistical models that either included BMR (logf H ∼ logM + logBMR) or did not include BMR (logf H ∼ logM). The best model was chosen on the basis of its Akaike weight (w i, the probability that a model is the best of a candidate set, given the data). The best model included logBMR and logM (w i = 0.19) with a maximum likelihood λ of 0.64 and all branch lengths set equal to one (a punctuated model of evolution). Summed over all evolutionary models (dated, punctuated, Grafen's, Nee's and Pagel's), the probability that the best model for logf H includes logBMR in addition to logM is 0.66 (see legend to Figure 3 for full details of analysis procedure). Residuals are shown to account for the relationships between logM and both logBMR and logf H, but the analysis of the relationship between f H and BMR was not assessed using residuals; the solid line is the parameter estimate for logBMR from the best model (logf H ∼ logM + logBMR) plotted through the bivariate mean of residuals.


Figure 9. (A) scaling of basal metabolic for 148 species of murid rodent. Data adapted, with permission, from White and Seymour (443) synonymized to match the supertree of Bininda‐Emonds et al. (28); see White et al. (432) for details. Solid blue line is the ordinary least squares relationship; dashed blue lines are the 95% prediction interval of the ordinary least squares (OLS) relationship. Solid red line is the relationship for Notomys alexis estimated by independent contrasts (PIC) (110) following Garland and Ives (128); dotted red lines are the 95% prediction interval for the phylogenetically informed regression. (B) BMR measured for 11 individual Notomys alexis (mean mass 33 g) using indirect calorimetry (440) shown ± SEM and compared with predicted BMR for a 33 g murid rodent (shown ± SEE) for the OLS and PIC regressions presented in (A). Error bounds of BMR value predicted by OLS encompass the measured value, but absolute BMR is overestimated by 29%, and the OLS relationship estimates BMR with considerable uncertainty. The PIC estimate of BMR for Notomys alexis is more accurate and overestimates BMR by only 4%, an error similar to the measurement error associated with experimental determination of metabolic rate by indirect calorimetry (e.g., 439), but the error bounds associated with the PIC estimate of BMR are wider than those of OLS. Note that the error bars for predicted BMR are asymmetric because of back‐transformation from log‐transformed data.


Figure 10. Mean coefficient of determination (r 2, filled diamonds and solid line) and standard error of estimate (filled squares and dashed line) for 10,000 scaling exponents (b, where Y = aMb ) calculated for mass ranges of 0.2 to 6 orders of magnitude. For each of the 10,000 simulations, 100 values of log(M) spanning the appropriate range were randomly generated and values of Y were calculated as M 0.75 plus a normal deviate with a mean of 0 and a standard deviation equal to 20% of M 0.75. The value of b was then calculated as the slope of the relationship between log(Y) and log(M). For a given quantity of residual variation, the coefficient of determination increases with mass range but the standard error of estimate does not.


Figure 11. Scaling of (A) maximum metabolic rate ( V · O 2 max, filled diamonds and solid black line), (B) field metabolic rate (FMR, filled and unfilled circles, blue and red solid lines), and (C) basal metabolic rate (BMR, unfilled diamonds, blue and red dashed lines) with body mass (M) for mammals. V · O 2 max = 3300 M 0.87 (see also 93 for a recent analysis including additional species that reported a scaling exponent of 0.85, data adapted, with permission, from 445). Data for FMR are shown for terrestrial (filled circles) and aquatic (unfilled circles) animals (2,289,385); blue and red solid lines in (B) are maximum sustained metabolic rates predicted by the Heat Dissipation Limit theory for endotherms at 10 and 30°C, respectively (385). Blue and red dashed lines in (B) and (C) are predicted BMR for tropical/xeric/desert and widespread species, respectively, and are curvilinear on log‐log axes (429).


Figure 12. Size dependence of (A) factorial aerobic scope ( = maximum aerobic metabolic rate, V · O 2 max, divided by basal metabolic rate, BMR) and (B) factorial activity scope ( = field metabolic rate, FMR, divided by BMR) for bats (unfilled symbols) and other mammals (filled symbols). Both aerobic and activity scope show a triangular pattern, with a wider range of aerobic scopes observed for large mammals than small ones, and a wider range of activity scopes observed for small animals than large ones. Aerobic and activity scopes were calculated using published data for V · O 2 max (198,234,412,445), FMR (385), and BMR (101,267).
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Article Title: Metabolic Scaling in Animals: Methods, Empirical Results, and Theoretical Explanations
 

How to Cite

Craig R. White, Michael R. Kearney. Metabolic Scaling in Animals: Methods, Empirical Results, and Theoretical Explanations. Compr Physiol 2014, 4: 231-256. doi: 10.1002/cphy.c110049