Allometric exponents support a 3/4 power scaling

law

Catherine C. FarrellNicholas J. Gotelli

Department of BiologyUniversity of VermontBurlington, VT 05405

Gotelli lab, May 2005

Allometric Scaling

• What is the relationship metabolic rate (Y) and body mass (M)?

Allometric Scaling

• What is the relationship metabolic rate (Y) and body mass (M)?

• Mass units: grams, kilograms

• Metabolic units: calories, joules, O2 consumption, CO2 production

Allometric Scaling

• What is the relationship metabolic rate (Y) and body mass (M)?

• Usually follows a power function:

• Y = CMb

Allometric Scaling

• What is the relationship metabolic rate (Y) and body mass (M)?

• Usually follows a power function:

• Y = CMb

• C = constant

• b = allometric scaling coefficient

Allometric Scaling: Background• Allometric scaling equations relate basal

metabolic rate (Y) and body mass (M) by an allometric exponent (b)

0

2

4

6

8

10

12

0 20 40 60 80 100 120

M

Y

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.5 1 1.5 2 2.5

Log (M)

Log (Y)

Y = YoMb Log Y = Log Yo + b log M

Allometric Scaling: Background• Allometric scaling equations relate basal

metabolic rate (Y) and body mass (M) by an allometric exponent (b)

0

2

4

6

8

10

12

0 20 40 60 80 100 120

M

Y

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.5 1 1.5 2 2.5

Log (M)

Log (Y)

Y = YoMb Log Y = Log Yo + b log M

b is the slope of the log-log plot!

Allometric Scaling

• What is the expected value of b?

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.5 1 1.5 2 2.5

Log (M)

Log (Y)

??

Hollywood Studies Allometry

Godzilla (1954)

A scaled-up dinosaur

Hollywood Studies Allometry

The Incredible Shrinking Man (1953)

A scaled-down human

Miss Allometry

Raquel Welch

Movies spanning > 15 orders of magnitude of body mass!

1 Million B.C. (1970)

Fantastic Voyage (1964)

Alien (1979) Antz (1998)

Hollywood (Finally) Learns Some Biology

Hollywood’s Allometric Hypothesis:

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.5 1 1.5 2 2.5

Log (M)

Log (Y)

b = 1.0

Surface/Volume Hypothesis

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.5 1 1.5 2 2.5

Log (M)

Log (Y)

b = 2/3

Surface area length2 Volume length3

Surface/Volume Hypothesis

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.5 1 1.5 2 2.5

Log (M)

Log (Y)

b = 2/3

Surface area length2 Volume length3

Microsoft Design Flaw!

New allometric theory of the 1990s

• Theoretical models of universal quarter-power scaling relationships– Predict b = 3/4– Efficient space-filling energy transport

(West et al. 1997)– Fractal dimensions (West et al. 1999)– Metabolic Theory of Ecology (Brown

2004)

Theoretical Predictions

• b = 3/4– Maximize internal exchange efficiency– Space-filling fractal distribution networks (West et al.

1997, 1999)

• b = 2/3– Exterior exchange geometric constraints– Surface area (length2): volume (length3)

Research QuestionsMeta-analysis of published exponents

1. Is the calculated allometric exponent (b) correlated with features of the sample?

2. Mean and confidence interval for published values?

3. Likelihood that b = 3/4 vs. 2/3?

4. Why are estimates often < 3/4?

Allometric exponent

Species in sample

Taxon Source

0.71 391 mammals (Heusner 1991)

0.713 321 mammals (McNab 1988)

0.69 487 mammals (Lovegrove 2000)

0.737 626 mammals (Savage et al. 2004)

0.74 10 mixed (Kleiber 1932)

0.76 228 mammals (West et al. 2002)

0.724 35 passerine birds

(Lasiewski and Dawson 1967)

Research Questions

1. Is the calculated allometric exponent (b) correlated with features of the sample?

2. Calculate mean & confidence interval for published values?

3. Likelihood that b = 3/4 vs. 2/3

4. Why are estimates often < 3/4?

Question 1

• Can variation in published allometric exponents be attributed to variation in– sample size– average body size– range of body sizes measured

Allometric exponent as a function of number of species in sample

Other

P = 0.6491

0 100 300 500 7000.60

0.65

0.70

0.75

0.80

0.85

0.90

Number of species in sample

Mammals

Allo

met

ric E

xpon

ent

Allometric exponent as a function of midpoint of mass

P = 0.5781

Weighted by sample size P = 0.565

0 500 1000 1500 20000.60

0.65

0.70

0.75

0.80

0.85

0.90

Midpoint of mass

Mammals

Other

Allo

met

ric E

xpon

ent

Allometric exponent as a function of log(difference in mass)

P = 0.5792

Weighted by sample size: P = .649

Mammals

Other

0 1 2 3 4 5 60.60

0.65

0.70

0.75

0.80

0.85

0.90

Log(difference in mass)

Allo

met

ric E

xpon

ent

Non-independence in Published Allometric Exponents

• phylogenetic non-independence – species within a study exhibit varying levels of

phylogenetic relatednessBokma 2004, White and Seymour 2003

• data on the same species are sometimes used in multiple studies

Independent Contrast Analysis

• Paired studies analyzing related taxa (Harvey and Pagel 1991)

– e.g., marsupials and other mammals

• Each study was included in only one pair• No correlation (P > 0.05) between difference in the

allometric exponent and– difference in sample size,

– midpoint of mass

– range of mass

Question 1: Conclusions

• Allometric exponent was not correlated with– sample size– midpoint of mass– range of body size

• Reported values not statistical artifacts

Research Questions

1. Is the calculated allometric exponent (b) correlated with features of the sample?

2. Calculate mean & confidence interval for published values?

3. Likelihood that b = 3/4 vs. 2/3

4. Why are estimates often < 3/4?

Question 2: What is the best estimate of the allometric

exponent?

Mammals Birds Reptiles

Allo

met

ric E

xpon

ent

Mammals Birds Reptiles

0.60

0.65

0.70

0.75

0.80

0.85

0.90

b = 3/4

b = 2/3

Allo

met

ric E

xpon

ent

Mammals Birds Reptiles

0.60

0.65

0.70

0.75

0.80

0.85

0.90

b = 2/3

b = 3/4

Allo

met

ric E

xpon

ent

Mammals Birds Reptiles

0.60

0.65

0.70

0.75

0.80

0.85

0.90

b = 2/3

b = 3/4

Question 2: Conclusions

Reptiles

Variation is due to small sample sizes and variability in experimental conditions

Mammals and Birds

Results suggest the true exponent is between 2/3 and 3/4

Research Questions

1. Is the calculated allometric exponent (b) correlated with features of the sample?

2. Calculate mean & confidence interval for published values?

3. Likelihood that b = 3/4 vs. 2/3?

4. Why are estimates often < 3/4?

Question 3: Likelihood Ratio

b = 3/4 : b = 2/3

All species 16 074

Mammals 105

Birds 7.08

Reptiles 2.20

Research Questions

1. Is the calculated allometric exponent (b) correlated with features of the sample?

2. Calculate mean & confidence interval for published values?

3. Likelihood that b = 3/4 vs. 2/3?

4. Why are estimates often < 3/4?

Allo

met

ric E

xpon

ent

Mammals Birds Reptiles

0.60

0.65

0.70

0.75

0.80

0.85

0.90

b = 3/4

b = 2/3

Question 4: estimates often < 3/4?

Linear Regression

• Most published exponents based on linear regression • Assumption: x variable is measured without error • Measurement error in x may bias slope estimates

Measurement Error

• Limits measurement of true species mean mass

• Includes seasonal variation

• Systematic variation

• “Classic” measurement errors

Simulation: Motivatione.g. y = 2xtrue

0 20 40 60 80 100

0

50

100

150

200

True measurement

Slope = 2.0

Slope = 1.8

0 20 40 60 80 100 120

0

50

100

150

200

Error in measurement

Simulation: Assumptions

Assumed modelYi = mi 0.75

Add variation in measurement of mass

Yi = (mi + Xi)b

Simulate error in measurement

Xi = KmiZ

Z ~ N(0,1)

Y = met. Rate

m = mass

X = error term (can be positive or negative)

b = exponent

K = % measurement error

Z = a random number

Circles: mean of 100 trialsTriangles: estimated parametric confidence intervals

Allo

met

ric E

xpon

ent

0.05 0.10 0.15 0.200.70

0.71

0.72

0.73

0.74

0.75

0.76

Proportion Measurement Error

Question 4: Conclusions

• Biases slope estimates down

• Never biases slope estimates up

• Parsimonious explanation for discrepancy between observed and predicted allometric exponents for homeotherms.

Slope Estimates Revisited

• Other methods than least-squares can be used to fit slopes to regression data

• “Model II Regression” does not assume that error is only in the y variable

• Equivalent to fitting principal components

Ordinary Least-Squares Regression

• Most published exponents based on OLS • Assumption: x variable is measured without error • Fitted slope minimizes vertical residual deviations

from line

Reduced Major Axis Regression

• Minimizes perpendicular distance of points to line • Does not assume all error is contained in y variable • “Splits the difference” between x and y errors

Reduced Major Axis Regression

• Slope of Major Axis Regression is always > slope of OLS Regressions

• Major Axis Regression slope = b / r2

increasing b

Re-analysis of Data• Adjusted slope for n = 5 mammal data sets

Conclusions

• Measured allometric exponents not correlated with features of sample

• Published exponents cluster tightly for homeotherms – values slightly lower than the

predicted b = 3/4.

• Published exponents highly variable for poikilotherm studies

Conclusions

• Body mass measurement error always biases least-squares slope estimates downward

• Observed allometric exponents closer to 3/4 than 2/3

Acknowledgements

Gordon Research Conference Committee

Metabolic Basis of Ecology

Bates College

July 4-9, 2004