Size and Scaling

Updated 16 August 2005

Note to students: If you rely on printed copies of these notes, make sure that you have the latest copy (check date stamp) and remember to study the linked figures and graphs online.

If you have not yet taken Biol. 254 (Biostatistics), please read Appendices E (Fitting Lines to Data) and G (Allometric Equations) to help you understand this lecture.

A. Consequences of having different body sizes

1. “You can drop a mouse down a thirty-yard mine shaft, and on arriving at the bottom, it gets a slight shock and walks away - a rat is killed, a man broken, and a horse splashes.” Haldane (1928)

2. SIZE MATTERS!

3. What is the biological significance to physiologists?

a. almost every physiological variable is a function of body size, often more so than adaptation or phylogeny

b. fundamental question: how does body size affect the physiological process being considered?

B. Importance of scaling – e.g., Tusko’s bad trip

1. Calculate dose of LSD for elephant based on known cat dosage (by mass): 0.26 mg LSD/2.6 kg cat = 0.1 mg LSD/kg; 2970 kg elephant x 0.1 mg/kg = 297 mg LSD

2. Results: “The elephant immediately started trumpeting and running around, then he stopped and swayed, five minutes after the injection he collapsed, went into convulsions, defecated, and died.”

3. Conclusion: Elephants are particularly sensitive (cf cat) to LSD – what do you think?

4. Calculate dose for elephant based on known human dosage (by mass): 0.20 mg LSD/70 kg human = 0.003 mg LSD/kg; 2970 kg elephant x 0.003 mg/kg = 8.9 mg LSD – perhaps the dose was too big?

5. Alternatively, calculate dose based on metabolic rate (Table):

Animal	Mass (kg)	MR/mass (lO2/kg/h)	Dose (mg LSD)
Cat	2.6	0.53	0.26
Human	70	0.23	0.20
Elephant	2970	0.09	2.72

a. 297 mg vs. 2.72 mg LSD; a 100-fold OVERDOSE!

b. relationship of mass and metabolic rate is not what it seems!

c. SCALING MATTERS!

C. Animal size ranges ~18 orders of magnitude (10^-10 g bacteria - 10⁸ g blue whale)

1. Structure and function change predictably with size (among and within phylogenetic lines)

2. Change in size may involve changes in:

a. dimensions – e.g., bone thickness

b. materials – e.g., soft vs. hard tissue (skeletal) support of body mass

c. design – e.g., respiration - simple diffusion vs. mass convection (ventilation and circulation)

3. Scaling: quantitative structural and functional consequences of changes in size (dimension) among similarly structured organisms (within phylogenetic lines)

D. Uses of scaling

1. Allows comparison within and among species at any taxonomic level (phylogeny: control for size variation within species)

2. Evaluate existing hypotheses; e.g., differences once attributed to species adaptations often simple consequences of scaling effects (e.g., SN2Fig. 3.2; dinosaur brain size)

3. Use as null hypotheses to evaluate deviations

a. e.g., diving mammals have larger blood volumes than most mammals; adaptation?

b. e.g., HFig. 7.9

E. Types of scaling: Isometric scaling (geometrically similar objects)

1. Simple relationships (Y/X ratio constant; linear relationships have same relative proportions); WFig. 3.1; KFig. 4.13

Example: surface area and volume

a. surface Area = Length²

b. volume = Length³

c. SA = V^2/3 or SA = V^0.67

d. conventional: plot on log-log scale to produce straight-line (WFig. 3.2)

- log SA = log a + 0.67log V (logarithmic formula)

- SA = aV^0.67 (conventional power/exponential formula)

- intercept (a) varies with different shapes

- slope (b) is a constant (=isometric scaling exponent); applies to cells, tissues, organs, and organisms (regardless of shape!)

e. SA/V ratio (SA/V =aV^-0.33) is a major consideration in physiological comparisons (WFig. 3.2); e.g., exchange rates (oxygen, heat, etc.)

F. Types of scaling: Allometric scaling (geometrically dissimilar objects)

1. More complex relationships (Y/X ratio varies with body size); WFig. 3.3; KFig. 4.9

2. Most common independent variable used in scaling is body mass

a. many physiological parameters vary allometrically with body mass

b. general formula: Y = aM^b

c. intercept (a) = proportionality constant (intercept at unity body mass, i.e., M=1)

- useful in comparing the level of a physiological parameter between groups (e.g., metabolic rate of marsupials 30% lower than placentals; must have statistically same slope to compare)

c. slope (b) = mass exponent (how variable of interest changes with body size)

- example: SN2Fig. 2.6;  

Slope = 1 (simple proportion)	Slope <1.0
heart wgt = 0.006M^1.0 (heart wgt = 0.6% body mass)	O2 consump = 0.676M^0.75
Lung vol = 0.063M^1.02 (lung vol = 6.3% body mass)	lung ventilation = 20.0M^0.75
blood vol = 0.055M^0.99 (blood vol = 5.5% body mass)	heart rate = 241M^–0.25

G. Example: Scaling of metabolic rate (one of the most commonly studied scaling questions; Hill et al., pp. 137-144)

1. Measure by oxygen consumption; endotherms

2. Body size explains 90% variation in MR (r²)

3. Must factor out body size to understand effects of environmental/biological factors that affect MR (10%)

a. physically hold body size constant (experimental design); not always practical or possible

b. statistically control for effect of body size

4. Statistical approach (determine log-log regression equation); WFig. 3.5

a. general form: log Y = log a + b(log X)

b. add actual variables/values: log MR = -0.167 + 0.75(log M)

c. rearrange: log MR = -0.167 + log M^0.75

d. take inverse logs: MR = 0.68M^0.75

5. MR/unit body mass (permits direct comparisons among taxa); WFig. 6.23 (semi-log)

a. conventional log-log: Specific MR = 0.68M^0.75/M

b. rearrange: Specific MR = 0.68M^-0.25; HFig. 5.10 (log-log)

6. The metabolism-weight relationship pervades almost every aspect of an animal’s physiology.

H. Scaling at higher (and lower) levels - Some biologists think that scaling relationships reflect fundamental properties affecting life processes across a broad range of scales, not just individual physiology, but from molecular up to landscape levels. For current arguments on these ideas, READ the articles, “Life on the Scales“ and “Ecology’s Big Hot Idea“

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