
Size and Scaling


Updated 1 April
2010
A. Consequences of
having different body sizes
1.
“You can drop a mouse down a thirtyyard 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 (lO_{2}/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 100fold
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  10^{8 }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 = Length^{2}
b. volume =
Length^{3}
c.
SA = V^{2/3 }or SA = V^{0.67}
d. conventional:
plot on loglog scale to produce straightline (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) 
O_{2} 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)
1.
Measure by oxygen consumption; endotherms
2.
Body size explains 90% variation in MR (r^{2})
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 loglog 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 (semilog)
a.
conventional loglog:
Specific MR = 0.68M^{0.75}/M
b. rearrange: Specific MR = 0.68M^{0.25};
HFig. 5.10 (loglog)
6.
The metabolismweight 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, see Ecology’s Big Hot
Idea