Lecture 20: an introduction to scaling in biology

1. Lecture 20: an introduction to scaling in biology Do anatomy or physiology change as organisms change their overall size? If so, why? If so, are there general principles or trends? NOTE: MathBench module is very helpful-- Please use it!

2. different relative dimensions are common Ex: length/width ratios not constant

3. physiology can also have weird scaling Fig. 41.10

4. Q: why not just scale everything up with constant proportions? A: this often introduces problems. recall our spherical sailor’s eyeball: r =5 vol=166.7π units3 r =1 vol=1.33π units3

5. if adequate air can only diffuse into the first 1 unit of depth: anoxia!! happy

6. a shape change in larger forms could fix this: still happy! happy h =2

7. another example: strength • cross-sectional area: how much material exists to bear the load at a given spot • a good predictor of strength (ability to resist compression, tension, shearing, etc.) 1 cm cube of water: bears 1g/cm2 on bottom 3 cm cube of water: bears 3g/cm2 on bottom

8. isometry: objects in a size series have the same geometry (e.g. spheres, cubes) allometry: objects have distinct geometries as they change size (e.g. sphere vs. ovoid). Because isometric forms of different sizes can have very different properties (see previous two examples), we might expect allometry to be common. How do we detect allometry?

9. when b = 1, y is a linear function of x: y = 0.2x a general way to describe scaling y = axb x typically a measure of overall size, like mass a a constant b the scaling power y dependent measure, e.g. a body part size, metabolic rate, etc. • non-linear scaling: • 0 < b < 1: y increases at a slower rate than x(plateaus) • b >1: y increases faster rate than x(accelerates)

10. cross-sectional area and volume of cubes of varying sizes are related by b=2/3: log X-sec. area log V = 0.67(log A) log Volume to discover b easily, we can log-transform the scaling equation: logy = log a + b(log x) This is in the form y=mx+b (linear equation) y-intercept of a, slope of b.

11. in the previous plot, b = 2/3 • 2 in numerator: exponent from y-axis variable –area • 3 in denominator: exponent from x-axis variable – volume For traits that are about area (e.g. gut or wing area), isometry predicts not b=1, but b=2/3 for scaling with mass. Similarly, traits that are about length (e.g. femur width or wing length), isometry predicts a b = 1/3 scaling with mass. Finally, traits measured in the same order of dimension should relate with b = 1 (e.g. body length vs. width). However, biology often deviates from these expectations: • Wing span ~ (mass)1/3 – b is actually 0.37 • Wing area ~ (mass)2/3 – b is actually 0.62

12. human allometric growth • head mass is 43% of body mass @ wk 8 of gestation • only 6% at adulthood

13. body mass vs. head mass-- isometry predicts b = 1 • How far off is it? • two points define a line: x1,y1 (fetus), x2,y2 (adult) • slope of line (b) is “rise over run”(∆y/∆x) • (3.52-1.93)/(4.74-2.30)=1.59/2.44 • So, b = 0.65