3.3. Allometric scaling laws3.3.1 Phenomenology

More general: how are shape and size connected?

Fox (5 kg)Elephant (6000 kg)

Simple scaling argument (Gallilei)

Load is proportional to weight

Weight is proportional to Volume ~ L3

Load is limited by yield stress and leg area; I.e. L3 ~ d2 Y

This implies d ~ L3/2

Or d/L ~ L1/2 ~ M1/6

Similar for the size of the stem in trees – the bigger the tree the bigger its stem

This is also connected to the development of the individual

Look at the heart rate of different animals

...or the lifespan as a function of weight

i.e. There’s only a constant number of heart beats

However,

This can also be looked at in the same individual

But also for populations of different people to basically determine the ideal weight in terms of size...

Plot this on a double logarithmic scale and it becomes simpler – and you can see where the BMI comes from…

Metabolic rate is conveniently measured by oxygen consumption.

Plot the metabolic rate for many different animals

Works over many decades...

Kleiber’s Law: 43MB(M) /

3.3.2. Fractals and scalingAn example of a scaling argument – Flight speeds vs.mass

Cruise speeds at sea level Mass (grams)

Cruising speed (m/s)

Boeing 747

crane fly

goose

house wren

Beech Baron

100

101

102

103

damsel fly

starling

-5 -3 -1 1 3 5 710 10 10 10 10 10 10 10

9

F-16

hummingbird

fruit flybee

sailplane

eagle

dragonfly

Cruise speeds at sea level Mass (grams)

Cruising speed (m/s)

Boeing 747

crane fly

goose

house wren

Beech Baron

100

101

102

103

damsel fly

starling

-5 -3 -1 1 3 5 710 10 10 10 10 10 10 10

9

F-16

hummingbird

fruit flybee

sailplane

eagle

dragonfly

W

26

3 2 2ˆL

S S

W L C AVcV M

A C W C M

W Weight A Area

V Speed L Lift

A=Area

L

ˆ, functions of geometry (shape, angle of attack)L SC C

Consider a simple explanation

Mass (grams)

Cruising speed (m/s)

Boeing 747

crane fly

goose

house wren

Beech Baron

100

101

102

103

damsel fly

starling

-5 -3 -1 1 3 5 710 10 10 10 10 10 10 10

9

F-16

hummingbird

fruit flybee

sailplane

eagle

dragonfly

Fits pretty well!

6 1/ 6CV M V cM

Mass (grams)

Cruising speed (m/s)

Boeing 747

crane fly

goose

house wren

Beech Baron

100

101

102

103

damsel fly

starling

-5 -3 -1 1 3 5 710 10 10 10 10 10 10 10

9

F-16

hummingbird

fruit flybee

sailplane

eagle

dragonfly

Short wings,maneuverable

Long wings,soaring and gliding

What do variations from nominal imply?

A famous example: The energy of a nuclear explosion

US government wanted to keep energy yield of nuclear blasts a secret.

Pictures of nuclear blast were released in Life magazine

Using Dimensional Analysis, G.I. Taylor determined energy of blast and government was upset because they thought there had been a leak of information

• Radius, R, of blast depends on time since explosion, t, energy of explosion, E, and density of medium, , that explosion expands into

• [R]=m, [t]=s,[E]=kg*m2/s2, =kg/m3

• R=tpEq k

12q 3k

0 p 2q

0 q k

q=1/5, k=-1/5, p=2/5

R (E /)1/ 5 t 2 / 5 E R5t 2

We’re looking for a similar argument to explain the scaling of metabolic rate

Metabolism works by nutrients, which are transported through pipes in a network. This forms a fractal structure, so what are fractals?

A fractal looks the same on different magnifications...

This is not particularly special, so does a cube...

What’s special about fractals is that the “dimension” is not necessarily a whole number

Consider the Koch curve

Or the Sierpinski carpet

How long is the coast of Britain?

Vessels in nutrient transport (veins, xyla, trachaea) actually have a brached fractal structure, so consider this for explaining metabolic rate

3.3.3. Physical model

• Branching, hierarchical network that is space filling to feed all cells

• Capillaries are invariant of animal size

• Minimization of energy to send vital resources to the terminal units (pump blood from the heart to the capillaries)

three basic assumptions

Modelling the network of tubes

All vessels of the same level can be considered identical. Define scale-free ratios

k1l

kl 1/ 3n

k1r

krand

k1l

kl

The network is space-filling to reach the whole body

Allows us to relate one level to the next.

n is branching ratio

rk is radius of vessel at kth level lk is length of vessel at kth level

I T

T

R

Minimize Energy Loss through Natural Selection

Dissipation (Important for small vessels, Poiseuille flow)

n 1/ 2Area Preserving

Reflection at junctions (Important for larger vessels, pulsatile flow)

Metabolic Rate, B, and Body Mass, M

M bV knlevels

k2r kl TN 4 / 3

TV B4 / 3

Follows fromEnergy Min.

Use scale factorsto relate each levelto terminal units.

Invariance ofterminal unitsB=NTBT

B 3 / 4M

Blood volume

Number ofTerminal units

Volume of Terminal units

Mass

Met Rate

Other predictions from the model

l0 T lT nT

3 lT NT1/ 3lT (M 3 / 4 )1/ 3 M 3 / 8

r0 M 3 / 8

ZTOT 1

NT

M 3 / 4

p Ý Q 0ZTOT M 3 / 4 M 3 / 4 M 0

u 0 Ý Q 0

r02

M 3 / 4

M 3 / 4 M 0

Speed of flow through capillaries is invariant

Invariant blood pressure

Remember tree stems from beginning

One slightly couterintuitive conclusion: Each capillary feeds more cells in larger organisms

capillary

tissue

But in fact this is the case, cells in vivo have less consumption the bigger the animal (but constant

in vitro…)

Transport happens through fractal networks even in subcellular instances – the law can be extended…

This implies that the ¾ rule is true over 25-30 decades!!

Can this help in understanding how an organism grows?

growthResting metabolism

Energy input

Kleiber's rule tells us:

With the stationary Solution:

Solving this gives a sigmiod curve:

Compare to experimental data for M and

Scaling collapse:

This does not cover the growth spurt during puberty

But remember that proportions do change as well during life

In fact the weight curve shows less of a puberty anomaly

The model also implies how much energy is used by an individual (tree)

This implies that population densities can be predicted via the usage of land for each tree as a function of tree size

r0 M 3 / 8

This works pretty well

Recap Sec 3.3Small animals live faster than large ones – and have very different structures

This follows a very general law, which is summarised by the dependence of metabolic rate on mass

The fractal structure of the nutrient transport is instrumental in explaining the exponent of this law

Given the dependence of metabolic rate on size, population densities and also growth of organisms can be treated

This gives predictions that have been tested experimentally