Post on 18-Dec-2015
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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