What if animals were fractals?
University of Utah
ACCESS 2009
universal laws in biology?
In 1917, D’Arcy Thompson
began his book
On Growth and Form
with the quote:
“chemistry… was a science
but not Science… for that
true Science lay in its
relation to mathematics.”
He then goes on to say:
• math + chemsitry = Science
• biology + fluffy = science
universal laws in biology?
Do biological phenomena obey underlying
universal laws of life that can be mathematized
so that biology can be formulated as a
predictive, quantitative science?
 “Newton’s laws of biology”
allometric scaling laws
• Allometry is the study of changes in characteristics of organisms as body
sizes grow.
•
Can we quantify how body mass/size affect other physiological aspects
such as metabolic rate, life span, heart rate, or population density?
• A typical allometric scaling law is usually written in the form of
Y = Y0Mb
where Y is the biological variable of interest, M is the mass.
Both Y0 and b are numbers to be determined from experimental data, and
the scaling exponent is of particular interest as it characterizes how Y
specifically changes as the mass is varied.
size matters
Metabolic rate: rate of energy consumption if the animals are
at rest in a neutrally temperate environment with digestive
system inactive (Wikipedia definition)
some examples
Allometric scaling exponents for various biological
variables as a function of mass:
Scaling Exponent
Metabolic rate
Heart beat rate
Life Span
Radius of aortas/ tree trunks
Genome length for unicellular organism
Brain mass
¾
-¼
¼
3/8
¼
¾
scaling of heart rate and life span
Calculate the number of heart beats among the following animals…
animal
heart rate
life span (wild)
mouse
elephant
gorilla
500 beats/min
28 beats/min
70 beats/min
2 years
60 years
30 years
# of heart beats
metabolic rate scaling law
How should metabolic rates depend on mass? It may be the case that…
• All animals are made up of cells, so
mass a number of cells
• Each cell is consuming energy at a certain rate so
metabolic rate a mass
FACT: Aerobic metabolism is fueled by oxygen, whose concentration in
hemoglobin is fixed.
Here is a thought: maybe there is a relationship between surface area used to
dissipate heat/waste and the metabolism of the animal…
metabolic rate (R) a surface area (SA)
mass (M) a volume (V)
metabolic rate scaling law
Compare a spherical mouse of radius r with
a spherical cat who is 3 times larger.
=
r
=
3r
metabolic rate scaling law
A 10 lb. goose needs 300 calories per day to survive. What about a 160 lb
person?
derivation of the ¾ exponent
West, Brown and Enquist proposed a
derivation of the ¾ scaling exponent
based on the idea of space filling
fractals filling up the body (Nature
276(4),1997).
www.bodyworlds.com
derivation of the ¾ exponent
Suppose the body is supplied by a network of tree-like structures. Let L be the
length scale of the network. The volume V served by the entire network is
proportional to L3.
V = mL3
derivation of the ¾ exponent
Let’s fill a ball with a branch. Find m.
(Hint: V = 4/3pr3)
The volume served by the entire
network is the sum of volumes
served by each of the branches…
mL3 = Sml3
derivation of the ¾ exponent
Unlike real fractals, the tree-like structure of the network will end somewhere.
For the circulatory system, it ends at the capillary levels and for trees, at the
leaf structures.
 terminal nodes
L3 = Nl3
The metabolic rate R should be proportional to N. Why?
R = wN
where w is the energy consumption of cells supplied by a
terminal node. Then
L3
R=w 3
l
what were we doing again?
Remember we are trying to find R = R0Mb.
• The mass should be proportional to the volume of the network. In
particular, thing about the fluid flowing within the structure.
V a Mblood a M
• The amount of fluid within the structure must be conserved
Amount flowing in = amount flowing out
vin Ain = vout Aout
where v is the average speed and A is the cross sectional area.
• Assume that the flow is steady.*
Then vin= vout . What does the mean in terms of the cross sectional area?
umm…
circulatory system
respiratory system
From the assumption that the cross sectional
area is independent of any sectional cut,


M = V = LA


where
A = cross sectional area of the network
= density of fluid
= proportion of blood/fluid to body
Also assume A = aN, where a is the cross
sectional area of the terminal node.
The Final Stretch
Let’s put everything together now
to get the ¾ scaling exponent…
Conclusions?
We have found that b = ¾, which
matches our data…