what if animals were fractals?
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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 S cience… for that true S cience lay in its relation to mathematics.” - PowerPoint PPT PresentationTRANSCRIPT
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What if animals were fractals?University of Utah
ACCESS 2009
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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
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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”
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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.
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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)
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some examples
Allometric scaling exponents for various biological variables as a function of mass:
Scaling Exponent
Metabolic rateHeart beat rateLife SpanRadius of aortas/ tree trunksGenome length for unicellular organismBrain mass
¾ -¼¼3/8¼¾
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scaling of heart rate and life span
animal heart rate life span (wild) # of heart beats
mouseelephantgorilla
500 beats/min28 beats/min70 beats/min
2 years60 years30 years
Calculate the number of heart beats among the following animals…
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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 number of cells• Each cell is consuming energy at a certain rate so
metabolic rate 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) surface area (SA) mass (M) volume (V)
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metabolic rate scaling law Compare a spherical mouse of radius r with
a spherical cat who is 3 times larger.=
=
r
3r
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A 10 lb. goose needs 300 calories per day to survive. What about a 160 lb person?
metabolic rate scaling law
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www.bodyworlds.com
derivation of the ¾ exponentWest, 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).
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derivation of the ¾ exponentSuppose 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 = L3
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derivation of the ¾ exponentLet’s fill a ball with a branch. Find .
(Hint: Vr3)
The volume served by the entire network is the sum of volumes served by each of the branches… L3 = l3
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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
derivation of the ¾ exponent
3
3
lLwR
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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 Mblood M
• The amount of fluid within the structure must be conservedAmount 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?
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respiratory system
circulatory system
umm…
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From the assumption that the cross sectional area is independent of any sectional cut,
where
A = cross sectional area of the network
density of fluid
proportion of blood/fluid to body
Also assume A = N, where is the cross sectional area of the terminal node.
LAVM
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The Final StretchLet’s put everything together now to get the ¾ scaling exponent…
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Conclusions?We have found that b = ¾, which
matches our data…