Here I have plotted these two power laws,
y=x^2/3 and y=x^3/4, note that this is a
standard plot and not a log-log plot, you
can see that these are non-linear
relationships. What can be gleaned from
this is that the metabolic rate of
organisms have somehow evolved to be more
efficient than we would expect, in the
sense that higher metabolic rate
represents a higher distribution of
nutrients to cells, so more efficient in
that sense. Thus, if this scaling
relationship with exponent 3/4 is indeed
correct, then evolution has somehow
allowed organisms to overcome the
limitation implied by the ratio of surface
area to volume. Interestingly, scientists
have observed other biological scaling
laws that have 4 in the denominator of
the exponent. Heart rate scales to body
mass to -1/4, that is, the lower your body
mass, the higher your heart rate.
Blood circulation time scales to body mass
to the 1/4, life span scales with body
mass to the 1/4, etc. What's going on
with this 1/4 or 3/4 exponent. People
have talked about these so-called 1/4
powers, scaling laws. In the 1990's a
group of scientists at the Santa Fe
Institute formed a collaboration to try
and understand these scaling laws and what
caused them. Geoffrey West is a
theoretical physicist, Jim Brown who is an
ecologist, and Brian Enquist a graduate
student working with Jim Brown. This
inter-disciplinary group approached this
subject in a new way, asking the question,
what is the structure of the distribution
networks inside organisms and what effect
does that have on metabolic rate?
Their general idea is summed up like this:
metabolic scaling rates (and other
biological rates) are limited not by
surface area but by rates at which energy
and materials can be distributed between
surfaces where they are exchanged and the
tissues where they are used. The idea
here is that surface area shouldn't be
seen as a limitation. The limitation
should be seen as the structure of the
distribution system. How are energy and
materials distributed? Just to show some
pictures, here is a picture of the
circulatory system, in humans. A picture
of the lungs, with these so-called bronchi
which have a similar tree-like structure
to the circulatory system. Here is an
electron-micrograph that shows a highly
magnified view of the vascular system, and
you can really see this kind of fractal
tree structure that makes up these
distribution networks.
West, Brown and Enquist developed a theory
which they called metabolic scaling theory
in order to explain the scaling
relationships that were seen in the data.
Their theory involved some assumptions
about distribution networks whether they
be airways in the lungs or the vascular
system bringing blood to cells. The idea
is that these distribution networks have
a fractal tree-like structure with
branches that reach all parts of the 3D
organism. They have to be as space
filling as possible in order to optimally
deliver nutrients to all parts of the body
to all cells. Also they assume that the
terminal units in the branches of these
structures, which are the capillaries,
don't vary with size with organisms and
that seems to be the case, and they assume
that these networks have evolved to
minimize the total energy required to
distribute resources. They conclude that
because the distribution network has a
fractal branching structure that
Euclidean geometry is the wrong way to
view scaling in this case. Euclidean
geometry was what gave rise to this 2/3
exponent in the surface hypothesis, but
West, Brown and Enquist asserted that one
should use fractal geometry instead.
Their theory involved a considerable
amount of physics and mathematics, and I
won't go into it here, but the result was
with their detailed mathematical model
using the three assumptions I mentioned
they were able to derive Kleiber's law
that metabolic rate is proportional to
body mass to the 3/4, where the
explanation for this lies in the fractal
geometry of the distribution networks.
I realize that this discussion about
metabolic scaling has been somewhat
complicated, and in fact metabolic
scaling is a very complicated topic.
It may have been unsatisfying for some of
you, people who don't have a mathematical
background may have found the mathematics
here a little challenging and people who
do have a strong mathematical background
might be frustrated because I didn't
really talk about how the model works.
I have put up some papers on the course
materials page, some of which give a
description of this model in completely
non-mathematical terms and some of which
give technical explanations of how the
model works. You can choose the level of
paper that you're most interested in, if
you want to follow-up on this. To finish
with, I want to talk about one of the
things I found most interesting in reading
about this model which was the
interpretation of the model described by
the West, Brown and Enquist team. I will
call that the surface hypothesis which
turned out not to match the data was based
on the idea that surface area scales with
volume to the 2/3 power. What West, Brown
and Enquist say is that metabolic rate
indeed scales with body mass like surface
area scales with volume, but not in 3D,
but rather, geometric scaling is in 4D.
Let's talk about what that means.
West, Brown and Enquist say, "Although
living things occupy 3D space, their
internal physiology and anatomy operate as
if they were four-dimensional...Fractal
geometry has literally given life an added
dimension." Ok, so let me show a picture
to illustrate what this means. Earlier on
we idealized organisms as spheres, and
the surface hypothesis argued that since
surface area is proportional to volume to
the 2/3 power, metabolic rate is
proportional to body mass to the 2/3 power
Ok, that's if we assume that we are 3D,
but what West, Brown, and Enquist are
saying is that because we have these
fractal branching distribution networks,
internally we aren't 3D, but rather we
have some kind of fractal dimension,
that's between 3 and 4 dimensions, and
it's approaching 4 dimensions because of
the space filling aspect of these fractals
and that metabolic rate scales with volume
or mass to the 3/4, completely in analogy
with this 3D idea, that this would be as
if we are being approximated by a 4D
sphere, so that's very intriguing, the
idea that we are actually, we behave as
though we are 4D creatures due to this
fractal structure of our distribution
networks. Well, as you can imagine this
idea, and in fact West, Brown, and
Enquist's entire theory has been very
controversial and has gotten a lot of
criticism in the biological literature.
Some people have argued that 3/4 is not
the correct exponent, if there even is a
single exponent. Others have questioned
the mathematical correctness of the West,
Brown, and Enquist model, and there have
been many other criticisms as well, and a
lot of back and forth between supporters
and critics of this model. For our
purposes I think the bottom line is that
this model is interesting, it is very
elegant, but both the explanation and the
underlying data are controversial, and I
should also note that there have been many
updated versions of the model developed by
various groups since the original set of
papers by West, Brown, and Enquist.
If you are really interested in this and
you have a technical background, there is
much in the literature for you to explore
that's been done on this in recent years.
Geoffrey West has gone in a different
direction. He's now taken up the subject
of urban scaling. He's working with Luis
Bettencourt on how attributes of cities
such as crime and so on scale the city
population size, and they ask the question
- can this kind of scaling behavior also
be explained via fractal distribution
networks? This is the topic of our next
subunit. After we hear a guest spot with
Geoffrey West talking about metabolic
scaling.