here i've plotted these two power laws
y equals x to the two thirds and y
equals x to the three-fourths
note that this is a standard plot not a
log-log plot so you can see that these
are non-linear
relationships what can be gleaned from
this
is that the metabolic rates of organisms
have somehow evolved to be
more efficient than we would expect
in the sense that a higher metabolic
rate represents a higher rate of
distribution of nutrients to cells
so more efficient in that sense thus if
this scaling relationship with an
exponent of three
fourths 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 a 4 in the denominator of the
exponent
so heart rate scales like body mass to
the minus one fourth
that is the lower your body mass the
higher your heart rate
blood circulation time scales with body
mass to the one-fourth
span scales with body mass to the
one-fourth
etc what's going on with this one-fourth
or three-fourths exponent
people have talked about these so-called
quarter power scaling laws
in the 1990s a group of scientists at
the santa fe institute
formed a collaboration in order to try
and understand
these scaling laws and what caused them
jeffrey west is a theoretical physicist
and he started working together with jim
brown who is a
ecologist and brian enquist who was at
that time a graduate student working
with jim brown
this interdisciplinary 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
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
so the idea here is now that surface
area shouldn't be seen as a limitation
the limitation should be seen as the
structure of the distribution
system so how are energy and materials
distributed
well just to show some pictures here's 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's a 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
that 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 network 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 three-dimensional
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 these branching
structures
which are the capillaries don't vary
with size among 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 the fractal
branching structure
that euclidean geometry is the wrong way
to view
scaling in this case euclidean geometry
was what gave rise to this
two-thirds 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 is that metabolic
rate
is proportional to body mass to the
three-fourths
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 sort of unsatisfying
for some of you
people who don't have a mathematical
background may have found the
mathematics here a little bit
challenging
and people who do have a strong
mathematical background
might be frustrated because i didn't
really talk about how the model works
so i 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 go and choose the paper at the
level 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 the most
interesting
in reading about this model which was
the interpretation of the model
described by the west brown and enquist
team now recall 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 two-thirds power
what wes brown and enquist say is that
metabolic rate indeed scales with body
mass
like surface area scales with volume but
not in three dimensions
rather the geometric scaling is in four
dimensions
okay so let's talk about what that means
west brown nyquist
say although living things occupy a
three-dimensional space
their internal physiology and anatomy
operate as if they were
four-dimensional fractal geometry has
literally given life
an added dimension okay so let me show a
picture to illustrate what this means
so earlier on we idealized organisms
as spheres and
the surface hypothesis argued that since
surface area is proportional to volume
to the two-thirds power
metabolic rate is proportional to body
mass to the two-thirds power
okay that's if we assume that you are
three-dimensional
but what west brown and enquists are
saying
is that because we have these fractal
branching distribution networks
internally we aren't three-dimensional
but rather we have some kind of
fractal dimension that's between three
and four dimensions
and it's approaching four dimensions
because of the space filling aspect of
these fractals
and that metabolic rate scales with
volume
or mass to the three-fourths
completely in analogy with this
three-dimensional idea
but this would be as if we were being
approximated by a four-dimensional
sphere
so that's very intriguing the idea that
we are actually
or behave as though we are four
dimensional 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
three-fourths
is in fact not the correct exponent
if there even is a single exponent
others have questioned the mathematical
correctness of the west brown and
enquest model
and there's 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's 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 inquist so if you're
really interested in this
and you have a technical background
there's much in the literature for you
to explore that's been done on this in
recent years
jeffrey 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 with 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 sub-unit
after we hear a guest spot with jeffrey
west talking about metabolic scaling