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.