Hello and welcome to Unit 7

This is the first of two units in which

the ideas of fractals and scaling are

applied to particular types of complex systems.

In this unit we’ll look at scaling in metabolic systems.

And in the next unit we’ll look at

scaling in urban systems in cities.

I’ll begin this unit by presenting Kleiber’s law

which is an empirical relationship between

an organism body mass and its metabolic rate.

I’ll then present the West Brown Enquist theory

that posits a mechanism or explanation for

why this relationship is true.

And then look at some other scaling relationships

that follow from this main metabolic scaling result.

This unit also includes interviews with a couple of

researchers doing work in metabolic scaling.

So, let’s get started by looking at Kleiber’s law

So the starting point in this unit

will be to think about Kleiber’s law.

Kleiber’s law is a statement about

how metabolic rate is related to body mass.

So let me say a little bit about this

and then we’ll look at some data.

So metabolic rate refers to the rate

at which an organism consumes resources.

Food or equivalently the rate at which

it generates heat. Because that’s where

the food that it consumes.

that energy ultimately goes into heat.

So metabolic rate could just be how many calories

the creature needs to consume in a day

in order to live, in order to carry out

its basic life function whatever those are.

And then the question is

how is that related to body mass.

Well, we would expect that

bigger creatures need eat more.

Because there is more of them to keep alive.

But the question is what is this relationship.

And that is what Kleiber started to look at,

in, I think maybe around 1930, 1940,

it's sometimes in the 30’s

And subsequently and he discovered

certain regular relationship.

And subsequent to that more data was gathered.

And this relationship proved to be rather robust.

So here’s a version of what this looks like.

This is from a more recent paper.

I’ll put the reference down here.

I think it’s West and Brown

So let me draw these labels more clearly

what we have here on a logarithmic scale is mass

And on the vertical axes is metabolic rate.

Ok, so we have mass on a log scale.

And metabolic rate on a log scale.

And it’s little hard to see,

but there are lots of data points a bunch here,

and many many many in here.

So let’s focus of this portion of the graph first.

These dots, I know they just look like a cloud.

These are lots and lots of mammals.

So take a mammal.

Weight a mammal to get its mass.

Watch how much it eats

or do something maybe little more sophisticated that

And you can get the metabolic rate.

And so there is a pretty strong linear relationship.

This data point here is an elephant.

And this here is a shrew.

One of he smallest mammals.

Here these are cells.

So this is an average mammalian cell.

And mitochondria.

These are resting cells.

and let’s ignore those two points.

But the main result is that,

there is a linear relationship here

and a linear relationship here.

and we’ve learned that a linear relationship

on a log-log plot is an indication of

some sort of power-law or scale free behavior.

So we expect that there’s some scaling relationship

more careful statistics has been done on this.

And it has been shown that

yes, this is indeed very well described by power law.

These two power laws have the same slope

but they have different intercepts,

but the slope is the same.

And it’s very very close to 3 quarters.

So this poses a number of questions.

That’s often the case I think

with power law research one starts doing

this gathering data and doing some empirical work

and noticing hey ! there is a pattern

There is pattern here that seems to fit a power law

And then from that a number of questions flow.

So I want to, before we dig into the theory

Say little bit about, other shapes up

this graph could have but didn’t.

So first of all,

Note that for given mass

there is a relatively small range of metabolic rates.

The world may be didn’t have to be that way

or certainly any generically

a relationship like this doesn’t have

to follow along the line at all.

It could be the case

that you have creatures that are the same size

but with very different metabolic rates.

You still expect some upward trend I would guess.

Because big creatures need more foods

to keep their mass alive.

But may be it could have then

that this data look like this.

In other words, a big cloud around this.

So, it doesn’t really seem to fill a line at all

any curve at all, right

so that this would mean that

this mass you could have a creature like this

and may be this would be some really

some creature that sleeps all the time

and really doesn’t do much

so it doesn’t need a lot of food

and maybe this is a creature that lives

in the snow and runs around all the time

so it needs enormous amount of food

but it happen to be same amount, same size.

So there is no reason,

at least, in certain maybe upper alright

to expect that these were followed

along a curve at all.

I think it could just be a big,

a big spread of this.

So additionally the fact that they do fall along

where appears to be a curve a line and it’s noteworthy

but then it turns out that the fact that

these slopes are the same,

it’s also interesting so it could be that

we looked at some data

and instead of this being a line

it had some concavity like this

or something like that who knows.

Right so it doesn’t mean you have a bunch of data

It doesn’t automatically have to follow along at all,

and of course it doesn’t automatically

if there is a line or curve

that curve doesn’t have to be straight.

So this is sort of a remarkable and noteworthy thing

that makes one pay attention to

hey! what’s going on here

We’re seeing a straight line on a log-log plot

We didn’t have to see that you don’t always see that

But we did it in this case.

And so that means some sort of power law relationship.

And then we’ll also learn that power laws are scale free.

So that’s a little bit surprising.

Because may be that says in some sense

shrews are scaled up or scaled down versions of elephants

They didn’t look the same

But when you think about the metabolic rate,

maybe there’s some similarity across here

some similar processes that are working across many scales.

We see a similar scaling here for cells of many many different sizes

So that suggest to us there is something interesting

going on and something that is scale free.

some maybe sort of broader principle

that’s connecting all of these, all these data points.

So Kleiber’s law was pretty well under

not well understood, but well established as

a relationship between metabolic rate and mass.

What was lacking was a clear mechanistic explanation for

why this might be so

and the West Brown Enquist scaling theory

which I’ll get to in a few videos

provides an explanation for why would expect to see this

linear behavior and more interestingly

why we end up with this slope of 3 quarters,

and not something else.

So that’s an overview of Kleiber’s law

that sets the stage what’s gonna follow.

In the next couple videos, I am gonna talk

about scaling a lit bit more generally.

And we’ll think about surface area to volume ratio

and see that, that doesn’t explain this relationship

and then we’ll look at the West Brown Enquist theory

and see how thinking about this is a fractal network

can provide a good potential explanation for why we see

an exponent of 3 quarters.