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.