3/4 and Kleiber's Law - Numberphile


Uploaded by numberphile on 31.01.2012

Transcript:

THOMAS WOOLLEY: OK, so today we're going to be talking
about the number 3/4, or 0.75.
Unusual, I know, because it's not a whole number,
like 0, 1, 2, 3, 4.
But this one is very important in biology.

So Max Kleiber in the 1930s posited a graph of mass of
animal against metabolic rate.
Down here we have the mouse, he'll have his little legs
there and little ears.
MALE SPEAKER: Is that a mouse?
THOMAS WOOLLEY: That's a mouse.
MALE SPEAKER: That's the worst mouse I've ever seen.
THOMAS WOOLLEY: Wait until you see my elephant.
So somewhere up here, 1,000 times heavier,
you'll have an elephant.
So let's try this one.
MALE SPEAKER: That's not bad.
THOMAS WOOLLEY: That's not bad.
Well, you know, it's kind of like a big
mouse, to be honest.
And then all the way up here, you have your blue whale.
OK, so he plotted the mass against the metabolic rate of
all the animals.
So mouse, elephant, blue whale, there was
also mayflies in there.
And as you might expect, with the bigger animals, they need
more energy to survive.
And that's what metabolic rate essentially measures.
How much energy do you need to survive?
So small animals need a small amount of energy, bigger
animals need more energy.
So one way it could happen is that since the elephant is
1,000 times heavier than the mouse, it would need 1,000
times much more energy.
And if you had that sort of scaling, it would be linear.
And so it would have just a straight line between the
metabolic rate and the mass.
So the whale, that's a million heavier than the mouse, so it
would need a million times more energy.
Or it could be super linear.
So the elephant is 1,000 times heavier than the mouse, so it
needs, say, a million times more.
It actually needs more than the line originally, the
linear line.
What Max Kleiber found was that it's neither of those.
The curve actually tapers off, and so it's below
the straight line.
What that means is that the bigger you get, although you
need more energy, you can spend it more efficiently.
So down here, with your mayflies and your spiders and
all your insects, they need very little energy, but they
don't use it very efficiently.
And this scale crosses all sizes of animal, right down to
cells, all the way up to the blue whale.
It's amazing how this curve describes all the animals.
And how does this curve link in the 3/4?
Well, what Max found was that this curve, is that the
metabolic rate is proportional to the mass to power 3/4.
Now, statistically, it was around 3/4, so it's something
like 0.74 or 0.76.
But on average, that's what biologists now take.
So this is Kleiber's rule, or the 3/4 rule.
Well, when they found 3/4, one thing we want to understand is
where does that number come from?
Is it just a coincidence that it's 3/4, or is this some
reasoning behind it that we can get at?
So originally when they were looking for this 3/4, they
were trying to link metabolic rate and the mass.
And so your metabolic rate is how much you use energy.
And you use energy, they thought, mostly through
heating yourself.
And so what affects the heat?
Well, your surface area.
The bigger your surface area, the more you lose heat.
And the bigger you are in terms of your mass, again, the
more heat you will have.
So they looked at spherical animals.
So just for the sake of mathematics, we're going to
treat these animals as spheres, because your main
body is sort of spherical.
And you can at least get an approximation of what you'd
expect to get at the end.
We draw our sphere, and we give it a radius, r.
So that's the body of the animal, OK?
So even if you're a mayfly, you'd be a tiny sphere.
If you're a blue whale, you'd be an enormous sphere.
So the area, the surface area of this sphere, that's
proportional to r squared.
As you increase r, if you double r, your surface area
increases by 4.
And the volume of the sphere is proportional to r cubed.
So if you double the radius, your volume
increases by eight.
So using this idea that the metabolic rate is proportional
to these two guys, you'd expect it to be metabolic rate
is proportional to r 2/3.
So you'd get a 2/3 power out.
So although 2/3 is quite close to 3/4, we wanted to
understand where that difference is coming from,
because they're close.
So it's a good first approximation, but there has
to be some reason that we're not getting 3/4.
And so what mathematicians have looked at is using the
fractal structure inside us to deliver that energy.
So the circulation system that we have, the veins, the
arteries, and the capillaries, is
self-similar all the way down.
As you zoom into the veins, they look
like the whole system.
And that's what a fractal is.
And using that fractal structure, you can link the
mass to the elephant to the blue whale.
And that does spits out the 3/4 power law
at the end of it.
So as I said, yeah, the bigger the animal, the more efficient
you're using your energy.
But using this idea of the circulation system being a
fractal, you can then start using it to look at cities,
because what are cities?
Well, they've got their own circulation system, because
they have roads that deliver cars to places.
They have water pipes, they have electricity cables.
And you can plot those certain things against population, and
you get very similar graphs that follow this power rule.
It may not be 3/4, but you'll get that it's proportional to
the population, and it's less than linear.
And they usually are following a quarter structure.
So whatever this number may be on top, you'll usually find a
4 on the bottom.