In this subunit we're going to talk about
the issue of metabolic scaling in biology
and, in particular, one theory that has
made a lot of waves recently in the
complex systems community and throughout
biology. Let's first start with some
definitions. The metabolic rate of an
organism is the amount of energy that
organism expends per unit time. You
probably know that all of your cells are
at all times under going metabolic
reactions in which food is turned into
energy and metabolic rate of an organism
can be measured by the amount of heat that
it emits per unit time. It has been known
for a long time that metabolic rate scales
with body mass, but there has been some
controversy about what exactly the scaling
law is. The theories of metabolic scaling
are a bit complicated and involve a little
bit of math. I will go through it very
slowly, step-by-step, and hopefully, it
won't be too complicated. First, there
are some assumptions that were made. We
know that the body is made of cells.
Metabolic reactions are constantly taking
place. As a simplifying assumption
scientists sometimes approximate body mass
by assuming that the body is a sphere of
cells with radius r. Here is our little
hampster who we're going to assume is a
sphere of cells and here is the radius r.
We're going to use some geometric
arguments. Now let's look at a variety of
sizes of organisms, starting out with our
little mouse. Say it has radius r.
Geometrically we know that the surface
area of this sphere is proportional to r^2
and the volume is proportional to r^3
That is similar to the arguments I was
making earlier about the bedroom scaling
Now suppose we have a hampster. Let's
assume that the radius of the hampster is
twice the radius of the mouse. That would
mean that the surface area of the hampster
this sphere would be proportional to the
square of the radius, that is, (2r)^2,
that is, 4r^2. Ok, where r is the mouse's
radius. The volume would be the radius
cubed or 8 times r^3. Now, let's look at
a hippo, much bigger, let's assume that
it's radius is 50 times the radius of the
mouse which means that it's surface area
would be 50^2 times r^2 which would be
2500 times r^2, and its volume would be
50^3 times r^3 which is 125000 times r^3
so it's getting pretty big. We're also
going to assume that the mouth of the
organism is proportional to the volume of
the sphere. That's a reasonable
assumption. Our simplest hypothesis might
be that metabolic rate, which is the
amount of energy or heat given off by the
organism scales with body mass directly,
that is, directly proportional to body
mass, where body mass is proportional to
volume. Ok. Well, there is a problem.
The problem is that while the mass is
proportional to the volume heat can only
radiate from the surface. What does that
mean? That means that the amount of heat
is proportional to volume, a huge number,
but that heat can only radiate over a much
smaller number, the surface area. In
our hippo we have 125000 times the heat of
the mouse, that is, proportional to the
volume, radiating over an area that is
only 2500 times the surface area of the
mouse, so a huge amount of heat radiating
over a relatively smaller area can produce
only a very hot hippo, a hippo that is
burning up. Fortunately for hippos and
the rest of us, evolution did not make
metabolic rate scale directly with body
mass, so that hypothesis is wrong. We can
argue geometrically that surface area is
proportional to volume raised to the 2/3
power. That is because the surface area
we know is proportional to r^2 and we can
actually write r^2 as (r^3)^2/3, the 3's
cancel out and we are left with r^2, but
r^3 is proportional to the volume so the
surface area is proportional to the volume
raised to the 2/3 power. What you might
expect is a second hypothesis being true
is that metabolic rate scales with body
mass to the 2/3 power. That is, it
doesn't scale with body mass directly, it
scales with a smaller number, that is,
the surface area. That's called the
surface hypothesis and it was believed for
many years. It seems reasonable to assume
that we would like metabolism to produce
as much energy as possible which means
that it would radiate heat proportional to
surface area of the organism. Ok, so
here's some data. This is a log-log plot.
Body mass is plotted here, and metabolic
rate is plotted here. You can see these
different organisms fall pretty well on a
straight line, and if you measure the
slope of this line, it is not 2/3, but
rather it is 3/4. Unexpectedly, while
the geometric argument would argue for the
exponent being 2/3, the actual data shows
that it is 3/4. This is called Kleiber's
law after the person who discovered it.
For sixty years nobody really understood
why metabolic rate scaled with body mass
to the 3/4. At this point, let's stop and
have a brief quiz to make sure you
understand what we've done so far.