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.