Miscellaneous: The ¾ Scaling Law

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www.mathbench.umd.edu The ¾ scaling law May 2010 page

The ¾ Scaling Law

URL: http://mathbench.umd.edu/modules/misc_scaling/page01.htm

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The Power Function

Scaling studies examine how form and function change as organisms get larger - in other words, how do biological features scale across size? Do they change in meaningful ways as organisms get bigger or smaller? Of course, you can't even ask these types of questions without having a way of measuring how relationships change mathematically. So, our main goal in this section is to introduce you to the idea of scaling studies - and also the mathematical function used to measure them: the power function! What are scaling relationships?

How do organisms change as body size increases or decreases? This is the fundamental question behind studies that measure scaling. Most scaling studies have focused on how physical structures (such as body shape) or physiological factors (such as metabolism or heart rate) changes with size. There are lots of types of factors that could be examined in relation to size. Here are a few: Why study these relationships? Well, if you understand how form or functions change as organisms get bigger or smaller, it is possible to learn something fundamental about what underlies the processes or learn about what factors place evolutionary limits on organismal growth and adaptations. For instance, determining at what size arthropods can no longer support the weight of their exoskeleton gives us clues about the limits of their growth.

Scaling studies have a long history in biology, but the physiological relationship that has received the most attention is how metabolic rate changes as organisms get larger. Therefore, we are going to focus on that example as we go through some of the mathematical concepts that underlie the science of scaling. So, what do we really mean by scaling? Let's use a concrete example so you'll know what we mean.

Here is some data on body size and metabolic rate for mammals. Later, we're going to show you an expanded data set of over 600 mammals so we can examine this relationship more rigorously, but for right now, we'll start with just four: OK, a couple of things that we want you to notice. The first is that metabolic rate increases as animals get bigger. That's because we are specifically interested in total energy consumed (here measured through oxygen consumption). Of course, bigger animals will use more oxygen than smaller ones (think about how big a breath a lion takes compared to a mouse). But look at the values adjusted for body size (the last value listed for each species). Mice use a lot more oxygen per gram than a lion. This means that lions use oxygen more efficiently than mice. There is something else to notice. As mammals get bigger, this increase in efficiency is not linear (notice how the steepness of the slope decreases as size increases). This means that metabolism does not scale linearly with body size.

Right now, you may be thinking to yourself: "Who cares?" Well, it turns out that how metabolism (and other factors) scales with body size can give important information about which factors are most important in limiting these biological functions. If we can understand that, we understand a lot more about biology! All right, now that you're convinced of the importance of these studies - let's look at these relationships in more depth by seeing how people model them mathematically!

The Power Function

What equation could we use to describe these scaling relationships mathematically? Well, a linear relationship won't work for these data (remember from the last page that the relationship wasn't linear). So we can't use the equation Y = mX + b (the equation for a straight line). That's too bad, because that equation is SOOOO easy to work with.

No, we need an equation that can show us all kinds of scaling relationships. Ones where the process rates could increase with size, scale linearly, or allow a decrease in rate (as in the last example). This means that we want a function that is flexible enough to capture all these potentially biologically interesting behaviors. Well, it turns out that although there are many functions that can do this, the most common function that is used in scaling studies is the power function:

Y = aXb

There are several good reasons to use this function, and we're going to spend a little bit of time examining its properties. By the time we're done, you'll see why its so commonly used, and you will come to love it as much as we do!

This function is a general equation and could apply to any situation. X and Y are both variables, meaning they take on a range of values. As is usually the case, Y is the dependent variable and X is the independent variable, meaning that the value of Y is dependent on the value of X. We used Y and X to make life simple, because when we are ready to graph this function (and we will ALWAYS want to graph the function), the dependent variable (Y) goes on (where else?) the Y-axis. That leaves the independent variable (X) to go on the X-axis. Finally, something in mathematics that makes sense!

X and Y could represent anything, although for most scaling studies, X is generally related to size and we are specifically interested in how metabolic rate scales with size, so lets rewrite the general equation Y = aXb in a more specific way to fit our main example:

Metabolic Rate = a * (size)b

Now we want to think a little bit about how this function behaves. Luckily, we already know something about size and metabolic rate, right? They are both always going to be positive, non-zero values (you've surely never heard of anything with a negative body size - and if your metabolic rate is 0, unfortunately that means you're dead!). That information should help us think about these functions.

Next, we'll examine how "a" and "b" influence the way the two quantities are related to each other.

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