Table of Contents

• Prequiz...
• The Power Function
• 1: Introduction
• 2: What are scaling relationships?
• 3: The power function
• 4: The shape of the power function
• 5: The conversion factor "a"
• 6: The power factor "b"
• Log Transformations
• 7: Introduction
• 8: What are logarithms again?
• 9: Graphing on a log scale
• 10: Visualizing the data
• 11: Getting to a straight line
• 12: The key parameter is "b"
• The value of "b"
• 13: Introduction
• 14: A geometric approach
• 15: Scaling surface area of cubes
• 16: Scaling surface area of cubes
• 17: A "2/3 scaling law"?
• Confronting ideas with data
• 18: Introduction
• 19: An unexpected result
• 20: A "3/4 scaling law"?
• 21: Internal transport systems
• 22: Branching and supply networks
• 23: Deriving the "3/4 scaling law"
• 24: So what's the real value of "b"?
• Review
• 25: Review of Concepts

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.

Now, 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 = aX^b

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 = aX^b 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.