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

Review of Concepts

 

• Scaling relationships in biology describe how form or function change as organisms get larger. By using a power function (Y = aX^b), we can determine if the relationship is isometric (scales linearly) or allometric (one factor changes at a different rate than the other). Our entire module focused on the relationship between metabolic rate and body size - one of the most well-studied scaling relationships there is!
• For this relationship, the only biologically plausible range for the power parameter "b" is greater than zero and less than or equal to 1. That is because it doesn't make sense that metabolic rate would show large increases for very small increases of body size (as would be the case if b>1), or that metabolic rate would approach zero as organisms get larger (as would be the case if b<0).
• Most scaling studies present and analyze log-transformed data. This is because the data are easier to visualize (the logarithmic scale draws out the values at the smaller end of the scale). Also, log-transforming a power function (Y = aX^b) results in a linear function: log(Y) = b*log(X) + log(a). And linear functions are much easier to work with statistically!
• For more than a century, people hypothesized that the relationship between metabolic rate and body size would scale with a 2/3 power rule. That's because they believed that metabolism was limited by surface area to volume ratios that related how heat is lost through the skin (measured in surface area) relative to the weight of the body (measured in volume). For all shapes, surface area to volume scales at a 2/3 power rule.
• When people actually measured the relationship, they found the surprising result that metabolism actually scales to body weight with a 3/4 power rule. This led to a new (and very complicated) theory that relates the geometry of the internal transport systems (i.e., circulatory, respiratory, tracheal, xylem) to the limits on metabolic rate. This new theory is controversial, and now people are even questioning the actual value of the scaling relationship (is "b" equal to 2/3 or 3/4). The controversy remains to be worked out. How exciting!

 

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If you want a printer-friendly version of this module, you can find it here in a Microsoft Word document. This printer-friendly version should be used only to review, as it does not contain any of the interactive material, and only a skeletal version of problems solved in the module.