# Metabolic rate and Kleiber’s law

*power law*. In this step we look at metabolism and Kleiber’s law.

## The metabolism of an animal

Metabolism refers to the chemical processes that keep an animal alive. This involves breaking down incoming nutrients, and building up required compounds. It is a very complicated story. Since most chemical processes require oxygen, scientists can use oxygen consumption as a way of measuring metabolic rate. This is usually done when the animal is at rest.We know that the mass of an animal scales cubically with its linear size, while the surface area scales quadratically. Since metabolism relates to the rate of input and output of materials (i.e. oxygen) of an organism, one might expect that metabolism scales proportionally to surface area, or to cross-sectional area.This would suggest that the metabolic rate of an animal ought to be roughly proportional to the \(\normalsize{\frac{2}{3}}\) power of its mass. Some early versions of Kleiber’s law stated the relation in this form.## Kleiber’s law

But the modern form of Kleiber’s law states that an animal’s metabolic rate scales to the \(\normalsize{\frac{3}{4}}\) power of its mass, that is \[\Large{\text{metabolic rate} \approx \operatorname{constant} \times \left(\operatorname{mass}\right)^\frac{3}{4}}.\]So a cat, having a mass \(\normalsize{100}\) times that of a mouse, will metabolise roughly \(\normalsize{100^\frac{3}{4} = 31.6227766\cdots \approx 32}\) times more energy per day than a mouse.**Q1.**(E): Does this imply that a cat ought to consume 32 mice per day?There is also a version of the law valid for plants, but there the exponent is close to \(\normalsize{1}\); that is metabolic rate and mass are almost directly proportional.Here is a diagram from Kleiber’s work. Note that to determine the power law nature of the relation the logarithm of metabolic rate is plotted against the logarithm of the mass. This is a standard technique when working with power laws. By using log scales on the axes, we can use our understanding of lines to understand more complicated power laws.So the power law above can be rewritten as: \[\Large{\ln\left(\text{metabolic rate}\right) \approx \frac{3}{4}\ln\left(\operatorname{mass}\right)} + \ln\left(\operatorname{constant}\right).\]

^{Kleiber 1947 By Max Kleiber, Public domain, via Wikimedia Commons}

Q2(E): Can you see where the slope of \(\frac{3}{4}\) appears in the above graph?

## Why does the exponent \(\normalsize{\frac{3}{4}}\) appear?

## Discussion

## Answers

A1.That would not be good for mice, nor for cats. Probably a couple of mice a day would be adequate.A2.In terms of the squares, if you start on the red line and go over four squares and up three, you arrive back on the line.

#### Maths for Humans: Linear, Quadratic & Inverse Relations

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