Application of Power Laws to Biology
Allometry is the study of the “shapes” of organisms, the relative sizes of their organs, their relative metabolic rates, their lifespans, among other measurable quantities. When some of these two quantities are plotted against one another with log – log axes, a linear relationship emerges. One of these relationships is the relationship between body mass and metabolic rate:
It is useful to ask why this pattern emerges from measuring and comparing such simple quantities, and what use there is to be obtained. When this relationship is found, a “steep” or “shallow” slope indicates a disproportionate change in one variable with respect to another. This yields insight into the physical constraints (or lack thereof) that govern the fitness or development of an organism. One example of such a comparison (in article linked below) is the size of the human heart and head compared to overall body mass.
The heart grows proportionally with the rest of the body while the head grows early in life, reaches a certain size, and stops growing. This suggests that head growth is subject to very different growth conditions compared to the heart. One hypothesis for this difference is that if the relationship between two variables displays a log-log relationship with high statistical confidence, the two variables are linked by a stringent external constraint. In the case of heart size, this constraint is the fulfillment of its function economically. But what is also significant is that the absence of highly statistically significant log-log relationship points to the lack of a stringent constraint, or the influence of an external variable. An example of this is the relationship between brain size relative to total body mass. The relationship vaguely follows a power law, but it is not as pronounced as the relationship between body mass and metabolic rate shown above:
This graph shows the strong influence of an external variable (probably body temperature) and shows many outliers. So although not everything follows a power law, the marginal instances can still yield insight into the constraints that govern them.
http://www.nature.com/scitable/knowledge/library/allometry-the-study-of-biological-scaling-13228439