Complexity_ A Guided Tour - Melanie Mitchell [127]
Brown, Enquist, and West realized that the circulatory system is not just characterized in terms of its mass or length, but rather in terms of its network structure. As West pointed out, “You really have to think in terms of two separate scales—the length of the superficial you and the real you, which is made up of networks.”
In developing their theory, Brown, Enquist, and West assumed that evolution has produced circulatory and other fuel-transport networks that are maximally “space filling” in the body—that is, that can transport fuel to cells in every part of the body. They also assumed that evolution has designed these networks to minimize the energy and time that is required to distribute this fuel to cells. Finally, they assume that the “terminal units” of the network, the sites where fuel is provided to body tissue, do not scale with body mass, but rather are approximately the same size in small and large organisms. This property has been observed, for example, with capillaries in the circulatory system, which are the same size in most animals. Big animals just have more of them. One reason for this is that cells themselves do not scale with body size: individual mouse and hippo cells are roughly the same size. The hippo just has more cells so needs more capillaries to fuel them.
The maximally space-filling geometric objects are indeed fractal branching structures—the self-similarity at all scales means that space is equally filled at all scales. What Brown, Enquist, and West were doing in the glass-walled conference room all those many weeks and months was developing an intricate mathematical model of the circulatory system as a space-filling fractal. They adopted the energy-and-time-minimization and constant-terminal-unit-size assumptions given above, and asked, What happens in the model when body mass is scaled up? Lo and behold, their calculations showed that in the model, the rate at which fuel is delivered to cells, which determines metabolic rate, scales with body mass to the 3/4 power.
The mathematical details of the model that lead to the 3/4 exponent are rather complicated. However, it is worth commenting on the group’s interpretation of the 3/4 exponent. Recall my discussion above of Rubner’s surface hypothesis—that metabolic rate must scale with body mass the same way in which volume scales with surface area, namely, to the 2/3 power. One way to look at the 3/4 exponent is that it would be the result of the surface hypothesis applied to four-dimensional creatures! We can see this via a simple dimensional analogy. A two-dimensional object such as a circle has a circumference and an area. In three dimensions, these correspond to surface area and volume, respectively. In four dimensions, surface area and volume correspond, respectively, to “surface” volume and what we might call hypervolume—a quantity that is hard to imagine since our brains are wired to think in three, not four dimensions. Using arguments that are analogous to the discussion of how surface area scales with volume to the 2/3 power, one can show that in four dimensions surface volume scales with hypervolume to the 3/4 power.
In short, what Brown, Enquist, and West are saying is that evolution structured our circulatory systems as fractal networks to approximate a “fourth dimension” so as to make our metabolisms more efficient. As West, Brown, and Enquist put it, “Although living things occupy a three-dimensional space, their internal physiology and anatomy operate as if they were four-dimensional … Fractal geometry has literally given life an added dimension.”
Scope of the Theory
In its original form, metabolic scaling theory was applied to explain metabolic scaling in many animal species, such as those plotted in figure 17.2. However, Brown, Enquist, West, and their increasing cadre of new collaborators did not stop there. Every few weeks, it seems, a new class of organisms or phenomena is added to the list covered by the theory. The group has claimed