growth process and the flowing liquids that move nutrient-laden water past the organism.

How generally can these principles be extended beyond these two fairly limited examples? This question is going to occupy most of the rest of the book, so I will have to leave it largely unaddressed for now. I offer one example now, however—a teaser, if you will, that suggests strongly that external physiology is a pervasive phenomenon. Perhaps it will give you pause for thought.

The example concerns corals, which, you will recall, are fractal objects. Corals are commonly denizens of coastlines, which can be thought of as an interface between the nutrient pools of the open ocean and the organisms that live along its margins. For organisms to obtain these nutrients, the nutrients must cross this boundary. In this sense, a coastline is like the exchange barrier between, say, blood and air in the lung.

A lot of conventional physiology is concerned with exchange across boundaries, and physiologists often find themselves faced with the question: what are the structural attributes of a "good" exchange surface? If you refer back to Fick's law (equation 5.4), you can see in that equation one important "design principle" for good exchangers: make the ratio of the surface area, A, and the thickness of the diffusion barrier, x, as large as possible. Fick's law deals with nice, differentiable exchange barriers, though. If a coastline is a fractal boundary, what can "fractal thinking" tell us about exchange of nutrients across it?

Consider a box separated into two compartments by a membrane, like the simple example used to illustrate Fick's law (Fig 5.10). In that example, I implicitly assumed that flux across the boundary was limited by diffusion. This is not the case for diffusion-limited accretion: flux is limited by a relatively slow movement of a molecule from the space above the membrane to the membrane itself. The "design principles" implied by Fick's law are of limited help here.

Here is where fractal thinking becomes useful. If movement to the boundary is limiting, the boundary's capabilities as an exchanger would be improved by putting a "kink" into it so that every molecule in both compartments is brought a little closer to the boundary (Fig 5.16). Just a single kink, though, still leaves some molecules farther from the boundary than others, so some inefficiency of design remains. This problem is easily got around, though, simply by doing it again: put an additional kink into the kinks you put in previously. In other words, you make the boundary grow as if it were a Koch curve (Fig 5.9).

You can probably see where this is leading.

Figure 5.16 Increasing the fractal dimension of a boundary to diffusion increases the flux rate across the boundary by decreasing the distance between the boundary and all points on either side of it.

Efficiency of the exchange between the compartments can be maximized by ensuring that every molecule in both compartments is brought as close as possible to the membrane separating them. This is done by repeatedly convoluting the membrane so it becomes a fractal boundary, forcing the dimension, D, of the cross-section to a value greater than one.6 Now we have a different design principle: a good exchanger

6. Of course, real boundaries in the lungs are areas (D = 2), not curves (D = 1). Remember that we are dealing here with cross-sections, and that the cross-section of a surface is a curve. Convoluting an actual membrane in the way described in the text would increase its dimension from 2 (a surface) toward 3 (a volume). The cross-section would increase in dimension from 1 to 2.

is one whose fractal dimension is maximized. Still, however, dimension is limited by the fact that a curve traced by the boundary can never become an area— that is, D must be less than two. Maximum efficiency is achieved by making the fractal dimension of the boundary as close to two as possible. Indeed, in cross-sections of the highly efficient biological gas exchange membrane of the mammalian lung, the fractal dimension of the lungs' folded surfaces is between 1.9 and 2.0.

If a coastline is a boundary separating the nutrient pool of the ocean and the intertidal organisms living on its margin, how efficiently would exchange across this boundary operate? Ordinary coastlines, with their fractal dimensions of 1.2 to 1.4 or so, would seem to be not very efficient. When the coastline is fringed with a coral reef, however, the boundary becomes much rougher, and the coastline's fractal dimension increases, approaching 1.8 to 1.9 or so. This suggests that coral turns the coastlines it occupies into more efficient exchangers of nutrients and minerals than they would otherwise be. This seems to me to be external physiology on a grand scale.

What Happened with Animals?

I wanted to use this chapter to bolster the link between transient phenomena like bioconvection cells and more substantial animal-built structures, like coral reefs. Perhaps I have accomplished this, perhaps not. Nevertheless, I am now ready to launch into the "real biology" part of the book, discussions of animal-built structures that I believe are acting as external organs of physiology. Before doing so, however, I want to explore just one more question relating to the origin of animals and to what role external physiology might play in it.

I think most biologists would agree that radical differences in embryogenesis and development divide the metazoan animals from the problematic sponges and corals. Sponges' and corals' body forms are ruled by epigenetic factors. Development among animals seems much more strongly controlled by genetics, the epigenetic influences relegated to a relatively minor role. Why did this radical difference arise?

Much of the early diversification of animal body plans resulted from an increasing degree of what we might call "physiological in-sourcing." If we examine the simplest of organisms, we find that many of the major innovations in body plan exist to accommodate ever more complex internal organs of physiology. So, for example, sponges and coelenterates have no organs or organ systems whatsoever. The platyhelminthine (flat) worms possess simple organ systems for digestion and nervous function, but little else. Still higher animals are equipped with numerous and complex organ systems for water balance, digestion, circulation, gas exchange, and so forth. The development of each of these organ systems necessitated new body plans because the relatively simple bodies of coelenterates had to be folded in special ways to fit them in. Although anatomical details of folding patterns differ in the various body plans we know about, all the higher animals have fit these organs in, taking on board an increasing range of physiological function. What drove them to do this?

In all likelihood, the value of this increasingly internalized physiology was increased reliability and flexibility. External physiology has numerous advantages, among them that physiological work can be done at scales many times larger than the animals themselves. What makes this possible, of course, is the linkage between external physiology and positive feedback. If an animal positions itself in a large-scale gradient in physical potential energy, it can use positive feedback to tap enormous reservoirs of energy to do its physiological work (Fig. 5.17).

Along with the advantages of positive feedback go certain disadvantages, however. One of the biggest might have been that physiology powered in this way would have been either unreliable or inflexible or both. If, for example, you are powering external physiology with wind, and the wind dies from time to time, so does your physiology, and so, perhaps, do you. Also, if the positive feedback loop is modulated somehow, any external physiology driven by it may be limited in the range of things it can do. Suppose, for example, protozoans could derive some hypothetical advantage from inhabiting smaller or more rapidly circulating bioconvection cells. If the size and circulation rate of the cells is limited by the water's viscosity and density, any advantages to smallness would be moot— physically, they couldn't exist.

Somehow or other, animals seem to have evolved toward a strong reliance on chemical potential energy to power their physiology. Why this happened is uncertain, but several possible reasons come to mind. Perhaps glucose is a more predictable fuel. Its availability might vary, but it is easy to store in relatively stable physical energy stream physical energy stream

Im growth I—► reproduction

Im growth I—► reproduction

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