MARGINALIA

# The Biology of What Is Not There

Is it only natural selection that guides the shapes seen in nature?

# Shells Imagined

The most accessible example of how to examine the shapes of the possible comes from work on shell forms carried out in the 1960s by American paleontologist David Raup and his collaborators *(see the first figure). *Shell shapes are beautiful; they are also geometrically well behaved and can be described relatively easily in mathematical terms. The mathematical model of shell shape, at its most basic, includes the rate at which a shell grows around its central axis, the rate at which it descends down that same axis and the rate at which the shell opening expands as its architect grows. With these three parameters we can depict a three-dimensional shape space: Any point in that space defines a single shell shape that may or may not occur in the living world. Real shell-bearing organisms are varied, abundant both in the current biosphere and in the fossil record, and their shapes can be plotted onto the shape space we have defined. When we do so, the pattern is striking. Certain regions of the cube, shown in color, are densely occupied, but most of the cube is uninhabited.

The challenge of understanding the distribution of biological objects in shape space thus consists of two separate, but interrelated, problems. The first of these problems involves the definition of the space of possibilities. For snail shells, we are fortunate, because the shape is mathematically tractable (a logarithmic spiral is a reasonable first approximation). The shape also involves a limited number of variables (in this case, three, which allows us to depict the space as a three-dimensional cube). For other biological shapes at other scales, defining the shape space may be considerably more challenging. How, for instance, do we define the space of all possible protein or RNA shapes? What mathematical expression will capture all of the possible architectures of tree shape? Not only are these spaces likely to be of a higher dimensionality, making them harder to depict, but the underlying mathematical function that can generate the shapes is likely to be far more complex.

The second problem with understanding shape distributions involves mapping the real objects of the world onto the shape space that we have constructed—and then accounting for the distribution of these realized forms. And it is here that we find an astonishingly consistent result. In every case studied, the distribution of realized shapes is nonisotropic: Some parts of the cube are densely populated with realized shapes, whereas others are virtually, if not entirely, empty. Such patterns in biology demand an explanation. What is this strange occupancy of shape space at all scales telling us about the forces that give form to the living world?

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