Free Arrival of the Fittest: Solving Evolution's Greatest Puzzle by Andreas Wagner Page B

immediate neighbors. But there is a problem: To buy or build shelving for this library would be a real pain.
    In a human library, every book has two immediate neighbors, one to the left and one to the right, or maximally four, if you want to count the volumes on the shelf above and below as well. How many neighbors would any one text in the metabolic library have? Recall that a string of five-thousand-odd ones and zeroes describes a metabolic genotype. Any neighbor would differ in exactly one of these letters, one chemical reaction that may be either present or absent. (It cannot possibly differ in less than that, and if it differed in more, it would no longer be a neighbor.) There is one neighbor that differs in the first letter of this string, another that differs in the second letter, one that differs in the third letter, and so on, until the very last of these letters. In other words, each metabolic text has not two, not four, but thousands of neighbors, as many as there are biochemical reactions, each of these neighbors differing in a single letter and reaction. Shelves that can hold this sort of inventory aren’t easy to find.
    To see how peculiar they would have to be, imagine a much simpler world than ours, the simplest possible chemical world with only one chemical reaction. In this world the metabolic library has only two texts. One of them consists of the letter 1, containing the one and only reaction, the other of the letter 0—it lacks this reaction. Figure 8a shows these texts as the endpoints of a straight line.
    A slightly larger universe with two reactions would be big enough for 2 × 2 = 4 possible metabolic texts. One of them has both of these reactions (11), two of them have one reaction but not the other (10, 01), and the fourth metabolism has no reaction (00). Figure 8b shows these metabolisms as the corners of a square.
    You may already see where this is going. The next larger reaction universe would have three reactions and 2 × 2 × 2 = 8 possible metabolisms that form the corners of a cube (figure 8c). For a universe with four reactions, we have 2 × 2 × 2 × 2 = 16 possible metabolisms. But which geometric object would correspond to it? As our reaction universe increased from one to two to three reactions, its metabolic texts occupied the endpoints of a line, a square, or a cube, which exist in a one-, two-, and three-dimensional space. Taking it one step further, we need an object in a four-dimensional space. Spaces with four or more dimensions are hard to visualize, but mathematicians routinely work with them, because we can extend our geometrical laws to them. 37 Just as in a square and a cube, the edges of the object we are looking for have to be equally long, and adjacent edges would have right angles to one another. Such an object is a four-dimensional
hypercube
. Figure 8d uses a geometric trick to show this hypercube on paper. It has sixteen corners, each one corresponding to one metabolic text—from 0000 to 1111—that is no longer shown in the figure.
    FIGURE 8. Hypercubes
    This trick no longer works in five dimensions, much less higher ones. But although it is hopeless to imagine higher-dimensional spaces, they follow the same laws as our three-dimensional space: The edges of a hypercube are equally long, adjacent edges are at right angles to one another, and each corner corresponds to a possible metabolism. And such cubes in high-dimensional space turn out to have curious properties well suited to house the metabolic library.
    The number of corners in a square is four, in a cube it doubles to eight, and in a four-dimensional hypercube it doubles again to sixteen. With every added dimension, it doubles, and by the time you have reached 5,000 dimensions, this number has become the hyperastronomical 2 5000 , the size of the metabolic library. In other words, we can arrange the library’s metabolic texts on the corners of a hypercube in a 5,000-dimensional space. This is why