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 Biomathematics Patterns: Spiral Slime MATHEMATICAL RECREATIONS by Ian Stewart Finding mathematics in creatures great and small Scientific American, November 2000 This past summer I attended a conference in Portugal on the mathematics of pattern formation, and one of the lectures reminded me of my second favorite animal. My first favorite animal is the tiger, partly because I like its dramatically striped fur. I like my second favorite because of its patterns, too, but this creature isn’t as elegant as the tiger. It is the slime mold, or more specifically, a species of cellular slime mold known as Dictyostelium discoideum. Biologists find the slime mold fascinating because it lies on the borderline between single-celled protozoa and multicellular organisms. The slime mold also illustrates a biological truth that the explorers of the human genome should take to heart: it’s not just your genes that matter but what you do with them. Despite its lowly position on the tree of life, Dictyostelium manages to create astonishingly beautiful spiral patterns. To what extent are these patterns encoded in the slime mold’s genes? Is there, in fact, a gene for spirals? To answer that question, we need to know how the slime mold makes its spirals. The pattern is actually the result of collective activity. The life cycle of Dictyostelium begins with a microscopic spore wafting along on the winds. If the spore happens to land on a nice, moist resting spot, it germinates into a single-celled amoeba and starts hunting for food (mostly bacteria). When the amoeba gets big enough, it reproduces by splitting in two. Pretty soon there are lots of amoebas. The artistry appears when the food runs low. The amoebas clump together, and as they make their way toward the center of the clump they sometimes form an elegant spiral. The crowd of amoebas gradually becomes more dense and the spiral more tightly wound. At some point it breaks up into “streaming patterns” that look like roots or branches extending from the center. The streams thicken, and as more and more amoebas try to get to the same place, they pile up in a heap known as a slug (not to be confused with the mollusk of the same name). The slug is a colony of amoebas, but it moves as if it were a single organism. Once it finds a dry place, it attaches itself firmly to the ground and puts up a long stalk. At the top of the stalk is a round blob called a fruiting body. The amoebas in the fruiting body turn into spores and blow away on the wind, thus continuing the cycle. Thomas Höfer, a biophysicist at Humboldt University, Berlin, has discovered a simple system of mathematical equations that reproduces both the slime mold’s spirals and its streaming patterns. Cornelis J. Weijer of the University of Dundee has shown that very similar equations can model the movement of the slug. The main factors that determine the patterns are the density of the amoeba population, the rate at which the amoebas produce a chemical known as cyclic AMP and the sensitivity of individual amoebas to this chemical. Roughly speaking, each amoeba “shouts” its presence to its neighbors by sending out cyclic AMP. The amoebas then head in the direction from which the shouts are loudest. The spiral pattern is a mathematical consequence of this process. It forms when the amoebas at the center of the clump are rotating as they send out waves of cyclic AMP. It therefore seems that most of the slime mold’s genes simply tell it how to be an amoeba. The genes tell the cells how to send out chemical signals, how to sense them and how to respond to them—but the spiral patterns they produce are not specified in the genes. Instead the patterns emerge from the mathematical rules that the amoebas are obeying. Mathematics may define the life cycle of the slime mold as much as genetics does. The equations that lead to this far-reaching (and controversial) conclusion are modifications of equations devised nearly 50 years ago by English mathematician Alan Turing, who is best known as one of the founders of computer science. Turing was also interested in morphogenesis—the formation and differentiation of biological tissues and organs. In 1952 he postulated that ordered patterns in living creatures don’t need an ordered precursor. He argued that the patterns could arise from chemical substances called morphogens that react with one another as they diffuse through tissue. When Turing first published his ideas, they were purely theoretical, but a striking example of “Turing patterns” soon appeared: the Belousov-Zhabotinsky (BZ) chemical reaction. Russian scientist B. P. Belousov and later his compatriot A. M. Zhabotinsky discovered that mixing just a few ordinary chemicals—including sodium bromate, sulfuric acid and malonic acid—in a petri dish will produce concentric rings and spirals very similar to those made by the slime mold. Similar reactions can produce stripes, spots, dappling and many other patterns that are common in the animal kingdom. Nevertheless, Turing’s ideas were rejected by biologists. A major problem with his thesis was that the patterns that appear spontaneously in the BZ reaction are not fixed—they move across the petri dish. The same is true for all the other Turing patterns observed by chemists. In contrast, the patterns in most living creatures are fixed. We don’t see zebras with moving stripes or leopards with moving spots. Turing had shown theoretically that his equations can produce both stationary and moving patterns, but laboratory experiments seemed to create only moving ones. Later on, the chemists discovered why: if you carry out the reactions in a gel rather than a liquid, the patterns become stationary. Living organisms resemble gels more than they do liquids. But by the time this distinction became clear, biologists had lost interest in the debate. Mathematicians, though, continued to ponder Turing’s ideas. Although his equations were far too simple to model real biological phenomena, they did produce the same kinds of patterns typically seen in animals. If pigments are deposited according to the peaks and troughs of parallel waves, you get stripes. More complex waves produce spots. The challenge for mathematicians was to flesh out Turing’s scheme using theoretical models that more closely simulate the workings of biology. In 1995 two Japanese scientists found the first convincing evidence of Turing patterns in living things. Shigeru Kondo and Rihito Asai of Kyoto University observed growing angelfish over several months and noticed a gradual rearrangement of their stripes. In mammals, the skin patterns simply enlarge as the creature grows, but in maturing angelfish new stripes are constantly forming as the older stripes split in two. What is more, the changes can be predicted by mathematical equations very similar to Turing’s. A computer simulation of molecular interactions in a one-dimensional array of cells yielded a wave pattern that closely matched the rearrangement of angelfish stripes. The movement of the stripes is rather slow, which is why we don’t generally notice it. But, as Galileo said, “It moves all the same.” Mathematics changes the appearance of the angelfish just as it influences the life cycle of the slime mold—and perhaps our own as well.