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A monthly survey of math news
Computing an Organism. The e-mail journal Science-Week for May 25 2001 picked up an item from the March 27 PNAS (98:3879): Stan Marée and Paulien Hogeweg, of the University of Utrecht, published an account of their simulation of the culmination behavior of the social amoeba (``slime mould'') Dictyostelium dicsoideum. ``Computing an organism'' is the title of the accompanying commentary by Lee A. Segal. As the Science-Week editors note, ``The D. discoideum morphogenesis cycle is one of the great puzzles of biology.'' Briefly, the ``normal'' stage of this organism is an amoeba, an independent unicellular organism. It eats bacteria and reproduces by binary fission. But when a population of these creatures is starving, they aggregate to form a slug 2 to 4 mm long which moves (``migration'') as a single organism towards light. There (``culmination'') the slug puts up a stalk approximately 1 cm high bearing at its tip a fruiting body containing spores, which eventually disperse over a wide area, each becoming a new ``normal'' amoeba. Marée and Hogeweg were able to construct a mathematical model of part of this amazing behavior, and to use it to run computer simulations of the process. Their model is a ``a two-dimensional simulation using a hybrid stochastic cellular automata/partial differential equation schema'' in which ``individual cells are modeled as a group of connected automata: the basic scale of the model is subcellular.'' Here are two sample printouts. On the left is a simulation of the phototactic migration, from an earlier work with their colleague Alexander Panfilov (Proc. R. Soc. Lond. B (1999) 266, 1351-1360); on the right is a simulation of the culmination behavior.
The Science-Week editors conclude: ``...viewing the simulation produced by the mathematical model of Maree and Hogeweg will no doubt startle many biologists. Perhaps the most important consideration is that this work provides evidence that computer modeling involving recognized subcellular dynamic entities may soon be used to predict (and explain) specific tissue development and tissue morphology. The implications for both basic and medical biology are profound.''
Proof on NPR. On April 20, 2001 the National Public Radio's Online News Hour aired Terence Smith interviewing David Auburn, author of the Pulitzer-prizewinning play Proof, ``a family drama that unfolds against the backdrop of mathematical theory.'' Smith asked: ``Did you feel any special burden to explain or make accessible the world of the mathematician to the audience?'' Auburn's answer: ``The real trick of writing the play was figuring out how much math to put in it. This ended up being constrained by the story. Since there is a mystery as to who wrote the mathematical proof, I sort of had to withdraw information when I could so that I didn't give away the solution to the mystery but I did try to get in as much lore about the mathematical profession as I could. In that I was helped a lot by reading popular books and spending time with mathematicians. We even had some come in to meet with the cast and talk to them. So that was really the fun part of doing the play: getting as much of the world of mathematics into the play as possible and putting it up on stage.'' The segment (which can be read and heard online) includes an excerpt from the New York performance (HAL was KATHERINE's father's graduate student):
| HAL: Some friends of mine are in this band - they're playing at a bar up on Diversity, way down the bill, they probably go on around 2, 2:30, I said I would be there.|
HAL: They're all in the math department. They're really great. They have this good song, you'd like it, it's called "I." Lower case I. They just stand there and don't play anything for three minutes.
KATHERINE: Imaginary number.
HAL: Math joke. You can see why they are way down the bill (laughter).
KATHERINE: A long drive to see some nerds in a band.
HAL: God, I hate it when when people say that. It's not that long a drive.
KATHERINE: So they are nerds.
HAL: They are raging geeks but they are geeks that, you know, can dress themselves (laughter) -- hold down a job at a major university. Some of them have switched from glasses to contacts. Play sports. They play in a band. In that sense they make you question the whole set of terms, geek, nerd, dweeb, Dilbert, paste eater.
KATHERINE: You're in this band, aren't you?
HAL: Okay yes, I play the drums.
Flat and Flatter. Ian Stewart's new book ``Flatterland: Like Flatland Only More So'' (Perseus) is reviewed in the 17 May 2001 Nature. The review, by Lisa Lehrer Dive (Department of Philosophy, Sydney) and Andrew Irvine (Department of Philosophy, UBC) praises Stewart's versatility in conjuring up examples to show ``just how fundamental mathematics is, and how everything around us, indeed everything we can think of, may be formed and discussed in mathematical terms.'' Stewart follows Edwin Abbott's lead in using the adventures of human-like characters (Vicki Line meets the Space Girls: Curvy Space, Bendy Space, Pushy Space, Squarey Space and Minny Space) to lead the reader through unfamiliar mathematical territory. ``Cleverly and enganingly, he opens our minds to the possibility that, just as the Flatlanders' conception of their world was inadequate for an understanding of what space is really like, so too, our view of space as three `flat' dimensions and a single temporal one is only the tip of the iceberg.''
Why don't we have six limbs? This is one of the questions considered by David Ruelle in a ``concept'' piece ``Here be no dragons'' in the 3 May 2001 Nature. The piece is mostly an introduction to chaos, but he goes on to digress ``on the tantalizing topic of historical probabilities.'' ``To take a concrete example, instead of asking when biological evolution decided that terrestrial vertebrates have four limbs, we might ask what the probability was that at some prescribed earlier time, say the end of the Cambrian, they would develop six limbs.'' He mentions possibly useful adaptations of the extra limbs for manipulation (centaurs) or for flying (dragons). More generally, he brings up the question of ``how life on Earth would have evolved if the great cataclysm and extinction at the end of the Cretaceous had not taken place,'' and suggests that some day we may understand enough about biological evolution to estimate the probabilities involved. Nature illustrated the piece with Leonardo's ``Fight between a dragon and a lion,'' but Leonardo had voted for four: his dragon is organized along the lines of a bat.
Complexity too simple? The May 11, 2001 Chronicle of Higher Education ran a long piece by Richard Monastersky reporting on the controversy raging on the borders of paleobiology and mathematics. It starts with Per Bak (Department of Mathematics, Imperial College of Science, Technology, and Medicine, London), who published How Nature Works: The Science of Self-Organized Criticality (Copernicus) in 1996. The title alone was enough to ruffle feathers. Bak claimed that populations of species are governed by the dynamical theory of Self-Organized Criticality (S.O.C.), and that large fluctuations in the fossil record can be explained by the same kind of laws that govern the birth and death of avalanches in a sand pile. His evidence is that these fluctuations obey a power law. Monastersky quotes Bak: "I do think that's how it works, because it has the fingerprint of this phenomenon. That's not a proof, but at the very least people should think about [the possibility] that mass extinctions are intrinsic to the way that evolution works and do not need an external cataclysmic effect." The reaction has been bitter. ``That's logic on the same level as saying, `Bears like honey, my wife likes honey, therefore my wife is a bear.'" This from Mark Newman, another professor at the Santa Fe Institute, where Bak also holds a chair. But Newman concedes that the impact of S.O.C. has stimulated new angles of research in paleobiology. Monastersky concludes with the thought that even if S.O.C. theory does not survive in paleobiology, ``it has left behind some intellectual progeny that continue to compete in the harsh world of evolutionary studies.''
Newton vs. Leibniz on the air. If you were listening to WSHU/WSUF in Fairfield CT Friday May 25 you would have heard an episode of John Lienhart's radio column Engines of our Ingenuity devoted to Leibniz (dx/dt) and Newton (x-dot) and their priority controversy over the invention of the Calculus (which has been called the war between deity and dotage). Lienhart bases his story on the episode as recounted in Hal Hellman's Great Feuds in Science (Wiley, 1998,1999). Newton's forces won the war: ``Leibniz died poor and dishonored, while Newton was given a state funeral.'' But Leibniz' conception of Calculus, encapsulated in the brilliantly apposite notation he invented and that we use today, has flourished alongside Newton's. His reputation survived his propaganda defeat and his lampooning by Voltaire; he ``gradually finds his place as one of the greatest thinkers of all time.'' This episode, #1375, is available online.
Physiological rhythms. Nature for 8 March 2001 featured a review article ``Synchronization and rhythmic processes in physiology'' by Leonard Glass of the McGill Physiology Department. (The item was also picked up in the April 13 Science-Week.) Glass gives a wide survey of the area, with many detailed examples, and giving special emphasis to the important roles played by chaos and noise. He calls attention to the existing lag between the development of mathematical understanding of dynamical systems and the implementation of that understanding in physiological and clinical practice, but also to the size of the theoretical task ahead. ``In the physical sciences, scientific understanding has been expressed in elegant theoretical constructs and has led to revolutionary technological innovation. If the advances in understanding physiological rhythms will follow the same trajectory, then we are still just at the beginning.''
Beyond the spherical cow is not the title of a surrealist painting but of a ``news and views'' piece by John Doyle (CalTech) in the 10 May 2001 Nature. The spherical cow comes from the old joke about the theoretical physicist presenting his solution to the problem of increasing milk production. ``First,'' he begins, ``we assume a spherical cow.'' Doyle is reporting on the First International Symposium on Computational Cell Biology (March 4-6, 2001) and concludes: ``A prominent theme was that biology needs more theory, in additional to modeling and computation, to make sense of complex networks.''