Interface Between Biology And Mathematics Expanding Rapidly

Biologists and mathematicians alike are finding their respective disciplines meeting at a very busy crossroads. The rapid advances in both fields, they say, have resulted in an explosion of information in the life sciences, and the creation of sophisticated mathematical tools for handling complicated biological systems. Studies over the entire range of the life sciences--from the molecular level (DNA and proteins) to entire organisms and ecosystems--benefit from insights derived from the mathematical sciences, encompassing mathematics, statistics, computer sciences, and theoretical physics, note researchers from these disciplines. One highly publicized mathematical concept that has found wide use in interpreting biological phenomena is game theory, for which Reinhard Selten, a mathematician at the University of Bonn in Switzerland, and economists John F. Nash of Princeton University and John C. Harsanyi of the University of California, Berkeley, won the 1994 Nobel Prize in Economics. "Game theory provides a mathematical way to find the best strategies in any situation where there are different payoffs associated with different choices," explains Princeton University biology professor Simon Levin. "Its applications in evolutionary theory involve dealing with simple and abstract 'games' as ways to gain insight into the evolution of behaviors of living organisms." Levin, who holds a Ph.D in mathematics, cites "predator-prey relationships, seed-dispersal strategies in plants, and the shading behavior of some plants" as examples of ecological patterns that can be explained by this approach. In studying the evolution of a species, for example, game theory would treat those within a species as players, with their modes of interaction representing strategies used for their survival and propagation. A comprehensive look at the subject is provided by Karl Sigmund, a professor of biological mathematics at the Institute of Mathematics of the University of Vienna, Austria, in his book Games of Life: Explorations in Ecology, Evolution, and Behavior (New York, Penguin Books, 1995). Another particularly active hub of interaction between the mathematical and life sciences has been molecular biology. "In recent years, [molecular] biologists have come to recognize that they are dealing with large amounts of information, and that processing these large amounts of data are the types of problems math and the computer sciences can help with," notes Fred Roberts, a professor of mathematics at Rutgers University. "The genetic code, for instance, deals with many possibilities of sequences." "A branch of mathematics that is coming to fruition for its value in this area is discrete mathematics," comments Martin Farach, an assistant professor of computer science at Rutgers. This subdiscipline, according to Roberts, is concerned with the study of "patterns, designs, strings, and sequences, where the data comes in finite or discrete packets, and not as a range of continuous values. "DNA can be seen as a long string of symbols where each unit or nucleotide takes one of four values--there are no smooth transitions from one to the other, and no range of values," says Farach, pointing out that these properties make it an ideal raw material for this branch of mathematics. Both Roberts and Farach are researchers at the Center for Discrete Mathematics and Theoretical Computer Science (also called DIMACS), located on Rutgers' New Brunswick campus. This year, both scientists are the organizers of a year-long program entitled the "DIMACS 1994-95 Special Year on Mathematical Support for Molecular Biology." The project features workshops, lectures, and seminars that, according to Roberts, "bring biologists and mathematical scientists together to identify ways in which discrete mathematics and theoretical computer science can address various questions in molecular biology." As an example of a discrete mathematical analytical tool, he offers graph theory, which has formed the basis for various algorithms that have been developed to address such problems as gene mapping and the construction of evolutionary trees. "A graph is basically a set of nodes joined by a set of connections," Farach explains. Gene mapping, for instance, involves cutting a piece of DNA with enzymes that recognize and cleave specific arrangements of base pairs to generate discrete segments, and using the sequence information to align these fragments. "If we consider the fragments as the nodes and overlapping sequences as the connections, then we have a final path or graph in the form of one long chain of sequence information," relates Farach. Thus, he adds, various mathematicians have fashioned algorithms "to glean the correct sequence by determining the correct order of the fragments." Using sequence data from specific common regions of the genomes of different species to build evolutionary relationships among them is a very fundamental use of graph theory, whereby "the species are the nodes connected by evolutionary distances between them," Roberts points out. Farach recently developed algorithms for the comparison of phylogenetic trees. The algorithms are being used in evolutionary studies by David Swofford, a biologist at the Smithsonian Institution in Washington, D.C. While discrete mathematicians explore the intricacies of molecular biology, others trained in more classical mathematical areas like differential calculus and dynamical systems theory are also developing new tools and making headway in modeling biological systems. Levin and Princeton biology professor Steven Pacala, for instance, use these tools to construct models for the workings of ecosystems. "What we are trying to do is understand the deterministic, macroscopic properties of a system that is composed of smaller, less predictable units," Levin elaborates. For instance, he offers, while a forest containing several thousand trees would exhibit certain broadly uniform or predictable responses--like growth--to a change in climate, every single tree does not respond the same way. "In order to understand this system properly, we need to interface models of forest growth--how individual trees affect the dynamics of the forests--to those of climate change," he says. Levin points out that an important aspect of such models is assessing how much detail in the smaller units is important in understanding the larger system: "Earlier models of ecological systems often tended to ignore the spatial structure of the system--they were based on the assumption that every individual in the population is as likely to interact with every other one. But obviously this is not always true--especially in a forest, where there would probably be clusters of different species. Thus, we have to consider the spatial relationships of the units to each other. Too much detail can be as crippling as too little, and finding the right level of detail is the essence of scaling," he declares. Falling somewhere between the minutiae of molecular sequences and the magnitude of ecosystems are numerous biological systems that are also under the scrutiny of mathematicians interested in the life sciences and biologists seeking to understand them better via mathematics. The workings of the nervous system have lent themselves particularly well to various mathematical analyses. Neuroscientists observe that signal transmission in the nerves as well as information processing by the brain are especially suited for description in mathematical terms. "As far back as 1952, neuroscientists did fundamental experiments on the giant axon of squid and came up with the Hodgkin-Huxley theory--a mathematical model for nerve-impulse conduction," notes Jonathan Bell, a professor of mathematics at the State University of New York, Buffalo. In fact, both the eponymous scientists--Alan L. Hodgkin of Cambridge University and Andrew F. Huxley of London University--went on to win the 1963 Nobel Prize in physiology or medicine, along with Australian neuroscientist John C. Eccles, for this work. Today, math has much wider applications in studying the nervous system. With sophisticated, computer-based tools at their disposal, neuroscientists are able to attempt simulations of how the nerves and brain receive and process different types of sensory and cognitive information. Bell, for example, collaborates with Mark Holmes, a professor and chairman of mathematics at Rensselaer Polytechnic Institute, to develop mathematical equations for modeling the sense of touch. "I look at a physiological phenomenon and try to understand it in mathematical terms--I work in equations," he comments. Holmes says that tactile sensation can be looked at in terms of two components: "First, we are interested in how the mechanical signal is received and converted into the neural signal, and next how the brain uses this information to figure out what the body is feeling." Currently, Holmes and Bell are working on modeling events at the receptors. Physically, the touch receptor in the skin, called the Pacinian corpuscle, consists of a "large, bulb-like structure--with multiple layers of elastic membrane separated by fluid--with a nerve ending piercing it like a lollipop," Bell says. When the skin surface is touched, he explains, the bulb is compressed, transmitting a mechanical force to the nerve, which in turn is activated and sends electrical signals to the brain. The scientists use partial differential equations to describe the system, with each equation accounting for a different component, relates Holmes: "Basically, we are trying to boil the events at the Pacinian capsule down to a few equations." One advantage in choosing touch over the other senses, Bell suggests, is that "we have an experimental system to test out our [mathematical] theories. The Pacinian corpuscle is physically very large, and we can actually stimulate it and record the nervous responses in the lab," which allows the scientists to see how closely their equations imitate the events at the corpuscle. Mathematical simulations such as these offer a deeper understanding of the physiological processes, Holmes maintains. For example, he says, "earlier we did not know what was going on inside the corpuscle as the layers of liquid and membrane were too small for us to see what was happening." By applying the mathematical branches of fluid dynamics and elasticity to derive the equations, they were able to obtain a better grasp of the events. Biologists and mathematicians concur that the potential overlap between mathematics and biology is immense in its scope--and is continuing to accelerate. The disciplines of topology and geometry have been applied in understanding molecular structure, while computational scientists are bringing their expertise to bear upon designing drugs against various diseases. Chaos theory is another area with application in the life sciences that has also received its fair share of attention from the popular press in recent years. Scientists from different fields in the life sciences are beginning to apply the methods of chaos in their studies (N. Sankaran, The Scientist, Aug. 22, 1994, page 3). For example, Gary Burlingame, a group n psychotherapist, uses chaos theory to try to determine fundamental behavioral patterns among his patients. Based on evidence that bone structure displays chaotic patterns, Raj Acharya, an associate professor of electrical and computer engineering at SUNY-Buffalo, has developed a new computational tool with fractal geometry--a branch of chaos theory--to make early diagnoses of osteoporosis. "It is only now--in the past decade or so--that we have mathematical tools to meet the conceptual complexity of biological systems," observes Sally Goerner, director of the Triangle Center for the Study of Complex Systems in Chapel Hill, N.C. Sidebar:Where Math, Biology Meet (The Scientist, Vol:9, #14, pg.14 , July 10, 1995) (Copyright, The Scientist, Inc.) garfield@aurora.cis.upenn.edu

Interface Between Biology And Mathematics Expanding Rapidly

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