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Date: 01/29/2001
Writer:
Jack King
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"Imagine this model is the physical representation of a dynamic process," New Mexico State University mathematics professor Reinhard Laubenbacher said recently, holding up a paper polyhedron that looked like a giant, awkward Christmas tree ornament. "Only the real model wouldn't be just three dimensions. It could be 23 dimensions."

The "process" could be any of a number of things, Laubenbacher added -- a breeding population of coyotes, a busy anthill or the metabolism of a cell regulated by a dozen different genes. Traditionally, biological processes, or "systems," have been studied by picking one part of the system, studying it, then picking another part and studying it, then, eventually, putting all the parts together and hoping that hundreds, or thousands, of individual studies will yield knowledge of the whole. But increasingly scientists have become aware that many systems are greater than the sum of their parts, he said.

"If you look at an anthill as a social organization, each ant is an actor in this complex organization. Each ant acts on a set of relatively simple instructions, yet what comes out is an extremely complicated system. But an ant has no idea of the complicated structure which depends on its individual actions and you cannot understand how an anthill works by studying individual ants or even individual ant interactions," he said.

This is why the National Science Foundation has given Laubenbacher a $99,000 grant to study whether the properties of such complex systems can be modeled using a discipline called "algebraic combinatorics."

Combinatorics, Laubenbacher explained, began as the mathematical study of counting. A classic exercise in combinatorics is to ask, "Into how many pieces can I cut a block of cheese with 15 cuts of a knife?" But Laubenbacher's use of the discipline is more elaborate.

"Think of the knife cuts as a series of planes intersecting in three dimensional space, creating a number of chambers," he said. "What you have is a geometrical object in space and the 'counting' problem you had becomes a problem about the geometry of the chambers."

The illustration is a good one, he added, because what he is trying to do in his study is turn a complex of movements, such as the action of taking 15 slices, and the relationships between the strokes to each other, into a static geometrical model. By using abstract algebra to measure the geometrical features of the model, he hopes to be able to discern the rules governing the interactions in a complex system.

For centuries, scientists have used algebraic functions to describe the relationships between objects they have observed, but for complex systems such a description is usually not effective, Laubacher said.

"What you cannot describe with a function is the dynamics of the system as a whole," he said.

What gives Laubenbacher's work its pioneering quality is his effort to develop a mathematical method for describing complex systems in action, he said. And the geometrical models that arise from that effort look like infinitely more complicated, multi- dimensional, versions of his paper "ornament."

"Different processes produce different geometrical objects and we want to know what inferences we can draw by knowing the properties of these objects," he said.

"Of course," he added, "we don't know which are the good ways to construct these models. There are a number of different ways. Some might be useless; some might be good. That's part of the study."

Once researchers have such models and once they understand which parts of the models relate to interesting properties of living systems, they might be able to create computer programs to do a variety of research, Laubenbacher said.

He originally developed the methods he is now exploring as part of a joint project with researchers at NMSU's Physical Science laboratory on human decisions systems, but there are many other possibilities, he said.

"A computer program could allow you to understand if one gene network is highly connected, if it is more highly centralized, or if only a few genes are involved. You could study what would happen if you removed a gene. Would the dynamics of the system stay the same?"

Photo is available at http://kiernan.nmsu.edu/newsphoto/biomath.jpg.

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CUTLINE: NMSU mathematics professor Reinhard Laubenbacher holds up a paper geometrical model that represents the models he constructs mathematically to study complex, dynamic systems. Laubenbacher said someday scientists could use the models to study anything from a busy anthill to the genetic regulation of living cells. (NMSU photo by Michael Kiernan)

Jack King

Jan. 29, 2001

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