Mathematics Awareness Week 1994

Mathematics and Medicine

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Mathematical Algorithms are Key to Molecular Modeling
(One in a series of six articles on Mathematics and Medicine being distributed by the Joint Policy Board for Mathematics in celebration of Mathematics Awareness Week 1994.)

At the University of Houston, Dr. Ridgway Scott, Professor of Computer Science and Mathematics, and Director, Texas Center for Advanced Molecular Computation, is trying to make a computer think like a molecule acts. He and his colleagues are taking mathematical descriptions of elementary atoms and forces and writing them into programs that mimic molecular systems on the computer. If successful, the mathematical models developed by Scott and his co-workers may lead to discoveries of important new chemicals, such as life-saving pharmaceuticals.

The idea is to give the computer the capacity to design and analyze molecular systems, thereby offering alternatives to time- consuming "wet lab" methods of chemistry. The mandate of this national project -- part of the National Science Foundation's "Grand Challenge" program -- is to create the systems on powerful supercomputers that employ parallel processing. In the long run, parallel systems offer low cost and high efficiency.

In their contribution to the project, Scott and fellow mathematician, Dr. Roland Glowinski, develop algorithms, or mathematical rules, that govern or express atomic behaviors. It comes as no surprise that the number of algorithms needed to describe complex molecular structures and activities is large. And although different algorithms can lead to the same results (like different map-routes to the same place), Scott and Glowinski must determine which strategies will be most efficient.

Parallel computers have anywhere from tens to tens of thousands of processing units that work simultaneously. (In contrast, the personal computer or laptop has a single processor.) The following graph provides an example of performance analysis. Researchers must program each processor with algorithms that mimic any of a variety of qualities of the molecule to be modeled. The biological molecules of interest in this project have upwards of tens of thousands of atoms.

(graph omitted here; it may be obtained by surface mail)

Researchers describe molecular systems on three strategic levels. The first level, electrostatics, considers the electrical forces surrounding large molecules -- such as proteins. Algorithms for such models employ equations for statistical distributions, including the Poisson-Boltzmann equation.

A second and more common level, molecular dynamics, uses mathematical techniques that describe forces between atoms; the atoms are considered unitary particles, like planets. Algorithms for these models employ Newtonian or Hamiltonian mechanics.

A third level, the quantum mechanical level, considers forces and particles inside the atom. In this case, the mathematics of quantum mechanics -- in particular, the Schrodinger equation -- is employed.

Once algorithms are constructed, Scott and colleagues face the daunting task of incorporating them into computer software. Scott points out that the state of parallel computing software is "in flux." Also, computers don't understand mathematical notation, so that algorithms have to be translated into programming languages on an "ad hoc" basis. The fields of mathematics and computer science truly merge at this point.

One example of the potential benefits of modeling is the case of superoxide dismutase. In the human body, this natural enzyme binds to and destroys superoxide molecules, harmful species that can arise during heart attacks and cause secondary damage. Computer modeling of superoxide dismutase may lead to designs of artificial molecules that can rid the body of harmful superoxides more efficiently. A separate research group has already used an electrostatic model and a nonparallel system to design such a molecule.

Back to Math Awareness Week 1994

Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area.