Mathematics Awareness Week 1994

Mathematics and Medicine

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Poster: Mathematics and DNA
Much interest and optimism surrounds mathematics-intensive research into DNA. The MAW theme poster displays two computer-generated views of tenfold B form DNA, produced by the Computer Graphics Laboratory, University of California, San Francisco (copyright Regents, University of California).

Mathematical simulation and modeling help researchers visualize and understand DNA structure and energetics. The double-stranded DNA is examined using techniques of topology and differential geometry, as well as computer simulation. A basic property of circular double-stranded DNA is its linking -- the two strands cannot be separated without breaking one or the other. The fundamental equation (proven by James White of UCLA) is

Lk = Tw + Wr

which states that the linking number results from the sum of the twist of the DNA strands around each other (Tw) and the writhing of the DNA in space (Wr). The enzymes in cells that control the level of these three quantities turn out to be excellent targets of anticancer drugs, demonstrating the vital importance of DNA topology.

From a quite different mathematical perspective, human genome research is producing a vast amount of data. If compiled in telephone book form, the information in the human genome would fill 200 volumes at 1,000 pages each. The databases for this research are so complex that mathematical approaches such as combinatorics, pattern recognition, and sequence comparisons are required.

Postcard: Mathematics and Cardiac Cells

Researchers in the Applied Mathematics program at the University of Arizona are integrating equations that simulate the propagation of electrical impulses within a flat sheet of cardiac cells. In cardiac muscle, these electrical signals tell the cells to contract. The sheet of cells is treated as a continuous "excitable medium" whose electrical signals can be modeled mathematically to predict what electrical stimulus is needed to trigger fibrillation. The equations used to describe the propagation of electrical signals in tissues such as cardiac muscle or nerve fibers are usually referred to as the "cable equations." They are part of a group of nonlinear partial differential equations called "reaction- diffusion equations," and can be written down as

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In this equation e is the amplitude of the electrical signal and v represents all the other variables specific to the particular model used to represent the electrical properties of the cardiac cells. The equation describes how the values of e and v evolve in time (t). The three-dimensional colored surface is obtained by using elevation to represent the amplitude e of the electrical signal at each point of the sheet of cells at a selected time. Color is used to simultaneously represent the value of the v components in the equation.

Postcard: Mathematics and MRI Contrast Agents

Magnetic resonance imaging (MRI) is a tool for viewing organs in the body non-invasively. Simply put, the method measures the differences in the concentration of water in different regions of the body which show up more or less bright on an MRI image. Researchers Richard Judson and Carl Melius at Sandia National Laboratories in Livermore, California are developing mathematical methods for designing better agents to enhance the contrast. These agents are commonly used to pinpoint the cause of stroke in the brain and abnormalities in the blood flow in the kidneys.

The view of a model agent shows the results of calculating the electrostatic potential about the compound when it is emersed in a water-like medium. In the figure, gadolinium is a silver ball, oxygen atoms are red, carbon atoms are light blue, nitrogen are dark blue, and hydrogen are white. Hamilton's derivations of Newton's equation are used to calculate the agent's behavior.

Back to Math Awareness Week 1994

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