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
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.
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
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.
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