(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.)
One of the most active scientific fields today is the effort to construct a map of the human DNA. This involves detailing the structures and functions of discrete segments of huge DNA molecules, which are the basis of life. It requires cutting-edge techniques and offers the promise of revolutionary results, especially cures for inherited diseases. Many people might not realize that this effort employs techniques of not only biology and chemistry, but also mathematics. Why mathematics? Because a key strategy of genetic mappers is to establish probabilities of links between segments of DNA molecules and the traits they produce in people.
Eric Lander is one of the many researchers working to unlock the secrets of human DNA. A professor at M.I.T. and Director of the Center for Genome Research at the Whitehead Institute for Biomedical Research, Lander has a doctoral degree in mathematics. In Lander's words, "Genetic mapping -- trying to find out where the genes are that cause disease traits -- is quintessentially a mathematical problem. What you are looking for is to show the correlated inheritance pattern of markers in a family with a disease."
A marker is any segment of DNA that is different from what is normally expected. A gene can be a marker. Genes are sections of DNA that contain codes or instructions for a cell to manufacture important substances known as proteins. By coding for proteins, genes set off the chains of events that produce the body's traits - from brown eyes, to pointed noses, to allergies. When genes are passed on through generations, traits are passed on as well. When genes are altered, traits are altered, sometimes with harmful results.
Genetic mappers study generations of one large family or a number of families, noting the co-occurrence of a certain marker and a certain trait, such as a disease. If a correlation between marker, gene, and physical trait can be made, then at least one function of a DNA segment is established. Part of the DNA is mapped.
The problem is to establish the correlation fully. For that, researchers use statistics. Professor Jurg Ott, of Columbia University, New York, is a statistical geneticist whose specialty is the analysis of linkage between genes and traits. Says Ott: "What we use is statistical mathematics. And we almost always use one estimation procedure -- the 'maximum likelihood' estimation." From the family data, researchers produce two mathematical probabilities known as maximum likelihoods -- one that assumes gene and trait are linked, and one that assumes they are not linked. If the ratio of these two likelihoods is at least 1000 times, then a linkage is established.
The mathematics of genetic linkage is complex and researchers rely on computer programs to carry out the analyses. Newer programs are being developed to deal with complex inheritance, such as when multiple genes act together to cause a disease. Mathematics also is used to analyze error in studies of linkage and mapping.
Many aspects of genetic research related to gene mapping also involve mathematics. For example, researchers study the mathematics of the organization of the genome; that is, possible patterns that occur in DNA's sequence of nucleotides, the molecules that string together to form DNA. Statistical procedures that distinguish complex patterns from noise or randomness will aid genetic researchers, including those mapping human genes.
Note: Computer-generated views of tenfold B form DNA may be seen
on the 1994 Mathematics Awareness Week theme poster.
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