Biological Puzzles : Math Offers Way to Untie Gordian Knots

According to Greek mythology, Gordius, king of Phrygia, had a very big problem--the great Gordian knot.

The pole of his wagon was fastened to the yoke with a knot so entangled that it defied all efforts to unwind it. The catch, of course, was that whoever managed to undo the knot would rule Asia, an honor eventually achieved by Alexander the Great, who solved the problem by cutting the knot with his sword.

There are probably few things more confounding than knots--knots in yarn, knots in wire, string or even DNA, the master molecule of life.

Scientists now think that they are on the brink of finding a hidden meaning in knots with the development of new equations that will help solve--at least mathematically--the thousands of Gordian problems that people encounter daily.

Phone Cords, Molecules

Knot theorists, as they are professionally known, are trying to develop equations for all the different kinds of entanglements that can occur in one’s telephone cord or in complicated molecules much too small to see with the naked eye.

So far the efforts have been successful. Knot theorists have identified 12,965 distinct entanglements, but they say there are virtually countless others.

“Knotting occurs in nature,” said University of California, Santa Barbara, mathematician Kenneth C. Millett who is studying the phenomenon of knotting in the natural sciences. “But knots represent very, very big mysteries.”

Among the mysteries in nature are the geometry of DNA molecules, the basic units of heredity in every cell of every organism. Structurally, DNA is a double helix, or spiral, that folds into a complicated knot inside the nucleus of cells.

“In our studies of DNA and its replication process, we have found that enzymes act on these molecules in a way that is sensitive to the knotting of DNA,” he said of a natural unraveling process. Enzymes are proteins that speed up or cause chemical reactions in living matter.

The enzymes know exactly when to begin the unwinding of the DNA while maintaining the integrity of the molecule, and it is necessary for this to occur in all our cells for the cells to reproduce themselves.

“If the DNA remains knotted, then the cell will die,” Millett said.

Secrets inherent in the enzymes, however, may ultimately reveal solutions to quickly unwind a piece of wire or telephone cord because “the enzymes are nature’s solution to the Gordian problem,” Millett said.

“Enzymes know a great deal about knotting and we hope to use mathematics to understand more about the enzymes and DNA.”

Number of Crossings

Of the 12,965 knots scientists have identified, including knots in biological molecules, many have fewer than 14 crossings. But even when one of these knots is encountered in a length of fishing line, for example, it can pose perplexing challenges.

To mathematicians, a knot is a closed loop that is twisted and tangled in various ways. A loop without any twists is called an “unknot,” said Millett, who analyzes knotted structures by studying their shadows.

Until 1984, mathematicians relied on a formula known as the Alexander polynomial to distinguish one knot from another. But, Millett said, the formula was so imprecise in some instances that it mathematically “could not always distinguish a knot from an unknot.”

In 1984, University of California, Berkeley, mathematician Vaughn Jones developed another algebraic formula, or polynomial, to cull more information from knots. His work was quickly followed by that of a number of other theorists who discovered the “oriented polynomial.”