Locked But Not Knotted

Science  12 Feb 1999:
Vol. 283, Issue 5404, pp. 931d
DOI: 10.1126/science.283.5404.931d

Once scientists had assumed that DNA and other long-chain biomolecules are as floppy as a strand of overcooked spaghetti. A more true-to-life model, however, says these biopolymers are more like chains or loops of linked rigid pieces, like bits of uncooked spaghetti joined by hinges.

This rigidity affects the shapes a molecule can assume, specialists in knot theory have now found. A newly discovered six-sided shape cannot be untangled—that is, manipulated to form a simple, flat loop—if its sides are rigid. But the twisted shape can be undone if its sides are made supple enough to bend.

Jason Cantarella, a differential geometer at the University of Pennsylvania, Philadelphia, had heard speculation that there might exist a stiff-sided polygon that can be “locked” without being knotted. (Knotted polygons can be untied only by breaking their loops, while a locked one, in theory, could be untied by letting its sides go limp.) Then one day, he says, “I was just doodling hexagons on a pad, and I said, ‘That's it!’” He had drawn a hexagon with two long “wings” joined by three segments that twist around a horizontal collar, forming a loop at the top (above). The loop can't be unhooked from the collar without moving the wings, and the wings can't be spread without pulling the loop through the collar. “It's a Catch-22,” says Cantarella, who unveiled this novel shape last month at a meeting of the American Mathematical Society in San Antonio.

Scientists say this is the first proof that polygons with flexible joints are prevented by geometry from being untangled. According to mathematician Ken Millett of the University of California, Santa Barbara, Cantarella's simple polygon is a step toward understanding how small-scale rigidity influences the shape of DNA and other complex molecules.

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