DNA is typically much longer than its host cell, and so must wrap and coil around itself in order to fit. The "writhing number" is a numerical measure of this wrapping and coiling, and is important to molecular biologists because it is closely connected with the interlinking of the two strands of DNA on the double helix...an interlinking which must be overcome by the action of cut-and-paste enzymes during replication/reproduction.
Our goal is to obtain a sharp upper bound for the writhing number in terms of length and thickness, and to describe the patterns of coiling which achieve this upper bound.
To begin, we have already derived an upper bound for the writhing number in terms of a 4/3 power growth law: The writhing number of a knot of length L and thickness R is bounded by a universal constant times the 4/3 power of L/R . A "knot" is simply a mathematical representation of the wrapped and coiled strands of DNA.
To investigate whether this 4/3 power is optimal or can be reduced, it is natural to seek a family of knots of increasing L/R, with as much writhing as possible. This is where the key idea of the project enters.
A knot of great complexity can be visualized as a large ball of tightly wound wire. Imagine a current flowing through the wire, compute the corresponding magnetic field, and let the magnetic field lines again remind us of a large, complex knot.
By the laws of physics, a magnetic field writhes beautifully around the current distribution which produces it. But what we want is a knot writhing beautifully around itself. So we blur in our minds the distinction between current distributions and magnetic fields, pretend that the magnetic field is a new current distribution, and compute its magnetic field. Then we iterate this procedure, time and again, hoping that it converges...the limit should be a field whose field lines writhe maximally around themselves.
Miraculously, this happens. We modelled the original current-distribution-knot as a field of arrows defined on the vertices of a 10 x 10 x 10 grid. Computing the corresponding magnetic field required a million sets of calculations. On the math department NeXT station, this took about seven minutes. We ran the program overnight, fifty iterations, and got convergence.
Successive runs on larger and larger grids achieved similar results, and put us in position to estimate the growth of the writhing number of the limiting configurations as the grid size increased. To our surprise and our pleasure, these limiting fields exhibited a 4/3 power growth, with better than 1% accuracy.
But we remain cautious: the discretization of the problem may be changing it in a way we don't yet understand, and our interpretation of the results so far obtained may be misguided. Thus we are currently proceeding along three lines, which mix computer-aided and analytical approaches.
In the first approach, we're trying to understand the limiting fields well enough to replace the computer results with an analytical estimate of the growth of the writhing number.
In a second approach, we're trying to construct a family of highly writhing knots by "collective common sense". That is, we have consulted molecular biologists and chemists and asked them how they would organize DNA in order to maximize its writhing number. The answer seems to be an iterated helical supercoiling. But there is a surprising obstruction to this procedure.
In the third, purely mathematical approach, we are seeking the limiting fields which maximize the writhing number by solving appropriate differential equations and investigating the solutions both computationally and analytically. At present, this seems to be our best bet.
Tea. High tea will be served at 3:30pm in the Lounge.
Emmy's. Certain refreshments will be available at the Emmy's after the talk.
Host. David Webb is the host. Anybody who is interested in having dinner with the speaker should contact David at 646-1271.