A few months ago a paper popped up on the archive about tangles and biochemistry. I’ve always wondered if analyzing knotted DNA is more useful for understanding biochemistry or for coming up with interesting topology problems. It seems that in this case the lab scientists found a way to turn a chemistry problem into a topology problem, which they solved under extra assumptions. The authors of this paper improved the mathematical analysis enough to remove these extra assumptions.
Based on my minimal understanding of the chemistry involved, here’s what seems to be going on: There is a protein (Bacteriophage Mu) that grabs three arcs of DNA and holds them in some configuration (a tangle). The goal is to determine what tangle it holds them in. The protein always grabs three arcs that have three specific sequences, so one can control where the protein grabs the DNA by constructing DNA with different sequences.
The scientists construct unknotted circular DNA with a given sequence, then put it in with the Bacteriophage Mu. The Mu binds to the DNA. They then throw in a second protein (Cre) that takes two specific identical sequences in the DNA, cuts the loop at those sequences and glues it back together, but switching their ends. They then somehow release the DNA from the Bacteriophage Mu and look at what the new knot type is. (I think they can determine the knot type using an electron microscope, but I didn’t find any details of the exact methods in the paper.)
Since the Bacteriophage Mu and the Cre proteins each bind to specific sequences in the DNA, the experimenters can control how the arcs trapped in the Bacteriophage Mu are separated by the two arcs that are switched by the Cre. They thus get a collection of knot types and are able to deduce from this what the inside tangle looks like.
I wonder to what extent this can be generalized. It seems like it should be relatively straightforward to come up with some definitions and axioms that would model this type of behavior for arbitrary proteins that grab onto strands of DNA. (Though I don’t yet understand the paper or the methodolgy well enough to work it out myself.) I should mention that Dorothy Buck and Erica Flapan have written about the knot theory behind site specific recombination, (which I think is what the Cre protein does) for loops that aren’t necessarily bound by other proteins.