A slightly more complex knot problem (for advanced boy scouts)


I heard a pretty great talk today by Jennifer Mann from Baylor College of Medicine. Her talk was on a natural phenomenon of the long DNA strands called knotting. It is well known that DNA must be compacted in order to avoid taking up an unreasonable amount of space within the nucleus of a cell. However one can envision that this compaction process is not without its intricacies.

Imagine taking a long string of some sort, not unlike your headphone cord. When you hastily throw your headphones into your pocket, there’s a good chance that when you pull them out later on, they are in a headache of a knot. It’s somewhat of a mystery how the cord ended up in such a tangled configuration, and every once in awhile, this happens in the compacting process of DNA, which is a long strand of nucleic acids strung together that eventually give rise to the amino acids that make proteins, which are largely the stuff of life. (Forgive my really rudimentary biological descriptions.)

Several interesting properties arise, and one can describe the topology of the resultant knots, if they do form, within the DNA, as pictured here. Though there are an infinite number of minor changes within a given conformational structure, the topology can still be quite similar and thus create tractable problems. This was termed “knot invariant.”

Well, without getting into what amounted to some heavy an fanciful biology, one can imagine a multitude of ways to untangle the knot of DNA, which is quite a problem for a cell to undertake naturally. Essentially, Mann was providing evidence that the cell’s mechanism to untangle the DNA knot was via the easiest (read: lowest energy) change that would then allow the entire molecule to relax.

The computational approaches that such a system presents are vast. For one, there is undoubtedly a conformational strain on the natural configuration of the twisted, knotted DNA strand, and those changes are likely to have some energy stored in them, not unlike a spring, that is eager to be released into a lower energy “rest” configuration.

From a dynamical systems perspective there might be many such stable states of relaxation, and small perturbations from the knotted conformation (i.e. untangling) might allow one to predict – based on optimized bond length and energy parameters – to what stable fixed point the molecule might end up.

Among the several other interesting questions that exist, minimization of energy is among my favorite. But from a biological point of view, it would be nice to be able to selectively target a site at which just one small change (a cut in the loop, twist, and rejoin) can greatly reduce or eliminate the knotting.

For more information, see Z. Liu, J. Mann, E. Zechiedrich, and H. Chan. Topological Information Embodied in Local Juxtaposition Geometry Provides a Statistical Mechanical Basis for Unknotting by Type-2 DNA Topoisomerases. J Mol Biol, 2006.

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