A Brief Review of Mechanical DNA Models

On Friday some of us wondered why Nelson included such a large amount of material on DNA when he could have happily truncated the discussion much earlier (from a pedagogical standpoint). The answer is simple... Nelson is into DNA stretching!

I have been doing some reading around to try and resolve just what the precise definition of the stretch (u) is, but have had little success because the majority of studies don't bother to include it! I strongly suspect that it is simply the elastic strain, given as a function of the applied stress by some appropriate constitutive relation (e.g. Hooke's Law relates the stress F/L to the strain x/L, where x is the change is the difference between the actual and equilibrium lengths (L), by the spring constant).

As I mentioned, most studies appear to completely neglect the stretch and focus on the wormlike chain (here's a nice blog on this topic), freely-jointed chain or discrete persistent chain models. The first two are discussed in the text, whereas the last is a generalisation of the freely-jointed chain in which there is an energy cost associated with introducing an angle between neighbouring segments.

The discrete persistent chain is a mathematical beast! This paper, one of Nelson's own, gives a lot of detail on this model, and demonstrates the use of the transfer matrix and variational methods (this link goes through many different variational methods e.g. mean-field approach, Bayesian estimation, graph theoretical approaches, etc.). It seems to give good results in the study of B-to-S DNA transitions: S-DNA being an overstretched state.

It also seems that Nelson was a little disingenuous: he didn't actually start with the most general model and simplify down to the FJC... the beginning model was actually already dramatically simplified! Extensions include sequence-dependent parameters (bend and twist persistence length, etc, which can even be asymmetric about the axis to account for the pitch of the molecule) which account for possible mechanical differences between different base arrangements (believed to be important in protein binding to dsDNA, even more so than the actual chemical properties of the bases), cyclisation (making small loops of DNA), phase-dependent parameters, finite-length effects and the influence of histones and other DNA-binding and packing proteins.

It turns out that cyclisation occurs in much shorter segments than people expected, and no-one knows why yet. There's also a whole great big world of knot theory and DNA which was left unexplored, and different types of nucleic polymers (dsDNA, ssDNA, RNA, etc).

There's still a lot of research to be done, and a lot of reading in the meantime!