The writhe distribution in DNA plasmids as derived from the free energy of supercoiling

In theoretical work on the molecule, DNA is often treated, approximately, as a naturally straight, inextensible, isotropic elastic rod of circular cross section. It is shown that, consistent with this level of approximation, there exists a general connection between the free energy of supercoiling of plasmids formed by the DNA, and the writhe distribution in plasmids having a given value of the linking number difference, DeltaL(k). In particular, the writhe distribution in a collection of torsionally relaxed (DeltaL(k)=0), but non-nicked, plasmids is completely determined once the free energy of supercoiling as a function of DeltaL(k) is known. The writhe distribution in the supercoiled plasmids characterized by any other value of DeltaL(k), we shall also show, is simply related to the distribution in the relaxed plasmid, and, therefore, it, too, is completely determined. These general results are illustrated for two cases: Large plasmids for which the measured free energy of supercoiling, a quadratic function of DeltaL(k), implies a normal writhe distribution, and miniplasmids for which a theoretical expression for the free energy of supercoiling involving the frequencies of the normal modes of vibration of a circular elastic ring has recently become available. In this latter case, the writhe distribution for supercoiled plasmids is not normal, but shows a skewness related to a property of elastic rings, namely, the loss of stability of the circular equilibrium configuration of the rings when they are twisted beyond a critical value. Such a skewed writhe distribution for miniplasmids is, according to the model, associated with a free energy of supercoiling which is not, as has been assumed, a rigorously quadratic function of DeltaL(k). (C) 2000 American Institute of Physics. [S0021-9606(00)50140-8].