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

In theoretical work on the molecule, DNA is often treated, approximately, a s a naturally straight, inextensible, isotropic elastic rod of circular cro ss section. It is shown that, consistent with this level of approximation, there exists a general connection between the free energy of supercoiling o f plasmids formed by the DNA, and the writhe distribution in plasmids havin g 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 distributio n 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 r esults 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 expres sion for the free energy of supercoiling involving the frequencies of the n ormal modes of vibration of a circular elastic ring has recently become ava ilable. In this latter case, the writhe distribution for supercoiled plasmi ds is not normal, but shows a skewness related to a property of elastic rin gs, 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 ri gorously quadratic function of DeltaL(k). (C) 2000 American Institute of Ph ysics. [S0021-9606(00)50140-8].