Statistical mechanics of DNA topoisomers: The helical worm-like chain

Abstract

Recent experimental data of Shore & Baldwin (1983 b) and of Horowitz & Wang (1984) for the apparent twisting coefficient K, which determines the breadth of the Gaussian distribution of DNA topoisomers with different linking numbers N, show that the product of K and n bp (the number of base-pairs) is nearly a constant for n bp ≳ 2000 , but that it increases sharply with decreasing n bp for n bp≲ 2000 . The main purpose of the present paper is to explain theoretically such behavior of K as a function of n bp . Thus the statistical mechanics of DNA topoisomers in general is developed on the basis of a twisted worm-like chain, i.e. a special case of the helical worm-like chain. The previous treatments of the N-dependent ring-closure probability, i.e. the distribution of N, which are valid only for small chain length L, are extended to the range of larger L. The variance of N is then shown to be exactly the sum of those of the writhe Wr and the twist Tw. For small values of L, the distribution of Wr is not Gaussian, and its variance or moment 〈 Wr 2 〉 increases rather steeply with increasing L. With these and known Monte Carlo results for freely jointed chains, an empirical interpolation formula for 〈 Wr 2 〉 is also constructed to be valid for all values of L. It predicts that 〈 Wr 2〉 L increases monotonically, with increasing L to its coil-limiting value. On the other hand, the distribution of N is actually Gaussian in the practical range of N for all values of L. The conditional distribution of Wr with N fixed is also evaluated. Finally, K is expressed in terms of the torsional constant C, the stiffness parameter λ −1 (which is equal to the Kuhn segment length and twice the persistence length for this special case), and 〈 Wr 2 〉. The derived equation predicts that n bpK decreases monotonically to its coil-limiting value with increasing n bp . This decrease arises from the fluctuation in Wr and its neglect leads to an underestimate of C by 7 to 10%, even for short DNA with n bp ∼- 200 . From an analysis of the experimental data of the two groups, the estimates of C = 3.1 to 3.2 × 10 −19 erg cm and λ −1 = 1000 to 1200 A ̊ are obtained.