Shimada J, Yamakawa H
J Mol Biol. 1985 Jul 20;184(2):319-29. doi: 10.1016/0022-2836(85)90383-3.
Recent experimental data of Shore & Baldwin (1983b) 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 nbp (the number of base-pairs) is nearly a constant for nbp approximately greater than 2000, but that it increases sharply with decreasing nbp for nbp approximately less than 2000. The main purpose of the present paper is to explain theoretically such behavior of K as a function of nbp. 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 (Wr2) increases rather steeply with increasing L. With these and known Monte Carlo results for freely jointed chains, an empirical interpolation formula for (Wr2) is also constructed to be valid for all values of L. It predicts that (Wr2)/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 lambda-1 (which is equal to the Kuhn segment length and twice the persistence length for this special case), and (Wr2). The derived equation predicts that nbpK decreases monotonically to its coil-limiting value with increasing nbp. 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 nbp approximately equal to 200. From an analysis of the experimental data of the two groups, the estimates of C = 3.1 to 3.2 X 10(-19) erg cm and lambda-1 = 1000 to 1200 A are obtained.
肖尔和鲍德温（1983b）以及霍洛维茨和王（1984）关于表观扭曲系数K的近期实验数据表明，K决定了具有不同连接数N的DNA拓扑异构体高斯分布的宽度。对于碱基对数（nbp）约大于2000的情况，K与nbp的乘积几乎是一个常数，但对于nbp约小于2000的情况，该乘积会随着nbp的减小而急剧增加。本文的主要目的是从理论上解释K作为nbp函数的这种行为。因此，一般DNA拓扑异构体的统计力学是基于扭曲的蠕虫状链发展而来的，即螺旋蠕虫状链的一种特殊情况。先前仅对小链长L有效的关于N依赖的闭环概率（即N的分布）的处理方法被扩展到更大L的范围。然后表明N的方差恰好是缠绕数Wr和扭曲数Tw的方差之和。对于小的L值，Wr的分布不是高斯分布，其方差或矩（Wr2）随着L的增加而相当急剧地增加。利用这些以及自由连接链的已知蒙特卡罗结果，还构建了一个关于（Wr2）的经验插值公式，该公式对所有L值都有效。它预测（Wr2）/L随着L的增加单调增加，直至达到其卷曲极限值。另一方面，在所有L值的N的实际范围内，N的分布实际上是高斯分布。还评估了固定N时Wr的条件分布。最后，K用扭转常数C、刚度参数lambda - 1（对于这种特殊情况，它等于库恩链段长度且是持久长度的两倍）以及（Wr2）来表示。推导的方程预测，随着nbp的增加，nbpK单调减小至其卷曲极限值。这种减小源于Wr的波动，即使对于nbp约等于200的短DNA，忽略这种波动也会导致C被低估7%至10%。通过对这两组实验数据的分析，得到C = 3.1至3.2×10^(-19)尔格·厘米和lambda - 1 = 1000至1200埃的估计值。
