Torsional Rigidity of DNA and Length Dependence of the Free Energy of DNA Supercoiling

By analyzing the Boltzmann populations of DNA topoisomers that differ only in their linking numbers, the dependence of the free energy delta G tau of DNA supercoiling on the linking number alpha has been determined for DNA rings as small as 200 base-pairs (bp) in length. All experimental data can be fitted by the relation delta G tau = K (alpha-alpha)2, where alpha is a constant for a given DNA at a given set of conditions and K is a DNA length-dependent proportionality constant. For DNA rings with length N larger than 2000 bp, K is inversely proportional to N and the product NK is nearly a constant around 1150 RT X bp. For rings smaller than 2000 bp NK increases steadily with decreasing N; for a 200 bp ring NK is 3900 RT X bp. The increase in NK when N decreases can be interpreted as a result of the decrease in the contribution of the fluctuation in the writhing number to the equilibrium distribution in alpha. Assuming that the writhing contribution approaches zero for DNA rings 200 bp in size, the torsional rigidity of the DNA double helix is calculated to be 2.9 X 10(-19) erg cm. In addition, the large value of K for the small circles allows precise calculation of the helical repeat of DNA. For the 210 bp rings, the repeat is measured to be 10.54 bp.