Biopolym. Cell. 1988; 4(2):85-90.
Structure and Function of Biopolymers
Topological characteristics of circular DNA: excluded volume effects
1Klenin K. V., 1Vologodskii A. V., 1Anshelevich V. V., 2Dykhne A. M., 1Frank-Kamenetskii M. D.
  1. Institute of Molecular Genetics, Academy of Sciences of the USSR
    Moscow, USSR
  2. I. V. Kurchatov Institute of Atomic Energy
    Moscow, USSR


The Monte Carlo method is used to calculate the probability p of knots formation, the variance of the writhing number <Wr2> and the expansion factor α for a closed polymer chain as a function of its effective diameter d. The results are presented in a form of simple interpolation equations and applied for DNA. The dependence of DNA super-helix energy on its effective diameter is evaluated. Theory predicts significant dependence of the superhelix energy on ionic strength.


[1] Brian AA, Frisch HL, Lerman LS. Thermodynamics and equilibrium sedimentation analysis of the close approach of DNA molecules and a molecular ordering transition. Biopolymers. 1981;20(6):1305-28.
[2] Yarmola EG, Zarudnaya MI, Lazurkin YuS. Osmotic pressure of DNA solutions and effective diameter of the double helix. J Biomol Struct Dyn. 1985;2(5):981-93.
[3] Stigter D. Interactions of highly charged colloidal cylinders with applications to double-stranded. Biopolymers. 1977;16(7):1435-48.
[4] Hagerman PJ. Investigation of the flexibility of DNA using transient electric birefringence. Biopolymers. 1981;20(7):1503-35.
[5] Kirste RG. Radius of gyration of stiff chain molecules as a function of the chain length and the interactions with the solvent. Discuss Faraday Soc. 1970; 49:51-59.
[6] Yamakawa H, Stockmayer WH. Statistical Mechanics of Wormlike Chains. II. Excluded Volume Effects. J Chem Phys. 1972; 57(7):2843-2854.
[7] Webman I, Lebowitz JL, Kalos MH. Excluded-volume expansion of polymer chains: a Monte Carlo study of the scaling properties. Phys Rev B Solid State. 1980. 21(12):5540-5543.
[8] Slonitskii SV, Frisman EV, Valeev AD, El'iashevich AM. Calculation of the intrinsic viscosity of synthetic and biological polyelectrolytes of various rigidity. Mol Biol (Mosk). 1980;14(3):484-95.
[9] Manning GS. A procedure for extracting persistence lengths from light-scattering data on intermediate molecular weight DNA. Biopolymers. 1981; 20(8):1751-5.
[10] Post CB. Excluded volume of an intermediate-molecular-weight DNA. A Monte Carlo analysis. Biopolymers. 1983;22(4):1087-96.
[11] Vologodskii AV, Anshelevich VV, Lukashin AV, Frank-Kamenetskii MD. Statistical mechanics of supercoils and the torsional stiffness of the DNA double helix. Nature. 1979;280(5720):294-8.
[12] Le Bret M. Monte Carlo computation of the supercoiling energy, the sedimentation constant, and the radius of gyration of unknotted and knotted circular DNA. Biopolymers. 1980;19(3):619-37.
[13] Shimada J, Yamakawa H. Ring-closure probabilities for twisted wormlike chains. Application to DNA. Macromolecules. 1984; 17(4):689-698.
[14] Shimada J, Yamakawa H. DNA-topoisomer analysis on the basis of the helical wormlike chain. Biopolymers. 1984;23(5):853-7.
[15] Frank-Kamenetskii MD, Lukashin AV, Anshelevich VV, Vologodskii AV. Torsional and bending rigidity of the double helix from data on small DNA rings. J Biomol Struct Dyn. 1985;2(5):1005-12.
[16] Depew DE, Wang JC. Conformational fluctuations of DNA helix. Proc Natl Acad Sci U S A. 1975;72(11):4275-9.
[17] Pulleyblank DE, Shure M, Tang D, Vinograd J, Vosberg HP. Action of nicking-closing enzyme on supercoiled and nonsupercoiled closed circular DNA: formation of a Boltzmann distribution of topological isomers. Proc Natl Acad Sci U S A. 1975;72(11):4280-4.
[18] Shore D, Baldwin RL. Energetics of DNA twisting. I. Relation between twist and cyclization probability. J Mol Biol. 1983;170(4):957-81.
[19] Shore D, Baldwin RL. Energetics of DNA twisting. II. Topoisomer analysis. J Mol Biol. 1983;170(4):983-1007.
[20] Horowitz DS, Wang JC. Torsional rigidity of DNA and length dependence of the free energy of DNA supercoiling. J Mol Biol. 1984;173(1):75-91.
[21] Frank-Kamenetskii MD, Vologodskii AV. Topological aspects of the physics of polymers: theory and its biophysical applications. Usp fiz nauk. 1981; 134(4):641-73.
[22] Vologodskii AV, Lukashin AV, Frank-Kamenetskii MD, Anshelevich VV. Problem nodes in statistical mechanics of polymer chains. Zh Eksp i Teor. Fiziki. 1974; 66(6):2153-2163.
[23] des Cloizeaux J, Mehta M. L.Topological constraints on polymer rings and critical indices. J Phys Lett. 1979; 40(7):665-70.
[24] Chen Y-D. Monte Carlo study of freely jointed ring polymers. II. The writhing number. J Chem Phys. 1981; 75(5):2447-53.
[25] Michels JPJ, Wiegel FW. Probability of knots in a polymers ring. Phys Lett A. 1982; 90(7):381-384.
[26] White JH. Self-Linking and the Gauss Integral in Higher Dimensions. Amer J Math. 196; 91(5):693-728.
[27] Fuller FB. The writhing number of a space curve. Proc Natl Acad Sci U S A. 1971;68(4):815-9.
[28] Spengler SJ, Stasiak A, Cozzarelli NR. The stereostructure of knots and catenanes produced by phage lambda integrative recombination: implications for mechanism and DNA structure. Cell. 1985;42(1):325-34.
[29] Vologodskii AV, Frank-Kamenetskii MD. Theoretical study of cruciform states in superhelical DNAs. FEBS Lett. 1982;143(2):257-60.
[30] Frank-Kamenetskii MD, Vologodskii AV. Thermodynamics of the B-Z transition in superhelical DNA. Nature. 1984 Feb 2-8;307(5950):481-2.
[31] Peck LJ, Wang JC. Energetics of B-to-Z transition in DNA. Proc Natl Acad Sci U S A. 1983;80(20):6206-10.
[32] Singleton CK, Klysik J, Stirdivant SM, Wells RD. Left-handed Z-DNA is induced by supercoiling in physiological ionic conditions. Nature. 1982;299(5881):312-6.
[33] Singleton CK. Effects of salts, temperature, and stem length on supercoil-induced formation of cruciforms. J Biol Chem. 1983;258(12):7661-8.
[34] Nordheim A, Rich A. The sequence (dC-dA)n X (dG-dT)n forms left-handed Z-DNA in negatively supercoiled plasmids. Proc Natl Acad Sci U S A. 1983;80(7):1821-5.