Equilibrium Distributions of Topological States in Circular DNA: Interplay of Supercoiling and Knotting

Two variables define the topological state of closed double-stranded DNA: the knot type, K, and Δ LK, the linking number difference from relaxed DNA. The equilibrium distribution of probabilities of these states, P(Δ LK,K), is related to two conditional distributions: P(Δ LK|K), the distribution of...

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Bibliographic Details
Published in:Proceedings of the National Academy of Sciences - PNAS 1999-11, Vol.96 (23), p.12974-12979
Main Authors: Podtelezhnikov, Alexei A., Cozzarelli, Nicholas R., Vologodskii, Alexander V.
Format: Article
Language:English
Subjects:
Bending
Circular DNA
Deoxyribonucleic acid
DNA
DNA, Circular - chemistry
Enzymes
Free energy
Gaussian distributions
Genetic equilibrium
Knot tying
Knots
Mathematics
Models, Chemical
Molecules
Monte Carlo Method
Nucleic Acid Conformation
Physical Sciences
Topology
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Summary:Two variables define the topological state of closed double-stranded DNA: the knot type, K, and Δ LK, the linking number difference from relaxed DNA. The equilibrium distribution of probabilities of these states, P(Δ LK,K), is related to two conditional distributions: P(Δ LK|K), the distribution of Δ LK for a particular K, and P(K|Δ LK) and also to two simple distributions: P(Δ LK), the distribution of Δ LK irrespective of K, and P(K). We explored the relationships between these distributions. P(Δ LK,K),P(Δ LK), and P(K|Δ LK) were calculated from the simulated distributions of P(Δ LK|K) and of P(K). The calculated distributions agreed with previous experimental and theoretical results and greatly advanced on them. Our major focus was on P(K|Δ LK), the distribution of knot types for a particular value of Δ LK, which had not been evaluated previously. We found that unknotted circular DNA is not the most probable state beyond small values of Δ LK. Highly chiral knotted DNA has a lower free energy because it has less torsional deformation. Surprisingly, even at $|\Delta LK|>12$, only one or two knot types dominate the P(K|Δ LK) distribution despite the huge number of knots of comparable complexity. A large fraction of the knots found belong to the small family of torus knots. The relationship between supercoiling and knotting in vivo is discussed.
ISSN:0027-8424
1091-6490
DOI:10.1073/pnas.96.23.12974