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 DeltaLk, the linking number difference from relaxed DNA. The equilibrium distribution of probabilities of these states, P(DeltaLk, K), is related to two conditional distributions: P(DeltaLk|K), the distribution of DeltaLk for a particular K, and P(K|DeltaLk) and also to two simple distributions: P(DeltaLk), the distribution of DeltaLk irrespective of K, and P(K). We explored the relationships between these distributions. P(DeltaLk, K), P(DeltaLk), and P(K|DeltaLk) were calculated from the simulated distributions of P(DeltaLk|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|DeltaLk), the distribution of knot types for a particular value of DeltaLk, which had not been evaluated previously. We found that unknotted circular DNA is not the most probable state beyond small values of DeltaLk. Highly chiral knotted DNA has a lower free energy because it has less torsional deformation. Surprisingly, even at |DeltaLk| > 12, only one or two knot types dominate the P(K|DeltaLk) 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.