Equilibrium distributions of topological states in circular DNA: Interplayof supercoiling and knotting

Two variables define the topological state of closed double-stranded DNA: t he knot type, K, and Delta Lk, the linking number difference from relaxed D NA. The equilibrium distribution of probabilities of these states, P(Delta Lk, K), is related to two conditional distributions: P(Delta LK\K), the dis tribution of Delta Lk for a particular K, and P(K\Delta Lk) and also to two simple distributions: P(Delta Lk), the distribution of Delta Lk irrespecti ve of K, and P(K), We explored the relationships between these distribution s. P(Delta Lk, K), P(Delta LK), and P(K\Delta Lk) were calculated from the simulated distributions of P(Delta Lk\K) and of P(K), The calculated distri butions agreed with previous experimental and theoretical results and great ly advanced on them. Our major focus was on P(K\Delta Lk), the distribution of knot types for a particular value of Delta Lk, which had not been evalu ated previously. We found that unknotted circular DNA is not the most proba ble state beyond small values of Delta 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 Delta\ 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 kn ots. The relationship between supercoiling and knotting in vivo is discusse d.