DNA supercoiling inhibits DNA knotting

Abstract

Despite the fact that in living cells DNA molecules are long and highly crowded, they are rarely knotted. DNA knotting interferes with the normal functioning of the DNA and, therefore, molecular mechanisms evolved that maintain the knotting and catenation level below that which would be achieved if the DNA segments could pass randomly through each other. Biochemical experiments with torsionally relaxed DNA demonstrated earlier that type II DNA topoisomerases that permit inter- and intramolecular passages between segments of DNA molecules use the energy of ATP hydrolysis to select passages that lead to unknotting rather than to the formation of knots. Using numerical simulations, we identify here another mechanism by which topoisomerases can keep the knotting level low. We observe that DNA supercoiling, such as found in bacterial cells, creates a situation where intramolecular passages leading to knotting are opposed by the free-energy change connected to transitions from unknotted to knotted circular DNA molecules.

Figure 1.

Topology and numerical simulation of supercoiled DNA. (A) Crossing sign convention. When the axis of circular DNA molecules is considered as an oriented curve (see B), each perceived crossing can be given a sign according to the rotation direction needed to align the imaginary arrow on the overlying segment with the imaginary arrow on the underlying segment while the rotation can not exceed 180°. Positive crossings require a counter-clockwise rotation to align the arrows while the opposite applies to negative crossings (2). In writhe (Wr) calculations, positive crossings score as 1 and negative as −1. (B) Writhe of a given projection (2D writhe) and a global Wr. The same rigid configuration of supercoiled DNA can have different 2D writhe depending on the direction of the projection. In a lateral projection the molecule reveals two negative crossings, but in a ‘tilted’ projection one observes an additional positive crossing. The 3D global writhe is usually denoted as Wr and is the average over all 2D writhe values. (C) Differences between topological and physical consequences of Topo II-mediated passages and those occurring in standard Monte Carlo simulations. In standard Monte Carlo simulations, the linking number of modeled DNA does not change after intersegmental passage. This contrasts with the physical and biological fact that such intersegmental passages change the linking number by 2. Energetic and topological consequences of intersegmental passages in standard Monte Carlo simulations (upper pathway) are compared to consequences of real Topo II-mediated passages (lower pathway). Notice that although the minimal move that results in an intersegmental passage changes the Wr value by nearly 2, the writhe change is different between an equilibrium state before the passage and an arbitrary state after the passage.

Figure 2.

Comparison of probabilities of knotting in simulations that keep the ΔLk constant and those that maintain the same effective level of DNA supercoiling. (A) Conditional probability profiles of various knots obtained in numerical simulations where ΔLk was kept constant. (B) Snapshot of knot 10₁₂₄ that has a torsionally relaxed appearance despite having ΔLk = −12. (C) Snapshot of the unknot with ΔLk = −12 reveals that supercoiling is present. (D) Conditional probability profiles of various knots obtained in numerical simulations where ΔLke was kept constant. The logarithmic scale shows that knotting is many orders of magnitude lower than knotting in simulations shown in (A). Inset in (D) shows how changes of the effective diameter of modeled DNA affect the probability of trefoil knot formation for a given ΔLke.

Figure 3.

Energetic and topological consequences of a Topo II- mediated strand passage leading to conversion of supercoiled unknot into knot (upper pathway) as compared to corresponding knotting event in standard Monte Carlo simulations (lower pathway).

Figure 4.

Comparison of energy difference between supercoiled unknot and left-handed trefoil knots with corresponding ΔLk. The light blue arrow indicates the energy difference between unknot and trefoil knot with the same ΔLke. The dark blue arrow indicates the energy difference between the unknot and a trefoil knot resulting from one round of a Topo II-like action. The black arrow indicates the energy difference between the unknot and a trefoil knot formed under the unrealistic assumption that topoisomerase II could mediate an intramolecular passage reaction without changing the linking number. The energy values were obtained by simulations of 3 kb DNA molecules that formed unknotted circles or left-handed trefoil knots, respectively. Note that left-handed trefoil knots reach their minimal energy state for ΔLk ≈ −3.5 and that this closely corresponds to the average writhe of torsionally relaxed left-handed trefoil knots (40–42) which by definition would have ΔLke = 0.