Local selection rules that can determine specific pathways of DNA unknotting by type II DNA topoisomerases

Abstract

We performed numerical simulations of DNA chains to understand how local geometry of juxtaposed segments in knotted DNA molecules can guide type II DNA topoisomerases to perform very efficient relaxation of DNA knots. We investigated how the various parameters defining the geometry of inter-segmental juxtapositions at sites of inter-segmental passage reactions mediated by type II DNA topoisomerases can affect the topological consequences of these reactions. We confirmed the hypothesis that by recognizing specific geometry of juxtaposed DNA segments in knotted DNA molecules, type II DNA topoisomerases can maintain the steady-state knotting level below the topological equilibrium. In addition, we revealed that a preference for a particular geometry of juxtaposed segments as sites of strand-passage reaction enables type II DNA topoisomerases to select the most efficient pathway of relaxation of complex DNA knots. The analysis of the best selection criteria for efficient relaxation of complex knots revealed that local structures in random configurations of a given knot type statistically behave as analogous local structures in ideal geometric configurations of the corresponding knot type.

Figure 1.

Geometry of juxtapositons. Juxtaposition of two two-segment-long subchains with central vertices A₁ and A₂. To describe the geometry of the juxtapositions we use the distance d, the opening angles α₁ and α, the facing angle γ (between the bisectrices b₁ and b₂) and the geometrical chirality angle θ between unoriented tangential lines passing through the vertices A₁ and A₂ (see the Inset). To consider juxtapositions as inter-hooked the following three conditions have to be fulfiled: (i) each subchain intersects with a sector of the plane extending from angle a defined by the other subchain. (ii) The angle γ > 105°. (iii) the mean opening angle (α₁ + α₂)/2 < 22.5°. Inset: Geometric chirality between two unoriented lines in space. The two lines are projected on the plane that is parallel to both the lines while the projection conserves the information which of the lines was closer and which was further from the observer. Perpendicular lines are considered as achiral while the geometric chirality angle is the smallest angle of rotation applied to the overlying line in order to make it perpendicular to the underlying line. In the case of positive (right-handed) chirality, a clockwise rotation of the overlying line is required to make it perpendicular to the underlying line (shown), while a counterclockwise rotation would be required for lines with negative (left-handed) geometric chirality. The bigger the angle of the rotation, the higher the absolute value of the geometric chirality.

Figure 2.

Topological consequences of strand-passage reaction for random and hooked juxtapositions in trefoil knots (A) unknots (B) and 5₂ knots (C). Insets show standard diagrams of simple knots analysed here. As a notation, we use a modification of Alexander–Briggs designation where the main number indicates the minimal number of crossings a given knot can have in a projection and the subscript number indicates the tabular position of a given knot among knots with the same minimal crossing number in standard tables of knots (48,49). The _R and _L letters indicate left- or right-handed form of a given chiral knot. The observed topological outcomes were consistent with the theoretically predicted strand-passage distances between various knots (50,51). The error bars denote the SD (in case of all juxtapositions, the big statistical sample resulted in error bars smaller than the thickness of the lines demarking the histogram bars).

Figure 3.

Effects of various geometric characteristics of juxtapositions on the topological outcome of strand passages occurring at these juxtapositions in 5_2L knots. Notice the qualitative difference between effects of mean opening angle (A), the geometric chirality (B) and the distance between the vertices A₁ and A₂ (C).

Figure 4.

Inferring local geometric characteristics of random configurations of knots from the topological consequences of intersegemental passages. (A and B) 0₁/3₁ ratio arising from intersegmental passages within 5_2L knot as a function of the opening angle at both subchains forming the juxtaposition at which the strand passage occurs (A) or the mean opening angle and the chirality of the juxtaposition (B). (C) Ideal configuration of 5_2L knot with indicated topological domains within which intersegmental passages lead to unknotting (dark blue background) or to formation of 3_1L knots (yellow background) and to the formation of 5_1L knots (light blue background).