Simulation of DNA catenanes

Abstract

DNA catenanes are important objects in biology, foremost as they appear during replication of circular DNA molecules. In this review we analyze how conformational properties of DNA catenanes can be studied by computer simulation. We consider classification of catenanes, their topological invariants and the methods of calculation of these invariants. We briefly analyze the DNA model and the simulation procedure used to sample the equilibrium conformational ensemble of catenanes with a particular topology. We consider how to avoid direct simulation of many DNA molecules when we need to account for the linking-unlinking process. The simulation methods and their comparisons with experiments are illustrated by some examples. We also describe an approach that allows simulating the steady state fraction of DNA catenanes created by type II topoisomerases.

Figure 1

Links of two circular contours with less than seven intersections in the standard form.

Figure 2

Definition of the linking number of two closed contours. We stretch an imaginary surface on one contour and then calculate the algebraic number of intersections between this surface and the other contour. Lk = 1 for the shown example.

Figure 3

On the calculation of an Alexander polynomial for links. The projection is shown with all generators and crossing points.

Figure 4

The two types of crossings. The directions of the segments are defined by the arbitrarily chosen directions for each of the circular chains.

Figure 5

The model of double stranded DNA. The length of the cylinders can vary, although it usually equals 30 base pairs of the double helix (1/5 of DNA persistence length). The diameter of the cylinders corresponds to the effective diameter of the double helix that exceeds DNA geometrical diameter.

Figure 6

The chain displacement in the course of the Metropolis procedure. The subchain is rotated by a random angle around the axis passing through two randomly chosen vertices.

Figure 7

The links between nicked circular DNA molecules. (a) Typical conformations of DNA molecules 4 kb in length forming link 2₁. In this simulation one straight segment of the model chain corresponded to 10 bp. (b) The computed fractions of overlapped molecules (gray circles), and molecules forming link 2₁ (filled black circles) and 4₁ (opened circles). The calculation corresponds to the concentration of monovalent salt of 0.2 M.

Figure 8

Measured and simulated probabilities of catenation as a function of supercoiling. The experimental values of B₂₁ (open symbols) are shown together with calculated results (filled symbols) for NaCl concentrations of 0.02 M (◇, ◆) and 0.2 M (○, ●). The large changes of conformational properties of supercoiled DNA with ionic conditions result, in good approximation, from the change of intersegment electrostatic interactions, specified by DNA effective diameter.

Figure 9

Topological interaction of circular polymer molecules. Simulated second virial coefficient of two unlinked chains is plotted as a function of the chain length (filled circles and the solid line). To emphasize the topological nature of the chain repulsion the modeled chains had zero thickness. For comparison, the second virial coefficient of rigid beads in which diameter equals the average radius of gyrartion of the corresponding circular chains is also shown (dotted line).

Figure 10

Supercoiling of circular DNA induced by the catenation. In the experiment, circular DNA molecules that form torus catenanes with the average linking number Ca were nicked to remove all torsional stress from each molecule and then treated by DNA ligase to reseal the nicks. After nick ligation one DNA of each link was linearized and the supercoiling of the intact circular molecule, ΔLk, was measured. The measured values of ΔLk (opened circles) are plotted together with the simulated values of the average induced writhe in the torus links, 〈Wr〉. The simulation results were taken from ref. (30); they correspond to DNA molecules 3.5 kb in length used in the experiment.

Figure 11

The model of type IIA topoisomerase action. The enzyme (green) bends the G segment (light brown) of DNA into a hairpin-like conformation. The entrance gate for the T segment of DNA (dark brown) is inside the hairpin. Thus, the T segment can pass through the G segment only from inside to outside the hairpin. Different stages of the reaction are separated by the black arrows.