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- Summary
- Introduction
- Classification of Links
- Identification of Topological States
- Computation of Δ(s,t)
- Simulation of the Conformational Ensemble of Circular DNA Molecules
- Modeling Topological Equilibrium
- Comparing the Simulation Results with Experimental Data
- Topological Interaction Between Circular Molecules
- Conformational Properties of Linked Circular DNA
- Topology Simplification by Type II Topoisomerases
- Concluding Remarks
- References

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Phys Chem Chem Phys. Author manuscript; available in PMC 2010 December 7.

Published in final edited form as:

Published online 2009 October 23. doi: 10.1039/b910812b

PMCID: PMC2845312

NIHMSID: NIHMS183243

corresponding author: Email: ude.uyn@iiksdogolov.xela

The publisher's final edited version of this article is available at Phys Chem Chem Phys

See other articles in PMC that cite the published article.

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.

DNA topology can be divided into three levels: Supercoiling of individual circular molecules, knots in circular DNA, and formation of links, or catenanes, between two or more DNA molecules. DNA catenanes, as objects, are very important in biology. The complementary strands of circular DNA are highly linked and cannot be replicated without the permanent unlinking of newly synthesized daughter molecules by topoisomerases.^{1}^{–}^{4} Despite the highly potent topoisomerase activity, however, catenanes of the double-stranded daughter DNAs are formed at the end of the replication, when unreplicated region of circular DNA becomes too small for the topoisomerase action on the parental double helix.^{1} The catenanes between circular DNA molecules can serve as a model of very long, interwound linear DNA molecules which appear during the replication of linear chromosomes. Catenanes also appear as reaction products of site-specific recombination.^{5}^{–}^{6} A good understanding of catenane conformational properties is important in studying topoisomerases.^{2}^{–}^{3}^{,}^{7} A system of braided DNA molecules, as a model of catenanes, has been developed in single-molecular studies.^{8}^{–}^{11} Over the years, various computational methods have been developed to simulate the conformational and thermodynamic properties of DNA catenanes. These methods are a topic of this review.

This review will focus on DNA catenanes, with an emphasis on problems specific to their simulation. We will start from a brief description of mathematical classification of links between two circular contours and from methods of topology determination for a given conformation of two circular chains. We will then consider simulation of the equilibrium conformational ensemble of circular DNA molecules. One of the challenging aspects of simulation related to DNA catenanes arises from the fact that catenation is a bimolecular reaction and therefore its direct modeling is very time consuming. We consider how this problem can be bypassed in the simulation of the equilibrium fraction of catenanes and interaction between unlinked circular chains. Finally, we address the calculation of steady state fraction of catenanes which can be produced by type II DNA topoisomerases.

There is an infinite number of topologically distinguished links between two closed contours. For classification, links must be deformed to obtain their standard projection onto a plane. In its standard form, a link projection is such an image when the minimum number of intersections on the projection is achieved. Of course, there should be no self-intersection of the chains in the course of the link deformation. Deformations of this kind are known in topology as isotopic deformations. Two links which can be transformed into each other by way of isotopic deformation belong to the same isotopic type. There are tables of all types of two-contour links with a particular number of crossings in their standard form.^{12} The number of different types of links grows very fast when this minimum number of crossings increases. The table which includes all types of links with up to six crossings in the standard form is shown in Fig. 1.

There is one class of links which is especially important in biology—the class of torus links.^{13} The torus links can placed on the surface of a torus without any intersections of the contours. The links between complementary strands of the double helix in closed circular unknotted DNA belong to this class. Correspondingly, the links of this class are formed between daughter DNAs during replication. Although the fraction of torus links among more complex links is very small, three links shown in Fig. 1, ,22_{1}, 4_{1} and 6_{1} belong to the torus class. Links of the torus class can be distinguished among themselves by a single descriptor, *the linking number*. In DNA-related literature, two different notations are used for the descriptor: *Lk* for the linking number of DNA complementary strands and *Ca* for the linking number of two double-stranded circular molecules. We will follow this rule throughout this review but will use *Lk* for the linking number of two curves as mathematical objects. One way to define *Lk* is to calculate the number of turns one contour makes around the other when the first contour has a flat conformation.

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.

There is a more general definition of *Lk* which does not require a flat configuration of either contour. Let us imagine a surface defined by the edge of one of the contours (any such surface gives the same result). *Lk* is the algebraic (i.e., sign-dependent) number of intersections between the other contour and this spanning surface (Fig. 2). It is clear from the definition that the value of *Lk* is always an integer. By convention, *Lk* of a closed circular DNA formed by a right-handed double helix is defined as positive.

*Lk* can be also defined through the Gauss integral:

$$Lk=\frac{1}{4\pi}\underset{{C}_{1}}{\oint}\underset{{C}_{2}}{\oint}\frac{(\text{d}{\mathbf{r}}_{1}\times \text{d}{\mathbf{r}}_{2})\xb7{\mathbf{r}}_{12}}{{r}_{12}^{3}}$$

(1)

where **r**_{1} and **r**_{2} are vectors, starting in an arbitrary chosen point, whose ends run, upon integration, over the first and second contours, C_{1} and C_{2} respectively, **r**_{12}= **r**_{2}−**r**_{1}.

Suppose we want to determine the type of link formed by two highly coiled closed chains. To achieve this, we need an algorithm of verification of the topological identity of the conformation in question. Topologists developed many of such algorithms based on invariants of the topological states, characteristics that remain unchanged with any isotopic deformations of the chains, which are possible without the disruption of their integrity^{14}. The simplest invariant of a topological state is the linking number of two chains. Of course, to be efficient in the classification of the chains a topological invariant must assume different values for different topological states. Not a single topological invariant meets this requirement in full measure, but there are very powerful ones among them that help identify many elementary types of links and distinguish them from more complex ones. The linking number is a fairly weak topological invariant because it does not distinguish many linked states of two chains from the unlinked state. Still, the linking number is very useful, especially in DNA-related applications, since it uniquely identifies all links of the torus class. It is also convenient for theoretical treatment of catenane statistical properties, and has been widely used there.^{15}^{–}^{27}

A more powerful invariant, which is very convenient for computer simulation, is the Alexander polynomial.^{28} In the case of links between two circular contours, it is a polynomial of two variables. The great majority of the simplest links have different Alexander polynomials, Δ(*s*,*t*), and thus the invariant allows us to distinguish all of the simplest links from one another and from the unlinked chains. For unlinked chains Δ(*s*,*t*)= 0, the polynomials for the simplest links are given in Table 1. Δ(*s*,*t*) for all links with less then 10 crossings in the standard form can be found in ref. (^{12}). A few Monte Carlo studies have used the Alexander polynomial to identify the topological states of the two chains.^{4}^{,}^{29}^{–}^{34} There are more powerful invariants than Δ(*s*,*t*) but their calculations are very time-consuming.^{35}

The value of Δ(*s*,*t*) for a particular conformation of two circular chains can be calculated in the following way.^{29} First, one projects both chains on to a plane along an arbitrarily chosen axis, while drawing breaks at the crossing points in the part of the chains that lies below the other part (Fig. 3). The projection of the chains amounts to the set of curves, which are called the generators. Let us arbitrarily choose the directions of chains and renumber the generators correspondingly, selecting arbitrarily the starting points on both contours. The crossing that separates the *k-*thand (*k* + l)-th generators will be called the *k*-th crossing. Renumbering of the generators, *x _{k}*, and the corresponding crossings starts in one contour and continues in the other. We denote the number of crossings on the first contour by

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

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

All elements of the matrix except *a _{kk}*,

*k*≤*M, M*> 1- for
*i*=*k*or*i*=*k*+ 1*a*= −1,_{kk}*a*_{kk}_{+1}= 1 independent of the type of crossing;

- for
*i*≠*k, i*≠*k*+1,*i*≤*M**a*= 1,_{kk}*a*_{kk}_{+1}= −*s; a*=_{ki}*s*−1 for type I crossing,*a*= −_{kk}*s*,*a*_{kk}_{+1}= 1;*a*=_{ki}*s*−1 for type II crossing;

- for
*i*>*M**a*= 1,_{kk}*a*_{kk}_{+1}= −*t; a*=_{ki}*s*−1 for type I crossing,*a*= −_{kk}*t*,*a*_{kk}_{+1}= 1;*a*=_{ki}*s*−1 for type II crossing;

*k*=*M*= 1*; i*>*M**a*= 1−_{kk}*t*,*a*=_{ki}*s*−1 independent of the type of crossing;

*k*>*M; N*>*M*+ 1- for
*i*=*k*or*i*=*k +*1*a*= −1,_{kk}*a*_{kk}_{+1}= 1 independent of the type of underpass;

- for
*i*≠*k, i*≠*k*+1,*i*>*M**a*= 1,_{kk}*a*_{kk}_{+1}= −*t; a*=_{ki}*t*−1 for type I crossing,*a*= −_{kk}*t*,*a*_{kk}_{+1}= 1;*a*=_{ki}*t*−1 for type II crossing;

- for
*i*≤*M**a*= 1,_{kk}*a*_{kk}_{+1}= −*s; a*=_{ki}*t*−1 for type I crossing,*a*= −_{kk}*s*,*a*_{kk}_{+1}= 1;*a*=_{ki}*t*−1 for type II crossing;

*k*=*N; N*=*M +*1,*i*≤*M**a*= 1−_{kk}*s*,*a*=_{ki}*t*−1 independent of the type of crossing;

These relationships hold under conditions that for *k* = *M* one makes the substitution (*k*+1)→1 and for *k* = *N* the substitution (*k*+1) → *M*+1. It is also assumed that there is at least one crossing in each contour formed by generators from different contours – in other case the contours are certainly unlinked.

To obtain Δ(*s*,*t*) one has to calculate a minor *A _{kj}* of order

It is usually sufficient to calculate Δ(−1, −1) for the majority of problems related to circular DNA molecules. The analysis shows that Δ(−1, −1) is an essentially more powerful topological invariant than *the linking number* (which equals Δ(1,1)).

Checking topology of a circular chain may be the most time consuming part of the whole calculation. Therefore this part of the computer program deserves the most attention in terms of its efficiency. In this connection a procedure of reducing the order of the Alexander matrix before calculating the polynomial by eliminating trivial intersections is very useful (for details, see refs. (^{36}^{–}^{38})).

Nearly any problem related to large-scale conformational properties of DNA requires simulation of the equilibrium conformational ensemble for the molecules. The computational methods for such simulation are well developed and carefully tested (reviewed in ref. (^{39})). We only briefly describe the simulation approach here.

The simplest model of double-stranded DNA that quantitatively describes large-scale DNA conformational properties represents a discrete analog of the wormlike chain. The model also accounts for DNA torsional rigidity, physical volume of the DNA segments and intersegment electrostatic interaction.^{40}^{–}^{42} A DNA molecule composed of *n* Kuhn statistical lengths (≈ 300 bp) is modeled as a closed chain consisting of *kn* rigid segments that are cylinders of equal length, *l*. The value of *k* specifies the number of the rigid segments per Kuhn length and is a computational parameter (Fig. 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 **...**

The bending elastic energy of the chain, *E _{b}*, is computed as

$${E}_{b}=1/2{k}_{B}Tg\sum _{i=1}^{kn}{\theta}_{i}^{2},$$

(2)

where the summation extends over all the joints between the elementary segments; θ* _{i}* is the angular displacement of segment

$$g=a/l.$$

(3)

Eq. (3) assumes that *l* *a;* in general case the relation between *l* and *a* is described in ref. (^{40}).

Replacement of the continuous wormlike chain by the discrete chain consisting of *kn* hinged rigid segments is an approximation that improves as *k* increases. Since the computer time needed for a simulation increases approximately as (*kn*)^{2}, it is necessary to choose a value of *k* that is large enough to ensure reliable results but small enough to keep the computational time reasonable. The minimal value of *k* that provides the limiting properties of the wormlike chain depends on the property of interest. It has been found that the majority of the simulated properties of DNA links do not depend on *k* if *k* ≥ 10.^{30}^{,}^{42} If *k* equals 10, one straight segment of the model chain corresponds to ≈30 bp and *g* equals 4.81.

The excluded volume effect and electrostatic interactions between DNA segments are taken into account in the model via the concept of the effective diameter, *d*. This is the diameter of impenetrable uncharged cylindrical segments of the model chain. The quantitative definition of *d* is based on the concept of the second virial coefficient.^{43} It was shown that approximation of the electrostatic interaction by this hard core potential and by more realistic Debye-Hückel potential give very similar results in the Monte Carlo simulations, given that the second virial coefficients of the chain segments corresponding to the potentials have the same values.^{44}

The model’s features specified above are sufficient to simulate large-scale DNA conformational properties, both for linear DNA and circular DNA with a single-stranded nick (nicked DNA), because in these cases bending and torsional deformations of DNA are independent. However, dependence of the chain energy on the DNA twist, *Tw*, is crucial for properties of closed circular DNA. These simulations were described in detail in ref. (^{42}).

There are three parameters of the model that specify equilibrium properties of the double helix; each of these has been determined in independent studies. The first parameter, the DNA persistence length that defines the bending rigidity constant, is close to 50 nm for solutions containing more than 0.01 M monovalent ions or more than 1 mM multivalent ions.^{45}^{–}^{47} The second parameter is the DNA torsional rigidity, *C*. The value of 3·10^{−19} erg·cm for *C* seems to be the most reliable for this range of ionic conditions.^{48}^{–}^{52} The third parameter, the DNA effective diameter, *d,* depends strongly on ionic conditions. Accurate values of *d* have been determined in the experimental and theoretical studies.^{43}^{,}^{53}^{–}^{55} For near-physiological ionic conditions, *d* is close to 5 nm.

The model described above is the simplest one that can provide a quantitative description for a large number of DNA conformational properties. It can be easily extended to cases in which a group of the chain segments forms a specific conformation induced by a protein binding. However, a more elaborated model that explicitly accounts for the torsional orientation of each segment is required for the analysis of DNA molecules with two or more groups of bent segments distributed along the chain contour. Such a model and the corresponding simulation procedures have been developed.^{56}^{–}^{57} Small variations of the basic model were used in some studies,^{58}^{–}^{60} foremost to adjust it for the dynamic simulations. These variations do not change the simulation results, however, if the parameters are properly chosen (see ref. (^{39}) for review). Of course, the model accounts only for elastic deformation of the double helix. This restriction means, essentially, that it cannot describe the properties of strongly negatively supercoiled DNA molecules (|σ| < −0.06). Indeed, under strong negative supercoiling, DNA undergoes structural transitions that drastically alter its conformational properties (see refs. ^{61}^{–}^{62} for reviews). Such cases are beyond the scope of this review.

Sampling the equilibrium ensemble of DNA catenanes is performed in the same way as for isolated circular molecules. The Metropolis Monte Carlo procedure is used to achieve this goal. The procedure consists of consecutive steps of displacements of certain parts of each chain. Usually the displacement consists of the rotation of an arbitrary number of adjacent segments by a random angle within the interval (−ϕ_{0},+ϕ_{0}) around the straight line connecting two randomly chosen vertices, *m*_{1} and *m*_{2} (Fig. 6).^{50} The ϕ_{0} value is adjusted during the simulation in such a way that about one half of the steps would be successful.

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.

Whether or not the new conformation is accepted is determined by two rules: (a) If the energy of the new conformation *E _{new}* is lower than the energy of the previous conformation,

In many cases, we need to preserve the topology of the circular molecules during the simulation. Clearly, the chain displacement, as described, can alter the topology of the chains. If the topology is changed during displacement, the new conformation has to be rejected. The topology test is performed by calculating Δ(−1, −1) for the pair of chains.

In a solution of closed DNA molecules we always have a certain distribution of topological states: each molecule can be knotted or unknotted, linked with another molecule(s) or unlinked. Among different distributions there is one of special importance—the equilibrium distribution. The definition of the equilibrium distribution of topological states does not differ from those for any other property. This is the most probable distribution, which is reached by random exchange between different states in solution. Thus, the distribution minimizes the free energy of the polymer molecules. We would get the equilibrium distribution of topological states if our circular molecules had phantom backbones so that their segments could pass through one another during thermal motion. Thus, to calculate the distribution, we should forget that the exchange between different topological states is forbidden and use the usual rules of statistical mechanics. This means that, regardless of the topology, the probability of state *i* of circular DNA molecules in solution, *P _{i}*, follows to the Boltzmann distribution:

$${P}_{i}\propto exp(-{E}_{i}/{k}_{\text{B}}T),$$

(4)

where *E _{i}* is the energy of both of the circular chains in state

To calculate the equilibrium distribution we can choose a volume of the solution that includes many DNA molecules and sample the states of these molecules. Then we can find the fractions of different links by analyzing the simulated states. This way of addressing the problem requires very extensive computation, however. If the solution of DNA molecules is sufficiently diluted, the problem can be greatly simplified. In this case, the states of each circular molecule are independent of the states of other molecules, so the conformational distribution for individual molecules corresponds to the distribution for a single isolated DNA. Here we consider a case of moderate DNA concentrations, so that at equilibrium a small fraction of the molecules is topologically linked to another circular DNA. We will neglect the possibility of links between three or more circular molecules, which appear at higher DNA concentration. Therefore, we can analyze only pairs of circular molecules.

Under this condition, the equilibrium fraction of circular molecules forming a link of type *i* with another molecule, or equilibrium probability that a molecule forms a type *i* link, *P _{i}*, can be calculated as

$${P}_{i}=\underset{0}{\overset{\infty}{\int}}\frac{{cN}_{A}}{M}{p}_{i}(R)4\pi {R}^{2}dR,$$

(5)

where *c* is the DNA concentration in (w/v); *p _{i}* (

To calculate *p _{i}* (

Let us consider an example of such a calculation performed for nicked circular DNA molecules 4 kb in length at ionic conditions close to the physiological (≈ 0.2 M of NaCl). Typical simulated conformation of two model chains which form 2_{1} link is shown in Fig. 7a. Under these ionic conditions many mutual orientations of the molecules are forbidden due to the overlap between segments at small values of *R*. The corresponding probability, *p*_{int} (*R*) is shown in Fig. 7b together with the probabilities to form links 2_{1} and 4_{1}, *p*_{21} (*R*) and *p*_{41} (*R*), respectively. The calculations were performed using a set of 20 random conformations; each pair of conformations was placed in 144 mutual orientations obtained by all possible 90° rotations of each conformation around the X, Y, and Z directions.

The links between nicked circular DNA molecules. (a) Typical conformations of DNA molecules 4 kb in length forming link 2_{1}. In this simulation one straight segment of the model chain corresponded to 10 bp. (b) The computed fractions of overlapped molecules **...**

It is convenient to introduce the equilibrium constant for formation of type *i* link, *B _{i}*:

$${B}_{i}=\underset{0}{\overset{\infty}{\int}}\frac{{N}_{A}}{M}{p}_{i}(R)4\pi {R}^{2}dR.$$

(6)

This allows us to express *P _{i}* as

$${P}_{i}={B}_{i}c.$$

(7)

The value of *B _{i}* does not depend on DNA concentration and reflects only the properties of a particular circular DNA and the solution conditions. Of course, Eq. (7) is valid only under the condition that

Recently this computational approach was applied in the extensive study of equilibrium catenation between circular polymer chains.^{34} The authors found, in particular, that the linking probabilities weakly depend on *R* if plotted as a function of *R*/*R _{g}*, where

Placing circular chains at a particular distance between their centers of mass is the most rigorous way to analyze the topological equilibrium. A simpler, although less rigorous way was widely used in the analytical studies of the equilibrium catenation. In this approach two circular chains are tethered by a short rigid linker.^{18}^{–}^{19}^{,}^{21}^{–}^{22}^{,}^{24}^{–}^{26} Clearly, the linker simplifies the analysis since the tethered chains, when unlinked, cannot diffuse away from one another. Recently Marko combined analytical treatment of the latter system with Monte Carlo simulations.^{27} In particular, he studied in detail how the free energy of the tethered circular chains depends on their *Lk*.

It is important that the values of *B _{i}* can be measured experimentally. We can obtain thermodynamic equilibrium over topological states by slow cyclization of linear molecules with long cohesive ends.

Rybenkov et al. used equilibrium catenation to probe the conformations of supercoiled DNA.^{31} In these experiments, linear DNA with cohesive ends (which was used at a very low concentration) was cyclized in the presence of relatively high concentration of unlinked supercoiled molecules. The cyclization resulted in the appearance of catenanes formed by supercoiled and relaxed circular DNAs. The equilibrium constants *B*_{21} for these catenanes were measured for various ionic conditions and superhelix density of the supercoiled molecules, σ. The same constants were calculated using the simulation described above. An important aspect of the calculation was the use of simulated equilibrium conformational sets of supercoiled molecules.

The measured and simulated values of *B*_{21} for solutions of two different concentrations of NaCl (0.02 M, 0.2 M) are shown in Fig. 8 as a function of σ.^{31} One can see from the figure that the simulated and measured values of *B*_{21} agree very well over the entire range of σ for both NaCl concentrations studied, even though the range of *B*_{21} values exceeds two orders of magnitude. The data make it clear that conformations of supercoiled DNA vary greatly over this range of sodium ion concentrations. This work proved that the simulation is capable of predicting conformational properties of both relaxed and supercoiled DNA molecules with very good accuracy.

There is an entropic repulsion between unlinked circular DNA molecules in solution. They repulse each other because many of their mutual conformations correspond to linked states that are forbidden to them. Similar repulsion exists between linear (and circular) polymer molecules because many of their mutual conformations would result in segment overlapping.^{66} Thus, we can consider the topological excluded volume of unlinked circular molecules.^{29} This excluded volume is specified by the second virial coefficient of two interacting chains,

$$B=\frac{1}{2}\underset{0}{\overset{\infty}{\int}}{p}_{\mathit{lin}}(R)4\pi {R}^{2}dR,$$

(8)

where *p _{lin}* (

Clearly, conformational properties of topologically linked DNAs differ from those of isolated circular molecules. The simulation methods described above allow one to assess the extent of such differences in a very straightforward way. To this end, one needs to prepare an initial conformation with a desired topology and keep this topology unchanged during the Monte Carlo sampling of the equilibrium ensemble. To achieve the latter goal a topological invariant of the circular chains has to be calculated during each Monte Carlo step and the trial conformation has to be rejected if the chain topology was changed. Such simulation was used to estimate the values of DNA writhe induced in the nicked circular molecules by torus links with other circular DNA molecules.^{30} The system was studied experimentaly,^{67} and therefore the simulation results could be compared with the measurements. In the experiment, torus links with spesific values of *Ca* between two DNAs were prepared by the site-specific recombination. These linked DNA molecules were then nicked to remove any torsional stress from each circular molecule. These linked DNAs kept some non zero writhe, however, which was induced by the chirality of torus links. This writhe was converted into supercoiling, Δ*Lk,* by ligating the nicks in the linked molecules. The values of the catenation-induced Δ*Lk* were measured after one DNA molecule of each link was linearized by a restriction enzyme. The results of these experiments are plotted in Fig. 10 together with the simulated values of writhe induced in the torus links. One can see from the figure that the calculations are in very good agreement with the measurements.

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 **...**

Recently Martínez-Robles et al. used this kind of simulation to calculate the free energy of DNA supercoiling for circular DNA molecules forming torus catenanes with nicked circular DNA.^{4} They found that the supercoiling free energy increases substantially when the linking number of catenanes grows.

We have discussed above how we can simulate equilibrium properties of DNA catenanes. Intriguingly, enzymatic reactions do not always produce equilibrium distribution of DNA topological forms. The computational methods described above can be also used to analyze such cases, when DNA topology is changed by enzymatic reactions.^{68}^{–}^{70} A challenging problem of this kind is related to the phenomenon of topology simplification by type II topoisomerases.^{7} Type II DNA topoisomerases catalyze passage of one DNA segment of the double helix through another (reviewed in ref. (^{71})). They bind a DNA segment (gate or G segment) and introduce a transient double-stranded break in this segment, then capture another segment of the same or another DNA molecule (T segment) and transport it through the break. The break is resealed after the passage of the T segment. Thus, these enzymes can change topology of circular DNA molecules, create knots and catenanes from circular DNA molecules as well as unknot and unlink them, and also change DNA supercoiling. It was discovered that the steady state fractions of knotted and linked circular DNA molecules created by these enzymes are many times smaller than those at thermodynamic equilibrium. This property of the enzymes makes clear biological sense, since removing links between DNA molecules is one of their major functions.

The discovery posed a puzzle that has attracted much attention in the biophysical community in the last 12 years. The problem is that topology is a global property of circular DNAs, and thus cannot be recognized by enzymes that can only interact with DNA locally. Therefore, there is no way for the topoisomerases to *uniquely* determine topological consequences of a particular strand-passing event. However, the *probability* of a particular local conformation of two juxtaposed segments, which participate in the strand-passing event, can depend on the DNA topology, and the enzymes can use this dependence. If some local conformation appears more often in knotted/linked molecules than in unknotted/unlinked and the enzyme uses it as a substrate, it would lead to simplification of DNA topology. Several models were proposed to explain this topology simplification.^{70}^{,}^{72}^{–}^{74} Essentially all of these models were based on an assumed correlation between local conformation of the DNA segments in the reaction complex and topology of the circular molecules (see ref. (^{75}) for review).

The following treatment of the problem was proposed for the analysis of *the model of the hairpin-like G segment*,^{70} although it can be used for other models of this kind as well. The model assumes that the enzymes create a sharp bend in the G segment, with a specific orientation of the enzyme relative to the bend. This complex with the bent G segment can provide a unidirectional passage of the T segment from the inside to the outside of the hairpin formed by the G segment (Fig. 11).^{70} This last assumption has strong experimental support, since it had been known that the enzymes catalyze the passage of a T segment in one direction relative to themselves.^{76} Of course, the directionality of the strand passage is only local, since the hairpin can have any orientation relative to the DNA chain.

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 **...**

Below we describe a way to calculate the steady state concentration of catenanes for a model of this kind. It is quite clear that direct simulations of many circular molecules in a compartment would be very time consuming. There is, however, an efficient way to perform the calculation that is based on the following treatment of the problem.

Let us assume that the reaction has reached its steady state, which is the condition when the concentration of 2_{1} catenanes, *c _{c}*, does not change (no other catenanes were observed in the experiments). We also assume that the rate of the strand passage reaction is not limited by diffusion, so that the equilibrium conformational distribution defines the rates of catenation and decatenation.

$$\frac{d{c}_{c}}{dt}=k{c}_{u}P(j\mid u)-k{c}_{c}P(j\mid c)=0,$$

(9)

where *k* is the enzyme-dependent rate constant, *c _{u}* is the steady state concentration of the unlinked circular molecules,

$$\frac{{c}_{c}}{{c}_{u}}=\frac{P(j\mid u)}{P(j\mid c)}=\frac{P(j,u)P(c)}{P(j,c)P(u)}.$$

(10)

Here *P*(*j*,*u*) and *P*(*j*, *c*) are the equilibrium unconditional (joint) probabilities that two events occur together, i.e. that the T-segment is properly juxtaposed with the enzyme-bound protein, and that this segment belongs to an unlinked or linked DNA, correspondingly; *P*(*u*) and *P*(*c*) are the equilibrium probabilities that a chosen circular molecule with the enzyme-bound G segment is unlinked or linked with another DNA. Since conformation of the protein-bound G segment does not notably affect global DNA conformations, we can approximate *P*(*u*) and *P*(*c*) with their values for the protein-free circular molecules. Thus, the ratio of these two probabilities can be related to the equilibrium constant of catenation (see Eq. (7)):

$$P(c)/P(u)=P(c)/(1-P(c))\cong P(c)={B}_{{2}_{1}}c.$$

(11)

Combining Eqs. (11) and (12), we obtain a relationship between the steady-state and equilibrium catenation levels:

$$\frac{{c}_{c}}{{c}_{u}}=R{B}_{{2}_{1}}c,$$

(12)

where $R=\frac{P(j,u)}{P(j,c)}$ is the proportionality constant that relates the steady state and equilibrium fractions of catenanes.

Using again the general relation between conditional and unconditional probabilities we can obtain a relationship which suggests a way for the efficient simulations:

$$R=\frac{P(j,u)P(j)}{P(j,c)P(j)}=\frac{P(u\mid j)}{P(c\mid j)}.$$

(13)

To calculate the conditional probabilities *P*(*u*|*j*) and *P*(*c*|*j*), we have to place a segment of the second circular chain in the proper juxtaposition with the protein-bound G segment of the first chain and calculate the ratio of linked and unlinked conformations while keeping the juxtaposed segments immobilized during simulations. This approach does not require long computations and allows one to calculate the value of *R* with high accuracy.

We considered here various methods developed for computer simulation of catenanes formed by circular DNA molecules. It is important that despite system complexity these methods allow one to perform the simulation by using regular desktop computers. We described a few problems in which these methods have been used in the studies of DNA and enzymes that change DNA topology. More such problems continue to emerge as we write this review.

Of particular interest in this respect are the problems related to the chirality of protein-DNA interactions. The ability of DNA-binding proteins to distinguish between left- and right- handed DNA crossings has long been recognized. Recent studies raised the intriguing possibility that the chirality-sensing mechanisms could also serve for topology recognition. The best-documented case here remains type 2 topoisomerases. These enzymes are far more efficient in relaxation of positively than negatively supercoiled DNA due to their preferred interaction with right-handed DNA nodes.^{9}^{,}^{11}^{,}^{79} Structural studies implicate the C-terminal domain of topoisomerases in both chirality sensing and topology simplification,^{80}^{–}^{81} indicating that the two activities might be inherently related. Curiously, another class of enzymes with a major role in global DNA organization, condensins, also exhibits a marked preference for right-handed DNA crossings.^{82}^{–}^{83} This raises the intriguing possibility that the coupling between topology- and chirality-sensing might be a common phenomenon in biology. The approach outlined in this review can serve as a solid foundation for a quantitative analysis of such problems.

The majority of experimental studies of topological links in polymer chains have been done with DNA molecules. There are two reasons of this. First, DNA catenanes often appear in biology and therefore represent an important biological object. Second, well-developed biochemical methods allow a level of sophistication in manipulation with DNA molecules that is hardly imaginable with other polymers. Thus, DNA catenanes can serve as a good model system for catenanes in polymer chains in general. However, many theoretical studies of catenanes address properties of polymer chains in general. These studies use many approximations and simplifications that are not necessary in the numerical analysis. The methods described in this review can be useful to one who wants to evaluate the importance of a particular approximation. A recent study by Marko gives a good example of this approach.^{27}

The work was supported by the National Institutes of Health grant GM54215 to AV and a grant from the Oklahoma Center for the Advancement of Science and Technology to VVR.

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