## Abstract

By performing strand-passages on DNA, type II topoisomerases are known to resolve topological constraints that impede normal cellular functions. The full details of this enzyme–DNA interaction mechanism are, however, not completely understood. To better understand this mechanism, researchers have proposed and studied a variety of random polygon models of enzyme-induced strand-passage. In the present article, we review results from one such model having the feature that it is amenable to combinatorial and asymptotic analysis (as polygon length goes to infinity). The polygons studied, called Θ-SAPs, are on the simple-cubic lattice and contain a specific strand-passage structure, called Θ, at a fixed site. Another feature of this model is the availability of Monte Carlo methods that facilitate the estimation of crossing-sign-dependent knot-transition probabilities. From such estimates, it has been possible to investigate how knot-reduction depends on the crossing-sign and the local juxtaposition geometry at the strand-passage site. A strong relationship between knot-reduction and a crossing-sign-dependent crossing-angle has been observed for this model. In the present article, we review these results and present heuristic geometrical arguments to explain this crossing-sign and angle-dependence. Finally, we discuss potential implications for other models of type II topoisomerase action on DNA.

- crossing-sign discrimination
- DNA topology
- knot-reduction
- self-avoiding polygon
- strand-passage model
- type II topoisomerase

## Introduction

Type II topoisomerases are enzymes found in every organism and they play essential unknotting, unlinking and supercoiling simplification roles that help to facilitate important cellular processes [1–4]. They accomplish this by passing one DNA segment through another (strand-passage). This strand-passage mechanism is reviewed in [5–7]. In fact, from the DNA experiments of Rybenkov et al. [8], “the steady-state fraction of knotted or catenated DNA molecules produced by prokaryotic and eukaryotic type II topoisomerases was found to be as much as 80 times lower than at thermodynamic equilibrium”. In this, the term steady-state refers to the distribution of knots and catenanes after a topoisomerase-catalysed reaction has reached its steady state, whereas the term equilibrium refers to the corresponding distribution after random cyclization of a linear duplex DNA with cohesive ends. Identifying the specific details of the topoisomerase strand-passage mechanism that lead to such high efficiency in knot or catenane reduction remains an open problem. In the present article, we focus on knot-reduction studies only.

In order to identify potential mechanisms, various random polygon models of strand-passage have been studied [9–19]. In [10], using a random lattice polygon model, numerical evidence indicates that knot-reduction depends on the local juxtaposition geometry at the strand passage site, with tighter hook or clasp-like juxtapositions leading to greater knot-reduction than other juxtaposition geometries. Further lattice and off-lattice model studies [11,13,15,17,18] indicate similar dependencies on the local juxtaposition geometry. Most recently in [20], using a lattice polygon model, numerical evidence is presented that indicates that knot-reduction can also depend on the crossing-sign (chirality) and the crossing-angle at the strand-passage site, with knot-reduction being greatest for acute crossing-angles. This complements experimental results [21] on the crossing-angle dependence of positive supercoil relaxation by type II topoisomerases. However, whether or not the crossing-sign and angle actually play roles in the topoisomerase strand-passage mechanism remains an open question. In the present article, we review the recent results of [20] and present heuristic geometrical arguments explaining the crossing-sign and angle-dependence of knotreduction. The implications of this for models of type II topoisomerase action on DNA are also discussed.

## The Θ-structure lattice strand-passage model

In order to compare with the experiments of Rybenkov et al. [8], one goal for random polygon model studies has been to estimate the knot-reduction factor, *R _{K}* [10], which is the ratio of knots-to-unknots at equilibrium divided by the ratio of knots-to-unknots at an enzyme- or strand-passage-induced steady-state. Note that a knot-reduction factor

*R*>1 indicates that the ratio of knots to unknots at the enzyme-induced steady state is smaller than at equilibrium and in this case we say that knotting has been reduced. For the simplest two-state model where DNA is either unknotted (φ) or knotted (

_{K}*K*),

*R*can be written as where

_{K}*t*

_{a}_{→b}is the one-step transition probability for going from state

*a*∈ {φ,

*K*} to state

*b*∈ {φ,

*K*} via a topoisomerase strand-passage, and

*P*

_{K}

^{eq}/

*P*

_{φ}

^{eq}is the ratio of knots-to-unknots at equilibrium. In [20], the focus was on comparison factors obtained by taking the ratio of two

*R*values obtained from two different strand-passage models that share the same equilibrium (this eliminates the common factor

_{K}*P*

_{K}

^{eq}/

*P*

_{φ}

^{eq}). The comparison factor can thus be used to determine which strand-passage mechanisms are better at knot-reduction.

The knot-reduction studies in [20] are based on a SAP (self-avoiding lattice polygon) model. On the macroscopic scale, circular DNA can be viewed simply as a ring polymer, and, for ring polymers in dilute solution, SAPs are known to be a good model [22]. Furthermore, self-avoiding lattice polygon models with an interaction term added to take into account the effect of salt concentration [23,24] have been shown to yield equilibrium knot distributions comparable with those obtained in the DNA experiments of [25,26]. Hence, despite the bond-angle limitations of the lattice, lattice models can yield results comparable with those in DNA experiments. In the strand-passage model of [20], a ring polymer's configuration is represented by a lattice polygon composed of consecutive nearest-neighbour edges on the simple cubic lattice ℤ^{3}. In the model, it is assumed that two strands of the polygon have already been brought close together at a pre-specified strand-passage site. Specifically, the ring polymer configurations are represented by a SAP which contains a specific structure, the so-called strand-passage structure Θ (Figure 1), and such polygons are called Θ-SAPs. (The structure of Θ was chosen initially because it is similar to the shape of the topoisomerase–DNA complex as first proposed in [27].) One-step knot-transition probabilities are calculated by performing strand-passage (as shown in Figure 1) on each Θ-SAP and determining the resulting knot type. Estimates of various *R _{K}* ratios (the comparison factors) can then be obtained from the one-step knot-transition probabilities. One advantage of the Θ-SAP model is that it is amenable to combinatorial and asymptotic analysis (as polygon length goes to infinity). Another feature of the model is the availability of a Monte Carlo method (the so-called Θ-BFACF algorithm) that facilitates estimation of crossing-sign-dependent knot-transition probabilities. Symmetry arguments also allow for crossing-sign-independent estimates from the same data.

For simplicity, the initial studies [19,20,28,29] of the Θ-SAP model considered ring polymers in a good solvent, and, in this case, ‘equilibrium’ corresponds to the case that each SAP with the same length (number of polygon edges) is an equally likely conformation. Different strand-passage models were then considered which were all based on the Θ-structure and the local juxtaposition geometry immediately adjacent to it. The crossing-sign determined by Θ's occurrence in a specific SAP was also taken into account. In particular, if Θ occurs in a SAP such that, using a right-hand rule, a positive (negative) crossing is formed at the strand-passage site, then the SAP is called a Θ^{+}-SAP (or Θ^{−}-SAP). To illustrate the model, Figure 2 shows two examples of juxtaposition geometries (denoted *S* for straight-top and *Z* for Z-top respectively) along with their positively and negatively signed forms. A juxtaposition's geometry is defined by the edges not in Θ, but immediately adjacent to the vertices *A*, *C*, *D* and *H* of Θ, ignoring any orientation on the polygon in which Θ is embedded. The sign (or chirality) of a juxtaposition is determined by the sign of the crossing at Θ in a SAP containing the signed juxtaposition. A crossing-angle, called the opening angle α, which depends on both the juxtaposition geometry and the crossing-sign has been defined, as illustrated in Figure 2. Figures 2(a) and 2(b) illustrate respectively that the opening angle for *S*^{+} is 45°, whereas, for *S*^{−}, it is 135°. For both *Z*^{+} and *Z*^{−}, the two straight lines used to define the angle are parallel, but, as indicated in Figures 2(c) and 2(d), the opening angle is 0° and 180° respectively for *Z*^{+} and *Z*^{−}. From the definition, the opening angle reflects the shortest distance between the end points of the juxtaposition according to how they are joined within the polygon.

With these definitions, consider any particular local juxtaposition geometry *J* (either signed or unsigned). Then, for the set of *n*-edge Θ-SAPs, for example, one can compare knot-reduction between a strand-passage mechanism in which strand-passage is always performed at the strand-passage structure Θ (regardless of the local geometry adjacent to it) and another mechanism in which strand-passage is only performed at Θ if the local juxtaposition geometry is *J*. The comparison factor for this case then reduces to the polygon-length and *J*-dependent comparison factor,
(1)
where *K* denotes all knots except for the unknot, and where *p _{n}*(

*K*

_{1}→

*K*

_{2}) is the number of

*n*-edge Θ-SAPs which are transformed from knot type

*K*

_{1}to knot type

*K*

_{2}after a single strand-passage and is the number of these that have the juxtaposition geometry

*J*. In order to calculate the numerator ratio in eqn (1), one would need to consider all possible knot types (other than the unknot) as an initial knot type for a Θ-SAP. However, for polygon lengths ≤5000 and the non-interacting Θ-SAP model studied, the predominant knot type observed at equilibrium is the trefoil (see Table 4 in [30], Table 1 in [31], [32], and Tables 4 and 5 and Conjecture in [33]). Hence the numerator ratio can be approximated by replacing

*K*with 3

_{1}as indicated above.

Estimates for the approximate *R*^{(J)}_{K,n} were obtained from Markov chain Monte Carlo computer simulations that generated variable length, but fixed knot type Θ-SAPs. With a large sample of unknotted Θ-SAPs, strand-passage is performed and the HOMFLY polynomial is used to determine their after-strand-passage knot types. The results are then used to estimate the denominator term in eqn (1) for various choices of *n*. Similarly, from a large sample of trefoil Θ-SAPs, an approximation to the numerator term in eqn (1) has been estimated.

## Review of results and discussion

In this section, we review the results from [19,20], present heuristic geometric arguments to explain the results and finally discuss the implications for modelling the topoisomerase–DNA interaction mechanism.

Overall in [20], it was found that the comparison factors were strongly dependent on the juxtaposition geometry and the crossing-sign. For example, for the two juxtapositions *S* and *Z*, when compared with always performing strand-passage at Θ, knot-reduction decreased respectively with the juxtapositions in the order *S*^{+}, *S*, *S*^{−}, *Z*^{+}, *Z*, *Z*^{−}, and this was fairly consistent over polygon lengths up to 600 (see Figure 4 in [20]). The first four yielded comparison factors which were, on average, greater than 1, whereas the last two were well below 1. The results, for example, for the unsigned juxtapositions *S* and *Z* show that the local juxtaposition (independent of the strand-passage crossing-sign) influences knot-reduction. But the results for the signed juxtapositions indicate that the crossing-sign can affect knot-reduction more. Specifically, whereas the juxtaposition geometry at the strand-passage site is exactly the same for *Z*^{+} and *Z*^{−}, a strand-passage model which takes into account the crossing-sign leads to greater knot-reduction if a positive crossing-sign change is preferred. On the other hand, by symmetry, there is a juxtaposition geometry which is a mirror image of *Z* and such that the opening angle and comparison factor for is exactly the same as that of *Z*^{+} (*Z*^{−}), thus a strand-passage model based on that also takes into account the crossing-sign leads to greater knot-reduction if a negative crossing-sign-change is preferred. Altogether, the general trend (based on an observed negative correlation) was observed that knot-reduction decreases as opening angle increases and that, on average, those signed juxtapositions with opening angles 90° or less favoured knot-reduction (see Figure 5 in [20]). However, the highest knot-reductions were found to correspond to opening angles between 45° and 90°.

In the present article, we provide a heuristic argument to explain these results. A Θ-SAP which contains a particular local juxtaposition geometry *J* can be thought of as consisting of the strand-passage juxtaposition structure plus two self-avoiding walks (represented, for example, by the curved lines on the right of Figure 2a). For Θ-SAPs, as polygon length grows, one of these two walks is generally much smaller than the other [29]; call this the small-walk and call the other the large-walk. Consider the loop formed by joining the ends of the small-walk. If the large-walk never interpenetrates this loop, then a strand-passage performed at Θ is equivalent to a Reidemeister Type I move and the knot type will not change. One expects that the probability that the large-walk interpenetrates the small-walk loop will increase with the size of the small-walk. Since the opening angle puts a limit on how small the small-walk can be, we expect that smaller opening angles will correspond to lower probabilities of knot type change. In particular, one expects the probability of knotting an unknot to increase with opening angle and indeed this was found to be the case (see Figure 11(b) in [19]). We propose that this explains the overall observed negative correlation between knot-reduction and opening angle. However, it does not explain why a juxtaposition with the smallest possible opening angle does not yield the highest knot-reduction comparison factor. This issue is discussed next.

To explain this, consider again the juxtapositions *Z*^{+} and *S*^{+}. For *Z*^{+} (and its mirror image ), α=0 and the small-walk can be smaller than for any other juxtaposition. Thus one expects that the chance the knot type remains unchanged after strand-passage is greater for *Z*^{+} (and its mirror image ) than for any other juxtaposition. Thus the denominator term in eqn (1) is as small as possible; however, at the same time, the numerator term is also expected to be small. That is, although strand-passage at *Z*^{+} is not likely to knot an unknot, it is also not likely to unknot a knot. So knot-reduction does not achieve the best possible value at *Z*^{+}. Meanwhile, *S*^{+} (and its mirror image ) with an intermediate opening angle has the better knot-reduction properties since its structure strikes a better balance at allowing for knot type change; in particular, among all of the juxtapositions studied, it yielded the highest probability of unknotting the trefoil and, at the same time, a relatively low probability of knotting an unknot (see Figure 11(a) in [19]). Thus for the highest knot-reduction, an opening angle that is not too small (that it limits the chance to unknot a knot) and not too large (that it increases the chance to knot an unknot) is needed; our results indicate that this corresponds to opening angles between 45° and 90°.

We expect that the general trends described above regarding the relationship between knot-reduction and the opening angle α will hold for other strand-passage models. In particular, if topoisomerases bind preferentially using opening angle information, then the results of [20] indicate that angles of less than 90° favour knot-reduction. However, this opening angle definition is dependent on the juxtaposition geometry and the crossing-sign (chirality), or, equivalently, the way that the two strands at the strand-passage site are joined to create a polygon. Thus this angle cannot necessarily be determined only on the basis of local geometric information; so, can this really play a role in the actual topoisomerase strand-passage mechanism? Neuman et al. [21] proposed a model to explain chirality discrimination in supercoil relaxation by type II topoisomerase. In their model, the topoisomerase binds differently depending on the chirality of the supercoiling and, as a result, acts processively in the positive supercoiling case and distributively in the negative supercoiling case. In Figure S4(B) of [21], they suggest a hypothetical topoisomerase–DNA interaction mechanism in which the topoisomerase initially binds at a crossing near the distal end of a plectonemic structure. If it is binding near the distal end, then the opening angle α can be determined from ‘local’ information. Thus this points to a potential way for topoisomerases to ‘detect’ the opening angle. Note that the opening angle definition given in the present article only corresponds to that of [21] in the positive supercoil case. For example, the opening angles, by the definition here, for the crossings closest to the distal ends in Figure S4B of [21] are identical in both the positive and negative supercoil cases shown; hence knot-reduction is expected to be the same for both cases. In their model, the topoisomerase binds differently depending on the chirality; the Θ-SAP model does not address this. The Θ-SAP results, however, do indicate that if topoisomerase is binding positive supercoiled DNA preferentially with an angle of less than 90°, then there will be greater knot-reduction than if the binding angle is greater than 90°. Equally, if it is binding negative supercoiled DNA preferentially with an opening angle of less than 90°, then there will be greater knot-reduction than for higher binding angles. Furthermore, such chirality biases can be used mathematically to explain knot-chirality biases [34].

The opening angle definition and heuristic arguments discussed in the present article only apply to a local strand-passage within a single polygon and hence only to knot-reduction properties. We have not investigated decatenation properties with this model.

## Funding

C.S. acknowledges financial support from Natural Sciences and Engineering Research Council (NSERC), a Compute Canada WestGrid resource allocation from Compute Canada, and local support from the Isaac Newton Institute for Mathematical Sciences while participating in the workshop Topological Aspects of DNA Function and Protein Folding (3–7 September 2012).

## Acknowledgments

We acknowledge useful discussions with M. Vazquez and C.S. also acknowledges useful discussions with K. Shimokawa and M. Vazquez during the workshop at the Isaac Newton Institute for Mathematical Sciences.

## Footnotes

Topological Aspects of DNA Function and Protein Folding: An Independent Meeting held at the Isaac Newton Institute for Mathematical Sciences, Cambridge, U.K., 3–7 September 2012, as part of the Isaac Newton Institute Programme Topological Dynamics in the Physical and Biological Sciences (16 July–21 December 2012). Organized and Edited by Andrew Bates (University of Liverpool, U.K.), Dorothy Buck (Imperial College London, U.K.), Sarah Harris (University of Leeds, U.K.), Andrzej Stasiak (University of Lausanne, Switzerland) and De Witt Sumners (Florida State University, U.S.A.).

**Abbreviations:**
SAP, self-avoiding lattice polygon

- © The Authors Journal compilation © 2013 Biochemical Society