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New York University, Department of Chemistry, New York, New York 10003
Correspondence: Address reprint requests to Alexander Vologodskii, E-mail: alex.vologodskii{at}nyu.edu.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODS OF COMPUTATIONRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODS OF COMPUTATIONRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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). It belongs to the type II DNA topoisomerases that catalyze passing one double-stranded DNA segment through another one. The free energy required for supercoiling comes from ATP hydrolysis coupled with the strand-passing reaction. The enzyme is important for maintaining a certain level of DNA supercoiling inside bacterial cells and for DNA replication (Levine et al., 1998; Reece and Maxwell, 1991; Wang, 1996).
The active form of the Escherichia coli enzyme consists of two A and two B subunits (reviewed by Reece and Maxwell, 1991). For many years it was accepted that the tetramer binds a segment of double-stranded DNA 
140 bp in length that is supposed to wrap around the enzyme forming approximately one turn of the right-handed helix (Fisher et al., 1981; Kirkegaard and Wang, 1981; Liu and Wang, 1978a). The bound DNA segment, named the gate (G) segment, is cleaved approximately in the middle, and the 5' ends of the broken strands are covalently attached to the protein (Fig. 1 A). Another DNA segment, the so-called transporting (T) segment, then passes through the double-stranded break and the break is resealed (Reece and Maxwell, 1991; Wang, 1998). This model of the G-segment conformation is supported by the results from nuclease protection experiments (Liu and Wang, 1978a; Morrison and Cozzarelli, 1981) as well as by DNase I (Fisher et al., 1981; Kirkegaard and Wang, 1981) and hydroxyl radical footprinting (Orphanides and Maxwell, 1994). It has been shown that binding DNA gyrase to nicked DNA in the absence of ATP increases the linking number of the DNA strands, measured after ligation of the nicks (Kampranis et al., 1999; Liu and Wang, 1978b; Peng and Marians, 1995). Quantitative analysis of these experiments showed that each complex between the enzyme and DNA increases the linking number in the ligated molecules by 0.5–0.8. These data are considered to provide additional support for the model. If both the G- and T-segments belong to the same closed circular DNA, each strand-passing reaction results in the change of the linking number (Lk) of DNA complementary strands by two (Brown and Cozzarelli, 1979). If the T-segment belongs to second circular DNA, the reaction can link or unlink the molecules.
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, is the ability of the enzyme to introduce supercoiling in the very small DNA circles, 174 bp in length. Clearly, in this case, the strand passing cannot follow the model simply because the contour length of the circles is insufficient to form the starting conformation (Fig. 1 A). Indeed, if 140 bp are wrapped around the enzyme, there are only 40 bp available to make the external loops. Recent data obtained by atomic force microscopy are also in certain disagreement with the model (Heddle et al., 2004). For us, however, this is an example of a model in which the G- and T-segments are separated by a free DNA loop. The major conclusion of the work will be essentially the same for any model of this type. In the alternative type of models the G- and T-segments belong to one DNA stretch whose conformation is completely specified by the interaction with DNA gyrase.
Although DNA gyrase can catenate, decatenate, and unknot circular DNA molecules (Kreuzer and Cozzarelli, 1980; Mizuuchi et al., 1980), the efficiency of these reactions is very low (Marians, 1987; Ullsperger and Cozzarelli, 1996). The major catalytic activity of the enzyme is selective introduction of negative supercoils into circular DNA. The right-handed helix turn, formed by G-segment, has been assumed to provide this selectivity (Fig. 1 A). However, DNA molecules of a few thousand base pairs in length adopt many different conformations in solution, and it is hard to believe that a small chiral loop can provide the required selectivity. Indeed, other types of conformations (with a potential T-segment juxtaposed with the enzyme gate) would result in different topological changes. In particular, the outcome from two conformations shown in Fig. 1, B and C, would be the introduction of two positive supercoils and formation of a positive trefoil. None of these products has been observed as a reaction product under conditions optimal for supercoiling. Thus, we suggested that the model might not account for the experimentally observed selectivity of the enzyme action. In this article we analyze this question quantitatively by computer simulation.
There is convincing evidence that computer simulations provide accurate quantitative description of the large-scale conformational properties of DNA. Simulations reproduce experimental data on hydrodynamic properties of DNA molecules (Hagerman, 1981; Hagerman and Zimm, 1981; Rybenkov et al., 1997d), DNA cyclization (Hagerman, 1990; Levene and Crothers, 1986; Taylor and Hagerman, 1990; Vologodskaia and Vologodskii, 2002), equilibrium distributions of topological states (Klenin et al., 1988, 1989; Rybenkov et al., 1993, 1997c; Shaw and Wang, 1993; Vologodskii and Cozzarelli, 1993), elasticity of single molecules (Vologodskii, 1994; Vologodskii and Marko, 1997), and light and neutron scattering data on supercoiled DNA (Gebe et al., 1996; Hammermann et al., 1998, 1997; Klenin et al., 2000). The simulations are based on the statistical-mechanical treatment of a well-established model of the double helix. There are only three parameters in the model and all of them have been reliably determined for various solution conditions. Thus, the simulations are able to provide reliable quantitative information on many DNA properties that are hard to measure experimentally. The simulations are very useful for the analysis of enzymatic reactions that involve formation of DNA loops (Grainge et al., 2002; Klenin et al., 2002; Vologodskii et al., 2001). Here we apply the simulation to estimate the distribution of the reaction products corresponding to the model of DNA gyrase shown in Fig. 1. The results leave no doubt that the model is inconsistent with selective formation of negative supercoils by DNA gyrase.
The simulation results allow us to formulate a general requirement for a correct model of the enzyme. Models that satisfy this requirement were suggested by Kampranis et al. (1999) and recently by Corbett et al. (2004). We discuss and elaborate important details of such a model.
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METHODS OF COMPUTATION |
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TOPABSTRACTINTRODUCTIONMETHODS OF COMPUTATIONRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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rigid cylinders of equal length with harmonic potentials with respect to the angles between adjacent segments and the value of torsional deformation, 
Tw (Klenin et al., 1991; Vologodskii et al., 1992). Electrostatic interactions between the segments were taken into account through a hard core potential specified by DNA effective diameter (Vologodskii and Cozzarelli, 1995; Vologodskii et al., 1992). The equilibrium ensemble of chain conformations was simulated by the Metropolis procedure (Vologodskii et al., 1992). Each simulation was performed for the chains with a particular topology and Lk value. To preserve the chain topology during a simulation run we used a topological invariant, the Alexander polynomial, 
A topological invariant has the same value over all conformations with a particular topology. We calculated 
for 
and 
after each move of the Metropolis procedure (Vologodskii et al., 1974). If the value of 
or 
were different for the current conformation and trial conformations, the trial conformation was rejected. A rigid subchain of five straight segments that form nearly one turn of the right-handed helix corresponded to the G-segment bounded with DNA gyrase (Fig. 2 A). The geometry of the subchain did not change during the simulation.
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Parameters
All computations were performed for a DNA persistence length of 50 nm (Hagerman, 1988), a torsional rigidity, C, of 3.0 x 10–19 erg·cm (Horowitz and Wang, 1984; Klenin et al., 1989), and a DNA effective diameter, d, of 5 nm. The later value of d corresponds to a 0.2 M solution of NaCl (Brian et al., 1981; Rybenkov et al., 1993, 1997b; Stigter, 1977). The length of the straight segments was equal to 10 nm.
Analysis of the simulated conformations
A segment of the chain was considered to be juxtaposed with the G-segment if all of the following conditions were satisfied:
(Fig. 2 A); 
(0° 
90°) between the tested segment and the axis of the helix, which specifies the orientation of the G-segment, did not exceed 
(Fig. 2 A); 
for 
(Vologodskii et al., 1975). This test excluded conformations when the tested segment was inside the loop formed by the G-segment. 
If the link did not appear, we considered that there is no juxtaposition. Such cases were very rare, however.
Although the probabilities of juxtaposition depend on the values chosen for and 
for sufficiently small and the product distributions do not depend on their choice. In most of the calculations we used 
nm and 
If a particular conformation of the chain passed all tests for the juxtaposition, we determined the potential reaction product by calculating the change of chain writhe, 
Wr, and the values of or after the extension of the G-segment shown in Fig. 2 C. The Alexander polynomial, however, does not distinguish between a knot and its mirror image. To solve this problem, we used Wr (Kleninet al., 2002). It can be proven that, if the substrate is an unknotted conformation and the product is a trefoil, then the "sign" of the product coincides with the sign of Wr. Similarly, if the substrate is a trefoil and the product is an unknot, then the "sign" of the substrate is opposite to the sign of Wr (see, e.g., Murasugi, 1996; in particular Example 6.4.2 and Theorem 6.4.7).
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RESULTS |
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Lk, and by the topology of DNA axis, which can be unknotted or form various knots. Averaging the results of such analysis over a large volume of sampled conformations gives us the desired distribution of the reaction outcomes.
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Lk = 0) unknotted chains. Fig. 4 A shows the results obtained for DNA molecules of different lengths. One can see from the figure that introduction of two (–) supercoils would be the dominant reaction outcome only for DNA molecules shorter than 2000 bp. For molecules of 3000 bp or longer, formation of positive trefoils should be comparable with the introduction of negative supercoiling. Clearly, for the relaxed molecules of these lengths the chirality of the G-segment does not provide the selectivity of reaction outcomes comparable with what is observed experimentally.
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Fig. 5 shows the product distributions obtained from negatively supercoiled molecules. The results were obtained for a circular DNA 3500 kb in length. Although for this DNA length negative supercoiling is the major reaction outcome for all tested initial values of supercoiling, the total fraction of other products remains close to 30%. The distribution of the other products changes, however, with the increase of (–) supercoiling. Surprisingly, the fraction of positive trefoils, which was the second largest fraction for the relaxed chains (Fig. 4 A), declines rapidly as (–) supercoiling increases. Instead, an increase of Lk by (+2) becomes the second major outcome of the reaction.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODS OF COMPUTATIONRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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and recently by Corbett et al. (2004). Below we elaborate on a version of such a model with some additional features.
What reliable experimental data have to be explained by a realistic model of the enzyme action? Wrapping the G-segment around the enzyme seems to be an established feature of the complex. A model also has to account for the fact that the double-stranded break is introduced nearly in the middle of DNA segment bound to the enzyme (reviewed in Reece and Maxwell, 1991). Additional strong restriction for a model imposes the finding that the enzyme is able to introduce supercoiling into very small DNA circles, 174 bp in length (Bates and Maxwell, 1989). We believe that the simplest and perhaps the only way to fit all of the above requirements for DNA minicircles corresponds to the diagram shown in Fig. 6. Here the T-segment enters the gate from inside rather than from outside the loop formed by the G-segment. This allows the G- and T-segments to be combined into a single stretch that interacts with the enzyme along its entire length. If the enzyme has twofold symmetry, as is usually assumed, the pathway of the short DNA circle keeps (or nearly keeps) this symmetry during the strand-passing reaction in the model. To generalize the model for longer DNA molecules we extend one of the two short internal loops formed by the G- and T-segments outside the enzyme surface (Fig. 7). Conformations of the DNA stretch interacting with the protein during the strand-passing reaction are completely specified by this interaction, and this provides a unique reaction outcome. It is important to note that even a small free loop between the G- and T-segments suggested by Kampranis et al. (1999) as an alternative to the bound loop, cannot exclude other outcomes of the reaction. Indeed, there is no way to restrict the size and conformations of the free loop if its pathway is not controlled by DNA-protein interaction.
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; Heddle et al., 2004; Kirkegaard and Wang, 1981; Liu and Wang, 1978a; Morrison and Cozzarelli, 1981; Orphanides and Maxwell, 1994). A model that assumes that both the G- and T-segments belong to a continuous DNA stretch interacting with the enzyme cannot explain, however, catalysis of the catenation/decatenation reaction by DNA gyrase. Indeed, in this case the G- and T-segments belong to different DNA molecules and cannot form a continuous stretch of the double helix. Thus, we have to conclude that there are two modes of the enzyme action. In the major mode DNA gyrase interacts with a continuous stretch of DNA and introduces negative supercoiling into closed circular DNA molecules. In the minor mode the second part of the stretch, which corresponds to the T-segment, is substituted by another DNA segment. The later segment can belong to a remote part of the same DNA or another molecule. In this second mode the enzyme is able to knot/unknot DNA molecules and catenate/decatenate DNA circles. The catalytic activity of the major mode is at least 1000 times greater than that of the minor mode. It cannot be excluded, however, that the activity of the minor mode is related with a small fraction of damaged enzymes.
Formation of knots in circular DNA has never been observed in the reaction catalyzed by DNA gyrase. According to the computer simulations performed for the model shown in Fig. 1, trefoils have to make up a large fraction of reaction products if relaxed circular molecules serve as a substrate (see Fig. 4 A). On the other hand, the fraction of trefoils that corresponds to the thermodynamic equilibrium does not exceed 3% for 7-kb DNA molecules under close to physiological ionic conditions (Rybenkov et al., 1993). Thus, if DNA gyrase works according to this model, it would greatly increase the fraction of knots over the equilibrium level on nicked circular DNA (see Fig. 4 A). This effect would be the opposite of the property of other type II topoisomerases that strongly decrease the steady-state fraction of knots relative to the equilibrium level (Rybenkov et al., 1997a). This is not surprising because the model shown in Fig. 1 is exactly opposite to one that was suggested to explain topology simplification by the topo II enzymes (Vologodskii et al., 2001). The computational analysis showed that a topoisomerase reduces the fraction of knots and links below the equilibrium level if the T-segment passes through the G-segment from inside to outside the loop formed by the G-segment upon binding with the enzyme (Vologodskii et al., 2001). Conversely, the fraction of knots and links will be increased above the equilibrium level if the T-segment passes through the G-segment from outside to inside the loop, as it is shown in Fig. 1. The data presented in Fig. 4 A reflect this feature of the gyrase model. In the model shown in Fig. 7 the T-segment passes through the break from inside to outside the loop. Thus, the minor mode of the enzyme action has to reduce the steady-state fraction of knots below equilibrium level. The fact that formation of knots by DNA gyrase has been never observed experimentally gives additional support to the suggested model. Unfortunately, the very low level of activity of the enzyme in the minor mode makes it difficult to study the issue quantitatively.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMETHODS OF COMPUTATIONRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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The work was supported by National Institutes of Health grant GM 54215.
Submitted on March 17, 2004; accepted for publication August 26, 2004.
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REFERENCES |
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and the angle 
