Alexander V. Vologodskii, Stephen D. Levene, Konstantin V. Klenin, Maxim Frank-Kamenetskii and Nicholas R. Cozzarelli- Conformational and Thermodynamic Properties of Supercoiled DNA

J. Mol. Biol. (1992) 227, 1224-1243 -

Conformational and Thermodynamic Properties of Supercoiled DNA

Alexander V. Vologodskii’, Stephen D. Levene2, Konstantin V. Klenin3 Maxim Frank-Kamenetskii’ and Nicholas R. Cozzarelli4

‘Institute of Molecular Genetics Russian Academy of Science Moscow 123182, Russia, CIS

2Program in Molecular and Cell Biology University of Texas at Dallas

Richardson, TX 75083, U.S.A.

3Branch of I. V. Kurchatov Institute of Atomic Energy, Troitsk Moscow region 142092, Russia, CIS

4Department of Molecular and Cell Biology 401 Barker Wall

University of California, Berkeley, CA 94720, U.S.A.

(Received 16 December 1991; acce.pted 9 June 1992)

We used Monte Carlo simulations to investigate the conformational and thermodynamic properties of DNA molecules with physiological levels of supercoiling. Three parameters determine the properties of DNA in this model: Kuhn statistical length, torsional rigidity and effective double-helix diameter. The chains in the simulation resemble strongly those observed by electron microscopy and have the conformation of an interwound superhelix whose axis is often branched. We compared the geometry of simulated chains with that determined experimentally by electron microscopy and by topological methods. We found a very close agreement between the Monte Carlo and experimental values for writhe, superhelix axis length and the number of superhelical turns. The computed number of superhelix branches was found to be dependent on superhelix density, DNA chain length and double-helix diameter. We investigated the thermodynamics of supercoiling and found that at low superhelix density the entropic contribution to superhelix free energy is negligible, whereas at high superhelix density, the entropic and enthalpic contributions are nearly equal. We calculated the effect of supercoiling on the spatial distribution of DNA segments. The probability that a pair of DNA sites separated along the chain contour by at least 50 run are juxtaposed is about two orders of magnitude greater in supercoiled DNA than in relaxed DNA. This increase in the effective local concentration of DNA is not strongly dependent on the contour separation between the sites. We discuss the implications of this enhancement of site juxtaposition by supercoiling in the context of protein-DNA interactions involving multiple DNA-binding sites.

Keywords: DNA supercoiling; tertiary structure of DNA; Monte Carlo simulation; thermodynamics of supercoiling; loop formation in DNA

1. Introduction

A detailed understanding of the structure of supercoiled DNA has assumed greater importance with the recognition that supercoiling plays key roles in DNA replication, transcription and recombination. Xearly all of the DNA in cells is negatively supercoiled in one of two fundamentally different forms. In a predominantly storage form of super-

coiling, the path of the DNA is dict.ated by the surface of the proteins to which it is bound. The most notable example is the solenoidal winlding of DNA around nucleosomes in eukaryotic cells (Richmond et al., 1984), which affords considerable compaction of the DNA. Supercoiled DNA free from protein-binding constraints adopts a second, plectonemic (interwound) form that is better adapted to an active role in cell physiology (Weintraub, 1985;

0022-28X/92/20122&20 $08.00/O 1224

0 1992 Academic Press Limited

Conformation and Thermodynamics of Gupercoiled DNA 1226

Figure 1. Model of plectonemically supercoiled DNA. A DPU’A double helix is depicted in a plectonemic (interwound), negative superhelical conformation. The superhelix is idealized so that the interwinding is completely regular. The superhelix diameter, D, and winding angle, y, are indicated. The number of superhelical turns, the number of times the DNA winds about the superhelix axis, is close to the number of crossings (nodes) in a projection such as shown, and this value multiplied by sin y equals - Wr (Cozzareili et al., 1990). The molecule shown has 1 branchpoint and 3 ends. Node sign is determined by a convention implicit in the drawing; the mirror image model would be positively supercoiled.

Cozzarelli et a,Z., 1990). Although no regular DNA structure such as chromatin has been identified in prokaryotic cells, it is estimated that about half of the supercoiled DNA is physically constrained by binding to proteins and that the remainder has the same basic physical properties as supercoiled DNA free in solution (Bliska & Cozzarelli, 1987; Zacharias et al., 1988; McClellan et al., 1990; Dayn et aZ., 1991). In some cases, the energizing effect of negative supercoiling arises from the promotion of unwinding of the double helix (Wang et al., 1982; Frank-Kamenetskii, 1990). Examples include the unwinding of DNA at promoters for transcription and at origins for DNA replication (Kornberg & Baker, 1992). In other cases, it is the conformation of supercoiled DNA that promotes specific reactions. For example, the interwinding of supercoiled DNA promotes recombination by bringing into proximity two or more DNA sequences distant along the DNA contour; this conformational role of supercoiling can be mimicked by catenation or

knotting (Craigie & Mizuuchi, 1986; Dr6ge $ Cozzarelli, 1989; Kanaar et al., 1989; Benjamin & Cozzarelli, 1990). The role of supercoiling in the juxtaposition of DNA sites has also been implicated in the action of eukaryotic topoisomerases (Zechiedrich & Osheroff, 1990).

Despite the many key roles played by uncon- strained, negatively supercoiled DNA, its conformation has not been completely established because supercoiling is difficult to study by standard physical methods. Significant supercoiling of DNA requires a circular molecule at least 1 kbt in length, which is too large for high resolution techniques such as X-ray crystallography and n.m.r. spectro- scopy. Measurements of hydrodynamic and related properties, using techniques such as sedimentation, diffusion and light scattering, give information only about the average dimensions of DNA. By far the most useful methods for studying supercoiled DNA have been electron microscopy and the measurement of topological properties such as linking number and the catenation or knotting of products of site-specific recombination. A recent analysis (Boles et al., 1990) using these methods has resulted in a very good model for the average structure of negatively supercoiled DNA, which is illustrated schematically in Figure 1. In this model, the supercoils are extended and plectonemic, and the superhelix axis is branched. Boles et al. (1990) assessed the variation with superhelix density of writhe, twist, superhelix axis length, number of superhelical turns and the frequency of superhelix branches. However, to estimate quantities that could not be measured directly, an idealized regular geometry for the superhelix had to be assumed. Moreover, any analysis relying on electron microscopy is limited by the need to make assumptions about the relation- ship between structural features of molecules in solution and on the microscope grid surface. These assumptions are normally difficult to test.

Theoretical modeling of supercoiling should be extremely useful because it can overcome these limitations in the experimental methods and provide the equilibrium distribution of eonforma- tional and energetic properties. Three numerical approaches to DNA supercoiling have been proposed recently. Tan & Harvey (1989) used a detailed model for double-helical DNA that allows determination of the effect of local changes in DNA structure on superhelical conformations. However, because their model involved a large set of structural parameters, the calculations were restricted to small DNA circles. Hao & Olson (1989) developed the first Monte Carlo treatment of highly supercoiled DNA that involved an elastic model for the double helix. Both of these approaches sought to determine the minimum elastic-energy conformation of the molecule and each reproduced the interwound structure of the superhelix. However, a

t Abbreviations used: kb, 10’ base-pairs; n.m.r., nuclear magnetic resonance; c.p.u., central processing unit; bp, base-pair(s); e.u., entropy units.

226 A. V. Vologodskii et al.

complete picture of the behavior of supercoiled DNA in solution cannot be obtained from a single conformation corresponding to the elastic energy minimum. In solution, DNA adopts a large number of conformations that are close in elastic energy but differ considerably in geometry. Estimates of the average properties of supercoiled DNA can be obtained only by generating an ensemble of DNA conformations at thermal equilibrium. This was achieved by the Monte Carlo approach of Klenin et al. (1991).

In the present paper we used an improved version of this approach to analyze a number of properties of DNA molecules over the physiologically significant range of superhelix density. We calculated the average length, diameter, writhe and number of turns of the superhelix as functions of superhelix density. We found that the geometry of the highly supercoiled DNA molecules in the simulation is in very good agreement with available experimental data (Roles et al., 1990). We showed that the equilibrium branch number distribution of the superhelix axis is a function of DNA length, superhelix density and effective double-helix diameter. We investigated the thermodynamics of supercoiling and found that the relative contributions of entropy and enthalpy to the free energy of supercoiling varied strongly with superhelix density. Finally, we estimated the probability of an encounter between two sites that were separated along the contour of supercoiled NA and showed that this probability was markedly increased by supercoiling.

A novel technical aspect of our approach is a modification of the simulation procedure used by Klenin et al. (1991). We found that the motions of tbe chain used in that procedure rapidly led to equilibrium values of writhe and elastic energy, but that equilibrium distributions for the number and length of superhelix branches could not be obtained. We overcame this problem by incorporating into the simulation an additional motion, in which a portion of the DNA chain translates along the superhelix axis.

2. Methods

(a) Basic procedure

In this section we give a simplified overview of the simulation procedure, and in the following sections we describe the methods in detail. The degree of supercoiling of DNA can be expressed in terms of the linking number (Lk) difference, ALk, which is equal to Lk-Lk,, where Lk, is the Lk for relaxed DNA. Lk, equals the number of base-pairs in a molecule divided by the helical repeat. It is often more convenient to use the length-independent quantity, specific linking number difference, or superhelix density, 0, given by:

ALk

“=K (1)

Most supercoiled DNA molecules isolated from either prokaryotes or eukaryotes have u values between -0Tk5 and -067 (Bauer, 1978). To simulate the structure of such highly supercoiled DNA, we used an extension of the

Metropolis-Monte Carlo procedure developed by Klenin et al. (1991). The DNA chain is modeled as a polygon of cylindrical segments. The parameters determining the characteristics of DNA in this modei are the DNA bending and torsional rigidity and the effective DNA diameter. The effective DNA diameter accounts for excluded volume effects and, as a consequence of electrostatic repulsion, is larger than the geometric diameter.

At each step of the simulation procedure, a trial conformation of the chain is generated by displacement from the previous conformation. The starting conformation is chosen arbitrarily and the final results are independent of the choice. A trial conformation is rejected if segments of the chain interpenetrate or if the chain is kno,tted. The energy of a trial conformation that passes these tests is then compared with the energy of the current conformation and the probability of acceptance ‘of the move depends on the difference in energy between trial and current conformations. This procedure is repeated numerous times to obtain an ensemble of conformations that, in principle, is representative of the equilibrium dist,ribution.

(b) The DNA w~odei

A closed DNA molecule composed of n Kuhn statistical lengths is modeled as a closed chain consisting of kn rigid, cylindrical segments of equal size. The elastic energy of the chain, E, is computed as the sum of the bending and torsional elastic energies, Eb and E,.

The bending energy term is:

(2)

where the summation is done over all the joints between the rigid segments, R is the gas constant, T is the absolute temperature (297 K), Bi is the angular displace:ment (in radians) of segment i relative to segment i - 1 and a is the bending rigidity constant. The bending constant a is defined so that the Kuhn statistical length corresponds to k rigid segments (Frank-Kamenetskii et al., 1985).

The torsional energy, E,, is a quadratic function of the displacement of the chain twist from equilibrium, ATw. We do not compute ATw directly because TU is a property of the double-helical structure of DNA, which is not specified in this simple polymer model. However, for a closed chain, ATw may be computed from the expression:

ALk = ATul+ Wr, (3)

where Wr (writhe) is a property of the chain axis alone, and we fix the particular value of ALk in each sirnulation (White, 1969, 1989; Fuller, 1971). Hence the t)orsional elastic energy term may be expressed as:

E, = (2z2C/L)(ALk- W#, (4)

where C is the torsional rigidity constant and L is the DNA length.

There are several assumptions and approxima.tions in the model. First, we approximate DNA as an i,sotropic elastic chain. Thus, local sequence-dependent variations in structure are not taken into account. Second, we do not consider supercoiling-driven structural transitions such as the introduction of cruciforms and Z-DNA (Frank- Kamenetskii, 1990). Third, we assume that torsional and bending rigidity are independent. This assumption is the simplest starting point and is supported by the available experimental data on closed circular DNA with low values of Wr (see Klenin et al., 1989). Even at the relatively high 1~1 values considered here, there is little distortion of the double helix as long as structural transitions have been

Conformation and Thermodynamics of Supercoiled DNA 1227

Cb)

Figure 2. Trial motions of the DNA chain during Monte Carlo simulations. The current conformation of the chain is shown by the thick crosshatched line and the trial conformation by the thin continuous line. (a) Crankshaft move. A portion of the chain is rotated by an angle, 4, about an axis connecting 2 randomly chosen vertices. In the example shown, 3 elementary segments are rotated. (b) Reptation move. The sub-chain between the vertices i, and i, is translated by 1 segment length along the chain contour. The 4-segment section of the chain between i, and i, -4 is deformed by changing angles $r, $z and $a by a constant amount so that the section is incorporated between vertices i, and i,+3. Reciprocally, the 3-segment section between vertices i, and i, +3 is deformed by changing angles *a and G5 so that it can be incorporated between i, and i, -4.

excluded. Fourth, we approximate the electrostatic interactions between DNA segments by cylinders whose effective diameter accounts for excluded volume effects.

(c) Simulation procedure

(i) Types of displacement

The 2 types of displacement used are diagrammed in Fig. 2. In particular, 1 type of motion was discussed by Klenin et al. (1991) and can be described as a crankshaft rotation (Fig. 2(a)). A sub-chain containing an arbitrary number of adjacent segments is rotated by a randomly chosen angle, 4, around the straight line connecting the vertices bounding the sub-chain. The value of I$ was uniformly distributed over an interval ( - 40, 4,,) that was continuously adjusted during the simulation so that the probability of accepting a crankshaft move was about 0.5.

Moves of the 2nd type, a new feature of these simulations, involve a sub-chain translation or reptation, along the local chain axis. This deformation involves 3 operations and is illustrated in Fig. 2(b) for chains with k = 10. First, 2 vertices, i, and i,, were randomly chosen as the ends of the sub-chain. Translation of the sub-chain can be carried out only if the distance between i, and i, - 4 is less than 3 segment lengths. Second, the 4segment chain section between vertices i, and i,--4 was deformed by changing each of the angles $i; i/~~ and G3 by an amount, 6, so that this chain section could be incorporated

between vertices i, and i, + 3. This displacement was done with conservation of the dihedral angles that are determined by adjacent triplets of vertices between i, and i, -4. Similarly, the a-segment section between vertices i, and i, + 3 was deformed by changing the angles ti4 and ti5 so that the section could be incorporated between vertices i, and i, - 4. Third, the 2 deformed chain sections were exchanged. The manner in which the exchanged sections of the chain were oriented about the axes spanning i, and i, -4 and spanning i, and i,+3 was important. The acceptance probability of sub-chain reptation was greatly increased if the direction of vectors pointing from these axes to the centers of masses of the exchanged chain sections was preserved. For chains with k = 5, the acceptance probability was greater if portions consisting of 2 and 3 segments were exchanged. Otherwise, the procedure was the same. Both types of moves satisfy the basic requirement of the Metropolis procedure of microscopic reversibility; namely the probability of trial conformation B when the current conformation is A must be equal to the probability of trial conformation A when the current conformation is B.

The net result of this 2nd type of motion was a translation of the sub-chain by 1 segment along the original conformation. Even though these moves were accepted with low frequency in the Monte Carlo algorithm, they substantially increased the probability of extrusion and resorption of superhelix branches without changing the average values of E and Wr.

The a priori probabilities of selecting the 2 types of moves were equal.

(ii) Excluded-volume effects

Excluded-volume effects in supercoiled DNA were incorporated into the simulation via the concept of effective DNA diameter, d. This is the actual diameter of the cylindrical segments of the DNA model. The value chosen depends not only on the actual physical size of DNA, but also on electrostatic repulsion. In the Monte Carlo procedure, a chain has infinit,e energy if any point on the axis of a segment lies within a distance d of any point on the axis of a non-adjacent segment. Therefore, the minimum distance between all pairs of non-adjacent segments of a trial conformation was calculated. If any distance was less than d, the trial conformation was rejected.

(iii) Knot checking

The starting conformations were unknotted. However, this does not guarantee that the chain will remain unknotted, because segments of the chain were allowed to cross during trial moves. With such a phantom, or incor- poreal, chain, knot-checking is necessary even for short chains because the compaction resulting from supercoiling sharply increases the probability of knotting (Frank- Kamenetskii & Vologodskii, 1981). In principle, knotting could be avoided by forbidding crossings of segments during Monte Carlo moves; i.e. by using an impenetrable chain model. The 2 approaches are identical from the standpoint of statistical mechanics, but the phantom- chain approach with subsequent knot checking is much more computationally efficient. Knots were detected by evaluating the Alexander polynomial A(t) for the trial conformation at t=-1 (Frank-Kamenetskii & Vologodskii, 1981). If a trial move knotted the chain, it was rejected.

(iv ) Energy If a trial conformation passed the tests of chain overlap

and knotting, its elastic energy, E, was calculated as a

A. V. Vologodskii et al

6

& iI4

2

k

Figure 3. Dependence of the simulated value of writhe on the number of cylindrical segments per Kuhn length, k. The average writhe, (WV), was obtained from simulation runs consisting of about I million trial moves for a chain extending 6 Kuhn lengths (1% kb) and with cr = -0.054.

sum of 4 and E, (eqns (2) and (4)). This requires calculating the writhe of t’he trial conformation, which is most conveniently done by the method of Le Bret (1980). The writhe is the sum of 2 terms in this method, 1 of which, the directional writhing number, we obtained simultaneously with the calculation of the Alexander polynomial.

(v) The Metropolis-Nonte Carlo procedure

The probability of accepting a trial conformation was obtained by applying the classical rules of Metropolis et aZ.

(1953). If the elastic energy of the trial conformation, E new, was lower than that of the previous conformation, E old, then the trial conformation was accepted. If the energy of the trial conformation was greater than the energy of the previous conformation then the probability of acceptance of the trial conformation was equal to

exp(@b --K,)/~~).

(d) Choice of the k value

The global conformations of the DNA double helix are very well described by the wormlike chain model (Kratky & Porod, 1949). Our replacement of the continuous wormlike chain with a semiflexible chain consisting of kn hinged rigid segments is an approximation that improves as k increases. The computer time needed for a simulation increases approximately as (En)‘. It is therefore necessary to choose a value of k that, is large enough to ensure reliable results but small enough to keep the computation time within reasonable bounds. Figure 3 shows the dependence on k of the average Wr, ( Wr), for a 6 Kuhn length Dn‘A chain with cr = -0.054. Even for such a highly supercoiled molecule. a k value equal to 10 was sufficient for reliable determinations of Wr and so was generally used. The bending constant CI (eqn (2)) for k = 10 equals 2.403.

(e) Calculation of average values over the ensemble of conformations

We found that after about IO4 trial moves, a chain of 100 cylindrical segments and 0 = -905 adopted the con-

I I I

I I I

16

12

5 10 15

Number of trial moves (~10~~)

20

Figure 4. Dependence of the elastic energy (0) and writhe (0) of a simulated DNA chain on t,he number of trial moves. The chain was 11% Kuhn lengths and had d = -905; its initial conformation was a random coil. Each point represents an average of over 500 successive trial moves in the Monte Carlo simulation.

formation of a branched, interwound superhelix. Over this range, (E) and (WV) approached asymptotic values around which they fluctuated slightly (Fig. 4).

The number of branches varied at a much lower frequency than fluctuations in (E) and (wr). Thus, to obtain reliable average values of quantities sensitive to the number of branches, we had to carry out much longer simulations. The number of steps necessary to calculate statistically reliable average quantities sharply increases with increasing segment number and superhelix density. For a chain of 100 segments at @ = -096. about 10’ trial moves are required.

Most of the simulations described were run on the Gray Y-MP/8 at the San Diego Supercomputer Center or on the Cray X-MP/l4 a,t the UC Berkeley Computer Center. On the Cray Y-MP/S, the program executes about 3 x IO6 trial moves per c.p.u. hour for a chain composed of 100 cylindrical segments. We have recently adapted the code to run on a Silicon Graphics 4Dj25 workstation.

(f) Data analysis

To compare directly the Monte Carlo results to the electron microscopy data on DNA supercoils, the chain conformations were flattened to mimic the shape of molecules deposited on the surface of electron microscope grids. A plane projection in which the molecule was well spread out was selected visually. using the graphics display on the computer monitor. A uniform potential, U,, of the form:

was then added to the elastic energy of the chain, where the z-axis is normal to the plane of projection, and [ is a constant or order of magnitude I. After several thousand steps, the conformation of the projected chain was plotted for further analysis. The chain’s average final dimension in the z-direction was close to the diameter of the superhelix, and the (WV) for the flattened conformations


i

I I I I I 1

/

T

+

2 ++

+ +

0 20 40 60 80 100 120

Chain co-ordinate

Figure 5. An algorithm for detecting branches in a superhelix. Branching of the superhelix was measured by the attendant increase in the number of superhelix ends. The number of ends, in turn, was determined from the value of the Wr of the sub-chain between vertices i and i + I, denoted as Wr(i, i + 1). The sub-chain Wr is shown as a function of the chain vertex co-ordinate, i, along with the conformation of the corresponding chain. Local maxima in - Wr(i, i+ 1) occur whenever the sub-chain contains an end of the superhelix, indicated here by arrows drawn from the corresponding ends of the superhelix. The algorithm counted the number of maxima in - Wr(i, i + I) with amplitudes greater than an empirically

predetermined value of 1.15 (broken line). The chain in this Figure is 3.5 kb in length with 0 = -@05. The value of I depends on (r and was equal to 24 in the example shown.

differed from that of the 3-dimensional structures by no more than 5%.

We determined the projected chain contour length, the superhelix axis length and the number of crossing, or nodes, of the flattened chains (see Fig. 1). The contour length and the superhelix axis length were determined by tracing the chain using a Numonics digitizer. The superhelix axis was defined, as described by Boles et al. (1990), as the curve that passes through the nodes and bisects the area enclosed by the DNA between adjacent nodes. Values of the superhelix axis length were normalized to the projected chain contour length. The number of superhelical turns for each chain was determined as the sum of all signed crossings of the chain.

We also compared the experimental results with values of structural parameters obtained by computations on the ensemble of 3-dimensional chain conformations. Branching of the superhelix was most conveniently measured by the attendant increase in the number of superhelix ends. We define the number of branch points as 2 less than the number of ends. To determine the number of superhelix ends corresponding to a given superhelical conformation. we calculated the function Wr(i, i + Z), equal to the Wr for the sub-chain between vertices i and i + 1. This function was calculated using the Gauss integral for the sub-chain (Frank-Kamenetskii & Vologodskii, 1981; Braun, 1983). In the case of plectonemic DNA superhelices, the function is close to 0 for the extended portions of the chain and has extrema when the chain loops back at the ends. An example of the dependence of

Wr(i, i+Z) on i is shown in Fig. 5 along with the corresponding chain conformation. The number of superheiix ends is defined as the number of - Wr(i, i + 1) peaks whose height exceeds a certain discrimination value, Wrcontr. The values of Z and of Wrcontr were chosen to obtain the best agreement with visual inspectiop of hundreds of chain conformations. The values used were WT,,,,~ = 1.15 and z = 400a + 44.

The Wr of each DNA conformation in the ensemble is calculated during the simulation and (Wr) is therefore known quite accurately. The average superhelix diameter, CD)> was estimated by considering the axis-to-axis distance between all cylindrical segments of the chain except those in the terminal loops. We defined (D) as the average distance between a vertex and the closest point on a chain segment located more than a minimal contour distance from the vertex. This contour separation (40 nm) was chosen so that there is a high probability that the nearest segment to a vertex will be across the superhelix axis from the vertex. Other superhelix parameters (the average values of the superhelix axis length, the superhelix winding angle and the number of superhelical turns) were then estimated from the values of (WY) and (D) and the relationships between the parameters of the idealized superhelix model (Fig. 1; CozzareIli et al., 1990).

(g) DNA parameter values

The properties of DNA in our model are completely determined by 3 parameters, the Kuhn statistical length, b (or the persistence length, b/2), the torsional rigidity, C and the DNA effective diameter, d. The Kuhn length of DNA is known, quite accurately, to be 100 nm (294 bp). It is nearly independent of electrolyte concentration above approximately 10 mM monovalent ion (for a review; see Hagerman, 1988). The value of C is known with less certainty, but is probably in the range of 1.5 x lo-” to 3.8 x 10-I’ erg.cm (1 erg = 10e7 J; see discussions, Taylor & Hagerman, 1990; Crothers et al., 1992). We used a value of 30 x lo-l9 erg. cm. The effective diameter. $, depends strongly on ambient conditions, most notably ionic composition (Stigter, 1977, 1985; Brian et al., 1981; Yarmola et al., 1985). Unless stated otherwise. the results presented here were obtained using a value of 3.5 nm for d. This is a high salt concentration value for d, corresponding approximately to 05 M-NaCl. In the cases noted, we used a value of 10 nm for d. This choice is appropriate for DNA in solutions that contain low concentrations of monovalent counterions, about 0.02 M.

3. Results

(a) Examples of simulated conformations of superhelical DNA

The simulations produce a large ensemble of supercoiled DNA conformations, which, in principle, corresponds to the equilibrium distribution. Stereoscopic views of typical examples of relaxed and supercoiled 3.5 kb circular molecules are shown in Figure 6. The relaxed molecules in Figure 6(a) and (a’) are irregular in form and have occasional random foldovers in projection. However, with increasing Joj, the structures become much more uniform. By 101 2 @03, the molecules are all clearly plectonemically wound and many are branched. Examples of branched (Fig. 6(b’), (c’) and (d’)) and unbranched (Fig. 6(b), (c) and (d)) conformations


(b)

(d ) (d’)

Figure 6. Stereoscopic views of typical conformations of 35 kb DNA molecules with increasing superhelix density. (a) and (a’), B = 0; (b) and (b’), G = -@03; (c) and (c’), 0 = -005; (d) and (d’), CJ = -@07. The superheliees in I@‘), (c’) and (d’) each have a single branchpoint. Each chain consists of 118 cylindrical segments.

are displayed. We did not observe DNA with extensive solenoidal supercoiling; the conformations were either irregular or plectonemic. The superhelices were elongated, gently curving structures whose diameter decreased as (01 increased.

(b) Comparison of the Monte Carlo and experimental values of writhe, superhelix axis length, and number

of superhelical turns

In order to test the validity of the simulations, we compared the values of superhelix structural parameters determined by simulation with those derived from experimental data. In particular, we used in our comparison the values obtained by Boles et al. (1990), which are based on the electron microscopy of supercoiled DNA and the topology of the products of site-specific recombination of super-

coiled DNA. They analyzed their results in terms of the idealized, branched, plectonemic model for the superhelix shown in Figure 1. In this model, know- ledge of any two of the following parameters, writhe, superhelix axis length, superhelix diameter, number of superhelical turns and superhelix winding angle, allows calculation of the remaining parameters.

The value of Wr is calculated in each step of our procedure and the average writhe, ( Wr>, is known with an uncertainty of only about 1%. To obtain the additional dimensions of the simulated supercoiled molecules, we used two independent methods.

In one method, we used an algorithm to flatten our three-dimensional simulated molecul.es until the superhelix axes were nearly planar. This mimias the effects of sample preparation for the electron microscope and allows for a direct measurement of the


Figure 7. Comparison of an electron micrograph of a 7 kb supercoiled DNA molecule with a simulated chain of identical length and superhelix density (o = -0.045). The simulated DNA was flattened to mimic the effect of spreading for microscopy. The simulated chain consisted of 236 elementary segments. The micrograph was provided by T. C. Boles and is of plasmid pAB4 (Boles et az., 1990).

length of the superhelix and of the number of superhelical turns. Figure 7 shows a flattened, simulated 7 kb supercoiled molecule and an electron micrograph of a DNA molecule of the same size and supercoiling. The similarity is striking.

In the second method, an algorithm was developed for computing the average superhelix diameter, (D), from the complete population of simulated three-dimensional conformations. From the values of (D) and (Wr), we calculated the average values for the length of the superhelix and the number of superhelical turns in terms of the idealized model.

The average number of superhelical turns for the simulated 3-5 kb molecules determined by either method of analysis and the values derived empirically show the same linear dependence on ALk (Fig. 8). The slope of the line is equal to 087.

Figure 9 shows as a function of supercoiling, the experimental and Monte Carlo values of the average length of the superhelix axis of a 3.5 kb molecule. For both data sets, the superhelix axis length is, within error, independent of 0 and about 40% of the DNA contour length.

In Figure 10 we compare the supercoiling dependence of (Wr)jALk for the simulated conformations

I I I I I 0 10 20 30

-AL&

Figure 8. Comparison of the Monte Carlo and experimental values for the average number of superhelical turns as a function of ALk. The data are for 35 kb chains. Two procedures were used to determine the Monte Carlo values for the average superhelix axis length. In one (0). axes were measured using a projection of 5 to 15 flattened superhelix conformations. In the other (A), the results were calculated from the (Wr) and average diameter of the complete population of 3-dimensional chain conformations. The experimental data (0) of Boles et al. (1990) were obtained from Int recombination assays carried out on various DNA substrates. The continuous line indicates the linear dependence of the number of superhelical turns on ALk. Error bars indicate 1 standard deviation.

and the values calculated from the experimental measures of the superhelix axis length and the number of superhelical turns. The value of (Wr)/

-loo0 I

0’ I I I I I 0.00 O-03 0.06 0*09 0.12

--B

Figure 9. Comparison of Monte Carlo and experimental determinations of the length of the superhelix axis as a function of superhelix density. The Monte Carlo results were for flattened (0) and 3-dimensional (A) conformations of 35 kb chains. The experimental values (0) show the electron microscopy results of Boles et al. (1990). The continuous line shows the average value of the superhclix axis length determined from all 3 data sets. Error bars indicate 1 standard deviation.


o-0 ‘A 0.00 0.02 0.04 0.06

-0

0.08

Figure 18. Comparison of Monte Carlo (a) and experimental values (continuous horizontal line) of the reduced average writhe, (Wr)/ALk, as a function of superhelix density. The data are for 35 kb chains. The experimental results are those of Boles et al. (1990), which gave a value for (Wr)/ALk of 072. This value was independent of 101 within experimental uncertainty, indicated by the hatched area.

ALk obtained from the experimental results is independent of a(902 < 101 < 0.12) and is equal to 0.72 kO.09 (Boles et al., 1990). A similar ratio was obtained from cryoelectron microscopy of highly supercoiled DNA (Adrian et al., 1990). The simulations revealed that the value of ( Wr)lALk increased slightly from 072 at low 161 to 0.80 at high /cr(, but once again the differences between the experimental and computed values are within the error margins of the methods.

In summary, there is very good qualitative and quantitative agreement between the results of simulation and experiments. This agreement gives us confidence in the outcomes of both approaches and justifies extending the simulations into areas where few experimental data are available.

(c) Equilibrium number of branches

Branching of the superhelix axis has been observed many times by electron microscopy and has been directly implicated in the apposition of three separate DNA sites in key intermediates of recombination (Kanaar et al., 1989) and transposition (Mizuuchi & Mizuuchi, 1989). Branching also has a large effect on the topology of the products of recombination (Boles et al., 1990) and of type-2 topoisomerases (Wasserman & Cozzarelli, <991; White, 1991). Despite their importance, the properties of superhelix branches are relatively unexplored. Also, the extent of branching observed by electron microscopy has for unknown reasons varied greatly within and among separate studies (Laundon & Griffith, 1988; Boles et al., 1990; Adrian

et al., 1990). A key goal of this study Wats, therefore, to provide at least a first-order, quantitative treatment of branching. The primary problem in studying branching by simulation is the low frequency of changes in the number of superhelix branches. We alleviated this difficulty in two ways. First, we included sub-chain reptation moves (Fig. 2(b)) in the Monte Carlo program in addition to the crankshaft rotations (Fig. Z(a)) that had been used previously. Both motions are described fully in Methods. This improvement in the algorithm acce- lerated the approach to equilibrium with respect to branched conformations. Second, we used vlery long simulations, up to 4 X 10’ trial moves.

We measured branching by counting the number of superhelix ends rather than the number of branch points for computational convenience. The number of superhelix ends in any given conformation was determined by an algorithm in which a short sleeve was systematically moved along the chain contour and its Wr computed continuously. Local extrema in this function occurred whenever the sleeve spanned an end of the superhelix (Fig. 5).

Even with the improved algorithm and the large number of trial moves, the number of transitions in branch number was not large, 20 to 50 per simulation. Consequently, the standard deviations in the probability that a molecule has i ends, FQ, were relatively high, 10 to 20% of the mean for pi > O-2.

The probability distribution for the number of superhelix ends of a 3.5 kb molecule as a function of g is plotted in Figure 11. It is clear that the probability of branching decreases with increasing supercoiling. For example, the probability that the superhelix has at least one branch decreases from 058 at o = -0.04 to 0.20 a,t o = -907. This decrease in branching with increasing supercoiling can be readily explained. The formation of each branch requires bending energy because of the additional DNA curvature at the branch Ipoint and, particularly, at the hairpin turn at the end of the branch. As the superhelix diameter decreases with increasing 1~1, the terminal hairpin turn will be sharper and, therefore, require more bending energy.

The energetics of branching as a function of 0 is shown in Figure 12. We have plotted the free energy and enthalpy difference between molecules with two ends (unbranched) and three ends (one branch); these are designated AGb, and AHbr, respectively. The value of AGb:,, is given by:

(6)

The branching enthalpy, AHbr, equals AE,,, the difference in (E) between singly branched and unbranched structures. AHbr has a substantial positive value, but AGbr is near zero. For example, at cr = -0.06, AHbr = 4.7 kcal/mol (I cal = 4.184 J) and AGbr = 0.7 kcal/mol. This implies that the entropy change is positive and hence is the driving force behind branching. Both AGi,:,, atnd AHbr increase with supercoiling, but AGbr incre.ases more slowly than AHbr (Fig. 12). Therefore, the entropy

Conformation and Thermodynamics of Supercoiled DNA ‘1233

I

O-8 1

u= a*04

Figure 11. The distribution of the number of superhelix ends as a function of superhelix density. The distributions were obtained by applying the branch detection algorithm described in Fig. 5 to I x 10’ to 4 x lo7 conformations in the simulation. The chains were 3.5 kb in length and d = 3.5 nm.

contribution to branching, equal to (AH,,-AG,,,)IT, increases as a function of J(T~ (see Appendix).

Using a statistical-mechanical analysis, we derive in the Appendix (eqn (A2)) that AS,,, the entropy of forming a single branch in an unbranched molecule, is given by:

AS,, = AX,+2R ln(L-3Z), (7)

0.8 t

u= -0.06

2 3 4 5 6

Number of ends

2

2 :/

0 0

0

7 * I I I I I

CM4 Mb5 0.06 0~07 -0

i

Figure 12. Gibbs free energy and enthalpy of branching of a superhelix as a function of superhelix density. Shown are Monte Carlo values for the free energy (0) and enthalpy (0 ) for introduction of a 3rd superhelix end into an unbranched 35 kb molecule. The continuous lines are the best least-square fits to the data.

where L is the total DNA length, 1 is the observed minimal DNA length of a branch (about equal to the length of a terminal loop plus 1 superhelical turn) and AS, is a constant which is independent of CJ, 1, and d. This equation takes into account the fact that branching leads to new chain conformations for DNA, and that this expanded opportunity increases with total DNA length and diminishes with the amount of DNA devoted to the branch. By inspec- tion of numerous superhelical conformations, we estimated that 1 equals 300, 250, 200 and 150 nm for c equal to -004, -0.05, -0.06 and -@07, respectively. Substitution of these values into equation (7) gives a dependence of AS,, on n, which is similar to the relation derived from the variation of both AG,, and AHb, with CJ that is shown in Figure 12. It follows from equation (7) that t#he change in AX,, between cr = -0.04 and o = -0.07 is 4 e.u. and from Figure 12, this change in AS,, is 6 e.u.

We evaluated the dependence of AGbr on DNA length that is implied by equation (7). If the value of AGbr is known for one DNA length, L,, it follows from equation (7) that for DNA of length L,,

where AGb,(L) is the free energy of branching for DNA of length L. For a 35 kb molecule with rr=--005, we found that AG,,(L,) equals 048 kcal/mol. To test the validity of equation (8) we simulated a 5.25 kb DNA with g = -0.05 (Fig. 13(a)). By substituting the values of AGbl(L1)> L, and L, into equation (8), we calcuiate that AC&,,(&) equals -0.58 kcal/mol. The value of AG,,(L,) determined from the simulated conformations (Fig. 13(a)) and from equation (6) is -0.83 kcal/mol. Given the high value of the standard deviation of


2 3 4 5 6

Number of ends

(a)

2 3 4 5

Number of ends

(b) %;igure 13. Variation of the distribution of the number of superhelix ends with values of DNA length and effective

TINA diameter. The distributions were obtained from simulated conformations of molecules with o = -005. (a) Distributions for chains 3.5 kb and 5.25 kb in length; d = 3.5 nm. (b) Distributions for 3.5 kb chains with d = 3.5 nm and d = 10.0 nm.

the Monte Carlo estimates of branching frequency, these two values are not significantly different. The results of the simulations illustrate the high sensi- tivity of branching to DNA length. Whereas the unbranched conformation is by far the more probable conformation for a 3.5 kb molecule, conformations with three or even four ends are more proba,ble for the 5.25 kb DNA (Fig. 13(a)).

The dependence of the branch number distribution on the effective DNA diameter is also striking. The results for d = 3.5 nm and d = IO nm for a 3.5 kb DNA with D = -905 are presented in Figure 13(b). The ratio of molecules with one branch to those with none increases six times at the higher d value. Presumably branching increases with effective DNA diameter because of the attendant decrease in the curvature of the terminal loop of the superhelix. A practical consequence of the dependence of branching on d is that branching should be strongly dependent on ionic conditions.

(d) Thermodynamics of supercoiling

An important advantage of the Monte Carlo approach to DNA supercoiling is that it permits the investigation of thermodynamic as well as conformational properties. Here we have investigated the enthalpy contribut,ion to the supercoiling free

energy, AGs:,,, as a function of g. The value of AGSc was determined using an umbrella sampling procedure (McCammon & Harvey, 1987). as described by Klenin et al. (1991).

The elastic energy of the chain, IZ, is recorded continuously during the simulations. The enthalpy of supercoiling, AH,,, is equal to the difference in (E) between supercoiled and relaxed chains. (E) depends strongly on Ic, AHsC does not. Even the slight dependence of the calculated value of AH,, on k vanishes for sufficiently large values of L. The calculated value of AHsC changed by no more than 7% between k = 5 and k = 10, and we used exclu- sively t.he higher value of k.

Figure 14 shows the supercoiling dependence of AH,, and AGSc for a 3.5 kb circular DNA. The dependence of AG,, at low /cl is essentially quadratic (Rauer & Vinograd, 1970; Hsieh & Wang, 1975); however, there is a departure from this ;second-order dependence at high 101 (Klemn et al., 1991). Unexpectedly, we found that the enthalpy contribution to AGS, decreases from 100% at Low /cj to 50% at c = -0.06. There is, therefore, little or no entropic penalty associated with the loosely coiled molecules at low levels of supercoiling. However, as supercoiling approaches physiological levels it sharply restricts the number of available chain conformations.

Conformation and Thermodynamics of Supercoiled DNA h235

80 I I I

0 0.00 0.02 0.04 0.06

-0

Figure 14. Gibbs free energy and enthalpy of supercoiling as a function of superhelix density. The free energy of supercoiling ( - - - ) for the 3.5 kb molecules was calculated numerically (Klenin et al., 1991). The value of AH,, (0) is the difference between the elastic energies of supercoiled and relaxed molecules obtained from the Monte Carlo analysis. The continuous curve is the least- squares fit of a polynomial to the enthalpy data.

(e) Radial distance distribution in supercoiled DNA

Supercoiling is expected to have a dramatic effect on the spatial distribution of sites within a single DNA molecule. In the ensemble of supercoiled chain conformations, any two sites should at some time be close together in space when they are positioned across the superhelix axis from each other. Such close juxtaposition should decrease the free energy of the interactions with DNA-binding proteins that bind simultaneously to two or more sites.

We used a Monte Carlo simulation to test this prediction in a quantitative fashion. We determined the radial site concentration, W(R), which is the concentration of two sites separated by a through- space distance, R. The method of calculation is illustrated in Figure 15. Site 1 is placed at the center of uniformly spaced spheres, thereby forming a set of concentric spherical shells. The radial site concentration, W(R), is the concentration of site 2 within a certain shell of mean radius R. We calculated radial concentration profiles for ensembles of chains at given values of superhelix density and site contour separation, S. The function W(R) is expressed in units of concentration (mol sites/l) and is equal to the bulk site concentration that would be needed to match the probability that sites 1 and 2 are separated by R if the sites were on separate molecules. Figure 16 is a plot of radial concentration as a function of R for sites separated by 50, 100, 200 or 400 nm along the chain contour. The values for supercoiled (a = -0.06) and relaxed DNA are compared as well. For sit,es close together in space, W(R) is much greater for supercoiled than for relaxed DNA. The curves meet and even cross as R

Figure 15. The radial concentration of sites along a DKA chain. We consider all pairs of vertices 1 and 2 of simulated chains that are separated by a given distance, S, along the chain contour. As shown in the schematic, the 1st vertex (site) is chosen as the origin of a spherical polar co-ordinate system, and the space is partitioned into a succession of spherical shells. The simulation maintains a count of the number of occurrences of site 2 in a given spherical shell. This value was used to calculate the radial concentration of site 2 in mol/l from the expression 1000fiiViN~,, where j”; is the fraction of conformations with site 2 in shell i, Vi is the volume of this shell in cma, and NAV is Avogadro’s number. The radial concentration is reported as a function of the through-space distance, R, which is equal to the mean radius of a shell.

increases; thus the concentrating effect of supercoiling disappears for large values of R.

A particularly important issue biologically is the local site concentration, W(RO). This we define as W(R) for sites that are separated in space by less than some small distance, R,, but are not close along the chain contour. We have set R, at 10 nm for several reasons. First, the calculations do not depend strongly on the choice of R, as long as it is not much greater than the average superhelix diameter; at 0 = -00-06, the superhelix diameter is about 10 nm. Second, given that the geometric diameter of DNA is 2.2 nm, a 10 nm spacing between two DNA sites is easily close enough for typical proteins bound at each site to close off a DNA loop. Third, with smaller values of R,, volume exclusion of DNA segments can obscure key results of the analysis.

The local concentration of one site within a distance R, of another site is plotted in Figure 17 as a function of (T for four different values of 8. Tbe local concentration increases by two orders of magnitude as rr goes from 0 to -0.07, with most of the change between 0 = 0 and o = -0.04. This is the interval over which the regular interwound superhelix structure is established. An important feature of these plots is the absence of a large effect of S. As S increases from 100 to 400 nm the local

1236 A. I’. Vologodskii et al.

I I I I

-w 0

/

-/

I I I I I l \S=lOOnm

‘\ 1

B \

‘i

1

I I 1 I .

L

S=200nm

- \. \

0 40 80 120 160 200 0 50 100 150 200

R, nm R, nm

‘0

Figure 16. Radial concentration profiles for supercoiled and relaxed DXA as a function of through-space distance and contour separation. The radial concentration was calculated for simulated 3-5 kb chains with CT = -PO6 (0) or 0 (0). The variation with through-space separation, R, and 4 values of the contour separation, 8, are shown: 50 nm: 100 nm, 200 nm and 400 nm.

concentration changes minimally, about two-fold. An informative exception to this near independence from S occurs when S is small. In this case, the bending elastic energy is high since small DNA loops must be formed. The formation of small DNA loops is also favored by supercoiling, because of the high curvature of DXA at the ends and branch points of the superhelix. The increase in W(R,) by supercoiling for S = 50 nm is close to three orders of magnitude (Fig. 17).

4. Discussion

(a) The sim&ation procedure

We present here an improved Metropolis-Monte Carlo treatment of the structure of DNA molecules containing physiological levels of supercoiling. The Metropolis-Monte Carlo procedure allows faithful calculation of the statistical-mechanical properties of a system given an infinite number of steps. With a finite number of steps, however, no universal criteria exist that guarantee that all regions of con-

formation space are adequately sampled. Some properties require many more steps for equilibration than do others. Thus, the reliability of bhe procedure for computing some properties d’oes not guarantee it for all properties.

The Monte Carlo simulation precedure used previously to study supercoiling (Klenin et al., 1991) relied solely on local crankshaft motions of the DNA chain (Fig. 2(a)) and did not equilibrate branched conformations. It is likely that equilibrium was not achieved because crankshaft rotations ar’e often perpendicular to the overall direction of the chain, but branching requires the sliding of chain segments past each other within the interwound superhelix. Therefore, in this study we employed the additional motion of sub-chain reptation (Fig. B(b)): which greatly accelerates the transitions between conformations with different numbers of branches.

We believe that the simulation procedure we adopted accurately reproduces many of the key physical properties of supercoiled DXA. The results were independent of the choice of starting conformation and there was very good agreement


10-9 4 0.00 0.02 0.04 0.06 0.08

--d

Figure 17. Dependence of the local concentration of sites along a DKA molecule as a function of super-helix density. The concentration of DNA sites within 10 nm of another site in 3.5 kb simulated chains is plotted. Data for 4 values of the contour separation, X, of the sites are shown: 50 nm (a); 100 nm (0); 200 nm (0); 400 nm (V). The local concentration within a 10 nm sphere was determined as described in Fig. 16.

between the Monte Carlo results and the available experimental data. The values of the adjustable parameters of the DNA model were chosen at the outset; not fitting to the experimental results was done. In the following sections, we analyze in detail the values of the adjustable parameters and the comparison with experimental data.

(b) The choice of DNA model parameters

Only three parameters are needed to specify the properties of DNA in the model: the bending rigidity constant, the torsional rigidity constant and the effective diameter of DNA. There is general agreement on the value of the bending rigidity of DNA, which is conveniently expressed in terms of a Kuhn length of 100 nm or of a persistence length of 50 nm (Hagerman, 1988). The Kuhn length is relatively insensitive to ionic conditions in the presence of a least 10 mM monovalent ion or of even lower amounts of multivalent cations. The ion concentrations present in vivo and in typical in vitro experiments are much higher than the minimal values.

The measured values of the torsional rigidity constant, C, vary from 1.5 x 10-l’ erg. cm to 3.8 x lo-l9 erg. cm (for a review, see G-others et al., 1992). We used the value of 30x 10P”erg.em, which is in the range of values obtained from DNA topoisomer distributions and DNA ring closure kinetics (Shore & Baldwin, 1983a,b; Horowitz & Wang, 1984; Shimada & Yamakawa, 1984, 1988; Levine & Crothers, 19866; Klenin et al., 1989; Taylor & Hagerman, 1990). Lower estimates for C (1.5 to 2.0 x 10-l’ erg.cm) have been obtained by fluores- cence depolarization and similar techniques

(Shibata et al., 1985). We found, however, that use of a value of C from the lower range does not dramatically change our conclusions. For example, if C is reduced from 3 x IO-l9 to 2 x 10-‘g erg.cm, the decrease in average writhe is only 10%. The value of C, like that of b, is insensitive to ionic conditions above a minimal salt concentration (Taylor & Hagerman, 1990).

The choice of the value for the effective helix diameter, cl, is more complicated that the choice of the other two parameters. The simulation results are sensitive to the value of d. When cl was increased from 3.5 to 10 nm, the value of (Wr) decreased by 20% and the fraction of molecules that are branched increased by more than twofold. There is no single correct choice for d because its value is strongly dependent on ionic conditions. The value of d has been estimated provisionally to vary from 3.5 to 20 nm over the range of [Na+] from 0.5 M t’o 6005 M (Stigter, 1977, 1985; Brian et al., 1981; Yarmola et al., 1985). We used primarily a value of 3-5 nm for d in the simulations. This is the appropriate choice for high electrolyte concentrations and is about equal to the limiting geometric diameter of DNA plus monovalent cations. Occasionally a ci: value of 10 nm was used to explore the structure of supercoiled DNA in dilute salt solutions. Experimental measures of DNA structure are usually carried out in conditions where d is between these extreme values.

Unfortunately, we cannot specify the effective helix diameter in the extensive experimental studies of DNA supercoiling by conventional electron microscopy. The ionic conditions are not well defined in such studies because of various changes in the ionic environment after the DNA is deposited on a grid (see discussion in Boles et al., 1990). In cryoelectron microscopy of DNA imaged in a vitreous glass, the addition of IO mM-MgCl, to a low ionic-strength buffer decreased the superhelix diameter significantly (Adrian et al., 1990), which is consistent with a decrease in cl.

(c) Comparison of Monte Carlo and experimental values for superhelix structure

The (Wr) for simulated supercoiled DNA is known with precision, because its value is computed continuously during the Monte Carlo runs. We used two procedures to measure the number of superhelical turns as well as the axis length, diameter and winding angle of the superhelix. The assumptions and limitations of the two procedures are different? yet the calculated values were the same within error (Figs 8 and 9). This substantiates the validity of both procedures.

We compared the values of the parameters derived from simulation with those deduced by Boles et aZ. (1990). They measured directly the number of superhelical turns and the length of the superhelix axis and calculated the remaining parameters by assuming the model of an idealized superhelix (Fig. 1). The variation with cr of the number of


superhelical turns (Fig. S), the length of the superhelix axis (Fig. 9) and (Wr) (Fig. 10) are, within error, the same for both the Monte Carlo and the experimental results.

We conclude, therefore, that the main features of superhelical DNA structure have been established for molecules at least 2.5 kb in length, with cr between -0.03 and -@07, and in solutions of moderate or high salt concentration. For DNA molecules in this molecular weight range, changes due to superhelicity are distributed between twist and writhe. In significantly smaller molecules superhelicity is absorbed primarily by twist and their structure is expected to differ significantly from the DNA analyzed here (Shore & Baldwin 19833; Horowitz & Wang, 1984).

To summarize the principal features of the structure of supercoiled DNA: (1) superhelical DNA conformations are plectonemic and frequently branched. Extensive solenoidal supercoiling is completely absent. (2) The (Wr) of the structures is nearly proportional to cr and is about three times (ATw). (3) The superhelix diameter varies inversely with superhelix density. For u = -006, it is equal to 10 to 12 nm. The superhelix winding angle is about 55” and does not depend significantly on superhelix density. (4) The superhelix axis lengt,h is independent of 0 and is about 40% of DNA contour length. (5) The number of supercoils is directly proportional to jcr/ and equals about @9 ALlc.

(d) EquiZibrium number of branches

We have determined some of the basic features of superhelix branching, including its energetics and dependence on DNA length, supercoil density and effective helix diameter. For a 3.5 kb DNA molecule with 0 = -0.06 and d = 3.5 nm, the free energy of branching, AGbr; is small so that the fraction of molecules with one branch to those with none, p,/p2, is about l/4 (Fig. 11). The low value of AGbr comes about because the opposing enthalpy and entropy contributions to branching are nearly equal in this case. Because branching requires significant energy to bend DNA at the branch point and particularly at the termini, AHbr is large; it equals 4~7 kcal/mol for adding a single branch to an unbranched DNA with cr = -0.06. A consequence of the increase in the number of chain conformations allowed by branching is the substantial value of TAS,, of about 4 kcal/mol.

The nearly equal probabilities of branched and unbranched conformations in our standard simulations helped the detection of fact’ors that influence branching. Branching increases markedly with chain length. We stimulated the structures of 3.5 and 525 kb molecules with ts = -@05 and found values of p,/pz of 0.44 and 4, respectively. We derived a relation, equation (7), which estimates &p2 for molecules of any length and it was in accord with the length dependence of p,/p2 determined numerically. The simulations show that

increased chain length also leads to an increase in the probability .of multiple branches (Fig. 13(a)).

We found a clear dependence of branching on superhelix density (Fig. 11) and on effective DNA diameter (Fig. 13(b)). Both decreases in 101 and increases in effective DNA diameter may increase branching by increasing the average superhelix diameter and terminal loop radius. The latter effect reduces the elastic energy needed to form a branch and thereby makes branching more probable. Intrinsic bends in the DNA (Crothers et al., 1990), either sequence-directed or protein-induced, would be expected to have the same efiect and to be localized in the terminal loops of superhelix branches. Laundon & Griffith (1988) have demon- strated that intrinsically bent regions of the DNA are, indeed, preferentially localized at the ends of branches.

The frequency of branching in superhelical DNA as measured by electron microscopy has varied greatly among and within separate investigations. This precludes a ready comparison with1 the simulations. In one study by microscopy (Boles et al., 1990) the fraction of molecules that are branched (97% for a 3.5 kb molecule) was much higher than that found in our simulations (about 4O”/d for c~ = -905). In a second study (Laundon & Griffith, 1988), the fraction of branched molecul~es (22% for a 6 kb DNA) was lower than expected from our calculated length dependence. In a third investigation which used a 2.7 kb plasmid, branching was even more rare (Adrian et al., 1990), but only a small fraction (about 6% for cr = -0.05) is expected for such a short DNA. We believe that t,his complex pattern of experimental results has emerged because branching is a sensitive function of the solution composition and DNA length, 0 and nucleotide sequence. The studies cited used very different sample preparation procedures and DNA molecules, and u was measured in only one of the investigations. A systematic examination of branching under controlled experimental conditions is needed.

(e) The energetic8 of supercoding

Monte Carlo procedures have been used previously to calculate the equilibrium distribution of linking number topoisomers (for a review, see Hagerman, 1988) and with improvements in the DNA model, excellent agreement with experimental data has been obtained over a wide range of DNA sizes (Shimada 8: Yamakawa, 1988; 1989). An important advance was the development of a method for calculating the free energy of supercoiling, AGs,, for large values of /cr/ (Klenin et al., 1991). Using this method, we found that AC& was nearly a quadratic function of superhelix density (Fig. 14). The second order proportionality constants determined from the Monte ‘Carlo and experimental results (Frank-Kamenetskii, 1990) are within 10% of each other, which rein§oroes the validity of the simulations.

A quadratic dependence is typical of systems that

Conformation and Thermodynamics qf Supercoiled DNA 1239

obey Hooke’s Law, and the simplest assumption would be that AH,, and A& are also proportional to the second power of 0’. If so, then their relative contributions to AGs, would be independent of 0, but this is not even approximately correct. The enthalpy contribution to AGs, is close to lOOo/’ for nearly relaxed DNA but diminishes to about half at d = -0.06 (Fig. 14). We conclude that the number of available conformations in closed circular DNA is not restricted significantly by low amounts of supercoiling, but that this restriction becomes important as supercoiling increases. AH,, is not proportional to g2, presumably because the change in DNA curvature decreases with increasing jcrl. The changing contributions of AH,, and A& to AGs, suggest that the quadratic dependence of AG,, on cr may be coincidental and not due to fundamental elastic properties of DNA.

A calorimetric study of DNA supercoiling concluded that the value of AH,, is about twice that of AGsc and, therefore, that AS,, is large and positive (Seidl & Hinz, 1984). Clearly, these results do not agree even qualitatively with ours. The discrepancy could be due to solvent effects, which are not explicitly contained in the Monte Carlo treatment. It seems unlikely, however, that supercoiling is highly entropically favorable, and it is difficult to measure accurately the excess heat of DNA supercoiling. Further experimental work is needed to reconcile the large differences between the estima- tions of AHs,.

(f) Effect of supercoiling on site juxtaposition

One of our primary goals is the quantitative analysis of properties of superhelical DNA that are difficult to measure experimentally yet have important implications for interactions with DNA-binding proteins. Tn DNA replication, transcription and recombination multiple DNA sites are often bound simultaneously by protein(s) to form a specialized nucleoprotein complex (Echols, 1986). The free energy of such protein-induced looping in DNA is expected to be a strong function of the effective local concentration of one DNA-binding site near another. We find that the local concentration is about loo-fold higher in supercoiled DNA (Ial 2 0.04) than in relaxed DNA. Moreover, the enhancement does not vary greatly with an increase in contour separation of the two sites and is therefore a generally important feature of supercoiled conformations. For sites that are close along the contour (S Z 50 nm), the concentrating effect of supercoiling goes up nearly to three orders of magnitude due to the increased curvature at ends of a superhelix.

In viva; a number of factors could modulate the near distance independence of the concentrating effect of supercoiling. Limiting factors include domain boundaries and proteins that bind to the branchpoints or terminal loops of supercoiled DNA. Therefore, the stretch of DNA over which a cis- acting protein, such as a transcriptional enhancer, is

functional can be regulated by proteins that limit the conformational freedom of DNA.

It is informative to compare the absolute values of the local concentration of two sites on the same molecule (in cis) with the equivalent bulk concentration of the same sites on separate molecules (in tram). We consider a 3.5 kb DNA present at 5 pg/ml, a typical concentration for in vitro enzymo- logy experiments. If the two interacting sites are 300 bp apart on a linear molecule, then the local radial concentration, as determined from the probability of ring closure (Shimada & Yamakawa, 1984; Levene & Crothers, 1986a), is equal to a bulk concentration of 80 pg/ml. This is a I6-fold enhancement by the eis arra,ngement. If the molecule is then closed and supercoiled to o = -0.06, the local concentration is now 10,000 pg/ml, corresponding to a prodigious cis effect. At such a high bulk concentration, DNA may not be in solution but in the form of a gel.

We can provide a precise physical interpretation of the enhancement of site juxtaposition by DNA supercoiling. The enhancement is basically the result of the difference between a one-dimensional and a three-dimensional search. We first consider any two sites on a supercoiled DNA molecule. On the left side of Figure 18(a), these sites [filled circles) are separated by any distance through space, R. On the right side of Figure IS(a), they are juxtaposed; i.e. they are within a small distance, R,, of each other. The probability of juxtaposition is the ratio of the number of conformations in which both sites are within a sphere of radius R, to the total number of conformations. We set R, at 10 nm for the reasons stated previously in Results. Because we are interested in the general result and not in the specific effects of the hairpin ends of the superhelix, we exclude sites that are very close along the chain contour. With this exclusion, the two sites will be juxtaposed when they are across the superhelix axis from each other (Fig. 18(a)). The fraction of supercoiled molecules in which two sites are juxtaposed is just the average length of DNA that is within the sphere of radius R, centered at one site, divided by the total DNA length, L. The length of DNA within the sphere will fluctuate depending on the particular conformation, but should vary between 8 and 2&. Thus, it is of the order of R, and the probability of juxtaposition is about R,/L. The 3.5 kb molecules that we have simulated have a length of 1180 nm, and thus the probability that any two sites are juxtaposed is about 0.01. The results of a simulation of 3.5 kb molecules with cr = -606 and S = 260 nm is that the probability of juxtaposition is 09P I i in accord with the calculation.

The particular assumptions that we made in t’his calculation are unimportant when we compare the probability of site juxtaposition in supercoiled and relaxed molecules. We consider in two stages the juxtaposition of two sites in relaxed DNA separated by some distance S along the contour (Fig. 18(b)). In the first stage, any two points that are separated by S are brought within R, of each other to create a


(b)

Figure 18. Enhancement of site juxtaposition by supercoiling. Two sites (filled circles) separated by a dista,nce, 8, along the same D?U’A molecule are indicated. One site is t’he center of a sphere (shaded circle). We define site juxtaposition as the placement of the 2nd site within the sphere around the first. (a) In highly supercoiled DNA, juxtaposition of the 2 sites could be brought about by sliding the chain along its particular conformation. The probability of juxtaposition, P, is about the ratio of the DNA length inside t.he shaded sphere to the total DNA length, L. (b) In relaxed DNA, it is convenient to consider juxtaposition as occurring via 2 operations. In the lst, the conformations are changed so that any 2 points separated by the same distance S along the contour as the sites in question are juxtaposed. The probability, P, that these intermediate conformations have the 2 sites juxtaposed in the 2nd operation is the same as in (a). Therefore, the overall difference in juxtaposition probability for relaxed and supercoiled DiYA is due to the low probability of the intermediate states in (b) in which the 2 loops must be of sizes, S and L-S. This analysis shows that site juxta- poskion is enhanced thermodynamically by supercoiling. The consequences for particular reactions may depend on the rate determining steps and the kinetic consequences of suoereoiling. 1

figure-of-eight molecule. In the second stage, we maintain the same chain conformation but slither the sites toward each other until they both occupy the sphere. The fraction of juxtaposed conformations in stage 2 for relaxed DNA should be about the same as the corresponding fraction for supercoiled molecules. Both events can be viewed as a result of one-dimensional diffusion with analogous starting and end points. Therefore, the difference between site juxtaposition in relaxed and supercoiled DNA reduces to the probability of stage 1 for relaxed DNA. Stage 1 involves a significant entropy decrease because the two loops in the figure-of-eight product m-ust be equal to S and L-S. This product has a supercoil, but this conformation is much less probable than the introduction of a supercoil in the

absence of such a length restriction. We determined numerically that for a relaxed 3.5 kb DNA, the probability of stage 1 is 0016. The Monte Carlo results (Fig. 17) show that the ratio of juxtaposed configurations for relaxed and supercoiled DNA is 6012, which is in very good agreement. The free energy of supercoiling limits chain conformations to extended interwound forms. Thus the free energy cost for site juxtaposition has been prepaid for a supercoiled molecule, but not for a relaxed one.

We give three examples in which superhelical conformation is thought to play a role in site juxtaposition. First, many DNA regulatory proteins in Escherichia coli bind to pairs of sites on DNA to form a protein-mediated DNA loop (Reitzer & Magasanik, 1986; Hochschild, 1990; Lobe11 & Schleif, 1990). Perhaps the best-known example is the lac repressor. Supercoiling stabilizes complexes of Zac repressor with DNA molecules that, contain multiple operator sites. Eismann & Muller-Hill (1990) recently showed that for a supercoiled DNA with c = -0-05, the half-life of complexes containing two operator sites approximately 400 bp apart was about tenfold longer than that for relaxed molecules.

Second, a recent study of transcription from the Zac promoter by Gartenberg & Crothers (1991) showed that intrinsically bent $-tract-containing sequences can substitute for the protein-induced bending associated with the binding OF CAP protein. The stimulation of transcription initiation required a supercoiled template in addition to the appro- priately phased DNA bends. They propos,ed that the formation of the open transcription complex is favored by wrapping DNA around RNA polymerase at a terminal loop in a superhelix.

Third, site-specific recombination by the Tn3/$ family of resolvases requires a (-) supercoiled substrate (Hatfull & Grindley, 1988). Pa,rker & Halford (1991) have shown that recombination site synapsis mediated by resolvase is remarkably fast. They estimate that it is too fast to occur by a ra,ndom collision of the sites and suggest that it is assisted by the restrict)ed one-dimensional search afforded by supercoiling, such as proposed in the slithering model of Benjamin & Cozzarelh (1986). In our simulations we only rarely observed site juxtaposition brought about by a bending of the superhelix axis as required by a random collision mechanism in synapsis. Instead, ilae juxtaposed sites are positioned across the superhelix axis from each other.

Appendix

The Entropy of Superhelix Branching

We consider the transition from an unbranched supercoiled DNA (2 ends) to one with a single branchpoint (3 ends). We wish to estimate the change in the number of stat,es associated with this transition. We ignore those degrees of freedom that are the same in both forms, including t&c: bending of

Conformation and Thermodynamics qf Supercoiled DNA 3241

the superhelix axis and the internal translation along the superhelix axis. The location of the branchpoint defines the length of the branches that emanate from it, and the variable location of the branchpoint is the reason for the increased number of states in a branched superhelix. To evaluate this increase we need to calculate the number of different combinations of branch lengths that can be realized in any particular DNA chain.

We measure DNA length in arbitrary units because we are interested only in differences in entropy and we designate the minimum length of DNA that is necessary to produce a superhelix branch as 1. The length of 1 can be defined only approximately as about equal to the terminal loop length plus the length of one interwound turn of the superhelix. The length of each branch, Zi, may vary from 1 to L- 21, where L is the total DNA length. Taking into account that 1, +l, +1, = L, one can represent the total number of different states, Q, as:

Lp21 L-21 L-21

Q=CCCx liZI 12=1 Is=1 6(L-I,-Z1,--1,) = (L-31)‘/2, (Al)

where the function 6(lc) equals, by definition, 0 for k # 0 and 1 for I% = 0. Using this equation we can express the entropic contribution to the free energy of branching, AS,,, as:

AS,, = R In Q = AS,+ZR ln(L-31). (A21

The value of AS, in this equation depends on neither L nor 1; it depends only on the choice of length units.

We thank J. Langowski, B. H. Zimm and J. C. Wang for fruitful discussions. We gratefully acknowledge support from NIH and NSF and the facilities at the San Diego Supercomputer Center and the Berkeley Computation Center. We are also indebted to the TAaboratory of Chemical Biodynamics at the Lawrence Berkeley Laboratory for the use of their computer graphics facilities.

References

Adrian, M., ten Heggeler-Bordier, B., Wahli, W., Stasiak, A. Z., Stasiak, A. & Dubochet, J. (1990). Direct visualization of supercoiled DNA molecules in solution. EMBO J. 9, 4551-4554.

Bauer, W. R. (1978). Structure and reactions of closed duplex DNA. Annu. Rev. Biophys. Bioeng. 7, 2877313.

Bauer, W. $ Vinograd, J. (1970). The interaction of closed-circular DNA with intercalative dyes. II. The free energy of superhelix formation in SV40 DNA. J. Mol. Biol. 47, 419-435.

Benjamin, H. W. & Cozzarelli, N. R. (1986). DNA-directed synapsis in recombination: slithering and random collision of sites. In The Proceedings of The Robert A. Welch Foundation Conferences on Chemical Research, vol. 29, pp. 1077126, The Robert A. Welch Foundation, Houston, TX.

Benjamin, H. W. & Cozzarelli, N. R. (1990). Geometric arrangements of Tn3 resolvase sites. J. Biol. Chem. 265, 6441-6447.

Bliska, J. B. & Cozzarelli, N. R. (1987). Use of &e-specific recombination as a probe of DNA structure and metabolism in viwo. J. Mol. Biol. 194, 205-218.

Boles, T. C., White, J. H. & Cozzarelli, N. R. (1990). Structure of plectonemically supercoiled DNA. J. Mol. Biol. 213, 931-951.

Braun, W. (1983). Representation of short and long-range handedness in protein structures by signed distance maps. .I. Mol. Biol. 163, 613-621.

Brian, A. A., Frisch, H. L. & Lerman, L. S. (1981). Thermodynamics and equilibrium sedimentation analysis of the close approach of DNA molecules and a molecular ordering transition. Biopolymers, 20, 1305-1328.

Cozzarelli, N. R., Boles, T. C. & White, J. (1990). Primer on the topology and geometry of DNA supercoiling. In DNA Topology and Its Biological Eflects (Cozzarelli N. R. & Wang, J. C., eds), pp. 139184, Cold Spring Harbor Laboratory Press, Cold Spring Harbor, NY.

Craigie. R. & Mizuuchi, K. (1986). Role of DNA topology in mu transposition: mechanism of sensing the relative orientation of two DNA segments. GelZ, 45, 793-800.

Crothers, D. M., Haran, T. E. 85 Nadeau, J. 6. (1990). Intrinsically bent DNA. J. Biol. Chem. 265, ‘7093-7096.

Crothers, D. M., Drak, J., Kahn, J. D. $ Levene, S. 13. (1992). DNA bending, flexibility, and helical repeat by cyclization kinetics. Methods Enzymol. 1n the press.

Dayn, A.; Malkhosyan, S.; Duzhy, D., Lyamichev, V.. Panchenko, Y. & Mirkin, S. (1991). Formation of (dA-dT), cruciforms in E. coli cells under different environmental conditions. J. Bacterial. 173. 2658-2664.

Droge, P. & Cozzarelli, N. R. (1989). Recombination of knotted substrates by Tn3 resolvase. Proe. Nat. Acad. Sci., U.S.A. 86, 6062-6066.

Echols, H. (1986). Multiple DNA-protein interactions governing high-precision DNA transactions. Science, 233, 1050-1056.

Eismann, E. R. & Muller-Hill, B. (1990). Lac repressor forms stable loops in vitro with supercoiled wild-type Zac DNA containing all three natural Zac operators. J. Mol. Biol. 213, 763-775.

Frank-Kamenetskii, M. D. (1990). DNA supercoiling and unusual structures. In LXNA Topology and Its Biological Effects (Cozzarelli. N. R. & Wang, J. C., eds), pp. 185-216, Cold Spring Harbor Laboratory Press, Cold Spring Harbor, NY.

Frank-Kamenetskii, M. D. & Vologodskii, A. V. (1981). Topological aspects of the physics of polymers: the theory and its biophysical applications. Sov. Phys.-Usp. 24, 679-697.

Frank-Kamenetskii, M. D., Lukashin, A. V., Anshelevich, V. V. & Vologodskii, A. V. (1985). Torsional and bending rigidity of the double helix from data on small DNA rings. J. Biomol. &ruct. Dynam. 2, 1005%1012.

Fuller, F. B. (1971). The writhing number of a space curve. Proc. Nat. Acad. Sci., U.S.A. 68, 815819.

Gartenberg, M. R. & Crothers, D. M. (1991). Synthetic DNA bending sequences increase the rate of in vitro transcription initiation at the Escherichia coli Fat promoter. J. Mol. Biol. 219, 217-230.

Hagerman, P. 5. (1988). Flexibility of DNA. Annu. Rev. Biophys. Biophys. Chem. 17, 265-286.

Hao, M.-H. & Olson, W. K. (1989). G!obal equilibrium

Ii242 A. V. Vologodskii et al.

configurations of supercoiled DNA. Macromolecules, 22, 3292-3303.

Hatfull, 6. F. & Grindley, N. D. F. (1988). Resolvases and DNA-invertases: a family of enzymes active in site- specific recombination. In Genetic Recombir&on (Kucherlapati, R. & Smith, G. R., eds), pp. 357-396, American Society for Microbiology, Washington, D.C.

Hochschild. A. (1990). Protein-protein interactions and DNA loop formation. In DNA Topology and: Its Bio&,%cal Effects (Cozzarelli, N. R. & Wang, J. C., eds), pp. 107-138, Cold Spring Harbor Laboratory Press, Cold Spring Harbor, NY.

orowitz, D. S. & Wang, J. C. (1984). The torsional rigidity of DNA and the length dependence of the free energy of DNA supercoiling. J. Mol. Bkol. 173, 76-91.

Hsieh; T.-S. $ Wang, J. C. (1975). Thermodynamic properties of superhelical DNA. Biochemistry, 14, 527-535.

Kanaar, R., van de Putte, P. & Cozzarelli, N. R. (1989). Gin-mediated recombination of catenated and knotted DNA substrates: implications for the mechanism of interaetion between cis-acting sites. Cell, 58, 147-159.

Klenin, K. V., Vologodskii, A. V., Anshelevich, V. V., Klisko, V. Y.; Dykhne, A. M. & Frank-Kamenetskii, M. D. (1989). Variance of writhe for wormlike DNA rings with excluded volume. J. Biomol. Struct. Dynam. 6, 707-714.

Klenin, K. V., Vologodskii, A. V., Anshelevich, V. V., Dykhne, A. M. & Frank-Kamenetskii, M. D. (1991). Computer simulation of DNA supercoiling. J. Mol. Biol. 217, 413-419.

Kornberg, A. & Baker, T. A. (1992). DNA Replication, 2nd edit.. W. H. Freeman & Co., New York.

Kratky, 0. & Porod, G. (1949). RGntgenuntersuchung geloster fadenmolekule. Rec. Trav. @him. 68, 1106-1122.

Laundon, C. H. & Griffith, J. C. (1988). Curved helix segments can uniquely orient the topology of super- twisted DNA. Cell, 52, 5455549.

Le Bret, M. (1980). Monte Carlo computation of supercoiling energy, the sedimentation constant, and the radius of gyration of unknotted and circular DNA. Riopolymers, 19, 619-637.

Levene, S. D. & Csothers. D. M. (1986a). Ring closure probabilities for DNA fragments by Monte Carlo simulation. J. MOE. Biol. 189, 61-72.

Levene, S. D. & Crothers, D. M. (1986b). Topological distributions and the torsional rigidity of DNA: a Monte Carlo study of DNA circles. J. Mol. Biol. 189, 73-83.

Lobell, R. B. & Schleif, R. F. (1990). DNA looping and unlooping by AraC protein. Science, 250, 5288532.

-McCammon, J A. & Harvey, S. C. (1987). In Dynamics of Protei?as and Nucleic Acids, pp. 68-70, Cambridge University Press, Cambridge, UK.

McClellan, J. A., Boublikova, P., Palecek, E. & Lilley, D. LM. ,J. (1990). Superhelical torsion in cellular DNA responds directly to environmental and genetic factors. Proc. Nat. Acad. Sci., U.S.A. 87, 8373- 8377.

Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. & Teller, E:. (1953). Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087-1092.

Mizuuchi, M. & Mizuuchi, K. (1989). Efficient Mu transposition requires interaction of transposase with a

DNA sequence at the Mu operator: implications for regulation. Cell, 58, 399-408.

Parker, C. N. & Halford, S. E. (1991). Dynamics of long- range interactions on DNA: the speed of synapsis during site-specific recombination by resolvase. Cell, 66, 781-791.

Reitzer, L. J. & Magasanik, B. (1986). Transcription of gllzA in E. coli is stimulated by activator bound to sites far from the promoter. Cell: 45. 7S5-792.

Richmond, T. J., Finch, J. T., Rushton, B., Rhodes, D. & Klug, A. (1984). Structure of the nucleosome core particle at 7 A resolution. Nature fLondon), 311. 532-537.

Seidl, A. & Hinz, H.-J. (1984). The free energy of DNA supercoiling is enthalpy-determined. f’r#nc. iliat. Acad. Sci., U.S.A. 81, 1312-1316.

Shibata, J. M., Fujimoto, B. S. & Schurr, J. M. (1985). Rotational dynamics of DNA from lQ-ia to lo-* seconds: comparison of theory with optical experiments. Biopolymers, 24, 1909-1930.

Shimada, J. & Yamakawa, H. (1984). Ring-closure probabilities for twisted wormlike chains. Application to DNA. Macromolecules, 17, 689-698.

Shimada, ,J. & Yamakawa, H. (1988). Moments for DNA topoisomers: the helical wormlike chain. Biopolymers, 27, 657-673.

Shore, D. & Baldwin, R. L. (1983a). Energetics of DNA twisting. I. Relation between twist and cyclization probability. J. Mol. Biol. 170, 957-981.

Shore, D. $ Baldwin, R. L. (1983b). Energetic5 of DNA twisting. II. Topoisomer analysis. J. .MoZ. Biol. 170, 983-1007.

Stigter, D. (1977). Interactions of highly charged colloidal cylinders with applications to double-stranded DNA. Biopolymers, 16, 1435-1448.

Stigter, D. (1985). Ionic charge effects ori the sedimentation rate and intrinsic viscosity of polyeleetrolytes. Macromolecules, 18, 1619-1627.

Tan, R. K. Z. & Harvey, S. C. (1989). IMolecular mechanics model of supercoiled DNA. ,I. Mol. Biol. 205, 573-591.

Taylor, W. H. & Hagerman, P. J. (1990). Application of the method of phage T4 DNA ligase-catalyzed ring- closure to the study of DNA structure. II. NaCl-dependence of DNA flexibility and helical repeat. J. Mol. Biol. 212, 363-376.

Wang, J. C., Peck, L. J. & Becherer, K. (1982). DNA supercoiling and its effects on DNA structure and function. Cold Spring Harbor Symp. Quant. Biol. 47, 85-91.

Wasserman, S. & Cozzarelli, N. R. (1991 I. Supercoiled DNA-directed knotting by T4 topoisomerase. J. Biol. Chem. 266, 20567-20573.

Weintraub, H. (1985). Assembly and propagation of repressed and derepressed chromosoms.1 states. Cell, 42, 705-711.

White, J. H. (1969). Self-linking and the Gauss integral in higher dimensions. Amer. J. Math. 91, 693-728.

White, J. H. (1989). An introduction to the geometry and topology of DNA structure. In Math,ematical Methods

j’or DNA Sequences (Waterman, M S.; ed.), pp. 225-253, CRC Press, Boca Raton, U.S.A.

White, J. H. (1991). Appendix. A formula for the average number of knot interlinks caused by type 2 topoisomerase action. J. Biol. Chem. 266; 20574--20575.

Yarmola, E. G., Zarudnaya, M. I. & Lazurkin, Y. S. (1985). Osmotic pressure of DNA solutions and effective diameter of the double helix. ,J. Biomol. Struct. Dynam. 2; 981-993.

CoFformation and Thermodynamics qf Supercoiled DNA 1243

Zacharias, W., Jaworski, A., Larson, J. E. & Wells, R. D. (1988). The B- to Z-DPU’A equilibrium in viva is perturbed by biological processes. Proc. Nat. Acad. Sci., U.S.A. 85, 7069-7073.

Zechiedrich, E. L. & Osheroff, N. (1990). Eukaryotic topoisomerases recognize nucleic acid topology by preferentially interacting with DNA crossovers. EMBO J. 9, 4555-4562.

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Alexander V. Vologodskii, Stephen D. Levene, Konstantin V. Klenin, Maxim Frank-Kamenetskii and Nicholas R. Cozzarelli- Conformational and Thermodynamic Properties of Supercoiled DNA