Valentin V. Rybenkov, Alexander V. Vologodskii and Nicholas R. Cozzarelli- The Effect of Ionic Conditions on the Conformations of Supercoiled DNA. I. Sedimentation Analysis

8/3/2019 Valentin V. Rybenkov, Alexander V. Vologodskii and Nicholas R. Cozzarelli- The Effect of Ionic Conditions on the Con

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The Effect of Ionic Conditions on the Conformationsof Supercoiled DNA. I. Sedimentation Analysis

Valentin V. Rybenkov1, Alexander V. Vologodskii2

and Nicholas R. Cozzarelli1*

1Department of Molecular andCell Biology, University ofCalifornia at Berkeley, BerkeleyCA, 94720, USA

2Department of ChemistryNew York University, NewYork, NY 10003, USA

We studied the conformations of supercoiled DNA as a function ofsuperhelicity and ionic conditions by determining its sedimentation coef-cient both experimentally and by calculation. To cancel out unknownparameters from both calculations and experiments, we determined the

ratio of the sedimentation coefcient, s, to that of open circular DNA, soc.Calculations of the sedimentation coefcient were based on direct sol-ution of the Burgers-Oseen problem for an equilibrium set of DNA con-formations generated for each condition by the Metropolis Monte Carloprocedure. There were no adjustable parameters in the Monte Carlosimulations because all three parameters of the DNA model used, bend-ing and torsional elasticity of DNA and DNA effective diameter specify-ing electrostatic interactions, were known from independent data. Thegood agreement between measured and calculated values ofs/soc allowedus to interpret the sedimentation results in terms of DNA conformations,with particular emphasis on the marked effect of ionic conditions. AsNaCl concentration decreases, s/soc increases because the superhelixbecomes less regular and more compact. In the presence of just 10 mMMgCl2, supercoiled DNA adopts essentially the same set of confor-

mations as in moderate to high concentrations of NaCl. Our simulationsshowed that s is a strong function of the superhelix branching frequency.At near physiological ionic conditions, there are about four branches inthe 7 kb DNA molecule used in this work. We found no indication ofsuperhelix collapse in any ionic conditions even remotely approachingphysiological ones. For all ionic conditions studied, we conclude that theelectrostatic interaction of DNA segments specied by the DNA effectivediameter is the primary determinant of supercoiled DNA conformations.

# 1997 Academic Press Limited

Keywords: DNA supercoiling; Monte Carlo simulations; DNA tertiarystructure; sedimentation; DNA collapse*Corresponding author

Introduction

Thirty years ago Vinograd and co-workers discov-ered the supercoiling of DNA (Vinograd et al.,1965). Since that time, it has become clear that theDNA in virtually all organisms is supercoiled andthat supercoiling is essential for the functions ofDNA. Recently, the conformational aspects ofDNA supercoiling have been analyzed by a varietyof experimental and theoretical methods. The ex-perimental studies principally employed electronmicroscopy, hydrodynamic measurements, and to-

pological methods (Laundon & Grifth, 1988;Adrian et al., 1990; Boles et al., 1990; Bednar et al.,1994; Langowski et al., 1994), while the theoreticalapproaches involved either computer simulations(Hao & Olson, 1989; Klenin et al., 1991; Schlick &

Olson, 1992a,b; Vologodskii et al., 1992; Bauer et al.,1993; Schlick et al., 1994a,b; Tesi et al., 1994; Gebeet al., 1995; Klenin et al., 1995; Vologodskii &Cozzarelli, 1995; 1996) or analytical treatment(Hearst & Hunt, 1991; Hunt & Hearst, 1991; Marko& Siggia, 1994, 1995; Shi & Hearst, 1994). As a re-sult of these studies, many conformational proper-ties of supercoiled DNA have been claried (for areview see Vologodskii & Cozzarelli, 1994). Weknow that the linking number difference, Lk, thequantitative measure of DNA supercoiling, is dis-tributed between the changing of the double helixtwist, Tw, and the folding of the DNA axis,which is quantitatively specied by writhe, Wr. Atphysiological salt conditions, the ratio Wr/Tw isabout equal to 3 and depends on neither supercoil-

J. Mol. Biol. (1997) 267, 299311

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ing density, s, nor DNA length for DNA longerthan 3 kb (Adrian et al., 1990; Boles et al., 1990;Vologodskii et al., 1992; Vologodskii & Cozzarelli,1994). The supercoiled DNA adopts interwound,

or plectonemic, conformations, which are often branched, as determined by electron microscopy(Rhoades & Thomas, 1968; Laundon & Grifth,1988; Adrian et al., 1990; Boles et al., 1990; Bednaret al., 1994), Monte Carlo simulation (Klenin et al.,1991; Vologodskii et al., 1992; Vologodskii &Cozzarelli, 1994), and the analysis of the productsof topoisomerases (Wasserman & Cozzarelli, 1991)and recombinases (Boles et al., 1990). Electron mi-croscopy studies determined that the averagesuperhelix winding angle is about 60 and doesnot depend on s (Adrian et al., 1990; Boles et al.,1990). An analysis of the products of site specicrecombination demonstrated that within Escherichia

coli cells, supercoiled DNA is plectonemic (Bliska& Cozzarelli, 1987; Bliska et al., 1991).

Despite the good agreement between these basicconformational features of supercoiled DNA ob-tained by different techniques and different groupsof investigators, for two aspects of supercoilingthere are troubling differences between various ex-perimental and computational data. The rst con-cerns the branching of the superhelix. According tothe simulation results, the average number ofsuperhelix branches should be proportional toDNA length if the length is larger than 3 kb (Volo-godskii & Cozzarelli, 1994; 1996) with a propor-

tionality of one branch per 1.7 kb. The branchingfrequency measured by electron microscopy, how-ever, differs greatly in different studies, with branching values both greater and lesser than oneper 1.7 kb (Laundon & Grifth, 1988; Boles et al.,1990).

The second discrepancy concerns the effect of ionicconditions on the diameter of the superhelix. Bothexperiments and theory indicate that ionic con-ditions are important determinants of the confor-mational properties of topologically constrainedDNA (Klenin et al., 1988; Adrian et al., 1990; Hunt& Hearst, 1991; Klenin et al., 1991; Vologodskiiet al., 1992; Rybenkov et al., 1993; Shaw & Wang,

1993; Bednar et al., 1994; Schlick et al., 1994a; Tesiet al., 1994). Recently, Dubochet and co-workersobserved by cryoelectron microscopy (cryo EM) adramatic effect of ionic conditions on confor-mations of supercoiled DNA. In the presence of atleast 10 mM MgCl2 or 0.1 M NaCl, this group ob-served a collapse of the plectonemic superhelix(Adrian et al., 1990; Bednar et al., 1994), resulting ina structure in which there is no visible space be-tween the opposing segments of the interwoundsuperhelix. Thus, the superhelix diameter is justtwice the double helix diameter. Because in vivothe concentrations of ions are higher than the

threshold for collapse, they argued that the col-lapsed form of the superhelix is the physiologicallyrelevant conformation of DNA. The collapse of thesuperhelix was never observed by conventionalelectron microscopy. Nor was it observed in com-

puter simulations of supercoiling because of ther-mal uctuations and electrostatic repulsionbetween DNA segments. However, we have onlylimited knowledge about the interaction between

DNA segments separated by very short distances.An attraction of DNA segments such as that pro-vided by di- and polyvalent cations (Rau &Parsegian, 1992; Shaw & Wang, 1993; Ma &Bloomeld, 1994; Bloomeld, 1996) could conceiva- bly change the simulation results enough to causecollapse of the superhelix.

These questions cannot be solved by theoreticalanalyses or by further EM studies alone. Theoreti-cal analysis requires the use of a simplied modelof DNA, so the results will depend on the initialchoice of model. The major limitation of EM is thatlabile features of superhelix conformations may besubstantially altered during sample preparation,

and it is not known when during these manipula-tions the DNA is xed in its nal observed form(discussed by Vologodskii & Cozzarelli, 1994).Conventional electron microscopy is limited bysample deformation during changes in the solutioncomposition, dehydration, staining, shadowing,and adhesion to the grid surface. In cryoelectronmicroscopy, it is not known if the cooling is rapidenough to prevent conformational changes, andstrong surface forces are also clearly present andmay deform the DNA. Therefore one needs ad-ditional methods in which the DNA is in true sol-ution.

Hydrodynamic and optical methodshave clear ad-vantages, because they measure structure-depen-dent features in well-dened solutions whileperturbing conformation only minimally. Sedimen-tation analysis, the rst physical probe of super-coiled DNA conformations, showed that thesedimentation coefcient, s, increases with the ab-solute value of superhelix density, s, reaches amaximum, and then slightly decreases to a localminimum (Wang, 1969, 1974; Upholt et al., 1971).The maximum value of s is greater than that ofopen circular DNA by 30 to 60%, depending onDNA length. Since the sedimentation coefcient re-ects an overall compactness of the molecule, these

results showed clearly the range ofs in which theformation of superhelix structure takes place.However it was difcult to make additional struc-tural conclusions from the data themselves. Thevalue of such data is augmented greatly whencombined with a theoretical analysis that allows astructural interpretation of the experimental re-sults. This dual approach has been used in thestudy of the conformational properties of linearDNA molecules (for example, see review by Ha-german, 1988) and of short supercoiled DNA mol-ecules (Langowski et al., 1994).

Therstattempttoapplyacombinedexperimental

and theoretical approach to supercoiled DNA wasmade 18 years ago by Camerini-Otero andFelsenfeld (1978). Only recently, however, have ad-vances in computational opportunities allowed thestatistical-mechanical analysis of supercoiled DNA

300 Sedimentation of Supercoiled DNA


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conformations, the method which must be used insuch an approach. One can now choose a model ofthe DNA double helix and calculate a measurableproperty such as the sedimentation coefcient orthe radius of gyration of supercoiled DNA fromthe simulated conformational distribution. A com-parison of the calculated and measured data showsthe degree of agreement between the simulatedconformations and the actual ones, and allows spe-cication of the difference in conformational terms.Here, we performed an extensive study of super-coiled DNA conformations by comparing themeasured and computed sedimentation coefcientsof these molecules. In the accompanying paper(Rybenkov et al., 1997a), we describe a dual theor-etical and experimental analysis of supercoiling based on measures of catenation probability. Incontrast to the sedimentation coefcient, which de-pends primarily on the overall size and the shapeof the molecule, the catenation probability reectsmore local features of superhelix conformations,and therefore these two methods complement eachother.

Here, we studied the sedimentation coefcient of

supercoiled DNA as a function of superhelical den-sity, s, in solutions containing 0.01 M to 3.0 MNaCl. The same dependencies of s on s and onNaCl concentration were then computed and com-pared with the experimental data. We used 7 kbmolecules rather than the smaller ones often usedpreviously, because they are branched and there-fore have more of the essential features of super-coiled DNA. We found a good agreement betweenthe experimental and theoretical results. This al-lowed us to interpret in conformational terms theobserved changes in sedimentation coefcient. Wealso measured the s values over a range of MgCl2

and spermidine concentrations, including thosewhich were reported to promote the collapse of in-terwound superhelix. However, we found no indi-cation of such a collapse under any conditionseven remotely approximating physiological ones.

The results of the accompanying paper reinforceand extend these conclusions, and we defer to thelatter a general discussion of the results.

Results and Discussion

Sedimentation of supercoiled DNA inNaCl solutions

For each particular ionic condition and s value we

measured the sedimentation coefcient at severalDNA concentrations and then extrapolated thevalue to a zero DNA concentration. The values ofthe DNA sedimentation coefcient reect both theDNA conformations in a particular solvent and thephysical-chemical properties of the solution, suchas solvent viscosity and the DNA partial specicvolume. The values of the DNA partial specic vo-lume in NaCl and CsCl solutions have beenmeasured previously (Cohen & Eisenberg, 1968).However, no such accurate data are available forMgCl2 or spermidine-containing solutions. There-fore, we chose to monitor the relative sedimen-tation coefcient, s/soc, the ratio of sedimentation

coefcients of supercoiled and open circular DNA,so that the solution physical-chemical propertieswould cancel.

The extrapolated values ofs for the open circularDNA in all ionic conditions studied are shown inTable 1. For NaCl solutions, the values of sedimen-tation coefcient corrected to standard conditions,s20,w, are also shown. The change of s20,w reectsthe change in conformations of open circular DNAwith salt concentration and similar changes of s20,whave been observed for linear DNA (Rinehart &Hearst, 1972). The standard deviation in the s va-lues averages to only 0.5% of the mean, which is

much lower than the change in soc with ionic con-ditions that we measured.We measured the sedimentation coefcients of

7 kb supercoiled DNA as a function of superhelixdensity, s, in 0.2 M NaCl and in 0.01 M NaCl

Table 1. Sedimentation coefcient of 7.0 kb open circular DNA at different ionic conditions

Ionic conditions Measured Calculated[Spermidine]

[Na](M) [Mg2](M) (mM) soc s20,w s20,w

3.0 11.33 0.11 22.1 20.00.2 19.24 0.05 21.0 19.70.07 19.88 0.16 20.6 ND0.03 19.67 0.09 19.8 19.30.01 18.68 0.03 18.7 18.80.1 0.01 20.45 0.08 ND ND0.01 0.01 20.95 0.09 ND ND0.001 0.01 21.22 0.10 ND ND0.001 0.1 20.18 0.02 ND ND0.02 0.01 3.5 20.89 0.05 ND ND

The sedimentation coefcients for open circular DNA are given in Svedberg units without ( soc) and with (s20,w) correction to standardconditions. The correction of s to standard conditions was done as described previously (Cohen & Eisenberg, 1968; Rinehart &Hearst, 1972). The calculated values of s20,w are shown for the chain with the bead diameter, a, equal 3.18 nm (Hagerman & Zimm,1981). Both measured and calculated values of s20,w are rather sensitive to the values of the DNA buoyant density, dradCmemployed for correction. Therefore, throughout the paper we compare the values of the relative sedimentation coefcient, s/soc,which are independent of the correction procedure. The errors indicate standard deviations. ND, not determined.

Sedimentation of Supercoiled DNA 301


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(Figure 1) and found that the conformations ofsupercoiled DNA are different for these two ionicconditions. Supercoiling compacts DNA and there-fore lowers its frictional coefcient and increasess/soc for both ionic conditions, but this effect isgreater at the low salt concentration. Moreover, thecompaction reaches a maximum at s of 0.04 to0.05 for the high salt concentration, but no maxi-mum is observed at the low concentration of NaCl.The dependence ofs/soc on s that we measured in0.2 M NaCl is in good agreement with the resultsobtained previously for DNA of a similar size in3 M CsCl (Wang, 1974).

We calculated the dependence of s/soc on s for

both 0.2 M NaCl and 0.01 M NaCl. The calcu-lations were made for random samples of DNAconformations obtained by Monte Carlo simu-lation, a computational method that generates theequilibrium conformational distribution for amodel DNA chain. In the range of ionic conditionsused in this study, the only parameter of themodel that varied as a function of ionic conditionswas the DNA effective diameter. This parameterspecies the electrostatic intersegment interactionand its values have been accurately dened (seeMethods for details).

Figure 2 shows the results of the calculations to-

gether with experimental data of Figure 1 demon-strating good agreement, within 5%, between theexperimental data and the calculations. Through-out this and the accompanying paper, themeasured data are represented by open symbols

and the calculated data by lled symbols. Thedifferences between calculated and measured va-lues are greater than the error of the methods, butare signicantly less than the changes of s/soc withNaCl concentration. The calculated values repro-duce the shapes of the curves even to the nding

of a maximum ofs/s

ocat high but not low salt con-

centration. This agreement is especially gratifyingbecause there were no adjustable parameters in thecalculations. This convergence of values gives uscondence that the picture of supercoiled DNA

Figure 1. The measured relative sedimentation coef-cient of DNA as a function of superhelix density andsalt concentration. The measurements were for 7.0 kbDNA in 0.2 M NaCl (*) and 0.01 M NaCl (!). The rela-tive sedimentation coefcient, s/soc, is the value of thesedimentation coefcient normalized to that of open cir-cular DNA for the corresponding conditions. s, super-helix density.

Figure 2. Comparison of the measured and calculatedrelative sedimentation coefcients for 7 kb DNA. Thedata correspond to NaCl concentrations of 0.2 M (a) and0.01 M (b). Throughout this and the accompanying

paper, measured data are shown by open symbols andcalculated data are depicted with lled symbols. Theerror bars for simulation data correspond to one stan-dard deviation and are shown only when larger thanthe symbol size.



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conformations we obtained from computer simu-lations faithfully reects the actual conformationalproperties of supercoiled DNA in solution, and al-lows us to make a detailed interpretation of the ex-perimental results in conformational terms.

Structural interpretation

The maximum of s/soc near a s value of 0.05 athigh ionic strength (Figure 2) has previously been

observed (Wang, 1969,1974; Upholt et al., 1971).The interpretation initially suggested was that thesuperhelix changed from a toroidal form to themore extended plectonemic form in this range ofs. Our simulation results support a different in-

terpretation of the maximum: they show that theplectonemic form is the only regular conformationof supercoiled DNA. This superhelix is, however,often branched (Figure 3), which compacts themolecules and therefore increases the value of s.Because branching frequency decreases for highersupercoiling (Vologodskii et al., 1992), s shouldalso decrease, resulting in a maximum in s causedby a maximum in branching.

Neither the experimental data nor the simulationsshow a maximum of s in 0.01 M NaCl (Figure 2b).The simulated conformations are more open andloose at this ion concentration (Figure 3b). They areso irregular at all values ofs that even a direct in-spection of a conformation does not lead to an un-

Figure 3. Typical simulated conformations of supercoiled DNA in solution containing 0.2 M NaCl (a) and 0.01 MNaCl (b). The conformations of the model chains correspond to DNA 7 kb in length and a s value of 0.05.



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ambiguous denition of the number of branches.Thus, it is not surprising that s monotonicallychanges with s (Figure 2b).

Among the parameters describing specic featuresof supercoiled DNA, we were particularly inter-ested in branching frequency and in the averagesuperhelix diameter. We developed a special com-putational experiment to determine the depen-

dence of the calculated values of s on theseparameters. We separated simulated conformationsof supercoiled DNA (s 0.05) according to thenumber of the superhelix branches and then calcu-lated the average values ofs/soc as a function of the branch number for both 0.2 M and 0.03 M NaClsolutions (Figure 4). For both salt concentrations,we found that the sedimentation coefcient is pro-portional to the number of branches, whichstrongly affect the average size of supercoiled mol-ecules. However, we did not detect an effect of thesuperhelix diameter on the value ofs. The value ofthe average superhelix diameter increases 30%when NaCl concentration is reduced from 0.2 M to0.03 M (Table 2). This increase would be predictedto cause a decrease in s (see equation 4 inMethods). Instead we found a small increase in s/soc at the lower salt concentration (Figure 4) which

can be explained by small changes in the globalsize of open circular and supercoiled DNA. Thus,the value of s/soc depends primarily on the globalsize and shape of supercoiled molecules and ismuch less sensitive, if at all, to the superhelix di-ameter. Figure 4 also shows that a small difference

Figure 4. The calculated relative sedimentation coef-cient of supercoiled DNA as a function of superhelix

branch number. The set of simulated conformations wasdivided into subsets according to the number of super-helix ends of the simulated conformations, and thevalues of s/soc were calculated separately for the eachsubset. The number of branches is approximately twoless than the number of ends. The calculations corre-sponded to supercoiled DNA 7 kb in length, s 0.05,in 0.2 M NaCl (*) or 0.03 M NaCl (!). For 0.2 M NaCl,the average value of s/soc for all simulated confor-mations (Computed s/soc) and the measured value

(Measured s/soc) of s/soc are also shown.

Table 2. Variation of DNA effective diameter and super-helix diameter (s 0.05) as a function of NaClconcentration

NaCl concentration

DNA effective

diameter Superhelix diameter(M) (nm) (nm)

0.01 15 170.02 11 140.03 9 130.2 5 101.0 3 9.5

The values of DNA effective diameter used in the simulationsare shown (Stigter, 1977; Brian et al., 1981; Rybenkov et al.,1993; Shaw & Wang, 1993). The superhelix diameter, dsh, isdened as the diameter of an idealized regular superhelixwhich has the same DNA contour length, L, superhelix pitchangle, a, number of superhelix ends, ne, and Wr as the averagevalues of irregular superhelices (Boles et al., 1990). We calcu-lated superhelix diameter using:

dsh 2Lpne 4Wra sin2a

Y

where Wr, a, and ne were the average values obtained fromcomputer simulations.

Figure 5. The measured relative sedimentation coef-cients of 7 kb DNA in solutions containing MgCl 2. Theopen symbols correspond to solutions containing 10 mMMgCl2 and different concentrations of NaCl: 1 mM (!),10 mM (^), and 100 mM (~). The continuous line cor-responds to a solution containing 0.2 M NaCl only

(reproduced from Figure 1). The broken line corre-sponds to the data measured for a solution containing100 mM MgCl2 plus 1 mM NaCl (&) or 3.5 mM spermi-dine plus 10 mM sodium phosphate and 10 mM MgCl2(,).



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in branching of actual and simulated confor-mations could be responsible for the discrepancybetween measured and simulated values ofs.

Sedimentation of supercoiled DNA in solutionscontaining di- and trivalent cations

It is well known that many di- and trivalent cat-ions affect DNA conformations much morestrongly than monovalent ions. We studied rstthe effect of a moderate concentration of MgCl2(10 mM) on the sedimentation of supercoiledDNA. We found that the dependence ofs on s in10 mM MgCl2 is very similar to that in 0.2 M NaCl(Figure 5). Addition of 0.01 M or 0.1 M NaCl to asolution of 10 mM MgCl2 did not change this de-pendence of s on s. Thus, the presence of 10 mM

MgCl2 eliminated the change of supercoiled DNAconformations between 0.01 M and 0.2 M NaCl.We conclude that within the physiological range ofsalt conditions, supercoiled DNA adopts essen-tially the same set of conformations.

We next studied the sedimentation of supercoiledDNA in extreme ionic conditions, in which we ex-pected some attraction between double helix seg-ments. We observed that 100 mM MgCl2 loweredvalues of s/soc for supercoiled DNA (Figure 5). Wefound a very similar dependence of s/soc on s inthe presence of 3.5 mM spermidine, a concentrationthat results in aggregation of approximately 50% of

the DNA (Krasnow & Cozzarelli, 1982). The valuesof s/soc for the non-aggregated fraction of DNA areshown in Figure 5. A lower concentration of sper-midine, 1.5 mM, did not cause as signicant a re-duction in the value of s/soc as did 3.5 mMspermidine (data not shown). The change of thesedimentation rate under these extreme ionic con-ditions can be explained by the elimination ofsuperhelix branching. As shown in Figure 4, an un- branched superhelical DNA would have s/soc of1.27, similar to the value obtained. One interpret-ation of the reduction in branching is that thesuperhelix has collapsed. Indeed, Dubochet andco-workers observed by cryo EM a signicant re-

duction of superhelix branching for collapsed con-formations of the superhelix (Adrian et al., 1990;Bednar et al., 1994). In the accompanying paper,though, we show that the superhelix has not col-lapsed under even these extreme ionic conditions.This conclusion is supported by the data of Ma &Bloomeld (1994) who found no indications ofDNA condensation at any concentrations ofMgCl2. Unfortunately, we cannot use computersimulations to determine the superhelix confor-mations under these conditions, which limits ourinterpretation of the corresponding data. The hardcore approximation of the electrostatic interaction

used in the simulations can fail for ionic conditionsunder which DNA segments attract one another asis the case at 100 mM MgCl2 and at 3.5 mM sper-midine (Shaw & Wang, 1993; Rybenkov et al.,1997).

Electrostatic intersegment interaction definesthe dependence of supercoiled DNAconformations on ionic conditions

OurresultsshowthatconformationsofsupercoiledDNA depend strongly on ionic conditions. Thiswas predicted by computer simulations (Kleninet al., 1991; Vologodskii et al., 1992; Vologodskii &Cozzarelli, 1994) and by theoretical analysis (Hunt& Hearst, 1991) and was also supported by cryoEM (Adrian et al., 1990; Bednar et al., 1994). Thisdependence results primarily from the electrostaticrepulsion of DNA segments in the dense confor-mation of the interwound superhelix. We specify

the value of the intersegment electrostatic repul-sion in terms of DNA effective diameter, d (Stigter,1977); a parameter whose value changes from 3 nmfor 1 M Na to 15 nm for 0.01 M Na (Stigter,1977; Brian et al., 1981; Yarmola et al., 1985;Rybenkov et al., 1993; Shaw & Wang, 1993). Thesechanges cause signicant alterations in the confor-mations (Figure 3) and the sedimentation (seeFigure 1) of supercoiled DNA.

We recently measured the values ofd for solutionscontaining mixtures of sodium, magnesium andspermidine ions (Rybenkov et al., 1997b). This al-lowed us to compare the calculated and measured

results for all ionic conditions studied. The exper-imental and calculated values of s vary onlyslightly near a s of 0.05 (Figure 2), and thus wechose s/soc at this s as a characteristic value for par-ticular ionic conditions. Both the measured and

Figure 6. The measured and calculated relative sedimen-tation coefcients of supercoiled DNA as a function ofthe DNA effective diameter, d. The DNA had a s of0.05. The measured values are for solutions containingdifferent amounts of NaCl (*), solutions containing10 mM MgCl2 and different concentrations of NaCl (^),100 mM MgCl2 plus 1 mM NaCl (&), and 3.5 mM sper-midine plus 10 mM sodium phosphate and 10 mMMgCl2 (,). Calculated values are shown with the lledcircles (*).



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simulated values of s/soc showed, within error scat-ter, monotonic dependence on d (Figure 6). There-fore, we conclude that the DNA effective diameteradequately species the conformational properties

of supercoiled DNA not only for NaCl solutions but also for solutions with di- and trivalent ions.This issue is discussed in greater detail in the ac-companying paper (Rybenkov et al., 1997a).

Methods

DNA samples

The 7.02 kb plasmid, pAB4 (Wasserman et al., 1988) waspuried by the Triton lysis method (Ausubel et al., 1989).To vary the superhelicity of the plasmid, the DNA wasrelaxed by wheat germ topoisomerase I (Dynan et al.,1981) in the presence of different amounts of ethidium

bromide. It was then extracted successively with phenol,phenol:chloroform and chloroform, precipitated withethanol, resuspended in TE and ltered through a Sepha-rose CL-4B column. Seven different samples of closedcircular DNA were prepared and their superhelical den-sity was measured by band counting (Keller, 1975) rela-tive to a sample relaxed in 0.1 M NaCl. To determine thesuperhelical densities of the samples at the ionic con-ditions of interest, we took into account the change inDNA helical repeat with temperature and ion concen-tration. The temperature coefcient, s/T, has beenshown to be 3.1 104 degree1 for all ionic conditionsstudied (Wang, 1969; Upholt et al., 1971; Depew &Wang, 1975; Duguet, 1993). Variation of the superhelicaldensity with NaCl and MgCl2 concentration is alsoaccurately known (Anderson & Bauer, 1978; Rybenkovet al., 1997b). We used the value of s/lg[Na] 2.5 103 found for the 0.01 to 0.2 MNaCl solutions (Rybenkov et al., 1997). For all samplesand ionic conditions studied, the value of s was deter-mined with an accuracy of at least 0.001. The only ex-ception to this is the value in 3 M NaCl which wasestimated by extrapolation and therefore could be inerror by as much as 0.005.

The extent of DNA aggregation by spermidine was de-termined by centrifugation (Krasnow & Cozzarelli,1982). Singly nicked circular DNA was prepared by lim-ited digestion by DNase I in the presence of ethidium

bromide (Barzilai, 1973).The nucleotide sequence of pAB4 DNA contains no long

regions with a high AT-content, alternating purine-pyri-midine stretches, or palindromic sequences which couldform open regions, Z form segments, and cruciforms, re-spectively, under low torsional stress. We also used astatistical-mechanical analysis (Anshelevich et al., 1988)with the same objectives, which showed the absence ofstable non-canonical structures at any s between 0 and0.06.

Experimental measurement ofsedimentation coefficients

The analytical ultracentrifuge Optima XL-A (Beckman)was used to measure the sedimentation coefcient, s. Foreach sample we measured the value of s for four to eight

DNA concentrations in the range of 10 to 60 mg/ml andextrapolated the data to a zero DNA concentration. Theextrapolated sedimentation coefcients were obtainedwith a precision of 0.5%. All measurements were done at20C. The extrapolated values of s for open circular

DNA at all ionic conditions studied are shown in Table 1,together with their standard deviations. For MgCl2-con-taining solutions, these values are not corrected to stan-dard conditions, s20,w, because we lack accurate data onthe corresponding values of the DNA partial specic vo-

lume. However, this correction is unnecessary becausewe report exclusively the relative sedimentation coef-cient for supercoiled DNA, i.e. the ratio of the sedimen-tation coefcient of supercoiled DNA to that of opencircular DNA.

Each DNA sample contained a mixture of topoisomerswith an average value of the superhelix density, hsi. Asa result, the measured value of s, hsi was slightly differ-ent from the value of s at the center of the distribution,s(hsi), which was the calculated value. This difference,however, was small. The distribution of topoisomers can

be described by a Gaussian with the average value hsiand the variance h(ds)2i. Using this we can approximatehsi as:

hsi shsi 12

d2ssds2

hds2iX

The value of h(ds)2i was equal to (1.2 0.3) 105 forour samples, and

d2ss

ds2

did not exceed 2 104 for all experimental points. Thusjhsi s(hsi)j was always less than 0.2 S.

Calculations of sedimentation coefficient

To calculate the value of s we used the usual replace-

ment of exible DNA molecules by an ensemble of rigid,randomly selected conformations. (For a discussion ofthis approximation, see Zimm, 1980.) To use this ap-proach we rst simulated the sets of conformations ofsupercoiled DNA for different conditions. These sets cor-responded to the equilibrium conformational distri-

butions of the model chains. In the second step, wecalculated the average sedimentation coefcients for thesets. We next describe the model of the double helixused in the simulations.

The DNA model

The model is a discrete analog of the worm-like chain

which includes additional volume interactions. A DNAmolecule composed of n Kuhn statistical lengths is mod-eled as a closed chain consisting of kn rigid segmentsthat are cylinders of equal length. The elastic energy ofthe chain, Eb is computed as:

Eb RTakni1

y2i 1

where the summation extends over all the joints betweenthe elementary segments, R is the gas constant, T is theabsolute temperature, yi is the angular displacement ofsegment i relative to segment i 1, and a is the bendingrigidity constant. The bending constant a is dened sothat the Kuhn statistical length corresponds to k rigid

segments (Frank-Kamenetskii et al., 1985).We showed before that the simulated properties of

supercoiled conformations do not change, within the ac-curacy of the simulations, if k5 10 (Vologodskii et al.,1992). We used a kvalue of 10 in this work.



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The model also takes into account the volume inter-actions between DNA segments. We incorporated theseinteractions into the model via the concept of the effec-tive diameter, d, the actual diameter of the cylindricalsegments of the model. Its value takes into account not

only the physical diameter of DNA, but also the electro-static repulsion between DNA segments. The effectivediameter of DNA is dened as the diameter of an un-charged model chain that mimics the conformationalproperties of actual DNA. Its quantitative denition is

based on the concept of the second coefcient of virialexpansion (Stigter, 1977). A more accurate approxi-mation of electrostatic interactions based on the Debye-Hu ckel potential yields similar results (Vologodskii &Cozzarelli, 1995).

The energy of torsional deformation is considered to beindependent of bending deformation and proportional tothe displacement of DNA twist from its equilibriumvalue, Tw. The torsional energy was calculated as (Hao& Olson, 1989):

Et 2p2CaLLk Wr2 3

where Wr is the writhe of the chain (White, 1989), Lk isthe linking number difference of the supercoiled DNA, Cis the torsional rigidity constant, and L is the DNAlength. The value ofLk was a parameter in each simu-lation and Wr is calculated for each conformation.

Thus, there are three parameters of the model. All arewell known from independent studies. The rst par-ameter, the Kuhn statistical length, is about 100 nm forsolutions containing more than 0.01 M monovalent ionsor more than 1 mM multivalent ions (Hagerman, 1988).We use a value of 3 1019 erg cm for C (Hagerman,1988; Klenin et al., 1989; Rybenkov et al., 1997). This

value of C gave better agreement between experimentaland computed results than the lower value, 2 1019 ergcm, accepted by some other workers (Gebe et al., 1996;Heath et al., 1996). For the lower value of C, the samevalue Wr is achieved at higher Lk. Since we considertorsional and bending distortions of DNA independentfrom each other, the conformations of supercoiled DNAare dened entirely by the value of Wr. Therefore, forthe lower value of C, the calculated values of the sedi-mentation coefcient would correspond to the DNAwith the higher degree of supercoiling. This increase insuperhelical density that yielded the same value of Wrdepended on ionic conditions and varied over the rangeof 10 to 15% of the value found for C equal 3 1019 ergcm. The third parameter, DNA effective diameter, d, de-

pends strongly on ionic conditions. For solutions ofmonovalent ions the values of d are known with accu-racy from both experimental and theoretical methods(Stigter, 1977; Brian et al., 1981; Rybenkov et al., 1993;Shaw & Wang, 1993; Stigter & Dill, 1993). We used thevalues ofd shown in Table 2.

Generation of an equilibrium set of conformations

We used the Metropolis Monte Carlo procedure to simu-late the equilibrium set of conformations of supercoiledmolecules as previously detailed (Vologodskii et al.,1992). In this procedure the set of system states is the re-sult of successive deformations from an initial state. The

deformations included two types of motions. The rsttype can be described as a crankshaft rotation (Kleninet al., 1991). Here we modied this type of motion to in-crease the rate of sampling of different conformations ofthe interwound superhelix. A portion of the chain con-

taining an arbitrary number of adjacent segments wasrotated by an angle, f, around a line connecting theends of the sub-chain. The value off was uniformly dis-tributed over a range (f0, f0). If the end-to-end dis-tance of the sub-chain, R, exceeded the length of fourchain segments, no rotation was made and the currentconformation was counted as a trial one. The value ofthe f0 was chosen so that the probability of acceptancefor a crankshaft move, for R < 4, was about 0.5. By ex-cluding rotations for large values of R (which arestrongly restricted in an interwound superhelix) wecould increase by a factor of 10 the average amplitude ofrotation for small values of R. Some of these rotationscorrespond to a bending of an interwound superhelixand thus are important in obtaining an equilibriumsampling of global superhelix conformations. This typeof move was R-independent if jsj4 0.02, conditionsunder which DNA conformations are loose, and onlyscattered elements of an interwound superhelix are pre-

sent.The second type of motion, reptation, was introducedinto the simulations to increase the rate of change of thesuperhelix branch number (Vologodskii et al., 1992).There were about 20 attempts of rotation per one at-

Figure 7. Variation of the calculated number of superhe-

lix ends (a) and relative sedimentation coefcient (b)during a simulation run. Each point corresponds to anaverage of over 106 trial moves. The data show thateven for these most slowly changing features of simu-lated conformations, there was no notable correlation

between successive conformations separated by morethan 107 steps.



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tempt of reptation for moderate to highly supercoiledDNA. We used only R-independent crankshaft rotationfor low jsj.

We tested each trial conformation for overlap of non-ad-jacent segments. A trial conformation was rejected if the

distance between two non-adjacent segments was smal-ler than d. The starting conformations were unknotted.However, this did not guarantee that the chain remainedunknotted, because segments of the chain were allowedto cross during trial moves. Therefore we checked thetopology of the trial conformation by evaluating theAlexander polynomial, (t), at t 1 (Frank-Kamenetskii & Vologodskii, 1981). If a trial conformationwas knotted, it was rejected.

If a trial conformation passed the tests of absence ofchain overlap and knotting, its elastic energy was calcu-lated as a sum of Eb and Et (equations 1 and 3). This re-quired calculating the writhe of the trial conformation,which is most conveniently done by the method of LeBret (1980). The probability of accepting a trial confor-mation was obtained by applying the classical rules ofMetropolis et al. (1953).

The current conformation did not change after moststeps of this algorithm for jsj > 0.02, because R wasgreater than four for these steps. Although one mustconsider these steps as unsuccessful moves, they tookvery little computer time. Therefore we were able to useas many as 4 108 steps for sampling highly supercoiled7 kb DNA for each particular condition. Each run tookabout 100 hours of processor time on a Silicon GraphicsIndigo2 workstation. There was no notable correlation

between successive conformations separated by morethan 107 steps. This is illustrated in Figure 7, in whichthe changes of the branch number and the sedimentation

coefcient, the most slowly changing features of super-coiled DNA, are shown for a quarter of a simulationrun.

There are several reasons to conclude that the algorithmdescribed provides equilibrium sampling of DNA con-formations. We found that our algorithm gave the sameresults when starting from very different initial confor-mations. Our previous simulation results were conrmed

by the experimental data (Vologodskii et al., 1992;Vologodskii & Cozzarelli, 1994). Some of our earliersimulation results were also conrmed recently by Gebeet al. (1995) who used a slightly different DNA modeland different moves in their Metropolis algorithm.

Weusedanalgorithmpreviouslydescribed(Vologodskiiet al., 1992) to calculate the number of superhelix

branches in a particular conformation.

Calculation of sedimentation coefficient

To calculate s values for simulated conformations, wesubstituted for a chain consisting of kn straight cylindri-cal segments a chain consisting of 3kn touching beads.The diameter of the beads can be varied by changing thesegment length. Values close to 3 nm were used for the

bead diameter to approximate the hydrodynamic prop-erties of the double helix (Kovacic & van Holde, 1977;Hagerman & Zimm, 1981; Langowski et al., 1994). Wefound that the ratio of s values for supercoiled and opencircular DNA, s/soc, does not depend on bead diameter if

it is between 2.5 and 3.33 nm. Thus, to reduce the num-ber of beads in the chain, we used a diameter of 3.33 nmfor most of the simulations.

WeusedtheKirkwood-Risemanapproximationtocalcu-late the sedimentation coefcient of each individual

chain conformation, sm, from the constructed equilibriumset (Bloomeld et al., 1974):

sm

dr

dC

m

M

NA

1

N

1

3pZ0a

1

r

N

Ni1

Ni1

r1ij

4

where NA is Avogadro's number, M is the molecularweight of DNA, dradCm is the DNA buoyant density, Nis the number of the beads in the chain, Zo is the vis-cosity of solution, a is the diameter of the beads, and rijis the distance between beads i and j. We then calculated

Figure 8. Comparison of the sedimentation coefcientscalculated using the Kirkwood-Riseman approximation,sKR, with that obtained from a direct solution of the Bur-gers-Oseen problem, sBO. The calculations were done for

supercoiled DNA 7 kb in length. For each value of sand a NaCl concentration of 0.2 M (a) or 0.01 M (b), wecalculated the average values of sKR/sBO over threeorientations of 25 randomly chosen conformations.



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the average value ofs for the conformational set. It is im-portant that the only term in this expression that de-pends on DNA conformation is r1ij; all other termscancel when we calculate s/soc.

Although we avoided the pre-averaging of the hydro-

dynamic interaction between chain segments throughdifferent chain conformations, equation (4) still givesonly an approximate solution of the Burgers-Oseen pro-

blem. To evaluate the quality of the approximation, wecompared, for a part of the conformations, the results ofthe Kirkwood-Riseman approximation with a direct sol-ution of the Burgers-Oseen problem as proposed byZimm (1980).

For any particular conformation of a chain described bythe position ofNbeads, the Burgers-Oseen problem is re-duced to a system of (3N 4) linear equations. The coef-cients of these linear equations are composed mainlyfrom the tensor of hydrodynamic interaction. We usedthe Rotne-Prager approximation (Rotne & Prager, 1969)for the tensor:

Tij 1

8pZ0rij

I

rijrij

r2ij

a2

2r2ij

1

3I

rijrij

r2ij

!5

where rij species the interparticle distances, rij ri rj,and I is the unit tensor. The rst term in the sum, whichis proportional to 1/rij, is the commonly used Oseen ten-sor, and the second term accounts for nite bead size.

The value of Nwas equal to 708 in our case (when thediameter of the beads was equal to 3.33 nm) and it took20 minutes to solve the system for one orientation of oneparticular conformation of the chain on the SiliconGraphics Indigo2 workstation.

We used both the Kirkwood-Riseman approximationand the exact solution to calculate the sedimentationcoefcient for three different orientations of 25 uncorre-lated conformations for each set of superhelix densityand ionic conditions. We found that the Kirkwood-Rise-man equation overestimates the sedimentation coefcient

by 5 to 10% for our model chain, depending on DNAsuperhelix density (Figure 8). Using these data we ob-tained a correction coefcient for the Kirkwood-Risemanapproximation. The value of the coefcient, Kcorr(s),changes from 1.0 for s equal 0 to approximately 0.96 fors equal 0.06 (Figure 8). Thus, to calculate the sedimen-tation coefcient for each s and salt concentration, theaverage s/soc obtained by the Kirkwood-Riseman ap-proximation for a large array of simulated conformationswas multiplied by Kcorr(s).

Acknowledgements

We thank C. Benham, H. Schachman, D. Stigter and B.Zimm for helpful discussions and H. Schachman for useof his ultracentrifuge. The work was supported by NIHgrant GM31657 to N. R. C. and NIH grant GM54215 toA. V. V.

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Edited by D. E. Draper

(Received 15 August 1996; received in revised form 12 December 1996; accepted 12 December 1996)


Documents

Valentin V. Rybenkov, Alexander V. Vologodskii and Nicholas R. Cozzarelli- The Effect of Ionic Conditions on the Conformations of Supercoiled DNA. I. Sedimentation Analysis