Tamar Schlick *The Howard Hughes Buckling Transitions in

Medical Instituteand New York University

Chemistry DepartmentCourant Institute of

Mathematical Sciences

Superhelical DNA: Dependenceon the Elastic Constantsand DNA Size

251 Mercer StreetNew York, New York 10012,

USA

Buckling trunsi!ions In superhe!itd DNA are sudden chwges in shape [hal accompmr]’ asmooth varia[ion in a k(~?,purame[er, sut-h as superhelical densi[>’.Here ~te twplore the depen-dence of’[hese transitions on [he elastic constant. f,fi)rbending and twisling, A and C, impor(an(c‘Izarac’teri.<[ics oj’DN,4 ‘.~bending and m,is[ing persistence lengths The large range we exploreatend.s [o o[her elastic ma[erials kvith .sel~con[act interactions, modeled here by a Debye–Hiickel eiectro.s[a(icpo[cntial.

Our (olleciivc dmcription of DN.4 shapes and energies over a wide range Qfp = A/C revealsa dramati[ dependenw of DN.4 shape and associated conflgurationa] transitions on p: transi-tions are sharpjiw large p but rnasked,fiw small p. In particular, at small p, a nonplanar circular/hmil~’ emerges, in agreement with Jiilicher’s recent analytical predictions: a continuum of./i)rm.s(and a,ssocia(cd writhing numbers) is ai.ro observed.

The relevance qfthcse buckling transitions to DNA in solution is examined through siudiesof size dependence and [herrna{ effects. Buckling transitions smoo[h considerabt~, as size in-creases, and this can he t’.rplained in part b?’the loner curvature in larger plasrnids. This trend.suggmts tba( blick[ing !ransilions should not be detectable fbr isolated (i. e., unbound) DNAplasmid.r of biological interest, t’.~ceptpossibl~v,for vet-y large p. Buckling phenomena wouldnonetheless bCrele~,ant,{iwsmall DN.4 loops, particularly. for higher values (!fp, and might havea role in regulator)’ mechanisms: a smali change in superhelica[ stress could lead to a largecorrfiguratiorral change.

U‘rithe di,rtributions as afi~nction ofp, generated b~ .Langevin d~wamics simulations,reveal [he importance of therm al,fluc(ua[ions, Each distribution range (and multipeakedshape) can he interpreted by our buckling pr~files. Sign[~cantl~, [he distribution .~for mod-era[e to high .superbelical densities are most sensitive to p, isolating djflercrr[ distributionpatterns. [f(his cflt’ct could be captured e.rperimentall~ ,ji~rsmall plasrnid.s b~’currentlyava ilahlc imaging (echniques, such resui[s suggest a .~lightl~’dlferent e.rperimental proct’-durc ji~r estirna[ing the torsional .~t~fness of supercooled DN,4 than considered to date.@ i997 John 14’ile~’& Sons, lnc

Received September 12.1995: accepted December 15, 1995.* To whom correspondence should he addressed.

Biopolymers. Vol. 41. 5-25( 1997)LC]997 John Wile! & Sons, Inc. CCC 0006-3525/97/01 0005-2 I

5

6 Rutnachandran and Schlick

INTRODUCTION

Closed-circular DNA adopts a variety of higher or-der forms in which the DNA twists and bendsabout the global double-helical axis. These formsare known as supercooled or superhelical DNA. Su-percooling condenses the DNA significantly andalso stores energy that can be readily available foressential biological functions such as replication,recombination and transcription. This compactstate also facilitates further packaging of DNA withproteins. Thus, DNA supercools define an impor-tant functional state of the hereditary material. Un-derstanding the structural response of DNA to su-perhelical stress is therefore a subject of great im-portance.

In closed-circular DNA systems, the closure ofthe ring determines the linking number of theDNA, Lk. This topological invariant is changedonly by breaking one or two of the DNA strands.Lk describes the number of times that a DNAstrand winds about the closed circular axis of theDNA (for a rigorous definition, see Ref. 1). In re-laxed closed-circular DNA, Lk is directly related tothe number of primary turns of the DNA: LkO= N/h, where N is the number of base pairs (bp)and h is the number of bp per turn (about 10.5 forB-DNA ). When superhelical stress is imposed, Lk

# Lko, and the DNA is said to be underwound oroverwound with respect to the relaxed state. Thisnonzero difference ALk = Lk – Lko, known as thelinking number di~erence, is relieved by bendingand twisting about the global double-helical axis,thereby partitioning ALk into twist and writhe:ALk = W’r + ATw.’ In this fashion, interesting newgeometrical states of DNA that are functionally im-portant arise depending upon the amount of tor-sional stress.

The diversity of forms that supercooled DNAassumes depends on many factors. These can beinternal, such as the degree of torsional stress,and external, such as the concentration of salt insolution. Electron-microscopy images of su-percooled DNA reveal that supercooled DNA isoften wrapped about itself in an interwound (orplectoneme) configuration.z If the DNA is longenough, branches can result.2 Other experiments[e.g., gel electrophoresis, 2 small angle x-rayscattering 3 scanning force microscopy (SFM ), 45and cryo-electron microscopy] can measurevarious properties that depend on the su-percooled form of DNA and therefore allow anal-ysis of the shape and possibly fluctuations ofDNA. However, since experimental resolution islimited, complementary theoretical modeling

can provide insight into important structural andenergetic aspects of supercooled DNA. System-atic modeling can also help analyze and interpretthe experimental observations.

One successful theoretical model of su-percooled DNA involves the Kirchhoff rod treat-ment based on elasticity theory and mechanics.Typical assumptions involve circular cross sec-tions and homogeneous bending. Though it iswell known that DNA exhibits preferential bend-ing into major and minor grooves, 7this assump-tion of uniform bending is reasonable for DNAthat has no intrinsic curvature ( i.e., is naturallystraight ). Still, several groups are developingmodels for intrinsically curved DNA within theelastic-rod formalism.xy

Many investigators have explored the minimumenergy configurations of the elastic-rod model forDNA by theoretical techniques. Hearst and Huntmodeled the DNA as an infinitely thin Kirchhoffelastic rod to study highly writhed states. ‘(1” Later,Shi and Hearst ‘z developed a new curvature-tor-sion coordinate framework to study DNA su-percooling on the basis of the nonlinear Schrtid-inger equation. Le Bret’ q and Benham’4 studiedthe first buckling transition, the abrupt changefrom the circle to the figure-8 interwound structurein the context of elasticity theory. In a later work,’sLe Bret also suggested that higher order bucklingtransitions may occur. Recently, Jiilicher ‘b pre-sented a detailed phase diagram (spanning ALk

values from Oto 3 ) for supercooled DNA by analyt-ically approximating the equilibrium profiles of ahomogeneous elastic rod model. Different DNAfamilies were studied (circles, nonplanar circles,figure-8 structures, and interwounds ), with transi-tions among them examined and stability issuesdiscussed. Other recent analytical works focus onthe role of thermal fluctuations and entropiceffects.’”x

A variety of computer simulations have also beendeveloped and applied to obtain minima as a functionof important parameters. These include simulatedannealing ‘Yz’)Newton minimization, “~zz~ and fi-nite-element analyses (FEA ).24z~In addition to theseminimization studi~ statistical and kinetic modelshave also been developed. Monte Carlo approachesprovide collective descriptions of configuration space(e.g., Ref. 26). More recent dynamic approaches by

27 ~nge~n dynami=molecular dynamics, 21.2?.28

Brownian dynamics, 29 and those based on the dy-namic theory of elastic mds9-Ware providing comple-mentary information on the kinetic aspects of su-percooled DNA (e.g., writhe n4axation, folding

Buckling Transitions in DN.4 7

diffusion ). For a recent review of these exciting devel-opments, see Ref. 31.

While our major interest lies in dynamic simu-lations of supercooled DNA, we became interestedin buckling catastrophes that emerged from FEAand our deterministic minimization study.zq-~~These abrupt structural changes in global shape areof interest for several reasons. First, they are intnn-sic to the elastic rod model~~ applied to DNA andhence important. Second. they may be biologicallyrelevant. for example, as signals to enzymes thatregulate DNA by binding and altering Lk .3435Even if thermal fluctuations for long DNA maskthe sudden configurational changes expected theo-retically from the elastic model, buckling transi-tions may be important for small looped segmentsof DNA. Third. buckling transitions are sensitiveto the elastic constants ,.1and C for bending andtorsional rigidity, respectively: since there is sig-nificant uncertainty in the appropriate value of Cfor superhelical DNA. results as a function of p= .4/ [‘ might provide a slightly different directionfor comparing theoretical results with experimentto estimate the torsional stiffness.

In this paper, we present a systematic descri-ptionof the expected DNA configurations ( families )over a wide range of p values. The resulting profilesclarify the dependence of supercooling on the ratio.4/[”. Such relationships were examined in part inprior studies but not with realistic nonbonded po-tentials for DNA.’ ~‘4‘” Our potential energy in-cludes both elastic and electrostatic contributions,the latter in Debye-Hiickel form.~~ All terms areadapted for a macroscopic treatment. Both energyminimization and Langevin dynamics simulationsare used to determine the writhe and energy pro-files and to examine them in light of thermal fluc-tuations. The relevance of these transitions to longDNA (thousands of bp) is also examined. We findthat thermal effects mask these transitions, exceptfor very large p values. Finally, we suggest how thedata might be used with experiment to estimate Cand explore the size and system dependence of C.

In the following section we present the detailsof our model and the methods used. In the thirdsection. we present and discuss minimization pro-files as a function of ALk: we consider four differ-ent values of P for closed-circular DNA systems of1000 bp. Next. variations with size are examinedby comparing results for DNA systems of 1000 and2000 bp. Finally, distributions of writhe as a func-tion of P and ALA are also presented, as obtainedfrom Langevin dynamics simulations, to evaluatethe role of thermal fluctuations. The last section

summarizes and discusses these findings and sug-gests possible applications.

NUMERICAL MODEL AND SIMULATIONDETAILS

Energy Model

We use the B-spline model to represent the DNAduplex curve in terms of a small numlxr of controlpoints~’ (e.g., 14 per 1000 bp). From these controlpoints we generate curve points (e.g., 8 or 16 perspline segment), which are then used for energyevaluation. The potential energy E consists of elas-tic and electrostatic components. The elastic termsinclude homogeneous bending (El] ) and twisting( E7 ) potentials. The Coulombic term ( E( ) is inDebye-Huckel form and considers pairwise in-teractions among charged linear segments alongthe DNA curve. A simple harmonic term (El ) isalso included as a computational device to ensurethat the total chain length L remains very close toits target value L(). The energy components aregiven as follows:

(1)

27r2CE7=7 (ALk – Wrr)’ (2)(1

El = A’(L – L())z (3)

E{ =f3(/3) ff ‘- ““;df’d~ (4)1!

In these expressions, s denotes arc length, K denotescurvature, and the integrals are evaluated over thetotal length of the curve. The parameter K is astiffness constant chosen for computational rea-sons (i.e., not physical stretching, which might bea future addition). The writhing number Wr is ageometric descriptor of shape.’ In the electrostaticpotential, @is the Debye-Hiickel screening param-eter (0.33fi at room temperature, where c, is themolar salt concentration )~z; B( f?) is a salt-depen-dent coefficient’:; r,, is the distance between pointsi and j along the DNA chain; and /, denotes thelength of each charged segment along the chain.Full details of the electrostatic treatment are givenin Ref. 22, and details of the B-spline model andthe elastic-energy functions are given in Ref. 21.

Systems and Parameters

We use closed circular DNA systems of 1000 and2000 bp to generate energy minimization profiles

8 Ramachandran and Sch/ick

and 1000 bp for configurational distributions. TheLangevin dynamics simulations are started at anequilibrium configuration at the temperature T

= 300 K. A salt concentration of 1.0 M is used inall these studies. This is because profiles are moreinteresting at this value than at 0.1 M due to a largerrange of Wr observed over the same superhelicaldensity range.2~ At 1.0 M salt, E<. is much smaller(by an order of magnitude) than at lower salt sincerepulsive interactions between the polyelectrolytebackbone are more effectively Screened.*a How-ever, profiles (and energies) shift consistently atlower salt, 2? reducing the numerical values of W’rbut not changing the curve shapes. Furthermore,the position of the first buckling transition appearsto be dominated by the ratio of A/C and notaffected significantly by the salt concentration.2~

Algorithms

To find potential-energy minimized structures weuse the truncated-Newton scheme. 36For dynamicstrajectories, we use a Langevin dynamics protocolwith the damping constant y = 9.77 X 10 ‘()s-‘(program units ), as calibrated in Ref. 28 to giveboth an approximate representation of solventdamping and optimal configurational sampling.The Langevin equation of motion is integrated nu-merically by the Langevin / implicit-Euler (LI ) al-gorithm with a timestep 7 = 100 fs in programunits.~’ We have shown that the L] algorithmagrees well with the Verlet algorithm (with a pro-gram time step of H 5 fs) at this value of y, 38andmoreover, provides a computational advantage ofabout a factor of three. The program timestep of 100fs corresponds to an effkctive physical time step ofapproximately - 10 ns, 38since both the masses andthe value of ~ are scaled for computational reasons.

Full details concerning the algorithms are givenin Refs. 36, 37, and 39. The calibration of thetimestep is described separately in Ref. 38, a workthat extends Ref. 40. Extensions of the dynamicmodel to Brownian dynamics with hydrodynamicsare described separately (G. Ramachandran and T.schlick, in preparation).

Elastic Constants

Currently, there is a wide range of possible valuesfor C, from 1.5 to 4 X 10 -‘~ erg cm. In contrast, anaverage value for the bending constant A has beenmore narrowly bracketed to correspond to a bend-ing persistence length, p~, of around 55 nm at mod-erate salt, yielding the value A = 2.3 X 10- ‘gerg cm(A = kBTph, where k~ is Boltzmann’s constant).

Still, over the large sodium ion concentration rangeof 0.001-1 .OM, a broad range forph of 28-100 nmis noted, 4‘ corresponding to A between 1.12 and4.02 X 10-lq erg cm. In any case, the uncertaintyin C leaves a large possible range for p = A/C, bothgreater than and lower than unity. (See Ref. 31 fora discussion on the relation to the Poisson ratio.)

Early analyses of fluorescence depolarization ofDNA by Barkley and Zimm4~ suggested the broadrange for C from 1 X 10-’Y to 4 X 10-’q erg cm.Shore and Baldwin’s measurements of chaincyclization 4J4dsuggested a torsional constant of C= 2.4 X 10”9 erg cm. Measurements based on gelshifts analyzed by Horowitz and Wang45 pointedto a value of C = 2.9 X 10-’y erg cm as a lowerlimit. A similar value of C (around 3 X 10-” ergcm) was obtained 46 by a numerical treatmentbased on writhe distributions and experimentallybased ALk variances. This procedure was suggestedby Benham,47 developed by Volgodskii et al.,4s andrefined to include excluded-volume effects byKlenin et al.4fi

These values for DNA’s torsional stiffness aresomewhat higher than estimates based on time-resolved fluorescence and dynamic light scat-tering, 4g’s’)about C = 2 X 10-’q erg cm. Values inthis range were also suggested by earlier fluores-cence depolarization measurements of Millar etal.5’ (C = 1.43 X 10-’Y ergcm). More recent anal-yses based on static base-sequence variations in B-DNA crystal structures51S{ also point to a lower Cvalue ( C = 1.4 X 10- ‘gerg cm), which correspondsto observed rms twist fluctuations of 5.9° in the x-ray data.

From the estimates given above for the torsionalstiffness of the DNA, by taking A = 2.3 X 10- ‘yerg cm as the effective bending constant at 0.1 Msodium, we see that p might span the range 0.5-2.3. Variations in p are also expected with chemicalchanges in the environment, e.g., alcohol-inducedB to A-DNA transitions. Considering the sensitiv-ity of both analytical and numerical results to p, itis valuable to explore how this ratio might affectsupercooled DNA in the context of a simple model.

Ratio of the Elastic Constants in theContext of an Electrostatic Treatment

A point that requires clarification in our study isthe meaning of p in the context of a model thatincludes an explicit electrostatic potential in addi-tion to elastic bending and twisting terms. In thiscase, bending persistence length contributionscome from two sources the bending term and theelectrostatic potential. The latter contributes due

Buckling Trurrsi[ions in DNA 9

to the repulsive interactions from incompletely

screened backbone charges. Therefore, when com-paring results to experiment, the persistencelengths for bending and twisting are the relevantquantities to consider, not the p value per se.

The efli’c[ive persistence length in the context ofan elastic plus electrostatic model and its depen-dence on the input .4 is a complex subject on itsown (Jian, H.. Ramachandran, G., Schlick, T., &Vologodskii. A.. work in progress). Such studiesare best done for short linear DNA and must con-sider carefully excluded volume effects. However,it is clear that the added electrostatic term increase.$

the effective persistence length from that corre-sponding to the bending stiffness alone ( i.e., A/k,, T). The percentage of increase depends on thevalue of,4 used. Furthermore, a much wider in-crease is realized in the range of low sodium saltconcentration (e.g., 0.0 1–0. 1M), in comparison tothe range O.i– 1.OM. This is consistent with whatemerges from many experimental studies4’ but notall. s~Thus. in terms of parametrizing a numericalmodel, the chosen .4 can have a substantial rangeto correspond to any target experimental value.

In our work, we use .4 = 1.28 X 10-”) erg cm,corresponding to the nonelectrostatic contributionto the persistence length of approximately 31 nmsqand a sodium-ion concentration of 1.OM. Alterna-tively, a higher value of about 2 X 10 ‘“ erg cmhas been used in the context of a Debye-Hiickeltreatments’ Through rough comparisons of writhedistributions for nicked DNA we found an effectiveincrease in the persistence length of about 33% inour Debye–Hiickel treatment with respect to anelastic model (with an excluded volume term in theform of Lennard-Jonesz’ ) where,4 was varied forcalibration. For reference. when we set .4 to 2.3~ *O 1~,erg cm in our elastic plus electrostaticmodel the increase was smaller. about 26%. There-fore, we can very roughly consider the increase of

Table I Elastic Constants and Related Quantities for DNA=

persistence length in our electrostatic model with,4= 1.28X 10-’” ergcm to be about 33% correspond-ing instead to A = 1.7 X 10- ‘“erg cm in an electro-static-free model. This implies, for example, thatthe ratio p = 1 as used in our model (A = C = 1.28X 10-”) erg cm) corresponds to p’ = 1.7/ 1.28= 1.33= 1.33p (see Table 1).

Computational Performance

All computations are performed on a SiliconGraphics Indigo 2 Extreme workstation with a 150MHz IP22 processor. Minimizations for the DNAcircles of 1000 bp require from 2 seconds to 20minutes, depending on the starting structure. AnMD simulation of 10,000 iterations (roughly 100PS in physical time’s ) takes approximate] y 3 cen-tral processing unit hours for 1000 bp.

RESULTS

Presentation as DNA Profiles

In the work described below we fix A = 1.28X 10- ‘yerg cm and change the value of C to studydifferent A/C values. We consider four values ofp—3, 1, 0.71, and 0.43—for the following reasons.The values 0.43,0.71, and 1 lie within the biologi-cally relevant range for DNA. ~’ [n particular, thefirst two values correspond in our formulation (seediscussion above on elastic constants) to the com-monly used values of C = 3 and 1.8 X 10- ‘g ergcm, 4ssh respectively (see Table 1). We have alsoused the particular value of p = 0.71 recently .3’Thevalue p = 3 clearly lies outside of the experimen-tally expected range for DNA, but is useful here forextensive interpretation of our results and possiblyfor other elastic materials. (Note that by pure ap-plication of elasticit y theoty, 1 s p s 1.5, ” but de-

p (This Work) P’ P“ C(XIO-’9erg cm) ((?:)’”

3 4.() 5.40 0.43 10.3”1 1.33 1.80 \,~8 5,9°0.71 0.94 1,28 1.80 5.r().’43 0,57 0.77 3.00 3.9°

“The column heudings are defined as follows: p is the ratio ofthc elastic constants (.4/(’) as used in this work with.4 = 1,28 x 10 19erg cm corresponding to the nonelectrostatic contribution to the bending persistence length (see text): p’ is the associated A/C’ ratio it’(he etticcli\c.4 were 1.7>. 10 1’)ergcm (w estimated. sec text): P“ is ihe associated A/C’ifA is equal to 2.3 X 10 1“ergcm correspondingto the ct~ccti~cpersisicncc length at moderate salt (see text): C’is the torsional-rigidity constant: and (H, ,1‘1’2is the rms twist angle. The

‘ \ I z = ~~. wh~re d = 0.34 nm is Ihe distance between base pairs. are 8.4”. 7.3”, and 6.2” corre-rms bending deviations. !,flj,,spondinglo,.1 = 1,?8, 1,7, and ?.3 X 10 1“ergcm. respectively. The rms twist angle is given as a function of Cby (0?)1’2 = -.

10 Ramachandran and Schlick

viations are expected for real DNA, a nonuniformmaterial with noncircular cross section, wherebending is asymmetric and twisting is sequence de-pendent.)

Figures 1-4 display profiles obtained by energyminimization of closed-circular DNA of length1000 bp for our four p values. Figure 5 shows cor-responding views for a system of 2000 bp with p

= 0.71. The five parts of each figure show: the mag-nitude of the writhing number (part a), the totalenergy E (part b), the twist energy E7 (part c), thebending energy EB ( part d), and the Coulombic en-ergy E( (part e), all as a function of ALk. We useALk in the range Oto –7 and hence obtain negativevalues for Wr. (For display, we plot – Wr vs– ALk). Representative configurations for somefamilies are also shown. These profiles are con-structed by performing a large number of minimiz-ations from a wide range of starting points, such asthe circle, the figure-8 interwound structure, andother interwounds obtained by minimization. SeeRef. 32 for further details.

We use different symbols to distinguish amongrelated supercoil forms such as the circle (F’O), thefigure-8 interwound (F 1), the interwound withIWrl R 2( F2 ), and so on.3’ Families that are notpart of the global minimum energy are shown bydashed lines. The various families are most easilydistinguished according to the twist and Coulom-bic energy components (see Figure lC and 3e forexample ); we use the criterion of a sharp drop inE7 to detect another configurational family. Forsome configurational families, especially the highlywrithed and bent forms, there is a discontinuity inthe Coulombic energy within the family ( see Figure2e and 4e ). This occurs even when there is asmooth transition in Wr between forms within afamily. For example, in the F3 + family at P = 0.43(Figure 4a) there is a transition between bent in-terwounds and straight, more highly writhed, in-terwounds (see the two forms shown for the F3 +family in Figure 4a and compare to Figure 4e ).

Note from the Wr plots that more than onestructure can exist at a given value of ALk. Thecorresponding crossing of associated energy curves(part b) shows when one family becomes energeti-cally favored. In Figure 1b, for example, both F 1and F2 configurations are detected as minima for4< IALk I <5.2. F2 is energetically favored forIALk 1>4.5. F 1 configurations are energeticallyfavored over FO for IALk I >3.75. These bucklingtransitions occur at large IALk I in comparison tothose associated with the lower p values (e.g., 1,1.5) typically used.3z The dashed lines associatedwith higher energy forms often correspond to bent

interwound forms (see, for example, the imagescorresponding to the higher energy families in Fig-ure 2a).

OveraH Sensitivity of Profiles to P

The strong sensitivity of the profiles to p = A/C isapparent from the views. The extreme cases of p

= 3 and p = 0.43 (Figures 1 and 4) show the mostdramatic differences, but a systematic change aspis decreased is evident over this large range ofp thatwe explore. Namely, we see that as p is decreased:( 1) the Wr profiles smooth out (i.e., gaps in Wramong families become smaller); (2 ) the slope ofI Wrl vs IALk I curves ( if one draws a coarse line)increases, though in a complex way (see below);and ( 3) the profiles generally become more intri-cate overall, i.e., more families that are close in en-ergy and cannot be easily distinguished emerge atlower p. (In this connection, the p = 1 case maybespecial. )

For example, a gap of approximately one unit inWr separates both the FO and F 1 families and theF 1 and F2 families for p = 3 (Figure la). This im-plies that neither imperfect circles nor structureswith 1.2< I Wr I < 1.8 are detected as minima forthese p, These forms are unfavored due to the highcost of bending. When p = 1, a large unit gap stillseparates the writhe associated with the F’Oand F 1forms ( Figure 2a), but the second gap-notable forp = 3—has disappeared; instead, the figure-8 re-lated forms realize a much broader range of writhe,and I Wr I increases more steeply with IALk 1. Forour smallest p value, 0.43 (Figure 4a), the firstbuckling catastrophe disappeam altogether and anew family of imperfect circles emerges ( Figure 6).Indeed, this result is expected from elasticity the-ory, and Jiilicher predicted that these nonplanarrings become stable equilibria for p <0.61. ‘bNotethat this family of nonplanar circles is energeticallyequivalent to the circle at about IALk I = 0.8 andthen becomes energetically equivalent to the fig-ure-8 interwound forms at roughly IALk I = 1. Thep = 0.43 profile clearly shows that the families aremerged more smoothly to one another at low val-ues of p and that hardly any significant writhejumps are present.

Sensitivity of Profiles to DNA Size

By comparing the two profiles corresponding to p

= 0.71 for DNA systems of 1000 and 2000 bp(Figure 3 vs. 5), we clearly see how much simplerthe latter profile is: Families beyond F 1 are nearlyindistinguishable for the larger DNA. For this rea-

Buckfing Trun,sitions in DNA 11

son. we denote these configurations collectively byFN. Only the twist energy for the latter (Figure 5C)reveals nonsmooth behavior beyond IALk I = 2,with pronounced drops near IALk I = 2.4, and 5.6.These first two [ALk I values roughly correspondto emergence of F2 and F3 families for the systemof 1000 bp ( Figure 3a,c), but for the third we haveno analogue. Possibly. a sudden change in curva-ture for the larger DNA is responsible for this pat-tern in the bending energy (see below). Overall, thesmaller curvature in segments of the larger plasm idexplain the smoother profiles.

Another observation concerns the IALk I valuewhere buckling from the circle to the figure-8 in-terwound occurs. The critical value IALk, 1, wherethe energies of the two forms are equivalent, isabout 1.3 for both 1000 and 2000 bp at this p. in-deed, independence from length is expected fromtheory. ( Note that we plot – H ‘r vs – ALA ratherthan the superhelical density, c = ALk/I.kc,, inFigure 3a and 5a). The slope of the writhe curve isalso steeper for 2000 bp than for 1000 bp.

These results clearly show that buckling behav-ior is sensitive to DNA length and thus bucklingcatastrophes are not expected for plasmids of bio-logical interest (e.g., PBR322 is 436 I bp long). Ourstudies show that transitions may be relevant onlyfor large p and/or small DNA systems. DNA loops,for example, might exhibit such phenomena,which may be related to biological regulation. Abuckling transition for a DNA minicircle ( 178 bp)has indeed been observed expenmentally.s’ Wewill address the related question of the effect ofthermal fluctuations on these buckling phenomenain a separate subsection.

Analysis of Buckling in Terms of Curvature

A buckling catastrophe results when the DNA can-not accommodate a small change in IMkl by a

small change in bending (curvature) and writhe.We saw that buckling behavior is sensitive to bothp and DNA size. To further examine this sensitiv-ity, we analyze in Figure 7 the curvature distribu-tion along the DNA configuration for three valuesof p (3, 1, and 0.43) for two configurational fami-lies: F 1 and F2. A similar distribution for p = 0.7 Icomparing systems of 1000 and 2000 bp is shownin Figure 8. In each panel of the figure, we showtwo curvature (the variable Kin Equation 1) curvessuperimposed, corresponding to the first and laststructure of the F 1 (or F2 ) family analyzed. Thecolor configurations shown alongside the plot rep-resent the two corresponding forms, whose colorcoding matches the points on the curve. For thesepresentations. we use discretizations involving 112curve points for 1000 bp and 160 points for 2000bp. Note that the small, jagged peaks arise from thediscontinuity y of the curvature function in terms ofthe control points. For the 1000 bp plots ( Figures 7and 8), 14 jagged peaks corresponding to 14 con-trol points can be seen, and for 2000 bp ( Figure 8 ),20 peaks ( 20 control points) are evident.

From Figure 7 we see the natural characteristicsof the figure-8 curvature distribution most clearlyfrom the p = 3 panel: two minima and two broadmaxima. Curvature is smallest at the straighter seg-ments and largest at the apices of the bent struc-ture. This “signature” of the figure-8 curvaturedoes not change significantly as P is varied, thoughgreater variations within the F 1 family are notice-able as p is lowered (each of the two minima re-gions changes into two minima separated by ahigher curvature region; this indicates a movementtoward the F2 family). For the F2 interwoundfamily, four minima, two major maxima, and twominor maxima can be identified. Greater diversitywithin the family members again emerges as p islowered. This explains the “softer” buckling be-havior noted for lower p values.

FIGURE 1 Energy-minimized profiles for closed circular DNA of length 1000 base pairs,P = 3. The energy includes bending, twisting, and Debye–Huckel potentials, as described inthe text. The bending constant .4 = 1.28 X 10- ‘qerg cm is used (corresponding to the non-electrostatic contribution to the persistence length), and the torsional rigidity constant C isvaried to establish the desired p ( i.e., C’= A/p). The salt concentration is set in all cases to 1.Olfsodium ions. ( A lower salt concentration shifts the writhing numbers consistent] y. ) Shown asa function of – MA are as follows: (a) The negative of the writhing number ( – W’r), (b) thecorresponding total energy E (in kcal /mol ), (c ) the twisting energy E,, (d ) the bending energyEl,, and (c) the Coulombic energy E<. The various families (FO, F1, and F2 ) are distinguishedby different symbols (see text), and dashed lines rather than symbols are used for configura-tional families that are not part of the global energy minimum, The FO ( circle ) family has beenfollowed until the circle is no longer found as a minimum when starting optimization from thecircle. In (a). selected DNA configurations are also shown.

12 Ramachandran and Schlick

5 la1 1 1 1

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42 le1 1 1 I l“’’I’’ ’’1’ ’’’1 ’’’’1’” ‘1’’”

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FIGURE 1

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38 ~o..nu6a+”----0000066868800000000000000000000

3, ~o 1 2 3 4 5 6 7

-AUo 1 2 3 4 5 6 7

-ALk

FIGURE 2 Energy-minimized profiles forclo%d circular DNA of length 1000 base pairs,P= I. See Figure 1 caption.

BwklingTran.~i(ionsi nDNA 13

6 3aI 1 1 1

10001bp1 u 1 1l“’’l ’’’’ ’’i ’’l ’’’’i’” ‘ 15L, ,,, ,rlr,, 3C

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if

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o 1 2 3 4 5 6 7

42 3e1 I 1 1 [“ ’’1’ ’’’[ ’’’’ 1’’’ ’[’[”1 “r’

XX

o 1 2 3 4 5 6 7-AM

FIGURE 3 Energy -minimized proliles forclosed circular DNA of length 1000 base pairs,p= 0.71. Sce Figure 1 caption.

5 4a1 11 1 l“’’1 ’’’’1’’”

1000 bpA/c = 0.434 —

3 –

2 —

15L, 4C1 f 1 l“’’I’’ ’’1’ ’’’1 ’’’’1’” /’l’’’’x’-0 / ,,’ .x

o /’ .’ .x10 – //’ ,/’ #

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5 - 0 /“.’ -xx/o

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42 4e1 , t I I“’’l ’’’’ l’’r ’l ’’r ’l ’’dF}” ‘“-

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F

❑o,Bt#

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-ALko 1 2 3 4 6 6 7

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FIGURE 4 Energy-minimized profiles forclosed circular DNA of length 1000 base pairs,~=0,4~, See Figure I caPtion.

14 Rumuchatrdrun and Schlick

w

3i-.

0❑

5.3

0 I 1,11 (1, ,,,1<$1,1), ,,1, )1,0 1 2 3 4 5 6 7

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,~o 1 2 3 4 5 6 7

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FIGURE 5 Energy-minimized profdes forclosedcir-cular DNA of length 2000 base pairs, p = 0.71. See Figure1 caption. Note that there are nonstable writhe jumpsto distinguish the configurations beyond the~l family.Thus, allstructures outside of the Fl family aredesig-natedbyFN.

Now, the smoother profiles forlarger DNAsys-temscan alsobe understoodinterms ofthecurva-tureplots in Figure 8. For2000bp, the curvaturevariations along the DNA curve are much smallercompared to those associated with 1000bp. ThernagniZudeofcm-vature isalsosmaller incompari-son to the smaller systems and hence EB is lowerfor 2000 bp (about one half) for the same IALk 1.This fraction of one half remains when we increasethe mesh size for our B-spline representation of thesystem of 2000 bp, so as to generate 240 curvepoints (data not shown ), and is expected from theform of E~ [Eq. ( 1)].

Clearly, since curvature of a segment is inverselyproportional to the radius of the circle associatedwith it, curvature is smaller for longer DNA. Thus,as IALk I is varied, continuous changes in curva-ture can be sustained at a lower energetic cost andthus buckling catastrophes are more easily maskedfor longer plasmids.

Buckling Analysis

The behavior of the F 1 family (figure-8 forms) asa function of IALk 1 is particularly sensitive to p;hence, analysis of this behavior explains the moregeneral buckling trends noted above. At p = 3, thewrithe curve of the F 1family ( Figure 1a) is roughlya straight line with a near-zero slope, and this fam-ily extends over nearly four units of IALk 1. Bend-ing the DNA is energetically more expensive thantwisting it, and thus nearly planar forms (with oneor two crossings) are preferred over bent structures.When p = 1( Figure 2a ), the writhe of the F 1curveassumes a parabolic shape over a much smallerrange of IALk I (about 1.4). The value I Wr I risesslowly as IALk I is increased from 1.3 to 2.3, butthen rises steeply. Similar parabolic patterns areseen for p = 1 and p = 0.43, though the lowestIALk I where a figure-8 form is identified(beginning of the family) decreases with p, and theALk range associated with the family varies with p.

Examination of the bending and twisting energycurves clarifies these trends. For example, fromFigure Ic,d we see that E7 rises monotonically withIALk I for F 1, while EB hardly changes, mirroringthe Wr pattern, That is, the same bending (cur-vature )can be sustained more easily when p is largesince E7 increases more slowly with IALk 1. As pis decreased, the twist penalty is greater—note thegradual increase in slope of the E7 curve for FO asp is decreased. Thus, more families and larger vari-ations in Wr (over the same IALk I range) are seenas p is decreased. Note the small structural differ-ences among members of the F 1 family for p = 3

Bl[(k/in~ Trun.~i:i{wrsin DN.4 15

I ALk=-o.70

FI(2[!RK 6 The nonplanar-circle family identified at P = 0.43. These forms correspond tothe brunch of’() < I Ii ‘rl < 1 shown in Figure 4a by open star symbols. Note the progressionfrom a slightly bent circle at – AI.k = 0.7 to the nearly frgure-tl form at – AL.k = 1.().

( Figure la) in contrast to the large variations notedin FI when p = 0.43 (Figure 4a).

In particular, the circle to figure-8 buckling tran-sition occurs at lower IALk I as p is decreased sincethe critical IALAI where the lower curvature (ofthe circle ) is competitive with twist-energy penaltyinvolved ( lower I PI‘rl forms) decreases with p. InFigure 9 we show the critical IALk I value, IALk,. 1,where the energy of the circle is equal to that ofthe figure-8 form, as a function ofp. The change isdramatic over the range of p explored here: IALk{. I= 3.7, 1.6.1.3, and 1.0, forp = 3.1,0.71, and O.43,respectively (the point forp = 1.5. IALk, I = 2.1, is

taken from Ref. 22). The linear dependence ofALiic on P can be understood by equating the po-tential energies for the circle and the figure-8 struc-tures and using I 14‘rI = O and 1, respectively, forthese structures to evaluate the twist energy [Eq.( 2 )]. One then obtains the relation

where the tww?.s ( not actual energies) involvingbending and electrostatic energy differences be-tween the circle and tigure-8 ( c,,, C() forms are in-dependent of p and <’:

C/{=(*);[f, K’($’~’-f.OK’(’)ds]‘6)

~( = ()* [-z,.,-&, (,l (7)

For our model, we obtain the constants clj = 0.9706,c< = 0.0985 (both quantities are dimensionless).Note also from Figure 9 that the “stability” line,fip,lll+n has a greater slope and crosses theIALk, I line. For values of p beyond 0.8 (where thetwo lines cross) this implies that exchange of stabil-ity will occur for IALk I > IALk, 1, as expected.However, the stability line fip is most likely validonly beyond some critical p value. 1”

Buckling in View of Thermal Fluctuations

A central question in our application of an elastictreatment to DNA is the biological relevance ofthese buckling transitions. We already noted thatbuckling transitions are expected for DNA minicir-cles and loops, and that these phenomena may berelated to protein-regulated events.~~~f Further-more, the higher p is, the more pronounced thebuckling behavior will be. Can we offer additionalquantitative data on buckling in view of thermalfluctuations for DNA in solution?

Interactions between the DNA and solvent mol-ecules at ambient temperatures produce significantthermal motions.~~ Inertial forces become smaller(in comparison to vacuum), and many of the char-

16 Ramachandran and Schlick

FIGURE 7 (CUI%’ahIR? K at e

~urvature variations along the DNA for the F 1 and F2 families,ach curve point ( 112 are used) is plotted for each member of the

1000 bp. Thefamily (F1 or

Buckltng Trun.sitions in DNA 17

acteristic harmonic motions are overdamped bysolvent.zx This means that at a given superhelicaldensity. we expect a distribution of forms spanninga rungt’ of energies and geometries. If we knewthose distributions, we could analyze them in termsof the buckling profiles derived from potential energyminimization and understand the “signal vs noise”behavior. This important problem has been the re-cent focus of Harvey and co-workers.sX

To estimate such expected distributions, we per-formed Langevin dynamics simulations at physio-logical viscosity ‘Xlx for systems of 1000 bp at vari-

ous ALA ( – 1. –4. –6) and p (3, 1,0.43). Resultingdistributions from trajectories of 60,000 iterations(approximately 600 WSof physical time 3X) are dis-played in Figure 10. In each case, we started thesimulation at the lowest potential energy mini-mum identified (see profiles). The thick line in Fig-ure 10 corresponds top = 1 (C = 1.28 X 10-ig ergcm): the dashed and thin solid lines correspond,respectively. to p = 3 and p = 0.43 (C = 0.43 and3.0 X 10 ‘<’erg cm. respectively). These distribu-tions already reveal important distinctions as afunction ofp.

First, for higher IALk I (i.e., 4, 6), distributionsare sharper than at IALk I = 1. More peaks emergeat lower IM.k 1. Second, the larger IAl,k I is, themore sensitive the distribution is to p. This suggeststhat experimental data may reproduce distribu-tions that are sensitive top for relatively small plas-mids at hi,gh superhelical densities. Thus, forIALAI =6( Iu I - 0.063), distributions for C = 0.43,1.28, and 3.O(XIO 1”erg cm) center around Wr– – 1. – 3 and –4. respectively. while for smaller—

ALA the distributions shift toward smaller I W’rIvalues and are broader. The shifts are more pro-nounced for lower values of p (or higher values ofC): compare the progression of distributions atfixed C from the top to bottom panels for both theC’ = 1.28 and <’ = 3.0 series, as opposed to the C’= 0.43 case. For IALAI = 1, distributions for all pspan both the circle and the figure-8 structures. How-ever, the distributions for C = 0.43 and 1.28 showhigher peaks at the circle, as expected from the pro-files (see Figures 1.3, 4, and 9 ). Interestingly, the C= 3.0 profile spans three configurations: circles, bentcircles. and figure-8 like structures.

These distributions tie well with the minimiza-tion profiles. For example, the I Wr I vs IALk I plotof Figure 1a reveals two local minima at ALk = –4,with W’r= O and Wr - – 1 (cf. peaks in distribu-tion for C = 0.43, shown in middle panel), and noimperfect circle (dip in distribution). It is interest-ing that for ALk = –6 ( C = 0.43) structures withwrithe near –2 (minor peak) correspond to globalminimum but the writhe distribution clusters nearthe figure-8 family. This is consistent with ourtinding of the entropic lowering of the writhe.2’ ~’

For C = 1.28 (p = 1), the distribution for ALk

= –6 is again consistent with this lowering ten-dency, as forms associated with the F2 family arepreferred over the F3. Similarly, for ALk = –4 and– 1 the asymmetric distributions indicate a prefer-ence of more open, loosely coiled forms than thoseassociated with the potential-energy minimum.

For C = 3.0 or p = 0.43 (see Figure 4a), thesharp distribution near W’r = –4 for ALk = –6 cor-responds to the actual potential energy minimumidentified, but for ALk = –4, the preferred writheis around –2.5 rather than about – 3, indicatingagain a trend toward more open interwound struc-tures. For this lowest p (highest C) examined, allthree forms—planar circle, nonplanar circle, andfigure-8 interwound-are very close in energy atALk = – I (see Figure 4b ), and indeed the tnmodaldistribution displayed in the lower panel of Figure10 is consistent with the minimization profile.

Phase Diagrams of C/A vs IALk I

In Figure 11 we present a phase space diagram thatdelineates the areas where various families are en-ergetically favored as a function of both C/A ( 1/p)and IALk 1. This phase space diagram is motivatedby and contains data extracted from that presentedby Julicher. 1’ In his analytical treatment of su-percooled DNA using a simple elastic energy model,Jilicher predicted the areas where circles, nonplanarcircles, figure-8 forms, and interwound configura-tions are energetically favored. These regions areidentified here by the symbols FO, NP, F 1, and FN,respectively. In addition, Julicher uses dotted andsolid lines to distinguish among the continuous anddiscontinuous transitions between structures. 1’

F?) for p = 3, I and 0.43. The curvature curves corresponding to the first and last structures inthe family are superimposed. To the right of the plot, two color configurations are displayed:the upper image corresponds to the beginning of the family (curve of smaller curvature) andthe lower to the last structure in the family (curve of greater curvature), The color scheme usedfor the DNA points on the displayed configurations is color-coded to match that used in thecurvature curves.

18 Rumucilandran and Schlick

FIGURE 8 Curvature variations along the DNA for the F1 and F2 families, 1000 bp vs 2000 bp.The curvature Kat each curve point ( I I2 used for the DNA of 1000 bp, 160 for the DNA of 2(M0bp) is plotted for the first and last membem of the family (Fl or F2 ) for p = 0.71. See Figure 7caption.

Buckling Trun.si(i<m\ in 13N.4 19

0

-) I I

In

m

0

f?

oL flr3Q /c+

~“04“

0/

ElP ALkc

o 043 10071 1310 16

. 15 21

0 30 37

0 I000 05 10 1.5 20 25 30

P

FIGURE 9 The critical value AL.k( where the circleand figure-tl forms are energetically equivalent. Thevalue MA, is obtained from our profiles (Figures 1–4 ).The value at p = 1.5 is taken from Ref. 22, The solid lineis described hy Eq. ( 5 ). with the values of the constants61jand Ct as given in the text. The dots define the “stabil-ity” Iinc. l!~p. mentioned in the text

Our data is shown on the same plot as severalpoints corresponding to the critical values ofIALk 1, IALk, 1, at which two configurational fam-ilies are energetically equivalent: triangles ( A ) forNP - FO, open circles (0) for NP+ F 1, plus signs(+) for FO ~ Fl, and open stars for F1 ~ F2

transitions. Two interesting conclusions emerge:( I ) Theory and numerical data agree closely for theNP~ FO and FO ~ F I transitions. i.e., where self-contact is not important. This can be seen from theclose agreement of our triangular. open circular,and plus symbols with the analytical curves. (2)Theory and numerical data depart. as expected,when I M‘rI > 1. i.e., for structures with more thanone point of contact. In particular, our boundary sep-arating F’I from F2 is shifted significantly to a higher[AL/i I for the corresponding p value (e.g.. for p = 1,transition occurs for ALA = *2.5 rather than *2.O ).Thus. numerical simulations form an importantcomplement to theoretical analysis and an essentialtool for treating real DNA. Note also that in the phasespace diagram, Jtilicher predicts that the nonplanarcircle is stable at p E 0.6 I (C/.4 = 1.64). From aseries of minimizations of the tigure-8 at values ofALk near – 1, we found that indeed this familyemerges around P = 0.6 (see next section).

SUMMARY AND DISCUSSION

in this work we have examined, by energy minimi-zation and Langevin dynamics techniques, the

buckling transitions expected from an elastic mate-rial with homogeneous bending and twisting andrepulsive nonbonded interactions. Our focus wasthe sensitivity of these results to the ratio of bend-ing to torsional-rigidity constants, p = ,4/ C. Forthis reason. the high salt concentration of 1.OM so-dium was used. The relevance of these findings toclosed circular DNA in solution was exploredthrough examination of the size dependence ofbuckling transitions as well as the effects of thermalfluctuations. The latter results, presented as writhedistributions as a function of superhelical densityand p, are consistent with the minimization profilesand also help clarify the dependence on p. It is in-triguing to explore the possibility that these distri-butions might be compared to those generated ex-perimentally by a technique that can directly mea-sure the writhe of a plasm id (e.g.. cryo-electronmicroscopy ) to help to bracket the biological rangeOfp.

Our major findings can be summarized as fol-lows.

Writhe Profiles and Buckling TransitionsAre Very Sensitive to P

At large p, few and distinct families of DNA con-figurations are expected (Figure 1a), and the buck-ling transitions among those are both abrupt andlarge in terms of the spatial change. At small p,many smoothly related families emerge ( Figure4a), and larger I Wrrl values occur at lower p. Inparticular, the slope of the line I 14’rl -0.72 IALk Iobtained from analysis ofa few experimental struc-tures ( under assumptions of regular interwoundstructures )‘“ corresponds approximate y to our re-sults with p = 0.71.

Further, the family of nonplanar rings expectedat low p suggests that at sufficiently small valuesof p the first buckling transition (circle to figure-8form) will already be unobservable, even for verysmall DNA circles. This is also an interesting hy-pothesis to test by experiment. On the other hand,we predict that for high p buckling might be observ-able for small DNA circles or loops. In general. wefind that the critical IALk 1, ALk<., where the circleand the figure-8 interwound are energeticallyequivalent, decreases linear] y with p (Figure 9):this line lies below the stability line ALk = fip forp >0.8, which makes physical sense. Indeed, Jiilich-er’s analysis suggests that this stability line is rele-vant only for p >0.6.16

The overall sensitivity of DNA configurations tothe ratio of the elastic constants can be seen from

20 Ramachandran and Schlick

12 , )1A’

I , I , r I , I , Id

.1

:\

/ C=3.O

[(J = 0.43)

00 c = 0.43

/

\

c = 1.28 :, (p =3)06 — ,’/ ‘!,

,,

04:/ “

,,

\

,’ ‘v

02 – ,’ $,./ \

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-4 -3 -2 -1 0 1

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12I I I I I

1

00 —

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n I I I \ >-4 –3 -z –1 o 1

Wr

Writhe distributions from Langevin dynamics simulations of closed circularDNA, 1000bp. Shown are writhe distributions for three different values of ALk: – 1, –4, and–6 (bottom, middle, and top panels) at three different values of P—3, 1, 0.43—identified bydashed, thick solid, and thin solid lines, respectively. These three p values correspond to C= 0.43 x 10-’9 erg cm, 1.28 X 10-” erg cm, and 3.0 X 10-’S erg cm, respectively, and arelabeled in the top panel in units of 10-‘9 erg cm. The distributions are obtained from Langevindynamics trajectories of length 60,000 iterations (approximately 600 ~s). Each such trajectoryis started at the lowest energy structure found for the associated values of ALk and p. Thesimulations are performed at 1.0A4salt concentration and room temperature conditions, witha Langevin model that incorporates solvent damping effects as deseribed separately .2”

Figures 12a and 13a. Here, we show the I Wr I valuecorresponding to minimized structures of 1000 bpas a function of p for several values of IALk I: 0.5,0.98, 1.5, 2.5, and 5.5. These values of AL/c werechosen to explore representatives of different fami-lies. Clearly, the range of p illustrated here (O.1- 10;note the logarithmic scale in the abscissa) is largebut useful for extended analysis and for other elas-tic materials. These views might also be relevant toDNA when changes in the chemical environmentlead to changes in the elastic constants. To clarifythe issues mentioned earlier with respect to theeffective persistence length (see section on the ratioof elastic constants under Numerical Model), wegenerated these views for two values of A in ourmodel: 1.28 and 2.3 X 10-19 erg cm (Figures 12

and 13, respectively ). Results are overall very sim-ilar and reinforce the point that minima are sensi-tive to the ratio A/ C rather than to the specific val-ues ofzt or C.

We note from Figures 12a and 13a that eachcurve (associated with a different ALk) decreasesmonotonically with increasing p, with sharp de-creases at certain values of p, indicating a structuraltransition. A larger range of IWr I over the two or-ders of magnitude of p examined here is noted forhigher superhelical densities, with this range be-coming smaller as [ALk I decreases. When p issmall, twisting is penalized more severely, so [ Wrlis large relative to IALk 1, i.e., Wr - ALk. As pincreases, the partitioning of superhelicity betweentwist and writhe changes steadily until a critical

Buckling Tran.~ition.sin DN.4 21

0

“r

*+ *

0005101520253035404550ALk

FIGURE 1I A phase diagram (collective profiledescription ) showing behavior for C/.4 VSIMA 1. Thedata from J iilicher ‘hare shown for comparison with ourrcsuks, Specifically, the inner box delineates the bound-ary of Jiilicher’s reported results. The symbols FO. NP,F 1. and F2 separate regions where circles, nonplanarrings, tigure-8 forms, and interwound structures, respec-tively. are energetically favored.’h The solid circles arecritical points, and the solid and dotted curves denotediscontinuous and continuous transitions betweenforms. ” Our results regarding the critical IALA-I atwhich two families are energetically equivalent are over-laid for the following configurational transitions: FO tononplanar rings ( A), nonplanar rings to Fl forms (0),FO to F! forms(+), and F1 to F2 forms (open stars).

value of p is reached, at which point a sharp transi-tion to a more open form (different configurationalfami]y ) results. For MA = –5.5 at least two such

jumps can be seen; the first near p = 1 is smootherthan the second ( near p = 5), which reaches thefigure-8 interwound. For ALk = –2. 5, the twodrops in I V ‘rI as p increases reflect transitionsfrom interwounds with I H ‘r I = 2 to structureswith I J4’rl R 1 (near p = 1) and from figure-8 tocircle forms (near P = 8). Overall, we note that, as inthe minimization profiles, transitions are sharper forp > 1. Significantly, substantial differences in 113’rlare observed in the biological range of P (say 0.5-1.5). This again suggests that experimental data to-gether with theory might help bracket the relevant p

range for supercooled DNA.In Figures 12b and 13b we plot the energy

difference between the circle and the configura-tions identified in Figures 12a and 13a. The sim-ilarity between Figures 12b and 13b is striking.

Clearly, the p value where the energy difference iszero decreases with decreasing IALk 1. This meansthat at larger [ALk I more ( interwound ) forms areenergetically preferred in comparison to the circle.In other words, buckling from the circle occurs forsmaller IALk I when p is smaller. The more pro-nounced energy jumps at larger p are also evidentfrom the figure.

In addition to clarifying the trends presented inthis paper, the views of Figures 12 and 13 mighthave an interesting biological implication. [t hasbeen suggested by Schurr and co-workers thatchanges in the secondary structure of DNA occuras superhelical stress increases.5h These changeswould lead to an abrupt alteration of the elasticconstants and subsequently a sudden (puttingaside thermal fluctuations ) change in the three-di-mensional shape of the DNA. It is clearly inappro-priate to interpret changes in secondary structureon the basis of simulations involving a homoge-neous model unless the transition would be homo-geneous in some sense, but analogous data mightbe generated for an inhomogeneous model. Ourdata indicate, in particular, that the first transitionis most sensitive to p, so an experimental followingof DNA at low superhelical densities as physiologi-cal density is approached might exhibit the mostpronounced behavior. This is consistent with theobservations of Song et al. by fluorescence polar-ization anisotropy measurements. s(’

Buckling Transitions Are Maskedas DNA Size Increases

Our comparison of buckling profiles for DNA sys-tems of 1000 and 2000 bp ( Figures 3 and 5 ) clearlyshows that configurational and energetic evolu-tions as a function of ALk vary more smoothly asDNA size increases. These observations can be ex-plained in terms of lower curwiture for larger plas-mids ( Figures 7 and 8). Thus, buckling transitionsare expected to be masked for plasmids of biologi-cal interest ( several thousand bp ), except possiblyfor very large p. For DNA minicircles and smallloops, however, buckling phenomena, especiallythe circle to figure-8 transition, might be observ-able, as already suggested, 57and might be impor-tant in regulatory mechanisms. That is, the biolog-ical activity of an enzyme that changes the linkingnumber deficit of topologically constrained DNAmight be facilitated by an abrupt configurationalchange in response to a smooth deformation at acertain range of supercooling stress.~i’$

22 Ramachandran and Schlick

3f

k+

.4Lk=-2 5

2

Lk=-1 5

1

ALk=–O 5 ALk=-098

o

0

-5

–In

1 2 5 1 2 5 1012b

I I I I 1 , I I I I 1 1 I ,‘Energy of lnterwounds

‘(relat,ve to c,rcle)

,.1 2 5 2 5 10

p = lA/C

FIGURE 12 Structural transitions of closed circularDNA at fixed ALk as a function of p: A = 1.28 X 10-”erg cm. (a) The values – Wr for energy minimizedclosed-circular DNA of length 1000 bp are shown at fixedAL.kas P (plotted on a logarithmic scale) is vaned. Thevalue of A is labeled on the plot in units of 10–’9 erg cm.(b) The energies of the interwound structures relative tothe relaxed circle are plotted as a function of p (also on alogarithmic scale). The table inset indicates the valuesof p at which the energy difference between circle andinterwound AE is zero.

Buckling Transitions Are Masked/Softened by Thermal Fluctuations

Our writhe distributions from dynamic simula-tions of DNA in solvent indicate that a range ofconfigurations and energies is expected at a typicalsolution environment, rather than a sampling ofany one major family identified in the minimiza-tion profiles. While this basic result is expected, andalso demonstrated by Harvey and co-workem,’*the quantitative data presented in this connection(Figure 10) also reveal other trends. First, an en-tropic lowering of the writhe ( i.e., a tendency of theDNA to adopt more open forms in comparison tothe potential energy minimum) is evident. Thistendency is more pronounced for larger P andlarger IA.Lk 1. Second, the differences among thedistributions associated with different p are lamestfor greater superhelical densities. Both the top andmiddle panels of Figure 10, which correspond to

2 —

1 r

o1 2 5 1 2 5 10

13b

r ‘ “’’’” M

Energy of Interwounds

(relatlve to circle)

/1ALk=-55

/, ., wh,<hAt .0

.,o~1 2 5 1 2 5 10

P = A/C

FIGURE 13 Structural transitions of closed circularDNA at fixed ALk asa function of p:A = 2.3X 10-19 ergcm. Seecaption to Figure 12.

typical superhelical densities, reveal the displace-ments of the distributions from one another (fordifferent p), suggesting that these trends should beexperimentally observable. Third, the distributionlocations are most sensitive to IAL/cI for smaller p;that is, significant shifts in the distribution rangesare more dramatic for smaller p. This again sug-gests that the actual shifis observed in such labora-tory-generated distributions might help bracket theprevalent value of p.

Phase Diagrams of IVWI vs p Indicate theImportance of Self-Contact InteractionsBeyond the IWr I = 1 Structure andHence the Complement of Theoryand Numerical Work

We found from the analysis of Figure 11 that pre-dictions from an analytical treatment of the DNAas an infinitely thin elastic rod with no self-contact’6 are very accurate for structures with littleor no self-contact, i.e., conformations with I Wrlc 1. Indeed, our numerical results from a modelthat includes self-contact agree very closely withthose of Julicher for transitions between the circle,nonplanar circle, and figu~-8 families.’6 The theo-retical work of Julicher, moreover, can predict the

Btickiing Transitions in DNA 23

stability of the various configurations, ‘b an issuenot straightfonvard to address by our potential-en-ergy minimization approach. However, in the caseof highl y writhed structures, self-contact becomescritically important, and analytical treatmentsmust be complemented with numerical work thatcan easily model nonbonded, electrostatic poten-tials more realistically. Our comparison with theanalytic phase diagram, in particular, shows a pro-nounced shift in the line delineating the F1 and F2interwound regions ( Figure 11).

Possible Application to DNA’s TorsionalStiffness

It is interesting to explore how the results sum-marized above might be applied. Essentially, ourdata present a collective picture of a very stiff DNAat large p values. In this regime, the DNA tends tocluster around a rather small set of structures thatare similar in writhe for a rather large range of ALk

until some critical threshold of superhelical stressis reached, at which point there occurs an abrupttransition from one DNA structure into another( Figures 1a and 10). At the other extreme, when p

< 1, a very large difference in kl’r as ALk increases(see Figure 4a for p = 0.43) is noted. Thus, at verylow values of A/ C, the DNA is much more flexibleand can easily move from one writhe structure toanother with a small change in superhelical stress.While thermal fluctuations mask specific structuraltransitions, differences in writhe corresponding toaccessible DNA configurations at a given (fixed)ALk still exist. Figure 10 shows that such dilTer-ences are pronounced at higher ALk, in factaround values of physiological superhelical densi-ties. Writhe variances are expected to increase withsizeh(’ but might be quite useful in practice forDNA systems of 1000 bp or smaller.

Based on the above findings, a possible applica-tion arises concerning the torsional stiffness. As dis-cussed in the introduction, uncertainty exists re-garding the relevant value of C.43-454yf1Sbfl In-deed, there have been extensive discussions in theliterature regarding the dependence of thermalfluctuations on the elastic constants and interpreta-tion of this dependence in terms of various poly-mer models and experimental data. One procedurethat has been used to estimate C was first describedby Benham,47 explored by Vologodskii et al.4Handlater refined by Klenin et al., 4’ who also accountedfor excluded volume, and arrived at the value C= 3 x 10 “y erg cm. This technique estimates Cfrom the variance in twist in nicked DNA mole-cules as a difference of two terms: an experimen-

tally measured variance in ALk ( reflecting equilib-rium fluctuations) and a numerically derived var-iance in writhe:

(( ATw)2) = (( ALk)2) - (( Wr)2)

——()

< ~ (8)4X ‘

Indeed, many theorists have studied the generationof writhe moments on the basis of various polymermodelsG~-h4:see chapter 11Aof Ref. 60 for a sum-mary.

The above relation assumes statistical indepen-dence of the writhe and twist fluctuations, i.e.,Cov( ATw, Wr) = O. While it is possible that thisterm is smaller in magnitude than the variances inwrithe and twist, it is not known by how much andhow its magnitude might affect estimates of the tor-sional stiffness (see discussions of Benham”s andShore and Baldwin,4’ for example).

Now that experimental techniques are permitt-ing improved imaging of DNA, 4d”h including di-rect measurements of the writhe, ” perhaps aslightly different approach might be considered.Rather than relying on theoretically derived writhevariances for nicked DNA, a series of writhe distri-butions for various p values might be consideredand compared to writhe distributions derived byextensive imaging experiments.

Observing the writhe distribution of DNA plas-mids (at a given ALk) in solution without changeof structure due to the imaging process is not a triv-ial task. The reliability of this technique, however,is likely to increase in the coming years. Tech-niques in this class include cryo-electron micro-scopy 6’”and SFM.4 Both have received increasedattention recently and have been used to exploremany interesting properties of DNA.

The experiment might be used to producewrithe distributions of a specific mixed-sequenceDNA plasmid at a series of superhelical densities,and these could be compared to numerical datasuch as shown in Figure 10. However, it is not clearthat the biological regime of p will be sut%cient toproduce sensitive patterns. Furthermore, becauseof the expected sharp increase of writhe variancewith size, the plasmid should be as small as possi-ble, 500-1000 bp. From studies of nicked DNA, asharp variance in writhe is noted around 1000 bp,4Aso the writhe fluctuations should be easier to de-scribe below this size where thermal fluctuationsare much smaller. High salt conditions, as usedhere, would probably result in more discernible pat-terns, and other special experimental techniques

24 Ramacfmrrdran and .$chlick

(changes in the environment) might be developedto make measurements as sensitive as possible. Aprocedure to determine Wr from the direct imageswill also have to be perfected (e.g., Ref. 67) and,furthermore, the flat (rather than three-dimensional ) origin of the configurations consid-ered (i.e., perhaps by computing analogous distr-ibutions to those shown in Figure 10 for projectedplasm ids ). Clearly, it is not known if the resolutionof experiment will be sufficiently good for this pur-pose and, in any case, a large amount of reliabledata will have to be generated before experimentaldistributions can be constructed.

We hope further synergy will be possible be-tween experiment and modeling with respect to su-percooled DNA. Direct imaging techniques aresteadily improving,4f and advances in both com-puter hardware and software are increasing the re-liability and accuracy of simulation results. Cer-tainly, the elastic model approximation has beenuseful for a long time, but finer resolutions of nu-cleic acid models will undoubtedly be pursued inthe coming years to permit studies on new levels.The most recent improvements in treating thelong-range electrostatic interactions in atomic-res-olution DNA simulations, GX’byfor example, will un-doubtedly considerably increase the quality of all-atom nucleic acid simulations. A possible merging ofthe macroscopic with the microscopic treatments forDNA can be anticipated in the coming years.

This work is supported by the National Institutes ofHealth (Research Resource RR08 102) and an Alfred P.Sloan fellowship. Computational resources were pro-vided in part by the Cornell Theory Center. We thankDrs. Jerry Manning, Wilma Olson, and Alex Vologod-skii for valuable comments, We are grateful to HongmeiJian for related simulation work regarding the influenceof the Debye-Huckel potential on the effective persis-tence length. TS is an investigator of the Howard HughesMedical Institute.

REFERENCES

1.

2.

3.

4.

White, J. H. ( 1989) in Mathematical Melhods forDNA Sequences, Waterman, M. S., Ed., CRC Press,Boca Raton, FL, Chap, 9.Cozzarelli, N. R. & Wang, J. C., Eds. ( 1990) DNATopo/ogy and Its Biological Eflec(s, Cold SpringHarbor Laboratory Press,Cold Spring Harbor, NY.Brady, G. W., Fein, D. B., Lambertson, J., Grassian,V., Foos, D. & Benham, C. J. ( 1983) Proc, Natl.Acad. Sci. USA 80,741.Samon, B., Nigro, C., Armentano, V., Cimieri. S.,

5.

6.

7.

8.

9.

10.

11.

12.

13.14.15.16.17.18.19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.31.32.

33.

34.

Zuccheri, G. & Quagliariello, C. ( 1993) Angew).C’hern.In[. Ed. Engl. 32, 1461.Bustamante, C., Erie, D. A. & Keller, D. ( 1994)Curr. Opin. Struct. Bio/. 4,750.Adrian, M., Bordier, B. H., Wahli, W., Stasiak,A. Z.. Stasiak, A. & Duboehet, J. ( 1990) EMBO J. 9,4551.Zhurkin, V. B., Lysov, Y. P. & Ivanov, V. ( 1979)Nucleic Acid Rev. 6, 1081.Tobias, I. & Olson, W. K. ( 1993) Biopo/vmers 33,639.Dichmann, D. J., Li, Y. & Maddocks, J. H. ( 1996)in Mathematical .4pplicat ions to BiornolecularSIructure and Dynamics, IMA Volumes in Mathe-matics and its ,4pplications, Vol. 82, Mesirov, J. P.,Schulten K. & Sumners, D. W., Eds., Springer-Ver-lag, New York, pp 71-113.Hearst. J. E. & Hunt, N. G. ( 199 I ) 1 Chern. Phys.95,9322.Hunt, N. G. & Hearst, J. E. ( 1991 ) 1 Chem. Phys.95,9329.Shi, Y. & Hearst, J. E. ( 1994) J. Chem. Ph.r’s.101,5186.Le Bret. M. ( 1979 ) Biopol~’mers 18, 1709.Benham. C. J. ( 1989) Hrys. Rev. A 39,2852 ( 1989 ).Le Bret, M. ( 1984 ) Biopolyrners 23, 1835.Ji,ilicher, F. ( 1994 ) Phy.x Rev. E 49,2429.Lahiri. A. ( 1994) Biopolvrners 34,799.Marko. J. F. & Siggia, E. D. ( 1994 ) Science 265,506.Hao, M.-H. & Olson, W. K. ( 1989)Biopo/ymers 28,873.Hao, M.-H. & Olson, W. K. ( 1989 ) Macromo/ecu/es22,3292.Schlick, T. & Olson, W. K. ( 1992 ) J, Moi. Biol. 223,1089.Schlick, T., Li, B. & Olson, W. K. ( 1994) Bioph.vs.J. 67,2146.Liu, G., Olson, W. K. & Schlick, T. ( 1995) Comp,Polvrn. Sci. 5,7.Yang, Y., Tobias, I. & Olson, W. K. ( 1993) .1.Chem. Ph~’.s.98, 1673.Bauer, W. R., Lund, R. A. & White, J. H. ( 1993)Proc. Natl. Acad. Sci. USA 90,833.Vologodskii, A. V. & Cozzarelli, N. R. ( 1994) Ann.Rev. Biophys. Biomol. S(ruct. 23,609.Tan, R. K. Z. & Harvey, S. C. ( 1989) J. Mol. Biol.205,573.Ramachandran, G. & Schlick, T. ( 1995) Phys. Rev,E51,6188.Chirico, G. & Langowski, J. ( i994) Biopolymers 34,415.Klapper, 1.( 1996) J. Comp. Phys.,125, 325.Schlick, T. ( 1995 ) Curr. Opin. Strut. Bio[. 5,245.Schlick, T., Olson, W. K., Westcott, T. & Greenberg,J. P. ( 1994 ) Biopol~’mers 34,565.Zajac, E. E. ( 1962) Trans. ASME Ser. E J. App/.Mech. 29, 136.Zhang, P., Tobias, 1.& Olson, W. K. ( 1994 ) J. Mol.Biol. 242,271.

Biwklirrg Trun.siiiotrs in D,V.4 25

35. Tohias. l,. Coleman, B. & Olson, W. K. ( 1994) J( ‘bcn7. P/n’\. 101, 10990.

M. Schlick. T.. Hingerty, B, E,, Peskin, C. S,, Overton.

M. 1.. & Broyde, S. ( 199 I ) in Tbeowtica/ Bioc/wm-{\[rI’ <1/d .\/~J[>ci//t~rBi~@l~’.~ic.s,Vol. 1, Bevendge,D. L. & Lava-y, R., Eds.. Adenine Press, Schenec-tady. N1’. pp. 39–58.

37. Peskin. C. S. & Schlick. T. ( [989) (’{~)nn?.Pure ..lpp,,ll(t//I, 42, 1001.

38. Romachandran. G. & !jchlick. T, ( 1996) in Globu/,\litlitjljxl{tvj 1~/,VfJtI-C ‘cw(I.Vk’twr<~jF[mc(i(ms. J!ok’c-ld~~rC‘(m/(w))7wIrM7 utd Pro(c>itl FfMin~ ( DIMACS Se-ries in Discrete Mathematics and Theoretical Com-puter Science. Vol. 23), American Mathematical So-ciet}, Providence, RI. pp. ? 15-231.

39. Zhang. G. & Schlick. T. ( 1993) J Cfnnp. (’bwn, 14,1212,

40, Schlick, T,, Li. B.& Hao, M.-H, ( 1994 ) in S(rIw/I/ra/Bi,d(j,qj .Yt(llc 01”1/11’Irl 199.f, l’rcxct’ding~ (!/”1/11’I<i,yhtb ( ‘(jt?~’(’r$c~li(~tl$,[“(iMI)Jf) [, !jarma, R. &Sarma. M.. Eds.. Adenine Press. Schenectady. NY.pp. 157-174,

41. Hagcrman. P, J. ( 198X).-tnn. Ret’. Bi~@lj. Bi~yrbls(“/!1’1)1 17, 265.

42. Barkle>, M. D. & Zimm, B. H. ( 1979) J Cbem.1’/7)’$ 70,2991.

43. Shore. D,& Baldwin. R. L. ( 1983 ) Y..!1{)/ f3/~~/.170,957.

4-1. Sh~we.D.& Baldwin, R. L. ( 1983 ) J .if[d. Bio/. 170,983.

45. }{{)r(jwl(z.D. S. & Wang. J. (“. ( 1984) .1. ,tffd, Bi(d.

173,75.46. Klenin. K. V.. Vologodskii, A. V., Anshelevich,

V. V,, Klisko. V. Y., Dykhne, A. M, & Frank-Kamen-etskii, M. D, ( 1989) J. lli~m~i .Stru(l. D~wun?.6,707.

47. Benham. C’,J. ( 1978) Y. .tf~J/.BI(J. 123,361.48. Vologodskii, .A. V., Anshele~ich. V. V.. Lukashin,

A. V. & Frank-Kamenetskii, M. D, ( 1979) Na[urc280.294.

49. Clcndenning, J. B.. Naimushin. A. N.. Fujimoto.B. S.. S~ewwt, D. W. & Schurr. J. M. ( 1994) Bi{)-p/~I\. ( ‘/7(>n7.52, 191.

50. Cicbc. J, A.. ,Allison, S. A., Clendenning, J. B. &Schurr. J. M. ( 1995 ) Bi(~p//~).J 68,619.

51. Millar, D. P., Robbins. R. J. & Zewail, A. H. ( 1982)J. C17mr. P/?,Is.s.76,2080.

52. Gorin. A. A., Zhurkin, V. B. &Olson, W. K. ( 1995)J. M(J. Bi(l 247, 34,

53. Olson, W. K,, Babcock. M. S.. Gorin, A., Liu,G.-H., Markey, N. L., Martino, J. A., Pedersen,S. C., Srinivasan, A. R.. Tobias, I., Westcott. T. P.&Zhang, P. ( 1995) Biopb~.s Cbrtn. 55,7.

54. Manning, G. S. ( 198 I ) Biopo/~wtcr.s 20, 1751.55. Vologodskii, A. V. & Cozzarelli, N. R. ( 1995) Bio-

pol~wuv-v35,289.56. Song. L.. Fujimoto, B. S., Wu. P.. Thomas, J. C.,

Shibata, J. H. & Schurr, J. M. ( 1990) J. M(JI. Biol.241,307.

57. Bednar, J., Furrer, P., Stasiak. A., Dubochet. J., En-gelman, E. H. & Bates, A. D. ( 1994) J. Md. Bio/.235,825.

58. Sprous, D.. Tan, R. K.-Z. & Harvey, S. C. ( 1996)Biopolytncr.f 39,243-258.

59. Boles. T. C., White, J. H.& C’ozzarelli. N. R. ( 1990)J. ,~f{J. Bifd. 213,931.

60. Vologodskii. A. ( 1992) T{tpf~/qg~and P/11’.YK.i~~/Cir-cw/ar D.V.I, CRC Press, Boca Raton, FL.

61. Naimushin. A. N., Clendenning. J. B., Kim, U..Song, L.. Fujimoto. B. S., Stewart. D. W. & Schurr,J. M. ( 1994) Bif@~~$. ~’hcln. 52,219,

62. Le Bret, M. ( 1980 ) Bif~p~Jjmm\ 19,619.63. Chen. Y. D. ( 198 I ) J. Cbctn. Pb.I’.!.75,2447.64. Shimada, J. & Yamakawa. H. ( 1988) f?iopo/~wwr.\

27,657,65. Benham. C. J. ( 1990) in Slnwlurc and .tfcrb~tds I ‘(J/-

II))7C .I: DN.4 untl RV’.4. Sarma. R. H. & Sarma,M. H., Eds., Adenine Press. Guilderland. NY, pp. 17-38.

66. Dubochet, J.. Adrian, M.. Chang, J.-J., Homo, J. C..Lepault, J., McDowall. A. W. & Schultz, P. ( 1988)Q~w[. Rm. Bif@~i. 21, 129.

67, Zuccheri, G.. Raneri, G. A.. Nigro, C, & Samon, B.( 1995) Y. 1‘(w.Sci. Tccbn~~/.B 13, 158.

68. York. D. M.. Yang, W.. Lee, H., Darden, T. & Ped-erson. L. G. ( [995 ) J. .fm. Cbmn. Sm. 117, 5001.

69. Cheatham. T. E.. Ill. Miller. J. L., Fox, T., Darden,T. A. & Kollman. P. A. ( 1995 ) J .lm. Cbcm .S{x117,4193.