Elastic property of single double-stranded DNA molecules ...

PHYSICAL REVIEW E VOLUME 62, NUMBER 1JULY 2000Elastic property of single double-stranded DNA molecules:Theoretical study and comparison with experimentsHaijun Zhou, 1,2, * Yang Zhang, 1 and Zhong-can Ou-Yang 1,31 Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing 100080, China2 State Key Laboratory of Scientific and Engineering Computing, Beijing 100080, China3 Institute for Advanced Study, Tsinghua University, Beijing 100084, ChinaReceived 7 December 1999This paper aims at a comprehensive understanding of the novel elastic property of double-stranded DNAdsDNA discovered very recently through single-molecule manipulation techniques. A general elastic modelfor double-stranded biopolymers is proposed, and a structural parameter called the folding angle is introducedto characterize their deformations. The mechanical property of long dsDNA molecules is then studiedbased on this model, where the base-stacking interactions between DNA adjacent nucleotide base pairs, thesteric effects of base pairs, and the electrostatic interactions along DNA backbones are taken into account.Quantitative results are obtained by using a path integral method, and excellent agreement between theory andthe observations reported by five major experimental groups are attained. The strong intensity of the basestacking interactions ensures the structural stability of DNA, while the short-ranged nature of such interactionsmakes externally stimulated large structural fluctuations possible. The entropic elasticity, highly extensibility,and supercoiling property of DNA are all closely related to this account. The present work also suggests thepossibility that negative torque can induce structural transitions in highly extended DNA from the right-handedB form to left-handed configurations similar to the Z-form configuration. Some formulas concerned with theapplication of path integral methods to polymeric systems are listed in the Appendixes.PACS numbers: 87.15.By, 05.50.q, 64.60.Cn, 75.10.HkI. INTRODUCTIONThe DNA molecule is the primary genetic material ofmost organisms. It is a double-helical biopolymer in whichtwo chains of complementary nucleotides the subunitswhose sequence constitutes the genetic message wind usuallyright-handedly around a common axis to form a doublehelicalstructure 1. Because of this unique structure, theelastic property of DNA molecule influences its biologicalfunctions greatly. There are mainly three kinds of deformationsin a DNA double helix: a stretching and bending of themolecule and a twisting of one nucleotide chain relative toits counterpart. All these deformations have vital biologicalsignificance. During DNA replication, hydrogen bonds betweenthe complementary DNA bases should be broken, andthe two nucleotide chains should be separated. This strandseparationprocess requires a cooperative unwinding of thedouble helix 2. In a DNA recombination reaction, RecAproteins recombination proteins of Type A polymerizealong DNA template and the DNA molecule is stretched to1.5 times its relaxed contour length 3,4. It is suspected thatthermal fluctuations of the DNA central axis might be veryimportant for RecA polymerization 5. Another importantexample is the process of chromosome condensation duringprophase of the cell cycle, where a long circular DNAchain wraps tightly onto histone proteins and is severely bent2. Furthermore, in living cells the DNA chain is usuallyclosed, i.e., the two ends of the molecule are linked togetherby covalent bonds and the molecule becomes endless. Withthis chain-closing process, all the quantities characterizing*Electronic address: zhouhj@itp.ac.cnthe topological state of the chain are fixed and can only bechanged externally by topoisomerases, which are capable oftransiently cutting one DNA strand or both and making onestrand pass through the other at the cutting point in the caseof type I topoisomerases 6 or one segment of DNA passesthrough another in the case of type II topoisomerases 7,8.It is possible that this kind of enzyme-induced topologychangingprocess is also closely related to the particular mechanicproperty of the DNA molecule. For example, the frequencyof collisions between two distant DNA segments in acircular DNA molecule is influenced by the different knottypes and different linking numbers for a definition of thisquantity, see below and in Sec. II C. A thorough investigationof the deformation and elasticity of DNA will enable usto gain a better understanding of many important biologicalprocesses concerned with life and growth.A detailed study of DNA elasticity has now become possibledue to recent experimental developments, including,e.g., optical tweezer methods, atomic force microscopy, andfluorescence microscopy. These techniques make it possibleto manipulate single polymeric molecules directly, and torecord their elastic responses with a high precision. Experimentsdone on double-stranded DNA dsDNA have revealedthat this molecule has very interesting elastic properties9–16. When a torsionally relaxed DNA is pulled with aforce of less than 10 picoNetwon pN, its elastic responsecan be quantitatively understood by regarding the chain as aninextensible thin string with a certain bending rigiditynamely, the wormlike chain model 9,17,18. However, ifthe external force is increased up to 65 pN, the DNA chainbecomes highly extensible. At this force, the molecule transitsto an overstretched configuration termed S-DNA, whichis 1.6 times longer than the same molecule in its standard1063-651X/2000/621/104514/$15.00 PRE 62 1045 ©2000 The American Physical Society

1046 HAIJUN ZHOU, YANG ZHANG, AND ZHONG-CAN OU-YANGPRE 62B-form structure 11,12. In addition to external forces, it isalso possible to apply torsional constraints to a DNA doublehelixby external torques. The linking number of DNA, i.e.,the total topological times one DNA strand winds around theother, can be fixed at a value larger less than the molecule’srelaxed value. In such cases we say that the DNA molecule ispositively negatively supercoiled. It was shown experimentally13 that when the external force is less than a thresholdvalue of about 0.3 pN, the extension of DNA molecule decreaseswith increasing twist stress and the elastic responseof positively supercoiled DNA is similar to that of negativelysupercoiled DNA, indicating that the DNA chain might beregarded as achiral. However, if the external force is increasedto be larger than this threshold, negatively and positivelysupercoiled DNA molecules behave quite differently.Under the condition of a fixed external force between 0.3 and3 pN, while positive twist stress keeps shrinking the DNApolymer, the extension of negatively supercoiled DNA is insensitiveto the supercoiling degree 13,14. In the higherforce region, it was suggested by some authors that positivelysupercoiled DNA may transit to a configuration calledPauling-like DNA (P-DNA) with exposed nucleotide bases15, while negative torque may lead to strand separation inthe DNA molecule the denaturation of the DNA double helix14. A very recent systematic observation performed byLéger et al. 16, on the other hand, suggested another possibility:that negative supercoiling may result in a lefthandedZ-form configuration in DNA.The above-mentioned complicated elastic property revealedby the experiments may be directly related to theversatile roles played by the DNA molecule in living organisms.Theoretically, to understand the DNA elastic propertyis of current interest. Models concerned with one or anotheraspect of DNA elasticity were proposed, and valuable insightswere obtained see, for example, Refs. 19,17,18,20–28, and now it is widely accepted that the competition betweenDNA bending and torsional deformations deservesconsiderable attention in order to understand the elastic propertyof DNA molecules. However, it is still a great challengeto understand systematically and quantitatively all aspects ofthe DNA mechanical property based on the same unifiedframework. What is the intrinsic reason for the DNA molecule’sentropic elasticity and high degree of extensibility, aswell as its supercoiling property? Is it possible for negativetorque to stabilize left-handed DNA configurations? Theseare just some examples of unsolved questions.In the present work, we have tried to obtain a comprehensiveand quantitative understanding of DNA mechanicalproperties. We have postulated that the double-stranded natureof the DNA structure should be extremely important toits elastic property, and therefore we have constructed a generalelastic model in which this characteristic is properlytaken into account via the introduction of a structural parameter:the folding angle . The elastic property of longdsDNA molecules was then studied based on this model,where the base-stacking interactions between DNA adjacentnucleotide base pairs, their steric effects, and the electrostaticinteractions along DNA backbones were all considered.Quantitative results were obtained by using the path integralmethod, and an excellent agreement between theory and theexperimental observations of several groups was attained. Itwas revealed that, on the one hand, the strong intensity of thebase-stacking interactions ensures the structural stability ofthe DNA molecule, while, on the other hand, the shortrangednature of such interactions makes externally stimulatedlarge structural fluctuations possible. The entropic elasticity,highly extensibility, and supercoiling property of thedsDNA molecule are all closely related to this fact. Thepresent work also revealed the possibility that negativetorque can induce structural transitions in highly extendedDNA from right-handed B-form-like configurations to lefthandedZ-form-like configurations. Some discussions on thisrespect were performed, and we suggested that a possibledirect way to check the validity of this opinion is to measurethe values of the critical torques under which such transitionsare anticipated to take place according to the present calculations.This paper is organized as follows: In Sec. II we introducethe elastic model for dsDNA biopolymers. At the action ofan external force, the elastic response of dsDNA molecules isinvestigated in Sec. III, and compared with experimental observationsof Smith and co-workers 9,12 and Cluzel et al.11. Section IV focuses on supercoiled dsDNA molecules,where the relationship between the extension and supercoilingdegree is obtained numerically and compared with theexperiment of Strick et al. 13. From the calculated foldingangle distribution, we infer that negative torque can causestructural transitions in dsDNA molecules from right-handeddouble helixes to left-handed ones. Section V is reserved forconclusions. Two appendixes are also presented: in AppendixA we review some basic ideas on the application of pathintegral method in polymer physics, and in Appendix B welist the matrix elements of the operators in Eqs. 15 and23. Some parts of this work were briefly reported in a previouspaper 29.II. ELASTIC MODEL OF DOUBLE-STRANDED DNAMOLECULEAs already stressed, the DNA molecule is a doublestrandedbiopolymer. Its two complementary sugarphosphatechains twist around each other to form a righthandeddouble helix. Each chain is a linear polynucleotideconsisting of the following four bases: two purines (A andG) and two pyrimidines (C and T) 1,2. The two chains arejoined together by hydrogen bonds between pairs of nucleotidesA-T and G-C. Hereafter, we refer to the two sugarphosphatechains as the backbones, and the hydrogenbondedpairs of nucleotides as the base pairs. In this sectionwe discuss the energetics of such an elastic system see Fig.1. First, the bending energy of the backbones of suchdouble-stranded polymers will be considered; then we willdiscuss the interactions between DNA base pairs and energyterms related to external fields.A. Bending and folding deformationsBackbones can be regarded as two inextensible wormlikechains characterized by a very small bending rigidity k B Tl p , where k B is Boltzmann’s constant, T is the environmentaltemperature, and l p 1.5 nm is the bending persistencelength of single-stranded DNA ssDNA chains 12.The bending energy of each backbone is thus expressed as

PRE 62 ELASTIC PROPERTY OF SINGLE DOUBLE-STRANDED . . .1047as half the rotational angle from t 2 to t 1 , with b being therotational axis Fig. 1. We call the folding angle, and itcan vary in the range (/2, /2), with 0 correspondingto right-handed rotations and hence right-handeddouble-helical configurations, and 0 to left-handed ones.With the help of Eq. 1, we know thatanddbds t 2t 1 sin 2R R n 2FIG. 1. Schematic representation of the double-stranded DNAmodel used in this paper. The right part demonstrates the definitionof on the local t-n plane, where t, t 1 and t 2 are the tangentialvectors of the central axis, and the two backbones, respectively; is the folding angle; and the unit vector btn is perpendicular tothe t-n plane.(/2) 0 L (dt i /ds) 2 ds, where t i (s) (i1 and 2 is the unittangent vector at arclength s along the ith backbone 30,31,and L is the total contour length of each backbone. The positionvectors of the two backbones are expressed as r i (s) s t i (s)ds.Since there are many relatively rigid base pairs betweenthe two backbones in many cases the lateral distance betweenthe backbones can be regarded to be constant andequal to 2R. 1 In this subsection we focus on the bendingenergy of the backbones; therefore, for the moment, we regardeach base pair as a rigid rod of length 2R linking betweenthe two backbones and pointing along a direction denotedby a unit vector b from r 1 to r 2 Fig. 1. Then r 2 (s)r 1 (s)2Rb(s). In B-form DNA the base pair plane isperpendicular to the DNA axis; therefore, in our model therelative sliding of the two backbones is not considered andthe base pair rod is thought to be perpendicular to both backbones31, with b(s)•t 1 (s)b(s)•t 2 (s)0. The centralaxis of the double-stranded polymer can be defined as r(s)r 1 (s)Rb(s) r 2 (s)Rb(s)(r 1 (s)r 2 (s))/2, andits tangent vector is denoted by t. In consistence with actualDNA structures, the central axial tangent t is also perpendicularto b, i.e., b(s)•t(s)0. Notice, however, that t(s)dr/ds; in this paper, s always refers to the arclength of thebackbones.Since all the tangent vectors t 1 , t 2 , and t lie on the sameplane perpendicular to b, we can write thatt 1 stscos snssin s,t 2 stscos snssin s,where n is also a unit vector, and nbt, and is defined1 In our present work, we have not taken into account the possibledeformations of the nucleotide base pairs. In many cases this maybe a reasonable assumption. However, under some extreme conditionssuch kinds of deformations may turn out to be important. Forexample, when a DNA double helix is stretched and at the sametime has a large positive torque applied to it, the nucleotide basepairs may collapse see Ref. 15 for a description.1drds 1 2 t 1t 2 t cos .Equation 3 indicates that cos measures the extent towhich the backbones are ‘‘folded’’ with respect to the centralaxis. Based on Eqs. 1–3, the total bending energy of thetwo backbones can be expressed in the following form:dt 1dsdt 22dsdsE b 2 0LL ds0 dtL ds0 dsdt2ds 20L2cosds2 sin 2dnds2dsd 2 R 2 sin 4 .2dsd 2The bending energy is thus decomposed into the bendingenergy of the central axis the first term of Eq. 4 plus thefolding energy of the backbones the second and third termsof Eq. 4. The physical meanings of these two energy contributionsare very clear, and Eq. 4 is very helpful for ourfollowing calculations. In Eq. 4, the bending energy of thecentral axis is very similar with that of a wormlike chain18, both of which are related to the square of the changingrate of the axial tangent vectors. But there are two importantdifferences: a in the derivative dt/ds of Eq. 4, thearclength parameter s is measured along the backbone, notalong the central axis; and b in the wormlike chain modelthe central axis is inextensible, while here the central axis isextensible.In deriving Eq. 4, the base pairs are models just as thinrigid rods of fixed length, i.e., the DNA molecule is viewedas a ladderlike structure see Fig. 1. Actually, however, basepairs form disklike structures and have a finite volume. Thesteric effects caused by the finite volume of base pairs areanticipated to hinder considerably the bending deformationof the central axis, and hence will increase its bending rigiditygreatly 1. Furthermore, the dsDNA molecule is a strongpolyelectrolyte, with negatively charged groups distributedregularly along the chain’s surface. The electrostatic repulsionforce between these negatively charged groups will alsoconsiderably increase the bending rigidity of the dsDNAchain 1. To quantitatively take into account the above mentionedtwo kinds of effects is very difficult. Here we treat thisproblem phenomenologically by simply replacing the bendingrigidity in the first term of Eq. 4 with a quantity *.It is required that *, and the precise value of * will34

1048 HAIJUN ZHOU, YANG ZHANG, AND ZHONG-CAN OU-YANGPRE 62then be determined self-consistently by the best fitting withexperimental data, as shown in Sec. III.B. Base-stacking interactions between base pairsIn Sec. II A, we have discussed in detail the bending energyof dsDNA polymers, which is caused by a bending ofthe backbones, as well as steric effects and electrostatic interactions.In the dsDNA molecule there is another kind ofimportant interaction, namely, the base-stacking interactionbetween adjacent nucleotide base pairs 1,2. Base-stackinginteractions originate from the weak van der Waals attractionbetween the polar groups in adjacent nucleotide base pairs.Such interactions are short ranged, and their total effect isusually described by a potential energy of the Lennard-Jonesform 6-12 potential 1. Base-stacking interactions play asignificant role in the stabilization of the DNA double helix.The main reason why DNA can but RNA cannot form a longdouble helix is as follows 32: Because of the steric interferencecaused by the hydroxyl group attached to the 2carbon of RNA riboses, the stacking interaction between adjacentRNA nucleotide base pairs is very weak and cannotstabilize the formed double-helical structure; while in theDNA ribose, a hydrogen atom is attached to its 2 carbon,and serious steric interference is avoided fortunately.In a continuum theory of elasticity, the summed totalbase-stacking potential energy is converted into the form ofthe integrationN1E LJ i1U i,i1 0Lds,where U i,i1 is the base-stacking potential between the ithand (i1)th base pairs, N is the total number of base pairs,and the base-stacking energy density is expressed asr 0 cos 120cos 2 cos 60cos for 06cos 12 r 0 2 cos 6 0 for 0.0In Eq. 6, the parameter r 0 is the backbone arclength betweenadjacent bases (r 0 L/N); 0 is a parameter related tothe equilibrium distance between a DNA dimer (r 0 cos 03.4 Å ); and is the base stacking intensity which is generallybase-sequence specific 1. In this paper we focus onmacroscopic properties of long DNA chains composed ofrelatively random sequences; therefore, we just consider inthe average sense and take it as a constant, with 14.0 k B T averaged over quantum mechanically calculatedresults on all the different DNA dimers 1.The asymmetric base-stacking potential Eq. 6 ensuresthat a relaxed DNA molecule takes on a right-handeddouble-helix configuration i.e., the B form, with its foldingangle 0 . To change the local configuration of DNA considerablyfrom its B form generally requires a free energy ofthe order of per base pair. Thus, the DNA molecule will bevery stable under normal physiological conditions, and thermalenergy can only make it fluctuate very slightly around itsequilibrium configuration, since kT. Nevertheless, althoughthe stacking intensity in dsDNA is very strong5compared with the thermal energy, the base-stacking interactionby its nature is short ranged, and hence sensitive to thedistance between the adjacent base pairs. If a dsDNA chainis stretched by large external forces, which cause the averageinter-base-pair distance to exceed some threshold value determinedintrinsically by the molecule, the restoring forceprovided by the base-stacking interactions will no longer beable to offset the external forces. Consequently, it will bepossible that the B-form configuration of dsDNA will collapse,and the chain will become highly extensible. Thus, onthe one hand, the strong base-stacking interaction ensuresthat the standard B-form configuration is very stable uponthermal fluctuations and small external forces this is requiredfor the biological functions of DNA molecule to beproperly fulfilled 2; however, on the other hand, its shortrangedness gives it considerable latitude to change its configurationto adapt to possible severe environments otherwise,the chain may be pulled apart by external forces, forexample, during DNA segregation 2. This property ofDNA base-stacking interactions is very important to thedsDNA molecule. As we will see in Secs. III and IV, themechanical properties of a DNA chain are indeed closelyrelated to the above-mentioned insight.C. External forces and torquesIn the previous two subsections, we have described theintrinsic energy of the DNA double helix. Experimentally, toprobe the elastic response of linear DNA molecule, the polymerchain is often pulled by external force fields and/or untwistedor overtwisted by external torques. To study the mechanicresponse of the dsDNA molecule, in this subsectionwe consider the energy terms related to external forces andtorques in our theoretical framework.For the external force fields, here we constrained ourselvesto the simplest situation where one terminal of a DNAmolecule is fixed and the other terminal is pulled with a forceF f z 0 along the direction of the unit vector z 0 9. In fact,hydrodynamic fields or electric fields are also frequentlyused to stretch semiflexible polymers 33, but we will notdiscuss such cases in this paper. The end-to-end vector of aDNA chain is expressed as L 0 t(s)cos (s)ds according to Eq.3. Then the total ‘‘potential’’ energy of the chain in theexternal force field isE f 0Lt cos ds•F 0Lf t•z0 cos ds.In the experimental setup, external torques can be appliedon a linear dsDNA molecule by the following procedure:first, DNA ligases are used to ligate all the possible singlestrandednicks; then the two strands of a dsDNA molecule atone end are fixed onto a template, while the two strands atthe other end are attached tightly to a magnetic bead; afterwards,torques are introduced into a DNA double helix byrotating the magnet bead with an external magnetic field11,15. The torque energy is then related to the topologicalturns caused by the external torque on a DNA double helix.The total number of times one DNA strand winds topologicallyaround the other, which is usually termed the total linkingnumber Lk 34–36, is expressed as the sum of the twist-7

PRE 62 ELASTIC PROPERTY OF SINGLE DOUBLE-STRANDED . . .1049ing number Tw„r 1 (r 2 ),r… of backbone r 1 or r 2 ) around thecentral axis r and the writhing number Wr(r) of the centralaxis, i.e., LkTwWr. According to Refs. 35–37 and Eq.2, we obtain thatTwr 1 ,r2 1 L dbtb•0 ds 2 ds1 Lsin 0 Rds. 8The writhing number of the central axis is generally muchmore difficult to calculate. It is expressed as the followingGauss integral over the central axis 34:Wrr4 1 drdr•rr. 9rr 3In the case of linear chains, provided that some fixed directionfor example, the direction of the external force, z 0 ) canbe specified, and that the tangent vector t never points toz 0 i.e., t•z 0 1), it was proved by Fuller that the writhingnumber Eq. 9 can be calculated alternatively accordingto the following formula 38:Wrr2 1 L z 0 t•dt/dsds. 100 1z 0 •tThe above equation can be further simplified for highly extendedlinear DNA chains whose tangent t fluctuates onlyslightly around z 0 . In this case, Eq. 10 leads to the approximateexpression thatWrr4 10Ldt xt yds t xdt ydsds,11where t x and t y are the two components of t with respect totwo arbitrarily chosen orthonormal directions (x 0 and y 0 )onthe plane perpendicular to z 0 , respectively.The energy caused by the external torque of magnitude is then equal toE t 2 Lk2TwWr.12To conclude this section, the total energy of a dsDNAmolecule under the action of an external force and an externaltorque is expressed as*ds2 dtdsd 2 R 2 sin 4 EE b E LJ E f E t 0L f t•z 0 cos R sin z 0t•dt/dsds1z 0 •t0L *dsdt2dsd 2 R 2 sin 4 f t•z 0 cos R sin 2 t dt xyds 2 t dt yxdsds.1314Note that Eq. 13 can be applied only in the case of highlyextended DNA. In the following two sections, we will studythe mechanical property of single dsDNA molecules basedon the model energy Eqs. 13 and 14. The theoreticalresults will be compared with experimental observations anddiscussed.III. EXTENSIBILITY AND ENTROPIC ELASTICITYOF DNAIn this section we investigate the elastic responses ofsingle DNA molecules under the actions of external forcesbased on the model introduced in Sec. II. There is no externaltorque acted upon; thus 0 in Eq. 13. The particularform of the energy function Eq. 13 of the present modelmakes it convenient for us to study its statistical property bythe path integral method. In Appendix A a detailed descriptionof the application of the path integral method to polymerphysics is given 39. Our calculations in this section and thenext section are based on this method.For a polymer whose energy is expressed in the form ofEq. 13 with 0, according to the technique outlined inAppendix A see Eqs. A1 and A7, the Green equationgoverning the evolution of the ‘‘wave function’’ (t,;s)of the system is obtained to be of the formt,;ss 24l*t 22 p4l p 2 f cos k B Tt•z 0 k B T l pR 2 sin 4 t,;s,15where l p**/k B T and l/k B T. The spectrum of theabove Green equation is discrete and, for a long dsDNAmolecule according to Eqs. A10 and A13, its averageextension can be obtained either by a differentiation of theground-state eigenvalue g 0 of Eq. 15 with respect to f,Z0Lt•z0 cos dsLk B T g 0 f ,16or by a direct integration with the normalized ground-stateeigenfunction, 0 (t,), of Eq. 15:ZL 0 2 t•z 0 cos dtd.17Both g 0 and 0 (t,) can be obtained numerically throughstandard diagonalization methods and identical results areobtained by Eqs. 16 and 17. Here we just briefly outlinethe main procedures in converting Eq. 15 into the form of amatrix.First, for our convenience, we perform the transformation˜ 2 ;18hence the new argument ˜ can change in the range from 0 to. Then we choose a combination of Y lm (t) and f n (˜ ) as thebase function of the Green equation Eq. 15:t,˜ ;slmnC lmn sY lm tf n ˜ .19

1050 HAIJUN ZHOU, YANG ZHANG, AND ZHONG-CAN OU-YANGPRE 62FIG. 2. Force-extension relation of torsionally relaxed DNAmolecule. Experimental data are from Fig. 2A of Ref. 11 symbols.The theoretical curve is obtained by the following considerations:a l p 1.5 nm and 14.0k B T; b l p*53.0/2cos f0 nm, r 0 0.34/cos f0 nm and R(0.3410.5/2)tan f 0 nm; and c the value of 0 is adjusted to fit the data.For each 0 , the value of cos f0 is obtained self-consistently.The present curve is drawn with 0 62.0° in close consistencywith the structural property of DNA, and cos f0 is determinedto be 0.573840. The DNA extension is scaled with its B-form contourlength Lcos f0 .In the above expression, Y lm (t)Y lm (,) (l0,1,2, ...;m0,1,...,l) are the spherical harmonics40, where and are the two directional angles of t, i.e.,t(sin cos ,sin sin ,cos ); andf n ˜ 2 a sin n a ˜ n1,2,...20are the eigenfunctions of one-dimensional infinitely deepsquare potential well of width a. 2With the wave function being expanded using theabove mentioned base functions, the operator acting on i.e., the expression in the square brackets of Eq. 15 canalso be written in matrix form under these base functions.This matrix, whose elements are listed in Appendix B, isthen diagolized numerically to obtain its ground-state eigenvalueand eigenfunction. To simplify the calculation, we furthernote that in the present case of Eq. 15, the ground stateis independent of , i.e., m can be set to m0 in Eq. 19.The resulting force vs extension relation obtained fromEqs. 16 or 17 is shown in Fig. 2 in the whole relevantforce range, and compared with the experimental observationof Cluzel et al. 11. The theoretical curve in this figure isobtained with just one adjustable parameter see the captionof Fig. 2; the agreement with experiment is excellent. Figure2 In the actual calculations, the right boundary of the square well ischosen to be slightly less than to avoid flush off of computermemory caused by the divergence of the base-stacking potentialEq. 6 at ˜ . Weseta0.95 in this paper. However, wehave checked that the results are almost identical for other values ofa, provided that a165°.FIG. 3. Folding angle distribution for torsionally relaxed DNAmolecules under external forces.2 demonstrates that the high extensibility of the DNA moleculeunder large external forces can be quantitatively explainedby the present model.To further understand the force-induced extensibility ofDNA, in Fig. 3 the folding angle distribution of a dsDNAmolecule is shown, with the external force kept at differentvalues. Here, according to Eq. A13 the folding angle distributionP() is calculated by the formulaP 0 t, 2 dt.21Figures 2 and 3, taken together, demonstrate that the elasticbehaviors of the dsDNA molecule are radically differentunder the condition of low and large applied forces. In thefollowing, we will discuss these separately.The low-force region. When the external force is low(10 pN), the folding angle is distributed narrowly aroundan angle of 57°, and there is no probability for thefolding angle to take on values less than 0° Fig. 3, indicatingthat the DNA chain is completely in the right-handedB-form configuration with small axial fluctuations. Thisshould be attributed to the strong base-stacking intensity, aspointed out in Sec. II B. Consequently, the elasticity of theDNA is solely caused by thermal fluctuations in the axialtangent t Fig. 2, and the DNA molecule can be regarded asan inextensible chain. This is the physical reason why, in thisforce region, the elastic behavior of the DNA can be welldescribed by the wormlike chain model 17,18,41. Indeed,as shown in Fig. 4, at forces 10 pN, the wormlike chainmodel and the present model give identical results. Thus wecan conclude with confidence that, when external fields arenot strong, the wormlike chain model is a good approximationof the present model to describe the elastic property ofdsDNA molecules; the bending persistence length of themolecule is 2l p*cos , as indicated by Eqs. 3 and 4.The large-force region. With a continuous increase of externalpulling forces, the axial fluctuations become more andmore significant. For example, at forces 50 pN, althoughthe folding angle distribution is still peaked at 57°, thereis also a considerable probability for the folding angle to bedistributed in the region 0° Fig. 3. Therefore, at thisforce region, the DNA polymer can no longer be regarded asinextensible. At f 65 pN, another peak in the folding angledistribution begins to emerge at 0°, marking the onset of

PRE 62 ELASTIC PROPERTY OF SINGLE DOUBLE-STRANDED . . .1051In Sec. III, we have discussed the elastic response of longDNA chains under the action of external forces. In thepresent section, we study the elasticity of a supercoiled DNAdouble helix. For this purpose, in the experimental setup, allthe possible nicks in the DNA nucleotide strands are ligated13, and a torque as well as an external pulling force act onone terminal of the DNA double helix, which typically untwistsor overtwists the original B-form double helix to someextent and makes its total linking number refer to Sec. II Cfor the definition of the linking number less or greater thanthe equilibrium value. We say that such DNA moleculeswith deficit excess linking numbers are negatively positivelysupercoiled, and define the degree of supercoiling as LkLk 0Lk 0, 22FIG. 4. Low-force elastic behavior of DNA. Here the experimentaldata are from Fig. 5B of Ref. 9, the dotted curve is obtainedfor a wormlike chain with bending persistence length 53.0nm, and the parameters for the solid curve are the same as those inFig. 2.a cooperative transition from B-form DNA to overstretchedS-form DNA 11,12. This is closely related to the shortrangednature of the base-stacking interactions 1 see Sec.II B. At even higher forces ( f 80 pN), 3 the DNA moleculechanges completely into an overstretched form, with its foldingangle peaked at 0°.The force-induced axial fluctuations in the DNA doublehelix can be biologically significant. For example, it has beendemonstrated that axial fluctuations in dsDNA considerablyenhance the polymerization of RecA proteins along the DNAchain 5,42,43. A quantitative study of the coupling betweenRecA polymerization and DNA axial fluctuation is anticipatedto be helpful.It seems that in experiments 11,12 the transition toS-DNA occurs even more cooperatively and abruptly thanpredicted by the present theory see Fig. 2. This may berelated to the existence of single-stranded breaks nicks inthe dsDNA molecules used in the experiments. Nicks inDNA backbones can lead to strand separation or to a relativesliding of backbones 11,12, and this can make the transitionprocess more cooperative. However, the comprehensiveagreement achieved in Figs. 2 and 4 indicates that such effectsare only of limited significance. The elasticity of theDNA is mainly determined by the competition between thefolding angle fluctuation and the tangential fluctuation,which are governed, respectively, by the base-stacking interactions() and the axial bending rigidity (*) in Eq. 13.IV. ELASTIC PROPERTY OF SUPERCOILED DNAwhere Lk 0 represents the linking number of a relaxed DNAmolecule of the same contour length. In living organisms,DNA molecules are often negatively supercoiled, with alinking number deficit of about 0.06. Thus, a detailedinvestigation on the mechanical property of supercoiledDNA molecules is not only of academic interest, but can alsohelp us to understand the possible biological advantages ofnegative supercoiling.A. Relationship between extension and supercoiling degreeWe focus on the property of highly extended DNA moleculeswhose tangent vectors fluctuate only slightly aroundthe force direction z 0 . According to what we mentioned inSec. II C, in this case an approximate energy expression Eq.14 can be used. The external stretching force is restrictedto be greater than 0.3 pN to make sure that the end-to-enddistance of the DNA chain approaches its contour length seeFig. 5. Based on Eqs. 14 and A7, the Green equation forhighly stretched and supercoiled dsDNA is then obtained tobet,;ss 24l*t 22 p4l p 2 f cos k B Tt•z 0 k B T l pR 2 sin 4 sin Rk B T 4k B Tl*p 216l*k p B T 2sin2 t,;s,23where (,) are the two directional angles of t, as mentionedin Sec. III. Similar to what we did in Sec. III, we cannow express the above Green equation in matrix form usingthe combinations of spherical harmonics Y lm (,) andf n () as the base functions. The ground-state eigenvalue andeigenfunction of Eq. 23 can then be obtained numericallyfor given applied force and torque and the average extensionbe calculated through Eq. 16 or through the formulaZL 0 t,t•z 0 cos 0 t,dtd, 24where 0 (t,) is the ground-state left-eigenfunction of Eq.23. 4 The writhing number Eq. 11 is calculated accordingto Eq. A16 to be3 This threshold f t of the overstretch force is also consistent with aplain evaluation from a base-stacking potential of f t r 0 , i.e., f t90 pN.4 As remarked in Appendix A, because the operator in the squarebrackets of Eq. 23 acting on (t,;s) is not Hermitian, the resultingmatrix form of the operator may not be diagonalized byunitary matrices. Consequently, in general, 0 (t,) 0*(t,).

1052 HAIJUN ZHOU, YANG ZHANG, AND ZHONG-CAN OU-YANGPRE 62FIG. 5. Extension vs supercoiling relations at fixed pullingforces for torsionally constrained DNA. The parameters for thecurves are the same as in Fig. 2, and experimental data are fromFig. 3 of Ref. 13 symbols.WrL16l*k p B T 0 t,sin 2 0 t,dtd,and the average linking number is then calculated to beLkTwWr2R L 0 sin 0 dtdL16l*k p B T 0 sin 2 0 dtd.2526Thus, after we have obtained the ground-state eigenvalueas well as its left and right eigenfunctions numerically, wecan calculate numerically all the quantities of our interest,for example, the average extension, the average supercoilingdegree, and the folding angle distribution see also AppendixA. The relation between the extension and supercoiling degreecan also be obtained by fixing the external force andchanging the value of the applied torque. To calculate theground-state eigenvalue and eigenfunctions of an asymmetricmatrix turns out to be complicated and time consuming.Fortunately, as we have calculated in Appendix B, eacheigenfunction of Eq. 23 shares the same quantity m; thematrix for m0 is still Hermitian, and can be diagonalizedby unitary matrices. The ground-state eigenvalue for m0 islower by several orders than those for m0 in the wholerelevant region of external torque from 5.0k B T to5.0k B T. Thus, actually, we only need to consider the case ofm0, and in this case we still have 0 (t,) 0*(t,). Thewhole procedure we performed in Sec. III can safely be repeatedin this section, and the relationships between forceand extension and torque and linking number can consequentlybe calculated.To make the calculation further easier, and also to makesure that the above-mentioned calculation is indeed correct,here we introduce an approximate method which reduces thecomputational complexity considerably. It turns out that thecalculated results using this method are in considerableagreement with the above-mentioned precise method. In theexperiment of Ref. 13, the applied external forces changein the region of 0.3–10 pN. In this region, as demonstrated inFigs. 3 and 4, both the tangential (t) and folding angle ()fluctuations of dsDNA molecules are small. Taking this factinto account, in Eq. 14 the energy term *(dt/ds) 2 can beapproximately calculated to be *„(dt x /ds) 2 (dt y /ds) 2 …,and f t•z 0 cos f cos fcos (t x 2 t y 2 )/2. Thus Eq. 14 isdecomposed into two ‘‘independent’’ parts. The first part isrelated only to . At each value of f and , we can calculatethe average quantities cos and sin based on this energyusing the path integral method. The second part is quadraticin t x and t y , therefore the average values of t•z 0 andWr Eq. 11 can be obtained analytically. Using this decompositionand preaveraging technique, the average extensionand average supercoiling degree can both be calculatedat each value of the external force and torque, and the relationbetween the extension and linking number at fixedforces can be then obtained.The theoretical relationship between extension and supercoilingdegree is shown in Fig. 5 and compared with theexperiment of Strick et al. 13. In obtaining these curves,the values of the parameters are the same as those used inFig. 2, and no adjustment has been made to fit the experimentaldata. We find that in the case of negatively supercoiledDNA, the theoretical and experimental results are inquantitative agreement, indicating that the present model iscapable of explaining the elasticity of negatively supercoiledDNA; in the case of positively supercoiled DNA, the agreementbetween theory and experiment is not so good, especiallywhen the external force is relatively large. In thepresent work, we have not considered possible deformationsof the nucleotide base pairs. While this assumption might bereasonable in the negatively supercoiled case, it may fail fora positively supercoiled DNA chain, especially at largestretching forces. The work done by Allemand et al. 15suggested that a positive supercoiled and highly extendedDNA molecule can take on Pauling-like configurations withexposed bases. To better understand the elastic property ofpositively supercoiled DNA, it is certainly necessary for usto take into account the deformations of base pairs.For a negatively supercoiled DNA molecule, both theoryand experiment reveal the following elastic aspects: aWhen the external force is small, the DNA molecule canshake off its torsional stress by writhing its central axis,which can lead to an increase in the negative writhing number,and hence restore the local folding manner of DNAstrands to that of B-form DNA. b However, a writhing ofthe central axis causes a shortening of DNA end-to-end extension,which becomes more and more unfavorable as theexternal force is increased. Therefore, at large forces, thetorsional stress caused by negative torque supercoiling degreebegins to unwind the B-form double helix, and triggersthe transition of DNA internal structure, where a continuouslyincreasing portion of DNA takes on some certain configurationas supercoiling increases, while its total extensionremains almost invariant. Our Monte Carlo simulations havealso confirmed the above insight 44.What is the new configuration? According to Ref. 13,such a configuration corresponds to denatured DNA segments,i.e., negative torque leads to a breakage of hydrogenbonds between the complementary DNA bases, and consequentlyto strand separation. The authors of Ref. 13 alsoperformed an elegant experiment in which short single-

PRE 62 ELASTIC PROPERTY OF SINGLE DOUBLE-STRANDED . . .1053FIG. 6. Folding angle distributions for negatively supercoiledDNA molecule pulled with a force of 1.3 pN.stranded homologous DNA segments were inserted into theexperimental buffer 14. They found, in confirmation oftheir insight, that these homologous DNA probes indeed bindonto negatively supercoiled dsDNA molecules. Recently,Léger et al. 16 also performed experiments on singledsDNA molecules. To explain their experimental resultqualitatively, they found that a left-handed Z-form DNAmolecule should be considered a possible configuration for anegatively supercoiled dsDNA chain, while the moleculesneed not be denatured. As seen in Fig. 5, although thepresent model has not taken into account the possibility ofstrand separation, it can quantitatively explain the behaviorof negatively supercoiled DNA. Therefore, it may be helpfulfor us to investigate the possibility of formation of lefthandedconfigurations based on our present model. This effortis made in Sec. IV B, where the energetics of such configurationswill also be discussed.B. Possible left-handed DNA configurationsWe mentioned in Sec. II A that left-handed configurationscorrespond to 0 in our present model see also Fig. 1.Based on the present model, then information about the newconfiguration mentioned in Sec. IV A can be revealed by thefolding angle distributions P(), as discussed in Sec. III. Inthe present case, P() is calculated asP 0 t, 0 t,dt, 27where, as mentioned in Sec. IV A, 0 (t,) and 0 (t,) are,respectively, the ground-state right and left eigenfunctions ofEq. 23, and, actually, 0 (t,) 0*(t,).The calculated folding angle distribution 45 is shown inFig. 6, which is radically different from that of the torsionallyrelaxed dsDNA molecules shown in Fig. 3. This distributionhas the following aspects: When the torsional stress issmall with a supercoiling degree 0.025), the distributionhas only one steep peak at 57.0°, indicating thatthe DNA is completely in B form. With an increase of torsionalstress, however, another peak appears at 48.6°and the total probability for the folding angle to be negativelyvalued increases gradually with supercoiling. Sincenegative folding angles correspond to left-handed configurations,the present model suggests that, with an increase ofFIG. 7. a The sum of the average base-stacking and torsionalenergy per base pair at a force of 1.3 pN. For highly extended DNAonly these two interactions are sensitive to torque. b The relationbetween the DNA supercoiling degree and the external torque at aforce of 1.3 pN.supercoiling, the left-handed DNA conformation is nucleatedand then elongates along the DNA chain as B-DNA disappearsgradually. The whole chain becomes completely lefthanded at 1.85.It is worth noting that, a as the supercoiling degreechanges, the positions of the two peaks of the folding angledistribution remain almost fixed; and b between these twopeaks, there exists an extended region of folding angle from0to/6 which always has only an extremely small probabilityof occurrence. Thus a negatively supercoiled DNA canhave two possible stable configurations: a right-handed Bform and a left-handed configuration with an average foldingangle 48.6°. A transition between these two structuresfor a DNA segment will generally lead to an abrupt and finitevariation in the folding angle.To obtain the energetics of such transitions, we have calculatedhow the sum of the base-stacking energy and torsionalenergy, (/R 2 )sin 4 ()(/R)sin , changes withexternal torque 45. Figure 7a shows the numerical result,and Fig. 7b demonstrates the relation between the supercoilingdegree and external torque. In both figures the externalforce is fixed at 1.3 pN. From these figures we can

1054 HAIJUN ZHOU, YANG ZHANG, AND ZHONG-CAN OU-YANGPRE 62TABLE I. For the torque-induced left-handed DNA configuration,the average rise per base pair (d), the pitch per turn of thehelix, and the number of base pairs per turn of the helix Num. arecalculated and listed under different external forces and torques.The last row contains the corresponding values for Z-form DNA.Force pN Torque (k B T) d(Å ) Pitch (Å ) Num.1.3 5.0 3.59 41.20 11.481.0 5.0 3.57 40.93 11.441.3 4.0 3.83 46.76 12.191.0 4.0 3.82 46.38 12.15Z form 3.8 45.6 125 Another possibility may be that we have overestimated the valueof the critical torque. It may be possible that the transition fromright- to left-handed configurations is initiated in the weaker ATrichregions, whose value of should be less than the average valuetaken by the present paper.infer that a for negative torque less than the critical value c 3.8k B T, DNA can only stay in B-form state; b nearthis critical torque, DNA can either be right or left handedand, as negative supercoiling increases see Fig. 7b moreand more DNA segments will stay in the left-handed form,which is much lower in energy (2.0k B T per base pairbut stable only when the torque reaches c ; and c fornegative torque greater than c , the DNA is completely lefthanded.Nevertheless, we should emphasize that the above calculationsare all based on our present model, which assumedthat nucleotide base pairs do not break. Figure 5 indicatesthat for negatively supercoiled DNA chains, the extension vssupercoiling degree relation can be quantitatively explainedby the present model. Figure 6 reveals the reason for thequantitative agreement is that the present model allows forthe possibility of the occurrence of left-handed DNA configurations.At the present time, to state that the negativelysupercoiled DNA will ‘‘prefer’’ left-handed configurationsrather than denaturation and strand separation is premature.To clarify this question, the present model should be improvedto consider deformations of DNA base pairs. On theexperimental side, it might also be helpful to measure preciselythe critical torque at which the elastic behavior ofnegatively supercoiled DNA changes abruptly, and to comparethe measured results with the value calculated in thepresent work. In the earlier experiment of Allemand et al.15, the critical torque was estimated to be 2k B T,byassuming the torsional rigidity of dsDNA to be 75 nm, andassuming that torsional stress builds up linearly along theDNA chain. This value, however, may not be preciseenough, since the torsional rigidity of dsDNA molecule isnot a precisely determined quantity, and the values given bydifferent groups are scattered widely. 5The structural parameters of the left-handed configurationsuggested by Figs. 6 and 7 are listed in Table I 45, andcompared with those of Z-form DNA 2. The strong similaritybetween these parameters suggests that the torqueinducedleft-handed configurations, if they really exist, belongto Z-form DNA 2,16,45.V. CONCLUSIONIn this paper, we have presented an elastic model fordouble-stranded biopolymers such as DNA molecules. Thekey progress is that the bending deformations of the backbonesof DNA molecules and the base-stacking interactionsexisting between adjacent DNA base pairs are quantitativelyconsidered in this model, with the introduction of a structuralparameter: the folding angle . This model has also qualitativelytaken into account the effects of the steric effects ofDNA base pairs and electrostatic interactions along DNAchain. In the calculation technique, the model is investigatedusing the path integral method; in addition, Green equationssimilar in form to the Schrödinger equation in quantum mechanicsare derived, and their ground-state eigenvalues andeigenfunctions are obtained by precise numerical calculations.The force-extension relationship in torsionally relaxedDNA chains, and the extension-linking number relationshipin torsionally constrained DNA chains are studied and comparedwith experimental results. A DNA molecule’s entropicelasticity and highly extensibility, as well as the elastic propertyof negatively supercoiled DNA, can all be quantitativelyexplained by the present theory. The comprehensive agreementbetween theory and experiments indicated that shortrangedbase-stacking interactions are very important in determiningthe elastic response of double-stranded DNAmolecules. The present work showed that highly extendedand negatively supercoiled DNA molecules can be lefthanded, probably in the Z-form configuration. A possibleway to check the validity of this opinion is to measure thecritical external torque at which the transition betweenB-form DNA and the new configuration takes place.The present work regarded DNA basepairs as rigid objects,and did not consider their possible deformations andthe possibility of strand separation. The comprehensiveagreement between theory and experiments indicates thatthis approach is well justified in many cases. However, asmentioned in Sec. II A, under some extreme conditions thisassumption may not be appropriate. For example, recent experimentalwork by Allemand et al. 15 showed that positivelysupercoiled DNA under high applied force can take onPauling-like configurations with exposed bases. In this casethe base pairing of DNA is severely distorted, and becausethe present model has not taken into account the possibledeformations of the base pairs, the theoretical results onpositively supercoiled DNA molecules are not in quantitativeagreement with experiment see Fig. 5. Furthermore, althoughthe present work showed that left-handed DNA configurationscan be stabilized by negative torques, much theoreticalwork is still needed to calculate the denaturation freeenergy and to compare with the free energy of left-handedDNA configurations.ACKNOWLEDGMENTSIt gives us great pleasure to acknowledge the help andvaluable suggestions of our colleagues, especially Lou Ji-Zhong, Liu Lian-Shou, Yan Jie, Liu Quan-Hui, Zhou Xin,Zhou Jian-Jun, and Zhang Yong. The numerical calculations

PRE 62 ELASTIC PROPERTY OF SINGLE DOUBLE-STRANDED . . .1055are performed partly at ITP-Net and partly at the State KeyLab of Scientific and Engineering Computing.APPENDIX A: PATH INTEGRAL METHODIN POLYMER PHYSICSIn this appendix we review some basic ideas on the applicationof the path integral method to the study of polymericsystems 39. Consider a polymeric string, and supposeits total ‘‘arclength’’ is L, and along each arclengthpoint s one can define an n-dimensional ‘‘vector’’ r(s) todescribe the polymer’s local state at this point. 6 We furtherassume that the energy density per unit arclength of thepolymer can be written in the general form e r,s m 2dsdr 2Ar• drds Vr,A1where V(r) is a scalar field and A(r) is a vectorial field. Thetotal partition function of the system is expressed by theintegrationLdr f f r f Gr f ,L;r i ,0 i r i dr i , A2where i (r) and f (r) are the probability distributions ofthe vector r at the initial (s0) and final (sL) arclengthpoint points, respectively. G(r,s;r,s) is called the Greenfunction, and is defined asGr,s;r,s rrDrsexp ssdse r,s ,A3where integration is carried over all possible configurationsof r, and 1/k B T is the Boltzmann coefficient. It can beverified that the Green function defined above satisfies therelation 39Gr,s;r,s drGr,s;r,sGr,s;r,ssss.The total free energy of the system is then expressed asA4systems. Suppose the value of at arclength point s is relatedto its value at s through the following formula suchthat 7r,s drGr,s;r,sr,s ss; A6then we can derive, from Eqs. A1, A3, and A6, that 39r,ss “ r 22m VrAr•“ r “ r•Arm 2m A2 r2mr,sĤr,s. A7Equation A7 is called the Green equation; it is very similarto the Schrödinger equation of quantum mechanics 40.However, there is an important difference. In the case ofA(r)0, the operator Ĥ in Eq. A7 is not Hermitian. Therefore,in this case the matrix form of the operator Ĥ may notbe diagonalized by unitary matrix.Denote the eigenvalues and the right eigenfunctions ofEq. A7 as g i and i i (r) (i0,1,...),respectively.Then it is easy to see, from the approach of quantum mechanics,thatsie g i (ss) isiss, A8where i i (r) (i0,1,...)denote the left eigenfunctionsof Eq. A7, which satisfy the following relation:ii dr i r i r ii .In the case where Ĥ is Hermitian i.e., A(r)0, then wecan conclude that i r i*r.From Eqs. A2, A6, A7, and A8 we know thatL dr drGr,L;r,s f r i rFk B T ln .A5i f ii i e g i L e g 0 L f 00 i To calculate the total partition function , we define anauxiliary function (r,s) and call it the wave function, becauseof its similarity to the true wave function of quantumfor L1/g 1 g 0 .A9Consequently, for long polymer chains the total free energydensity is just expressed as6 For example, in the case of a flexible Gaussian chain, r is athree-dimensional position vector; in the case of a semiflexiblechain, such as a wormlike chain 18,41, r is the unit tangent vectorof the polymer and is two dimensional.7 In fact, the choice of the wave function (r,s) is not limited.Any function determined by an integration of the form of Eq. A6can be viewed as a wave function.

1056 HAIJUN ZHOU, YANG ZHANG, AND ZHONG-CAN OU-YANGPRE 62F/Lk B Tg 0 ,A10and any quantity of interest can then be calculated by differentiationof F. For example, the average extension of a polymerunder external force field f can be calculated as ZF/ f Lk B Tg 0 / f .We continue to discuss another very important quantity,the distribution probability of r at arclength s, P(r,s). Thisprobability is calculated from the following expression:Pr,s dr f dr i f r f Gr f ,L;r,sGr,s;r i ,0 i r i . dr f dr i f r f Gr f ,L;r i ,0 i r i A11Based on Eqs. A6 and A8 we can rewrite Eq. A11 in thefollowing form:Pr,s dr f dr i dr f r f Gr f ,L;r,srrGr,s;r i ,0 i r i dr f dr i f r f Gr f ,L;r i ,0 i r i mn f mn i m r n rexpg m Lsg n sm f mm i expg m L. A12For the most important case of 0sL, Eq. A12 thengives that the probability distribution of r is independent ofarclength s, i.e.,Pr,s 0 r 0 r for 0sL.A13With the help of Eq. A13, the average value of a quantitywhich is a function of r can be obtained. For example,Qs drQrPr,s dr 0 rQr 0 rand0Q00LQ„rs…ds 0LQsdsL0Q0for L1/g 1 g 0 .A14A15Finally, we list the formula for calculating B(r)•dr/ds; here B(r) is a given vectorial field. The formulareads Br• drds 1 2 0r•ĤBB•Ĥr0for L1/g 1 g 0 .APPENDIX B: MATRIX FORMALISMOF THE GREEN EQUATION „23…A16In Eq. 23 and also Eq. 15 the variables t and arecoupled together. Denote the operator in the square bracketsof Eq. 23 as Ĥ. We express this operator in matrix form. Tothis end we choose the base functions of this system to becombinations of spherical harmonics Y lm (,) and f n (˜ )see Eq. 20.For m0, the base functions are expressed to beil•N nl;nY l0 ,f n ˜ ;B1and for m1,2,...,thebase functions are expressed to bei2lm•N 2n1kl;n;kY (k) lm ,f n ˜ .B2In the above two equations, n1,2,...,N , with N set tobe 60 in our present calculations; l0,1,...,N l 1 for thecase of m0 with N l set to 30); or lm,m1,...,mN l 1 for the case of m0 with N l set to 15). k1 and2, andY l0 , 2l14 P lcos ,Y (1) lm ,1 2l1 lm!m2 lm! P l m cos cosm,Y (2) lm ,1 2l1 lm!m2 lm! P l m cos sinm.With the above definitions, we then obtain that, for m0,

PRE 62 ELASTIC PROPERTY OF SINGLE DOUBLE-STRANDED . . .1057i p Ĥil p ;n p Ĥl;n ll p nn pll14l p* n2 24l p a 2fk B T l l1 apl,0 l1 lpa l1,0 n p sin ˜ nl lp n p ˜ k B T l pR 2 cos 4 ˜ nl lpRk B T n pcos ˜ n 2n np16l*k p B T 2 ll p1a l,02 2a l1,0 l2 lpa l,0 a l1,0 l2 lpa l1,0 a l2,0 ;B3and, for m0,li p Ĥil p ;n p ;k p Ĥl;n;k lp n knp kp ll14l*p n2 24l p a 2fk B T k k p l1 lpa l,m l1 lpa l1,m ln p sin ˜ n lp k kp n p ˜ k B T l pR 2 cos 4 l˜ n lpl lpn npk kp4l*k p B T kk pmRk B T n pcos ˜ n n 2knp kp16l*k p B T 2 ll p1a l,m2 2a l1,m l2 lpa l,m a l1,m l2 lpa l1,m a l2,m .In Eqs. B3 and B4, the following notations are used:a l,m l12 m 22l12l3 ,2 an p y˜ na0sin n p˜a y˜ sin n˜a ,B4where y(˜ ) is any function of ˜ .The ground-state eigenvalues of the matrices Eqs. B3and B4 have been calculated at force 1.3 pN in the wholerelevant region of from 5.0k B T to 5.0k B T. We havefound that the ground-state eigenvalues for the case of m0 are of the order of 10 nm 1 , while those for the caseof m0 are of the order of 10 6 nm 1 data not shown.Thus we can conclude with confidence that the ‘‘low energy’’eigenstates of the system all have the same m0.This can greatly reduce the calculation tasks.As a final point, here we demonstrate how to calculate theaverage values of the quantities of our interest. Supposey(t,˜ ) is a quantity whose average value we are interestedin. Denote U as the unitary matrix which can diagonalize thematrix Eq. B3. Because the matrix Eq. B3 is real symmetric,U is actually a real orthogonal matrix. Then its columnvector u(i) j U( j,i) j (i1,2,...) corresponds tothe ith eigenvector of Eq. B3. Consequently, we can calculatebased, on Eq. A14, thatyt,˜ i,ipUi p ,1Ui,1i p yt,˜ i. B51 W. Saenger, Principles of Nucleic Acid Structure Springer-Verlag, New York, 1984.2 J. D. Watson, N. H. Hopkins, J. W. Roberts, J. A. Steitz, andA. M. Weiner, Molecular Biology of the Gene, 4th ed.Benjamin/Cummings, Menlo Park, CA, 1987.3 T. Nishinaka, Y. Ito, S. Yokoyama, and T. Shibata, Proc. Natl.Acad. Sci. USA 94, 6623 1997.4 T. Nishinaka, A. Shinohara, Y. Ito, S. Yokoyama, and T. Shibata,Proc. Natl. Acad. Sci. USA 95, 110711998.5 J. F. Léger, J. Robert, L. Bourdieu, D. Chatenay, and J. F.Marko, Proc. Natl. Acad. Sci. USA 95, 122951998.6 L. Stewart, M. R. Redinbo, X. Qin, W. G. J. Hol, and J. J.Champoux, Science 279, 1534 1998.7 V. V. Rybenkov, C. Ullsperger, A. V. Vologodskii, and N. R.Cozzarelli, Science 277, 690 1997.8 J. Yan, M. O. Magnasco, and J. F. Marko, Nature London401, 932 1999.9 S. B. Smith, L. Finzi, and C. Bustamante, Science 258, 11221992.10 D. Bensimon, A. J. Simon, V. Croquette, and A. Bensimon,Phys. Rev. Lett. 74, 4754 1995.11 P. Cluzel, A. Lebrun, C. Heller, R. Lavery, J.-L. Viovy, D.Chatenay, and F. Caron, Science 271, 792 1996.12 S. B. Smith, Y. Cui, and C. Bustamante, Science 271, 7951996.13 T. R. Strick , J.-F. Alleman, D. Bensimon, A. Bensimon, andV. Croquette, Science 271, 1835 1996.14 T. R. Strick, V. Croquette, and D. Bensimon, Proc. Natl. Acad.Sci. USA 95, 105791998.15 J. F. Allemand, D. Bensimon, R. Lavery, and V. Croquette,Proc. Natl. Acad. Sci. USA 95, 141521998.16 J. F. Léger, G. Romano, A. Sarkar, J. Robert, L. Bourdieu, D.Chatenay, and J. F. Marko, Phys. Rev. Lett. 83, 1066 1999.17 C. Bustamante, J. F. Marko, E. D. Siggia, and S. Smith, Science265, 1599 1994.18 J. F. Marko and E. D. Siggia, Macromolecules 28, 87591995.19 M.-H. Hao and W. K. Olson, Biopolymers 28, 873 1989;Macromolecules 22, 3292 1989.20 B. Fain, J. Rudnick, and S. Östlund, Phys. Rev. E 55, 73641996.21 J. F. Marko, Europhys. Lett. 38, 183 1997; Phys. Rev. E 57,2134 1998.22 J. D. Moroz and P. Nelson, Proc. Natl. Acad. Sci. USA 94, 14418 1997.

1058 HAIJUN ZHOU, YANG ZHANG, AND ZHONG-CAN OU-YANGPRE 6223 A. V. Vologodskii and J. F. Marko, Biophys. J. 73, 123 1997.24 B.-Y. Ha and D. Thirumalai, J. Chem. Phys. 106, 4243 1997.25 P. Cizeau and J.-L. Viovy, Biopolymers 42, 383 1997.26 C. Bouchiat and M. Mézard, Phys. Rev. Lett. 80, 1556 1998;e-print cond-mat/9904018.27 P. Nelson, Phys. Rev. Lett. 80, 5810 1998.28 Zhou Haijun and Ou-Yang Zhong-Can, Phys. Rev. E 58, 48161998; J. Chem. Phys. 110, 1247 1999.29 Zhou Haijun, Zhang Yang, and Ou-Yang Zhong-Can, Phys.Rev. Lett. 82, 4560 1999.30 R. Everaers, R. Bundschuh, and K. Kremer, Europhys. Lett.29, 263 1995.31 T. B. Liverpool, R. Golestanian, and K. Kremer, Phys. Rev.Lett. 80, 405 1998.32 L. Stryer, Biochemistry, 4th ed. Freeman, New York, 1995.33 T. T. Perkins, D. E. Smith, and S. Chu, Science 276, 20161997, and references cited therein.34 J. H. White, Am. J. Math. 91, 693 1969.35 F. B. Fuller, Proc. Natl. Acad. Sci. USA 68, 815 1971.36 F. H. C. Crick, Proc. Natl. Acad. Sci. USA 73, 2639 1976.37 J. H. White and W. R. Bauer, Proc. Natl. Acad. Sci. USA 85,772 1988.38 F. B. Fuller, Proc. Natl. Acad. Sci. USA 75, 3557 1978.39 Zhou Haijun unpublished; also see M. Doi and S. F. Edwards,The Theory of Polymer Dynamics Clarendon Press,Oxford, 1986; H. Kleinert, Path Integrals in Quantum Mechanics,Statistics, and Polymer Physics World Scientific,Singapore, 1990.40 J. Y. Zeng, Quantum Mechanics, 2nd ed. Academic Press,Beijing, 1999 in Chinese, Vol. I.41 K. Kroy and E. Frey, Phys. Rev. Lett. 77, 306 1996.42 G. V. Shivashankar, M. Feingold, O. Krichevsky, and A.Libchaber, Proc. Natl. Acad. Sci. USA 96, 7912 1999.43 M. Hegner, S. B. Smith, and C. Bustamante, Proc. Natl. Acad.Sci. USA 96, 101091999.44 Zhang Yang, Zhou Haijun, and Ou-Yang Zhong-Can, Biophys.J. 78, 1979 2000.45 Zhou Haijun and Ou-Yang Zhong-Can, Mod. Phys. Lett. B 13,999 1999.