Reprinted fromCold Spring Harbor Symposia on Quantitative Biology, Volume XLVII

@ 1983 Cold Spring Harbor Laboratory

Computer Simulation of DNA Double-helix Dynamics

M. LEVITTDepartment of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel

The static structure of DNA has been known for 30years (Watson and Crick 1953). During the past 5 years,DNA has been shown to have a surprising degree ofconformational flexibility in that the number of basepairs per turn is not the same in solution and in fibers(Amott and Hukins 1972; Wang 1979; Rhodes and Klug1980), that the base and backbone atoms undergoangular motions of large amplitude ( > 25”) on a timescale of nanoseconds (Early and Kearns 1979; Boltonand James 1980; Hogan and Jardetsky 1980), and thatthere are cooperative conformational transitionsmediated by changing environment or binding of drugmolecules (Sobell et al. 1977; Hogan et al. 1979; Dat-tagupta and Crothers 198 1). Model building and com-puter calculations have considered the static deforma-tion of the DNA double helix by kinking (Crick andI$lug 1975; Sobell et al. 1977) or by smooth bending(Levitt 1978; Sussman and Trifonov 1978). Elegantmathematics has been used to analyze the dynamicbehavior of DNA by assuming that the moleculebehaves l ike an isotropic elast ic rod (Barkley and Zimm1979).

In the study described in this paper, the nature of thedynamics of the DNA double hel ix was invest igated us-ing the technique of molecular dynamics simulation,which proved so illuminating when used on globularproteins (McCammon et al. 1977; Levitt 198lb). Thistechnique s imulates the movement of a toms in the s ta t icX-ray structure and thus provides information about theamplitudes and frequencies of vibrations and the type,rate, and pathway of conformational changes.

Results are presented for simulations of room-temper-ature atomic motion of 12-bp and 24-bp DNA doublehelices for periods of more than 90 psec. The hydrogenbonds between base pairs are all found to be stable onthis t ime scale, and the motions of the torsion angles arefound to be of small ampli tude ( c loo). The length fluc-tuat ions of adjacent hydrogen bonds in the same basepair are weakly correlated, whereas the torsion angles ofeach nucleotide show stronger correlations that agreewith those seen in the stat ic X-ray structures. Both DNAfragments show cooperat ive overal l bending and twist-ing motions of large ampli tude that do not involve anymajor perturbation of the DNA torsion angles. Thissmooth bending differs from that expected of an isotro-pic elast ic rod in that (1) i t is asymmetric, always act ingto close the major groove of DNA, and (2) it consistspredominantly of a normal mode that has a wavelengthclose to the helical repeat length. The stack of base pairsis also seen to kink into the minor groove. The extent ofthis global motion is consistent with nuclear magnetic

resonance measurements and explains the observed sen-si t ivi ty of DNA conformation to local environment .

These calculat ions have implications for the way theDNA double helix may interact with repressors, poly-merases, and other cellular proteins (Anderson et al.198 1; McKay and Steitz 198 1). The marked contrast be-tween the bending flexibility and the stability of thehydrogen-bonded base pairs suggests that DNA mayprotect the integri ty of the genetic message by absorbingthermal perturbat ions in bending motions, rather than inbase-pair-opening motions.

METHODS

The two different base-paired complexes of DNAstudied were (1) the 12-bp fragment or dodecamer(CGCGAATTCGCG)2, whose static X-ray conforma-tion was recently solved by single-crystal diffraction(Wing et al. 1980; Drew et al. 1981), and (2) the 24-bpfwment O%(T)249 whose static X-ray conformationwas taken to be that found by fiber diffraction(Langridge et al. 1960; Amott and Hukins 1972). Allhydrogen atoms were added to the experimental heavy-atom coordinates, giving a total of 754 atoms for thedodecamer and 1530 atoms for (A)24(T)24 (two 5 ‘-ter-minal PO, groups were absent in both structures).

The potential energy of these molecules was calcu-lated as a function of the positions of the atoms using thesame type of empirical force field used in previousenergy calculations on nucleic acids (Levitt 1978) andproteins (Levitt 198 la). This potential has the usualterms that al low for bond stretching, bond angle bend-ing, hindered bond twisting, and van der Waals interac-tions (see Tables 1 and 2). Hydrogen bonds are modeledby a specific short-range directional interaction thatreproduces their expected strength (well depth is 5.0kcal/mole). Electrostatic interactions are neglected in al lbut one of the calculations presented here due to uncer-tainties about dielectric effects and polarizabil i ty . Thethousands of water molecules that surround these DNAfragments in solution are not included. This type ofpotential is only a first approximation to reality, but ithas been able to reveal a great deal about the atomic mo-t ion in proteins (McCammon et a l . 1977; Levi t t 198 lb) .

The forces on the atoms calculated from this potentialenergy function are used to solve i teratively the equa-tions of motion for a time step of 0.002 psec. Beforestart ing the dynamic trajectory, the X-ray conformationis relaxed by between 1000 and 2000 cycles of conjugategradient energy minimization. The temperature of thesystem is increased to 300°K (room temperature) by

251

The bond energy 1s calculated using K,(h - h,)‘. where K=h m kcal mole-’ A-’ and ho in A are given above. Theangle energy IS calculated usmg K,(8 -Oo)‘. where Ke m kcal mole-’ fadran-’ and (I0 m degrees are given above.

‘Atoms are indicated by a one-letter code: H is hydrogen; 0 is oxygen; Q is oxygen bound only to P; N is nitrogen;M is nitrogen that accepts hydrogen bonds; C is tetrahedral carbon; A, B, and G are trigonal carbons in different posi-ttons in the bases; and P is phosphorus. The symbolic chemical formulas of the four nucleotides are as follows: T is.OPQ(Q).OCH2CH(O)CH(CH2CH*NAONDAOA(CH3)*AH), C is .OPQ(Q).OCH2CH(O)CH(CH*NAOMA-(ND2)AH*AH). A is .OPQ(Q).OCH2CH(O)CH(CH2CH*NBHM/BA(ND2)MAHM/*G), and G is .OPQ(Q).OCH2-CH(O)CH(CHXH*NBHM/BBONDA(ND2)M/*G).

making small , random veloci ty changes during the f irs t200 t ime steps of the trajectories that are continued forabout 90 psec (45,000 time steps). In spite of the rela-tively large time step (0.002 psec) and the inclusion ofall the rapidly moving hYd rogen a toms, the sum ofpotential and kinet ic energy is well conserved. The ac-curacy of solution of the equations of motion is provedby the persistence of low-frequency motions that in-volve more than 10,000 time steps per cycle (seebelow). Any accumulated error would disrupt thesevery s low cooperat ive motions.

RESULTS AND DISCUSSION

Fluctuations of Hydrogen Bonds and Torsion Angles

The 32 hydrogen bonds between the 12 bp of thedodecamer are unexpectedly stable. The 0.H distanceis less than 2.2 A for al l hydrogen bonds but one (C, , ,H4*G1.,,06 is “broken” with an 0.H distance of morethan 2.5 A from 59.8 psec to 60.6 psec). The root-mean-square (rms) fluctuation in O*H distance is be-tween 0.07 A and 0.1 A, and the mean values of O*Hdistances are between 1.77 A and 1.84 A (see Fig. 1).When the same hydrogen bond funct ion was used in a

ble helix rather than to the nature of the potential func-t ion used .

The correlation coefficients of different hydrogenbond lengths were significant ( > 0.2 in absolute value)only for the adjacent hydrogen bonds in a given basepair. In C-G pairs, the first hydrogen bond G,Hl c,o2i s not correlated W ith the third hydrogen bondG,06C,H4. The values of the correlation coefficientswere between +0.2 and +0.4, with hydrogen bonds inG-C base pairs generally showing higher correlationsthan those in A*T base pairs .

Spectral analysis shows that the fluctuations IDfhydrogen bond length occur with a wide range of fs,e-quencies between 0.6 cm-’ and 333 cm-‘. There is noone characteristic frequency for this fluctuation. The 48hydrogen bonds in (A)24(T)24 behave in the same way asthose in the dodecamer, showing similar s tabil i t ies , cor-relat ions, and fluctuation frequencies.

The single-bond torsion angles fluctuate with rmsvalues of 6” to 12 O. There are no major changes of tor-sion angles other than those associated with equil ibra-tion and annealing of the dodecamer (see below). Thetorsion angles show a clear pattern of correlat ion that isessential ly the same for al l nucleotides in the dodecameror in w24m24. The significant correlations are all be-

molecular dynamics s imulat ion on a small protein (pan- tween the torsion angles of a single nucleotide ascreatic trypsin inhibitor), the interpeptide hydrogen follows: (6, x) 0.6, (6, <) -0.5, (E, x) -0.4, (a, y)bonds were much less stable than those found for DNA -0.4, (7, X) -0.3, (L X) -0.4, (P, 6) 0.3, and (P, xl(Levi t t 198 1 b) . This indicates that the high s tabi l i ty of 0.3, where the torsion angles CY to x are as defined inthese hydrogen bonds is due to the structure of the dou- Table 3 and the correlation coefficient is given after the

SIMULATION OF DNA DYNAMICS 253

Table 2. Nonbonded and Torsional Energy Parameters 27)

A t o m R”dl

H 2.6525 0.001 2.8525 0.011 1.10QQ 3.2005 0.18479 3.1005 0.18479 1 . 6 0N,M 3.2171 0.41315 3.8171 0.41315 1 . 6 5C,P 3.9150 0.07382 4.3150 0.07380 1 . 8 5

A,B,G 3.9202 0.03763 4.2202 0.03763 1 . 7 5

Atom pair

Hydrogen boruhb

H*O 1.7 5.0 1 . 7 5.0H*M 1.7 5.0 1.7 5.0

Cc)

Pair Kb

Torsion angles’

-c-o- 1 . 4 3.0 0.0-N-C- 0.3 6.0 0.0-c-c- 1 . 4 3.0 0.0-A-O- 20.0 2.0 1 8 0 . 0-A-N- 20.0 2.0 180.0-C-A- 0 . 1 3.0 0.0

-A-Asd 20.0 2.0 180.0-P-O- 2.0 3.0 0.0

Pair K.

(d) Out-of-plane angles

n d

-N- 2.0 2.0 180.0-C- 2.0 3.0 180.0-A- 2.0 2.0 180.0-B- 2.0 2.0 180.0-P- 2.0 3.0 180.0-G- 2.0 2.0 180.0

“The van der Waals energy is calculated using A/r’” - B/P. whereA = c (ro)12. B = -2~ (ro)6, E = (c’cJ)“~, and r. = (~-‘~e)“~ for an in-teraction between atoms of type i and j. The first set of c and r0 valuesare used for dynamics and have smaller r. values than the second set.which is used for energy minimization.

bathe hydrogen bond ne ergy is calculated using c{(rolr)‘l - 2(r,lr)“)exp (-&-&c?). where t and r, are given above and u = 20”. ~O-NHmeasures the linearity of the hydrogen bond. van der Waals interactionsand hydrogen bonds are calculated between all pairs of atoms separatedalong the chain by more than four bonds and separated m space by adistance of up to Rtd + & + 2 A.

‘The torsion angle and out-of-plane energies are both calculated us-ing K, [ 1 + cos (nd + d)]. where the parameters are as given above.The out-of-plane angles are improper torsion angles defined as i-j-k-l forthe group

‘\dj/

dThe torsion energy parameters used for A-A bond twisting are alsoused for B-N, B-A. B-B, G-N, G-B. M-A. M-B. and M-G bondtwisting.

pair of angles. The highest correlations involve the 6angle, which is a measure of ribose pucker, and the side-chain x angle, which is correlated to all backbone anglesexcept QI. The time variation of five typical torsionangles is shown in Figure 2. The frequencies of f luctua-t ion of torsion angles are more characterist ic than those

3 .I)2‘1

Figure 1. Time variation of 5 of the 32 hydrogen bonds connect-ing the base pairs in the dodecamer (CGCGAATTCGCG)2. Valuesare plotted every 0.2 psec for the period 24-72 psec of the 90-psectrajectory. Hydrogen bonds involve the following pairs of atoms:b, between G(W*WH4); B1,, btween GdHd*C,,(N,); Blz,between G4(H2)*Cz,(02); B,3, between A,(H,)*T,,(O,); B,4r be-tween A,(N,)*T,,(H,). The mean hydrogen bond lengths are be-tween 1.77 A and 1.84 A, and the rms fluctuations are between0.07 A and 0.1 A. The correlation coefficients are about 0.3 forthe following pairs of hydrogen bonds: (B,,,B,,), (B,,,B,,), and(BdW

of hydrogen bond lengths, with dominant frequencies of1, 2, 7, 10, and 15 cm?

Overall Bending and Twisting

The radius of gyration of the dodecamer was found tooscil late slowly (24 psec) and with large ampli tude (4%of the mean value of 13.3 A>. This motion can be at-tr ibuted to a smooth bending in the plane that containsthe dyad axis of the whole dodecamer (the dyad axis be-tween the central base pairs A6=T19 and T,*AJ. Themotion is not symmetr ical about the s t raight conforma-tion but tends to close the major groove relative tostraight DNA (see Fig. 3).

Quantitative characterization of this conformationalchange was obtained by first generating a standard B-form double helix with the dodecamer sequence (Amottand Hukins 1972) and then bending, twisting, andstretching this standard helix by different amounts togive a family of smoothly deformed helices (Levitt1978). Each of these helices was then fitted to a par-ticular dodecamer conformation as well as possible us-ing accepted methods (Kabsch 1976). The degree ofdeformation of the helix that gave the best f i t was takenby measuring the bending, twisting, and stretching ofthe dodecamer. The best-f i t rms deviation was typicallyabout 2 A ; for A-form DNA the best-fit rms deviation

254 LEVITT

Table 3. Torsion Angles and Other Calculated DNA Properties

Torsion angle (in degrees)” Baseb Helix’

ff 0 -r 6 c i x 7: 7, tH 0” C I lRppd

DodecamerX-ray 298 171 5 5 123 189 2 5 4 122 10 15 3 . 3 35.7 1 4 . 40 psec 2 9 4 166 60 112 186 2 6 3 111 10 15 3.5 3 2 . 1 1 3 . 948 psec 292 173 5 8 96 1 8 0 2 8 3 1 0 0 16 16 3.0 28.8 1 3 . 790 psec 295 173 55 1 0 0 1 8 0 2 8 1 105 15 12 2.8 28.7 1 4 . 0straight’ 296 173 56 95 1 7 6 282 98 10 14 3.6 30.5 -ben t ’ 2 9 2 172 60 9 6 1 8 0 284 104 10 15 2.7 29.3 -

NahumX-ray 3 1 3 171 3 6 1 5 6 155 2 6 5 142 6 4 3.4 36.0 46.70 psec 294 172 64 107 173 274 1 1 0 16 2 3 3.4 35.6 46.324 psec 296 1 7 4 57 90 1 7 9 286 9 3 2 3 12 3.4 27.7 45.648 psec 294 171 61 9 8 1 7 7 282 99 19 17 3.4 31.3 45.17 2 psec 294 173 5 7 9 4 1 8 0 284 102 21 15 3.1 28.8 46.896 psec 294 172 64 99 1 7 5 2 8 1 1 0 0 26 17 3.6 32.5 44.9

(NdLkinkf78 (8-psec 1 1 ) 298 171 5 9 94 171 2 8 1 9 3 17 12 3.4 31.0 -

(34-4 1) 294 172 6 5 99 178 281 91 16 12 3.4 31.0 -84 (8-psec 1 1 ) 3 0 3 1 7 6 6 3 1 0 6 173 2 7 7 94 3 0 20 4.0 36.5 -

(38-41) 2 9 5 172 7 5 1 0 6 1 7 6 2 7 5 9 7 31 20 4.0 36.5 -

The first and last base patrs are not Included when calcufattng mean valueb.“The torsion angles are defined to be zero in the (‘IS conformation and mcrease with clockwtse rotatton of the further bond.

The atoms defining each angle are Q (03’-P-05’~0’). /3 (P-05’-CS’-C4’). 7 (05’-C5’-C4’-C3’). 6 (C5’-C4’-C3’-03’).c (C3’-C3’-03’-P), < (C3’-03’-P-05’). and x: (C 2 ’ -C I ’ -N 1 -C2 for pyrimidmes. and C2 ’ -C 1 ’ -N9-C4 for purines). Notethat xo as defined by Drew et al. (198 1) is given approximately by XD = y - 240”. and XL as defined by Levitt ( 1978) isgiven approximately by XL = x - 60”.

hThe base orientation is defined by 7:. the angle between the normal to the base and the overall helix axis. and T,, the pro-peller twist angle between the base normals of the two bases in a base pair.

‘The local helix defined by the main chain atoms of 4 bp has a rise/bp of tH( A ) and a rotation/bp of 6~ (degrees).‘The electrostatic energy between the negatively charged phosphate groups depends on C l/Rpp (in A-‘). where Rpp is the

separation of the phosphate atoms. If each phosphate group carries a net charge of q electrons and the solvent dielectric constant. is D. the electrostatic energy is 332q’[C IIRppJID kcal/mole.

‘For straight. average over 8 nucleotides in the central 4 bp for 4-psec periods starting at 23. 48, and 74 psec. For bent.average over the same nucleotides for 4-psec periods starting at IO. 36. 6 I. and 86 psec.

‘Only the 4 nucleotides indicated are included in the calculation of mean values

was over 4 A, showing that the structure remains Bform.

The most significant deformation was bending (seeFig. 4). The bend angle per angstrom (6,) was found tovary with time, t, as

e,(t) = 1.75” - 1.18” cos (2d26)

The 26-psec period of the bending motion is the sameas that observed for the radius of gyration and cor-responds to a frequency of 33.4/26 = 1.28 cm?Although the t ime-averaged value of eB is not zero, thebase pairs do appear in Figure 3 to tilt symmetricallyabout the long axis. The radius of curvature is the in-verse of the bend angle (measured in radians) perangstrom. For the most-bent conformations (e. g . , at t =13 psec), Rc = 1/(2.93~/180) = 20 A, whereas forthe least-bent conformations (e.g., at t = 26 psec),

RC = 1/(0.57~/180) = 100 A.The bending motion persists for the entire t rajectory.

To eliminate all of the influence of the starting condi-tions, the conformation 24 psec along the trajectory wascooled by energy minimizat ion and then used to s tar t aparallel trajectory. The bending motion occurs asdescribed above (Fig. 4). Cooling after 36 psec gave asimilar result. Clearly, the Deriodic bendine is not sen-

sitive to the initial conditions and is the dominant slow-motion mode of the dodecamer.

There is a lso a s ignif icant twist ing motion that has anampli tude of about 0.32 “/ A but does not show the s im-ple periodic behavior observed for the bend angle (seeFig. 4). Whereas bending occurs with a single frequen-cy, twisting involves at least three different frequenciesthat are all of equal or higher frequency than that ofbending. Twisting and bending are correlated in that thetwist is about -2.6”/A when the dodecamer is bent and- 1.7’1 w when it is straight. This means that the bentstructure is wound less t ightly and has more base pairsper turn than the straight structure.

Normal Modes of Bending

The global motions of the 24-bp double helixw24m4 are similar to those of the dodecamer butshow greater diversi ty (Fig. 5) . The motion seen in the yprojection resembles that of two dodecamers stacked ontop of one another and has the same period ( - 24 psec).The bending is still into the major groove. It is as ifthere are pivots at the three points where the twophosphate chains intersect in the ,v projection. The mo-tion in the x projection follows the same principle.There are pivots at the two intersection points and themot ion i s asymmetr ic w i t h a period of about 48 psec.

SIMULATION OF DNA DYNAMICS 255

- ~- -.-! ~ - -i- -i-L--‘-- i..-36 48 60 ‘72

?‘ime (ps )Figure 2. Time variation of 5 of the 136 single-bond torsionangles in the dodecamer for the period 24-72 psec of the trajec-tory. All the torsion angles belong to nucleotide A6 and refer torotation about the following bonds: cy, about P-05 ‘; @, aboutOS-cs; y, about C5’-C4’; 6, about C4’-C3 ’ ; and x, aboutC 1’ -N9. The correlation of 6 and x and the anticorrelation of c~ andy are seen clearly. Mean values of these torsion angles are given inTable 3.

This complicated pattern of bending can be under-stood by considering a completely uniform elastic rod(Landau and Lifshitz 1959). At any time, t, the axis ofthe rod lying initially along the z axis will have x and 1coordinates for each normal mode of motion given by

x(Gz,k)= ~0s ad{ cos Kz + ( - l)‘k +‘)I’ cash &z/cosh m}

if k = 1,3,5= cos at{ sin Kkz + ( - l)(k’z sinh K,z/cosh dm}

if k = 2,4,6

where KA = (2k + 1)~/2L and or. = K&/m (Barkleyand Zimm 1979). The rod length is L, the bending forceconstant per unit length is C, and the mass per unitlength i s M. The overal l motion of the rod is a combina-t ion of al l modes, with the ampli tude of the mode beingproport ional to l/Kk. The mode of largest ampli tude wil lalways be the one with k = 1.

In the present s tudy of DNA, bending does not fol lowthis pat tern in tha t a s ingle mode dominates the x and ymot ions of both the dodecamer and (A)24(T)24. For thedodecamer the y-projection bending corresponds tok = 1 and has a period of 26 psec. For (A)24(T)24, thex projection motion corresponds to k = 2 and has aperiod of 48 psec, whereas the y-projection motion cor-responds to k = 3 and has a period of 24 psec. Thewavelength of these modes is given by X, = 2x/K,.For the dodecamer, X, = 4&/3 = 52 A (& = 12 x3.38 = 40 A ), whereas for (A)24(T)24, X, =4LJ5 =64 A CL24 = 24 x 3.3846 A.

= 81 A) and X3 = 4LJ7 =

For an isotropic elastic rod, the wavelength of the

Figure 3. Stereoscopic drawings of all 752 atoms in thedodecamer at three points of time along the second trajectory (from24 psec to 90 psec). The three molecular drawings are viewedalong the y axis at 37 psec (a ), 44 psec (b ), and 49 psec ( c) whenthe structure goes from most bent to almost straight (see Fig. 4).The symmetrical bending motion is seen to preserve the centraldyad and act to close the major groove. These and all othermolecular drawings were produced using PLUTO, a very versatileprogram written by Dr. S. Motherwell (The Chemical Laborato-ries, University of Cambridge, Cambridge, England).

dominant mode should increase with length, but here thesame wavelength of about 50 A for both DNA frag-ments is found. This occurs because the DNA doublehelix is not isotropic toward bending in a plane: It ismuch easier to bend the helix into the major groove than

256 LEVIYI’

0 1 1 i ’ - -- ---

1

- I -

-

Fiwre 4. Periodic bending and twisting of the dodecamerduring the entire 90-psec trajectory. (0) First trajectory(O-46 psec); ( 5) second trajectory (24-90 psec). Thebending and twisting measures of smooth deformationare calculated as described in the text. Note how the ben-ding motion is of a larger amplitude and more simplyperiodic than the twisting motion. During the 24-46-psecperiod, when both trajectories are propagated in parallel,the bending motions are very similar, whereas there aredifferences in the twisting motions.

into the minor groove or in any other direction. Thestanding wave due to bending will have the highest

not decrease with DNA length as expected for an

ampli tude when the weak points in the helix occur at theisotropic elast ic rod.

points of maximum curvature (pivot points in Fig. 5).The frequency of bending obtained from the molecu-

As these weak points are separated by one-half turn oflar dynamics simulation can be used to est imate values

helix, the wavelength of the dominant mode will befor the bending-force constant C. The frequency (in

close to the helix repeat for DNA of any length. As acm-‘) is given by vk = 108&/m, where M is the

consequence, the frequency of the dominant mode willmass per unit length (taken as 180 daltonsi A). For thek = 1 mode of the dodecamer, ZQ = 1.27 cm-!,

OPS 48~s 9opsFigure 5. Overall motion of the (A),,(T),, double helix for the entire 96-psec trajectory (48,000 time steps). ( x Projections of thebackbones (drawn as thick P-P virtual bonds) and the base pairs (drawn as thin Cl ‘-Cl ’ virtual bonds) at 6-psec intervals; (bortom ) cor-responding y projections. The motion seen in the x projection is an antisymmetric bending motion (k = 2) with a period of about 48 psec.The motion seen in the v projection is a symmetric bending motion (k = 3) with a period of about 24 psec that is like the motion seen in thedodecamer (Fig. 3). Although most of the overall motion appears to be smooth bending, after 80 psec the chain kinks between base pairsAg*TdO and AIO*TJ9.

SIMULATION OF DNA DYNAMICS 257

6 = h/X,,, givmg CA? For the k

= 102 kcai mole-’ radian-l= 2 mode of (A)24(T)24, vk = 0.69 cm-l,

K* = 57r/2&, giving C = 78 kcal mole-’ radian+ A-*.For the K = 3 mode Of (A)24(T)24, Yk = 1.38 Cm-‘,

K3 = 7a/2L,,, giving C = 81 kcal mole-’ radianm2A-‘. This suggests that the dodecamer may be a littlestiffer than (A)24(T)24.

Torsion Angles Are Insensitive to Overall Motion

After the ini t ial equil ibrat ion, the mean torsion anglesare essentially the same in both the dodecamer and(A)24(T)24 at any time along the trajectory (Table 3).The ini t ia l torsion-angle values in (A)24(T)24 correspondto high-energy, partially eclipsed conformations asfound by fiber diffraction (Arnott and Hukins 1972);they are relaxed by energy minimization and dynamicsto almost perfectly staggered values. In the X-ray con-formation of the dodecamer (Drew et al. 198 l), threenucleotides Gq, G 1o, and GZ2 have abnormal conforma-tions with mean values for 6, E, <, and x of 150°, 241 O,176 O, and 150 O, respectively. Energy minimizationdoes not relax these values, but during the f irst 20 psecof the trajectory, these irregular torsion angles aredynamically annealed to become like those of othernucleot ides .

For each of the seven torsion angles, the “static”variation for different nucleotides at a particular confor-mation is the same as the “dynamic” variation for dif-ferent times at a particular nucleotide. For example, inthe dodecamer at 90 psec, the rms static fluctuations are8’, 7’, 7”, loo, 6O, 8’, and 11” for cy, p, y, 6, E, <, andx, respectively. For nucleotide A6 the rms dynamic fluc-tuations (for 24 psec to 72 psec) are 7”, 6”, 7”, ll”, 7”,7”, and 11 o for (;Y& &r y6, 66, E6, 5-6, and x6, respective-ly. Any systematic variat ion in torsion angles due to se-quence or overall bending is smaller than the thermalfluctuat ion.

A more careful analysis did show that four of the se-ven single-bond torsion angles are significantly differentin the straight and bent dodecamer conformations (seeTable 3). On bending, CY changes by - 4”) y by 4”) E by2 O, and x by 6”. These changes are very small indeed,although they are calculated for the central 4 bp thatshould be most affected by the bending; the changeswere detected only by averaging over 240 values for thestraight structure and 320 values for the bent structure.Clearly, the DNA double helix can undergo large-scaleconformational changes that are not reflected by anysubstantial changes ( > loo) of torsion angles. This oc-curs because there are 14 flexible torsion angles perbase pair that can change in a concerted fashion.

Electrostatic Interactions

Stereoscopic drawings of four of the (A)24(T)24 con-formations are shown in Figure 6. The dramatic changesin conformation that occur spontaneously are seen veryclearly and demonstrate the high degree of conforma-tional flexibility available to the DNA double helix.

These different conformations must have energies thatare within a few RTof each other since they occur spon-taneously in this room-temperature s imulat ion. This in-sensi t ivi ty of the energy to the overal l conformation isnot surprising since the local conformation (as measuredby hydrogen bond lengths and torsion angles) changesvery slightly and the energy terms are all short-range(see Tables 1 and 2). Electrostatic interactions betweenthe negatively charged phosphate groups were not in-cluded and these long-range interactions might be ex-pected to be more sensi t ive to the overal l conformation.This was checked by summing l/Rpp over all pairs ofinterphosphorus distances, R,, (see Table 3). The elec-trostatic energy can now be approximated as332q*{ C l/Rpp} kcal/mole. The effective net charge onevery phosphate group, q, will be less than 0.5 electronsdue to t ightly bound ions (Manning 1978). The effect ivedielectric constant, D, will be close to that of water (80)(neglecting any Debye-Huckle-type shielding). Takingthe long-range electrostatic energy as 1.04C 1 lRppkcal/mole will give a useful upper-limit estimate. Forthe straight and bent form of the dodecamer, this givesenergies of 14.2 and 14.5 kcal/mole (at t = 48 psec andt = 90 psec, respectively, see Table 3), a difference ofonly 0.3 kcal/mole for 12 bp. For the straight and bentform of (A),,(T),,, this gives energies of 46.6 and 48.5k&/mole (at t = 96 psec and 72 psec, see Table 3), adifference of only 1.9 kcal/mole for 24 bp. Althoughthese differences are small, they are significant in thatthe bent structures always have higher electrostaticenergies; long-range electrostatic interactions will in-hibit global bending but wil l have a much smaller effecton the bending motions seen here.

In the calculations presented above, the average riseper residue varies from 2.8 A to 3.4 A, and the turnangle per base pair varies from 28.7” to 32.5 O. Thismeans that the number of base pairs per turn varies from11 to 12.5 over 20 psec, values that are significantly dif-ferent from the values of 9.6 to 10.6 bp per turn foundfor nat ive B-form DNA. At f irs t i t was thought that thecalculated conformations were closer to A-form DNA,but Figure 6 shows that the structures remain B form. I twas then thought that the disagreement was due to theomission of electrostat ic interact ions, especial ly thosebetween the partially charged atoms on the bases. Themolecular dynamics on (A)24(T)24 was thereforerepeated with addit ional electrostat ic interact ions usingatomic partial charges derived from an ab initio quan-tum mechanical calculation (Scordamaglia et al. 1977;Matsuoka et al . 1978). The dielectric constant was set to1 so that the electrostatic would be strong enough to giveinterbase hydrogen bonds and the directional hydrogenbond interact ion was omit ted. This gives r ise to the typeof electrostatic interactions thought to stabilize amidecrystal packing (Hagler et al. 1974). Although minimi-zation with this potential gave a reasonable double-helical structure, the structure was unstable and rapidlydenatured during the dynamics (see Fig. 7). The basepairs remain intact but the helix unwinds and then re-forms into a very distorted structure that remains to be

258

Figure 6. Stereoscopic drawings of some of the (A)24(T)Zb conformations studied here. The three views along the x axis (down the central

dyad) are the starting conformation as determined by fiber diffraction studies (Amott and Hukins 1972) (a ), after 24 psec of dynamics (6),and after 89.6 psec of dynamics ( c). The one view down the y axis (perpendicular to the central dyad) (d) is after 6 psec of dynamics. (b,d)Two dominant modes of motion close to their extremes; (c) kink formed between base pairs A9*Td0 and A,0*T39.

analyzed in detail . This dramatic conformational changeis not thought to represent any real process but providesa vivid example of the drastic effect of using inap-propriate energy parameters.

The defect of the calculat ion is probably the use of adielectric constant of 1. Analysis of the base-stackingenergy (Omstein et al . 1978) shows that ab ini t io part ialcharges give rise to a strong repulsion between base

SIMULATION OF DNA DYNAMICS 259

Figure 7. Overall motion of the (A),,(T),, doublehelix for the 56-psec trajectory that included elec-trostatic interactions. Conformations are drawn inthe same way as in Fig. 5. A dielectric constant of 1is used with atomic partial charges taken from an abinitio study (Scordamaglia et al. 1977; Matsuoka etal. 1978). Although the structure produced byenergy minimization is a normal double helix, thestructure is unstable and unwinds without breakingthe interbase hydrogen bonds, indicating a seriousdefect in the calculation.

pairs. With a dielectric constant of 1, the repulsion istoo strong and disrupts the double helix completely.These preliminary calculations suggest that the suc-cessful inclusion of short-range electrostat ic interactionswill depend on choosing a value for the dielectric con-stant that is small enough to increase the base-pair turnangle from 30” to 36” without disrupting the doublehelix. Inclusion of long-range electrostat ic interactionswill be more problematic as counterions provide ashielding atmosphere around the polyelectrolytic DNAmolecule. Even when DNA is treated as an isotropicrod, these interactions present considerable problems(Manning 1978; Le Bret and Zimm 1981; Schurr andAllison 1981).

Kinking of the Double Helix

Although much of the motion of the DNA doublehelix appears to be smooth bending, kinking occursonce during the 96-psec (A),,(T),, simulation. Inspec-tion of the x projection in Figure 5 for the period of 60psec to 96 psec shows that the kink is caused as the helixbends out of the major groove past the straight confor-mation. A detailed stereoscopic drawing of the kinkregion (Fig. 8) shows that the kink is formed within 6psec and persists for at least 16 psec (to the end of thetrajectory). Although the angle between the base pairs atthe kink is about 90”) none of the torsion angles in thekink region show a change greater than 30” (see Table3). This demonstrates once again how large changes inthe global structure do not require or result in largechanges of the torsion angles. When the double helixkinks, the minimum distance between PO, groups in thetwo strands drops from 11 A to 6 A, but the elec-trostatic energy as measured by C l/Rpp actually de-creases (compare the entries at 72 psec and 96 psec inTable 3). The kink is formed here as a natural conse-quence of room-temperature thermal motion; no path-way for this change of conformation had to be chosen.

The kink formed here does not correspond in detai l toeither of the kinks proposed by model building (Crickand Klug 1975; Sobell et al. 1977). The chain kinks by

about 90” into the minor groove as proposed by Crickand Klug, but the change in torsion angles found here ismuch smaller and not restricted to a single torsion angle.Kinking into the major groove proposed by Sobell e t a l .is not found here; bending into the major groove seemssmooth and involves torsion-angle changes of less than5”.

Influence of Solvent

The present calculations relate to DNA in vacuum,and i t is important to t ry to est imate the possible effectsof solvent. Water is expected to have three major roles:(1) Specific interactions can affect the DNA geometryby bridging between hydrogen-bonding groups that donot interact favorably in vacuum (this bridging has beenobserved in the X-ray structure by Drew and Dickerson[ 198 11). (2) The water molecules and counter-ions willhave a very large effect on the long-range electrostaticenergy between the negatively charged phosphategroups. (3) The water molecules will provide a viscousmedium that will damp out vibrations of the doublehelix.

Specific interactions are not expected to play a majorrole in determining the conformational dynamics ofDNA; if DNA itself is not rigid, how can water mole-cules bound to its surface make it so? Electrostatic ef-fects of water are more important, although the elec-trostat ic contr ibut ions to DNA st i ffness are thought tobe qui te small ( < 20%) for ionic strengths above 0.01 MNaCl (Harrington 1978; Borochov et al. 198 1; Schurrand Allison 198 1). Viscosity will have a major effect onthe in vacua results . The dominant bending mode can beapproximated by considering a chain that consists oflarge spherical particles, each corresponding to one-halfturn of helix. Each sphere will have a mass, M, of about3000 daltons and a radius, a, of 10 A, giving rise to aStokes’ frictional coefficient of 6naq. The momentumof the moving sphere will decay with a time constant7 -l = 6raqlM or less than 1 psec (for water, the viscos-ity q = 0.0 1 poise). The bending motion observed hereto have a 26-psec period in vacuum will therefore be

260 LEVITT

x view

a

b

y view

Figure 8. Detailed stereoscopic molecular drawings of the kinking that occurs at about 80 psec. (I!,@) x-Axis projection of the fragment(ABA9A,0A,,)*(T4,T40T39T38); (righr) corresponding y-projections. (a) At 78 psec, before the kinking occurs; (6) at 81 psec, during thekink formation; (c) at 84 psec, after the double helix has kinked. The mean values of torsion angles in the vicinity of the kink (nucleotides8- 11 and 38-41) given in Table 3 show that there is no major torsion-angle change associated with kinking.

damped. Instead, there wil l be a diffusive motion with arelaxation t ime that has been calculated by Barkley andZimm (1979) to be 110 psec (assuming C = 100 andA, = 50 A). The nature and amplitude of the motionwill not change. In part icular , conformations l ike thoseshown for (A)24(T)2, in Figures 5 and 6 may be impor-tant in water as they involve small changes of the long-range electrostatic energy and small overall movementsof the double hel ix .

Solvent could, in principle, be included in the calcula-

t ion by s imply surrounding the DNA double hel ix wi th afew thousand water molecules and using periodic boun-dary condit ions to give an extended system. Althoughabout 40 water molecules have been treated in a MonteCarlo calculation in which the DNA was rigid (Cor-ongiu and Clementi 198 1), addition of water to the DNAmolecular dynamics calculation would increase thecomputer requirements by between 20-fold andUK&fold. This is impract ical a t present , but i t should bepossible within the next 5 years , during which t ime the

SIMULATION OF DNA DYNAMICS 261

present in vacua resuits will heip us to Interpret thesefuture calculat ions.

and iow concentrat ions of a molecule that binds across agroove could have a major effect on the overall confor-mation.

Agreement with Experiment

The torsion-angle motions are found to be highly cor-related. The strongest correlation found here (0.6 be-tween angles 6 and x) has been observed in the X-rayconformation of the dodecamer (Drew et al. 1981). Thenext strongest correlation ( - 0.5 between 6 and {) wasnot detected ini t ial ly by Drew et al . , but i t was seen in amore extensive analysis by Fratini et al . ( 1982).

The force constants calculated above from the fre-quencies of smooth bending can be compared with thestiffness of DNA in solution as monitored by the per-sistence length, P. For DNA that always bends alongthe dyad axis, as found here, P = 2C/RT (Schellman1974) (where RT is the thermal energy, which is 0.6kcal/mole at T = 300°K). For the dodecamer, P iscalculated to be 340 OA, whereas for (A)24(T)24, P iscalculated to be 260 A and 270 A (for the x and y mo-tions, respectively). The experimental value in high salthas been estimated at between 200 A and 300 A (Harr-ington 1978; Borochov et al. 1981), in reasonableagreement with the calculated values.

The propeller twist of the bases is close to 15”) butdue to the overall motion in (A)24(T)24, the mean anglebetween the base normal and long axis (z) is greater than20°, in agreement with experimental values (Hogan etal. 1978). It is of interest that the apparent angle be-tween the base pair and the helix axis, as measured byelectric dichroism, decreases as the DNA is stretched byan electric field (Diekmann et al. 1982; Lee andChamey 1982‘) or aggregated into bundles (Mandelkemet al . 198 1). I t was proposed that DNA might be super-coiled under normal conditions (Lee and Charney1982). Bending like that observed here for (A)24(T)24could also explain these f indings since t h e ampli tude ofthe bending waves would be decreased by both stretch-ing and aggregation.

Nuclear magnetic resonance studies (Early andKearns 1979; Bolton and James 1980; Hogan andJardetsky 1980) have shown that there are nanosecondmotions of the base, ribose, and phosphate groups ofamplitude between 20” and 30”. For the central 14 bp of(A)24(T)24, the calculated rms fluctuation of the anglebetween the C8-H8 vector and the helix axis averages17” for a period of 20-96 psec. The correspondingangle between the two protons on C2’ has an averagerms f luctuat ion of 18 O. Thus, the f luctuations calculatedin vacuum on a t ime scale of 100 psec are smaller than,but comparable to , those observed in solut ion on a t imescale of 1 nsec.

The surprising sensitivity of DNA conformation todrug binding (Sobell et al. 1977: Hogan et al. 1979;Dattagupta and Crothers 198 1) can be understood on thebasis of the cooperative motion between the very dif-ferent conformations seen in Figure 5. The energies ofthese conformations are the same to within a few timesthe room-temperature thermal energy (0.6 kcal/mole),

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