Proc. Natl. Acad. Sci. USAVol. 85, pp. 1763-1767, March 1988Biochemistry

Calibration of DNA curvature and a unified description ofsequence-directed bending

(DNA bending/adenine tract/polyacrylamide gel electrophoresis/junction model/helix-axis trajectory)

HYEON-SOOK KOO AND DONALD M. CROTHERSDepartment of Chemistry, Yale University, New Haven, CT 06511

Contributed by Donald M. Crothers, November 16, 1987

ABSTRACT Chemically synthesized duplex oligodeoxy-nucleotides having different average numbers of adenine tracts(A6) per helix turn were ligated into multimers and analyzedby electrophoresis on polyacrylamide gels. The magnitude ofthe anomaly in gel mobility is found to be a quadratic functionof the curvature of the DNA molecule. Parameters thatdescribe intrinsic DNA bending, expressed as the tilt and rollcomponents of the helix-axis deflection at the junctions be-tween the adenine tract and adjacent B-DNA, were adjusted tofit the measured relative curvature of regularly repeated DNAbending sequences known from other studies and synthesizedfor this study. The model developed here retains the predom-inance of bending in the direction of tilt at the junctions butincorporates an appreciable roll component at the 5' end of anadenine tract, opening the minor groove there. This feature isconsistent with chemical "footprinting" experiments on mol-ecules containing adenine tracts. The overall direction ofbending is effectively toward the minor groove, viewed fromthe center of an A. or A6 tract. A possible underlyingstructure, which can also be described by a wedge bendingmodel, is that derived from fiber diffraction studies ofpoly(dA)-poly(dT). However, alternative models for the ade-nine tract, such as propeller twisted DNA, cannot be elimi-nated, although they do not lead to the correct direction ofbending. The results permit calculation of the helix-axis tra-jectory of natural DNA molecules containing adenine-tractbends.

Bending in DNA molecules is most commonly determinedfrom the anomalies in their electrophoretic mobilities onpolyacrylamide gels (1-10), although other techniques, in-cluding electron microscopy (11, 12), rotational dynamics(13, 14), and DNA cyclization efficiency (ref. 15 and H.-S.K., J. R. Rice, and D.M.C., unpublished data), have alsobeen employed. The anomaly in gel mobility is an increasingfunction of the extent of DNA bending, as expected fromtheory showing that gel mobility is proportional to themean-square end-to-end distance (16, 17). Here we seek toestablish an accurate empirical relationship between theanomaly in gel mobility and the curvature of DNA mole-cules. Oligodeoxynucleotides with various numbers of A6(or T6) tracts per helix turn were synthesized, and thehybridized oligodeoxynucleotides were ligated into multi-mers. By comparing the gel mobility for multimers of variouscurvatures, an empirical relationship was obtained betweengel mobility anomaly and DNA curvature.With a quantitative calibration of relative DNA curvature

in hand, it is possible to search for a simple model forA-tract-induced bending that accounts for the observed gelmobility of the known bending sequences, along with somevariants whose synthesis is described here. We present our

results in terms of the junction model (8), in which bendingis characterized by roll and tilt angles between the overallhelix axis of an A-T tract and the adjacent B-DNA helix axis.We deduced previously (8) that the bend angle is larger at the3' junction ofA tracts than at the 5' junction, and bending isprimarily toward the tilt direction rather than roll at thejunctions. However, from the gel mobility data on some bentsequences (9), whose properties could not be predicted byusing the earlier model incorporating tilt exclusively, wesurmised the presence of a roll component of bending at oneor more of the junctions. Bend angles at the junctions weredetermined by an iterative process, and the optimal bendangles were taken as those that minimized the deviationbetween the sets of calculated and observed mobility values.Adequate fit to experiment was achieved by incorporating aroll component in the bend at only the 5' A-tract junction.Quantitative values for the bend angles are based on anestimate of 220 of bend per A tract, consistent with the upperend of the range of values determined from the kinetics ofcyclization (H.-S.K., J. R. Rice, and D.M.C., unpublisheddata).

MATERIALS AND METHODSPreparation of the Multimers of Synthetic Duplexed Oligo-

deoxynucleotides and Measurement of Their Gel Mobilities.The oligonucleotides in Table 1 were made on a DNA syn-thesizer (Applied Biosystems, Foster City, CA) and purifiedas described in ref. 8. The oligonucleotides were labeled with[y-32P]ATP (Amersham) and T4 polynucleotide kinase andhybridized to their complementary strands. Duplexed oligo-nucleotides were self-ligated to produce multimers, whichwere electrophoresed on 8% polyacrylamide gels at roomtemperature and then located by autoradiography as de-scribed previously (8). The gel mobilities of the multimerswere compared with those of ligated products of 10-bp-longBamHI linker (New England Biolabs) used as size markers.The RL value is the ratio of apparent length, determinedfrom the size markers, to real length; (RL - 1) measures themagnitude of the anomaly in gel mobility.

Calculation of the Helix-Axis Trajectory of DNA MoleculesContaining Bends. A PC-based computer program analogousto that described by Levene and Crothers (18) was used tocalculate the helix-axis trajectory of DNA molecules. Eachincremental base pair is described in terms of helix screwand tilt and roll angles. No thermal variances were allowed,so the helix axis follows the minimal energy path. Theprogram was used to estimate relative curvature for multi-mers of sequences in our previous paper (8), and in Hager-man's papers (7, 9), using bend angles in terms of the junctionbending model. Helix screws of the sequences were assumedto be 10.0 bp per turn irrespective ofthe sequence, to producephase match between sequence repeat (10 bp in each case)and helix screw for convenience in calculating relative curva-tures. By inserting bending angles and positions for a se-quence, the trajectory of the corresponding DNA was calcu-

1763

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Table 1. Sequences of synthetic oligodeoxynucleotidesName Sequence (5'-.3') Length, bp

A6-1/1 GGGCAAAAAACGGCAAAAAAC 21A6-3/4 GGCTGGGCAAAAAACGGGCAAAAAACGGCAAAAAACGGCTCC 42A6-2/3 GGCTGGGCAAAAAACGGCAAAAAACGGCTCC 31A6-1/2 GGCTGGGCAAAAAACGGCTCC 21A6-1/3 GGCAGGGTCGGGCAAAAAACGGCTGGATCCC 31A6-1/4 GGCAGGGCGGTCGACGGGCAAAAAACGGCGTCGGGCGGATCC 42A5N5 GGCAAAAACG 10A6N4 GGCAAAAAAC 10A8N2 CCAAAAAAAA 10A5-T5 AAACCGGGTITII GGGCCAA 20A6-T6 AAAACGGGITITIGGGCAA 20A8-T8 AAAAACGTTI nTITGCAAA 20IHA GGGCAAAAACGCCAAAATIl-GCCGCGGGCC 31OHA GGGCAAAAACGGGCGGCCAAAATlTTGCCGC 31A10T1O AAAAAAA1TITITIT1TAAA 20A20 AAAAAAAAAAAAAAAAAAAA 20Only one strand is shown here, and the deoxy prefix d is omitted, as it is throughout this paper. All

oligodeoxynucleotide duplexes were designed to have 2-base-pair (bp) protruding 5' sticky ends so thatthey can be self-ligated to produce multimers with unique polarity. The sequence names A6-m/nindicate the average number of A6 tracts per helix turn, m/n.

lated. The distance from one end of the molecule to eachsuccessive base pair was followed, until it reached the firstminimum; the DNA segment of this size was approximated asa circle, whose circumference is inversely proportional to thecurvature of the sequence. The sequence A6N4 was used as astandard, and the relative curvature was obtained by calcu-lating the ratio of the circle size for A6N4 to that of a givensequence.

RESULTSRelationship Between the Curvature ofDNA Molecules and

the Anomalies in Their Gel Mobilities. Each calibrationsequence in Table 1 contains at least one A6 tract, and insequences with more than one tract, they are spaced atintervals of 10-11 bp. All the duplexed oligonucleotidesshould have overall helix screws ranging from 10.3 to 10.5 bpper turn, calculated from the nucleotide composition of thesequences and the fact that helix screws of B-DNA andpoly(dA)-poly(dT) are 10.5 and 10.1 bp per turn in solution,respectively (19-21). The lengths of the calibration se-quences (21, 31, and 42 bp) are close to integral multiples oftheir helix repeat, so their multimers contain A tracts thatare separated by integral multiples of one helical turn, andthe bends due to the A tracts add constructively. The averagenumber ofA tracts per helix turn ranges from 1 (in A6-1/1) to1/4 (in A6-1/4); relative curvature is defined as the averagenumber of A6 tracts per helix turn in the multimer molecules.RL values for the multimers of the sequences in Table 1

are plotted in Fig. 1, showing a continuous rise with frag-ment length. The value of RL is 1 for normal gel mobility,and (RL -1) is a measure of the net anomaly in gel mobility.When multimers of the various sequences are compared, it isevident that (RL - 1) is not a linear function of the curvatureof the DNA molecules. For example, the (RL - 1) value of147-bp multimers is 0.41 for the sequence A6-1/2 and 1.56 forA6-1/1. Hence, as the relative curvature was increased2-fold, the (RL - 1) value increased nearly 4 times.

In Fig. 2, RL values are plotted against the square of therelative curvature for the different multimer lengths. In eachcase, RL varies in a nearly linear manner with the curvaturesquared. Therefore we set RL = A(relative curvature)2 + B.The constants A and B calculated by using the least-squaresmethod are given in the legend to Fig. 2. Because thecalculated values ofB are close to 1.0 for all the multimers inFig. 1, B was approximated as 1.0. The value of the constant

A depends on the length (L) of the multimer, and an equationwas drawn from this relationship by using the least-squaresmethod, yielding A = 9.6 x 10-5L2 - 0.47. Combiningthese relationshjps, we obtain the final equation describingthe relationship between the anomaly in gel mobility and thelength and relative curvature of a DNA molecule (8%polyacrylamidie gel, 29:1 acrylamide/N,N'-methylenebis-acrylamide):

RL - 1 =(9.6 x 10-5L2 - 0.47)(relative curvature)2. [1]The DNA length L is specified in bp and the relative curvature

A3D / A6- 2

RL / a

2.0 /

/0-loo1502A06-02

)p1.5-/ / 9

/~~~~-/

60 100 150 200 250bp

FIG. 1. RL at room temperature vs. length of multimers for thesequences having different relative curvatures. RL for a multimer isthe ratio of apparent length (bp), determined from comparison withsize markers on an 8% polyacrylamide gel, to real length (bp). Thefractional number m/n in the name of each sequence indicates therelative curvature for the multimers of the sequence, defined as theaverage number of A tracts per helix turn, mr/n.

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Proc. Natl. Acad. Sci. USA 85 (1988) 1765

RL

bp

(RELATIVE CURVATURE)2FIG. 2. Dependence ofRL from Fig. 1 on the square of relative

curvature. Some data points here are obtained by interpolationalong the curves in Fig. 1. The data were fitted to the equation RL =A(relative curvature)2 + B, with (A, B) = (0.265, 1.04) for 84 bp;(0.658, 1.02) for 105 bp; (1.06, 1.02) for 126 bp; (1.55, 1.01) for 147bp; (2.10, 1.02) for 168 bp; and (3.08, 0.%) for 189 bp as determinedby least-squares analysis of the data.

is equivalent to the average number ofA6 tracts per helix turn.Applying this equation to the multimers used for calibration,we found that the relative curvature ofDNA molecules can bedetermined with an error ± 0.02 in the range of 120 - L (bp)- 170, RL>. 1.2. The absolute curvature is defined as theaverage bend angle per helix turn for a DNA fragment, and itcan be calculated by using the equation

Absolute curvature (degrees per turn)= (relative curvature)(bend per A tract), [21

where the bend angle for an A6 tract is estimated to liebetween 170 and 22.50 (H.-S.K., J. R. Rice, and D.M.C.,unpublished results).Bend Angles and Their Applications in Predicting Anoma-

lies in Gel Mobilities of Bent Multimers. In Fig. 3, gel mobilitydata for the multimers (A1 - T.) (j = 5, 6, or 8), in whichevery other A tract is inverted to a T tract, are comparedwith those of multimers (AON1o-)" having directly repeated Atracts of the same length. These results, along with data forthe multimers (A4T4N2)n and4T4A4N2)n studied by Hager-man (9), provide a substantial data set for testing A-tractbending models. It can readily be verified that the observedproperties cannot be explained by using a model that incor-porates bending exclusively by tilt at the junctions, and thata significant component of roll at the 5' junction is required.The multimers (A4T4N2)n have gel mobility anomalies simi-lar to the properties of (A6N4)n, and in contrast the mole-cules (T4A4N2)n show nearly normal gel mobilities, a strikingdifference that is particularly informative about the differ-ence in bend parameters at 3' and 5' A-tract junctions. Forthe calculated relative curvature of these sequences to agreewith their gel mobility data, there must be roll toward themajor groove, in addition to tilt, at the 5' junction of the Atract. For simplicity and uniformity, the T-A step in TnAncan be treated as a combination of the two steps 5'-Tn-B and5'-B-An (B stands for B-DNA region). The roll componentlocated at the 5' junction ofA tracts consequently results in

FIG. 3. RL of the multimers (Aj-T.),, in which every other Ajtract is dyad inverted to a Tj tract (j = 5, 6, or 8). Each solid curvefor (AjTj),, is compared with the broken curve for (AjN1,j)n con-taining A tracts of the same length. Because all the data in thecurrent figure are obtained from the same gel, the data for (AjN1_),,here are more valid for comparison with the results for (Af-T1)n thanare the slightly different previous data (8).

twice as much roll at the T-A step. In (T4A4N2)", the centralT-A roll bend works against the two flanking 3' junctionbends, and thus the calculated relative curvature is small, asobserved in their normal gel mobilities. In (A4T4N2)n, on theother hand, the A-T step has no roll component, and thecombination of roll and tilt at the 5' junctions producesoverall curvature similar to that of (A6N4)". The conse-quence of the 5' roll component is to open the minor grooveat the 5' end of an A tract, relative to the 3' end; thisphenomenon should be increased further at Tn-An junctions.

In the iterative process of refining the bending model,plausible initial roll and tilt components at the 5' and 3'junctions were assigned, and the curvature relative to A6N4was calculated from the helix axis trajectory. The 150-bpmultimer of the A6N4 sequence was used as a reference topredict the (RL - 1) values for other sequences from thequadratic dependence of (RL - 1) on relative curvature. The150-bp length was chosen because it is within the optimalrange for the application of Eq. 1. The gel mobility anomaliescalculated by using a given set of bend angles for the varioussequences were compared with the experimental values, andthe procedure was repeated after adjusting the angles, untilthe two anomalies became close. Positioning the junctionbends on the first and last bases of an A tract rather than onthe base step between an A tract and adjacent B-DNA regiongenerally gives better fits between calculated gel mobilityanomalies and experimental values, although this is notstrictly true for the AnN3O1 series.The estimated junction bend angles are listed in Table 2,

and gel mobility anomalies calculated by using these bendangles for the various sequences (all with 10-bp repeats) arecompared with experiment in Table 3. Agreement is remark-ably good, considering the size of the data set and thesimplicity of the model, with just two adjustable parametersexcluding the absolute curvature. Factors neglected includethe role of flanking sequence, variation in helix screw, andpossible special anomalies at An-Tn and Tn-An junctions.The procedure described here yields the relative tilt and rollcomponents of the helix axis deflection at the junctions; the

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Table 2. Bend angles in the junction model

Junction Tilt, degrees Roll, degrees

5'-B-An-3' -9.6 5.8Tn-B 9.6 5.8An-B 12.0 0.0B-Tn -12.0 0.0An-Tn 0.0 0.0Tn-An 0.0 11.6

To Apply the bend angles to DNA molecules, the sequences of theDNA ptrand should be read in the 5'-to-3' direction. A positive valueof tilt (or roll) means bending toward the strand that runs in the5'-to-1' direction (or toward the major groove). The B-An and Tn-Bjunctipns are the same 5' junctions ofA tracts, but the sequences ofoppolte strands are being followed. The same description can beattached to the An-B and B-Tn junctions, which are both 3' junctionsof A tacts. The bend is located on the first and last nucleotides ofA tracts for the 5' and 3' junctions, and in the middle of the T-A stepfor th* Tn-An junction. B stands for a B-DNA region.

absolute values of the bend components listed in Table 2 arebase4 on an estimate of 220 for the A6 bend angle, corre-sponging to the upper end of the range of values found fromcyclikation kinetic studies (H.-S.K., J. R. Rice, and D.M.C.,unpublished). The parameters in Table 2 can be linearlyrescaled to a different estimate of the A-tract bend angle ifdesired.

Different Phasing Between As and A4T4. These multimerswere inade to compare by experiment the bending directionsofA tracts and AnTn segments. In the multimers (IHA)n, thecenters of A5 and A4T4 segments are separated by 10.5 or

Table 3. Comparison of calculated gel mobility anomalies withexperijnental data for multimers (150 bp) of various sequences

Sequence Calc. circle RL - 1name size, bp Calc. Exp.

A6N4A3N7*A4N6*A5N5A7N3A8N2

A9Nl*A5-8*A5*

A5-T5A6-T6A8-T8

AIOTIOIHAA4T4N2tT4A4N2tA3T3N4t

161435248186155165197199196192167173621272161460188

(1.31)0.180.550.981.471.250.870.860.880.921.221.060.070.461.310.120.96

1.310.230.601.001.261.210.730.720.870.861.191.070.070.42[1.23][0.04][1.01]

Thq bend angles in Table 2 were used to calculate circle sizes forthe sequences. The calculated gel mobility anomalies (RL - 1) wereobtained from the circle sizes and the experimental (RL - 1) valuefor NN4 (1.31), taken as the standard. The sequences labeled with* and t are listed in our previous paper (8) and in Hagerman's papers(7, 9), respectively. The experimental (RL - 1) values in bracketswere obtained at different gel electrophoresis conditions [12%polyalrylamide gel (acrylamide/methylenebisacrylamide = 37:1)]and asr therefore not strictly comparable to the other results. How-ever, the trends within this data set are correctly predicted by themodel, as is the gel mobility of the IHA sequence, which has amixture ofA tracts and A4T4N2. Least-squares minimization was notused in determining goodness of fit of the model to the data set, so itis likely that small further improvements on the values in Table 2 arepossible; in our view the simplicity of the model and the potentialcontribution of other variables such as flanking sequence and helixscrew piake further refinement of the parameters unprofitable at thislevel 9f structural knowledge.

2.0

RL

1.5

1.0

6( 100 150 200250bp

FIG. 4. RL values at room temperature for the additionalsequences, with A20 (homopolymeric dA-dT) as a control for A10T10.

20.5 bp. In (OHA)", they are separated by 15.5 bp. As shownin Fig. 4, the in-phase set (IHA)" shows significantly retardedgel mobilities, while those of (OHA)" are nearly normal.According to thejunction model, in the DNA segment aroundA5 bending occurs primarily in the tilt direction at the junc-tions, producing overall bending toward the minor groove atthe center base of A5 (8, 22). In the DNA segment aroundA4T4 the resultant of the 5' junction bends is toward the minorgroove at the center of A4T4. When the centers of A5 andA4T4 are separated by integral multiples of the helix screw asin (IHA)", the bends by A5 and A4T4 add constructively toproduce optimized overall bending. However, in (OHA)" thecenters of A5 and A4T4 are out of phase, resulting in cancel-lation of adjacent bends. Agreement between predicted andobserved properties of these oligomers containing A-tractbends intermixed with the Hagerman (9) bending sequenceA4T4 confirms our assignment of their relative directions.

A10T10. The multimers (A10T10)n were synthesized to ex-amine the roll at T-A junctions. Up to 150 bp the multimershave normal gel mobilities, but those longer than 150 bp showsome anomaly, as seen in Fig. 4. Because the base composi-tion and sequence of A10T10 are close to those of A20, theseries (A20)n is probably a better reference in describinganomalies in gel mobilities of (A10T10),,. The multimers (A20)nhave faster gel mobilities than the marker DNA, multimers ofthe BamHI linker, perhaps because the helix rise ofpoly(dA)-poly(dT) is smaller than for DNA fragments ofrandom sequence (23). When the series (A20)n is used as areference for (A10T10),, these multimers have a noticeableanomaly in gel mobility, even for the short fragments (_ 150bp). In terms of the junction model, (A10T10),, has a bend atthe T-A step every 20 bp; successful prediction of its mobility(Table 3) indicates reasonable accuracy of the roll angle atT-A, assuming zero roll at ApT.

DISCUSSIONIn this work, by comparing gel mobilities for multimers thathave different A-tract densities, we derived an empiricalrelationship, of quadratic form, between the anomaly in gelmobility and the relative curvature, Eq. 1. The calculatedrelative curvature can be converted to the absolute curva-ture by using Eq. 2. Eq. 1 was obtained from comparison ofthe gel mobility data on multimers where bends are evenlydistributed along the helix axis of the DNA molecule andwell phased with the helix screw. The calculated relativecurvature will be most accurate for DNA fragments havingfeatures similar to those of the multimers.

Using the quadratic dependence on curvature and an esti-mate of the absolute curvature, we determined the bendangles listed in Table 2, which explain the main features of

IHA (e) /

I//.I

/

__ OHAA(O) /-- 0(- A2() -

~~I 11

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junction bending. At both 5' and 3' junctions, bending occursmainly by tilt, and at the 5' junction, there is a roll componentsmaller than the tilt angle. Tilt is directed toward the T base atthe 5' junction and toward the A base at the 3' junction; rollat the 5' junction is toward the major groove; the overalldirection of an A-tract bend is approximately toward theminor groove at its center. The 3'-junction bend is larger thanthe 5'-junction bend, which is a vectorial sum of the roll andtilt components. The T-A step in a T"A,, sequence can betreated as a combination of the Ta-B and B-As junctions, to anadequate level of approximation. The sum of the Ta-B andB-An bends results in only a roll in the T-A step, twice as largeas the roll in the T"-B or B-An bend. By the same logic, thecombination of the An-B and B-Tn junctions yields no bendingin the A-T step of AnTn sequences.Our model predicts roll that opens the minor groove at the

5' end of the A tract, relative to the 3' end. This is in accordwith the chemical "footprinting" observations reported byBurkoff and Tullius (24), showing that hydroxyl radicalattack on A tracts, presumably through the minor groove, ismore rapid at the 5' end than at the 3' end of the A tract.The junction bending model is useful for its simplicity and

for its lack of explicit assumptions about the underlyingstructure of the A tract, but it is not a unique description ofA-tract-induced bending. In the junction model, the A tract ischaracterized by a single helix axis, and the bend is assignedto the junctions where the A-tract axis intersects the axes ofadjacent B-DNA. One can equally well define a local helixaxis perpendicular to the plane of each base pair and describethe bend as a series of "wedge" angles between adjacentbase-pair planes. Ulanovsky and Trifonov (25) showed thatthe properties of some of the sequences in Table 3 could beexplained by using a model with a localized wedge bend in theA tract, with a predominant contribution from roll, in adirection that narrows the minor groove. Repeat of such anoperation with a helical screw advance produces a doublehelix with base pairs tilted relative to the overall helix axis,consistent with models deduced from fiber diffraction studiesof poly(dA)-poly(dT) (25, 26). It is this base pair tilt thatpredicts the predominance of tilt in the junction bendingmodel (8). While there are some differences in detail betweenthe junction and wedge bending models for DNA, they canboth be interpreted in terms of this underlying structure. Thejunction model, however, does not require this structure,since the bend could be entirely determined by structuraleffects at the junctions.The fiber diffraction model does not in an obvious way

predict one of the key findings of this work: the distinction interms of roll and tilt components between 3' and 5' A-tractjunctions. The recent calculations of Churprina (27) indicatethat the source may lie in destabilization of the minor groovehydration of poly(dA)-poly(dT) at a T-A step, which corre-sponds to the 5' junction. Since ordered hydration is thoughtby some to correlate with a narrow minor groove, its lossshould lead to roll at the 5'junction, opening the minor grooveas our model requires.Recent crystallographic studies of DNA molecules con-

taining APT tracts have revealed propeller twisted base pairswith bifurcated or three-center hydrogen bonds in the Atracts (28, 29). However, since the direction of bending inthese molecules is predominantly roll rather than tilt at thejunctions, probably due to the crystal packing constraints,the relationship of the propeller twisted structure to bentDNA is uncertain. Indeed, it is found that addition of dis-

tamycin, which is known to remove DNA bends (2), increasespropeller twisting (29). This raises the possibility that thepropeller twisted form may be the structure found in solutionat high temperature, a condition that is also known to elimi-nate bending (8, 30). Determination of the underlying struc-tural basis for DNA bending will require crystallographicstudy of A-tract-containing molecules in a lattice that permitsbending in the overall direction defined by the studies re-ported here.

We thank Tarmo Ruusala, James Nadeau, and Grace Sun fordiscussion and assistance. This work was supported by NationalScience Foundation Grant DMB-8405494 and by a Project Grant forParasitology and Tropical Medicine from the MacArthur Founda-tion.

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