dynamic bending rigidity of a 200-bp dna in 4 mm ionic...

Dynamic Bending Rigidity of a 200-bp DNA in 4 mM Ionic Strength:A Transient Polarization Grating Study

Alexei N. Naimushin, Bryant S. Fujimoto, and J. Michael SchurrDepartment of Chemistry, University of Washington, Seattle, Washington 98195-1700 USA

ABSTRACT DNA may exhibit three different kinds of bends: 1) permanent bends; 2) slowly relaxing bends due tofluctuations in a prevailing equilibrium between differently curved secondary conformations; and 3) rapidly relaxing dynamicbends within a single potential-of-mean-force basin. The dynamic bending rigidity (kd), or equivalently the dynamic persis-tence length, Pd 5 kd/kBT, governs the rapidly relaxing bends, which are responsible for the flexural dynamics of DNA on ashort time scale, t # 1025 s. However, all three kinds of bends contribute to the total equilibrium persistence length, Ptot,according to 1/Ptot > 1/Ppb 1 1/Psr 1 1/Pd, where Ppb is the contribution of the permanent bends and Psr is the contributionof the slowly relaxing bends. Both Pd and Ptot are determined for the same 200-bp DNA in 4 mM ionic strength by measuringits optical anisotropy, r(t), from 0 to 10 ms. Time-resolved fluorescence polarization anisotropy (FPA) measurements yield r(t)for DNA/ethidium complexes (1 dye/200 bp) from 0 to 120 ns. A new transient polarization grating (TPG) experiment providesr(t) for DNA/methylene blue complexes (1 dye/100 bp) over a much longer time span, from 20 ns to 10 ms. Accurate data inthe very tail of the decay enable a model-independent determination of the relaxation time (tR) of the end-over-end tumblingmotion, from which Ptot 5 500 Å is estimated. The FPA data are used to obtain the best-fit pairs of Pd and torsion elasticconstant (a) values that fit those data equally well, and which are used to eliminate a as an independent variable. When therelevant theory is fitted to the entire TPG signal (S(t)), the end-over-end rotational diffusion coefficient is fixed at its measuredvalue and a is eliminated in favor of Pd. Neither a true minimum in chi-squared nor a satisfactory fit could be obtained for Pd

anywhere in the range 500–5000 Å, unless an adjustable amplitude of azimuthal wobble of the methylene blue was admitted.In that case, a well-defined global minimum and a reasonably good fit emerged at Pd 5 2000 Å and ^dz2&1/2 5 25°. Thediscrimination against Pd values ,1600 Å is very great. By combining the values, Ptot 5 500 Å and Pd 5 2000 Å with aliterature estimate, Ppb 5 1370 Å, a value Psr 5 1300 Å is estimated for the contribution of slowly relaxing bends. This valueis analyzed in terms of a simple model in which the DNA is divided up into domains containing m bp, each of whichexperiences an all-or-none equilibrium between a straight and a uniformly curved conformation. With an appropriate estimateof the average bend angle per basepair of the curved conformation, a lower bound estimate, m 5 55 bp, is obtained for thedomain size of the coherently bent state. Previous measurements suggest that this coherent bend is not directional, orphase-locked, to the azimuthal orientation of the filament.

INTRODUCTION

It has been suggested that DNA exhibits three differentkinds of bends, namely 1) permanent bends, 2) slowlyrelaxing bends due to fluctuations in a prevailing equilib-rium between differently curved secondary conformations,and 3) rapidly relaxing bends within a single potential-of-mean-force basin (Song and Schurr, 1990; Schurr et al.,1992; 1997a,b). Although all three types of bends contributeto determine the total persistence length (Schurr et al.,1997a; Trifonov et al., 1987; Harrington, 1993), only thelast contributes to thedynamicsof DNA flexure on shorttime scales,t & 1025 s. The value of the total persistencelength, Ptot 5 500 Å, is reasonably well known from avariety of measurements, at least under standard conditions.However, much less is known about the separate contribu-tions of the different kinds of bends.

Permanent bends

Intrinsic curvature of certain sequences is well-established,and is directional in the sense that the bend is phase-locked(to some degree) to the azimuthal orientation of the filament(Trifonov et al., 1987; Harrington, 1993). However, themagnitude of that intrinsic curvature may reflect a tiltedequilibrium between intrinsically bent and intrinsicallystraight(er) states (Nelson et al., 1987), because it varies tosome extent with sequence context (Dlakic and Harrington,1996, 1998), and depends strongly on various environmen-tal factors, such as temperature (Diekmann, 1987; Chan etal., 1990), concentration of alcohols (Sprous et al., 1995),and concentration of Mg21 ions (Brukner et al., 1994;Dlakic and Harrington, 1995), all of which can be varied tocompletely remove the curvature of certain sequences. Theapparent absence of any effect of permanent bends onsingle-molecule force versus extension curves at moderateto high extension, including the very precise data recentlyreported (Wang et al., 1997), suggests that moderate ten-sion, far below that required for 30% extension, may alsoshift the secondary structure equilibrium in favor of intrin-sically straight states. Estimating the contribution of direc-tional permanent bends to the total persistence length is a

Received for publication 30 July 1999 and in final form 3 December 1999.

Address reprint requests to Dr. J. Michael Schurr, Dept. of Chemistry,University of Washington, Campus Box 351700, Seattle, WA 98195-1700.Tel.: 206-543-6681; Fax: 206-685-8665; E-mail: [email protected].

© 2000 by the Biophysical Society

0006-3495/00/03/1498/21 $2.00

1498 Biophysical Journal Volume 78 March 2000 1498–1518

delicate matter that requires sequence averaging of thepermanent bends and ensemble averaging of the nonperma-nent bends and twists, and the results in general dependupon both the anisotropy and noncentrosymmetry of thebending potential, which govern the coupling betweentwisting and bending (Schurr, 1985; Schurr et al., 1995;Allison and Schurr, 1997). The total persistence lengths forparticular sequences were simulated by using Trifonov’swedge model (Bolshoy et al., 1991) for the directionalpermanent bends and the simplest possible isotropicHookean twisting and bending potentials (Schellman andHarvey, 1995). These were found to obey an approximaterelation, which was later generalized to give

1/Ptot > 1/Ppb 1 1/Psr 1 1/Pd, (1)

wherePpb 5 1370 Å is the contribution of the permanentbends,Psr is the contribution of slowly relaxing bends dueto conformational fluctuations, andPd is the contribution ofrapid dynamic bends (Schurr et al., 1997a,b). Because theavailable experimental data rule out all extant nearest-neighbor permanent bending models (Dlakic and Har-rington, 1996, 1998), including the Trifonov wedge model,the particular value obtained forPpb must be regarded assomewhat tentative.

Dynamic bends

The dynamic persistence length,Pd, is related to the dy-namic bending rigidity (Ad) that governs the short-time (t &1 ms) flexural dynamics of DNA by

Pd ; Ad/kBT (2)

wherekB is Boltzmann’s constant andT the absolute tem-perature. Experimental estimates ofPd have come fromanalyses of the decays of the transient photoinduced dichro-ism (TPD) (Allison et al., 1989), and transient electricbirefringence (TEB) (Eden and Sunshine, 1989), and di-chroism (TED) (Diekmann et al., 1982; Po¨rschke et al.,1987), and more recently from EPR spin-label spectra (Hus-tedt et al., 1993; Okonogi et al., 1999).

The TPD data for a 209-bp DNA were compared withBrownian dynamics simulations of the optical anisotropiesof intrinsically straight model filaments with selected valuesof Pd and the torsion elastic constant (a) (Allison et al.,1989). It was concluded that the data couldnot be satisfac-torily fitted by Pd 5 500 Å. Among the trial values consid-ered, the best fit was obtainedover the span of the presenteddata by Pd 5 1000 Å anda 5 7.6 3 10212 dyne cm.However, the assumption thatPtot 5 Pd very likely yieldeda lower best-fitPd value than would have been obtained,had the effect of any permanent and/or slowly relaxingbends to diminishPtot and the end-over-end tumbling time(tR) been taken into account. The contribution of flexuraldynamicsto the fluorescence polarization anisotropy (FPA)

or TPD is governed entirely byPd, whereas the rate ofend-over-end tumbling reflectsPtot, which is generallysmaller thanPd. Thus, settingPd 5 Ptot is expected to yielda compromise best-fit value that underestimatesPd. Anapproximate analytical theory for treating weakly bendingrods allows for separate adjustment of the end-over-endtumbling time and thePd that governs the flexural dynamics(Song et al., 1990). Unfortunately, the TPD data for the209-bp restriction fragment at 5°C extend only to 23 1026

s, which is less than one-third of the expectedtR value, sothe quality of the fit at any longer times in the very tail ofthe decay curve could not be assessed, andtR could not bereliably extracted in an independent manner by fitting thattail. An equally good, if not superior, fit of the model to thedata could almost certainly be obtained by using a some-what smallera, a somewhat largerPd . 1000 Å, and alongest tumbling time corresponding toPtot . 500 Å , Pd.It is crucial to obtain precise optical anisotropy data extend-ing much farther into the tail of the curve so that thetR canbe obtained separately from the best-fitPd. A much moreextensive least-squares fitting of the model to the experi-mental data to determine the statistically “allowed” range ofPd values would also be desirable, but that is impractical toimplement via Brownian dynamics. The approximate ana-lytical theory for treating weakly bending rods and circleshas been tested against Brownian dynamics simulations(Song et al., 1990; Heath et al., 1995, 1996b) and found towork well, even in the presence of regular directional per-manent bends (Allison and Schurr, 1997), and provides ameans to rapidly calculatex2 at very many points of a gridsearch to map out the allowed range ofPd, as describedbelow.

Extracting a value ofPd from TED and TEB decay curvesis complicated by several factors. Unlike the FPA and TPD,which measure knownself-correlation functions, the TEDand TEB off-field decays monitormutualcorrelation func-tions of a kind that depends on the mechanism of orientationof the DNA in the electric field, which is not known withcertainty (Allison and Nambi, 1992). This uncertainty isespecially acute in regard to the nonlinear part of the re-sponse of the DNA to the electric field, which may beimportant, because nonlinear effects are evident in the ex-perimental data. For example, the shape of the decay curve,in particular the relative amplitude of the fast initial tran-sient, increases substantially (up to 20-fold) with electricfield strength from 1 to 20 kV/cm for DNAs containing95–250 bp (Eden and Sunshine, 1989; Diekmann et al.,1982; Po¨rschke, 1987). In addition, overshoot of the bire-fringence after the electric pulse is shut off is observed forDNAs as short as 300 bp, even at rather low electric fieldstrengths (1 kV/cm) (Eden and Sunshine, 1989). Thesefindings suggest that bending strain energy is stored in someof the DNA molecules during the pulse. Any distortion ofthe molecule or alteration of its response function must arisefrom a nonlinear effect. According to the theory developed

Dynamic Bending Rigidity of DNA 1499

Biophysical Journal 78(3) 1498–1518

for purely linear excitation mechanisms, the relative contri-bution of the collective bending modes to the decay of therelevantmutualcorrelation functions is much smaller thanin the case of the correspondingself-correlation functions,and the amplitude associated with the longest bending re-laxation time is negligibly small (Heath et al., 1995). Thesesomewhat surprising effects are due to extensive cancella-tion of terms in the particularmutualcorrelation functionsthat are expected forlinear orientation by induced dipole,correlated induced dipole, permanent dipole, and saturatedinduced dipole mechanisms. The limited available theoret-ical information aboutnonlineareffects comes from simu-lations of charged DNA-like rods with a bend-dependentorientation mechanism that more plausibly represents theeffects of ion-atmosphere polarization than any of thosenoted above (Elvingson, 1992). The results indicate thatfilaments, which are initially oriented perpendicular to theelectric field pulse, acquire a substantial horseshoe bend asthey undergo electrophoresis toward the closed end of thehorseshoe. This is tantamount to a very substantial enhance-ment of the initial amplitude of the longest bending mode.Of course, the bending will be asymmetrical when initiatedat any position other than the center, and the longer arm isexpected to be more aligned with the field than the shorter.Relaxation of an asymmetrically bent DNA after the electricfield is turned off almost certainly proceeds by realigningthe short arm on one side of the bend with the longer arm onthe other. This would be expected to produce an increase inmagnitude of the birefringence, which is a likely explana-tion for the overshoot during the off-field decays of theDNAs with N $ 300 bp. In any case, when the off-fieldTED and TEB decays for DNAs withN 5 120–250 bp arefitted to a double-exponential function, and the relaxationtime of the fast initial transient isassumedto correspond tothe longest bending normal mode, the optimum agreementbetween predicted and observed values is obtained forPd 52100 Å (Song and Schurr, 1990). This value is fairly pre-cisely determined by the data. However, until it can beunequivocally established thatnonlinearbending of trans-versely oriented DNAs during the pulse actually does suf-ficiently enhance the amplitude of the longest bendingmode, the validity of thisPd value is open to question. Infact, if the orientation mechanism is assumed to be exclu-sively linear, a much smaller value,Pd 5 500 Å, canaccount well for the relaxation times of the fast initialtransients, which are then dominated by higher bendingmodes. However, the predicted relative amplitudes of thefast initial transients are invariant to electric field strength,contrary to observation (Allison and Nambi, 1992). If sucha low value ofPd is ruled out by other experiments thatadmit a range of values containing 2100 Å, then the validityof such a high value becomes less doubtful, and it mayprovide a relatively precise guide to the actual value underthe prevailing 2 mM ionic strength conditions.

We have been exploring an alternative approach, inwhich the FPA of ethidium intercalated in 13 syntheticduplexes of different composition and length (12(3), 24(2),36(3), 48(3), 60(1), and 72(1) bp), in two native restrictionfragments containing 43 and 69 bp, and in several longnative DNAs was measured and analyzed (Fujimoto et al.,1993, 1994b). The objective is to fit all of these data with acommon value ofa and Pd, and eventually to assess theincrease of the effective hydrodynamic cylinder radius (dueto permanent bends) fromRH 5 10.16 0.2 Å for N 5 12and 24 bp up to an expected plateau value at largeN $ 200bp. According to Brownian dynamics simulations of 72-bpmodel DNAs with and without regular permanent bends(15.6° every 24.5 Å), the effect of the permanent bends is toincrease the effectiveRH for azimuthal spinning by only3%, from 10.0 to 10.3 Å (Allison and Schurr, 1997). If theincrease inRH from 12 to 72 bp isassumedto be #1.04-fold, then the FPA data imply thatPd $ 1500 Å for all 10DNAs with N 5 36–72 bp. This work is still ongoing anda full discussion will be presented elsewhere.

The EPR spin-label experiments (Hustedt et al., 1993;Okonogi et al., 1999) differ in several important regardsfrom the FPA and TPD experiments discussed above. 1)The principal axes of the hyperfine tensors of the spin-labelsmake a relatively small angle (;15–20°) with respect to thehelix-axis, so the EPR spectrum is much more sensitive toend-over-end tumbling and bending than to azimuthal spin-ning and twisting motions. In contrast, in the FPA and TPDexperiments the transition dipole makes a much larger angle(70–75°) with respect to the helix-axis, so those experi-ments are relatively more sensitive to azimuthal spinningand twisting than to tumbling and bending motions. 2) Theprimary effect of the bending motions of sufficiently shortDNAs on the EPR spectrum is a rapid pre-averaging of theelements of the hyperfine tensor. In such a case, the datareflect simply the equilibrium mean-squared amplitudes ofthe dynamic bending motions, which do not depend on thehydrodynamic radius. 3) The label is covalently attached toa particular base in the sequence, rather than distributedalong the DNA, as are the intercalated dyes used in the TPDand FPA experiments. By changing the position of thespin-label in the DNA, the relative sensitivity of the EPRspectrum to bending modes of even and odd symmetry canbe varied. In fact, the variations of the observed mean-squared amplitudes of internal angular motion of such spin-labels with the length of homologous DNA (12, 24, 36, 48,and 96–100 bp) and with the relative position of the label inthe DNA are in excellent accord with the analytical theory(Okonogi et al., 1999). The original experiments on onekind of sequence with a particular spin-label yieldedPd 52400 Å (Hustedt et al., 1993). More recent experiments ona different kind of DNA sequence with a new spin-labelyield Pd 5 1500–1700 Å (Okonogi et al., 1999). Theseresults pertain to 0.1 M ionic strength, and are almostcertainly the most reliable estimates presently available for

1500 Naimushin et al.


such standard conditions. Independent assessment ofPtot forthese same DNAs cannot be made in these EPR experiments.

Slowly relaxing bends associated with afluctuating conformation equilibrium

A great deal of evidence for slowly relaxing conformationalequilibria in both linear and circular DNAs has been re-ported previously by our and other laboratories, and was justrecently reviewed (Schurr et al., 1997a). NMR evidence forslowly relaxing conformational exchanges at certain sites insmall duplexes has also been reported (Kennedy et al.,1993; McAteer et al., 1995; McAteer et al., submitted forpublication; Gaudin et al., 1995, 1997; Paquet et al., 1996;Spielmann, 1998). However, the evidence that the involvedconformations exhibit different intrinsic curvatures is lessabundant and more indirect. A prevailing equilibrium be-tween a nearly straight secondary structure and one or moreintrinsically curved (or superhelical) conformations wouldnecessarily contribute both to the average extent of perma-nent bending and to slowly relaxing fluctuations about that.As noted previously, the sensitivity of sequence-dependentpermanent bends to various environmental factors is con-sistent with a prevailing equilibrium between two or morestates with different intrinsic curvature. The appearance oflong-lived structural isomers of multimers of a 147-bp in-trinsically curved DNA in electrophoretic gels is also con-sistent with a slowly relaxing equilibrium between differ-ently curved secondary conformations (N. C. Stellwagen,personal communication).

If slowly relaxing fluctuations between alternative con-formations with different intrinsic curvature contribute sig-nificantly to the mean-squared amplitude of bending of theDNA (away from its average curved conformation), then itfollows that an imposed bending strain must shift thatconformational equilibrium, and thereby change those prop-erties that are sensitive to average secondary structure, suchas the torsion elastic constant, the intrinsic binding affinityfor ethidium, and the circular dichroism (CD) spectrum.Such an experiment was performed by circularizing a181-bp DNA, and profound changes in those same proper-ties were actually observed (Heath et al., 1996a). In fact, theavailable data from our and other laboratories suggest that,as the bending strain is increased by decreasing the size ofthe DNA circle, a structural transition occurs between 360and 250 bp (cf. Fig. 1) (Shore and Baldwin, 1983; Clen-denning and Schurr, 1994; Horowitz and Wang, 1984; Shi-mada and Yamakawa, 1985; Taylor and Hagerman, 1990).Moreover, this transition may well be cooperative. Becausethe average bend per basepair required to induce the struc-tural transition (u1 # 1.5°) is much smaller than the root-mean-squared (RMS) bend between basepairs of an un-strained DNA (;6–7°), one may infer that coherence of thebend over many basepairs is more important than the abso-lute magnitude of the bend for stabilizing the curved state.

These observations suggest that the average domain size ofthe prevailing curved conformer might be considerablylarger than a single basepair. Because a modest degree ofcoherent bending strain suffices to shift the prevailing sec-ondary structure equilibrium toward the more curved state,fluctuations in that equilibrium are expected to contribute tothe equilibrium RMS curvature and 1/Ptot of the unstrainedDNA.

A bend-dependent conformational equilibrium mightplay a significant role in the activation of transcription. Anumber of transcriptional activators bind at specific sites inthe proximal promoter region, 50–250 bp upstream fromthe RNA polymerase binding site, and induce large bends inthe DNA either directly at their binding sites (Schultz et al.,1991; Passner and Steitz, 1997; Kim et al., 1994; Gross-chedl et al., 1994; Rice et al., 1996; Sjottem et al., 1997;Geiselmann, 1997; Perez-Martin and de Lorenzo, 1997;Diebold et al., 1998; Engelhorn and Geiselmann, 1998;Davis et al., 1999; Parkhurst et al., 1996, 1999) or indirectlyvia DNA looping (Mossing and Record, 1986; Gralla, 1991,1996; Schleif, 1992; Rippe et al., 1995). The activation oftranscription by DNA looping was formerly ascribed to anincrease in concentration of the activator (bound to one endof the loop) in the vicinity of the RNA polymerase bindingsite (at the other end of the loop), which was postulated tofacilitate activation by protein-protein interactions (Mossingand Record, 1986; Rippe et al., 1995). Now that the bendingstrain in small loops withN # 250 bp is known to alter theirsecondary structure, one could equally well postulate thatprotein-induced DNA bending, either direct or via DNAlooping, induces an allosteric transition of the DNA to an

FIGURE 1 Torsion elastic constanta versus DNA length (N) for variousDNAs. The crosses are linear DNAs and the open symbols are circularDNAs. SB denotes the data of Shore and Baldwin (1983) as analyzed byClendenning and Schurr (1994). HW denotes the data of Horowitz andWang (1984) as analyzed by Shimada and Yamakawa (1985) and Frank-Kamenetskii et al. (1985). TH denotes the data of Taylor and Hagerman(1990). The open circles and square denote the data of Heath et al. (1996a).



alternative secondary structure that somehow facilitates thebinding or kinetics of transcription initiation by the RNApolymerase. CAP and Sp1 are transcriptional activators thatinduce large contact bends,;90° (Kolb et al., 1983; Wuand Crothers, 1984; Liu-Johnson et al., 1986; Passner andSteitz, 1997) and 60° (Sjottem et al., 1997; Ikeda et al.,1993), at their respective binding sites. Evidence that bothspecific CAP binding to supercoiled plasmids and specificSp1 binding to a 1074-bp linear DNA induce long-rangealterations of the DNA secondary structure was recentlydiscussed (Schurr et al., 1997a).

Goals and strategy

Our primary goal is to assess the dynamic bending rigidityof DNA via time-resolved optical emission and absorptionanisotropy experiments. Specifically, we measure the FPAof intercalated ethidium in a 200-bp DNA at high timeresolution (;40 ps) from 0 to 120 ns, and the transientphotoinduced dichroism of intercalated methylene blue inthe same DNA at lower time resolution (10 ns) from 0 to 10ms by means of a new transient polarization grating (TPG)method (Naimushin et al., 1999b), which is an alternative todirect TPD experiment (Hogan et al., 1982, 1983; Allison etal., 1989). By detecting the photoinduced dichroism diffrac-tively with a single detector, the TPG experiment com-pletely eliminates certain systematic errors and faithfullymonitors the loss of dichroism out to sufficiently long timesthat thetR can be reliably determined by fitting just the verytail of the TPG data (Naimushin et al., 1999a,b). The best-fittR value of our most dilute sample is analyzed to givePtot >500 Å for this 200-bp DNA in the prevailing 4 mM ionicstrength. This best-fittR value is used in the subsequentanalysis of both FPA and TPG data. The FPA data areanalyzed to determine the empirical relation between thebest-fita- andPd values that fit the FPA data equally well.This empirical relation is then used to eliminatea as anadjustable parameter in the subsequent fitting of the TPGdata to determine the best-fit value ofPd, which is found tobe 2000 (2200,1300) in 4 mM ionic strength at 21°C.

Our secondary objective is to estimate the contribution,1/Psr, of slowly relaxing bends to 1/Ptot by using the bestavailable estimates ofPtot, Ppb, andPd in Eq. 1. To ratio-nalize the estimatedPsr 5 1300 Å, we adopt a very simplemodel, in which the DNA is divided into average domainsizes ofm basepairs, each of which exhibits an all-or-noneequilibrium between a straight and a uniformly curvedconformation. By assuming an intrinsic bend,u1 5 1.46°/bp, for the latter in accord with Fig. 1, we estimate a lowerbound for the average domain size,m $ 55 bp.

MODEL AND ANISOTROPY THEORY

The DNA is modeled by a chain ofN identical rigid sub-units, each of which is connected to its neighbors on either

side by Hookean torsion and bending springs (Schurr et al.,1992). These springs represent thedynamictorsional andbending rigidities, respectively. On the time scale of ourexperiment (t # 1025 s), each molecule exhibits 1) acom-mon intrinsic curvature due to permanent bends, whichcontains an average contribution from the prevailing equi-librium between intrinsically straight and curved conforma-tions, plus 2) avariable “intrinsic” curvature due to fluctu-ations in that same equilibrium, which are effectively frozenon the time scale of our experiment. As a consequence of itsintrinsic curvature, the average end-to-end distance andtR

of each molecule will be significantly smaller than reckonedsimply according itsPd. Moreover, because every moleculemay have a slightly different intrinsic curvature (due toslowly relaxing bends) that isnot averaged out by dynamicflexure on the time scale of the end-over-end rotation, itmay be necessary to consider adistribution of mean end-to-end distances (Schurr and Fujimoto, submitted for pub-lication) and end-over-end tumbling times, as will be donebelow. The effect of the dynamic bending rigidity is to resistthe deformation of any molecule from its particular intrin-sically curved state of mechanical equilibrium. The totalintrinsic curvature is assumed to be sufficiently modest thatthe dynamics of bending, twisting, and uniform azimuthalspinning of the vast majority of molecules are practicallyidentical to each other and to that reckoned for an intrinsi-cally straight rod, albeit one with an enhanced effectivecylinder radius to account for the increased friction ofazimuthal spinning of the bent filament (Schurr et al.,1997b; Allison and Schurr, 1997). The validity of thisassumption was demonstrated by Brownian dynamics sim-ulations of 72-bp model DNAs with regular permanentbends (Allison and Schurr, 1997).

We adopt the analytical theory developed for deformablefilaments with mean local cylindrical symmetry, in whichcase the optical anisotropy is given by Schurr et al. (1992),Schurr (1984), and Schurr and Fujimoto (1988),

r~t! 5Ii~t! 2 I'~t!

Ii~t! 1 2I'~t!5 ~ARF!~2/5! O

n50

2

In~`!Cn~t!Fn~t! (3)

For the FPA,Ii(t) and I'(t) are the fluorescence intensitieswith polarizations parallel and perpendicular, respectively,to that of an infinitely short exciting pulse. In the case ofTPD, Ii(t) andI'(t) are replaced byDAi(t) 5 Ai(t) 2 Ai

o andDA'(t) 5 DA'(t) 2 A'

o , respectively, whereAi(t) andA'(t)are the absorbances for probe beams with polarizationsparallel and perpendicular, respectively, to that of the pho-tobleaching pulse, andAi

o 5 A'o pertains to the unperturbed

solution prior to arrival of the photobleaching pulse. Themaximum initial anisotropy is 2/5,

ARF 5 ^P2~cosb!&2 > 1 2 6s2 (4)

is the amplitude reduction factor due to rapid unresolvedisotropic wobble of the dye relative to the DNA, ands2 is



the mean-squared amplitude ofsmall isotropic dye wobble.For the FPA of intercalated ethidium, ARF5 0.90 ands >7° (Schurr et al., 1992), and for the FPA of methylene blue(MeBl), ARF 5 0.73 ands > 12° (Fujimoto et al., 1994a).However, for the TPD of intercalated MeBl, Austin andco-workers often reported considerably lower ARF values,in some cases as low as 0.40 (Hogan et al., 1982, 1983;Allison et al., 1989), and this anomaly was more pro-nounced for GC-rich DNAs. The evidence that such lowinitial anisotropies of DNA/methylene blue complexes inTPD experiments stem primarily from enhanced azimuthalwobble is discussed in the Data Analysis section. TheIn(`)are internal correlation functionsafter the rapid relaxationof both isotropic and excess (beyond isotropic) azimuthaldye wobble (Schurr et al., 1992; Schurr, 1984; Schurr andFujimoto, 1988),

I0~`! 5 @~3/2!cos2« 2 1/2#2 (5a)

I1~`! 5 3cos2« sin2« exp~2^dz2&! (5b)

I2~`! 5 ~3/4!sin4« exp~24^dz2&! (5c)

where« is the polar angle between the transition dipole ofthe dye and the local symmetry axis of the DNA, and^dz2&is the excessmean-squared amplitude ofazimuthal dyewobble. The dye wobble manifested in the FPA of eitherethidium or MeBl is believed to be small in amplitude andisotropic, so we takedz2& 5 0 in those cases. However, forthe TPG of DNA/MeBl complexes,dz2& will be entered asan adjustable parameter into the fitting protocol. TheCn(t)are twisting correlation functions (Schurr, 1984; Schurr etal., 1992; Heath et al., 1996b)

Cn~t! 5 ~1/N! Om51

N

expF2S~n2kBTt/Ng!

1 n2 Ol52

N

dl2 Qml

2 ~1 2 e2t/tl!DG (6)

whereN is the number of basepairs,g is the friction factorper basepair for azimuthal rotation around the local sym-metry axis, anddl

2, Qml, andtl are the mean-squared am-plitude, projection onto themth basepair, and relaxationtime, respectively, of thelth normal torsion mode. Expres-sions for the latter quantities are given elsewhere (Schurr,1984; Schurr et al., 1992; Heath et al., 1996b; Allison andSchurr, 1979). Thedl

2, Qml, tl, andCn(t) are readily calcu-lated when N,a, RH, T, and the solvent viscosityh arespecified. TheFn(t) are tumbling correlation functions forthe weakly bending rod model (Song et al., 1990; Heath et

al., 1996b)

Fn~t! 5 ~1/N9! Om51

N9

expF2~6 2 n2!SDRt

1 ~b/Pd!Ol53

N9

~Sm11,l 2 Sm,l!2~1 2 e2t/Tl!DG (7)

whereN9 is the number of subunits,b 5 31.8 nm is thelength (diameter) of the contiguous spherical subunits ofthis weakly bending rod model, andSml and Tl are theprojection onto themth subunit and relaxation time of thelth bending normal mode, respectively. Formulas and nu-merical methods to calculateSml andTl from the dynamicalmatrix are described elsewhere (Schurr et al., 1992; Song etal., 1990; Heath et al., 1996b). The dynamical matrix de-pends only uponN9. The rotational diffusion coefficient forend-over-end tumbling (DR) is here regarded as an adjust-able parameter to be determined separately by fitting the tailof the TPG data. TheFn(t) are readily calculated whenN9,b, DR, Pd, T, andh are specified.

C0(t) 5 1.0 does not relax at all, whereasC1(t) andC2(t)relax much more rapidly thanF0(t), because uniform spin-ning of a 200-bp DNA is much faster than end-over-endtumbling. Consequently, then 5 0 term in Eq. 3 dominatesat long times, and for sufficiently long times that the bend-ing modes have all relaxed (i.e.,t . T3), the anisotropydecay becomes a single exponential with relaxation timetR 5 1/(6DR).

The rapid wobble of intercalated ethidium relaxes in;50ps (Schurr et al., 1992), whereas that of intercalated MeBl(in its FPA) relaxes in;200 ps (Fujimoto et al., 1994a).Negligible error is incurred by assuming that such motionsare infinitely rapid compared to the collective twisting andbending motions. Equations 6 and 7 forCn(t) andFn(t) werederived by assuming that no cross terms couple twisting andbending on the potential surface, and that any geometricalcoupling between such motions is also negligible. In thatcase, the twisting and bending motions are separable, andlinear Langevin equations apply in the molecular frame ineach case (Schurr et al., 1992; Song et al., 1990; Heath etal., 1996b; Allison and Schurr, 1979). The effect of hydro-dynamic interactions on the twisting dynamics was earliershown to be negligible (Allison, 1983). Hydrodynamic in-teractions were incorporated into the Langevin equations forthe bending dynamics, as described previously (Song et al.,1990). The 31.8 Å diameter of the contiguous sphericalsubunits is chosen to yield diffusion coefficients for uniformrotation and translation that agree with the predictions ofTirado and Garcia de la Torre (1979, 1980) for rigid cylin-ders withRH 5 12.5 Å, which in turn agree with measure-ments on short restriction fragments (Elias and Eden, 1981).The elementary subunit used in the calculation ofCn(t) isone basepair and its hydrodynamic cylinder radius is taken



to be 12 Å. This value, which significantly exceeds 10.1 Å,is intended to account for enhanced friction of azimuthalspinning of a 200-bp molecule when it exhibits intrinsiccurvature. Evidence that the effectiveRH for diffusionalspinning reaches a plateau value that might be;12 Å forN $ 170 bp was discussed previously (Schurr et al., 1997b).In any case, the bending and end-over-end tumbling mo-tions are very insensitive to the choice ofRH.

Although Eqs. 3, 6, and 7 take full account oftorquesarising from the torsion potential andforcesarising from thebending potential, they take no account of theforcesarisingfrom the torsion potential. Such forces stem from a geomet-rical coupling between twisting and bending, and generallyvanish unless the molecule is simultaneously both bent andtwisted (Heath et al., 1996b). It was demonstrated viaBrownian dynamics simulations that theseforces arisingfrom the torsion potential have no significant effect on ther(t) of a 196-bp model DNA along an equilibrium trajectory(Heath et al., 1996b), although they have a very substantialeffect on the rate of conversion of excess twist into writhein a small circular DNA (Chirico and Langowski, 1994).Equilibrium r(t) curves for 196-bp model DNAs, whichwere simulated by incorporating all forces and torques,could be successfully fitted and analyzed by using Eqs. 3–7(Heath et al., 1996b). This remained true even when regularpermanent bends were incorporated into the Brownian dy-namics simulations of a 72-bp model DNA provided thatDR

was adjusted separately andRH was allowed to increaseslightly to account for the enhanced friction of uniformazimuthal spinning (Allison and Schurr, 1997). Althoughthe analytical theory embodied in Eqs. 3–7 has been welltested against Brownian dynamics simulations, the afore-mentioned tests do not extend to timest . 200 ns.

Finally, we note that the model of a weakly bending rodthat is used to analyze the data is valid only for DNAlengths (L) such thatL & (0.6) Pd (Song et al., 1990;Fujimoto and Schurr, submitted for publication). It is ad-vantageous to work with the longest possible DNA subjectto this constraint so the amplitude of the longest bendingmode is as large as possible. Because of the prevailinguncertainty regardingPd we elected to study a 200-bp DNA,although one could certainly go as high as 250 bp ifPd $1500 Å, as appears to be the case.

Fixed parameters in the model

Cn(t) is calculated forN 5 200 subunits (1 subunit/bp),T 5294 K, h 5 0.009779 poise,RH 5 12.0 Å,h 5 3.4 Å, andg 5 4pRH

2 h h 5 6.02 3 10223 dyne cm s. All sums areevaluated numerically.Fn(t) is calculated forN9 5 21contiguous subunits with 31.8 Å diameter, using the samevalues ofT, h. DR is taken to be the measured value, exceptwhen a distribution ofDR is considered. In that case, anFn(t) curve is calculated for each subcomponent (j) of thepopulation using its appropriate value ofDRj, which is

estimated by a scaling procedure described in the DataAnalysis section, and the resulting (Fn(t))j curves are thenaveraged.

Pertinent times

Because of the very wide span of times accessed in thisstudy, it is pertinent to consider the relaxation times of someof the most relevant motions. The 50- and 200-ps relaxationtimes for the wobbles of excited ethidium and MeBl, re-spectively, in their intercalation sites have already beenmentioned. The relaxation time of the longest torsion nor-mal mode ist2 > [4hp RH

2 h/(4a sin2(p/2N))] > 40.7 ns fora 5 6.0 3 10212 dyne cm and the parameters given above(Schurr et al., 1992; Schurr, 1984; Allison and Schurr,1979). The relaxation time for the uniform mode of azi-muthal spinning inC1(t) is t11 5 4hp RH

2 h z (200)/kBT 5297 ns for the same choice of parameters. This is the longestrelaxation time inC1(t). The relaxation time for the uniformmode of azimuthal spinning inC2(t) is t21 5 t11/4 5 74 ns.This is the longest relaxation time inC2(t). If Pd 5 2000 Å,the longest bending relaxation time for a 200-bp DNA isT2 > 325 ns for the same choice of parameters. More thanhalf of the mean-squared amplitude of angular motion dueto bending is associated with this longest bending normalmode. The relaxation time for end-over-end tumbling isfound experimentally to be 4.26ms or 3.80ms, dependingon the DNA concentration. From these times one may inferthat then 5 2 and 1 terms in Eq. 3 are greatly attenuated by2t21 5 148 ns and 2t11 5 594 ns. Beyond 2t11 5 594 ns,the n 5 0 term in Eq. 3 begins to dominate.

MATERIALS AND METHODS

The DNA sample

The 200-bp DNA sample was prepared by copying the sequence fromposition 233 to 432 of pBR322 by polymerase chain reaction (PCR). Thiswas done using an ExpandTM Long Template PCR System kit fromBoehringer Mannheim. This 200-bp DNA contains the entire sequence ofthe 181-bp DNA (positions 236–417 of pBR322), whose response toimposed bending strain was studied previously by Heath et al. (1996b), andis 56% GC. Details of the sample preparation, purification, and character-ization are presented elsewhere (Naimushin, 1999). The purification pro-tocol includes 1) protein digestion with proteinase K and removal of theproducts by multiple extractions with buffered phenol; 2) exhaustive dial-ysis alternately versus high salt (HSTE) (500 mM NaCl, 10 mM Tris, 1mM Na2 EDTA, pH 8.5 at 21°C) and normal salt (NSTE) (100 mM NaCl,10 mM Tris, 1 mM EDTA, pH 8.5 at 21°C) buffers at 4°C, terminatingfinally in the latter; and 3) hydroxyapatite chromatography. The samplecharacterization includes 1) confirmation of the sequence by direct se-quencing; and 2) gel electrophoretic comparison with standards of knownlength to certify that the tertiary structure is normal with no significant gelretardation (e.g., due to large permanent bends). For the FPA and TPGstudies the samples contain 1.5 g/l DNA (i.e., 1.133 1025 M DNAmolecules or 2.3 mM basepairs). The buffer contains 5 mM Tris (but noNaCl or Na2 EDTA) at pH 7.5, and its ionic strength is 4 mM. Althoughhigher ionic strengths were studied up to 47 mM, uncertainties regardingthe nature of the MeBl binding at the higher ionic strengths preclude a full



analysis of those data at the present time. DNA concentrations weredetermined from the UV spectrum of an appropriately diluted sample byassumingA260 > 1.0 for a 0.05 g/l solution. The samples also contain 1added MeBl per 100 bp for the TPG studies, and 1 added ethidium per 200bp for the FPA studies. The MeBl was purified as described by Bergmannand O’Konski (1963).

Behavior of methylene blue

The binding constant of MeBl for a single intercalation site at 20°C in 4mM ionic strength was estimated from the data of Hagmar et al. (1992) inthe manner described earlier (Fujimoto et al., 1994a), and found to be1.9 3 105 M21. Under the prevailing conditions in our 4 mM sample, theMeBl is practically (99.8%) all bound. However, as shown previously,even under low ionic strength conditions, MeBl occupies (at least) threedifferent intercalation sites with very similar unwinding angles (Fujimotoet al., 1994a) and absorption spectra (Hagmar et al., 1992), but verydifferent fluorescence lifetimes (25, 130, and 620 ps). The transient pho-tobleaching of the MeBl (due to triplet formation) was found to ariseprimarily from the longest-lived of these intercalated species, which wasassociated with AT, TA, or AA steps (Fujimoto et al., 1994a). Theselong-lived sites should account for most of the photobleaching in thepresent TPG experiments. For calf thymus DNA in 8 mM ionic strength,such sites comprise only;20% of the total bound dye. Nevertheless, fromthe measured fluorescence lifetimes and relative fluorescence amplitudes,we estimate that the longest-lived sites are responsible for;75% of thetotal photobleach under 8 mM (or lower) ionic strength conditions.

At higher ionic strengths the situation is considerably more complex, forthe following reason. With increasing ionic strength, not only do therelative populations of the different intercalation sites shift, but the longest-lived fluorescent component, which still dominates the photobleaching,moves from an intercalated site to an outside bound site (Fujimoto et al.,1994a). The latter site exhibits a somewhat shorter fluorescence lifetime(430 ps) and a much greater amplitude of rapid local angular motion, orwobble, which reduces its anisotropy to;24% of its initial value with arelaxation time of;100 ps. In contrast, at the lowest ionic strengthexamined previously (8.3 mM), rapid dye wobble relaxes the FPA of thelongest-lived component only modestly to 73% of its starting value with arelaxation time of;200 ps. In fact, the observed residual anisotropy afterrelaxation by rapid dye wobble declines progressively with increasing ionicstrength from (0.73)(2/5) at 8.3 mM to (0.24)(2/5) at 207 mM (Fujimoto etal., 1994a). Because neither the equilibrium orientation (i.e.,«) nor thedistribution of excursion angles of this outside bound dye is known, it is notpossible to interpret thefull anisotropy decay at any but the two very lowestionic strengths in the present work (4 and 8 mM), for which the experi-mental decay curves are almost identical. Here we focus solely on the 4mM ionic strength data. We note that the equilibrium orientation anddistribution of excursion angles of MeBl strongly affect theamplitude, butnot the rate constant, of the terminal exponential decay ofr(t) due toend-over-end tumbling. Consequently, it is possible to determinetR as afunction of ionic strength to assess the effects of intermolecular electro-static interactions on self-rotational diffusion, and the results of that studyare discussed elsewhere (Naimushin et al., 1999a). Intermolecular interac-tions cause almost no change intR as the ionic strength is lowered from 47to 20 mM, but effect a 12% increase intR upon further lowering to 4 mM.This increase intR is removed simply by a twofold reduction of the DNAconcentration. Any effects of intermolecular interactions on the dynamicsof internal motions, such as bending, are expected to be much less than ontR, and therefore negligible.

Sample protocol

The samples prepared for TPG studies are deoxygenated by bubbling argonsaturated with buffer vapor through the solutions for 12–18 h before each

experiment. The argon is also blown gently across the top of each sampleduring the experiments.

The sample cuvette is 1 mm thick by 1 cm in breadth by 4 cm deep. Toameliorate any effects of transient heating and permanent photobleaching,the 300ml sample volume was stirred by periodically slowly withdrawingand reinjecting a small portion of the sample. This was accomplished usinga semi-collapsible tube that was immersed in the sample at one end, and theother contains an upstream N2 bubble that is alternately pinched andreleased by a peristaltic pump.

Fluorescence polarization anisotropymeasurements

The apparatus and experimental protocols for FPA measurements on the200-bp DNA/ethidium complexes are reviewed elsewhere (Schurr et al.,1992). However, the atypically high DNA concentration (1.5 g/l), thebuffer (5 or 10 mM Tris with no EDTA or NaCl), pH (7.5), and thetemperature (294 K) are peculiar to the present experiments. For thesemeasurements the instrumental width of the apparatus was;40 ps. Theanalyses of both the FPA and TPD data are discussed in the Data Analysissection.

Transient polarization grating measurements

A detailed description of the TPG instrument is provided elsewhere(Naimushin, 1999; Naimushin et al., 1999b), so only a brief account will begiven here. The grating is written in the sample by intersecting twocoherent (0.04 cm21 bandwidth) pulses (5 ns) of horizontally propagatinglaser light (690 nm), which haveorthogonal (vertical and horizontal)polarizations. Each beam is;2 mm in diameter and makes an angleu/2 520° with respect to the grating normal. As a consequence of this excitation,the sample is uniformly bleached by;1%. This is the fraction of MeBlmolecules that undergo intersystem crossing to their triplet states, fromwhich they return to the ground state with a relatively long lifetime of;10–20ms in these deoxygenated solutions. The repetition rate of thisexperiment is 10 s21. Although the intensity of the superimposed excitationpulses is uniform throughout the grating, thepolarizationof the light variesperiodically along the grating from right circular to linear (polarized at1p/4 from vertical) to left circular to linear (polarized at2p/4 fromvertical), and then back to right circular to begin another period. Theregions of the sample that experience circular polarization undergo isotro-pic (unpolarized) photobleaching in the plane of the grating, whereas thosethat experience linear polarization undergo anisotropic photobleaching.That is, in the linear polarization regions, those intercalated MeBl mole-cules with transition dipoles aligned along the local polarization of the lightare preferentially excited and become proportionately trapped in theirtriplet states. Because the MeBl triplet absorbance is negligible in thespectral region of the S0 3 S1 absorption band, those molecules areeffectively bleached for the duration of their lifetime in the triplet state. Inthese experiments the triplet state serves merely as a temporary reservoirfor optically silent molecules. Due to the anisotropic photobleaching-induced in the zones of linear polarization, the sample becomes uniaxiallydichroic in those regions with symmetry axes alternately at1p/4 and2p/4 from vertical, as one moves along the grating. The excitation pulseenergy (;0.2 mJ) is adjusted so that a MeBl molecule with its transitiondipole aligned along the local polarization in a linearly polarized region iscycled through its S03 S1 transition;10 times during the pulse, when itfails to reach its triplet state. Consequently, a majority of the photobleachedmolecules were excited previously during thesamepulse. A noteworthyfeature of the polarization grating is that the uniform intensity createsnoordinary thermal phase grating, which would relax by thermal diffusion onthe microsecond time scale of interest and potentially complicate the dataanalysis.



The amplitude of the photoinduced dichroism grating is monitored bydiffraction of a cw, horizontally propagating, vertically polarized, 632.8nm probe beam from a helium-neon laser. The angle (f) of incidence of theprobe beam and exit of the diffracted beam with respect to the gratingnormal satisfies the Bragg condition,

sin f 5632.8

690.0sin~u/2! (8)

so thatf 5 18.3°, and the polarization of the diffracted probe light isrotated tohorizontal (von Jena and Lessing, 1979).

A theoretical expression for the diffraction efficiency (h) of a polariza-tion grating was obtained by extending the theory developed by von Jenaand Lessing (1979), and is presented elsewhere (Naimushin et al., 1999b).When the total diffraction efficiency is small (h , 0.01), as is the case here(and in many other experiments), the result can be very accurately approx-imated by

h 5 expF2 ~2.3!Aprobe

cosf GS pT

lr cosfD2

z FSD@dk~t!#

2 D2

1 SD@dn~t!#

2 D2G (9)

whereA# probe is an “average” optical density adjusted so that the exponen-tial factor describes the absorptive loss of the probe,T is the sample (orgrating) thickness, andlr is the probe wavelength in the sample. Thequantitiesdk(t) 5 k1p/4(t) 2 k2p/4(t) and dn(t) 5 n1p/4(t) 2 n2p/4(t)denote thelocal imaginary and real parts, respectively, of the complexbirefringenceat a particular point in the sample.kV andnV denote imag-inary and real parts of the complexrefractive indexfor probe light withpolarization directionV in the plane of the grating. Hence,dk(t) is pro-portional to the photoinduceddichroismat a given point in the sample.That is,dk(t) } DAi(t) 2 DA'(t). The quantityD[dk(t)] is the peak-to-nullvariation indk(t), which varies periodically along the grating, wherein theprincipal axis of the dichroism is alternately rotated by1p/4 and2p/4with respect to vertical. A similar definition applies toD[dn(t)]. As notedabove, there is no ordinarythermalphase grating contribution toD[dn(t)].Any contribution of the 1% polarized photobleach (triplet formation) toD[dn(t)] is assumed to be negligibly small, so our polarization grating isthen purely absorptive. The diffraction efficiency of such an absorptivepolarization grating is 36 times smaller than that of the correspondingpurely absorptiveintensitygrating, in which both write beams have verticalpolarization (Naimushin et al., 1999b). However, because an absorptiveintensity grating inevitably also exhibits a thermal phase grating, its dif-fraction efficiency may exceed that of the corresponding polarizationgrating by considerably more than 36-fold. For one of our samples, themeasured ratio was 60-fold. The weak signal of the polarization gratingmakes alignment and optimization of the experiment both difficult andtime-consuming, and requires a great many (;100,000) shots for dataacquisition. After considerable effort, the observed initial diffraction effi-ciency was brought within a factor of 2.6 of the theoretical value calculatedfrom Eq. 9. TheD[dk(0)] value used in that calculation was evaluated fromthe measured change in absorbance of the probe beam following a writepulse, when one of the two intersecting write beams was blocked, asdescribed elsewhere (Naimushin et al., 1999b). A significant fraction ofthis 2.6-fold discrepancy is due to rapid unresolved dye wobble. Theremainder is attributed to partial multi-mode character of the pump laserand suboptimal alignment of the beams either at their intersection in thesample or on the most active region of the photomultiplier tube. Suchinstrumental defects diminish the signal amplitude in a proportional man-ner, but donot alter the time course of the relative decay of the signal.Because Eq. 9 does notin practiceyield the actual diffraction efficiency (orequivalently the amplitude of the diffracted signal), it isnot possible todetermine the absolute value of the peak-to-null variation in dichroism,

D[dk(t)], in the sample. This inability to provide theabsolute valueof thedichroism in the sample is the principal disadvantage of the TPG method.This requires an additional disposable parameter in the data analysis. In thecase of DNA/MeBl complexes, the initial anisotropy in TPD experimentswas often found to be anomalously low (Hogan et al., 1982, 1983; Allisonet al., 1989), so rather large adjustments of the ARF parameter in Eq. 3were required in any case. Hence, the relative disadvantage of TPGexperiments with respect to direct TPD experiments is perhaps not so greatin this case.

By writing the grating with 690-nm pulses and probing with a 632.8-nmbeam, and using appropriate narrow bandpass filters in the detection optics,we have effectively screened out the vast majority of scattered 690-nmlight and also the MeBl fluorescence, which is very short-lived (&630 ps)(Naimushin et al., 1999b; Naimushin, 1999). A polarizer in the detectionoptics passes the horizontally polarized diffracted beam and blocks anyvertically polarized probe light, whether scattered or diffracted from anadventitious thermal phase grating, such as might arise from the very smallinduced CD of the intercalated MeBl (Norde´n and Tjernfeld, 1982).

The diffracted probe light is detected by single-photon counting via aphotomultiplier and constant fraction discriminator (CFD). The output ofthe CFD is a nuclear instrumentation module (NIM) standard negativelogic pulse of adjustable width, which is set to match the channel width (10ns) of a transient digitizer-recorder, whose track and hold circuit samplesonce every 10 ns channel. In this way, every photon detected by thephotomultiplier is captured (provided that the maximum count rate doesnot exceed 0.01 photons/10 ns), and stored in the appropriate recorderchannel. Because the dichroism in the sample is monitored by a singledetector, both systematic and random errors are significantly reducedcompared to direct TPD experiments, in which the dichroism is measuredby using two detectors. In particular, systematic errors due to any mismatchin the calibration of the two detectors are completely eliminated, which iscrucial for obtaining precise data from the very tail of the decay curve.

The (slow) kinetics of photobleach recovery are measured by blockingone of the excitation pulses (so no grating is created) and monitoring theintensity of transmitted probe light, which is polarized at 54.7° fromvertical for these measurements. This is done using a photodiode and thesame transient recorder. The diffraction and absorbance signals are trans-ferred to a computer and accumulated between successive shots. Typically,the responses to 50,000 to 100,000 shots are accumulated to provideadequate diffracted signals, but only 1000 accumulated shots are needed toproduce excellent triplet decay curves (Naimushin et al., 1999b;Naimushin, 1999).

Least-squares fitting of the transient absorption curves to a double-exponential function typically yields a normalized triplet decay function(fT(t)) consisting of a major component with a lifetime;10–20ms, and aminor component with a much longer lifetime,;70 ms (Naimushin et al.,1999b; Naimushin, 1999). Because the kinetics of photobleach recoveryare rather slow compared to the decay of the diffracted signal, and theprecision of thefT(t) data is so great, the determination offT(t) contributesnegligibly to the error in our final results.

Rotations of the transition dipoles of the unbleached MeBl moleculestoward the bleached orientations diminish the local dichroism and cause adecay of the diffracted signal. Likewise, decay of the triplet T1 states of thebleached molecules back to their ground S0 states also diminishes thedichroism and diffracted signal. The signalS(t) is the number of accumu-lated photon counts in each channel and is proportional to the intensity ofdiffracted probe light, or equivalently to the diffraction efficiency in Eq. 8.Hence,S(t) is proportional to the square of the instantaneous dichroism inthe sample, and can be written as (Schurr, 1984; Naimushin et al., 1999b)

S~t! 5 W@rN~t!fT~t!#2, (10)

whereW is an unknown proportionality constant,rN(t) 5 r(t)/(ARF)(2/5)is thenormalizedanisotropy from Eq. 3, andfT(t) is thenormalizedtripletdecay function. All constants that affect the amplitude, but not the shape,of the decay are lumped into the adjustable parameterW. Translational



diffusion generally also contributes to the decay ofS(t), but in the presentexperiments on 200-bp DNA/MeBl complexes, relaxation of the grating bytranslational diffusion is several hundred times slower than by end-over-end tumbling, so it can be omitted from the data analysis with negligibleerror. A heterodyne term between a small amount of depolarized scatteredlight and the diffracted beam (Iscat/Idiff , 0.0006) was expected to con-tribute;5% of the initial signal, but was shown to vanish, presumably dueto shot-to-shot variations in phase of the grating and its diffracted beamrelative to that of the scattered light.

DATA ANALYSIS

Triplet decay function

The measured decay curve is fitted by a double exponentialusing an iterative nonlinear least-squares method. The fits,shown elsewhere (Naimushin et al., 1999b; Naimushin,1999), are excellent. The dominant component typically hasa lifetime in the range 10–20ms, and arises from the MeBltriplet state. We have not yet identified the origin of theminor slow component with a very long lifetime,;70ms. Itcould conceivably represent the reoxidation of a smallamount of reduced (so called leuko-) MeBl, generated bythe write pulses and subsequent photochemistry, back toground state MeBl. In any case, the kinetics of that processare so slow that it has no significant effect on our results.The best-fitfT(t) curve in each case is used in the subsequentfitting of Eqs. 10 and 11.

End-over-end tumbling times

As noted above,tR can be determined from the TPG data ina manner that is independent of the model or its inputparameters, except for the assumed exponential decay of theterminal relaxation process. In that case, we simply fit

S~t! 5 A@exp@2t/tR#fT~t!#2 (11)

to the experimental data by an iterative nonlinear least-squares procedure. The adjustable parameters aretR and theamplitudeA. Equation 11 is fitted to the TPG data fromdifferent beginning timestB in the tail of the decay curve outto a final time of 10ms. A typical fit is shown in Fig. 2, anda plot of the best-fittR vs. tB is presented in Fig. 3. Asexpected, the best-fittR rises gradually to a plateau value,and the errors increase astB moves progressively farthertoward the tail of the decay. The best-fit values oftR aretaken to be averages of the results along the plateau fromtB 5 2.4 to 3.4ms. The best-fittR values of all our samplesat different ionic strength and DNA concentration weredetermined and reported in a separate paper (Naimushin etal., 1999a). Those values most pertinent to the present workare collected in Table 1. Both increasing the ionic strengthabove 20 mM and decreasing the DNA concentration from1.5 to 0.75 g/l lead totR > 3.80 ms, which suggests thatsuch a value applies when intermolecular interactions aresufficiently weak. The effect of intermolecular interactionsto retard end-over-endself-rotational diffusion, which isprobed by the TPG experiment, is analyzed in the otherpaper (Naimushin et al., 1999a). It is expected thattR for thepresent DNA at much greater dilution in 4 mM ionicstrength would be 3.80ms or less.

Evaluation of Ptot

We can estimate the total persistence length of our DNA byfollowing the protocol of Hagerman and Zimm (1981).

FIGURE 2 S(t) vs. t for a 200-bp DNA in 4 mM ionic strength (top). Thesmooth line is the fit of Eq. 11 to the data at long times. The bottom panelis a plot of the residuals from that fit.

FIGURE 3 tR vs. tB for the 200-bp DNA in two different ionic strengths.

TABLE 1 Best-fit tR values of 200 bp DNA at pH 7.5 at 21°C

[DNA](g/l)

Ionic Strength(mM) tR (ms)

1.5 4 4.266 0.041.5 8 4.006 0.071.5 20 3.86 0.051.5 47 3.786 0.030.75 4 3.806 0.10



They analyzed the ratioR 5 tR(wc)/tR(rod), wheretR(wc) isthe average value, either measured or simulated, for a semi-flexible filament, andtR(rod) is the value calculated for acompletely rigid rod of the same contour length by using theBroersma relation (their Eq. 23) with an effective hydrody-namic radius,RH 5 13 Å. By using their recommendedvalue,h 5 3.4 Å, for the rise per bp, we calculatetR(rod) 55.53ms at 294 K in water. When combined with the exper-imental value,tR 5 3.80ms, observed for our least stronglyinteracting DNAs, the ratioR 5 (3.80/5.53)5 0.687 isobtained. Hagerman and Zimm simulated configurationsand computed thetR(wc) values for a large number ofwormlike chain filaments with appropriately chosen hydro-dynamic parameters over a fairly wide range of the ratio,X [ L/Ptot, of contour length to total persistence length.Their numerical results are summarized in a few simpleequations (their Eqs. 20–22) that enable one to readilycalculateR from X. If we assume thatPtot 5 500 Å, thenX 5 680/5005 1.36, and the predicted ratio isR 5 0.681,which is practically identical to the experimental ratio givenabove. Hagerman (1981, 1988) did not examine such shortDNAs in 4 mM ionic strength, but for longer restrictionfragments he obtainedPtot 5 500 Å in 4 mM and higherionic strength. Thus, the best available estimate ofPtot ofour DNA in 4 mM (and probably also higher) ionic strengthis ;500 Å. If intermolecular interactions significantly re-tard self-rotational diffusion in our sample, then the actualPtot of our DNA will be somewhatsmaller. The essentialpoint is thatPtot for our DNA in 4 mM and higher ionicstrength does not significantly exceed 500 Å if the analysisof Hagerman and Zimm is quantitatively accurate.

Our most precise and interpretable experimental datapertain to the 1.5 g/l sample in 4 mM ionic strength. Theaverage valueDR 5 1/(6 tR) 5 3.91 3 104 s21 measuredfor this sample is used in the subsequent analysis of both theFPA and TPD data. The intermolecular hydrodynamic in-teractions in this twofold more concentrated solution in-creasetR by ;12%, but are assumed not to significantlyaffect the dynamics of bending. If significant, their effectwould be to slow the bending modes and slightly decreasethe apparentPd below the actual value.

Analysis of the FPA data

The horizontally and vertically polarized fluorescence in-tensities are combined to form the sum (s(t) 5 iV(t) 12iH(t)) and difference (d(t) 5 iV(t) 2 iH(t)) decay curves.These are fitted by convolutions of the measured instrumentresponse function (e(t)) with appropriate theoretical modelfunctions:

s~t! 5 E0

t

dt9e~t 2 t9!S~t9! (12)

d~t! 5 E0

t

dt9e~t 2 t9!S~t9!r~t9! (13)

The theoretical sum function is taken to be:

S~t! 5 Asd~t! 1 S1e2t/t1 1 S2e

2t/t2 (14)

wherein the delta function accounts for Raman scattering.The best-fit relaxation times,t1 > 21.8 ns andt2 > 1.5–2.0ns, correspond to intercalated and nonintercalated ethidium,respectively. The contributions of Raman scattering andnonintercalated ethidium are suppressed by omitting thefirst 4 ns from the fit of the difference data. The anisotropyfunctionr(t) is given by Eqs. 3–7. The fit was performed fora wide range ofassumed Pd values from 500 to 5000 Å.Adjustable parameters in the fit were the torsion elasticconstanta and ARF. The fixed input parameters, in additionto DR for end-over-end tumbling, were listed above. The fitswere performed over four different time spans, 0–16, 0–32,0–70, and 0–120 ns. For each choice ofPd, the optimuma-value “varied” with time span in much the same way asreported previously for the linear 181-bp DNA (Heath et al.,1996a). The 0–16-ns fit and to a lesser extent the 0–32-nsfit gave somewhat highera values than the 0–70- and0–120-ns fits. The results from the latter two time spanswere averaged to obtain the best-fit value ofa for eachassumed value ofPd. Essentially the same reducedx2 valuesin the range 1.00–1.05 were obtained for all assumed valuesof Pd from 500 to 5000 Å (or more). The best-fita valuesare plotted versus 1/Pd in Fig. 4. This empirical relation isthe main result of our FPA measurements on the DNAethidium complexes, and serves to eliminatea as an inde-pendent variable in the subsequent analysis of the TPG data.

FIGURE 4 Best-fita vs. 1/Pd from fit of Eqs. 3–7 to the FPA data forthe 200-bp DNA in 4 mM ionic strength. For each assumed value ofPd abest-fit value ofa was obtained from the fit.



Redetermination of «

Hogan et al. obtained the value« 5 72° by measuring thelimiting value of the reduced linear dichroism, LDr (Hoganet al., 1982). Their limiting LDr value was analyzed withouttaking dye wobble into account. From the reported value« 5 72°, the unreported experimental datum, LDr 5 21.05,can be recovered by using the following relation withARF 5 1.0,

LDr 5 21.055 3~ARF I0~`!!1/2 (15)

A protocol for estimating both« and the root-mean-squaredamplitude of isotropic wobble,s 5 ^d«2&1/2, from LDr andthe ARF value derived from FPA measurements was de-scribed previously (Schurr and Fujimoto, 1988). The valueARF 5 0.75 for DNA/MeBl complexes in 4 mM ionicstrength is obtained by extrapolating the ARF versus ionicstrength data of Fujimoto et al. (1994a). The analysis pro-ceeds by determining numerically the pairs of values («,^d«2&) that simultaneously satisfy Eq. 15, and also the rela-tion, ARF 5 0.75. The results are shown in Fig. 5, fromwhich the solution« 5 75° is readily apparent. This valuewill be used in the subsequent analysis of TPG data. If thesmaller value,« 5 72°, were used, the best-fit value ofPd

would be somewhat larger than that obtained below.

Analysis of the entire TPG decay

The timing of the transient digitizer is such that the writepulses occur primarily within a single 10-ns channel withonly ;15% overlap into the channels on either side. Con-sequently, it is not actually necessary to deconvolute theTPG data, provided the data in the first two channelsafterthe peak of the signal are ignored. The TPG data from 30 nsto 10 ms after the peak of the excitation pulse are fitted byEq. 10, using the best-fitfT(t) and the normalizedrN(t)reckoned from Eqs. 3–7. The 0-ns channel was taken to be

that with maximum signal, which occurs 10 ns after thepeak of the excitation pulse. Thus, the fits begin with the20-ns channel, which occurs 30 ns after the peak of theexcitation pulse, and the theoretical value is appropriate forthe 30-ns delay of that channel. This delay of the start of thefit until 30 ns, where the slope ofrN(t) is considerablyreduced, serves to diminish the error arising from the vari-able location of the write pulses within the 10-ns channel inwhich they occur. Although not necessary, a convolution ofthe theoretical curve with a three-point excitation functionwas used in the final fits. Before fitting, a very smallfirst-order correction is applied to the data in each channelto account for missing counts that arrived during the 10-nsdead time following each photon. Three experimental decaycurves for the same conditions were normalized to the sametotal number of counts and fitted simultaneously with thesame theoretical curve. After determiningtR, the remainingadjustable parameters arePd, W, and the excess azimuthalwobble, ^dz2& in Eqs. 5a–c, which is initially taken to bezero. The fit is conducted by a grid search, beginning withselected values ofPd and their correspondinga values, asdetermined from the empirical relation in Fig. 4. Theoreticalvalues of (rN(t)fT(t))2 andS(t)th are then calculated for eachchannel, and the reduced chi-squared

xr2 5 ~1/~M 2 n!! O

j51

M

~S~tj!exp 2 S~tj!

th!2 (16)

is calculated for allM data points, wheren is the number ofdisposable parameters. A contour plot ofxr

2 vs.Pd andW isobtained for the isotropic wobble model with« 5 75° and^dz2& 5 0. In this case such a plot is not particularlyinformative, since instead of a more or less circular basin itis a descending canyon that drains toward higherPd nearlyparallel to thePd-axis. A more useful representation, espe-cially when three variables are adjustable, is to plot theminimum value ofxr

2 for each choice ofPd vs. Pd. Thisprojects the optimumxr

2 values along the bottom of thecanyon or basin onto thexr

2 2 Pd plane. For the isotropicwobble model with« 5 75°, the optimumx2 vs. Pd curvedescends monotonically with decreasing slope all the wayfrom one grid boundary (Pd 5 500 Å) to the other (Pd 55000 Å) (not shown). Throughout this regionxr

2 remains sohigh (*2.39344) that the fits are rather poor (not shown).For example,xr

2 5 4.42298 for the optimum fit atPd 52000 Å. The poor quality of the fit is confirmed by visuallycomparing the optimum theoretical curves for differentchoices ofPd with the experimental curves (not shown). Theessential problem is that the theoretical curve descends toorapidly from 30 ns to;800 ns, which implies that therelative amplitudes of then 5 1 and 2 terms in Eq. 3 are toolarge compared to that of then 5 0 term.

FIGURE 5 Root-mean-squared amplitude of isotropic wobble (^d«2&1/2)vs. «. The solid line is a plot of («, ^d«2&1/2) pairs that satisfy ARF5 0.75.The dashed line is a plot of («, ^d«2&1/2) pairs that satisfy LDr 5 21.05. Atthe intersection,« 5 75°.



Variation of Pd, W, and «

The possibility remains that the value« 5 75° is notappropriate for the TPG (or TPD) experiment. All boundMeBl molecules participate nearly equally in theabsor-bancedetected LDr, but only one particular kind of bindingsite dominates the photobleaching and TPG (or TPD) sig-nal, and that is a minority species. That species conceivablyhas a polar angle« that differs significantly from the aver-age value reckoned from the LDr. Hence, we variedPd, W,and « in the fitting protocol again via a grid search. Thecurve of optimumxr

2 vs. Pd again exhibits no minimumbetween the grid boundaries, but instead declines with de-creasing slope fromPd 5 500 Å to Pd 5 5000 Å (notshown). Throughout this region, the best-fit« exceeds76.4°. The optimumxr

2 values are now much lower and thefits greatly improved, but they are still not altogether satis-factory. A comparison of the optimum fit forPd 5 2000 Å(« 5 79.7°,W 5 7084) with the experimental data is shownin Fig. 6. The theoretical curve still descends too rapidlyfrom 30 to 800 ns. For this curve,xr

2 5 2.38418.xr2 rises

rapidly with decreasingPd, so the discrimination againstsignificantly lower values,Pd # 1500 Å, appears to be verygreat.

Unsuccessful attempts to improve the fit

Several modifications of the model were undertaken inunsuccessful attempts tosignificantly improve the fit.

1. The dye was assumed to undergo 7° of isotropic wobble,like ethidium, plus the necessary amplitude of excess

azimuthal wobble to attain the extrapolated ARF5 0.75for the FPA of DNA/MeBl complexes in 4 mM ionicstrength (Fujimoto et al., 1994a). Although this im-proved the fit slightly, it didn’t go nearly far enough;

2. A distribution of the polar angle« over a span of up to610° was admitted, but that had almost no effect oneither the quality of the fit or the optimum parameters.Hence this protocol was eschewed;

3. Two different binding sites (A and B) with very differentpolar angles were admitted. The polar angle«A of spe-cies A was allowed to vary from 65 to 90°, whereas«B

of species B was allowed to vary from 0 to 45°. SpeciesA would correspond to the normally intercalated MeBland B to an outside bound MeBl. The relative fraction ofspecies B was allowed to vary from 0 to 1.0. The fitimproved significantly only when both species com-prised roughly equal fractions of the dye. However, solarge a fraction of outside bound dye is incompatiblewith certain results of Fujimoto et al. (1994a). Specifi-cally, in 10 mM Tris, 0.01 mM Na2 EDTA, pH 7.5, at20°C both negatively supercoiled and linear DNA/MeBlcomplexes exhibit very similar relative amplitudes anddecay times of all three observed fluorescence compo-nents. However, the binding affinity of DNA for inter-calated MeBlrelative to outside bound MeBl is muchgreater in negatively supercoiled than in linear DNAs.Hence, the invariance of the relative populations to su-percoiling implies that all three fluorescence componentsmust arise from the same kind of site, either intercalatedor outside bound. Unwinding studies at low ionicstrength show that a majority of the MeBl is in factintercalated, so all three fluorescent sites must be inter-calated with similar unwinding angles. The slight differ-ences in relative fluorescence amplitudes between thelinear and supercoiled DNAs allow for at most only afew percent of outside bound MeBl. Such a small frac-tion of outside bound dye does not significantly improvethe fit, so this model was eschewed;

4. A distribution of tR times was admitted. This was notdone in earlier studies, because the bending motionswere believed to explore and average all relevant con-figurations in a time much less thantR. BecausetR . 4.3ms exceeds by;14-fold the longest dynamic bendingrelaxation time,T2 . 325 ns, for the present DNA, thisis a good approximationin regard to dynamic bending.However, the present DNA also exhibits sequence-de-pendent permanent bends and effectively frozen slowlyrelaxing bends. The former bends contribute to reducetR

by the same factor for every molecule. However, thefrozen slowly relaxing bends will vary from one mole-cule to another, and thereby generate a distribution oftR.A protocol for calculating a suitable distribution oftR isdescribed in Appendix B. Incorporation of this distribu-tion produced only a very slight improvement in the fitand very little effect on the best-fit parameters. Never-

FIGURE 6 TPG signalS(t) versus time. The three thin solid lines aresmoothed data for three different measurements on a 200-bp DNA in 4 mMionic strength. The dotted line is the best fit of Eq. 10 to the data, when itis assumed thatPd 5 2000 Å, but« andW are adjustable, and the MeBlwobble is isotropic. WhenPd 5 2000 Å, the optimum values are« 5 79.7°,W 5 7084,xr

2 5 2.38418. The dashed line is the global best fit of Eq. 10to the data when it is assumed that« 5 75°, butPd, ^dz2&1/2, andW are alladjustable. The globally optimum parameters arePd 5 2000 Å,^dz2&1/2 525°, W 5 11,100,xr

2 5 1.94862.



theless, this procedure was retained in the subsequentfitting protocol for the sake of self-consistency.

Admission of arbitrarily large excessazimuthal wobble

Finally, we admit arbitrarily large azimuthal wobble. Thispossibility was considered for three reasons. 1) Excessazimuthal wobblewill decrease the relative amplitudes ofthen 5 1 and 2 terms in Eq. 3, and thereby improve the fit.2) Anomalously low ARF values in TPD experiments onDNA/MeBl complexes were often observed, especially forGC-rich DNAs (Allison et al., 1989; Hogan et al., 1982,1983). The implied large amplitudes of rapid local angularmotion of the transition dipole more likely reflect azimuthalwobble (around the local symmetry axis) than polar wobble.3) In the case of the FPA of DNA/MeBl complexes, theARF value decreases significantly with increasing temper-ature (Fujimoto et al., 1994a), in contrast to DNA/ethidiumcomplexes, where ARF is practically independent of tem-perature. This suggests that the amplitude of (presumablyazimuthal) dye wobble in DNA/MeBl complexes is forsome reason surprisingly sensitive to thermal energy. Animportant difference between FPA and TPG (or TPD) ex-periments on DNA/MeBl complexes is that in the FPA onlya single photon is absorbed from the excitation pulse by theemitting MeBl, whereas in the TPG (and TPD) experiments,typically several photons have been absorbed from theexcitation pulse prior to the photon that leads to photo-bleaching. As most of the energy from previously absorbedphotons is internally converted, the photobleaching thentypically occurs in a locally preheated molecule. We suspectthat the intercalation site is temporarily enlarged in someway by this local preheating to allow a substantially largeramplitude of azimuthal wobble. If the lifetime of the en-larged intercalation site is.10 ms, then Eqs. 5a–c apply asis. However, if the enlarged intercalation site narrows backdown to the normal site typical of the FPA excited state(with ARF 5 0.75 and^dz2& 5 0) within 30 ns, as seemslikely, then the^dz2& in Eqs. 5b and c must be replaced by^dz2&/2, as described in Results and Discussion. In practicewe simply use equations 5a–c directly in the fitting routine,and interpret the best-fitdz2& in the appropriate way foreither scenario at the end. The fitting is again conducted bya grid search along the coordinates of the three adjustableparameters,Pd, ^dz2&, andW with « fixed at 75°. With thismodel, a true minimum occurs in the curve of the optimumxr

2 vs. Pd, as shown in Fig. 7. The minimum value,xr2 5

1.94862, is found forPd 5 2000 Å,a 5 5.543 10212 dynecm, ^dz2&1/2 5 25°, andW5 11,000. These optimum valuesare collected in Table 2. With this value of^dz2&1/2, the ratio,ARFtot/ARFiso 5 0.61, where ARFtot is the total ARF andARFiso is the factor due to isotropic wobble, which islumped in withW. A comparison of the optimum theoreticalcurve for this azimuthal-wobble model with the experimen-

tal signals is shown in Fig. 6. The azimuthal-wobble modelclearly provides a much more satisfactory fit than the iso-tropic wobble model withPd 5 2000 Å and« 5 79.7°. Thisis also evidenced by the substantial reduction in reducedxr

2

from 2.38418 to 1.94862. This difference is statisticallyoverwhelming when account is taken of the very largenumber of data points (N 5 2994) in the three experimentalcurves.

In the final fits of this azimuthal wobble model, thetheoretical curve was convoluted with a three-channel ex-citation function. The effects of such a convolution on theresults were exceedingly minor, and this was done primarilyfor the sake of completeness.

RESULTS AND DISCUSSION

Ptot and Pd

For the first time, practically independentandmore or lessunambiguous measurements pertaining toPtot and Pd areavailable for the same DNA molecule. These values applyfor this particular 200-bp DNA in 4 mM ionic strength at21°C. ThetR . 3.80ms of the 0.75 g/l DNA sample yieldsPtot . 500 Å. The fit of the isotropic-wobble model withvariable« to the TPG (and FPA) data yields only a lowerbound Pd $ 1500 Å. However, the fit of the azimuthal-wobble model to the same data yieldsPd 5 2000(2200,1300) Å, where the statistical errors are conservatively setat about twice the square root of the variance, which isestimated from the variation ofx2 with Pd along the opti-mum x2 vs. Pd curve.

FIGURE 7 Totalx2 vs.Pd for the fit of Eq. 10 to the data in Fig. 6 underthe assumption that« 5 75°, butPd, ^dz2&1/2, andW are all adjustable.

TABLE 2 Best-fit parameters for the azimuthalwobble model

N (bp) L (Å) Ptot (Å) Pd (Å) a 3 1012 (dyne cm) ^dz2&1/2 (deg)

200 680 500 2000 5.54 25



The ratio of the probabilities that any two optimumsolutions forPd, ^dz2&, andW with different values ofPd

give rise to the data is given approximately by

P~Pd2!/P~Pd1! 5 exp@~x22 2 x1

2!/2# (18)

whereinxj2 5 h xrj

2 is the total chi-squared for the solutionj(j 5 1, 2), xrj

2 is its reducedchi-squared, andh 5 M 2 nis the number of data points (M 5 2994) minus the numberof disposable parameters (n 5 2). For example, usingxr1

2 51.94862 forPd 5 2000 Å andxr2

2 5 1.95589 forPd 5 1600Å, we obtain P(1600)/P(2000) 5 1.9 3 1025. By thiscriterion,Pd 5 1600 Å is;50,000 times less likely to havegiven rise to the data than isPd 5 2000 Å. The optimalxr

2

increases and the probability ratioP(Pd2)/P(2000) declinesrapidly and monotonically asPd2 descends further. In apurely statistical sense, these data effectively preclude anyvalue of Pd outside the range 1600–2700 Å under theprevailing conditions.

The present optimum value,Pd 5 2000 Å, in 4 mM ionicstrength at 21°C is only slightly smaller than the valuePd >2100 Å, in 2 mM ionic strength at 21°C, which was ob-tained byassumingthat the rapid initial transient in the TEDoff-field decay represents the longest bending normal mode(Song and Schurr, 1990). Because the present data sostrongly precludePd values as low as 500 Å, and select avalue close to 2100 Å, that assumption now becomes themost likely interpretation of the TED data. This in turnsuggests that the electric field in the TED experiments actsin a nonlinear manner to enhance the longest bending nor-mal mode, presumably in the manner suggested by Elving-son (1992), because that is apparently the only way in whichit can appear with sufficient amplitude in the TED off-fielddecay (Heath et al., 1995).

Torsional rigidity

The optimum torsion elastic constant,a 5 5.54 3 10212

dyne cm, corresponds to a torsional rigidity,C 5 ha 51.9 3 10219 dyne cm2, whereh 5 3.4 Å is the rise per bp.This lies within the range of values obtained by Taylor andHagerman (1990) from cyclization kinetics of 350–360 bpDNAs, and is within experimental error of the valueC 52.0 3 10219 dyne cm2, which was shown to quantitativelypredict the supercoiling free energies, structure factors, andtranslational diffusion coefficients of a supercoiled DNA(Gebe et al., 1995, 1996; Delrow et al., 1997). The FPA dataon the present DNA/ethidium complexes indicate that thetorsional rigidity of the present sample is;5% lower thanthat of the 181-bp DNA studied by Heath et al. (1996a) in100 mM ionic strength, assuming that both samples have thesamePd.

RMS Wobble

In the TPG experiment the value^dz2&1/2 5 25° pertains toexcess (beyond isotropic) azimuthal wobble of the interca-lated MeBl in its S0 ground state. This value is too large tobe compatible with the measured ARF5 0.75 for interca-lated MeBl in its S1 excited state in the FPA experiments(Fujimoto et al., 1994a). Either the RMS amplitude ofazimuthal dye wobble is normally much larger for the S0

state than for S1, which seems unlikely, or else it is pecu-liarly enhanced in the TPG and TPD experiments. As notedabove, in a TPG or TPD experiment, the MeBl typicallyabsorbs several photons during the same 5-ns excitationpulse that internally convert before absorbing the photonthat leads to triplet formation and photobleaching. Conse-quently, the complex has been preheated. A rise in localtemperature due to such preheating that is accompanied bysome minor structural change provides a plausible explana-tion of the present results.

The expressions 5a–c are obtained under the assumptionthat dz(t) is a stationary Gaussian random variable, forwhich ^dz(0)2& 5 ^dz(t)2& 5 ^dz2&. In fact, theexp[2n2^dz2&] factors follow from

^exp@in~dz~0! 2 dz~t!!#& 5 exp@2n2~^dz~0!2& 1 ^dz~t!2&!/2#

5 exp@2n2^dz2&#,

whenevert sufficiently exceeds the dye wobble relaxationtime (tW . 200 ps) that the correlation functiondz(0)dz(t)& 5 ^dz2& exp[2t/tW] effectively vanishes. Use ofexp[2n2^d2&] requires the stationarity condition to hold forall times examined (up to 10ms). Because the heat depos-ited in the DNA surely diffuses away on a time scale muchless than 10ms (or even 5 ns) the existence of an enlargedbinding site with enhanced azimuthal wobble for so long atime could occur only if there were some persistence ofmetastable structure at the binding site. The likelihood ofsuch a scenario is difficult to assess.

At the other extreme, the perturbed binding site mightrelax toward its unheated equilibrium conformation at a ratethat is insufficient to equilibrate within the 5-ns excitationpulse, but suffices to equilibrate within the first 30 ns afterthe pulse. In this case, the variance^dz(t)2& of the excessazimuthal wobble is expected to decrease from^dz(0)2& to 0in a variance relaxation timetV in the range 5 ns# tV & 30ns. BecausetW ,, tV, it is reasonable to regarddz(t) as anonstationary, but still Gaussian, random variable with acorrelation functiondz(0) dz(t)& that still vanishes fort ..tW. In such a case, fort . tV one hasdz(0)2& 1 ^dz(t)2& 5^dz(0)2& 1 0 5 ^dz2&, and the exp[2n2 ^dz2&] factors in Eqs.5b and c must be replaced by exp[2n2 ^dz2&/2]. In thisscenario the RMS amplitude of wobble just before absorb-ing the photobleaching photon is (=2) z 25° 5 35°, but thatrelaxes to zero during the early stages of the anisotropydecay (t , tV . 30 ns). We have not fitted the TPG data to



an intermediate model in which the variance of the en-hanced azimuthal wobble relaxes with a timetV in the range5 ns& tV & 10 ms, because we are reluctant to introduceanother adjustable parameter, but that remains a possibility.

Psr and the slowly relaxing bends

When the present estimatesPtot 5 500 Å andPd 5 2000 Åare used along with the Schellman-Harvey (SH) prediction,Ppb 5 1370 Å, in Eq. 1, there resultsPsr 5 1300 Å. Theessential point is thatPsr is not infinite, and that all threekinds of bends make comparable contributions to 1/Ptot.This establishes, albeit indirectly, that 1) slowly relaxingfluctuations in an equilibrium between differently curvedconformationsdo contribute significantly to 1/Ptot, and 2)the bending rigidity of DNA is itself a relaxing quantity,such that its equilibrium value at long times,Aeq 5 kBT/(1/Pd 1 1/Psr), is only ;40% of its dynamic value at shorttimes,Ad 5 kBT Pd, and corresponds to a flexural persis-tence length,Peq 5 (1/Pd 1 1/Psr)

21 5 788 Å. Twoobservations suggest that 1300 Å likelyoverestimatestheactual value ofPsr. 1) Modeling the measuredj-factors forcyclization of small circles typically yieldsPeq 5 500 Å(Shimada and Yamakawa, 1984; Hagerman and Ramadevi,1990; Taylor and Hagerman, 1990; Hodges-Garcia andHagerman, 1995; Kahn and Crothers, 1998). CombiningPeq 5 500 Å with Pd 5 2000 Å then yieldsPsr 5 667 Å,which is only about half of the 1300 Å estimate. This (lessconservative) value assigns an even greater contribution ofslowly relaxing bends to the mean-squared bending of DNAand the reduction of its equilibrium bending modulus belowPd. 2) Ppb for the present DNA moleculeprobably exceedsthe SH prediction, which is obtained by averaging over amuch longer and more varied sequence than that of our200-bp DNA. In fact, the present 200-bp molecule comesfrom a region of pBR322 (position 233 to 432) that wasshownnot to exhibit anomalous electrophoretic mobilitiesin polyacrylamide gels (Stellwagen, 1983), and in electro-phoretic comparisons with different standards of knownlength also in polyacrylamide gels gave no evidence ofabnormal mobility or permanent bend. Thus, our DNA islikely to be somewhat straighter than the average sequencefor which the SH prediction applies, soPpb for our moleculeprobably exceeds the SH prediction. Hence, thePsr 5 1300Å estimated from Eq. 1 by using the SH prediction forPpb

likely represents an upper bound.We previously analyzed the effects of imposing asmall

uniform strain on a chain of twisting (or bending) springs,each of which fluctuates between two states with differentintrinsic twists (or bends) and torque constants, and exhibitsaverage equilibrium twistfo (or benduo) (Delrow et al.,1997). When theintrinsic bends of the two states differsignificantly, this model predicts that 1) asmall uniformbending strain (u–uo per spring) shifts the conformationalequilibrium to orderu–uo, so that 2) the bending rigidity

declines from the dynamic (frozen equilibrium) valueAd tothe equilibrium valueAeq on the time scale of the equili-bration process. This model accounts for both the observedalteration of the average secondary structure by imposedbending strain (Heath et al., 1996a), and the present findingthat Ad substantially exceedsAeq. When the two statesexhibit practically the sameintrinsic twist, but differenttorsion constants, asmall imposed twisting strain (f–fo perspring) doesnot alter the conformational equilibrium toorder f–fo (but instead to order (f–fo)

2), so the equilib-rium remains nearly frozen and doesnot reduce the equi-librium torsion constant significantly below the dynamicvalue (Delrow et al., 1997). This model accounts for the(near) equivalence of the dynamic and equilibrium values ofthe torsion constant, which is evident in Fig. 1, wherein bothequilibrium and dynamic methods yield comparable results.These considerations suggest that the two interconvertingstates exhibit significantly different intrinsic bends, but sim-ilar intrinsic twists.

Domain size

From the upper bound,Psr 5 1300 Å, we estimate a lowerbound for the average domain size,m, of the coherently bentconformation in the following way. Each domain of thecoherently bent state is assumed to exhibit a fixed planarbend (u1 per bp). The total bend per domain is then mu1. Forsimplicity, the duplex filament is divided into sections ofmbp. Within each section an all-or-none equilibrium betweencoherently bent (b) and straight (s) conformations prevails,that is, s^ b, with associated equilibrium constantK 5f b

o/f so, where f b

o 5 K/K 1 1 and f so 5 1/(K 1 1) are the

equilibrium fractions of the time that each section spends inits b and s states. This equilibrium is illustrated schemati-cally in Fig. 8. To estimate 1/Psr, the contributions ofdynamic bending must be omitted, so every section exhibitseither its intrinsically straight or coherently bent conforma-tion. The total contribution of slowly relaxingfluctuationsin the s^ b equilibrium toPsr is estimated by omitting thecontribution of that equilibrium to theaveragecurvature,which does not relax and must be counted as part of thepermanent bend contribution toPpb. We adopt the approx-imate expression given in Eq. A6 of Appendix A. WhenPsr

andu1 are known, Eq. A6 yields aminimumsolution form

FIGURE 8 Diagram of straightN bent equilibrium in domains.



as a function ofK, which provides a lower bound estimateof the domain size. The thresholdu1 required to stabilize thealternative secondary structure, which is induced by thebending strain in small circles (cf. Fig. 1), must lie between360/3605 1.0 and 360/2475 1.46°/bp. The largest esti-mates ofPsr andu1 yield the most conservative (smallest)estimate ofm. After Psr 5 1300 Å andu1 5 (1.46)p/18050.0254 radian are substituted in Eq. A6, the values ofmcorresponding to a series of assumedK values are deter-mined numerically via a grid search. Various assumedKvalues and their corresponding solutions form are indicatedin Table 3. The lower bound value is [m]min 5 55 bp, andoccurs betweenK 5 0.4 andK 5 0.6. There is no solutionfor K & 0.07, in which case there are evidently too few bentdomains to reducePsr from infinity to 1300 Å.

The present data imply a minimum average domain sizeof several turns. IfK & 0.10, as is not unlikely, then theactual domain size exceeds 100 bp.

Nature of the coherently bent state

Because the coherent bend of the alternative curved stateextends over several turns, it is almost certainly superhelicalrather than planar. The question arises whether this bend orsuperhelicity is directional or, equivalently, phase-locked tothe azimuthal orientation of the duplex. The meaning of thisquestion is illustrated in Fig. 9. If this curved state isidentified with that induced by the bending strain in smallDNA circles (N # 250 bp), then its curvature isnot direc-tional andnot phase-locked to the azimuthal orientation ofthe filament, because the uniform mode of azimuthal spin-ning is definitelynot frozen out in the 181-bp circles studiedby Heath et al. (1996a). We suspect that this kind of non-phase-locked superhelical curvature is the result of incom-mensuration and consequent frustration manifested by dif-ferential tensions and compressions among the foursubfilaments (two base columns and two backbone chains)of DNA. When such internal stresses are relieved by non-

linear (nonharmonic) restoring forces, quasi-periodic varia-tions in the structural parameters of successive basepairswill generally arise and cause a superhelicity that can betranslated anywhere along the helix, or equivalently canadopt any phase angle with respect to the azimuthual ori-entation of the duplex filament, at nearly constant potentialenergy.

APPENDIX A

An approximate expression for Psr

With each domain ofm bp we associate a bond vectorbj, j 5 1, . . . `,which has the lengthbs 5 mh, if the domain is in the straight (s)conformation, but is somewhat shorter if the domain is in the bent (b) state.When in the b state the entire domain is assumed to bend uniformly in asingle plane, which contains any entering or exiting straight section, abouteither of which the plane can be rotated by the full 2p provided the othermoves as the plane rotates. The angle between the tangent vectors at thebeginning and end of a bent domain ismu1. The bond vectorb of a bent

TABLE 3 Domain sizes for different values of K, whenPsr 5 1300 Å

K m (bp)

0.07 1870.08 1490.10 1160.20 710.30 600.40 560.50 550.60 560.70 580.80 601.00 652.00 1023.00 1405.00 18710.00 220

FIGURE 9 Model of intrinsically superhelical linear DNA. The localsymmetry axis of the DNA is held fixed at the position of the dotted line,while everywhere the duplex undergoes a uniform azimuthal rotationaround that, like the rotation of a speedometer cable in its curved sheath.If the variation of the potential-of-mean-force with phase angle (f) of thatazimuthal rotation over the interval 0 to 2p is small compared tokBT, thensuch a rotation can occur freely during Brownian motion and the azimuthalorientation of the duplex is not phase-locked to the orientation of thesuperhelix. In that case, the superhelical curvature is nondirectional. How-ever, if the potential-of-mean-force exhibits one or more barriers greaterthankBT asf varies from 0 to 2p, then such a rotation will be substantiallyhindered during Brownian motion and the azimuthal orientation of theduplex will be at least partially phase-locked to the orientation of thesuperhelix. In that case the superhelical curvature is directional.



domain is the chord spanning the arc of the bend, which has lengthbb 5(h/u1) z 2usin(mu1/2)u # mh. This bond vector of the bent state makes anglesmu1/2 with both the beginning and ending tangent vectors of the arc. Weassumethat relatively minor error is incurred by 1) averaging the bondvector lengths separately from the angular configurations of those samebond vectors, and 2) neglecting the correlations between adjacent pairs ofbends that occur in this model. Under these assumptions, this model shouldbehave as if its effective bending potential were isotropic and centrosym-metric, andPsr takes the form

Psr 5ûbu&

1 2 ^cos@bm 2 ^bm&#&(A1)

where

ûbu& 5 f so bs 1 f b

o bb

5 mh/~K 1 1! 1 ~K/~K 1 1!!~h/u1!2usin~mu1/2!u

(A2)

is the average bond vector length, which is computed usingf so 5 1/(K 1

1) andf bo 5 K/(K 1 1). The quantitybm is the actual angle betweenbj and

bj11, which depends on the states of thej and j 1 1 domains, andbm& isthe average ofbm, which contributes toPpb, and must be subtracted toisolate the contribution of fluctuations in the s b equilibrium. Theaveraging is performed over all four joint states of thej andj 1 1 domains:1) sj sj11; 2) sj bj11; 3) bj sj11; and 4) bj bj11, which occur with therespective probabilities 1)f s

o2 5 (1/K 1 1)2; f so f b

o 5 K/(K 1 1)2; 3) f bo f s

o

5 K/(K 1 1)2; 4) f bo2 5 (K/(K 1 1))2; and exhibit the respectivebm angles:

1) 0; 2)mu1/2; 3)mu1/2; 4)mu1ucos[f/2]u; wheref is the dihedral angle bywhich the plane of thej 1 1 bend is rotated with respect to that of thejbend. Expression 4) is valid whenmu1 is not too large. Before proceeding,we preaverageover f to obtain

ûcos@f/2#u& 51

p E0

p

df cos@f/2# 5 2/p (A3)

For simplicity the factorucos[f/2]u is replaced by 2/p in the sequel. Then,averaging over the four possible states at each bend yields

^bm& 5 S 1

~K 1 1!2D z 0 12~K/~K 1 1!2!mu1

2

1 S K2

~K 1 1!2D~2/p!mu1

2

5 mu1~K 1 2K2/p!/~K 1 1!2 (A4)

and

^cos@bm 2 ^bm&#&

5~1/~K 1 1!!2HcosFmu1~K 1 2K2/p!

~K 1 1!2 G12K cosFmu1S0.52

K 1 2K2/p

~K 1 1!2 DG1K2cosFmu1S2

p2

K 1 2K2/p

~K 1 1!2 DGJ (A5)

Inserting Eqs. A2 and A5 into A1 yields

Psr 5 H mh

K 1 11 S K

K 1 1DSh

u1D2UsinFmu1

2 GUJ4H1 2 ~1/~K 1 1!!2ScosFmu1~K 1 2K2/p!

~K 1 1!2 G12K cosFmu1S0.52

K 1 2K2/p

~K 1 1!2 DG1K2cosFmu1S2

p2

K 1 2K2/p

~K 1 1!2 DGDJ (A6)

APPENDIX B

Distribution of tR values

A rough estimate of the distribution oftR is obtained by making thefollowing assumptions, which though not strictly valid, may be reasonableapproximations on average. When the different kinds of bends are appliedin succession, 1) the common permanent bend reduces the end-to-enddistance of the rod slightly fromL to lpb by the factorlpb/L; 2) any givenslowly relaxing bend further slightly reduces the end-to-end distance tolpb,sr by the ratio lpb,sr/lpb 5 lsr/L, which is the same as if that slowlyrelaxing bend were applied directly to a rigid rod of lengthL to yield lsr;3) any given dynamic bend acting on an intrinsically curved filament withend-to-end distancelpb,srfurther slightly reduces the end-to-end distance tolpb,sr,dby the ratiolpb,sr,d/lpb,sr5 ld/L, which is the same as if that dynamicbend had been applied directly to rod of lengthL to yield ld. For a moleculewith the common permanent bend and a given slowly relaxing bend, it isassumed that the end-over-end tumbling time is given approximately bytR

} ^lpb,sr,d3 &d, wherein the averaging is takenonly over the dynamic bends.

Making use of the relations assumed above yields

tR } ~lsr/L!3~lpb/L!3^ld3/L3&dL

3 (B1)

By numerical comparison with the results of Hagerman and Zimm(1981), we have confirmed that^ld

3/L3&d is roughly equal toRc 5 tR(wc)/tR(rod), wheretR(wc) is the tumbling time computed for a wormlike coilwith no intrinsic curvature, andtR(rod) is that computed for the straightrigid rod, whenL/Pd # 1.36. In Eq. B1 the effect of both dynamic bendsand the common permanent bend is to reducetR by the same factor for allmolecules. Equation B1 can be written astR/tR

o } (lsr/L)3, wheretRo }

(lpb3 /L3)^ld

3/L3&d L3 is the same for all molecules. Hence, the probabilitydistribution of relativetR values is the same as the probability distributionof relative lsr

3 . The probability distributionP(lsr), is taken to be that of awormlike chain (without intrinsic curvature) with contour lengthL 5 680Å and persistence lengthPsr 5 1500 Å. That is the approximate valuereckoned from Eq. 3 by usingPtot 5 500 Å,Ppb 5 1370 Å, andPd 5 1700Å, and will prove to be approximately self-consistent, in the sense that thefinal estimate,Psr 5 1300 Å, is not very different. The protocol forsimulating chain configurations, which uses von Neumann’s method formodeling a continuous random variable (Sobol’, 1994), is described else-where (Schurr and Fujimoto, submitted for publication). The simulateddistribution is divided into fifths of the total population. The uppermostfifth spans a range of end-to-end distances fromlsr 5 L down to about thelsr value corresponding to the maximum inP(lsr). The measuredtR of theterminal relaxation process inS(t) is identified withtR1 of this uppermostfifth of the population, so we settR1 5 tR. It is then assumed that for eachother fifth (j 5 2, . . . 5) of the population,tRj/tR1 5 ^lsr

3 &j/^lsr3 &1, where^lsr

3 &j

is the mean cubed end-to-end distance for thejth subset of the totalpopulation. In this way the total population is subdivided into five com-ponents, each with its appropriately scaledtRj. TheFn(t) functions in Eq.



7 are reckoned separately for each fifth of the population by using theappropriateDRj 5 1/6tRj, and averaged together.

This work was supported in part by National Science Foundation GrantMCB 9607344.

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dynamic bending rigidity of a 200-bp dna in 4 mm ionic...

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