J . Phys. Chem. 1994,98, 8507-8518
8507
Field-Dependent 23NaNMR Relaxation of Sodium Counterions in Ordered DNA Johan Schultz, Bo Andreasson, Lars Nordenskiold,’ and Allan Rupprecht Division of Physical Chemistry, Arrhenius Laboratory, University of Stockholm, S - 106 91 Stockholm, Sweden Received: February 1 , 1994; In Final Form: May 9, 1994”
The quadrupolar NMR relaxation of 23Na+counterions has been studied in a solid sample of macroscopically oriented D N A fibers which was equilibrated a t relative humidities of 95 and 98%. The equilibrations resulted in a sample of relatively high water contents, corresponding to approximate distances between the D N A surfaces of 1.1 and 1.3 nm, respectively. Using a combination of relaxation experiments, including two-dimensional spin echo and two-dimensional double quantum quadrupolar echo techniques, the spectral densities Jo(O), J l ( w o ) , and J2(200) have been determined a t different orientations of the sample with respect to the external magnetic field. The high-frequency spectral densities, Jl(w0) and J2(2wo), were also determined a t two and four different magnetic field strengths, respectively. It was found that they are largely determined by fluctuations of the quadrupolar interaction that occur on a time scale of nanoseconds. The results indicate that the main contribution to Jl(w0) and J2(2wo) originates from local motions in the vicinity of the DNA molecule. Assuming that this contribution can be divided into two contributions, one fast (assumed to be a constant frequency-independent term) and one slow (assumed to be governed by a Lorentzian function), the experimental frequency dependence could be fitted. The effective correlation time for the slow local motion is in the range of 2-3 ns, depending on water content. It is suggested that this slow local motion is due to the relative motion of the sodium counterion in the vicinity of a charged phosphate group, caused by local diffusion of the sodium counterion and/or motion of the phosphate group itself.
Introduction Quadrupolar N M R of counterions with I > 1 has been extensively used to study counterion-DNA interactions in solutions of double helical DNA. Although the nucleus of choice most commonly has been 23Na+( I = 3/2),1-3 other counterions such as 7Li+( I = 3/2),39K+ ( I = 3/2), z5Mg2+( I = s/2), and 43Ca2+ ( I = 7/#4have also been studied. The origin of the interest and relevance of these investigations has its background in the importance of the long-range electrostatic interactions for determining many of the physical properties of this important biological polyelectrolyte in s ~ l u t i o n It . ~is~ also ~ of importance to elucidate to what extent other more specific effects such as localized interactions (oftencalled “site binding”) may be involved in the interactions between counterions and DNA. In addition, DNA, being a long stiff rodlike macroion, can serve as a good experimental model system for testing different theories of cylindrical polyele~trolytes.~JOne of the advantages of N M R is the possibility to obtain dynamic information on the counterionDNA interaction from nuclear spin relaxation studies, an advantage that has been utilized in some recent 23Na investigat i o n ~ .In~ the ~ ~ work by van Dijk et al.,3 field-dependent 23Na relaxation measurements were performed on semidilute DNA solutions. Recently, different forms of oriented DNA systems have received considerable attention and a number of investigations aimed at characterizing the behavior and physical properties of these kinds of systems have been published.9-16 The importance of these studies, which were mainly performed on model systems, is motivated by the fact that DNA in biological systems generally is arranged in an extremely compact and often ordered form, as e.g. in viruses, bacterial nuclei, chromosomes, and sperm heads. This interest has also stimulated quadrupolar ion N M R studies of oriented anisotropic DNA systems exhibiting quadrupolar splittings. In two papers, Strzelecka and Rill studied liquid crystalline phases obtained from concentrated solutions of DNA from nucleosome core particles (-150 base pairs), using 23Na NMR.17J8 Spin-lattice relaxation times and quadruplar split~
a
Abstract published in Advance ACS Abstracts, July 1, 1994.
0022-3654/94/2098-8507$04.50/0
tings were measured as functions of temperature, DNA and salt concentration. These spin relaxation measurements were performed in order to obtain static information on the counterionDNA interaction and information on the phase behavior of the systems, while no dynamic information on the ion-DNA interaction was extracted. Schultz et al.I9 studied the quadrupolar splittings of 7Li, 23Na, and l33Cs counterions in oriented fibers, prepared by the wet spinning method.20.21 Einarsson et al.,22 from this laboratory, measured the anisotropic Li+ counterion diffusion in oriented LiDNA fibers by means of the N M R pulsed field gradient spin echo self-diffusion method. Furthermore, the mechanism of the 7Li and 133Csspin relaxation in oriented Li- and CsDNA fibers was investigated in another c o n t r i b ~ t i o n enabling ,~~ some results on the dynamics of the interaction of these ions with DNA to be extracted from the quadrupolar relaxation. Detailed information could, however, not be evaluated from these measurements, due to the complication of contributions from multiple relaxation mechanisms. After the submission of the present work, Leyte and coworkers24 published a study on the field-dependent 23Na relaxation in liquid crystalline DNA which has considerable similarity with our investigation. The quadrupolar ion N M R method as applied to polyelectrolyte solutionsz5 is based on the fact that for a nucleus with a large quadrupole moment such as 23Na, the fluctuating interaction between the nuclear quadrupole moment and the field gradients, caused by surrounding charges at the site of the nucleus, will be the totally dominating mechanism for the spin relaxation. Direct measurements of the relaxation rates from inversion recovery experiments and/or line widths then enable information on the counterion-polyion interaction to be extracted. This is due to the fact that counterions associated with the polyelectrolyte will experience larger field gradients and fluctuations of the quadrupolar interaction that occur on a slower time scale, as compared to the counterions in the bulk aqueous phase outside the macroion. In anisotropic systems the static quadrupolar interaction is not averaged to zero on the N M R time scale, resulting in a quadrupolar splitting of the N M R resonance signal, the measurement of which may yield valuable information on the 0 1994 American Chemical Society
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counterion-polyelectrolyte interaction.Z5 The extraction of relaxation rates in these kinds of systems is, however, less straightforward than in the case of isotropic solutions, due to the inhomogeneous contributions to the relaxation and line width caused by large magnetic field inhomogeneities and inhomogeneities in the sample such as deviations from perfect macroscopic orientation of the system.26 In order to obtain dynamic information from the quadrupolar relaxation of counterions in oriented systems, Halle, and coworkers have recently developed sophisticated pulse techniques that make it possible to extract the homogeneous contribution to the relaxation of quadrupolar nuclei in these kinds of systems.2629 The relaxation rates are determined by the three spectral density functions Jo(O), J ~ ( w oand ) , J~(2wo).where wo is the Larmor frequency, and it is these functions that contain the interesting dynamic information on the counterion motions in the system. Halle and co-workers have applied these methods in 23Na counterion relaxation studies of lyotropic liquid crystals composed of charged amphiphilic molecules, sodium counterions and water, which form different cylindrical aggregates of hexagonal ordering.30Jl From these investigations it was revealed that the highfrequency spectral densities, J l ( w 0 ) and J z ( ~ w o )were , mainly determined by surface diffusion of the counterions in the vicinity of the ordered aggregates. On the basis of a theory of spin relaxation by translational diffusion in ordered the counterion surface diffusion coefficients in the systems could be extracted. In the present paper we have applied the above-referenced pulse methods to study the 23Na relaxation of the sodium counterions in macroscopically oriented DNA fibers of relatively high water contents, where the DNA is hexagonally oriented and in the B-form. The sample that was investigated was equilibrated at relative humidities (RHs) of 95 and 98%, corresponding to approximate axial DNA-DNA distances of 3.1 and 3.3 nm, with an aqueous phase between the DNA surfaces with a thickness of approximately 1.1 and 1.3 nm, re~pective1y.l~For 23Na,with a nuclear spin of I = 3/2, the spectrum is a triplet with a quadrupolar spitting VQ, defined by the frequency separation between the satellite peaks and the central peak. Here we have measured the transverse and longitudinal relaxation of the satellite and central peaks for different orientations of the sample (average orientation of the DNA helix axes) with respect to the magnetic field, at different magnetic field strengths. The measurements were specifically aimed at obtaining information on those dynamic processes of the fluctuating quadrupolar interaction, experienced by the ions interacting with DNA, that determine the highfrequency spectral densities, J1( W O ) and J2( 2 ~ 0 ) .These motions generally occur on a time scale of nanoseconds which is comparable to the inverse of the nuclear Larmor frequency. We also performed measurements of the zero-frequency spectral density Jo(0)which, apart from the motions affecting the high-frequency spectral densities, also reports on very slow motions on the time scale of microseconds and slower. Furthermore, the static and dynamic properties that determine the appearance of the spectrum have been discussed.
Experimental Section The Samples. The NaDNA sample was prepared with a wet spinning method from calf-thymus DNA (Pharmacia).20v21The NaCl content in the samples can be adjusted by equilibration of the spun deposit in a 75/25 (v/v) ethanol-water solution, containing a given excess of NaCI. In the present study, noexcess NaCl was used in the bath, resulting in a NaDNA sample containing less than 0.1 g of NaCl per 100 g of dry sample.33 During the subsequent drying procedure, the DNA fibers merge, forming a film of oriented NaDNA. This film was then folded several times back and forth perpendicularly to the helix axis, into a package of oriented DNA. The water content, which
Schultz et al.
TABLE 1: *3Na Quadrupolar Splitting, YQ, (0, = 0') and the Approximate Interhelied Distance, d, as a Function of RH RH (76) VQ ( k H 4 da (A) 85 95 98
18.2 1.30 0.175
23.8b 31.0 32.8
0 The interhelicaldistanceswere determined by Lee et al.3 on NaDNA fibers containingabout 1% NaCl which were prepared in our laboratory. Interpolated value.
determines the distance between the parallel DNA helix axes as well as the DNA double helix structure (A or B form), was controlled by equilibration at room temperature over saturated solutions of KCl (85% RH), Na2S04 (95% RH), and KzS04 (98% RH). The equilibration lasted for several weeks. Lee et al.13 studied the relationship between the R H used in the equilibration and the interhelical distance in wet-spun oriented DNA samples. For NaDNA containing 1% excess NaCl, the distances between centers of adjacent DNA double helices, d, and corresponding RHs are presented in Table 1. The water content of the sample is slightly dependent upon the salt content so that a higher salt content yields a higher water content.33As a consequence, the interhelical distances in our sample are somewhat smaller than in Table 1. When the sample is equilibrated in RHs C 92%, the NaDNA helix structure is predominantly in the A-form, otherwise it is in the B-form.34 In order to keep the watercontent of thesampleconstant, thesample was always stored in a desiccator together with the N M R tube between the experiments. During the experiments, the sample was placed in a plastic holder for mechanical support and then inserted into the N M R tube which was sealed with an airtight plug. All sample handling took place in a humidity-controlled chamber. NMR Methodology. Spectrometer Characteristics. The N M R measurements were performed on a Bruker MSL-200 spectrometer (4.69 T, 23Naresonance frequency 52.90 MHz), on a JEOL a-400 spectrometer (9.38 T, 23Naresonance frequency 105.70 MHz), and on a Bruker MSL-90 spectrometer with an electromagnet with adjustable magnetic field strength. All experiments were performed a t 290 f 0.5 K. In order to avoid temperature gradients over the sample, the temperature was controlled with an air flow, thermostated by a water bath. The 23Na90' pulse lengths were typically 5-8 ps (MSL-200), 13-16 ps (a-400), and 12-13 ps (MSL-90), which provided sufficiently uniform excitation in thequadrupole-split spectra. The 90' pulse lengths weredeterminedvery carefully by varying the pulselength in several (10-20) single-pulse experiments and combining them in a signal amplitude curve. The 90' pulse length was then fine adjusted by further experiments, so that the maximum amplitude of the Fourier-transformed spectra was determined. The known maximum signal amplitude was then used as a measure when determining the intermediate pulse lengths needed in some experiments (54.7', 45', 109.5'). The two-dimensional (2D) experiments were performed using the digital phase shifter (MSL200) or the anologue phase shifter (a-400). When the MSL-200 and MSL-90 was used, the angle betweeen the magnetic field and the fiber direction of the sample, BLP, could be adjusted by simply turning the tubecontaining the DNAsample to thedesired angle (to an accuracy within at least f2' as seen by the wellresolved quadrupolar splitting, VQ, of a sample that was equilibrated a t 85% RH). When using the a-400, a change in sample orientation was accomplished by opening the N M R tube in the humidity-controlled chamber, and turning the sample 90'. The quadrupolar splitting is a good measure of the water content in the sample19922and was therefore, when resolved, measured before and after the experimental sessions. The difference was always less than 4%. All relaxation measurements aimed at determining J l ( w 0 ) and Jz(2wo) were performed a t least three times and an average of the results was then taken.
Relaxation of N a Counterions in Ordered DNA
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Inversion Recovery Determination of J2(2w0,BLp). In an ordinary inversion recovery experiment, ( R ) ~ - T - ( R / ~ ) ~ - the ~C~, satellite peak intensity for a spin Z = 3/2 in an anisotropic system decays exponentially, the intensity following the relation27
The spectral densities Jk(k00,B~p)contain the dynamic information on the counterions in the system and will be further discussed in the following section. The decay of the central peak is biexponentialZ7and determined by both JI(WO,BLP) and J ~ ( ~ w o , ~ but LP), due to the numerical difficulty in multiexponential fitting, the inversion recovery experiment is only useful for accurate determination of the spectral density J2(2wo,B~p). In overlapped spectra like ours, a modified inversion recovery pulse sequence involving a magic angle detection pulse (IRMA),27 ( R ) ~ - T (54.7'),-acq, is preferable. Now the central peak recovers exponentially with a longitudinal relaxation rate identical to the onegiven above. When thespectrum was reasonably well resolved, we also performed the ordinary inversion recovery experiment on the satellites. The differences were small, always less than 3%, and no systematic difference was observed. The small discrepancies in the values obtained from the two different techniques were probably due to the difficulties in determining the 54.7O pulse lengths. 2 0 Spin Echo Determination of JI(w0,BLp). In a conventional 1D spectrum, the central line is inhomogeneously broadened by magnetic field inhomogeneities but is unaffected by broadening due to sample inhomogeneities. The satellite lines, however, are inhomogeneously broadened both by magnetic field inhomogeneities and by sample inhomogeneities. This precludes a direct extraction of relaxation rates from line widths. There are, however, at least two pulse techniques that can be used to determine the homogeneous central line width. These are 2D spin echo28J5 and triple quantum 2D spin echo.29 In the present study, we used the 2D spin echo sequence: (T/~)~-T-(R)#-Tacq, where the x pulse refocuses magnetic field inhomogeneities. The 2D spin echo experiment yields a homogeneous central line width, AucSE,in the F1 dimension which is obtained from Fourier transformation with respect to T of the peak amplitude of the inhomogeneous central line in the F2 spectrum. The associated relaxation rate is given by36
JI(OO,BLP)can then be evaluated using the value of J2(2w&p) obtained from an inversion recovery experiment as described above. The phase of the refocusing pulse was varied according to the "Exorcycle" ~cheme.3~ RcSEwas obtained from a Lorentzian fit to the homogeneous central line in the F1 cross section spectrum. Remaining satellites in the F1 cross section spectrum were only observed when the sample was equilibrated at 98% RH and BLp = 90°, Le., with the smallest value of VQ. In that case we performed a curve fit to a sum of three Lorentzians. The magnetic field inhomogeneity was =40 Hz at 4.69 T as seen from the difference - A@ ,'' JI(w0,BLp) can also be obtained using the Jeener-Broekaert pulse s e q ~ e n c e ,(x/2)#-~~-(~/4)#*=,r~-(~/4)11-acq, ~~.~~ with the phasecycle 4 = 0, x/2, R,3.rr/2.t7J0 The quadrupolar relaxation rate, R ~ Qof, the satellite lines is then given by R ~ = Q 2RcSE.27-30 The fixed delay time, 71, should be set to 1/4VQ in order to maximize the conversion of dipolar to quadrupolar polarization.27 Further, the acquisition delay should be set to a multiple of 1/ 2 u ~ in order to avoid baseline offsets due to first-order phase corrections. As a complement to RcSE,we thus determined R ~ Q where it was possible (Le., when the spectrum was reasonably well resolved). The agreement between the two experiments was good, the differences never exceeded 4% and no systematic difference was observed.
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2 0 Double-Quantum QuadrupolarEcho Determination of Jo(O,BLp). The homogeneous satellite line width, AuPm, is proportional to the transverse relaxation rate, RPm, and is given byz6
As was mentioned above, the satellite peaks in a conventional 1D spectrum are inhomogeneously broadened by magnetic field inhomogeneities and by sample inhomogeneities. R,homcan, in principle, be measured by a number of different 2D echo experiments,26-29J5the choice of which may depend on the actual system at hand. In the present system VQ is small compared to the homogeneous line widths resulting in a considerable overlap between the satellite peaks and the central line in the ordinary single pulse 1D spectra, particularly when the sample was equilibrated at 98% RH. In this situation, the 2D doublequantum quadrupolar echo (2D2QQE) experiment29was found to be the only experiment that gave satisfactory and reproducible results for the two water contents of our sample, as well as for a test sample of a nematic hexagonally ordered liquid crystal, obtained from a mixture of sodium dodecyl sulphate, decanol and ~ a t e r . * ~ * ~ ~ At first, however, we applied the single-quantum 2D quadrupolar echo (2DQE) e ~ p e r i m e n t 2 ~on9 ~our ~ DNA sample, but the slices in the F1 dimension (at thesatellite frequency in F2) suffered severely from the overlap and the obtained homogeneous satellite line widths were unreasonably small. The F1 slices at the central peak in F2, however, showed the same homogeneous line width of the central peak as the 2D spin echo experiment. In an attempt to make this failure of the 2DQE pulse sequence understandable, we applied the so-called 2D even rank quadrupolar echo (2DERQE) pulse sequence,29 a combination of a double-quantum filter and the single-quantum quadrupolar echo sequence, on the sample equilibrated at 95% R H at OLp = 0'. The double-quantum filter suppresses the central peak and makes the satellites appear in antiphase, thus reducing the overlap (still, the satellite peaks overlapped each other to some extent). Now the homogeneous satellite line width (found in the F1 dimension at the satellite frequency in F2) was reasonably close to the result gained from the use of the 2D2QQE pulse sequence, indicating that the failure of the 2DQE pulse sequence was due to the severe overlap in the spectra in the F2 dimension. In any case, it should be clear that theJo(0) valuesaresubject tosomeuncertainty. As an additional check we performed the three types of experiments on the nematic test sample. Within experimental uncertainty, the same result for the homogeneous satellite line width was obtained from these three experiments as in an earlier work.35 The 2D2QQE experiment consists of the following pulse sequence:29 (T / ~ ) ~ , - T - ( T / ~ ) ~/2-(B),,-t ,-~I I 12-( a/2)#,+-acq. When the correct phase cycle is used,29both the broadening due to the inhomogeneous magnetic field and the sample inhomogeneities will be refocused. By choosing the pulse B = 109.5', the optimal situation of maximum central peak and no satellites in the FI cross section spectrum can be realized. In order to obtain pure absorption Fz spectra, OQT (where WQ = 2RVQ)should be set to 180'. The F1 cross section spectrum at all F2 peaks consists of a single line, whose width yields the relaxation rate RPm. The 2D2QQE experiments were performed at one field (4.69 T) only. The pulse 8 was set to 109.5' and WQT = 180°, except in the 98% R H case where VQ < 200 Hz. Here T was set to 1 ms instead of the correct values 3-4 ms, because of the great loss of intensity due to relaxation that otherwise would render the experiment useless. qj3 was cycled according to 43 = m / 4 , with n = 0, 1, ..., 7. d4 and 4Rwere cycled according to 44 = nx/2 and 4R= na/2 with n = 0, 1,2, 3. Finally, = 42. The digital phase shifter was used and 64 or 128 experiments with different delay times T were acquired. Av,hom was obtained from a Lorentzian fit to the homogeneous line in the FI cross section
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spectrum which was zero filled to 1 K. Due to the higher signal to noise ratio, the F1 cross section spectrum a t the F2 central peak was used.
Schultz et al.
to the constraints imposed by the DNA-water interface, this contribution should be larger than for an ion in an isotropic aqueous solution. Second, for an ion associated to the DNA surface, the concentration of ion charges will be considerable and the Theory inhomogeneous ion distribution close to the DNA molecule will contribute to the efg. Third, the effect of the DNA phosphate The relaxation experiments described in the previous section charges, locally and by the electrostatic field they generate, will directly give the different laboratory frame spectral densities, also contribute. provided all the necessary relaxation experiments have been The molecular motions that can cause a fluctuation of the efg performed, so that the different Jk(kq,OLp) can be separated. time correlation function, leading to a contribution to the observed Below we will give the definition of these in terms of the field relaxation rates, may be divided into two classes based on their gradient autocorrelation function so that they, for a givendynamic time scales and length scales over which the relevant dynamic model, can give the desired dynamic information in terms of a processes occur.36*3941 First, we consider local motions, Le., given correlation time. However, we will first briefly discuss dynamic processes that are relevant for an ion associated to the some aspects of the static properties, i.e., the quadrupolar DNA and that are expected to occur on relatively fast time scales. splittings. Here two subclasses of motions can be distinguished. Based on Quardrupolar Splittings. For quadrupolar nuclei with a spin molecular dynamics simulations (MD),42 motions of the water Z L 1 there will be a coupling of the nuclear electric quadrupole molecules in the first hydration layer of the counterion (librations moment, denoted eQ, with the electric field gradients (efg’s) and intermolecular vibrations) are generally believed to be the generated by the surrounding charge distribution at the site of dominating fluctuation of the efg for Na+ ions in isotropicaqueous the nucleus. In highly ordered systems like the present, the solutions. A recent M D simulation of sodium ions in an anisotropic anisotropic nature of the fluctuation of this interaction, caused environment (an inverted micelle), showed that these dynamic by the thermal molecular motions, will result in a residual processes give a significant contribution also in that kind of quadrupolar splitting of the otherwise single resonance line. heterogeneous system.43 The time scale for these motions is Generally for I = 3 / 2 counterions in a system of cylindrical expected to be in the picosecond range. The second type of local symmetry with two site exchange between a bulk phase (where motions is local ion dynamics,a such as diffusion in the the splitting is zero) and a bound state, the quadrupolar splitting, neighborhood of a charged phosphate group, or local DNA UQ, between either of the two outer lines (satellites) and thecentral dynamics at the binding site. These motions are expected to line, is given by39 occur in the range between nano- and picoseconds. The second class of motions involves nonlocal ion dynamics. (4) Here we find the translational diffusion of the ion around (azimuthally, laterally) and out from (radially) the cylindrical where OLp is the angle between the direction of the applied external DNA molecule. These motions and the resulting efg correlation magnetic field and the average orientation of the DNA helix function have been treated in considerable detail by Halle within axes. ( XQ) is the residual quadrupolar coupling constant (qcc) the so-called two-step model for averaging of the quadrupolar averaged over all molecular motions taking place on a time scale i n t e r a c t i ~ (see n ~ ~below), and the model has been applied to Z3Na short compared to the inverse of the quadrupolar splitting which relaxation data from different lyotropic liquid crystal~.30J*~~~ This they average. ThepB is the fraction of counterions associated to class of motions is expected to occur on the nanosecond time scale the DNA molecule within the two-state model. It should be and should be most important for the spectral densities J I and noted that if more than one binding site with different qcc’s are considered, the above equations will be modified a c ~ o r d i n g l y . ~ ~ Jz. Finally, dynamics that are expected to be important contributors to the zero-frequency spectral density are those slow Spectral Densities and Counterion Dynamics. The timemotions (generally in the microsecond regime) that are of dependent fluctuations of the quadrupolar interaction will induce significant amplitude. Here we can mention slow exchange of relaxation in the spin system, the effectiveness of which is sodium ions between binding sites with different qcc’s, counterion determined by the different spectral densities that contribute to diffusion along the possibly curved DNA molecule, and migration the relaxation and line width in the different relaxation experiof counterions between domains of varying helix orientation ments as described in the Experimental Section above. The relative to the magnetic field. (generally orientation dependent) laboratory frame spectral Dynamic Model. The above considerations naturally lead to densities are given byz6 the introduction of a two-step model for the averaging of the quadrupolar i n t e r a ~ t i o n . ~ Due ~ , ~ ~to. ~the ~ assumption of the difference in time scale, and also on behalf of the statistical independence of the two classes of motions, the spectral densities may be decomposed into additive and independent contributions where Gk(t$Lp) is the efg time correlation function. The explicit from each of them36 expressions for the different Jk depend on the dynamic model chosen to describe the molecular motions that give the fluctuation of theefg experienced by the 23Nanucleus. Furthermore, motions occurring on varying time scales will contribute differently to the various spectral densities. This means that J1 and JZare sensitive Hereloc denotes the contribution from local processes and s that to relatively fast motions while Jo will be sensitive also to very from slow processes. In addition there is also a contribution to slow motions. Below, we will thus start with a discussion of ion the spectral density from those counterions that are not associated dynamics of varying time scales in heterogeneous systems, with with DNA, but rather “free” in the aqueous bulk phase, assuming particular emphasis on sodium ions associated with DNA,1s3s40 the N M R two-state model.’ However, this term is small and that in principle may contribute to the relaxation process. completely negligible in the present systems where the fraction Origin of the Field Gradients and Their Fluctuations. Before of “free” counterions can be neglected (see below). In Halle’s starting this discussion of dynamics, it is of relevance to briefly theory of spin relaxation of quadrupolar counterions due to consider the possible origin of the efg that a sodium nucleus in translational diffusion in locally ordered fluids, explicit expressions an oriented DNA system may experience. A considerable portion for Jk,a are given.32 This theory was recently applied to the of the efg is expected to be due to the deviation from a perfect interpretation of the high-frequency spectral densities, J I and Jz, symmetry of the water dipoles surrounding the counterion. Due of Z3Na relaxation in hexagonally aligned lyotropic liquid
Relaxation of Na Counterions in Ordered DNA crystalline phases.31J6 Since this system and our present DNA system both are composed of hexagonally oriented cylindrical charged polyelectrolyte rods with ions and water between them, this theory is the natural starting point for an interpretation of our data. For simplicity, we will restrict ourselves to the rigid rod version (no DNA c ~ r v a t u r e )although ,~~ this is not necessary and the general theory31932 would not alter our main conclusion as to the applicability of this model to ordered DNA (see below). In the model it is assumed that the spectral density J k , l N is frequency independent (extreme narrowing) and independent of k (weak anisotropy of the local motion). The resulting expression for J k , , contains contributions from both azimuthal and radial translational ion diffusion, but the contributions from the radial diffusion was shown to be of negligible magnitude for the present typeof ~ystems.3~ The resulting spectral density expression takes the form
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Here, (Xf,R) is a root mean square residual qcc averaged by the is the correlation time for slow local fast local motions and motions. The approximations within the above model as formulated by eqs 9 and 10, and their implications for the parameters (particularly TI,,,) obtained in the fitting of the experimental data, will be further discussed below.
Results and Discussion
Static Properties. In Figure 1 the conventional single-pulse 23Na spectra of the NaDNA sample equilibrated at three different RHs and at OLp = 0’ or 90’ are presented. Adherent data on the quadrupolar splitting, Y Q , and the approximate interhelical distancesl3 are presented in Table 1. The VQ’S were determined by performing a curve fit of the spectrum to a sum of three (7) Lorentzians. The magnitude of Y Q is proportional to the secondrank Legendre polynomial (3 cosz OLp - l), demonstrating an whereFk(OLP)is an orientation-dependent factor which may render averaging of the efg due to fast (compared to Y Q - ’ ) motion of the the experimental spectral densities highly dependent on the counterions around a uniaxial DNA system. This was also found macroscopic orientation of the sample with respect to the external in the exploratory work on oriented DNA fibers by Edzes et al.44 magnetic field. The spectral density Jazi is given by the r e l a t i ~ n ~ l , ~ ~In general, the spectra are characterized by relatively small VQ’S and broad satellite peaks which result in overlapping spectra when the sample is equilibrated at 95 and 98% RH. Therefore, in order to illustrate the discussion of the satellite peak line shape, where (XR)is the root mean sqaure residual qcc averaged by we have included the spectra for a sample equilibrated at 85% the local motions. The is the correlation time for the azimuthal R H which show a negligible overlap. When equilibrated at 85% diffusion which can be related to a surface diffusion constant, D,, RH, the NaDNA is predominantly in the A-form, while when through therelation D,= R2/(4~azi), whereR is a surfacediffusion equilibrated at RHs > 92%, the NaDNA is exclusively in the radius. PB is the fraction of counterions associated with the B - f ~ r m The . ~ ~large difference in YQ resulting from equilibration polyelectrolyte surface (“bound”), assuming the N M R two state at 85% R H compared to 95 and 98% RH, is mainly due to this model, the remaining fraction ( p ~ 1- ) being “free” in the aqueous difference in the DNA structure.19 Another apparent difference bulk phase outside the macroion. is that the line widths at half amplitude of the satellite peaks are As will be shown below, the above treatment is not capable of much broader when the sample is equilibrated at 85% RH. It reproducing the experimental data in the present ordered DNA should be noted, however, that qualitatively the effects discussed system. The azimuthal diffusion is rather expected to give a below on the satellite line shape are the same at all RHs but are negligible contribution to the spectral densities. As will be most clearly visible when the sample is equilibrated at 85% RH. discussed in more detail below, the results instead clearly indicate The origin of the inhomogeneous broadening of the satellite that the observed frequency dependencies of the spectral densities peaks may be divided into three different classes:36 (1) The J I and J2 are due to a frequency dependence in the spectral density magnetic field inhomogeneity, ( 2 ) an orientational distribution corresponding to the local motions. On account of the two of local director orientations, and (3) a distribution of motionally different subclasses of local motions discussed above, it is averaged qcc’s. The inhomogeneous broadening due to the reasonable to make an additional two-step model for the local magnetic field inhomogeneity is symmetric and affects the central averaging of the Z3Narelaxation in this ordered system. Acline as well. As mentioned in the Experimental Section, this cordingly, we assume that J I , can be written3 contribution is relatively small (-40 Hz at 4.69T). Contributions from the second and third sources are consequences of different (9) types of sample inhomogeneities. At this point it is convenient to introduce the time scale T on which the spread in W Q (AWQ= Here, f stands for fast local motions and s for slow local motions. ~ ~ A Vassociated Q ) with the sample inhomogeneities is averaged by molecular motions. As is most clearly seen in the spectra for It has been assumed that the orientational dependence in JlX is the equilibration at 85% RH, the line shapes of the satellite peaks small and can be neglected. Furthermore, it is assumed that the are asymmetric at OLP = 0” and 90’. This is a typical feature dependence on k , except the trivial one indicated by the argument for a static distribution of director 0rientations~~9~5 (thedistribution of the different J’s above, can be neglected as well. The physical of director orientations is expected to be static in these solid picture behind the model is that the quadrupolar interaction first samples). Thus T >> ~ / A w and Q the observed satellite peak is a is averaged by fast (compared to the inverse of the Larmor frequency), local and slightly anisotropic motions to a givenvalue. superposition of peaks, each corresponding to a different value This gives the frequency-independent contribution Jl,,f in eq 9. of W Q . Since the value of the second-rank Legendre polynomial ( 3 cos2 OLp - 1) is at an extreme point when BLp = 0’ and 90°, Thereafter, slower local motions (also anisotropic) on the time scale of the inverse of the Larmor frequency average the interaction the result will be an asymmetric broadening at those angles. The amount of broadening is, however, different for the two angles to a nonzero value. In the subsequent fitting of the experimental due to the difference in the spread in OLD (the local helix spectral densities obtained at different magnetic fields, an explicit expression for the slow local and frequency-dependent spectral orientation) between the two angles which stems from the fact that the perpendicular orientation is two-dimensional while the density is needed. In accordance with previous work, particularly the frequency-dependent 23Na relaxation in isotropic DNA parellel orientation is one-dimen~ional.~~ Thus, the satellite peak is subject to a more severe inhomogeneous broadening when OLp solutions,3 we assume that this can be described by a Lorentzian spectral density function. The slow local spectral density then = 0’. In addition, the spread in OLD causes a small shift of the takes the form3336 satellite peak to a smaller quadrupolar splitting.36
Schultz et al.
8512 The Journal of Physical Chemistry, Vol. 98, No. 34, 1994
40000
30000
10000
20000
0
-10000
COO00
HERTZ
40000
30000
20000
10000
0
-10000
-20000
-30000
-30000
-40000
-40000
HERTZ
6000
4000
2000
0 HERTZ
-2000
-4000
-6000
BOO
BBO
600
BOO
0
2000
4000
6000
400
-2000
HERU
200
0
-200
-6000
-4000
-400
-600
-401
-800
-600
L
HERTZ
400
200
0 HERTZ
-200
-800
Figure 1. Single-pulse 23Naspektra at 4.69 T of macroscopically oriented Na(B)DNA fibers equilibrated at three different RHs and at two sample orientations relative to the magnetic field, 6Lp: (from top to bottom, a-c on the left and d-f on the right) (a) 85% RH, 6Lp = Oo, vertically magnified 32 times (x32); (b) 85% RH, 6Lp = 90' (X32); (c) 95% RH, 6Lp = 0' (X4); (d) 95% RH, 6Lp = 90° (x4); (e) 98%RH, 6Lp = Oo (xl);and (f) 98% RH, 6Lp = 90' ( X l ) . The central peak is truncated in spectra (a)-(d).
Relaxation of Na Counterions in Ordered DNA
The Journal of Physical Chemistry, Vol. 98, No. 34, 1994 8513
TABLE 2: 23NaSpin Relaxation Rates'
RH (%)
OLP (de&
95
0 41 90
98
0
1.15 R I (s-') ~
2.11 R I (s-') ~
472
395
535 308 345
435 273 315
90 a
i
Rg2WE(s-') 2249 2357 1764 610 583
B( r ) 4.69 RlS(s-') 223 230 247 165 192
9.38
RcsE(SI) 304 309 307 216 228
RIS (s-9 147 167 124 143
RcSE(SI) 195 197 163 170
Estimated uncertainties based on reproducibility: R I ,&2%;RcsEi 2 % , R,*QQE &5%.
L
.
400
200
2000
1000
0
-200
-400
0
-1000
-2000
HERTZ
L
HERTZ
Figure 2. F1 slice at the central peak in the F2 dimension using (a, top) the 2D spin echo, and (b, bottom) the 2D double-quantumquadrupolar echo pulse sequences. The sample was equilibrated at 95% RH;OLp = OO.
If only the magnetic field inhomogeneity and the spread in OLD would contribute to the inhomogeneous broadening, the asymmetry of the satellite peaks at OLp = Oo and 90° would be more marked. That is so since the outer part of a satellite peak (from thesatellite maximum amplitudeand away from thecentral peak) would then be inhomogeneously broadened only by the relatively small magnetic field inhomogeneity. This means that the line widths at half amplitude of the outer part of the inhomogeneous satellite peaks would be only -20 Hz broader than the corresponding line width for the homogeneous satellite peak. A closer look reveals that, e.g., when the sample is equilibrated at 95% R H and OLp = OD (see Figure IC), the outer part of the inhomogeneous satellite line width, measured from the conventional spectrum, is -780 Hz, while the outer part of the
homogeneous satellite line width is 358 Hz (RS2QQE/2r = 358 Hz, see Table 2). This difference of -422 Hz minus -20 Hz (the broadening due to the magnetic field inhomogeneity) is then due to the third source that contributes to the inhomogeneous broadening, namely a distribution of motionally averaged qcc's within the sample which results in a symmetric broadening.36 Thus, the inhomogeneous broadening stemming from this source is -800 Hz. Since this is a static broadening, it follows that 7 >> 1 / 2 ~ 8 0 0 0.2 ms. In a self-diffusion study of the anisotropic 'Li+ diffusion in oriented fibers of LiDNA,464 / 0 0 was found to be 0.09 when the sample was equilibrated at 95% RH. Here Dll is the macroscopic diffusion coefficient in the direction parallel to the average helix axis orientation, and DO is the diffusion coefficient of lithium ions in a dilute aqueous solution. Assuming that this figure is valid also for Na+, and using DO= 13 X 10-10 m2s-l 47yieldsD(Na+)= 1.2 X 10-1°m2s-l. Fromtherelationship between the diffusion coefficient and the root mean square displacement during a time 7,it then follows that the inhomogeneity persists over at least 0.2 pm. At this point it is appropriate to briefly comment on the sample morphology as deduced from microscopy studies on samples from this laboratory equilibrated at low water contents (> J l ( w 0 ) > J2(200),which is the same trend as that in an isotropic system outside extreme narrowing. Furthermore, the magnitudes of J l ( q ) and J~(2wo)decrease as the magnetic field increases (Jo(0)was determined at one magnetic field only, 4.69 T). It should also be noted that the values of the different spectral densities Jz(2wo) at the magnetic field 4.69 T is, within experimental uncertainty, very similar to the spectral densities Jl(00) at twice that magnetic field, i.e., at 9.38 T. This implies no or only a very weak k dependence in the high-frequency spectral densities. The different spectral densities exhibit different orientation dependencies. In general, the orientation dependencies of J l ( w 0 ) and 52(200)are weak while the orientation dependence of Jo(O), especially when the sample was equilibrated at 95% RH, is more pronounced. The weak, opposite orientation dependencies of Jl(w0) and Jz(2wo) practically cancel, rendering RcSE independent of orientation. At 98% RH, J l ( w 0 ) does not exhibit an orientation dependence. In order to make absolutely sure that the orientation dependence is as weak as it appears to be, we performed a series of measurements where we determined Jl(wo)andJ2(2wo)at95%RHand4.69TforOLP=00, 18O,4lo, 60°, 75O, and 90" (not included in Table 3). As expected, J l ( w 0 ) and Jz(2wo) changed monotonously between the values for OLp = 0' and 90" which are given in Table 3. The fact that the zero-frequency spectral density is considerably larger than the high-frequency spectral densities shows that different dynamic processes contribute in the two cases. In the following we will concentrate the discussion on the dynamic processes affecting J I(00) and J2(200), and thereafter we will briefly consider the slower dynamics affecting Jo(0). The High-Frequency Spectral Densities Jl(w0) and Jz(2wo). In order to relate the obtained 23Na relaxation data to molecular
motions, a dynamic model is needed. As was mentioned in the Theory section, the model developed by Halle,32 which was successfully applied to the interpretation of the 23Narelaxation in amphiphilic lyotropic liquid crystals, is a natural starting point for an interpretation of the present data. A comparison of our 23Na laboratory frame spectral densities in Table 3 with the ones obtained in systems of hexagonal lyotropic liquid crystals of cylindrical ~ y m m e t r y ~ ~reveals ~ ~ ~ 3important 3~ differences. The orientation dependence of JI(OO)and J2(200) is both much stronger as well as opposite in the lyotropic liquid crystals compared to the present DNA system. In addition, the magnitude of the spectral densities is smaller in the lyotropic liquid crystals with the exceptionof reference 36 where the system was of a reversed cylindrical symmetry (the counterions were confined inside aqueous rods). In the diffusion model developed by Halle, the dynamics affecting Jl(w0) and Jz(2w0) is divided into fast local motions with a weak anisotropy and an azimuthal diffusion of the counterions (Le., around the cylindrical aggregates). The qualitative difference in the data indicates that the surface diffusion contribution to the high-frequency spectral densities may not contribute significantly to the counterion spin relaxation rates in ordered DNA systems. The magnitude of the surface diffusion spectral densities, Jazi, is given by eq 8 above. This relation is valid for a system of ordered cylinders with negligible curvature but a more general equation is given in the work by Quist et ~ 1 . 3However, ~ the straight cylinder model seems more appropriate in the case of ordered DNA and an application of the more general result would not alter the main conclusion from the present discussion. We will now make an order of magnitude calculation of the maximum contribution from Jazi to the spectral densities in our DNA system equilibrated at 95% RH. In order to do this we need values of the parameters PB,P(xR),and Tazi- P B is given its maximum value of one. The rms residual qcc, ( XR), can be set approximately equal to ~ Y Q ( O Orendering ),~~ ( XR) = 5200 Hz (YQ(OO)= 1300 Hz from Table 1). As discussed in the papers by Fur6 et al. and Quist et al., (XR)is smaller than 4q(0°), but the difference between these two values is not expected to be large.31J6 ~ , , i = R2/4D,can be estimated by using the value D, = 1.2 X 10-10 m2 s-I, given by using the above-mentioned value of the anisotropic parallel 'Li+ diffusion coefficient, 4, obtained from self-diffusion measurements on oriented LiDNA fibers equilibrated at 95% RH.46 The surface diffusion radius, R , is estiamted to be 12 X 10-10 m, which is slightly larger than the DNA double helix radius. The surface diffusion coefficient, D,, is smaller than D estimated above36 which has the consequence that the calculated value of the azimuthal surface diffusion correlation time, ~ , , i (=3 X le9s) is a minimum value. With these values inserted in eq 8 a value of Jazi of less than 1 Hz is obtained (depending on k and W O ) , which compared to the experimental spectral densities in the range 70-270 Hz (Table 3), is negligible. This conclusion holds also when the sample is equilibrated a t 98% RH, where the contribution from Jad can be calculated to be of even smaller magnitude. The main reason for the small contribution from Jazi in the ordered DNA systems, as compared to the systems of hexagonally oriented lyotropic liquid crystals, is the small magnitude of the residual qcc. This fact
The Journal of Physical Chemistry, Vol. 98, No. 34, 1994 8515
Relaxation of N a Counterions in Ordered DNA 300
TABLE 4: Resulting Values of the Parameters from the Nonlinear Fit According to Eqs 9 and 10, Assuming No Orientation Dependence and No k Dependence RH (%)
I
200
95
3 v
Jloc,f/s-'
61 f 1
(rZ/5)pe(xf,~)~/s-'
(8.9 f 0.1) X 10" 2.55 f 0.14
na,s/ns
v
*
98
53 3 (6.0* 0.4) X 10"
2.08 & 0.20
0
5-
100
t 0
i
1.0 l o 9 2.0 1 o9 w l rad s . '
3.0 1 o9
Figure 3. Nonlinear fit of spectral densities obtained when the sample
was equilibrated at 95%RH (open circles, solid line) and 98%RH (closed circles, broken line) to eqs 9 and 10. The result of the fit is presented in Table 4.
indicates that the quadrupolar interaction is effectively averaged by local motions occurring on a time scale comparable to the inverse of the Larmor frequency, which would then make the spectral density due to these local motions the dominant contribution to the experimentally observed spectral densities. Additional support for the hypothesis that the relaxation is dominated by local motions comes from the weak orientation dependence of the spectral densities. Since the azimuthal motion is constrained to occur around one single axis (the DNA helix axis) relative to the external magnetic field, the orientation dependence of this contribution will become large. The orientational constraint on the local motions, on the other hand, is naturally expected to be smaller since different local diffusive motions, occurring in different directions relative to the external magnetic field, are expected to contribute to the averaging of the quadrupolar interaction. Since there is a frequency dependence of the spectral densities ( J l ( w 0 ) # Jz(2wo)) the relaxation process is outside the extreme narrowing implying that, apart from a constant high-frequency contribution, there are also contributions from other slower local motions. As was briefly reviewed in the Theory section, the local motions may be divided into two subclasses: ( 1 ) fast (s), weakly anisotropic motions of the water molecules in the first hydration layer of the counterion, and ( 2 ) slower ( 10-9-10-'1 s ) motions of the counterion such as diffusion in the neighborhood of phosphate groups and motions due to the local DNA dynamics at the binding site. There is no strict theoretical treatment available for an analysis of the local motions in the present kind of systems and we are therefore going to resort to an approach similar to the "model-free" approach of Lipari and Szab0~~952 in our analysis of the relaxation data. This type of approach was also utilized by van Dijk et in their analysis of the 23Na relaxation in isotropic solutions of DNA. The approach was outlined in the Theory section (see expressions 9 and 10) and is a two-step model for the above-mentioned local motions with the assumption that the slow local motions can be described by a Lorentzian spectral density function. Proceeding with the twostep model of local motions we assume that there is neither orientation dependence nor a k dependence of Jl(00) and J2( 2 4 . Consequently, a mean value of the spectral densities at OLp = Oo and 90°, and additionally of the spectral densities J,(wo) and Jz(2wo) obtained at 9.38 and 4.69 T, respectively, was used. This approximation may not be entirely correct, as seen from the weak orientation dependence, but will be further discussed below. A nonlinear fit of three parameters was performed, namely Jloc,f, the product (?r2/5)pB(Xf,R)*, and qoF,s (see the expressions 9 and 10 in the Theory section). The result is presented in Figure 3 and N
Table 4. As can be seen, the quality of the fit is excellent. As expected, qoc,s is somewhat shorter (2.1 ns compared to 2.6 ns) when the sample contains more water due to the generally faster diffusive motions. We now consider the magnitude of the residual qcc. In the present dense DNA system, with surface to surface distances of 1 1 and 13 %, and with no excess salt, a simple approximation is to consider all counterions as bound @e = 1) and consequently the relaxation rates obtained here are for bound ions. The residual qcc, ( x f , ~ obtained ), from the fit, may then be calculated to 2 12 and 174 kHz when the sample is equilibrated at 95 and 98% RH, respectively. Comparison with Results in Isotropic Semidilute DNA Solution. It is of interest to compare the magnitude of 'Tloc,s (2.6 and 2.1 ns) with the Z3Na correlation time obtained in isotropic DNA solutions3where, using the same functional model in fitting the field dependent spectral densities, ~ l was~ found , ~ to vary between 3.6 and 4.0 ns depending on DNA type (these values were obtained at 290 K by extrapolating their temperaturedependent data). Clearly, there is some inherent uncertainty in these two sets of numbers as obtained within the model, and there areof course alsouncertainties due to the approximations involved within the model itself. In spite of this, the difference in the correlation time between the isotropic and the anisotropic systems, is large enough that there is reason to speculate on the possible physical origin of this difference. If the dynamic processes governing Jl(00) and Jz(2wo) are the same in the two systems, TI^,^ of the present dense DNA system is longer than in the more water-rich isotropic DNA solution. This does not seem to be plausible in view of the generally slower dynamics expected for the dense and ordered DNA system. It then follows that Jloc,s may have significant contributions from different motional processes in the two systems. van Dijk ef ala3suggested two possible contributions to J I ~ namely , ~ , local counterion diffusion around the DNA rod (Le., azimuthal surface diffusion) and/or correlated motion of the counterion and the phosphate groups on the DNA molecule. If so, then it seems likely that in solution long-range diffusional processes, such as azimuthal diffusion and perhaps also diffusion along the curved DNA molecule and diffusion between different DNA molecules,' may give significant contributions to the 23Na relaxation. These processes do not seem to be significant in ordered DNA. The DNA backbone is subject to fast internal motions on a time scale of 0.5-2 ns53-55and the motions of the DNA phosphates has been observed not to be strongly correlated with DNA concentration.17 The motions of the phosphate groups in the fibers could then very well be on the time scale mentioned above. Thus, it is plausible that a large portion of the Na+ ions in the ordered DNA system is spatially correlated to phosphate groups, at least partially during theintemal DNA motions, which therefore could be responsible for the dynamics leading to the slow component of the 23Na relaxation. Regarding the qcc, it is 100 kHz in dilute isotropic DNA sol~tions,1~~.56 compared to the values 212 and 174 kHz in the ordered DNA system. The fact that the values obtained in the ordered system are larger than in isotropic solutions is reasonable. The magnitude of the quadrupolar interaction is expected to be larger in the ordered system due to the proximity of the adjacent DNA molecules and the smaller amount of water, which should render the efg's experienced by the Na+ ion close to DNA larger
-
-
-
8516
The Journal of Physical Chemistry, Vol. 98, No. 34, 1994
Schultz et al.
TABLE 5 Resulting Values of the Parameters from the Nonlinear Fit of Jz(200) at Two Separate Orientations OLp = Oo and 90° According to Eqs 9 and 10 RH (%) 95 0
c
.
98 90 64 f 1.5 (9.5f 0.2)x 1010 2.53 f 0.10
200
v1
5 v
u 0
e- 100
0
I
1
I
0
1.0 1 o9
2.0 1 o9
I
3.0 1 o9
a / rad s . '
Figure 4. Nonlinear fit of spectral densities, J2(2wo), at two separate orientations, BLp = OD and OLp = 90°,obtained when the sample was equilibratedat95% RH (90': open circles, broken 1ine;O': closed circles, solid line) and 98% RH (90': open squares,dottedline; 0': solid squares, dashed line) to eqs 9 and 10. The result of the fit is presented in Table 5.
than in an isotropic solution. Furthermore, the averaged quadrupolar interaction as reflected in the residual qcc in the ordered system is, according to the proposed model, averaged only by very fast local motions. In the system of DNAin isotropic solution on the other hand, the residual qcc may very well be averaged by somewhat slower local motions as well (although not slow compared to the inverse of the Larmor frequency). In the very recent paper by Groot et al.24that appeared after this work was submitted, the 23Na relaxation in ordered liquid cyrstalline DNA was investigated by field-dependent measurements. A model identical to the one used in the present paper for fitting the frequency dependence of the spectral densities was used. The correlation time obtained for the slower motion was found to be 2.1 ns at 288 K and these authors also concluded that local counterion dynamics in the vicinity of DNA is responsible for this dynamic process. Regarding the comparison with isotropic DNA solution, some new data at higher DNA concentrations, compared to the study performed previously by the same group, were presented. The correlation time for the slower process was this time found to be longer than in the earlier paper, and was 8.6 ns at 288 K. The fact that the correlation time is longer in the isotropic DNA solution was attributed to premelting effects (absent in liquid crystalline DNA) on the averaging of the field gradient interaction caused by the slower motion. Consequently, these authors argue that the same type of dynamic process in solution as well as in their ordered system could be responsible for the averaging of the quadrupolar interaction causing the frequency dependence in the 23Na relaxation. In spite of the different opinion taken here, where we believe that it is likely that other and somewhat slower dynamic processes contribute in solution, it is gratifying that very similar results have been obtained for ordered fibrous DNA and liquid crystalline DNA. Effect of the Approximation of No Orientation Dependence. Finally, in the discussion of the high-frequency spectral densities, we turn to one of the main approximations invoked within the model used, namely that of assuming no k nor any orientation
0 47 A 0.5 (5.7 f 0.1)x 1010 2.16f 0.07
90 53 A 1.5 (6.6f 0.1)X 10'0 2.06f 0.06
dependence in the spectral densities. Within the set of experimental data at hand, it is, in fact, possible to exclude this assumption and test the consequences of it on the obtained fitted parameter values. Considering the data in Table 3, we can note that at both water content, there are a set of spectral densities, J2(2w0),at four different frequencies, 00, obtained at the two different orientations, OLp = Oo and 90°, respectively. Consequently, we can now fit these two sets of data to eqs 9 and 10 independently, i.e., a t the two separate orientations. The result is displayed in Figure 4 and Table 5. As can be seen from the figure the orientational dependence is small, although apparently systematic. Inspection of the obtained parameters in Table 5 reveals that the main origin of the weak orientation dependence can be traced to the frequency-independent term Jlw,f and the product (?r2/5)pe(x f , ~ ) Furthermore, ~. qwghas an orientation dependence that is insignificant compared to the uncertainty of these numbers caused by the experimental error in the data. It is also gratifying to note that the values of the correlation time, TI^,^, are very similar to the ones obtained by the full fitting procedure within the more approximate model assuming no k and orientation dependence. Zero-Frequency Spectral Density Jo(0). In contrast to Jl(w0) and J2(2wo),the Jo(0) values also report on motions considerably slower than the inverse of the Larmor frequency. From Figure 3 and Table 3, it can be deduced that the zero-frequency limiting values of the slow local motion spectral densities contribute only to a small extent to the measured Jo(0) when the sample is equilibratedat 95% R H (about 16%). However, when thesample is equilibrated a t 98% R H the contribution is significant and accounts for about 50% of the measured Jo(0) but not the orientation dependence. Therefore, we have to consider other motions that occur on such significantly slower time scales that their contributions to the high-frequency spectral densities may be neglected. The motions that are relevant are those slow motions, generally on a time scale of microseconds and slower, giving fluctuations of the quadrupblar interaction of large enough amplitude so that they may contribute to the relaxation. The treatment of these slow motions relevant for the 23Na+relaxation in aqueous systems of oriented charged cylinders is difficult, and no general treatment can be found in the literat~re.~'We will therefore only give a qualitative discussion of different possible contributions. On the basis of discussions in the literature, the following possible motions can be envisaged: Fluctuations of the director orientation due to31,36(1) surface diffusion along the slightly bent DNA molecule; (2) slow bending of the DNA molecule; and (3) diffusion between domains of different helix axis orientations. In addition, a fourth contribution due to exchange between different binding sites on the DNA molecule characterized by different quadrupolar coupling constants should also be c ~ n s i d e r e d . ~The ~ . ~first ~ two contributions do not seem likely for the present dense system where the DNA persistence length is expected to bevery long and where the rigidity of the DNA molecule should be considerable. The third contribution depends on the spread in DNA helix axis orientations. An rms angular distribution of up to 1Oo of the domain orientation has previously been reported for wet spun fibers prepared in this lab0ratory,4~~50 and diffusion of counterions between domains with different orientations may well give a contribution to Jo(0). The fourth proposed contributor to Jo(0) (which could be the
Relaxation of N a Counterions in Ordered DNA main contributor) is an exchange between sites with different residual qcc's. Schematically, one may consider the following site^:'^*^^ (1) free ions in an isotropic environment between DNA helices (probably nonexistant in this dense system), ( 2 ) %onspecifically" bound ions on the DNA helix, Le., counterions residing within -4 %, (one diameter of the hydrated counterion) from the DNA surface, and (3) specifically bound ions. The two important specific binding sites are the phosphate groups and the bases which may be reached in the minor groove. If the exchange occurs, e.g., between two well-defined sites and is slow compared to the relaxation time in the sites, separate lines (different VQ) for the counterions in the respective sites are will appear in the spectrum if the difference in VQ is large compared to the inhomogeneous satellite broadening. This has not been observed. In the very recent study by Groot et al.,24Jo(0) was found to be determined by the same motional process as Jl(w0) and J2( 2 4 , which is not the case in the ordered fibers. In that liquid crystalline DNA system any site exchange process should be similar to that in our system, while thecontributiondue to diffusion between domains is not expected to contribute toJo(0) since their system was found to be very well ordered. Therefore, we may conclude that Jo(0) in the ordered fibers is dominated by a contribution absent in the liquid crystalline system, and likely to be the diffusion between different domains. The diffusion between different domains may occur on a time scale of milliseconds. It should be noted that if this motion occurs on a time scale comparable to the homogeneous relaxation rates to which it contributes, then the perturbation regime within the standard Redfield relaxation theory may no longer be valid.
Conclusions A general picture of the averaging of the quadrupolar interaction responsible for the static residual quadrupolar interaction and 23Na relaxation in the ordered DNA systems may now be formulated. This picture is based on previous work on quadrupolar relaxation in polyelectrolyte systems in general, and on the present work in particular, and is the following: The static efg of a counterion associated with DNA is due to the asymmetric distribution of water dipoles around it, toother nearby charges such as phosphate charges and charges of other counterions, and to other efg's caused by the molecular nature of the DNA double helix (e.g., the bases). This efg will be averaged by different molecular motions occuring on different time and length scales. First, fast local motions occurring on a time scale faster than nanoseconds and mainly due to diffusional and vibrational motions of the water molecules in the hydration layer surrounding the ion will average the interaction to a value which we characterize by a residual qcc (Xf,R). These processes give a contribution Jlw,f to the 23Na relaxation. Thereafter, slower local motions, occurring on a time scale of nanoseconds, average the interaction to a value characterized by ( X R ) . These motions are believed to be due to local motions of the counterion in a binding site and/or the DNA backbone (mainly the nearby phosphate groups). These motions give a contribution Jlw,s to the relaxation. Then azimuthal diffusion around the DNA double helix, probably occurring on a time scale of several nanoseconds, averages the quadrupolar interaction with an additional factor of one-half to a value characterized by the quadrupolar splitting; see eq 4. This motion does not contribute significantly to the relaxation but is definitely averaging the efg since the diffusion around this axis determines the direction of the averaged quadrupolar interaction which results in the observed angular dependence of VQ as expressed in eq 4. The motions mentioned so far contribute to all spectral densities. Next, slow motions that do not contribute to J l ( w 0 ) and J2(2wo) give significant contributions to Jo(0). These motions are tentatively ascribed to slow counterion exchange between sites on the DNA molecule with different quadrupolar couplings and diffusion between domains characterized by different helix axis orientations.
The Journal of Physical Chemistry, Vol. 98, No. 34, 1994
8517
The slow local motion has been characterized by a correlation time, T ~ N , ~in, the model given by eqs 9 and 10, giving fitted experimental values of 2.6 and 2.1 ns for the sample equilibrated at 95 and 98% RH, respectively. The significance of these numbers is dependent on the model which assumes a Lorentzian form of the slow local spectral density function. This is clearly an approximation, but the excellent fit to the experimental data clearly motivates this assumption. Probably less accurate is the assumption of no orientation dependence in the slow local spectral densities. When fitting J2(2wo),the weak orientation dependence of the obtained parameters (Table 5 ) could be traced to Jlw,f and to the product (7r2/5)pa(x f , ~while ) ~ 71w,swas found independent of orientation. Finally, on the basis of comparison with the results obtained in the 23Na relaxation study of DNA,liquid crystals by Groot et al.,24 it was concluded that the zero-frequency spectral density is probably dominated by diffusion between domains of different director orientations.
Acknowledgment. We are especially grateful for valuable discussions and suggestions from Istvan Fur6 (particularly concerning the N M R experiments). We also thank Bertil Halle for valuable discussions and Ulf Henriksson for help with N M R experiments at the two low magnetic fields. This work has been supported by the Swedish Natural Science Research Council (NFR) and the Swedish Medical Research Council (MFR). References and Notes (1) Nordenskiold, L.; Chang, D. K.; Anderson, C. F.; Record, M. T., Jr. Biochemistry 1984, 23, 4309. (2) Braunlin, W. H.; Drakenberg, T.; Forstn, S. Curr. Top. Bioenerg. 1985, 14, 97. (3) Van Dijk, L.; Gruwel, M. L. H.; Jesse, W.; deBleijser, J.; Leyte, J. C. Biopolymers 1987, 26, 261. (4) Braunlin, W. H.; Drakenberg, T.; Nordenskibld, L. Biopolymers 1987, 26, 1047. (5) Braunlin, W. H.; Nordenskiold, L.; Drakenberg, T. Biopolymers 1989, 28, 1339. (6) Braunlin, W. H.; Drakenberg,T.; Nordenskibld, L. J.Biomol. Struct. Dyn. 1992, 10, 333. (7) Record, M. T., Jr.; Mazur, S. J.; Melanpn, P.; Roe, J.-H.; Shaner, S. L.; Unger, L. Annu. Rev. Biochem. 1981, 50, 997. (8) Anderson, C. F.; Record, M. T., Jr. Annu. Rev. Phys. Chem. 1982, 33, 191. (9) Rill, R. L.; Hilliard, P. R.; Levy, G. C. J . Biol. Chem. 1983,258,250. (IO) Rau, D. C.; Lee, B.; Parsegian, V. A. Proc. Natl. Acad. Sci. U.S.A. 1984, 81, 2621. ( I 1) Lewen, G. D.; Lindsay, S.M.; Tao, N. J.; Weidlich, T.; Graham, R. J.; Rupprecht, A, Biopolymers 1986, 25, 765. (12) Guldbrand, L.; Nilsson, L.; Nordenskiold, L. J . Chem. Phys. 1986, 85, 6686. (13) Lee, S. A.; Lindsay, S. M.; Powell, J. W.; et a1 Biopolymers 1987, 26, 1637. (14) Strzelecka, T. E.; Davidson, M. W.; Rill, R. L. Nature 1988, 331, 457. (15) Livolant, F.; Levelut, A. M.; Doucet, J.; Benoit, J. P. Nature 1989, 339, 724. (16) Nilsson, L. G.; Guldbrand, L.; Nordenskibld, L. Mol. Phys. 1991,72, 177. (17) Strzelecka, T. E.; Rill, R. L. Biopolymers 1990, 30, 803. (18) Strzelecka, T. E.; Rill, R. L. J . Phys. Chem. 1992, 96, 7796. (19) Schultz, J.; Nordenskiold, L.; Rupprecht, A. Biopolymers 1992,32, 1631. (20) Rupprecht, A. Biochem. Biophys. Res. Commun. 1963, 12, 163. (21) Rupprecht, A. Biotechnol. Bioeng. 1970, 12, 93. (22) Einarsson, L.; Eriksson, P.-0.; Nordenskiold, L.; Rupprecht, A. J . Phys. Chem. 1990, 94, 2696. (23) Einarsson, L.; Nordenskiold, L.; Rupprecht, A,; Fur6, I.; Wong, T. C. J . Magn. Reson., 1991, 93, 34. (24) Groot, L. C. A.;vander Maarel, J. R. C.; Leyte. J. C.J.Phys. Chem. 1994, 98, 2699. (25) Forstn, S.; Lindman, B. Meth. Biochem. Anal. 1981, 27, 289. (26) Fur6, I.; Halle, B.; Wong, T. C. J . Chem. Phys. 1988, 89, 5382. (27) Fur6, I.; Halle, B. J . Chem. Phys. 1989, 91, 42. (28) Fur6, I.; Halle, B.; Einarsson, L. Chem. Phys. Lett. 1991, 182, 547. (29) Fur6, I.; Halle, B. Mol. Phys. 1992, 76, 1169. (30) Quist, P.-0.; Halle, B.; Fur6, I. J . Chem. Phys. 1991, 95, 6945. (31) Quist, P.-0.; Blom, I.; Halle, B. J . Magn. Reson. 1992, 100, 267. (32) Halle, B. Mol. Phys. 1987, 60, 319. (33) Rupprecht, A,; Forslind, B. Biochim. Biophys. Acta 1970,204, 304.
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