- Email: [email protected]
[17] Determination of fast dynamics of nucleic acids by NMR
[I 7]
FAST DYNAMICS OF NUCLEIC ACIDS
413
is unaffected by added catalysts and where the exchange time is equal to the base pair lifetime. In the triple helix [d(TaCT4CT4)]2" d(A4GA4GA4), the lifetimes of the CH +. N7G Hoogsteen pairs of the two C*. G. C triplets are 3 s at 15e. The lifetimes of the Watson-Crick components are even longer, approximately 5 rain. 39 The lifetimes of internal C. C ÷ pairs of the four-stranded structure formed by intercalation of parallel oligo-dC duplexes (the i-motif48) are also long, in the hundreds of seconds. 1° Even when end-effects are manifest, as in the case of the internal C2. C2 pair of the tetramer formed by the trinucleotide d(TCC), the lifetime, 500 ms at 15°, is much longer than base pair lifetimes in B-DNA. Such lifetimes of C. C + pairs are part of the diagnostic in the search for the formation of the i-motif.49 48 K. Gehring, J. L. Leroy,and M. Gu6ron, Nature 363, 561 (1993). 49j. L. Leroy,M. Gu6ron,J. L. Mergny,and C. H61~ne,Nucleic Acids Res. 22, 1600 (1994).
[I 7] D e t e r m i n a t i o n
of Fast Dynamics by NMR
of Nucleic Acids
B y ANDREW N. LANE
Introduction The major sources of structural information in NMR are the three-bond coupling constants and the nuclear Overhauser enhancement (NOE). As coupling constants typically lie in the range of 0-10 Hz, any variation of the torsion angle on a time scale shorter than a few milliseconds leads to averaging. The NOE explicitly depends on the correlation time, which sets a characteristic time scale for motions that determine the rate of magnetization transfer. The dominant motion that determines this time scale is that of overall macromolecular rotation. For biopolymers in solution, other (internal) motions that are faster than the overall rotation generally tend to reduce the transfer rate. Motions slower than the molecular tumbling rate affect NOEs by averaging the transfer rates over all the conformation states that are sampled. Double-stranded nucleic acids behave hydrodynamically as cylinders and, therefore, are anisotropic so that magnetisation transfer rates via cross-relaxation may depend on the orientation of the vector to the long axis of the molecule. Hence, it is important to determine the overall rotation rate(s) of the molecule and use additional information METHODS IN ENZYMOLOGY, VOL. 261
Copyright © 1995 by Academic Press, Inc. All rights of reproduction in any form reserved.
414
NUCLEIC ACID DYNAMICS
[ 17]
that can be obtained from relaxation measurements to probe the extent to which a rigid rotor model is applicable. Because D N A and R N A provide a relatively low density of distance constraints, the precision of distances becomes a more important issue than it does for structure determination of proteins. If internal mobility is detected, then the structural refinement procedure needs to be more sophisticated than for a rigid body. Any refinement method that imposes a unique conformation as a constraint on the data under these circumstances must lead to a virtual structure of unclear significance. 1 Thus, refinements based on "back-calculations" of N O E data may be compromised by the averaging unless specific and appropriate motional models are directly incorporated into the refinement procedures. 2-9 Indeed, the concept of structure refinement against distances derived from NOE intensities becomes untenable and preferably should be performed directly against the N O E intensities themselves. Furthermore, the dynamic aspects become an intrinsic part of the description of the molecule. In this chapter, N M R methods for measuring overall tumbling rates and fast dynamic processes in nucleic acids are described. Relaxation Times and Molecular Tumbling Molecular tumbling rates enter into structure calculations because the proton-proton cross-relaxation rate constant, and, therefore, the NOE, is directly proportional to the rotational correlation time. Analysis of relaxation data requires that the effects of overall rotation be separated from those of internal motions. Indeed, for small amplitude internal motions, the relaxation rates are dominated by overall rotation. To analyze internal mobility, it is necessary to determine the contribution from overall molecular rotation. For a sphere there is a single rotational correlation time; however, short segments of duplex D N A can be expected to behave as essentially rigid cylinders as far as global rotation is concerned, for which at least two intrinsic correlation times are needed to describe the rotation. 1 0 . Jardetzky, Biochim. Biophys. Acta 612, 227 (1980). 2 B. A. Borgias, M. Gochin, D. J. Kerwood, and T. L. James, Prog. NMR Spect. 22, 83 (1990). 3 A. N. Lane, Biochim. Biophys. Acta 1049, 189 (1990). 4 T. M. G. Koning, R. Boelens, G. A. van der Marel, J. H. van Boom, and R. Kaptein, Biochemistry 30, 3787 (1991). 5 M. J. J. Bonvin, J. A. C. Rullman, R. M. N. J. Lamerichs, R. Boelens, and R. Kaptein, Proteins 15, 385 (1993). 6 N. B. Ulyanov, U. Schmitz, and T. L. James, J. Biomol. NMR 3, 547 (1993). 7 J-F. Lefbvre, A. N. Lane, and O. Jardetzky, Biochemistry 26, 5076 (1987). 8 p. Yip and D. A. Case, J. Magn. Reson. 83, 643 (1989). 9 A. M. J. J. Bonvin, R. Boelens, and R. Kaptein, J. Biomol. NMR 4, 143 (1994).
[17]
FAST DYNAMICS OF NUCLEIC ACIDS
415
This in turn means that the cross-relaxation rate depends on the orientation of the proton-proton vector within the molecule. Relaxation rate constants depend on the molecular tumbling rates via the spectral density function, which relates the power available at any frequency, including the Larmor frequency, to a correlation time for motion. In the simplest cases of isotropic motion, where the correlation function is exponential, the spectral density function is the familiar Lorentzian: J(o)) -~- "/'/(1 + 0$2T2),
(1)
where r is the correlation time and to is the angular frequency. R e l a x a t i o n Rate C o n s t a n t s
The relaxation rate constants, Ri, depend on the spectral density functions, which in turn depend in detail on the contributing relaxation mechanisms. For the dipolar mechanism, which is the most important mechanism for 1H and 13C, the relaxation rate constants (for unlike spins in the case of protons) for relaxation by a proton at a distance r are given byl°'lL12: R 1 "~- (a/r6)[J(Aoj) + R 2 = (0.5a/r6)[aJ(o)
3J(cox) + 6J(E~o)].
+ J(Aog) + 3J(oJx) + 6J(COH) + 6J(]~o9)].
o" = (a/r6)[6J(~oJ) - J(Aog)].
NOE = 1 + (TH/Tx)o'/R1.
(2) (3)
(4) (5)
X refers to the heteronucleus (e.g., ~SN, a3C, 31p). R1 is the spin-lattice relaxation rate constant, R2 is the spin-spin relaxation constant, o" is the cross-relaxation rate constant, and NOE is the nuclear Overhauser enhancement. a is a constant that depends on the gyromagnetic ratios of the dipolar coupled spins. If J(o~) is in ns and r is in ~ , the value of a is 56.95 for H : H relaxation, 9.33 for 31p:H, 3.60 for 13C:H, and 0.585 for 15N:H. It can be shown from Eqs. (2)-(4) that in macromolecules, R1 is dominated by the J(oJ) term for most heteronuclear interactions, whereas R2 is dominated by the zero-frequency term J(0), which is equal to the rotational correlation time. In the homonuclear case, the dominant term is J(&o) J(0) for all relaxation rate constants. In addition to the dipolar mechanism, relaxation by chemical shift anisotropy (CSA) is very important for 31p in phosphodiesters and to a lesser extent for t3C in aromatic systems. The CSA arises from anisotropic shielding at the nucleus due to a nonspherically symmetric magnetic envil0 A. Abragam, "The Principles of Nuclear Magnetism." Clarendon Press, Oxford, 1968. 11D. Neuhaus and M. P. Williamson, "The Nuclear Overhauser Effect." VCH Publishers,
Inc., New York, 1989. 12D. R. Kearns, CRC Critical Reviews in Biochemistry 15, 237 (1984).
416
NUCLEIC ACID DYNAMICS
[ 17]
ronment created by asymmetric bonding. The contribution to relaxation by CSA is given by Eqs. (6)-(9)1°,13: RI(CSA) = 133.33 X2tOx2J(tox).106, Rz(CSA) = 22.22 X2OJx2[4J(0) + 3J(OJx)].106, R2 - 0.5R1 = 88.88X2¢Ox2J(0).106, R2/Rt = [4J(0) + 3J(oJx)]/6J(oJx),
(6) (7) (8) (9)
where X 2 = Atr2(1 + ~2/3), Ao- is the chemical shift anisotropy and 77 is the asymmetry parameter. In these equations, Ri is in s -1, co is in G.rad s -1, J(oJx) is ns, and X is in ppm. For axially symmetric shift tensors, 77 is of the order --0.6.14 However, in solution, the asymmetry and chemical shift anisotropy terms cannot be separated; therefore, it is convenient to use the apparent anisotropy as given by X. For oligonucleotides, RI(CSA) is only weakly dependent on the magnetic field strength B0 because J(tox) ~ 1/tOx2r (Eq. (6)), whereas R2 increases with the square of the applied magnetic field strength because J(0) >>J(oJx) (Eq. (7)). In contrast, the dipolar contribution is independent of B0 for Rz or decreases with the square of B0 for R1 (Eqs. (2)-(4)). Hence, as the applied field is increased, the CSA mechanism eventually becomes dominant. However, the linewidth of a resonance for a system in fast intermediate chemical exchange also increases with increasing B0 for R2 as CSA. The exchange and CSA contributions to R2 for macromolecules can be distinguished by their different temperature dependencies. CSA should decrease with increasing temperature through the effects on the correlation time, whereas the exchange contribution to R2 may pass through a maximum as the temperature is increased. Fast Internal Motions Fast internal motions refer to fluctuations that are faster than overall rotation and, therefore, can contribute significantly to relaxation processes. In the presence of rapid internal motions, the relaxation rate constants can be considered as the sum of two parts, namely that due to overall rotation of the molecule, described by Eqs. (2)-(9), and contributions from internal rotations having a correlation time Ti.15'16 Thus, (R) = S2.R(TR) + (1 -- S2)R(%),
(10)
13M. J. Forster and A. N. Lane, Eur. Biophys. J. 18, 347 (1990). 14T. L. James, in "Phosphorus-31 NMR: Principles and Applications" (D. G. Gorenstein, ed.). Academic Press, Orlando, 1984. 15 G. Lipari and A. Szabo, J. Am. Chem. Soc. 104~ 4546 (1981). 16G. Lipari and A. Szabo, Biochemistry 20, 6250 (1981).
[17]
FASXDYNAMICSOF NUCLEICACIDS
417
where (R) is the observed (averaged) relaxation rate constant, for example, RI, R2, or o-. re = ' i ' R r i / ( r R + T i ) ~-" 'i'i for T i " ~ T R • R('rR) is the relaxation rate constant for overall molecular rotation, R(%) is the relaxation rate constant for the internal motion having an effective correlation time %, and S 2 is an order parameter that describes the spatial restriction of the motion. For a rigid body, S 2 = 1, and for a freely reorienting vector, S 2 = 0. Recently, an extension to Eq. (10) has been proposed to account for the lSN relaxation data in proteins a7 in which S 2 is expanded as a product of two uncoupled processes. T h e r e are insufficient data so far to determine whether this extension is required to describe relaxation in nucleic acids. The description of the dynamics of the molecule on the time scale of nanoseconds and shorter, therefore, is reduced to determining the values of S 2 and ri for each site and the overall correlation time(s), which should be the same for each site. The values of S 2 and ri can in principle be extracted from frequency-dependent measurements of R, although in many instances ri is poorly determined (discussed later). S 2 can also be calculated for specific motional models TM or calculated from molecular dynamics trajectories. 4,19 The amplitude of the fluctuations must be relatively large to have a significant effect on R. Consider the simple model of rotation of a vector within a cone of semiangle ~b, for which the order parameter is given by
Eq. (11):15'16 S 2 = 0.25 cos2qS(1 + cos qS)2.
(11)
For S 2 = 0.9, q~ = 15 °. A decrease in the relaxation rate constant of less than 10% is unlikely to be detected by relaxation methods, implying that angular fluctuations of less than about 15 ° are unlikely to be reliably detected by NMR. Whereas this may be seen as a limitation to the method, it should be realized that fluctuations of this size have little consequence for determining conformational features of nucleic acids, and it is a moot point whether such small rapid motions could have any functional significance. An order parameter of 0.9 indicates a maximum loss of entropy on complete rigidification of only 0.26 k J mole -1 at 298°K. Because R2 is proportional to J ( O ) = r ~> J(o~), it is clear from Eqs. (3), (7), and (10) that R2 will simply be scaled by the order parameter S 2 for rR > ri for all but the smallest values of S 2. The same principle holds also for the homonuclear cross-relaxation rate constant. In contrast, R~ and the ~vG. M. Clore, A. Szabo, A. Bax, L. E. Kay, P. C. Driscoll, and A. M. Gronenborn, J. Am. Chem. Soc. 112, 4989 (1990). is j. Tropp, J. Phys. Chem. 72, 6035 (1980). 19j. M. Withka, S. Swaminathan, D. L. Beveridge, and P. H. BoRon, J. Am. Chem. Soc. 113, 5041 (1991).
418
NUCLEIC ACID DYNAMICS
[171
N O E for heteronuclei show a greater dependence on the internal correlation time; for very large molecules, the relaxation will become dominated by the internal motion term in Eq. (10). This means that internal motions can be detected from proton measurements and a measure of the amplitude determined, but little information can be derived about the time scale other than the motions must be much faster than the overall tumbling rate. For heteronuclear relaxation, it is sometimes possible to determine a value of ri from the combination of R1, R2, and NOE data, possibly measured at different magnetic field strengths.
Proton Relaxation Correlation times can be measured using protons only if one deals with a single pair of protons whose separation is known and fixed. This means that only the cross-relaxation rate constant is really useful. Estimates of correlation times from the ratio of RE/R~ are not likely to be very accurate for protons for several reasons. As longitudinal relaxation is in principle nonexponential, there is a problem with the definition of R1. Thus, Eq. (2) is the correct expression for the autorelaxation rate constant of unlike spins, which can only be obtained experimentally from a measurement of the initial rate of recovery of magnetization of a particular spin following a selective excitation. In this case, the ratio R2/R1 tends to the fixed ratio of 2.5 for large correlation times 12 (i.e., it is independent of the correlation time). An alternative is to measure the initial slope following a nonselective inversion pulse for which the apparent rate constant for recovery of the magnetization is given by Eq. (12): Rl0 = 2.0(a/r6)[3J(co) + 12J(Eto)].
(12)
The ratio R2/Rlo then depends only on the spectral density function, and in principle an estimate of the correlation time for a rigid isotropic rotor could be determined. Because the recovery of the magnetization is not exponential, it can be difficult in practice to measure the true initial slope. The ratio can give the correlation time only if the relaxation is purely dipolar. There may be a significant contribution from nondipolar relaxation that affects the T1 value (e.g., from paramagnetic agents). Note also that because longitudinal relaxation is multiexponential, fitting a single exponential to the entire recovery curve following a nonselective pulse gives a physically meaningless number. Probably more important is the influence of internal motions. Eqs. (3) and (10) show that R2 is effectively scaled by S 2 for fast (Tint < 0.1 ns) and moderate order parameters (>0.5), whereas R10 will be dominated by the internal motion for to~-~ ~> 1. Hence, the ratio R2/R~o is not a simple function of the overall rotational correlation time alone.
[171
FAST DYNAMICS OF NUCLEIC ACIDS
419
The cross-relaxation rate constant, in contrast, is a measure of a pure dipolar interaction between a pair of protons, such that any correlation time that is determined refers to the motion of that particular vector. The H6-H5 vector of cytosine or uridine is very convenient for determining the overall rotational correlation times for the following reasons. First, the distance is fixed by the covalent geometry and is known (r = 2.45 ~). Second, the H5 is relatively isolated so that the measured NOE is unlikely to be significantly contaminated by spin diffusion. Third, the vector lies in the plane of the base, which is essentially perpendicular to the long axis of duplex D N A and within 20° of the perpendicular in RNA. The best experiment to use is the truncated driven 1D N O E experiment, 2°'21 in which a particular proton is rapidly saturated for a time t (i.e., the magnetization of one proton is selectively clamped to zero). For an isolated spin-pair, the magnetization transfer is described by a single exponential process (whereas the NOESY or transient NOE experiment requires two exponentials). Conditions for obtaining the best quality N O E difference spectra have been discussed by Neuhaus and Williamson. v Furthermore, if H6 is saturated, spin diffusion to other spins near H6 is not significant. The NOE on H5 is then given by Eq. (13): NOE(t) = (o'/0")[1 - exp(-p*t)].
(13)
This is a two-parameter equation, from which both 0" and p* can be recovered by nonlinear regression using standard routines. The parameter p* is relatively poorly determined unless very long irradiation times are used; however, this is not desirable because the relaxation becomes nonexponential under these conditions, p* is in fact only an apparent relaxation rate constant that accounts for nonlinearity in the NOE build-up curve. = Typically, five to six time points spaced essentially linearly adequately sample the N O E time course and provide a good estimate of 0". The apparent correlation time is then most easily found by iterative calculation according to Eq. (14): r = r60-/[(56.92)*(6/(1 + 4toZr2) - 1)].
(14)
In this equation, 0- is in s -1, r is in/~, and r is in ns. An initial estimate of r is found by ignoring the first term in the denominator (which should be near - 1 for macromolecules). Note that the measurement gives a value for the cross-relaxation rate constant. If there is internal motion, this value 20 G. 21 A. 22 C. 92
Wagner and K. WUthrich, J. Magn. Reson. 33, 675 (1979). N. Lane, J-F. Lef~vre, and O. Jardetzky, J. Magn. Reson. 66, 201 (1986), M. Dobson, E. T. Olejnicjak, F. M. Poulsen, and R. G. Ratcliffe, J. Magn. Reson. 48, (1982).
420
NUCLEIC ACID DYNAMICS
[1 7]
will be really the scaled cross-relaxation rate constant, at least for fast motions and modest order parameters. Eq. (14) then yields an apparent correlation time, which for larger molecules (rR > ca 2 ns) is S2ZR. The approximation of an isolated spin-pair is very good for the Cyt H6H5 vector. For example, for a fragment of D N A in the B conformation having a correlation time of 3 ns, the cross-relaxation rate constant for the Cyt H6-H5 vector is -0.78 s -1. Fitting a simulation of the NOE time course using an entire D N A spin-system gave a value of o- = -0.79 s -I in the isolated spin-pair approximation using times up to 0.5 s (NOE intensities up to -31%). The NOE intensities can usually be measured quite accurately because the H5 resonances are often resolved and, therefore, provide a direct calibration of the area in the off-resonance spectrum. In general, it is preferable to acquire the on and off resonance spectra in separate files and perform the subtraction of the free induction decays (FIDs) at the end of the experiment rather than acquire the difference FID in the computer memory. Apart from providing the necessary off-resonance spectrum for normalization, the separate files allow one to examine the quality of saturation. It is important that the system be stable, especially for the short irradiation times. Attention to temperature equilibration and stability is critical, as difference spectra magnify any imperfections. Interleaving the on and off resonance spectra is helpful (we generally interleave 8 or 16 acquisitions), and allowing sufficient time for the spins to relax is also important for accurate work. It is often possible to use a matched filter in the window function to optimize sensitivity, which is obviously important for the smallest NOEs. Good quantitation of NOEs at - 5 % is required to determine o" reliably. Provided that care is taken to measure the NOE intensities accurately, the main source of error in the correlation time is then the uncertainty in the bond lengths and bond angles needed to calculate the proton-proton distance. These uncertainties amount to +0.03 ~ for H6H5, or 1.2%. This translates to an uncertainty in o- and, therefore, in r of about 7%. Hence, one can expect to be able to determine the apparent correlation time by this method to better than about 10%. In principle, the magnitude of the NOE is affected by the pulse repetition rate, as intensities are attenuated by repeating experiments before the spin system has fully relaxed. However, the importance of this in nucleic acids is relatively small because the saturation factor for most protons is very similar after a 90° pulse and the error is proportional to the relative steadystate magnetizations. Thus, for a two-spin system A and B, the rate of magnetization transfer from spin B to spin A during the irradiation period is given by Eq. (15):
[17]
F A S T D Y N A M I C S OF N U C L E I C A C I D S
421
dMa/dt = -pa(Ma - Ma~) + °'Mb~.
(15)
After the observation pulse, all magnetization relaxes (nonexponentiaIly) toward equilibrium and will reach some steady-state value, M(ss), if the sum of the acquisition time and relaxation delay is not sufficiently long. In the off-resonance case, the relaxation during the irradiation period is dMa/dt = -p~(Ma - Ma~) - o'(Mb - Mb~).
(16)
For short irradiation times, when Ma = Ma(ss) and Mb = Mb(ss), the initial rate of change of the difference magnetization becomes: dAMa/dt = ¢rMu(ss).
(17)
Hence, the difference spectrum should be normalized to the on-resonance peak. The error obtained by normalization to other peaks of definable area is likely to be small for protons in nucleic acids as the steady-state magnetizations are all quite similar for recycle times of 4 s or more. The main exception is the Ade C2H and to some extent protons on the 5' nucleotide. Other vectors of fixed or nearly fixed length that can be used include the H2'-H2", HI'-H2", and H2'-H3'. The latter two vectors vary slightly in magnitude as the sugar pucker changes (sugar repuckering is the only motion that can affect the distance). The interpretation of the relaxation rates then requires application of the Tropp model, 18 which provides a recipe for calculating order parameters when the length of the vector is not fixed. For fast motions, the appropriate average is the square of the dipolar Hamiltonian, and the spectral density function is given by Eq. (18)18: J ( 0 ) = 0.2,R
piY2m/r? ,
(18)
m=-2
where Y2mis the normalized spherical harmonic, pi is the fractional contribution of conformational state i, and ri is the internuclear distance in state i. The averaging is done over each discrete conformational state, which requires that the coordinates in each state be known to calculate ri. A further complication that must be dealt with is spin diffusion within the sugar. This requires that all the magnetization transfer curves for the minimum spin system are fitted simultaneously. This is equivalent to finding the relaxation matrix. 23,24 Nonlinear regression to 1D time courses consisting of at least 5 time points and observation of the magnetizations on irradiation of the H I ' , H2', or H2" comprise the minimum effort. Initial 23 p. Koehl and J. F. Lefhvre, J. Magn. Reson. 86, 565 (1990). 24 A. N. Lane and M. J. Forster, Eur. Biophys. J. 17, 221 (1989).
422
NUCLEIC A C I D DYNAMICS
[ 171
estimates for the cross-relaxation and autorelaxation rate constants can be obtained using the known (or calculated) overall tumbling time and initially assuming a rigid body. The autorelaxation rate constants should also be treated as parameters, which then have no special physical significance. The precision of the cross-relaxation rate constant determined in this way is significantly lower than for the H6-H5 pair, but in favorable cases (e.g., where there is sufficient spectral resolution and good quality data), results can be obtained that compare favorably with those derived for heteronuclear relaxation measurements. If an independent measure of the molecular tumbling time is known, the cross-relaxation rate constant for a rigid rotor can be calculated. For protons in large macromolecules, where J(2to) ~ J(0) and Ti ":~ 'TR, the ratio of the measured and calculated crossrelaxation rate constant provides an estimate of S 2, as or ~ S20"(TR).
Alternatively, comparison of apparent correlation times for different sites can be used to show whether or not the rigid rotor approximation is adequate to describe the relaxation. Significant discrepancies that cannot be accounted for by alternative relaxation mechanisms or anisotropic motions have to be ascribed to additional internal motions.
13C NMR 13C relaxation is dominated by the dipolar interaction with the directly attached proton, except for carbonyl carbons and aromatic carbons, where there may be a significant contribution from CSA. The CSA for aliphatic carbons is only 20-30 ppm, which leads to an insignificant effect on relaxation (<1% of the dipolar rate at 11.75 T). For aromatic carbons, the CSA is approximately 150-180 ppm, which leads to approximately 20% contribution to the relaxation rate at 9.4 T, 25% at 11.75 T, and 35% at 14.1 T. In contrast, the contribution from nonbonded protons is up to about 5% for sugar carbons. Unless very detailed field-dependent measurements are made, it will not be possible to determine the CSA component on the sample under investigation. Therefore, it is necessary to calculate the CSA contribution for the correlation time as determined (discussed next) and subtract it from the observed rate. This will lead to some (unknown) error for the aromatic carbons. Figure 1 shows how the relaxation parameters vary as a function of overall tumbling time and spectrometer frequency. As Eq. (3) indicates, for macromolecules, R2 should be only weakly dependent on the spectrometer frequency and essentially linearly dependent on the correlation time, whereas R1 decreases strongly with increasing spectrometer frequency (in
[17]
FAST DYNAMICS OF NUCLEIC ACIDS J '
I
' ' I
' ~
I
423
~ ' ~ I
~
~,~ 6 5 4 3 2 °
1
Oo ,~
451
2
°
4
6
8
I0
, , , i , , , t , , , i , , , i , , ,
4°1 ~
35
-
ao! 25
2
4
6
8
I0
~ns
FIG. 1. Dependence of x3C relaxation rate constants on correlation time. Relaxation rates were calculated using Eqs. (2)-(4) for a rigid isotropic rotor at two magnetic field strengths (9.4 T and 14.1 T). The C-H bond length was taken as 1.095 A. (Top) R1 at 9.4 and 14.1 T and NOE at 9.4 T. (Bottom) R2 at 9.4 and 14.1 T. E],i R1; 0 , 0 NOE.
the limit as B0 -2 for very large molecules) and less strongly as the correlation time increases (~-0-1 in the limit of very large molecules). In the presence o f internal motions, R2 is essentially scaled b y S 2 for short internal correlation times, and the relaxation rate constant remains i n d e p e n d e n t of the s p e c t r o m e t e r frequency. I n contrast, Ra shows m o r e c o m p l e x b e h a v i o r (Fig. 2). F o r correlation times w h e r e COxr ~-- 1, the contribution f r o m overall t u m b l i n g dominates, and R1 is scaled by SL F o r ~Ox z > 1, the contribution f r o m internal m o t i o n b e c o m e s m o r e important, and R1 does n o t scale simply with the o r d e r p a r a m e t e r (except in the limit of infinitely fast internal motions). H e n c e , the ratio R2/R~ is n o t i n d e p e n d e n t of the o r d e r parameter.
424
NUCLEIC ACID DYNAMICS
w
[ 17]
5
o
3q 2 1
O0
4
/
0.2
'
'
'
!
0.4
'
'
'
I
0.6
S2 '
'
'
I
[
1
'
~
5 0.04
1
~ ~
~r/-
. I0
0.8
1
~ 0.08
0.12 'l;e/ns
0.16
0.2
FIG.2. Effect of internal motions on 13Crelaxation rate constants. Relaxation rate constants were calculated for an overall correlation time of 3 ns at a magnetic field strength of 11.75 T, using Eqs. (2)-(4) and Eq. (10). (Top) Dependence of R1 and NOE on the order parameter. The internal correlation time was 40 ps. (Bottom) Dependence of R1 and NOE on internal correlation time for S2 = 0.5. VI,IIR1; O,© NOE.
Furthermore, the N O E decreases with increasing order parameter. In the limit of a rigid spherical rotor, the N O E will be close to its minimum for correlation times larger than about 2 ns at magnetic field strengths of 9.4 T or higher. Therefore, most oligonucleotides of interest ( > 6 bp) would be expected to give rather small NOEs. N O E s significantly larger than expected would indicate the presence of internal motions. Furthermore, sites that give only the minimal N O E can be used to obtain an initial estimate of the correlation time from the ratio R2/R1, as the order parameter is near unity. The only unknown is the correlation time; rcr~ is 1.093 -+ 0.005/~ for the aliphatic carbons and 1.085 __+ 0.005 A for the aromatic
[ 17]
FAST DYNAMICS OF NUCLEIC ACIDS
425
carbons. Because R2 is dominated by J(0) for macromolecules, an initial estimate can also be determined from r 6- Rz/2a. The value of ~"is then obtained by iterative solution to R2/R1, which should then be adequate for predicting the values of R1, R2, and the NOE, or (equivalently o- = (R1.NOE-1)yx/TH) for all sites, and will be the same for all sites in the absence of motion. In practice, because of finite errors in measuring relaxation parameters, especially for 13C at natural abundance where low concentrations tend to give noisy spectra, a simultaneous fit to all three relaxation parameters is likely to give a more reliable estimate of the apparent correlation time at a particular site. When internal motions are present (the usual case), each site can be analyzed as a three parameter problem, namely rR, S2, and ri. This is straightforward in 13C NMR because all three relaxation rates are measured using nonselective experiments. Using the initial estimate for rR, or one derived independently from other measurements, these parameters can easily be found by nonlinear regression if at least three observations are available. In addition, frequency-dependent measurements of R~ are useful for sampling the internal dynamics component. Because there are several local minima in the target function, it is probably best to use a combined grid search with local optimization to find the best parameters. As rR will be known approximately, fairly narrow limits can be placed on this parameter. The order parameter can be adequately sampled in units of 0.1 from 0 to 1, and ri in units of approximately 25 ps up to 0.5 ns. This is in most cases already exceeding the limits of the validity of the L i p a r i - S z a b o m o d e l . 15,16 This represents perhaps 1000 grid points, with nonlinear optimization at each point. The optimization routine is not critical, although the Marquardt-Levenberg 25 algorithm works well for these systems. In addition, the search also provides detailed information about the precision of the "best" parameters. The total time required to calculate the parameters for all the hydrogen-bearing carbons in the molecule is quite small on modern personal computers. Such a program is also valuable for simulations for a range of nuclei. If the input is the proton frequency, all other frequencies can be straightforwardly generated by multiplying by the ratio of the gyromagnetic ratio to that of the proton (i.e., Yx/TH). This has the advantage that if the gyromagnetic ratios are stored correctly, the sign of the precession of the vector is automatically accounted for. This is an important consideration for 15N, which has a negative yN. The same equations can then be used for all nuclei on input of the range of correlation times, order parameters and internal 25 W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterlin, "Numerical Recipes." Cambridge University Press, Cambridge, New York, 1986.
426
[1 7]
NUCLEIC ACID DYNAMICS TABLE I EFFECT OF CSA ON 31p NMR RELAXATIONa Bo (T)
"rR (ns)
R1 (s -I)
R2 (s -1)
NOE
4.7
2 5 2 5 2 5
0.65/1.46 0.4/0.95 0.23/1.52 0.11/0.72 0.11/1.56 0.05/0.68
1.03/2.54 1.71/4.75 0.73/5.80 1.52/12.9 0.66/11.3 1.49/26.7
1.15/1.07 1.04/1.017 1.05/1.01 1.027/1.004 1.036/1,003 1.024/1.002
9.4 14.1
a Relaxation rate constants were calculated according to Eqs. (2) and (3) and Eqs. (6) and (7) using CSA = 155 ppm, r(eff) = 2.0/~. First entry dipolar terms only, second dipolar + CSA.
correlation times, and CSA. The results are presented in tabular form for plotting as in Figs. 1 and 2 or for display via a contour-plotting routine. Such plots can often be used to get a reasonably good idea of the range of parameters to use for the search/optimization routine.
sip NMR In contrast to 1H and a3C, 31p relaxation occurs by (at least) two mechanisms, namely, dipolar and chemical shift anisotropy (CSA). The former is difficult to analyze for phosphodiesters because the positions of all of the protons and the relative motions of the various P-H vectors cannot be known in advance. Table I shows the phosphorus relaxation rate constants as a function of overall tumbling time for two static field strengths. Ra shows a maximum as a function of correlation time that for practical magnetic field strengths is similar to or somewhat smaller than the correlation times of D N A fragments ranging from 6 to 20 base pairs (ca., 2-6 ns at 298 K). This means that R1 is a weak function of both the correlation time (or size) and the spectrometer frequency for such fragments. In contrast, owing to the J(0) term, R2 increases essentially linearly with increasing correlation time and is proportional to the square of the magnetic field strength. The {H}-P NOE is also small, not only because of the correlation time (the limiting value of the ({H}-P NOE is 1.022), but also because of the large CSA. In practice, because the CSA is 150-160 ppm in B-form DNA, ~3'z6-3° it 26 j. R. Williamson and S. G. Boxer, Biochemistry 28, 2819 (1989). 27 B. T. Nail, W. P. Rothwell, J. S. Waugh, and A. Rupprecht, Biochemistry 20, 1881 (1981). 28 A, N. Lane, T. C. Jenkins, D. J. S. Brown, and T. Brown, Biochem. J. 279, 269 (1991). 29 M. S. Searle and A. N. Lane, FEBS Lett. 297, 292 (1992). 30 H. Shindo, in ,,31p NMR: Principles and Applications" (D. G. Gorenstein, ed.). Academic Press, New York, 1984.
[17]
FAST DYNAMICS OF NUCLEIC ACIDS
427
is found that at B0 >9.4 T, the CSA accounts for >90% of the relaxation, whereas at 4.7 T, the CSA accounts for only 40% of the relaxation. Therefore, for the purposes of relaxation measurements, the highest possible magnetic field strengths should be used when the relaxation rates reduce to those of Eqs. (6) and (7). Furthermore, the combination rates R2 0.5R1 and R2/R1 allow both r and X2 to be determined. Thus, Eq. (9) can be rewritten for a rigid body 26 as = (1/OOx)[(3Rz/2R1) - 7/4] 1/2 = J(0),
(19)
which can be used in Eq. (8) to obtain X2. Note that the correlation time determined refers to the motion of the principal axis of the chemical shift tensor. Equation (8) is also useful to extract the CSA component from fielddependent measurements of R1 and R2 because the difference function R2 - 0.5R1 is essentially of the form a + bo)p2, with the slope directly proportional to the CSA. In the presence of modest amplitude internal motion, such that S2J(O) >> (1 - S2)j(0), the numerator and denominator in Eq. (9) simply become scaled by S 2 and, therefore, cancel so that the correct overall correlation time is recovered. In this case, only the scaled CSA (i.e., S2X 2) would be obtained from Eq. (8). Correlation times determined in this way appear to be comparable to those determined by dynamic light scattering techniques 3~ and larger than those obtained from proton cross-relaxation rate constants (discussed previously).
Effects of Anisotropy
Symmetric top rotors, such as cylinders, can be described by correlation times for rotation about two perpendicular axes ~'L and rs, which can be calculated from the dimensions of the molecule using the formula of Garcia de La Torre 32 or the Perrin equations using the equivalent ellipsoid of r e v o l u t i o n . 33'34 For a symmetric top rotor, the spectral density function includes an explicit dependence on the angle, q5, the dipole-dipole vector makes with the principal axis of rotation. J(to) = 0.25(3 cos 2 -- 1 ) 2 J ( t o , T1) -[- 3 COS2q5sin2qSJ(to, 7"2) + 0.75 sin46j(oJ, r3), 31 W. Eimer, J. R. Williamson, and S. G. Boxer, Biochemistry 29, 799 (1990). 32 M. M. Tirado and J. Garcia de la Torre, J. Chem. Phys. 73, 1986 (1980). 33 D. E. Woessner, J. Chem. Phys. 37, 647 (1962). 34 A. J. Birchall and A. N. Lane, Eur. Biophys. J. 19, 73 (1990).
(20)
428
NUCLEIC ACID DYNAMICS
[ 17]
where ra = ~'L, ~'2 = 6~'L~'s/(5Ts + TL), Z3 = 3~'L~'J(Zs + 2re). re and ~'s are related to the molecular radius (R) and the length (L) by 32
"rE = rrrlL3[ln(L/2R) + ~ ] / 1 8 k ~ T , 7"s = 3.842rr~TLR2[1 + 6s]/6kBT,
(21A) (21B)
where g~ = -0.662 + 0.917 ( 2 R / L ) - O.05(2R/L) 2 and 8s = 1.119 10 -4 + 0.6884(2R/L) - 0.2019(2R/L) 2. kB is Boltzmann's constant and T is the absolute temperature. F o r B-form D N A , the length is usually taken to be 3.38n/~, where n is the n u m b e r of base pairs. T o maintain the same volume as a hydrated sphere, the radius R is between 10 and 11 t~. 31 A rise of 2 . 6 / ~ per base pair and a radius of 12/~ would be m o r e appropriate for the A - f o r m usually adopted by double-stranded R N A . A vector parallel to the long axis of the D N A molecule (4~ = 0) is affected only by end-over-end tumbling, whereas one perpendicular to the helix axis is affected by rotation about both the long and short axes. Furthermore, as shown in Fig. 3, the dependence on 4, is very weak, in the neighborhood of & --- 90 °, so that the spectral density function for a vector that is nearly perpendicular to the helix axis reduces to two terms (Eq. 20). This applies to the H 6 - H 5 vector of cytosines, which lie in the x-y plane, nearly perpendicular to the helix axis in the B-form of D N A . For a macro-
lo.7O809 -"" : "':" :=61 6b'
8'o' ' 2' 1oo
angle/deg FIG. 3. Dependence of J(0) on orientation and axial ratio (nbp). J(0) was calculated usingoEq. (20) for correlation times calculated according to Eq. (21), using an axial rise of 3.38 A per base pair and a helix diameter of 20 A. J(0) is normalized for each curve to the value at an angle of zero (i.e., parallel to the long axis of the helix). The number of basepairs is shown on the right. [] = "IL;• = Teyt; • = "Is.
[17]
429
FAST DYNAMICS OF NUCLEIC ACIDS 3 0 ~ 1
,
,
~
I
'
'
'
~
I
'
i
,
,
I
'
'
'
'
I
'
~
'
'
2 55
15 IO o
-
5 ~
|
5
I0
i
i
i
i
i
,
i
15 20 no. base-pairs
,
i
25
i
i
i
30
FIG. 4. Dependence of correlation times on number of base pairs. The correlation times rL ([3) and rs (11) were calculated as for Fig. 3. Tcyt( ~ ) was calculated from re and rs using Eq. (22).
molecule, the most important component of the homonuclear cross-relaxation rate constant is J(0), which in this case reduces to J(0) = rE/4 + 2.25rL%/(2rL + rs).
(22)
rE and rs both depend on the axial ratio of the DNA, which also varies with the number of base-pairs, as shown in Fig. 4. If the ratio rL/% is denoted F, then rE = 2(2F + 1)J(0)/(F + 5).
(23)
J(0) can be obtained as the effective correlation time determined from the Cyt H6-H5 vector. Hence, in principle, both correlation times can be determined from the cross-relaxation rate constant. Furthermore, both rE and r~ are proportional to the ratio rt/kBT (cf., Eq. (21)), that is, the same dependence as the Stokes-Einstein equation for a sphere: rR = rtV/kBT = rIM(V + h)/kBT where ~ is the viscosity, V is the hydrated volume, M is the relative molecular mass, v is the partial specific volume of the D N A (ca., 0.5 ml g-l), and h is the product of the hydration (ca., 0.6 g/g in D N A ) and the partial specific volume of the bound water (assumed to be 1 ml g-l). Hence, a plot of J(0) versus ~7/T should be linear with a slope of V/ kB, and a plot of J(0) T versus 1/T should also be linear with a slope of Ev/R, where Ev is the effective viscosity for the solvent, which is 17-18 kJ mo1-1 for 020. 35 Both of these relationships have been verified for a range 35D. J. Wilbur, T. DeFries, and J. Jonas, J. Chem. Phys. 65, 1783 (1979).
430
NVCLEIC ACIDDYNAMICS
[ 1 71
of oligonucleotides. 21,26,34The ratio rL/~'Uincreases without limit as the axial ratio increases, whereas rS/rR approaches a constant value of 0.7. Hence, I' also increases without limit as the axial ratio increases. This means that the effective correlation time for vectors that are parallel to the long axis increase strongly with the number of base pairs. The correlation time for vectors that are perpendicular to the long axis (e.g., Cyt H6-H5) also increase with the number of base pairs (see Eq. (22)), with the ratio J(0) (parallel) :J(0)(perpendicular) reaching a limit of four. However, for very long molecules, the spectral density function for vectors oriented near the magic angle eventually becomes zero. This only becomes a significant complication for oligonucleotides longer than approximately 30 basepairs (Fig. 3). For comparison, many internucleotide H-H vectors are oriented nearly parallel to the long axis, whereas many intranucleotide vectors such as H8/H6-H2' lie essentially in the x-y plane. For an oligonucleotide of axial ratio 2 : 1, the error in distance that arises from neglect of anisotropy when using the cytosine H6-H5 correlation time to describe an effective isotropic rotor is at most 7%. For an axial ratio of 3 : 1 (20 bp), the maximum error increases to 12%, which translates to an error of 0.36/~ at a distance of 3 ~. Provided that internal motions are fast, such that they are effectively uncoupled from overall rotation, internal motions can be treated using Eq. (10) and the spectral densities given by Eq. (20). Sources o f Error
Relaxation experiments have been described in detail by many authors. Nonlinear regression to the appropriate equations is the most effective method of determining the relaxation rate constants. Thus, for inversion recovery, the FIRFT method 36 with data-fitting to M(t) = a + b exp(-Rat),
(24)
gives excellent estimates of the important parameter R1 if the delay times t are chosen sensibly. Similarly, R2 can be recovered using FRESCO 37 by fitting to the simple exponential M(t) = Mo e x p ( - R z t ) .
(25)
The choice of sensible delay times for sampling the magnetization decay curve naturally depends on the value of R1 and R2, which are not known a priori. However, approximate values can often be calculated using the equations given here, using a correlation time derived from the Stokes36R. K. Gupta, J. A. Feretti, E. D. Becker, and G. H. Weiss,J. Magn. Reson. 38, 447 (1980). 37M. J. Forster, J. Magn. Reson. 84, 580 (1989).
[17]
FAST DYNAMICS OF NUCLEIC ACIDS
431
Einstein equation (discussed previously). The variation of rotational correlation time with temperature is fairly well established, provided that there is no aggregation at low temperatures and high concentrations as has often been reported. 38 Detailed correlation times for duplex D N A of various lengths at different temperatures have been measured (shown previously). Once the expected relaxation rate constants have been calculated, exponential spacing of the delay times provides optimal sampling of the recovery time courses. Steady-state NOEs are very small for 31p, partly because of the limiting value approached for macromolecules (NOE = 1.31) and partly because at high magnetic field strengths, the CSA contribution to R1 effectively quenches the dipolar NOE. Only the specific {HJ-31pNOE, obtained by selective saturation of a single proton, would be in any way informative, but the magnitude of such an N O E will be extremely small and, therefore, rather imprecise. For 13C, the NOE can be measured as the steady-state value, which simply requires that the waiting time be sufficiently long (i.e., >5 Ta) to avoid attenuation of the NOE. Also, careful attention must be given to decoupling during the acquisition time such that significant differences in temperature gradients are not generated between the onresonance and control experiments. This is more likely to be a significant problem at the highest available spectrometer frequencies and in the presence of high concentration of salt, unless active dielectric shielding is used at the probe stage. An additional problem that is most acute for 31p is contamination with paramagnetic ions, particularly from divalent ions such as Mn 2÷. It is always wise to dialyze the nucleic acid fragment exhaustively against a buffer solution prepared from components that are especially low in paramagnetic species, and at least doubly deionized water, and containing 1 mM or more EDTA. A final dialysis can be made at lower EDTA, such that the concentration in the N M R tube is only a few hundred micromolar. Even then, it is worth checking the T2 measurement at two different concentrations of EDTA. In 13C NMR, one should also bear in mind residual error from incorrect treatment of the CSA component, which becomes most noticeable for the aromatic carbons at high magnetic field strengths. Also, the C5' and C2' (in DNA) are methylenes; therefore, the relaxation is twice as fast as for the other methine carbons. This is strictly true only if the two protons are uncorrelated, which of course is not possible. The magnitude of the error in ignoring correlation effects in this case is uncertain. There are many sources of error in determining correlation times. Some of the systematic sources of error have already been alluded to. Other 38 j. SantaLucia, Jr. and D. H. Turner,
Biochemistry, 32, 12612 (1993).
432
NUCLEIC ACID DYNAMICS
[ 17]
sources of systematic error arise from temperature artifacts, including inaccurate determination of the sample temperature. Systematic errors in the experiments themselves can be reduced by using composite pulses. The random error from a given experiment can be determined from the curvature of the error matrix in the nonlinear least squares routines, although one must bear in mind that these statistical errors do not necessarily equate to true variances. 25 More to the point, this only provides a measure of the random error present in that particular dataset and not to the precision of the actual derived parameter. This can really only be assessed by making repeat measurements, at least on the same sample, if not on independently prepared samples. Often the variance of a relaxation parameter determined in replicate experiments exceeds that derived from the error matrix of a given dataset. Large residual errors in a subset of the relaxation data indicate systematic errors for these sites or additional relaxation mechanisms. This is particularly likely when the calculated value of R2 is substantially smaller than the observed value and would signify the presence of slow motions such as chemical exchange. Determining the rate constants for exchange mechanism can be quite difficult, as it depends critically on whether the exchange is fast, intermediate, or slow. In the fast-intermediate regime, the spin-lattice relaxation rate constant in the rotating frame (R~p) can be useful. 39 Other slow dynamic events such as NH exchange and processes monitored by 2H NMR relaxation methods are discussed in Gurron and Leroy, Chapter [16]. Applications Measurements of the rotational correlation times of oligonucleotides of lengths ranging from 6 to 20 base pairs have been made using 1H, 31p, 13C NMR, and dynamic light scattering. 26'3~ Figure 5 shows the variation of the correlation time for Cyt H6-H5 vectors as a function of the number of base pairs in B-DNA. Although the theoretical function is nonlinear (concave up), the data over this modest range of base pairs can be reasonably well described by a straight line of slope 0.34 ns/bp. A comparison of correlation times determined by dynamic laser light scattering (DLLS) with those determined from cross-relaxation rate constants for cytosine H6-H5 vectors 3~ showed that the DLLS correlation time was larger than that of the NMR correlation time. Both experimentally derived correlation times obeyed the Stokes-Einstein law. For a hairpin molecule, which is expected to behave hydrodynamically as a sphere, the slopes of the plot were equal. 39 A. N. Lane and J-F. Lef~vre, Methods Enzymol. 239, 596 (1994).
[17]
FAST DYNAMICS OF NUCLEIC ACIDS ®
10
. . . . . . . . .
I
. . . . . .
'
'
'
I ' '
'
433 '
'
'
''
'~
8
6
5
10
15
20
no. base-pairs
FIG.5. Experimentalcorrelationtimes.Experimentalcorrelationtimes derivedfromcytosine H6-H5 cross-relaxationrate constants from the literature were converted to values in D20 at 298°K using the Stokes-Einstein relation as described in the text. ([0] calculated; [B] observed)
However, as the number of base pairs was increased, DLLS correlation times became increasingly larger than the NMR correlation times. This is partly because the DLLS method effectively measures the correlation time for end-over-end tumbling, whereas the NMR method measures the effect of tumbling about both the long and short axis of the essentially cylindrical molecule. Using a hydrodynamic model for cylinders,32 the DLLS data could be modeled assuming no effect of internal motion. In contrast, the same parameters applied to the NMR data overestimated the experimental values, indicating that the cytosines undergo internal motions on the subnanosecond time scale. Agreement could be made by assuming an order parameter of about 0.8. This is further supported by 31p NMR relaxation data, which indicate that the correlation time for the CSA component is typically 20% higher than that determined from the cytosine correlation times. These considerations show that the rotational correlation times for double-stranded nucleic acids can readily be accounted for within the general framework of the Stokes-Einstein formalism. Hence, a reasonable estimate of the appropriate correlation times can be calculated for a given solvent from the molecular weight. Conversion to different solvents is straightforward, as the correlation time is a linear function of the solvent viscosity. Thus, rotation in H20 is 20% faster than in D20. Correction to temperatures other than 25 ° is also simple, as the viscosity of D20 varies essentially exponentially over the temperature range 10°-60°, 35 with an
434
NUCLEICACIDDYNAMICS
S2
II
"
t'
i
i
i
'f
[ 171 i
I
0.8 0.6
0.41 0.2 o
C
A
C
T
A
G
T
G
FIG, 6. Variation of order parameters with sequence in d(CACTAGTG)2. Order parameters were taken from Lanefl ([11] 13C2'-H, [1~] 13C3'-H, [I-1] HI'-H2")
apparent activation energy of 18 kJ mole -1. Correlation times calculated in this way will not be exactly correct in practice because of the effects of salt (small) and nucleic acid concentration on viscosity. For detailed work, one should determine the correlation time experimentally on the sample of interest. Compared with other nuclei, 13C has not been extensively used in relaxation studies on synthetic oligonucleotides, presumably because of the low sensitivity at natural abundance (1.1%). Some work has been published on 13C relaxation of polynucleotides, 4° which indicated the presence of significant internal motions. More recently, Williamson and Boxer 26 have used 13C-enriched thymine C6 in a D N A hairpin to measure the mobility of the loop region. The apparent correlation times for the thymidine C6 carbons determined from 13Crelaxation were smaller than those determined from cytosine H6-H5 cross-relaxation rate constants, indicating the presence of mobility of the bases in the loop on the sub-nanosecond time scale. The order parameters for the C6-H vectors were between 0.6 and 0.7, and the effective internal correlation times were 30-50 ps. In addition, phosphates in the loop region showed anomalous T2 values, consistent with relatively slow exchange (k ~ 1600 s-l). These data dearly show the presence of substantial mobility of the unpaired bases over a range of time scales. In a multinuclear NMR study of a D N A octamer related to the trp operator, it was found that order parameters for both C-H and H-H vectors in the deoxyriboses were smallest at the termini and larger for the center of the molecule (Fig. 6).41 The bases showed the least evidence of mobility. 4oM. E. Hogan and O. Jardetzky, Biochemistry 19, 3460 (1980). 41 A. N. Lane, Carbohydr. Res. 221, 123 (1991).
I I 7]
FAST DYNAMICSOF NUCLEICACIDS
435
Borer et al. 42 have recently reached a similar conclusion based on 13C N M R relaxation measurements made at different field strengths (5.9 T, 8.5 T, and 11.8 T), with order parameters being largest for internal nucleotides and the sugars being more mobile than the bases. These results parallel restrained MD refinements of oligonucleotides, where terminal bases are generally less well defined than internal nucleotides, 43 and also the order parameters calculated for proton-proton vectors from long molecular dynamics simulations. 4 The effective internal correlation times were of the order 50 ps or less, but were poorly defined by the data and could be varied between 0 and 100 ps with little influence on the results. Summary Double-stranded oligonucleotides of <10 base pairs are adequately described as an isotropic rotor, using the correlation time for the cytosine H6-H5 vector. For longer fragments, the cylindrical model should be used for detailed analysis of NOEs. The appropriate correlation times can be calculated using the formulae of Tirado and Garcia de la Torre 32 or derived from measurements of the cross-relaxation rate constants for cytosine (or uridine) H6-HS. Order parameters describing the degree of motion of different vectors on the subnanosecond time scale vary substantially, with typical values of S 2 > 0.8 for base vectors and 0.5-0.8 for intrasugar and base-sugar vectors. Order parameters for terminal nucleotides are typically significantly smaller than for internal nueleotides, which may also mean that their conformation will be less well determined in the formalism of a unique structure. The CSA relaxation rates of the phosphodiesters appear to be insensitive to internal motions and may, therefore, provide the most accurate estimate of the overall tumbling time in nucleic acid fragments. Using a combination of relaxation data for different nuclei and different spectrometer frequencies may be expected to yield detailed information about fast motions in nucleic acid fragments. Acknowledgments This work was supported by the Medical Research Council of the United Kingdom. I thank Dr. J. Feeney for commentson the manuscriptand Dr. T. Frenkiel for helpful discussions.
42p. N. Borer, S. R. LaPlante, A. Kumar, N. Zanatta, A. Martin, and G. C. Levy, Biochemistry 33, 2441 (1994). 43U. Schmitz, I. Sethson, W. E. Egan, and T. L. James, J. MoL BioL 4, 510 (1992).
FAST DYNAMICS OF NUCLEIC ACIDS
413
is unaffected by added catalysts and where the exchange time is equal to the base pair lifetime. In the triple helix [d(TaCT4CT4)]2" d(A4GA4GA4), the lifetimes of the CH +. N7G Hoogsteen pairs of the two C*. G. C triplets are 3 s at 15e. The lifetimes of the Watson-Crick components are even longer, approximately 5 rain. 39 The lifetimes of internal C. C ÷ pairs of the four-stranded structure formed by intercalation of parallel oligo-dC duplexes (the i-motif48) are also long, in the hundreds of seconds. 1° Even when end-effects are manifest, as in the case of the internal C2. C2 pair of the tetramer formed by the trinucleotide d(TCC), the lifetime, 500 ms at 15°, is much longer than base pair lifetimes in B-DNA. Such lifetimes of C. C + pairs are part of the diagnostic in the search for the formation of the i-motif.49 48 K. Gehring, J. L. Leroy,and M. Gu6ron, Nature 363, 561 (1993). 49j. L. Leroy,M. Gu6ron,J. L. Mergny,and C. H61~ne,Nucleic Acids Res. 22, 1600 (1994).
[I 7] D e t e r m i n a t i o n
of Fast Dynamics by NMR
of Nucleic Acids
B y ANDREW N. LANE
Introduction The major sources of structural information in NMR are the three-bond coupling constants and the nuclear Overhauser enhancement (NOE). As coupling constants typically lie in the range of 0-10 Hz, any variation of the torsion angle on a time scale shorter than a few milliseconds leads to averaging. The NOE explicitly depends on the correlation time, which sets a characteristic time scale for motions that determine the rate of magnetization transfer. The dominant motion that determines this time scale is that of overall macromolecular rotation. For biopolymers in solution, other (internal) motions that are faster than the overall rotation generally tend to reduce the transfer rate. Motions slower than the molecular tumbling rate affect NOEs by averaging the transfer rates over all the conformation states that are sampled. Double-stranded nucleic acids behave hydrodynamically as cylinders and, therefore, are anisotropic so that magnetisation transfer rates via cross-relaxation may depend on the orientation of the vector to the long axis of the molecule. Hence, it is important to determine the overall rotation rate(s) of the molecule and use additional information METHODS IN ENZYMOLOGY, VOL. 261
Copyright © 1995 by Academic Press, Inc. All rights of reproduction in any form reserved.
414
NUCLEIC ACID DYNAMICS
[ 17]
that can be obtained from relaxation measurements to probe the extent to which a rigid rotor model is applicable. Because D N A and R N A provide a relatively low density of distance constraints, the precision of distances becomes a more important issue than it does for structure determination of proteins. If internal mobility is detected, then the structural refinement procedure needs to be more sophisticated than for a rigid body. Any refinement method that imposes a unique conformation as a constraint on the data under these circumstances must lead to a virtual structure of unclear significance. 1 Thus, refinements based on "back-calculations" of N O E data may be compromised by the averaging unless specific and appropriate motional models are directly incorporated into the refinement procedures. 2-9 Indeed, the concept of structure refinement against distances derived from NOE intensities becomes untenable and preferably should be performed directly against the N O E intensities themselves. Furthermore, the dynamic aspects become an intrinsic part of the description of the molecule. In this chapter, N M R methods for measuring overall tumbling rates and fast dynamic processes in nucleic acids are described. Relaxation Times and Molecular Tumbling Molecular tumbling rates enter into structure calculations because the proton-proton cross-relaxation rate constant, and, therefore, the NOE, is directly proportional to the rotational correlation time. Analysis of relaxation data requires that the effects of overall rotation be separated from those of internal motions. Indeed, for small amplitude internal motions, the relaxation rates are dominated by overall rotation. To analyze internal mobility, it is necessary to determine the contribution from overall molecular rotation. For a sphere there is a single rotational correlation time; however, short segments of duplex D N A can be expected to behave as essentially rigid cylinders as far as global rotation is concerned, for which at least two intrinsic correlation times are needed to describe the rotation. 1 0 . Jardetzky, Biochim. Biophys. Acta 612, 227 (1980). 2 B. A. Borgias, M. Gochin, D. J. Kerwood, and T. L. James, Prog. NMR Spect. 22, 83 (1990). 3 A. N. Lane, Biochim. Biophys. Acta 1049, 189 (1990). 4 T. M. G. Koning, R. Boelens, G. A. van der Marel, J. H. van Boom, and R. Kaptein, Biochemistry 30, 3787 (1991). 5 M. J. J. Bonvin, J. A. C. Rullman, R. M. N. J. Lamerichs, R. Boelens, and R. Kaptein, Proteins 15, 385 (1993). 6 N. B. Ulyanov, U. Schmitz, and T. L. James, J. Biomol. NMR 3, 547 (1993). 7 J-F. Lefbvre, A. N. Lane, and O. Jardetzky, Biochemistry 26, 5076 (1987). 8 p. Yip and D. A. Case, J. Magn. Reson. 83, 643 (1989). 9 A. M. J. J. Bonvin, R. Boelens, and R. Kaptein, J. Biomol. NMR 4, 143 (1994).
[17]
FAST DYNAMICS OF NUCLEIC ACIDS
415
This in turn means that the cross-relaxation rate depends on the orientation of the proton-proton vector within the molecule. Relaxation rate constants depend on the molecular tumbling rates via the spectral density function, which relates the power available at any frequency, including the Larmor frequency, to a correlation time for motion. In the simplest cases of isotropic motion, where the correlation function is exponential, the spectral density function is the familiar Lorentzian: J(o)) -~- "/'/(1 + 0$2T2),
(1)
where r is the correlation time and to is the angular frequency. R e l a x a t i o n Rate C o n s t a n t s
The relaxation rate constants, Ri, depend on the spectral density functions, which in turn depend in detail on the contributing relaxation mechanisms. For the dipolar mechanism, which is the most important mechanism for 1H and 13C, the relaxation rate constants (for unlike spins in the case of protons) for relaxation by a proton at a distance r are given byl°'lL12: R 1 "~- (a/r6)[J(Aoj) + R 2 = (0.5a/r6)[aJ(o)
3J(cox) + 6J(E~o)].
+ J(Aog) + 3J(oJx) + 6J(COH) + 6J(]~o9)].
o" = (a/r6)[6J(~oJ) - J(Aog)].
NOE = 1 + (TH/Tx)o'/R1.
(2) (3)
(4) (5)
X refers to the heteronucleus (e.g., ~SN, a3C, 31p). R1 is the spin-lattice relaxation rate constant, R2 is the spin-spin relaxation constant, o" is the cross-relaxation rate constant, and NOE is the nuclear Overhauser enhancement. a is a constant that depends on the gyromagnetic ratios of the dipolar coupled spins. If J(o~) is in ns and r is in ~ , the value of a is 56.95 for H : H relaxation, 9.33 for 31p:H, 3.60 for 13C:H, and 0.585 for 15N:H. It can be shown from Eqs. (2)-(4) that in macromolecules, R1 is dominated by the J(oJ) term for most heteronuclear interactions, whereas R2 is dominated by the zero-frequency term J(0), which is equal to the rotational correlation time. In the homonuclear case, the dominant term is J(&o) J(0) for all relaxation rate constants. In addition to the dipolar mechanism, relaxation by chemical shift anisotropy (CSA) is very important for 31p in phosphodiesters and to a lesser extent for t3C in aromatic systems. The CSA arises from anisotropic shielding at the nucleus due to a nonspherically symmetric magnetic envil0 A. Abragam, "The Principles of Nuclear Magnetism." Clarendon Press, Oxford, 1968. 11D. Neuhaus and M. P. Williamson, "The Nuclear Overhauser Effect." VCH Publishers,
Inc., New York, 1989. 12D. R. Kearns, CRC Critical Reviews in Biochemistry 15, 237 (1984).
416
NUCLEIC ACID DYNAMICS
[ 17]
ronment created by asymmetric bonding. The contribution to relaxation by CSA is given by Eqs. (6)-(9)1°,13: RI(CSA) = 133.33 X2tOx2J(tox).106, Rz(CSA) = 22.22 X2OJx2[4J(0) + 3J(OJx)].106, R2 - 0.5R1 = 88.88X2¢Ox2J(0).106, R2/Rt = [4J(0) + 3J(oJx)]/6J(oJx),
(6) (7) (8) (9)
where X 2 = Atr2(1 + ~2/3), Ao- is the chemical shift anisotropy and 77 is the asymmetry parameter. In these equations, Ri is in s -1, co is in G.rad s -1, J(oJx) is ns, and X is in ppm. For axially symmetric shift tensors, 77 is of the order --0.6.14 However, in solution, the asymmetry and chemical shift anisotropy terms cannot be separated; therefore, it is convenient to use the apparent anisotropy as given by X. For oligonucleotides, RI(CSA) is only weakly dependent on the magnetic field strength B0 because J(tox) ~ 1/tOx2r (Eq. (6)), whereas R2 increases with the square of the applied magnetic field strength because J(0) >>J(oJx) (Eq. (7)). In contrast, the dipolar contribution is independent of B0 for Rz or decreases with the square of B0 for R1 (Eqs. (2)-(4)). Hence, as the applied field is increased, the CSA mechanism eventually becomes dominant. However, the linewidth of a resonance for a system in fast intermediate chemical exchange also increases with increasing B0 for R2 as CSA. The exchange and CSA contributions to R2 for macromolecules can be distinguished by their different temperature dependencies. CSA should decrease with increasing temperature through the effects on the correlation time, whereas the exchange contribution to R2 may pass through a maximum as the temperature is increased. Fast Internal Motions Fast internal motions refer to fluctuations that are faster than overall rotation and, therefore, can contribute significantly to relaxation processes. In the presence of rapid internal motions, the relaxation rate constants can be considered as the sum of two parts, namely that due to overall rotation of the molecule, described by Eqs. (2)-(9), and contributions from internal rotations having a correlation time Ti.15'16 Thus, (R) = S2.R(TR) + (1 -- S2)R(%),
(10)
13M. J. Forster and A. N. Lane, Eur. Biophys. J. 18, 347 (1990). 14T. L. James, in "Phosphorus-31 NMR: Principles and Applications" (D. G. Gorenstein, ed.). Academic Press, Orlando, 1984. 15 G. Lipari and A. Szabo, J. Am. Chem. Soc. 104~ 4546 (1981). 16G. Lipari and A. Szabo, Biochemistry 20, 6250 (1981).
[17]
FASXDYNAMICSOF NUCLEICACIDS
417
where (R) is the observed (averaged) relaxation rate constant, for example, RI, R2, or o-. re = ' i ' R r i / ( r R + T i ) ~-" 'i'i for T i " ~ T R • R('rR) is the relaxation rate constant for overall molecular rotation, R(%) is the relaxation rate constant for the internal motion having an effective correlation time %, and S 2 is an order parameter that describes the spatial restriction of the motion. For a rigid body, S 2 = 1, and for a freely reorienting vector, S 2 = 0. Recently, an extension to Eq. (10) has been proposed to account for the lSN relaxation data in proteins a7 in which S 2 is expanded as a product of two uncoupled processes. T h e r e are insufficient data so far to determine whether this extension is required to describe relaxation in nucleic acids. The description of the dynamics of the molecule on the time scale of nanoseconds and shorter, therefore, is reduced to determining the values of S 2 and ri for each site and the overall correlation time(s), which should be the same for each site. The values of S 2 and ri can in principle be extracted from frequency-dependent measurements of R, although in many instances ri is poorly determined (discussed later). S 2 can also be calculated for specific motional models TM or calculated from molecular dynamics trajectories. 4,19 The amplitude of the fluctuations must be relatively large to have a significant effect on R. Consider the simple model of rotation of a vector within a cone of semiangle ~b, for which the order parameter is given by
Eq. (11):15'16 S 2 = 0.25 cos2qS(1 + cos qS)2.
(11)
For S 2 = 0.9, q~ = 15 °. A decrease in the relaxation rate constant of less than 10% is unlikely to be detected by relaxation methods, implying that angular fluctuations of less than about 15 ° are unlikely to be reliably detected by NMR. Whereas this may be seen as a limitation to the method, it should be realized that fluctuations of this size have little consequence for determining conformational features of nucleic acids, and it is a moot point whether such small rapid motions could have any functional significance. An order parameter of 0.9 indicates a maximum loss of entropy on complete rigidification of only 0.26 k J mole -1 at 298°K. Because R2 is proportional to J ( O ) = r ~> J(o~), it is clear from Eqs. (3), (7), and (10) that R2 will simply be scaled by the order parameter S 2 for rR > ri for all but the smallest values of S 2. The same principle holds also for the homonuclear cross-relaxation rate constant. In contrast, R~ and the ~vG. M. Clore, A. Szabo, A. Bax, L. E. Kay, P. C. Driscoll, and A. M. Gronenborn, J. Am. Chem. Soc. 112, 4989 (1990). is j. Tropp, J. Phys. Chem. 72, 6035 (1980). 19j. M. Withka, S. Swaminathan, D. L. Beveridge, and P. H. BoRon, J. Am. Chem. Soc. 113, 5041 (1991).
418
NUCLEIC ACID DYNAMICS
[171
N O E for heteronuclei show a greater dependence on the internal correlation time; for very large molecules, the relaxation will become dominated by the internal motion term in Eq. (10). This means that internal motions can be detected from proton measurements and a measure of the amplitude determined, but little information can be derived about the time scale other than the motions must be much faster than the overall tumbling rate. For heteronuclear relaxation, it is sometimes possible to determine a value of ri from the combination of R1, R2, and NOE data, possibly measured at different magnetic field strengths.
Proton Relaxation Correlation times can be measured using protons only if one deals with a single pair of protons whose separation is known and fixed. This means that only the cross-relaxation rate constant is really useful. Estimates of correlation times from the ratio of RE/R~ are not likely to be very accurate for protons for several reasons. As longitudinal relaxation is in principle nonexponential, there is a problem with the definition of R1. Thus, Eq. (2) is the correct expression for the autorelaxation rate constant of unlike spins, which can only be obtained experimentally from a measurement of the initial rate of recovery of magnetization of a particular spin following a selective excitation. In this case, the ratio R2/R1 tends to the fixed ratio of 2.5 for large correlation times 12 (i.e., it is independent of the correlation time). An alternative is to measure the initial slope following a nonselective inversion pulse for which the apparent rate constant for recovery of the magnetization is given by Eq. (12): Rl0 = 2.0(a/r6)[3J(co) + 12J(Eto)].
(12)
The ratio R2/Rlo then depends only on the spectral density function, and in principle an estimate of the correlation time for a rigid isotropic rotor could be determined. Because the recovery of the magnetization is not exponential, it can be difficult in practice to measure the true initial slope. The ratio can give the correlation time only if the relaxation is purely dipolar. There may be a significant contribution from nondipolar relaxation that affects the T1 value (e.g., from paramagnetic agents). Note also that because longitudinal relaxation is multiexponential, fitting a single exponential to the entire recovery curve following a nonselective pulse gives a physically meaningless number. Probably more important is the influence of internal motions. Eqs. (3) and (10) show that R2 is effectively scaled by S 2 for fast (Tint < 0.1 ns) and moderate order parameters (>0.5), whereas R10 will be dominated by the internal motion for to~-~ ~> 1. Hence, the ratio R2/R~o is not a simple function of the overall rotational correlation time alone.
[171
FAST DYNAMICS OF NUCLEIC ACIDS
419
The cross-relaxation rate constant, in contrast, is a measure of a pure dipolar interaction between a pair of protons, such that any correlation time that is determined refers to the motion of that particular vector. The H6-H5 vector of cytosine or uridine is very convenient for determining the overall rotational correlation times for the following reasons. First, the distance is fixed by the covalent geometry and is known (r = 2.45 ~). Second, the H5 is relatively isolated so that the measured NOE is unlikely to be significantly contaminated by spin diffusion. Third, the vector lies in the plane of the base, which is essentially perpendicular to the long axis of duplex D N A and within 20° of the perpendicular in RNA. The best experiment to use is the truncated driven 1D N O E experiment, 2°'21 in which a particular proton is rapidly saturated for a time t (i.e., the magnetization of one proton is selectively clamped to zero). For an isolated spin-pair, the magnetization transfer is described by a single exponential process (whereas the NOESY or transient NOE experiment requires two exponentials). Conditions for obtaining the best quality N O E difference spectra have been discussed by Neuhaus and Williamson. v Furthermore, if H6 is saturated, spin diffusion to other spins near H6 is not significant. The NOE on H5 is then given by Eq. (13): NOE(t) = (o'/0")[1 - exp(-p*t)].
(13)
This is a two-parameter equation, from which both 0" and p* can be recovered by nonlinear regression using standard routines. The parameter p* is relatively poorly determined unless very long irradiation times are used; however, this is not desirable because the relaxation becomes nonexponential under these conditions, p* is in fact only an apparent relaxation rate constant that accounts for nonlinearity in the NOE build-up curve. = Typically, five to six time points spaced essentially linearly adequately sample the N O E time course and provide a good estimate of 0". The apparent correlation time is then most easily found by iterative calculation according to Eq. (14): r = r60-/[(56.92)*(6/(1 + 4toZr2) - 1)].
(14)
In this equation, 0- is in s -1, r is in/~, and r is in ns. An initial estimate of r is found by ignoring the first term in the denominator (which should be near - 1 for macromolecules). Note that the measurement gives a value for the cross-relaxation rate constant. If there is internal motion, this value 20 G. 21 A. 22 C. 92
Wagner and K. WUthrich, J. Magn. Reson. 33, 675 (1979). N. Lane, J-F. Lef~vre, and O. Jardetzky, J. Magn. Reson. 66, 201 (1986), M. Dobson, E. T. Olejnicjak, F. M. Poulsen, and R. G. Ratcliffe, J. Magn. Reson. 48, (1982).
420
NUCLEIC ACID DYNAMICS
[1 7]
will be really the scaled cross-relaxation rate constant, at least for fast motions and modest order parameters. Eq. (14) then yields an apparent correlation time, which for larger molecules (rR > ca 2 ns) is S2ZR. The approximation of an isolated spin-pair is very good for the Cyt H6H5 vector. For example, for a fragment of D N A in the B conformation having a correlation time of 3 ns, the cross-relaxation rate constant for the Cyt H6-H5 vector is -0.78 s -1. Fitting a simulation of the NOE time course using an entire D N A spin-system gave a value of o- = -0.79 s -I in the isolated spin-pair approximation using times up to 0.5 s (NOE intensities up to -31%). The NOE intensities can usually be measured quite accurately because the H5 resonances are often resolved and, therefore, provide a direct calibration of the area in the off-resonance spectrum. In general, it is preferable to acquire the on and off resonance spectra in separate files and perform the subtraction of the free induction decays (FIDs) at the end of the experiment rather than acquire the difference FID in the computer memory. Apart from providing the necessary off-resonance spectrum for normalization, the separate files allow one to examine the quality of saturation. It is important that the system be stable, especially for the short irradiation times. Attention to temperature equilibration and stability is critical, as difference spectra magnify any imperfections. Interleaving the on and off resonance spectra is helpful (we generally interleave 8 or 16 acquisitions), and allowing sufficient time for the spins to relax is also important for accurate work. It is often possible to use a matched filter in the window function to optimize sensitivity, which is obviously important for the smallest NOEs. Good quantitation of NOEs at - 5 % is required to determine o" reliably. Provided that care is taken to measure the NOE intensities accurately, the main source of error in the correlation time is then the uncertainty in the bond lengths and bond angles needed to calculate the proton-proton distance. These uncertainties amount to +0.03 ~ for H6H5, or 1.2%. This translates to an uncertainty in o- and, therefore, in r of about 7%. Hence, one can expect to be able to determine the apparent correlation time by this method to better than about 10%. In principle, the magnitude of the NOE is affected by the pulse repetition rate, as intensities are attenuated by repeating experiments before the spin system has fully relaxed. However, the importance of this in nucleic acids is relatively small because the saturation factor for most protons is very similar after a 90° pulse and the error is proportional to the relative steadystate magnetizations. Thus, for a two-spin system A and B, the rate of magnetization transfer from spin B to spin A during the irradiation period is given by Eq. (15):
[17]
F A S T D Y N A M I C S OF N U C L E I C A C I D S
421
dMa/dt = -pa(Ma - Ma~) + °'Mb~.
(15)
After the observation pulse, all magnetization relaxes (nonexponentiaIly) toward equilibrium and will reach some steady-state value, M(ss), if the sum of the acquisition time and relaxation delay is not sufficiently long. In the off-resonance case, the relaxation during the irradiation period is dMa/dt = -p~(Ma - Ma~) - o'(Mb - Mb~).
(16)
For short irradiation times, when Ma = Ma(ss) and Mb = Mb(ss), the initial rate of change of the difference magnetization becomes: dAMa/dt = ¢rMu(ss).
(17)
Hence, the difference spectrum should be normalized to the on-resonance peak. The error obtained by normalization to other peaks of definable area is likely to be small for protons in nucleic acids as the steady-state magnetizations are all quite similar for recycle times of 4 s or more. The main exception is the Ade C2H and to some extent protons on the 5' nucleotide. Other vectors of fixed or nearly fixed length that can be used include the H2'-H2", HI'-H2", and H2'-H3'. The latter two vectors vary slightly in magnitude as the sugar pucker changes (sugar repuckering is the only motion that can affect the distance). The interpretation of the relaxation rates then requires application of the Tropp model, 18 which provides a recipe for calculating order parameters when the length of the vector is not fixed. For fast motions, the appropriate average is the square of the dipolar Hamiltonian, and the spectral density function is given by Eq. (18)18: J ( 0 ) = 0.2,R
piY2m/r? ,
(18)
m=-2
where Y2mis the normalized spherical harmonic, pi is the fractional contribution of conformational state i, and ri is the internuclear distance in state i. The averaging is done over each discrete conformational state, which requires that the coordinates in each state be known to calculate ri. A further complication that must be dealt with is spin diffusion within the sugar. This requires that all the magnetization transfer curves for the minimum spin system are fitted simultaneously. This is equivalent to finding the relaxation matrix. 23,24 Nonlinear regression to 1D time courses consisting of at least 5 time points and observation of the magnetizations on irradiation of the H I ' , H2', or H2" comprise the minimum effort. Initial 23 p. Koehl and J. F. Lefhvre, J. Magn. Reson. 86, 565 (1990). 24 A. N. Lane and M. J. Forster, Eur. Biophys. J. 17, 221 (1989).
422
NUCLEIC A C I D DYNAMICS
[ 171
estimates for the cross-relaxation and autorelaxation rate constants can be obtained using the known (or calculated) overall tumbling time and initially assuming a rigid body. The autorelaxation rate constants should also be treated as parameters, which then have no special physical significance. The precision of the cross-relaxation rate constant determined in this way is significantly lower than for the H6-H5 pair, but in favorable cases (e.g., where there is sufficient spectral resolution and good quality data), results can be obtained that compare favorably with those derived for heteronuclear relaxation measurements. If an independent measure of the molecular tumbling time is known, the cross-relaxation rate constant for a rigid rotor can be calculated. For protons in large macromolecules, where J(2to) ~ J(0) and Ti ":~ 'TR, the ratio of the measured and calculated crossrelaxation rate constant provides an estimate of S 2, as or ~ S20"(TR).
Alternatively, comparison of apparent correlation times for different sites can be used to show whether or not the rigid rotor approximation is adequate to describe the relaxation. Significant discrepancies that cannot be accounted for by alternative relaxation mechanisms or anisotropic motions have to be ascribed to additional internal motions.
13C NMR 13C relaxation is dominated by the dipolar interaction with the directly attached proton, except for carbonyl carbons and aromatic carbons, where there may be a significant contribution from CSA. The CSA for aliphatic carbons is only 20-30 ppm, which leads to an insignificant effect on relaxation (<1% of the dipolar rate at 11.75 T). For aromatic carbons, the CSA is approximately 150-180 ppm, which leads to approximately 20% contribution to the relaxation rate at 9.4 T, 25% at 11.75 T, and 35% at 14.1 T. In contrast, the contribution from nonbonded protons is up to about 5% for sugar carbons. Unless very detailed field-dependent measurements are made, it will not be possible to determine the CSA component on the sample under investigation. Therefore, it is necessary to calculate the CSA contribution for the correlation time as determined (discussed next) and subtract it from the observed rate. This will lead to some (unknown) error for the aromatic carbons. Figure 1 shows how the relaxation parameters vary as a function of overall tumbling time and spectrometer frequency. As Eq. (3) indicates, for macromolecules, R2 should be only weakly dependent on the spectrometer frequency and essentially linearly dependent on the correlation time, whereas R1 decreases strongly with increasing spectrometer frequency (in
[17]
FAST DYNAMICS OF NUCLEIC ACIDS J '
I
' ' I
' ~
I
423
~ ' ~ I
~
~,~ 6 5 4 3 2 °
1
Oo ,~
451
2
°
4
6
8
I0
, , , i , , , t , , , i , , , i , , ,
4°1 ~
35
-
ao! 25
2
4
6
8
I0
~ns
FIG. 1. Dependence of x3C relaxation rate constants on correlation time. Relaxation rates were calculated using Eqs. (2)-(4) for a rigid isotropic rotor at two magnetic field strengths (9.4 T and 14.1 T). The C-H bond length was taken as 1.095 A. (Top) R1 at 9.4 and 14.1 T and NOE at 9.4 T. (Bottom) R2 at 9.4 and 14.1 T. E],i R1; 0 , 0 NOE.
the limit as B0 -2 for very large molecules) and less strongly as the correlation time increases (~-0-1 in the limit of very large molecules). In the presence o f internal motions, R2 is essentially scaled b y S 2 for short internal correlation times, and the relaxation rate constant remains i n d e p e n d e n t of the s p e c t r o m e t e r frequency. I n contrast, Ra shows m o r e c o m p l e x b e h a v i o r (Fig. 2). F o r correlation times w h e r e COxr ~-- 1, the contribution f r o m overall t u m b l i n g dominates, and R1 is scaled by SL F o r ~Ox z > 1, the contribution f r o m internal m o t i o n b e c o m e s m o r e important, and R1 does n o t scale simply with the o r d e r p a r a m e t e r (except in the limit of infinitely fast internal motions). H e n c e , the ratio R2/R~ is n o t i n d e p e n d e n t of the o r d e r parameter.
424
NUCLEIC ACID DYNAMICS
w
[ 17]
5
o
3q 2 1
O0
4
/
0.2
'
'
'
!
0.4
'
'
'
I
0.6
S2 '
'
'
I
[
1
'
~
5 0.04
1
~ ~
~r/-
. I0
0.8
1
~ 0.08
0.12 'l;e/ns
0.16
0.2
FIG.2. Effect of internal motions on 13Crelaxation rate constants. Relaxation rate constants were calculated for an overall correlation time of 3 ns at a magnetic field strength of 11.75 T, using Eqs. (2)-(4) and Eq. (10). (Top) Dependence of R1 and NOE on the order parameter. The internal correlation time was 40 ps. (Bottom) Dependence of R1 and NOE on internal correlation time for S2 = 0.5. VI,IIR1; O,© NOE.
Furthermore, the N O E decreases with increasing order parameter. In the limit of a rigid spherical rotor, the N O E will be close to its minimum for correlation times larger than about 2 ns at magnetic field strengths of 9.4 T or higher. Therefore, most oligonucleotides of interest ( > 6 bp) would be expected to give rather small NOEs. N O E s significantly larger than expected would indicate the presence of internal motions. Furthermore, sites that give only the minimal N O E can be used to obtain an initial estimate of the correlation time from the ratio R2/R1, as the order parameter is near unity. The only unknown is the correlation time; rcr~ is 1.093 -+ 0.005/~ for the aliphatic carbons and 1.085 __+ 0.005 A for the aromatic
[ 17]
FAST DYNAMICS OF NUCLEIC ACIDS
425
carbons. Because R2 is dominated by J(0) for macromolecules, an initial estimate can also be determined from r 6- Rz/2a. The value of ~"is then obtained by iterative solution to R2/R1, which should then be adequate for predicting the values of R1, R2, and the NOE, or (equivalently o- = (R1.NOE-1)yx/TH) for all sites, and will be the same for all sites in the absence of motion. In practice, because of finite errors in measuring relaxation parameters, especially for 13C at natural abundance where low concentrations tend to give noisy spectra, a simultaneous fit to all three relaxation parameters is likely to give a more reliable estimate of the apparent correlation time at a particular site. When internal motions are present (the usual case), each site can be analyzed as a three parameter problem, namely rR, S2, and ri. This is straightforward in 13C NMR because all three relaxation rates are measured using nonselective experiments. Using the initial estimate for rR, or one derived independently from other measurements, these parameters can easily be found by nonlinear regression if at least three observations are available. In addition, frequency-dependent measurements of R~ are useful for sampling the internal dynamics component. Because there are several local minima in the target function, it is probably best to use a combined grid search with local optimization to find the best parameters. As rR will be known approximately, fairly narrow limits can be placed on this parameter. The order parameter can be adequately sampled in units of 0.1 from 0 to 1, and ri in units of approximately 25 ps up to 0.5 ns. This is in most cases already exceeding the limits of the validity of the L i p a r i - S z a b o m o d e l . 15,16 This represents perhaps 1000 grid points, with nonlinear optimization at each point. The optimization routine is not critical, although the Marquardt-Levenberg 25 algorithm works well for these systems. In addition, the search also provides detailed information about the precision of the "best" parameters. The total time required to calculate the parameters for all the hydrogen-bearing carbons in the molecule is quite small on modern personal computers. Such a program is also valuable for simulations for a range of nuclei. If the input is the proton frequency, all other frequencies can be straightforwardly generated by multiplying by the ratio of the gyromagnetic ratio to that of the proton (i.e., Yx/TH). This has the advantage that if the gyromagnetic ratios are stored correctly, the sign of the precession of the vector is automatically accounted for. This is an important consideration for 15N, which has a negative yN. The same equations can then be used for all nuclei on input of the range of correlation times, order parameters and internal 25 W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterlin, "Numerical Recipes." Cambridge University Press, Cambridge, New York, 1986.
426
[1 7]
NUCLEIC ACID DYNAMICS TABLE I EFFECT OF CSA ON 31p NMR RELAXATIONa Bo (T)
"rR (ns)
R1 (s -I)
R2 (s -1)
NOE
4.7
2 5 2 5 2 5
0.65/1.46 0.4/0.95 0.23/1.52 0.11/0.72 0.11/1.56 0.05/0.68
1.03/2.54 1.71/4.75 0.73/5.80 1.52/12.9 0.66/11.3 1.49/26.7
1.15/1.07 1.04/1.017 1.05/1.01 1.027/1.004 1.036/1,003 1.024/1.002
9.4 14.1
a Relaxation rate constants were calculated according to Eqs. (2) and (3) and Eqs. (6) and (7) using CSA = 155 ppm, r(eff) = 2.0/~. First entry dipolar terms only, second dipolar + CSA.
correlation times, and CSA. The results are presented in tabular form for plotting as in Figs. 1 and 2 or for display via a contour-plotting routine. Such plots can often be used to get a reasonably good idea of the range of parameters to use for the search/optimization routine.
sip NMR In contrast to 1H and a3C, 31p relaxation occurs by (at least) two mechanisms, namely, dipolar and chemical shift anisotropy (CSA). The former is difficult to analyze for phosphodiesters because the positions of all of the protons and the relative motions of the various P-H vectors cannot be known in advance. Table I shows the phosphorus relaxation rate constants as a function of overall tumbling time for two static field strengths. Ra shows a maximum as a function of correlation time that for practical magnetic field strengths is similar to or somewhat smaller than the correlation times of D N A fragments ranging from 6 to 20 base pairs (ca., 2-6 ns at 298 K). This means that R1 is a weak function of both the correlation time (or size) and the spectrometer frequency for such fragments. In contrast, owing to the J(0) term, R2 increases essentially linearly with increasing correlation time and is proportional to the square of the magnetic field strength. The {H}-P NOE is also small, not only because of the correlation time (the limiting value of the ({H}-P NOE is 1.022), but also because of the large CSA. In practice, because the CSA is 150-160 ppm in B-form DNA, ~3'z6-3° it 26 j. R. Williamson and S. G. Boxer, Biochemistry 28, 2819 (1989). 27 B. T. Nail, W. P. Rothwell, J. S. Waugh, and A. Rupprecht, Biochemistry 20, 1881 (1981). 28 A, N. Lane, T. C. Jenkins, D. J. S. Brown, and T. Brown, Biochem. J. 279, 269 (1991). 29 M. S. Searle and A. N. Lane, FEBS Lett. 297, 292 (1992). 30 H. Shindo, in ,,31p NMR: Principles and Applications" (D. G. Gorenstein, ed.). Academic Press, New York, 1984.
[17]
FAST DYNAMICS OF NUCLEIC ACIDS
427
is found that at B0 >9.4 T, the CSA accounts for >90% of the relaxation, whereas at 4.7 T, the CSA accounts for only 40% of the relaxation. Therefore, for the purposes of relaxation measurements, the highest possible magnetic field strengths should be used when the relaxation rates reduce to those of Eqs. (6) and (7). Furthermore, the combination rates R2 0.5R1 and R2/R1 allow both r and X2 to be determined. Thus, Eq. (9) can be rewritten for a rigid body 26 as = (1/OOx)[(3Rz/2R1) - 7/4] 1/2 = J(0),
(19)
which can be used in Eq. (8) to obtain X2. Note that the correlation time determined refers to the motion of the principal axis of the chemical shift tensor. Equation (8) is also useful to extract the CSA component from fielddependent measurements of R1 and R2 because the difference function R2 - 0.5R1 is essentially of the form a + bo)p2, with the slope directly proportional to the CSA. In the presence of modest amplitude internal motion, such that S2J(O) >> (1 - S2)j(0), the numerator and denominator in Eq. (9) simply become scaled by S 2 and, therefore, cancel so that the correct overall correlation time is recovered. In this case, only the scaled CSA (i.e., S2X 2) would be obtained from Eq. (8). Correlation times determined in this way appear to be comparable to those determined by dynamic light scattering techniques 3~ and larger than those obtained from proton cross-relaxation rate constants (discussed previously).
Effects of Anisotropy
Symmetric top rotors, such as cylinders, can be described by correlation times for rotation about two perpendicular axes ~'L and rs, which can be calculated from the dimensions of the molecule using the formula of Garcia de La Torre 32 or the Perrin equations using the equivalent ellipsoid of r e v o l u t i o n . 33'34 For a symmetric top rotor, the spectral density function includes an explicit dependence on the angle, q5, the dipole-dipole vector makes with the principal axis of rotation. J(to) = 0.25(3 cos 2 -- 1 ) 2 J ( t o , T1) -[- 3 COS2q5sin2qSJ(to, 7"2) + 0.75 sin46j(oJ, r3), 31 W. Eimer, J. R. Williamson, and S. G. Boxer, Biochemistry 29, 799 (1990). 32 M. M. Tirado and J. Garcia de la Torre, J. Chem. Phys. 73, 1986 (1980). 33 D. E. Woessner, J. Chem. Phys. 37, 647 (1962). 34 A. J. Birchall and A. N. Lane, Eur. Biophys. J. 19, 73 (1990).
(20)
428
NUCLEIC ACID DYNAMICS
[ 17]
where ra = ~'L, ~'2 = 6~'L~'s/(5Ts + TL), Z3 = 3~'L~'J(Zs + 2re). re and ~'s are related to the molecular radius (R) and the length (L) by 32
"rE = rrrlL3[ln(L/2R) + ~ ] / 1 8 k ~ T , 7"s = 3.842rr~TLR2[1 + 6s]/6kBT,
(21A) (21B)
where g~ = -0.662 + 0.917 ( 2 R / L ) - O.05(2R/L) 2 and 8s = 1.119 10 -4 + 0.6884(2R/L) - 0.2019(2R/L) 2. kB is Boltzmann's constant and T is the absolute temperature. F o r B-form D N A , the length is usually taken to be 3.38n/~, where n is the n u m b e r of base pairs. T o maintain the same volume as a hydrated sphere, the radius R is between 10 and 11 t~. 31 A rise of 2 . 6 / ~ per base pair and a radius of 12/~ would be m o r e appropriate for the A - f o r m usually adopted by double-stranded R N A . A vector parallel to the long axis of the D N A molecule (4~ = 0) is affected only by end-over-end tumbling, whereas one perpendicular to the helix axis is affected by rotation about both the long and short axes. Furthermore, as shown in Fig. 3, the dependence on 4, is very weak, in the neighborhood of & --- 90 °, so that the spectral density function for a vector that is nearly perpendicular to the helix axis reduces to two terms (Eq. 20). This applies to the H 6 - H 5 vector of cytosines, which lie in the x-y plane, nearly perpendicular to the helix axis in the B-form of D N A . For a macro-
lo.7O809 -"" : "':" :=61 6b'
8'o' ' 2' 1oo
angle/deg FIG. 3. Dependence of J(0) on orientation and axial ratio (nbp). J(0) was calculated usingoEq. (20) for correlation times calculated according to Eq. (21), using an axial rise of 3.38 A per base pair and a helix diameter of 20 A. J(0) is normalized for each curve to the value at an angle of zero (i.e., parallel to the long axis of the helix). The number of basepairs is shown on the right. [] = "IL;• = Teyt; • = "Is.
[17]
429
FAST DYNAMICS OF NUCLEIC ACIDS 3 0 ~ 1
,
,
~
I
'
'
'
~
I
'
i
,
,
I
'
'
'
'
I
'
~
'
'
2 55
15 IO o
-
5 ~
|
5
I0
i
i
i
i
i
,
i
15 20 no. base-pairs
,
i
25
i
i
i
30
FIG. 4. Dependence of correlation times on number of base pairs. The correlation times rL ([3) and rs (11) were calculated as for Fig. 3. Tcyt( ~ ) was calculated from re and rs using Eq. (22).
molecule, the most important component of the homonuclear cross-relaxation rate constant is J(0), which in this case reduces to J(0) = rE/4 + 2.25rL%/(2rL + rs).
(22)
rE and rs both depend on the axial ratio of the DNA, which also varies with the number of base-pairs, as shown in Fig. 4. If the ratio rL/% is denoted F, then rE = 2(2F + 1)J(0)/(F + 5).
(23)
J(0) can be obtained as the effective correlation time determined from the Cyt H6-H5 vector. Hence, in principle, both correlation times can be determined from the cross-relaxation rate constant. Furthermore, both rE and r~ are proportional to the ratio rt/kBT (cf., Eq. (21)), that is, the same dependence as the Stokes-Einstein equation for a sphere: rR = rtV/kBT = rIM(V + h)/kBT where ~ is the viscosity, V is the hydrated volume, M is the relative molecular mass, v is the partial specific volume of the D N A (ca., 0.5 ml g-l), and h is the product of the hydration (ca., 0.6 g/g in D N A ) and the partial specific volume of the bound water (assumed to be 1 ml g-l). Hence, a plot of J(0) versus ~7/T should be linear with a slope of V/ kB, and a plot of J(0) T versus 1/T should also be linear with a slope of Ev/R, where Ev is the effective viscosity for the solvent, which is 17-18 kJ mo1-1 for 020. 35 Both of these relationships have been verified for a range 35D. J. Wilbur, T. DeFries, and J. Jonas, J. Chem. Phys. 65, 1783 (1979).
430
NVCLEIC ACIDDYNAMICS
[ 1 71
of oligonucleotides. 21,26,34The ratio rL/~'Uincreases without limit as the axial ratio increases, whereas rS/rR approaches a constant value of 0.7. Hence, I' also increases without limit as the axial ratio increases. This means that the effective correlation time for vectors that are parallel to the long axis increase strongly with the number of base pairs. The correlation time for vectors that are perpendicular to the long axis (e.g., Cyt H6-H5) also increase with the number of base pairs (see Eq. (22)), with the ratio J(0) (parallel) :J(0)(perpendicular) reaching a limit of four. However, for very long molecules, the spectral density function for vectors oriented near the magic angle eventually becomes zero. This only becomes a significant complication for oligonucleotides longer than approximately 30 basepairs (Fig. 3). For comparison, many internucleotide H-H vectors are oriented nearly parallel to the long axis, whereas many intranucleotide vectors such as H8/H6-H2' lie essentially in the x-y plane. For an oligonucleotide of axial ratio 2 : 1, the error in distance that arises from neglect of anisotropy when using the cytosine H6-H5 correlation time to describe an effective isotropic rotor is at most 7%. For an axial ratio of 3 : 1 (20 bp), the maximum error increases to 12%, which translates to an error of 0.36/~ at a distance of 3 ~. Provided that internal motions are fast, such that they are effectively uncoupled from overall rotation, internal motions can be treated using Eq. (10) and the spectral densities given by Eq. (20). Sources o f Error
Relaxation experiments have been described in detail by many authors. Nonlinear regression to the appropriate equations is the most effective method of determining the relaxation rate constants. Thus, for inversion recovery, the FIRFT method 36 with data-fitting to M(t) = a + b exp(-Rat),
(24)
gives excellent estimates of the important parameter R1 if the delay times t are chosen sensibly. Similarly, R2 can be recovered using FRESCO 37 by fitting to the simple exponential M(t) = Mo e x p ( - R z t ) .
(25)
The choice of sensible delay times for sampling the magnetization decay curve naturally depends on the value of R1 and R2, which are not known a priori. However, approximate values can often be calculated using the equations given here, using a correlation time derived from the Stokes36R. K. Gupta, J. A. Feretti, E. D. Becker, and G. H. Weiss,J. Magn. Reson. 38, 447 (1980). 37M. J. Forster, J. Magn. Reson. 84, 580 (1989).
[17]
FAST DYNAMICS OF NUCLEIC ACIDS
431
Einstein equation (discussed previously). The variation of rotational correlation time with temperature is fairly well established, provided that there is no aggregation at low temperatures and high concentrations as has often been reported. 38 Detailed correlation times for duplex D N A of various lengths at different temperatures have been measured (shown previously). Once the expected relaxation rate constants have been calculated, exponential spacing of the delay times provides optimal sampling of the recovery time courses. Steady-state NOEs are very small for 31p, partly because of the limiting value approached for macromolecules (NOE = 1.31) and partly because at high magnetic field strengths, the CSA contribution to R1 effectively quenches the dipolar NOE. Only the specific {HJ-31pNOE, obtained by selective saturation of a single proton, would be in any way informative, but the magnitude of such an N O E will be extremely small and, therefore, rather imprecise. For 13C, the NOE can be measured as the steady-state value, which simply requires that the waiting time be sufficiently long (i.e., >5 Ta) to avoid attenuation of the NOE. Also, careful attention must be given to decoupling during the acquisition time such that significant differences in temperature gradients are not generated between the onresonance and control experiments. This is more likely to be a significant problem at the highest available spectrometer frequencies and in the presence of high concentration of salt, unless active dielectric shielding is used at the probe stage. An additional problem that is most acute for 31p is contamination with paramagnetic ions, particularly from divalent ions such as Mn 2÷. It is always wise to dialyze the nucleic acid fragment exhaustively against a buffer solution prepared from components that are especially low in paramagnetic species, and at least doubly deionized water, and containing 1 mM or more EDTA. A final dialysis can be made at lower EDTA, such that the concentration in the N M R tube is only a few hundred micromolar. Even then, it is worth checking the T2 measurement at two different concentrations of EDTA. In 13C NMR, one should also bear in mind residual error from incorrect treatment of the CSA component, which becomes most noticeable for the aromatic carbons at high magnetic field strengths. Also, the C5' and C2' (in DNA) are methylenes; therefore, the relaxation is twice as fast as for the other methine carbons. This is strictly true only if the two protons are uncorrelated, which of course is not possible. The magnitude of the error in ignoring correlation effects in this case is uncertain. There are many sources of error in determining correlation times. Some of the systematic sources of error have already been alluded to. Other 38 j. SantaLucia, Jr. and D. H. Turner,
Biochemistry, 32, 12612 (1993).
432
NUCLEIC ACID DYNAMICS
[ 17]
sources of systematic error arise from temperature artifacts, including inaccurate determination of the sample temperature. Systematic errors in the experiments themselves can be reduced by using composite pulses. The random error from a given experiment can be determined from the curvature of the error matrix in the nonlinear least squares routines, although one must bear in mind that these statistical errors do not necessarily equate to true variances. 25 More to the point, this only provides a measure of the random error present in that particular dataset and not to the precision of the actual derived parameter. This can really only be assessed by making repeat measurements, at least on the same sample, if not on independently prepared samples. Often the variance of a relaxation parameter determined in replicate experiments exceeds that derived from the error matrix of a given dataset. Large residual errors in a subset of the relaxation data indicate systematic errors for these sites or additional relaxation mechanisms. This is particularly likely when the calculated value of R2 is substantially smaller than the observed value and would signify the presence of slow motions such as chemical exchange. Determining the rate constants for exchange mechanism can be quite difficult, as it depends critically on whether the exchange is fast, intermediate, or slow. In the fast-intermediate regime, the spin-lattice relaxation rate constant in the rotating frame (R~p) can be useful. 39 Other slow dynamic events such as NH exchange and processes monitored by 2H NMR relaxation methods are discussed in Gurron and Leroy, Chapter [16]. Applications Measurements of the rotational correlation times of oligonucleotides of lengths ranging from 6 to 20 base pairs have been made using 1H, 31p, 13C NMR, and dynamic light scattering. 26'3~ Figure 5 shows the variation of the correlation time for Cyt H6-H5 vectors as a function of the number of base pairs in B-DNA. Although the theoretical function is nonlinear (concave up), the data over this modest range of base pairs can be reasonably well described by a straight line of slope 0.34 ns/bp. A comparison of correlation times determined by dynamic laser light scattering (DLLS) with those determined from cross-relaxation rate constants for cytosine H6-H5 vectors 3~ showed that the DLLS correlation time was larger than that of the NMR correlation time. Both experimentally derived correlation times obeyed the Stokes-Einstein law. For a hairpin molecule, which is expected to behave hydrodynamically as a sphere, the slopes of the plot were equal. 39 A. N. Lane and J-F. Lef~vre, Methods Enzymol. 239, 596 (1994).
[17]
FAST DYNAMICS OF NUCLEIC ACIDS ®
10
. . . . . . . . .
I
. . . . . .
'
'
'
I ' '
'
433 '
'
'
''
'~
8
6
5
10
15
20
no. base-pairs
FIG.5. Experimentalcorrelationtimes.Experimentalcorrelationtimes derivedfromcytosine H6-H5 cross-relaxationrate constants from the literature were converted to values in D20 at 298°K using the Stokes-Einstein relation as described in the text. ([0] calculated; [B] observed)
However, as the number of base pairs was increased, DLLS correlation times became increasingly larger than the NMR correlation times. This is partly because the DLLS method effectively measures the correlation time for end-over-end tumbling, whereas the NMR method measures the effect of tumbling about both the long and short axis of the essentially cylindrical molecule. Using a hydrodynamic model for cylinders,32 the DLLS data could be modeled assuming no effect of internal motion. In contrast, the same parameters applied to the NMR data overestimated the experimental values, indicating that the cytosines undergo internal motions on the subnanosecond time scale. Agreement could be made by assuming an order parameter of about 0.8. This is further supported by 31p NMR relaxation data, which indicate that the correlation time for the CSA component is typically 20% higher than that determined from the cytosine correlation times. These considerations show that the rotational correlation times for double-stranded nucleic acids can readily be accounted for within the general framework of the Stokes-Einstein formalism. Hence, a reasonable estimate of the appropriate correlation times can be calculated for a given solvent from the molecular weight. Conversion to different solvents is straightforward, as the correlation time is a linear function of the solvent viscosity. Thus, rotation in H20 is 20% faster than in D20. Correction to temperatures other than 25 ° is also simple, as the viscosity of D20 varies essentially exponentially over the temperature range 10°-60°, 35 with an
434
NUCLEICACIDDYNAMICS
S2
II
"
t'
i
i
i
'f
[ 171 i
I
0.8 0.6
0.41 0.2 o
C
A
C
T
A
G
T
G
FIG, 6. Variation of order parameters with sequence in d(CACTAGTG)2. Order parameters were taken from Lanefl ([11] 13C2'-H, [1~] 13C3'-H, [I-1] HI'-H2")
apparent activation energy of 18 kJ mole -1. Correlation times calculated in this way will not be exactly correct in practice because of the effects of salt (small) and nucleic acid concentration on viscosity. For detailed work, one should determine the correlation time experimentally on the sample of interest. Compared with other nuclei, 13C has not been extensively used in relaxation studies on synthetic oligonucleotides, presumably because of the low sensitivity at natural abundance (1.1%). Some work has been published on 13C relaxation of polynucleotides, 4° which indicated the presence of significant internal motions. More recently, Williamson and Boxer 26 have used 13C-enriched thymine C6 in a D N A hairpin to measure the mobility of the loop region. The apparent correlation times for the thymidine C6 carbons determined from 13Crelaxation were smaller than those determined from cytosine H6-H5 cross-relaxation rate constants, indicating the presence of mobility of the bases in the loop on the sub-nanosecond time scale. The order parameters for the C6-H vectors were between 0.6 and 0.7, and the effective internal correlation times were 30-50 ps. In addition, phosphates in the loop region showed anomalous T2 values, consistent with relatively slow exchange (k ~ 1600 s-l). These data dearly show the presence of substantial mobility of the unpaired bases over a range of time scales. In a multinuclear NMR study of a D N A octamer related to the trp operator, it was found that order parameters for both C-H and H-H vectors in the deoxyriboses were smallest at the termini and larger for the center of the molecule (Fig. 6).41 The bases showed the least evidence of mobility. 4oM. E. Hogan and O. Jardetzky, Biochemistry 19, 3460 (1980). 41 A. N. Lane, Carbohydr. Res. 221, 123 (1991).
I I 7]
FAST DYNAMICSOF NUCLEICACIDS
435
Borer et al. 42 have recently reached a similar conclusion based on 13C N M R relaxation measurements made at different field strengths (5.9 T, 8.5 T, and 11.8 T), with order parameters being largest for internal nucleotides and the sugars being more mobile than the bases. These results parallel restrained MD refinements of oligonucleotides, where terminal bases are generally less well defined than internal nucleotides, 43 and also the order parameters calculated for proton-proton vectors from long molecular dynamics simulations. 4 The effective internal correlation times were of the order 50 ps or less, but were poorly defined by the data and could be varied between 0 and 100 ps with little influence on the results. Summary Double-stranded oligonucleotides of <10 base pairs are adequately described as an isotropic rotor, using the correlation time for the cytosine H6-H5 vector. For longer fragments, the cylindrical model should be used for detailed analysis of NOEs. The appropriate correlation times can be calculated using the formulae of Tirado and Garcia de la Torre 32 or derived from measurements of the cross-relaxation rate constants for cytosine (or uridine) H6-HS. Order parameters describing the degree of motion of different vectors on the subnanosecond time scale vary substantially, with typical values of S 2 > 0.8 for base vectors and 0.5-0.8 for intrasugar and base-sugar vectors. Order parameters for terminal nucleotides are typically significantly smaller than for internal nueleotides, which may also mean that their conformation will be less well determined in the formalism of a unique structure. The CSA relaxation rates of the phosphodiesters appear to be insensitive to internal motions and may, therefore, provide the most accurate estimate of the overall tumbling time in nucleic acid fragments. Using a combination of relaxation data for different nuclei and different spectrometer frequencies may be expected to yield detailed information about fast motions in nucleic acid fragments. Acknowledgments This work was supported by the Medical Research Council of the United Kingdom. I thank Dr. J. Feeney for commentson the manuscriptand Dr. T. Frenkiel for helpful discussions.
42p. N. Borer, S. R. LaPlante, A. Kumar, N. Zanatta, A. Martin, and G. C. Levy, Biochemistry 33, 2441 (1994). 43U. Schmitz, I. Sethson, W. E. Egan, and T. L. James, J. MoL BioL 4, 510 (1992).