Truncated driven nuclear overhauser effect (TOE). A new technique for studies of selective 1H1H overhauser effects in the presence of spin diffusion

JOURNAL

OF

MAGNETIC

RESONANCE

33,

675-680

(1979)

Truncated Driven Nuclear Overhauser Effect (TOE). A New Technique for Studies of Selective ‘H-lH Overhauser Effects in the Presence of Spin Diffusion The potential of nuclear Overhauser effects (NOE) for studies of nonbonding nearest-neighbor interactions between protons in biological macromolecules has been recognized, and several applications of ‘H-iH NOES for improving the spectral resolution, assigning individual resonances, and characterizing local spatial structures were described (1-7). However, it was also pointed out that spin diffusion is of considerable importance in proteins (S-10), causing the conventional steady-state NOES (11) to be less specific and hence less useful. Theory shows that, in contrast, the initial buildup rates of NOES are simply related to the inverse sixth power of the distance between the observed and the presaturated proton (8, 9, 1 l-13). It is hence of great practical interest to develop techniques which, upon selective irradiation of individual resonance lines, enable one to observe spectral features which are in a simple way related to the initial buildup rates of the NOES. Recently we have demonstrated that transient NOE difference spectra obtained with the following pulse sequence are a particularly straightforward technique for measuring initial NOE buildup rates (13): [-180”(~~)-rl-observation

pulse-r2-180”(w,tr-,,,)-r1-observation

pulse-r2-1,.

Ill This experiment is initiated by a selective 180” pulse, with a pulse length of typically 10 msec. The observation pulse follows after a waiting time 71 during which the NOES are built up in the absence of a radiofrequency field. The reference spectrum without NOE is recorded after a waiting period r2 during which the spin system is allowed to recover. While experiment [l] proved to be very useful for studies of hemoproteins, where the ‘H NMR spectra contain numerous well-separated lines (13, 14), experience with other systems showed that a reasonable compromise between high selectivity of the presaturation pulse and a workable signal-to-noise ratio in the transient NOE spectra was difficult to obtain in crowded spectral regions (15). The experiment described in this paper, for which we suggest the name “TOE difference spectroscopy,” was found to be more useful in practice, although the interpretation may not be quite as straightforward as that for the transient NOE experiments (12,13). TOE difference spectra can be recorded with the experiment: [-ti(w*)-observation

pulse-t2-tl(w,ff~,,,)-observation

pulse-t2-I,.

PI

A selective low-power radiofrequency field is applied to resonance A for a period of time tl, which is followed immediately by the observation pulse. After a waiting time tZ, a reference spectrum without NOE is recorded. The free-induction decays with 675

0022%2364/79/030675-O6SOZ.OOiO Copyright 0 1979 by Academtc Press, Inc. All rights of reproduction in any form reserved Prmted in Great Britain

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and without NOE are stored in different parts of the memory. where they arc accumulated alternatively to minimize effects of instrumental drifts 151. ‘1~11~ difference spectrum is obtained by subtracting the FID with NOE from that without NOE. In contrast to transient NOE experiments, where no radiofrequency is applied while the NOES are built up and decay (13), the radiofrequency field WA in [2 ] is applied over the entire time span tl during which NOES in the spin system are built up. To distinguish experiments of type [2] from transient NOES, we shall use the term “radiofrequency driven NOES” or, for short, “driven NOES.” If the irradiation period tl in a driven NOE experiment is sufficiently long, a conventional steady-state NOE difference spectrum is obtained. On the other hand, if experiment [2] is used to record a series of “truncated driven NOES” (TOE) with different t,‘s which are all short compared to the irradiation time required for a steady-state NOE, the buildup of the driven NOES for individual resonance lines can be followed. To provide a basis for the interpretation of TOE difference spectra, the dependence of driven NOES on the irradiation time ti is investigated in the following. For simplicity, the complex situation encountered in a biological macromolecule is approximated by a three-spin system, where the location of the third spin relative to the irradiated spin and the observed spin is varied in order to simdlate the influence of the environment in a macromolecule. The time dependence of the magnetization of the nonirradiated spins i in a driven NOE experiment is determined by the equation (8):

where M, is the difference between the actual magnetization M,, of spin i and its equilibrium magnetization A%:. The quantities pi and ur, determine the spin-lattice relaxation and the spin diffusion, respectively (8):

[41 and h2y4 1 --

67c

1

ui;= 10 rfi 1+4(wTc)2-7c ’

[51

where w is the Larmor frequency, r;j the distance between spins i and j, 27rh the Planck constant, y the gyromagnetic ratio, and ‘TVthe effective rotational correlation time. In the three-spin system, we denote as spin 1 the irradiated nucleus, as spin 2 the observed nucleus, and as spin 3 the nucleus which represents the influence of additional spins on the driven NOE. Let us first consider spin 1. Following Torrey (36, 17), a damped precession of the magnetization about the effective magnetic field in the rotating frame is to be expected for the irradiated nucleus in a driven NOE experiment. When the irradiation is exactly at resonance, the time course of the z component of the magnetization is given by (16) B cos (s$!Cit)+Ssin

(syHrt)

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The constants B, C, and s have been calculated by Torrey (16). The exponent in the damping term is (16) b.yH,

=; (i+$-),

[71

and D accounts for the residual magnetization when the steady state is attained. In the situation of weak irradiation, the damping may be sufficiently fast to suppress the oscillations. In addition to [7], effective damping is further expected to result from inhomogeneities of HI and Ho (16). From these considerations we decided to replace the oscillatory term in [6] by a constant and to use the approximate description of the magnetization of spin 1 given by M,=O,

for r < 0.

=&ff(l

-eJ’),

for r 2 0,

[81

where M, = M, i -k:‘, and c includes [7] as well as the influence of field inhomogeneities. The quantity My is the difference between the equilibrium magnetization and the residual magnetization of the steady state (IM:i < lj@i). With Mi given by Eq. [8], the two coupled differential equations (Eq. [3]) for spins 2 and 3 can be solved readily. For the magnetization M2 of the observed spin we thus obtain: M2

M:‘-

tq21p3 (a-b)ab

-*(e-“‘-e-h’)+

(b-a)

flZlp3

-

(T23@3

+(a-b)(b-c)(c-a) fl21

1

q23c31)

(bepa’-ae?‘+(a-b))

((b-c)e~“‘+(c-a)e~h’+(a-b)e~C’)

PI

(c(a-b)e-“+a(b-c)e-“‘+b(c-a)embr),

-(a-b)(b-&-a) with U

=~{(p2+P3)-[(P2-P3)2+4~~31”2},

and b = : ((~2

+ ~3)

+ Lb2

~3)~

+

4d3

11'21.

With Eqs. [9] to [ll] the dependence of M,/M? on the irradiation time tl was computed for various situations of practical interest (Fig. 1). In Fig. 1A experimentally meaningful values for c in’Eq. [8] are compared with the hypothetical case c = co, where the initial slope would be equal to uzl and hence the distance rr2 could be calculated from Eq. [5]. From [7] one estimates a lower limit of c 2 10 see-’ for protons in a protein. Figure 1 shows that for c values of this order, the time course of a driven NOE experiment starts with a short induction period but is overall closely similar to that for c = co. In a previous analysis of data obtained with the basic pancreatic trypsin inhibitor [15], curves of the type of Fig. 1A provided a satisfactory description of the experiments, which indicates that the approximations made using Eq. [S] are reasonable even for small proteins.

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FIG. 1. Computed plots of -Ma/M:’ vs the pulse length rr for a three-spin system in driven NOE experiments (Eq. [9]). Spin 1 is the irradiated proton, spin 2 is the observed proton, and spin 3 represents the influence of additional protons. All computations are for 360 MHz and an effective correlation time 7, of 3 x 10m9 sec. (A) Variation of the constant c (Eq. [8]). The values of c are 00,20,10,5, and 1 set-’ from the top curve to the bottom curve. rIz = 2.2 A, rr3 = 2.9 A, and rz3 = 2.3 A. (B) Variation of ra3 for the case of a short distance rr2 = 2.2 A between irradiated and observed protons. From the top to the bottom curve, r2a is 5.2, 4.0, 3.0, and 2.0 A. rr3 = 3.0 A, c = 11 set-‘. (C) Variation of the distance between the irradiated and the observed spin. r r2 = 1.8, 2.2, 2.6, 3.0, and 3.4 8, from the top to the bottom curve. rr3=2.9A+rlz; c=llsec-‘. (D) Variation of rr3 and was for the case of a proton-proton distance rr2 = 5.0 A. Spin 3 was located on the line connecting spins 1 and 2. From the top to the bottom curve, rll and r2s are 2.2 and 2.8 A, 2.6 and 2.4 A, 3.0 and 2.0 A, 2.2 and 7.2 A, and 7.0 and 2.0 A. c = 11 sec.~‘.

In Fig. 1B the influence of the distance between spins 2 and 3 was investigated for the situation of a short distance r12 between the irradiated and the observed proton. The distances considered are quite typical for those to be found in proteins (15). It is seen that the TOES obtained with short irradiation tirnesti are affected only little by the choice of r23, so that they are sensitive to the distance r12 also in the presence of additional nearby protons. On the other hand it is readily apparent that the steady-state NOES obtained after long irradiation times tl are markedly smaller for shorter distances rz3. Figure 1C shows that for proton-proton distances ~3.5 8, the buildup rates of the driven NOES readily discriminate between protons at different distances r12 from the irradiated spin. This distance information is largely lost in the steady-state NOE. Figure 1D shows that for proton-proton distances r12 of the order of 5.0 A or more, the TOES obtained with short pulse lengths tl are so small that they will hardly be accessible to experimenta! observation. The signal intensities obtained after longer pulse lengths tl, on the other hand, can be strongly affected by spin diffusion via proton 3.

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679

Although the present treatment of a three-spin system is a rather drastic simplification of the situation in a biological macromolecule, comparison with experimental data indicated that for short tl values, i.e., tl G 500 msec, the curves of Fig. 1 provide a quite realistic description of the time course of driven NOES in proteins (15). In practice one aims for irradiation times r1 after which the signal intensities in the TOE difference spectrum are still simply related to the buildup rates of the driven NOES, yet which also allow for a workable signal-to-noise ratio. For example, for the system of Fig. lC, the relative line intensities in a TOE difference spectrum obtained with tl in the range 300 to 400 msec would be a reliable manifestation of the relative proton-proton distances. Since certain proton-proton distances in proteins are fixed within narrow limits by the covalent structure, these can in practice be used for a quantitative calibration of the line intensities in the TOE difference spectra obtained for a specific system. In contrast to the conventional steady-state NOE experiments, TOES are thus highly specific and are a reliable manifestation of the distances between irradiated and observed protons even in the presence of spin diffusion in macromolecules. Compared to transient NOES [ 11, TOES have two important advantages in practical use. One is that the amplitude of the preirradiation field can be considerably smaller, so that more selective irradiation in crowded spectral regions may be obtained. Second, Fig. 1C shows that with a suitable choice of tl the buildup rates of driven NOES can to a good approximation be obtained from a single TOE difference spectrum, which is considerably less time consuming than using a transient NOE experiment (13) for obtaining this information. It is to be expected that on the basis of a more thorough theoretical treatment, TOE difference spectra may in the future also be used for measurements of absolute internuclear distances without reference to the above-mentioned empirical calibrations. ACKNOWLEDGMENTS We would like to thank Drs. R. M. Keller and S. L. Gordon, R. Richarz, D. Picot, and A. Dubs for stimulating discussions on the subject of this paper. Financial support by the Swiss National Science Foundation, Project 3.0046.76, is gratefully acknowledged.

REFERENCES 1. P.BALARAM,A. A.BOTHNER-BY, AND J.DADOK, J. Am. Chem.Soc.94,4015 2. I.D.CAMPBELL,C.M.DOBSON, ANDR.J.P. WILLIAMS, J. Chem.Soc. Chem.

(1972). Commun.,888

(1974).

3. J.D.GLIcKsoN,S. L. GORDON,T.P.PITTNER,D.G.AGRESTI,ANDR.WALTER, Biochemistry 15, 5721 (1976). 4. R. M. KELLER AND K. W~THRICH, Biochim. Biophys. Acta 533, 195 (1978). 5. R. RICHARZAND K.WUTHRICH, J. Magn.Reson.30,147(1978). 6. K.W~~THRICH,G.WAGNER,R.RICHARZ,ANDS.J.PERKINS, Biochemistry 17,2253 (1978). 7. A.A.BOTHNER-B~,in "MagneticResonanceStudiesinBiology"(R.G.Shulman,Ed.),Academic Press, New York,in press. 8. A.KALKANDH.J.C.BERENDSEN, J. Magn.Reson.24,343 (1976). 9. W. E. HULL AND B. D. SYKES, J. Chem. Phys., 867 (1975). 10. J.D.STOESZ ANDA.G.REDFIELD, FEBSLert. 91,320 (1978).

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“The Nuclear Overhauser Effect.” Academic Press, Nc% J. H. NOCJGI.E AND R. E. SCHIRMEK, York, 1971. f?. I. SOLOMON, Phys. Reu. 99, 559 (1955). 13. S. L. GORDON AND K. W~THRICH, J. Am. Chem. Sot., in press. 14. K. W~IJTHRICH, R. M. KELLER, AND S. L. GORDON, in “Proceedings of the Johnson Foundanon Symposium on Frontiers of Biological Energetics: From Electrons to Tissues, Philadelphia, Julv 20-22, 1978," in press. 15. A. DUBS, G. WAGNER, AND K. W~THRICH, Biochim. Biophys. Am, in press. 16. H. C. TORREY, Phys. Rev. 76, 1059 (1949). 17. A. ABRAGAM, “The Principles of Nuclear Magnetism,” p. 68, Oxford Univ. Press (Clarendon). London, 1962. II.

GERHARD KURT

Institut fiir Molekularbiologie und Biophysik Eidgeniissische Technische Hochschule 8093 Ziirich-Hiinggerberg Switzerland Received August 14, 1978; revised November 29, 1978

WAGNER W~THRICH