G. wider Magnetic Resonance Spectroscopy 32(1998)193-275 203 increasing correlation times (Eq.(8), depends on the size of the coupling constants decreases monotonously with increasing molecular involved. Heteronuclear coupling constants are often weight. Low T2 values reduce the performance of much larger than proton-proton couplings(Fig. 5) NMR experiments with large molecules. However, The use of heteronuclear coupling constants requires relaxation depends not only on the size of a molecule the protein to be labelled withN and/or C isotopes but also on its internal motions. Two molecules with With C, N doubly labelled proteins, spin systems the same molecular weight may show quite different of individual amino acid residues can be connected relaxation behaviour depending on their particular J couplings across the peptide bond. Based on internal motions heteronuclear couplings a complete assignment can be obtained from through-bond correlations alone 2.1.5. Through-bond correlations Fig 5 summarizes some typical coupling constants The magnetic dipole-dipole interaction describes found for nuclei in proteins. a wide range of experi- the effect of the local magnetic fields associated ments for the determination of homo- and heteronu with the magnetic moments of surrounding nucl lear scalar coupling constants in proteins exists. An Two mechanism contribute to this effect: the"direct" excellent survey of these methods can be found in a (through-space)coupling and the indirect"spin- recent review [42] spin coupling or J coupling transmitted via polariza Two principally different mechanisms for the tion of bonding electrons. The complete analysis of through-bond correlation of spins are used.Eithe protein spectra is based on interactions between dif- individual spin pairs are correlated or all spins in a ferent spins, either mediated by electrons in through- spin systems interact simultaneously. The first case bond correlations or by direct interactions through often referred to as a COSY-type and the second as a pace.Through-bond correlations group individual TOCSY-type correlation TOCSY stands for total cor spins into spin systems [15] which are characteristic relation spectroscopy [43 also known under the for individual amino acids In proteins couplings over acronym HOHAHA for homonuclear Hartmann- more than three chemical bonds are not usually Hahn transfer [44]. Both types of correlation can observed. Consequently, only spin systems fo transfer magnetization between two nuclei. Thus the amino acid types can be obtained for unlabelled or sensitivity of a nucleus can be enhanced when the ISN-labelled proteins. The sequential arrangement of experiment starts with the polarization of a nucleus these spin systems relies on through-space correla- of a different species with a higher gyromagnetic ratio tions [ 5, 6] which may be ambiguous for larger (Figs. 2 and 3). Polarization transf proteins. The efficiency of through-bond correlations COSY-type sequence was given the acronym INEPT HH HH O 33 C H=C〓H C C-H H Fig. 5. Typical absolute values verage values are gi e bond; for multiple bond coupling constants a line drawn along the chemical bonds connects the two coupled nuclei
[45] which stands for insensitive nuclei enhanced by magnetization transfer through bonds, TOCSY mix polarization transfer Polarization transfer based on a ing sequences transfer magnetization through space as TOCSY-type sequence is usually referred to as an discussed in the next section. This pathway requires HEHAHA(heteronuclear Hartmann-Hahn) experi- special attention only for very sensitive nuclei such as ment(Section 4.2.3 protons and can safely be neglected for all other The COSY-type correlation can easily be rationa- nuclei. Detailed descriptions of the foundations of lized using the product operator formalism for two the TOCSY experiment can be found in the literature scalar coupled spins I and S(Eq(A7)in Appendix with both experimental [46] and theoretical treat B). Transverse 1, magnetization will evolve into anti- ments (43, 471 phase magnetization of the form 21_S, due to scalar In the more general case of the heteronuclear coupling. a 90 RF pulse with phase y on both spins TOCSY, magnetization is transferred between nuclei transforms this operator product into 21 Sr which can of different species. In this situation two RF fields B evolve into the observable operator S,. The crucial and Bis at two different nuclear resonance frequencies element in a COsY-type transfer is the 90 RF pulse have to be applied with But= y w and acting on an anti-phase state. If the two nuclei belong As e HOHAHA experiments pins must to two different nuclear species the polarization trans experience the same magnetic field strength to lose fer from spin I to spin S(Fig. 3)will change the their individuality and to form a strongly coupled gyromagnet te spin S by the ratio y/ys of the system Setting B u and Bis equal produces the well known Hartmann-Hahn condition [16, 19, 48] which In a homonuclear TOCS Y-type transfer a strong RF must be fulfilled to obtain a heteronuclear TOCSY field is applied to one nuclear species. Viewed in the transfer rotating frame this field locks the spins along the axis it is applied. In this spin-locked state individual pre (10) cession around B. is suppressed and replaced by a If, for example, I stands for a 'H and S for aN collective precession with the frequency of nucleus the locking RF field applied to N has to be applied RF field. Without their characteristic preces- almost 10 times larger than the one applied to protons sion frequencies the spins lose their individuality and (Table 1). Based on a quantum mechanical treatment can no longer be distinguished and behave as part of a the transformation properties of the operator I, of a strongly coupled spin system. The product operator heteronuclear two-spin system submitted to a formalism cannot describe such a state and an analysis HEHAHA sequence can be formulated in analytical is only possible using a quantum mechanical treat- form [16, 24] ment.a homonuclear two-spin system with scalar coupling J evolves under spin-locking for a period Ix Tm from the state I as follows (431 +I,S.-lS.] sin(TJT) (11) x→lI1+cos(2丌Jrm)2+Sl-cos(2丌Jrm)2 where J stands for the heteronuclear coupling constant +[,S.-l,S,I sin(2Tm) between the spins I and S, For a full transfer of the magnetization in an HEHAHA experiment the mixing For Tm =(JIs) complete in-phase magnetization time Tm must be I/us which corresponds to double the transfer from I, to S will occur. For more than two duration compared to the homonuclear transfer coupled spins different coupling constants will govern (Eq (9) the transfer and complete transfer from one spin to nother is usually not possible. The theoretical evalua- 2. 1.6. Through-space correlations tion leading to Eq (9)does not consider offset effects a nucleus with a spin different from zero generates of RF pulses. when the effective fields for two nuclei a magnetic dipolar field proportional to its magnetic are not aligned(Eqs. (A 3)and(A 4)in Appendix A) moment. As the molecule tumbles in solution, this the effective J coupling during the mixing sequence is field fluctuates and constitutes a mechanism of reduced resulting in a slower transfer. In addition to relaxation for nearby spins. Since the dipole-dipole
G, wider/Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)193-275 205 interaction involves a pair of spins, four states can These expressions are valid in the case of isotropic occur for a system with two spins 2. Due to the double tumbling and dipolar relaxation. ON can take positive and zero quantum transitions which are possible in or negative values depending on the value of the spec such a system, the longitudinal relaxation process tral density function in Eq (8)(Fig. 4)and has a zero for the two spins I and S are coupled [49] and the crossing(Fig. 6). For globular proteins measured at expectation values (1 )and (S )describing the z mag- high magnetic fields oN is negative. For protons K netization fulfil the equation equals 1.424"s. For example, in a glob- dd()-l0)=-pN(1)-l)-0N(S)-S)(12)uarproteinwitharotationacorelationtimeofl0ns, aN becomes approximately 10 s for two protons at a I((S, )-So)=-PN((S )-SO)ON(I, )-I) distance rof 0.2 nm and ON=0.04s-Iforr=0.5 nm Nuclear Overhauser enhancement(NOE)experiments where PN stands for the longitudinal relaxation rate make use of the cross-relaxation between spins and constant of the two spins I and S of the same nuclear allow detection of nuclei which are close in space, in species, oN for the cross-relaxation rate constant and Io or So are the equilibrium magnetizations. The practice at a maximal distance of about 0.5 nm for protons in a globular protein In a system with more coupling of the relaxation of the two nuclei will than two spins, consecutive cross-relaxation can occur alter the magnetization of one spin when the other leading to so-called spin diffusion [511 where two spin is not in its equilibrium state. For example, distant spins exhibit a larger apparent cross-relaxation when the equilibrium value So is selectively disturbed, due to the contributions of the intervening spins. This leading to a deviation AS from So, then the l magne- effect complicates the derivation of distances from tization will change from Io with an initial rate pro- Noe measurements. Since the initial NOE is propor- aNAS (Eq. (12)). The following tional to ON, spin diffusion becomes more important expression can be derived for aN and PN using two for larger proteins which exhibit a longer rotational spins of the same species without scalar coupling and correlation time T(Fig. 6). Practical implementations with an internuclear distance r[50] exploiting cross-relaxation use the NOESY segment discussed in Section 4.2.2 N=6(6(240)-J(0) (13) Based on the observation of cross-relaxation between 1, and S states, one may expect a similar pN=K(o)+3a)+61(2) effect with transverse magnetization, In general,a net magnetization transfer between transverse magnetization components does not occur because K (15) the spins precess with different frequencies and the continuously changing phase relationship between the oA[s-1 0010.020.0501020.5 Fig. 6. Plot of cross-relaxation rates o versus the rotational correlation time T, for a two-spin system. o is given for the NOE in the laboratory frame denoted by aN and in the rotating frame, aR, for two spectrometer frequencies: 800 MHz indicated with solid lines and 500 MHz drawn with dotted lines For these curves. the scale for o on the left- hand side of the figure applies. the dashed lines show ox and oR on a 20-fold maller vertical scale shown on the right-hand side. On this scale the curves for the two field strengths are indistinguishable. For the calculation. (8).(13).(15)and (16)were used, assuming two protons with a constant internuclear distance of 0.2 nm
206 G wider/Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)193-275 orresponding magnetization vectors prevents the with increasing mixing time twice as fast as noe and accumulation of a net transfer of magnetization. The in the opposite direction. When results from ROE and situation changes when different spins are forced to NOE spectra are to be directly compared, the mixing precess at the same frequency by applying a strong rF time for the roe experiment is often chosen to be half field. With the individual precession frequencies as long as for the NOE experiment. Both ROE and removed, a net transfer can be established which cou- NOE experiments not only detect cross-relaxation but ples the transverse relaxation process for two spins I also magnetization transfer by chemical or conforma- and S. By analogy to lation of relaxation rate constant, oR, and a relaxation rate con- the roe effect may be different, this is not the case ant, PR, can be obtained [50.52]: for exchange contributions which always have an the roe effect. This oR=(2/(0)+3/(0) (16) allows a separation of exchange contributions from NoE effects pR=-(5J(0)+9J(c)+6/(20) (17) 2.2. Radiofrequency pulses The NOE between spin-locked transverse magnetize tion components is usually referred to as NoE in the An NMR experiment consists of a series of RF rotating frame(ROE). Originally, the ROE experi- pulses and delays, the pulse sequence, followed by ment was called CAMELSPIN [53], but was later the measurement of the voltage induced by the result renamed RoESY for the two-dimensional ROE ing magnetization in an RF coil( Section 3.5.1).A experiment [54]. Fig. 6 illustrates that the cross- delay specifies a time period during which no external relaxation rate oR does not have a zero crossing and RF field is applied and the nuclear spin states evolve that for molecules with a correlation time T. larger due to their intrinsic properties(chemical shift, scalar than 0.05 ns, oR is larger than oN. A detailed theore coupling and relaxation). A pulse represents a time tical derivation of homonuclear cross-relaxation in the period during which RF is delivered to the coil in rotating frame can be found in an excellent review the probe. Four parameters describe an rf pulse: fre- [52] and a number of textbooks, e.g. [ 16, 24, 26,] quency, phase, duration and strength. The frequency The ROE and TOCSY mixing sequences both use a of the pulse is often called the carrier frequency since spin-lock field to suppress the individual precession generally, it is identical to the frequency used to frequencies of transverse magnetization components demodulate the detected NMR signal. For a rectangu- nd special care is required in the implementation to lar pulse the strength stays constant during its applica separate the two effects(Section 4.2. 1). Although tion. The magnetic field strength applied during such a there are experimental implementations which mini- pulse is often given in frequency units yB, which can mize the simultaneous occurrence of both effects, a be obtained with Eq. (3)by setting the pulse flip angle strict separation is not possible and both proces to 2T for a 360 pulse with the corresponding pulse contribute to correlations between scalar coupled length T 360 nuclei. Nevertheless. Roe and ToCSY mixing sequences find widespread applications because the residual interference between them can, in many practical cases, be distinguished since the two effects For example, a 90 pulse with a duration of 12.5 us is generate signals with different signs produced by a field strength yB, of 20 kHz Both the noe and roe enhancements for short deally, an rF pulse applied to a given nuclear mixing times are proportional to the cross-relaxation species will rotate all magnetization components rate which for globular proteins is dominated by the irrespective of their individual resonance frequencies spectral density function at zero frequency, J(O), and by the same Hip angle B about the axis in the rotating therefore from Eqs. (13) and(16)OR =2 oN. This frame defined by the phase of the pulse. However, relation indicates that the ROE effect builds up signal the performance of a real RF pulse degrades with
G. wide ss in Nuclear Magnetic Resonance Spectroscopy 32(1998)193-275 frequency C B 7180 For a 90 pulse the first null is at vgo= +0.97/T90 and Fig. 7. Excitation profiles of RF pulses represented by the normal. for a 180 pulse at v180=+0.87/7180- ized magnetization M, plotted against the offset given in units of Typically, several 180 pulses occur in a pulse nB where B, is the applied field strength. (A)M, after sequence and the signals created by their non-ideal tion pulse and(B)M: after a 180 inversion pulse applied to 2 behaviour may limit the spectral quality. with a magnetization.(C)M, after a 180@ refocusing pulse with phase x phase cycling scheme, EXORCYCLE [551, magneti- applied to x magnetization. (D)Same as (C) but M, is shown zation components not inverted by the 180 pulse and those which underwent a coherence transfer due to increasing offset of the nuclear precession frequencies off-resonance effects can be removed from the from the applied RF frequency. The precession axis detected signal. The EXORCYCLE consists of a introduced by an Rf pulse applied off-resonance four-step phase cycle where the phase of the 180 deviates from the direction of the Bi field of the pulse changes according to the scheme x, y,-x,-y pulse(see Appendix A). whereas for a 90 excitation together with the receiver phase cycling through x, pulse satisfactory performance can be obtained over a wide bandwidth, the efficiency of a 180 pulse Many applications require the inversion of z degrades rather rapidly. Fig. 7 shows excitation magnetization. In this case simple composite pulses profiles for 180 and 90 pulses dependent on the offset exhibit much broader inversion profiles than a single frequency. Best use of these excitation profiles can be 180 pulse. A composite pulse based on a 180 pulse nade when the carrier frequency sits in the middle of with phase y embraced by 90 pulses with phase x, in the spectral range of interest. A 90 pulse applied to z short notation 90o 18090 900, shows more than 80% magnetization brings most magnetization into the inversion over a bandwidth of +yB:[56]. The com- transverse plane when the offset frequency vor fulfils posite pulse 900 225 o 90o where the 180 pulse is the condition voff<yB. However, the magnetization replaced by a 225% pulse results in even better inver acquires an offset dependent angle g to the direction it sion of more than 98% but at the cost of a smaller would reach after an ideal 90 pulse with duration 79o. bandwidth of =0. 7y,. Analogous simple composite This angle e can be calculated to be 4Tgowofr in units of pulses for improved refocusing of transverse magne- radians [16, 24, 26) tization do not exist [571 A further consequence of the non-ideal behaviour When pulses or more complex pulse trains with low of RF pulses is specific offset frequencies at which ower are applied selectively to only a small fre the pulse does not perturb the resonances. Th quency range in a spectrum, the signals at an offset feature can be used to excite one group of resonances Av far from the irradiation frequency can still be while selectively avoiding excitation of another significantly affected. These off-resonance or non group, a technique that finds frequent applications in resonant effects can lead to a phase shift or and a heteronuclear experiments involving carbon nuclei, rotation p of the evolving magnetization and depend where carbonyl carbons and aliphatic carbons are on the strength v2(0)=yB2(0) of the applied magnetic excited separately. Using the Bloch equations(see field B2(t)[34, 35]. The rotation p is towards the posi Appendix A)the kth null in the excitation profiles e z axis around an axis perpendicular to the axis can be calculated for a 90 pulse with duration Tgo to along which the re pulse is applied. Both nr and p be at a frequency vgo and for a 180 pulse with depend on the length Tp of the pulse train applied and duration T180 at a frequency viso from the carrier on the average of the square of the field strength