4 DQF-COSY, Relayed-COSY, TOCSY Gerd Gemmecker, 1999

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1 44 4 DQF-COSY, Relayed-COSY, TOCSY Gerd Gemmecker, 1999 Double-quantum filtered COSY The phase problem of normal COSY can be circumvented by the DQF-COSY, using the MQC term generated by the second 90 pulse: 90 y I 1z cos(ω 1 t 1 ) cos(πjt 1 ) I 1 polarization double/zero quantum coherence single quantum coherence + 2I 1y I 2x cos(ω 1 t 1 ) sin(πjt 1 ) I 1 / I 2 + I 1y sin(ω 1 t 1 ) cos(πjt 1 ) I 1 in-phase + 2I 1z I 2x sin(ω 1 t 1 ) sin(πjt 1 ) I 2 anti-phase single quantum coherence Phase cycling can be set up to select only the DQC part at this time, which is only present in the 2I 1y I 2x term (leaving the cos(ω 1 t 1 ) sin(πjt 1 ) part away for the moment): 2I 1y I 2x = 2 -i / 2 (I I 1 - ) 1 / 2 (I I 2 - ) = -i / 2 (I 1 + I I 1 + I I 1 - I I 1 - I 2 - ) DQC ZQC ZQC DQC Only the DQC part survives (50 % loss!) and yields (after convertion back to the Cartesian basis): -i /2 (I 1 + I I 1 - I 2 - ) = -i / 2 {(I 1x + ii 1y ) (I 2x + ii 2y ) - (I 1x - ii 1y ) (I 2x - ii 2y )} = 1 / 2 (2 I 1x I 2y + 2 I 1y I 2x ) However, this magnetization is not observable, only after another 90 pulse: 90 y 1 / 2 (-2 I 1x I 2y - 2 I 1y I 2x ) 1 / 2 (2 I 1z I 2y + 2 I 1y I 2z ) Since we still have the cos(ω 1 t 1 ) sin(πjt 1 ) modulation from the t 1 time evolution, our complete signal at the beginning of t 2 is 1 / 2 2 I 1z I 2y cos(ω 1 t 1 ) sin(πjt 1 ) + 1 / 2 2 I 1y I 2z cos(ω 1 t 1 ) sin(πjt 1 ).

2 45 After 2D FT, this translates into two signals: - both are antiphase signals at Ω 1 in F1 (with identical absorptive/dispersive phase) and - both are y antiphase signals (i.e., identical phase) in F2, the first one at Ω 2 (cross-peak) and the second one at Ω 1 (diagonal peak). Characteristics of the DQF-COSY experiment: - the spectrum can be phase corrected to pure absorptive (although antiphase) cross- and diagonal peaks in both dimensions - both cross- and diagonal peaks are derived from a DQC term requiring the presence of scalar coupling (since it can only be generated from an antiphase term with the help of another r.f. pulse: 2I 1y I 2z 2I 1y I 2x ). Therefore, singulet signals e.g., solvent signals like H 2 O! should be completely suppressed, even as diagonal signals. Usually this suppression is not perfect (due to spectrometer instability, misset phases and pulse lengths, too short a relaxation delay between scans etc.), and a noise ridge occurs at the frequency of intense singulets. In addition, this solvent suppression occurs only with the phase cycling during the acquisition of several scans with for the same t 1 increment, i.e., after digitization! To cope with the

3 46 limited dynamic range of NMR ADCs, additional solvent suppression has to be performed before digitization (i.e., presaturation). If the DQ filtering is done with pulsed field gradients (PGFs) instead of phase cycling, then this suppresses the solvent signals before hitting the digitizer. However, inserting PGFs into the DQF-COSY sequence causes other problems (additional delays and r.f. pulses, phase distortions, non-absorptive lineshapes, additional 50 % reduction of S/N). With the normal COSY sequence, they result in gigantic dispersive diagonal signals obscuring most of the 2D spectrum. Intensity of cross- and diagonal peaks In the basic COSY experiment, diagonal peaks develop with the cosine of the scalar coupling, while cross-peaks arise with the sine of the coupling. Theoretically, this does not make any difference (FT of a sine wave is identical to that of a cosine function, except for the phase of the signal). While this is normally true for the relatively high-frequency chemical shift modulations (up to several 1000 Hz), the modulations caused by scalar coupling are of rather low frequency (max. ca. 20 Hz for J HH ), with a period often significantly shorter than the total acquision time. Time development of in-phase (cos πjt 1 ) and antiphase (cos πjt 1 ) terms, with Ω 1 = 50 Hz, J = 2 Hz, for T 2 = 10 s (left) and T 2 = 0.1 s (right). While the total signal intensity accumulated over a complete (or even half) period is identical for both in-phase and antiphase signals, an acquisition time much shorter than 1 / 2J will clearly favor the in-phase over the antiphase signal in terms of S/N. This difference in sensitivity is further increased

4 47 by fast T 2 (or T 1 ) relaxation, leaving the antiphase signal not enough time to evolve into detectable magnetization. This phenomenon can also be explained in the frequency dimension: short acquisition times or fast relaxation leads to broad lines, which results in mutual partial cancelation of the multiplet lines in the case of an antiphase signal. The simulation (next page) shows the dublet appearances for different ratios between coupling constant J and linewidth (LW). The linewidths were set constant to 2 Hz (at half-height), so that the different intensities of the dublet signal are only due to different J values. Obviously, the apparent splitting in the spectrum can differ from the real coupling constant, if the two dublet lines are not baseline separated: for in-phase dublets, the apparent splitting becomes smaller, for antiphase dublets it is large than the true J value. Ratio J/L: / 3 True J value [Hz} In-phase splitting n/a Antiphase splitting In the basic COSY experiment the diagonal signals are in-phase and the cross-peaks antiphase, so that signals with small J couplings and broad lines (due to short AQ or fast relaxation) will show huge diagonal signals, but only very small or vanishing cross-peaks. In the DQF-COSY, both types of signals stem from antiphase terms, so that both the cross- and diagonal peak intensity depends on the size of the coupling constants.

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6 49 Spins with more than one J coupling For spins with several coupling partners, all couplings evolve simultaneously, but can be treated sequentially with product operators (just as J coupling and chemical shift evolution). J 12 J 13 I 1x I 1x cos(πj 12 t) I 1x cos(πj 12 t) cos(πj 13 t) cos(πj 13 t) sin(πj 12 t) sin(πj 13 t) + 2I 1y I 3z cos(πj 12 t) sin(πj 13 t) + 2I 1y I 2z sin(πj 12 t) + 2I 1y I 2z sin(πj 12 t) - 4I 1x I 2z I 3z The double antiphase term 4I 1y I 2z I 3z develops straightforward from the I 1y factor in 2I 1y I 2z, according to the normal coupling evolution rules I 1y 2I 1x I 3z sin(πj 13 t). When we consider the time evolution of the single antiphase terms required for coherence transfer, such as 2I 1y I 2z sin(πj 12 t) cos(πj 13 t) and 2I 1y I 3z cos(πj 12 t) sin(πj 13 t), we find that their trigonometric factors (the transfer amplidute) always assume the general form 2I 1y I 2z sin(πj 12 t) cos(πj 13 t) cos(πj 14 t) cos(πj 15 t) with J 12 being called the active coupling (that is actually responsible for the cross-peak) and all other the passive couplings. When all J couplings are of the same size, then the maximum of these functions is not at t = 1 / 2J, but at considerably shorter times, between ca. 1 / 6J and 1 / 4J (depending on the number of cosine factors and relaxation).

7 50 However, in real spin systems the size of J varies considerably, for 2, 3 J HH from ca. 1 Hz up to ca. 12 Hz (or even Hz for 2 J and 3 J trans in olefins). The largest passive coupling determines when the transfer function becomes zero again for the first time (e.g., 1 / 2J = 35 ms for J = 14 Hz), and the maximum of single antiphase coherence the occurs at or shortly before ca. 1 / 4J for this coupling constant. With only one passive coupling constant and a very small active coupling, one could wait till after the first zero passing to get more intensity. However, with a large number of passive couplings of unknown size (as in most realistic cases), the only predictable maximum will occur at Hz for most spin systems.

8 51 The same considerations as for the creation of 2I 1y I 2z terms out of in-phase magnetization apply to the refocussing of these antiphase terms back to detectable in-phase coherence. In COSY experiments, the single antiphase terms are generated during the t 1 time and (after coherence transfer) refocus to in-phase during the acquisition time t 2. Since both are direct and indirect evolution times which are not set to a single value, but cover a whole range from t=0 up to the chosen maximum values, the functions shown in the above diagrams will be sampled over this whole range and always contain data points with good signal intensity (as well as some with zero intensity).

9 52 Relayed-COSY The considerations about transfer functions become more important in experiments with fixed delay, e.g., for coupling evolution. The simplest homonuclear experiment here is the Relayed-COSY, with the following pulse sequence: It allows to correlate the chemical shifts of spins that are connected by a common coupling partner, as in the linear coupling network I 1 I 2 I 3, with the coupling constants J 12 and J 23. After the t 1 evolution period and the second 90 pulse we get (cf. COSY): I 1z cos(ω 1 t 1 ) cos(πj 12 t 1 ) + 2I 1y I 2x cos(ω 1 t 1 ) sin(πj 12 t 1 ) + I 1y sin(ω 1 t 1 ) cos(πj 12 t 1 ) + 2I 1z I 2x sin(ω 1 t 1 ) sin(πj 12 t 1 ) During the period, chemical shift evolution is refocussed (180 pulse in the center!), but J 12 coupling evolution continues: J 12 I 1z cos(ω 1 t 1 ) cos(πj 12 t 1 ) (no coupling evolution, I z!) + 2I 1y I 2x cos(ω 1 t 1 ) sin(πj 12 t 1 ) (no coupling evolution, MQC!) + I 1y sin(ω 1 t 1 ) cos(πj 12 t 1 ) cos(πj 12 ) 2I 1x I 2z sin(ω 1 t 1 ) cos(πj 12 t 1 ) sin(πj 12 ) (evolution of antiphase) + 2I 1z I 2x sin(ω 1 t 1 ) sin(πj 12 t 1 ) cos(πj 12 ) + I 2y sin(ω 1 t 1 ) sin(πj 12 t 1 ) sin(πj 12 ) (refocusing to in-phase)

10 53 The last two terms, however, are spin 2 coherence, and spin 2 has two couplings, J 12 (which we have just considered) and J 23, the effect of which we have to calculate now. Just as with chemical shift and coupling, which evolve simultaneously, but can be calculated sequentially, we can here calculate the effects of J 12 and J 23 one after the other (the order doesn't matter). J 23 I 1z cos(ω 1 t 1 ) cos(πj 12 t 1 ) (not affected by J 23 ) 2I 1y I 2x cos(ω 1 t 1 ) sin(πj 12 t 1 ) cos(πj 23 ) 2I 1y I 2y I 3z cos(ω 1 t 1 ) sin(πj 12 t 1 ) sin(πj 23 ) + I 1y sin(ω 1 t 1 ) cos(πj 12 t 1 ) cos(πj 12 ) (not affected by J 23 ) 2I 1x I 2z sin(ω 1 t 1 ) cos(πj 12 t 1 ) sin(πj 12 ) (not affected by J 23 ) + 2I 1z I 2x sin(ω 1 t 1 ) sin(πj 12 t 1 ) cos(πj 12 ) cos(πj 23 ) + 4I 1z I 2y I 3z sin(ω 1 t 1 ) sin(πj 12 t 1 ) cos(πj 12 ) sin(πj 23 ) + I 2y sin(ω 1 t 1 ) sin(πj 12 t 1 ) sin(πj 12 ) cos(πj 23 ) 2I 2x I 3z sin(ω 1 t 1 ) sin(πj 12 t 1 ) sin(πj 12 ) sin(πj 23 ) From the evolution of the second coupling, J 23, we get a double antiphase term 4I 1z I 2y I 3z (J 23 does not refocus the original 2I 1z I 2x antiphase of spin 2 relativ to spin 1!) and another term 2I 2x I 3z, which is spin 2 antiphase coherence with respect to spin 3. The third 90 pulse has to be performed with the same phase setting as the second (i.e., either both from x or both from y)! After this 90 pulse, we get the folowing terms at the beginning of t 2 : 90 y I 1x cos(ω 1 t 1 ) cos(πj 12 t 1 ) 2I 1y I 2z cos(ω 1 t 1 ) sin(πj 12 t 1 ) cos(πj 23 ) 2I 1y I 2y I 3x cos(ω 1 t 1 ) sin(πj 12 t 1 ) sin(πj 23 ) spin 1 in-phase spin 1 antiphase MQC + I 1y sin(ω 1 t 1 ) cos(πj 12 t 1 ) cos(πj 12 ) spin 1 in-phase + 2I 1z I 2x sin(ω 1 t 1 ) cos(πj 12 t 1 ) sin(πj 12 ) spin 2 antiphase

11 54 2I 1x I 2z sin(ω 1 t 1 ) sin(πj 12 t 1 ) cos(πj 12 ) cos(πj 23 ) spin 1 antiphase + 4I 1x I 2y I 3x sin(ω 1 t 1 ) sin(πj 12 t 1 ) cos(πj 12 ) sin(πj 23 ) MQC + I 2y sin(ω 1 t 1 ) sin(πj 12 t 1 ) sin(πj 12 ) cos(πj 23 ) spin 2 in-phase + 2I 2z I 3x sin(ω 1 t 1 ) sin(πj 12 t 1 ) sin(πj 12 ) sin(πj 23 ) spin 3 antiphase From the observable terms during t 2 now, we get three types of peaks, all labeled with Ω 1 in F1: - diagonal peaks at Ω 1 in F2 (mixture of terms with different phases) - COSY peaks at Ω 2 in F2 (mixture of terms with different phases) - Relayed peaks at Ω 3 in F2 (pure antiphase in both dimensions) Relayed-COSY spectrum, only the Relayed peaks (in boxes) show pure (anti-) phase behaviour

12 55 For the interesting Relayed peak, the transfer amplitude part from the fixed delay is sin(πj 12 ) sin(πj 23 ), which would be at a maximum for = 1 / 2J (for J 12 = J 23 ). However, if there are more couplings to the relais spin 2, e.g., in a spin system topology then the transfer function, for going from 2I 1z I 2x at the beginning of to 2I 2x I 3z at its end (and 2I 2z I 3x after the final 90 pulse) would be sin(πj 12 ) sin(πj 23 ) cos(πj 24 ) and, because of the cosine factor, it would be zero at 1 / (2J24 ). Therefore, the delay should be set to no more than ms to avoid losing some Relayed peaks due to large passive couplings. The Relayed-COSY can be easily extended to a Double-Relayed-COSY experiment, just by adding another delay and another 90 pulse, to perform transfers I 1 I 2 I 3 I 4 : However, like in the simple Relayed-COSY, the phases of most of the peaks cannot be corrected to pure absorption, and the sensitivity decreases further, due to the inefficient transfers and the increasing length of the pulse sequence. Today, the Relayed experiments have been widely replaced by the TOCSY experiment.

13 56 TOCSY / HOHAHA The TOCSY / HOHAHA experiment does also create a multi-transfer step 1 H, 1 H correlation: It starts like any other 2D experiment so far, with a 90 pulse creating transverse magnetization (coherence) which then evolves during an incremented t 1 period to yield the indirect F1 dimension after 2D FT. Between the two evolution times t 1 and t 2 for the two 1 H dimensions, a mixing step has to perform the coherence transfer. While this is done with simple 90 pulses in all COSY type experiments, the TOCSY has a "black box" spinlock period instead. What happens during this time cannot really be understood in terms of vector models or even the product operators, because it relies on strong coupling. The term strong coupling applies to J coupled systems where the coupling actually dominates the spectrum: weak coupling J << Ω all signals occur at their proper individual chemical shift frequencies (dominating effect), but are split into multiplets with equally intense lines (small perturbation from J coupling) strong coupling J >> Ω J coupling no more "minor disturbance", but dominating: coherences of spins are "coupled" together; instead of individual resonance frequencies of individual spins, combination lines occur which cannot be assigned to just a single spin anymore Under strong coupling conditions, chemical shift differences between different spins become negligible, and in the energy level diagram for a two spin system the two states αβ and βα become identical in energy. Instead of transitions of single spins, the coherences now involve transitions of combinations of spins:

14 57 Under these conditions, a coherence / transition of one spin is actually in resonance with a coherence of its coupling partner(s) (all with the same frequency / chemical shift), and will oscillate back and forth between all coupled spins, like two (or more) coupled resonant oscillators (e.g., pendulums). For a two-spin system, the evolution of strong coupling can be described as follows: strong J I 1x 0.5 I 1x {1 + cos(2πj 12 τ)} I 2x {1 cos(2πj 12 τ)} + (I 1y I 2z I 1z I 2y ) sin(2πj 12 τ) So during the TOCSY spinlock, in-phase coherence of a spin is transferred directly into in-phase coherence of its coupling partner, and back, in an oscillatory way. The frequency of this oscillation is directly proportional to the coupling constant between the two spins, and complete transfer occurs (for the first time) at t = 1 / 2J (see following diagramm). As shown, another oscillatory component consists of "zero-quantum coherence" with respect to the spinlock axis, of the form (I 1y I 2z I 1z I 2y ), i.e., dispersive antiphase coherence of spins 1 and 2. This term is the reason for the (usually rather small) dispersive contributions to the mainly in-phase absorptive TOCSY cross-peaks.

15 58 1,00 0,80 0,60 0,40 Intensity 0,20 0,00-0,20-0, t [ms] -0,60-0,80-1,00 I1x I2x I1yI2z I1zI2y Locking spins with a B 1 field The spinlock needed for converting a weakly coupled into a strongly coupled spin system consists essentially of a continuing r.f. irradiation, e.g., in the x direction. Its field strength has to be much stronger than the z component corresponding to the chemical shifts of the spins. These chemical shifts usually cover a range of a few khz (= precession frequency of the spins relativ to the transmitter / receiver reference frequency). If the r.f. field strength is higher, then B 1 will dominate and the spins will start to precess about the B 1 (i.e., x ) axis instead of the z axis: the magnetization components aligned along the x axis (i.e., I x ) are frozen / spinlocked there, and no more chemical shift evolution occurs in the xy plane. With all chemical shifts reduced to zero, their differences also vanish, and the strong coupling condition J >> Ω is fulfilled. In praxi, the high power transmitters (ca. 50 W for 1 H, ca. 200 W for heteronuclei) can usually generate field strengths of ca khz, but only for a few hundred microseconds. Then the spectrometer usually turns itself of automatically (thankfully), to prevent damage to the amplifiers or the transmitting coils in the probe. Instead of just turning the transmitter on (CW mode), TOCSY spinlocks are therefore performed with composite pulse trains, consisting of a repetitive series of pulses with defined pulse lengths and phases. These spinlock sequences allow to effectively spinlock spins within a wide range of chemical shifts, with reasonable transmitter powers of a few khz. Some often used sequences are, e.g., MLEV-17, WALTZ16, DIPSI-2.

16 59 There are two different classes of spinlocks: isotropic and anisotropic. Isotropic spinlocks (like WALTZ or MLEV16) allow transfer of all magnetization components: I 1x I 2x / I 1y I 2y / I 1z I 2z At the end of the t 1 period in the TOCSY experiment, however, there are absorptive and dispersive magnetization components (as a result of chemical shift and J coupling evolution, cf. COSY). All these contribute to the TOCSY crosspeaks in the case of an isotropic spinlock, creating large dispersive contributions. Therefore, nowadays usually only anisotropic spinlock sequences are employed for TOCSY experiments (MLEV-17, DIPSI), which can transfer only one transverse component (e.g., I x ) and the z component: I 1x I 2x / I 1y // I 2y / I 1z I 2z This anisotropic TOCSY version was initially named the homonuclear HARTMANN-HAHN experiment, HOHAHA (today the terms TOCSY and HOHAHA are mostly used as synonyms). It leads to almost absorptive cross-peaks (at least in the 2D plots, as long as one does not look at rows or columns, or a 1D TOCSY spectrum). For really pure absorptive phases, a z-filtered TOCSY has to be performed: Here, all magnetization components except for I z are destroyed during the delays. This can be achieved in different ways: - by waiting. For larger molecules with T 1 >> T 2, the z components will relax only slowly (with T 1 ), while all transverse components will decay fast (with T 2 ).

17 60 - since the difference between T 1 and T 2 is usually not large enough for good suppression, the effect can be enhanced by greatly increasing the B 0 field inhomogeneities for a short time during. This is done either with homospoil pulses (i.e., DC pulses on the z shim coils) or with much better performance with pulsed field gradients (PFGs) from specifically designed z gradient coils directly in the NMR probe. - in additon, one can also vary the delays and then add scans acquired with different delay lengths. This does not affect the z components, but all transverse magnetization terms will evolve chemical shifts during. With many different settings, their positions will be always at different positions somewhere in the xy plane and cancel after coaddition. This requires, however, a large enough variation of (at least over ms), so that even the slowly rotating zero-quantum coherences can go around at least once (they evolve only with the difference of the chemical shifts of the two coupling protons). Also the coaddition of as many different settings as possible (at least 6-8) is needed for good cancelation, thus increasing the minimum experiment time considerably, since the variation has to be done on top of the phase cycling. The additional 90 pulses at the end of t 1 and beginning of t 2 are needed to convert transverse components into I z and then back to detectable magnetization again: 90 SL 90 I 1x cos Ω 1t 1 I 1z cos Ω 1 t 1 I 2z cos Ω 1 t 1 I 2x cos Ω 1 t 1 Since all other magnetization components containing any transverse terms are quickly dephased during the periods, the resulting spectrum shows pure in-phase absorptive lineshapes, for both the cross-peaks and diagonal peaks. Τ 1ρ Relaxation During the spinlock period, relaxation occurs according to a different mechanism, with a time constant T 1ρ which is neither T 1 nor T 2, but somewhere in between. This means that in cases with T 2 >> T 1 (slow tumbling limit) T 1 ρ is longer than T 2 (which is active, e.g., during the Relayed- COSY mixing sequence). Due to this and the in-phase nature of its cross-peaks, TOCSY can still be used for molecules up to ca kda.