Gradients. Effects of B0 gradients on transverse magnetisation Similar to figure 10 of Sattler review Progr. NMR 34 (1999), 93

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1 Gradients 1. What are gradients? Modern high-resolution NMR probes contain -besides the RF coils - additional coils that can be fed a DC current. The coils are built so that a pulse (~1 ms long) of DC current creates a gradient of the B 0 field along z (x or y, these refer to the lab coordinate system), as if the z (x/y) shims were mis-set so that the linewidths would be ~100 khz. A signal of 100 khz line width will be completely dephased after 1 ms. Effects of B0 gradients on transverse magnetisation Similar to figure 10 of Sattler review Progr. NMR 34 (1999), µs after the gradient pulse the B0 field is homogenous again, but the magnetisation is still dephased. But this holds only for the integrated signal over the whole active volume of the sample, the magnetisation is in a welldefined state in each volume element of the active volume. Therefore, if one applies another gradient after some time, the magnetisation can be refocussed if the gradients have the right ratio.. What gradients are good for. Experiment time, resolution and sensitivity: The time one needs to spend recording a multi-d experiment can depend on the desired sensitivity or on resolution. If the sensitivity is limiting, there is no way around long experiment time. But imagine that the first H N, CA/CB plane of a CBCA(CO)NH contains all expected signals. Then the S/N collected during this time is sufficient, and the 3D doesn t need to run longer than the first plane. But if one wants certain resolutions in the CA/CB and N dimensions of certain sweep widths, there is no way around collecting a number of points given by the Nyquist condition. How long the experiment takes depends then only on the number of scans per increment. Then one should keep this number to a minimum, which means that one can only afford a short phase cycle. Gradients allow to shorten phase cycles.

2 Reducing Phase Cycle by Gradients a) Phase-cycled HSQC 1 H x y t φ R X φ 1 t 1 φ φ 3 φ 1 = x -x A φ = x x -x -x φ 3 = x x x x -x -x -x -x φ R = x -x -x x 8-step phase cycle needed, subtraction artifacts. b) Gradient enhanced HSQC: reduced Phase cycle 1 H X x y φ 1 t 1 ε ε x x 1 1 y y x ε' ε' t φ R G z G 1 G φ 1 = x -x φ R = x -x -step phase cycle fully sufficient.

3 Phase cycles in standard HSQC: At the time A of the first carbon 90 pulse with phase cycle φ 1, 13 C-bound protons have formed the operators -H z C z (amplitude sin(πj)) and H y (amplitude cos(πj)). 1 C-bound protons form only H y. Of these, only the desired -H z C z is sensitive to φ 1 and is transformed into +- H z C y. The receiver phase follows, so that the signals originating from both scans add up. The signals of the H y operators from 1 C- or 13 C-bound protons are subtracted by the inversion of the receiver phase. The φ 1 phase cycle also achieves axial peak suppression: Signals originating from relaxation of the selected operator +- H z C y will not have opposite sign at t, and will therefore also get subtracted. The phase cycles on φ and φ 3 suppress signals that could create artefacts if the 180 pulses are imperfect. Phase cycle in the gradient-enhanced HSQC: Only the φ 1 phase cycle is kept, the gradients take care of pulse imperfections (details below). gradients have more advantages: phase cycling and gradients are not equivalent: With phase cycling, the suppression is achieved by subtraction after several scans. With gradients, the suppression is achieved in each scan. Two consequences: First, Suppression by phase cycling is imperfect, which can lead to t 1 noise. Second, unwanted signals (H O) which will get subtracted away can limit the RG, which can in the extreme reduce the sensitivity. 3. Understanding gradients coherence order: Are the operators I x, I y good to understand gradients? In NOESY one often uses a gradient during the mixing time to purge unwanted signals. It turns out that this purge gradient eliminates I x,y, but in coupled spin systems antiphase signals can survive that during the mixing time were of the form I 1 xi y. How these can survive is hard to see in the I x description. I x evolves as I x cos(ωt), this one can write as I x cos(ωt) = ½( I + exp(-iωt) + I - exp(iωt); ii y sin(ωt) = ½( I + exp(-iωt) - I - exp(iωt); (That cos and sin can be expressed as complex exponentials is just the math of trigonometric functions, another way to look at it is that the cosmodulated signal is the sum of two signals I +, I -, rotating clockwise and counterclockwise) I +, I - are single element operators of coherence order p = ±1. Then the operator I 1 xi y equals ½(I I 1 -) ½(I + - I -) = ¼ (I 1 + I + - I 1 + I - + I 1 - I + - I 1 - I -) I 1 + I +, I 1 - I - are ± quanta, I 1 + I - and I 1 - I + are 0 quanta. The single element operators I +,- rotate with frequency -,+ Ω. They are dephased by gradients proportional to the gyromagnetic ratio γ and the coherence order. 0 quanta

4 rotate with the difference frequency of I 1 and I and are not dephased by gradient pulses. So I x gets dephased, because it consists of +- 1 quanta, while the 0 quantum contribution of I 1 xi y survives a purge gradient. The single element operators are more useful to understand what gradients do. How to use gradients: In the above example, the second gradient simply undid the dephasing of the first. More useful are gradients surrounding a 180 pulse. Purge Gradient: HSQC 1 Purge Gradient suppresses C-bound protons. 1 H X x y t 1 t φ 1 φ φ 3 φ R G G 1 G G p H p X H: H z > -H y > H y > H: y suppressed C-H: H z > -H y> HC x z> HC: z z unaffected

5 The purge gradient G 1 and the pair of gradient pulses are two different categories of gradients: 1. Gradients that do not dephase the desired signal, such as the purge gradient G 1 : Purge gradient in HSQC. The gradient will only suppress the water well if the pulses are perfect. The exact gradient strength is not critical.. Gradients that dephase an operator that shall in the end contribute to the signal. The pair of gradients G around the 180 pulse belongs to this category. The 180 pulse inverts the 1 H coherence order. Therefore a gradient of the same sign will refocus the signal. One can write G p before + G p after = 0. The ratio of gradients must be exactly right, otherwise the desired signal also gets destroyed. In general, one can add up the effect of all gradients on operators in a pulse sequence. A coherence transfer pathway will give a signal if the condition Σγ i p i G i =0 is fulfilled. The magnetogyric ratio γ of the nuclei had to be introduced, because gradients dephase coherences proportional to the γ (or frequency) of the transverse nucleus (nuclei) of a product operator. 4. gradient-enhanced HSQC (sometimes called gradient coherence selection). To encode chemical shifts in t 1 (any incremented dimension), the signal that is detected must be modulated with t 1. How the signal is modulated is a characteristic of the pulse sequence. Figure 1 of review Progr. NMR 34 (1999), 93 amplitude-modulating pulse sequences: Traditional pulse sequences (such as the HSQC with purge gradient) record two transients for each t 1 increment, one cos(ωt 1 ) and one sin(ωt 1 ) modulated (Ruben States Haberkorn); that is the amplitude of the signal varies with t 1 (Fig. 1a). To cos and sin modulation is linked that these sequences select C x H z or C y H z for transfer in the two transients. Fourier transformation of the individual cos or sin modulated partial FIDs along ω 1 gives the signals displayed in figure 1a, because cos and sin modulation cannot distinguish ±Ω. In the Fourier transformation that one normally carries out, the four signals of both transients are merged, so that positive and negative frequencies can be distinguished and an absorptive line is obtained. 4 signals of intensity½ get added, but also noise of 4 signals contributes:

6 S/N ratio: 4 signals of intensity½ / 4 (noise of 4 signals) = 1 phase-modulating pulse sequences: These pulse sequences also record two transients per t 1 increment, but the signal is modulated with exp(iωt 1 ) or exp(-iωt 1 ) (echo-antiecho). The amplitude of the signal is not modulated with t 1, but the phase. Product operators during double INEPT of gradient enhanced HSQC The two INEPT steps of the (gradient) enhanced HSQC transfer C x H z and C y H z. As noted above are I x,y and I +,- related. So transferring (C x + ic y )H z or (C x - ic y )H z is equivalent to transferring I + H z or I - H z (figure 1c), that is to transfer selected coherence orders: Coherence Order Selective Coherence Transfer (COS-CT). Fourier transformation of these individual signals gives signals at frequencies - Ω or +Ω, because we are selecting pure (counter)clockwise rotation. In the full Fourier transformation of the HSQC, both signals are combined into a single absorptive line. signals of intensity 1 get added, noise of signals contributes: S/N ratio: signals of intensity 1 / (noise of signals) = Another way to look at this is as follows: The first increments of a standard HSQC and a (gradient) enhanced HSQC have the same intensities. But when t 1 is incremented, the later transients contribute less signal in the HSQC, but contribute equally in the gradient enhanced HSQC. Advantages of gradient enhancement: The (gradient) enhanced HSQC is for X-H fragments more sensitive than a standard HSQC. For X-H groups, the delay 1 can be reduced from 1/(J) to 1(4J), then the gradient enhanced HSQC has equal sensitivity for X-H, and still 0% more sensitive for X-H (Schleucher et al., JBN 4 (1994), 301). In multi-d experiments, such gains can in principle be achieved for every incremented dimension (Sattler et al., JBN 6 (1995), 11). This is because each incremented dimension with amplitude modulation reduces the sensitivity of the experiment - even in the absence of relaxation. In contrast, incremented dimensions with phase modulation do - in the absence of relaxation - not reduce sensitivity. Each transfer between amplitude-modulating incremented dimensions takes 1/(J), but 1/J for a transfer between amplitude-modulating dimensions, because two components are transferred each during a duration 1/(J). So because of relaxation, double enhancement is only useful in a few experiments, such as HCCH-TOCSY (figure 3b of review) and TOCSY- HSQC. The discussion of the sensitivity is a matter of the pulse sequence, whether or not one has gradients in it. The big advantage of gradient enhancement is that the pair of gradients during t 1 and before t gives extremely good water- and artefact suppression:

7 In alternate scans, the first gradient acts on coherence order +1 or -1 of the X nucleus, and the gradient before acquisition acts on coherence order -1 of 1 H (by definition the detected signal). To select these transfers the equation Σγ i p i G i = γ H (-1) G ± γ X G 1 = 0 must hold. So the gradient ratio G 1 : G must be ± γ H : γ X. All protons that are not bound to X cannot form coherences that get refocussed after two gradients of this ratio. They get suppressed in each scan, very little help from subtraction is needed to get pure spectra. Finally, it turns out that a gradient-enhanced HSQC without decoupling in t 1 or t automatically is a TROSY pulse sequence (Yang and Kay JBN 13 (1999), 3); in principle both the doubly-narrow and doubly-broad signals get selected, but the doubly-broad signal can be suppressed or simply relaxes away during longer pulse sequences.

8 Water suppression Problems with digitisation: The digitizer can handle 16-bit numbers, therefore the water ( mm) and protein resonances (1 mm) cannot be digitized at the same time. Exchangeable protons Exchange of NH and OH ( saturation transfer ): If the water polarization is saturated before the pulse sequence, chemical exchange of H O and sample reduces the polarization of the biomolecule. Saturating water attenuates iminos at 83 K. a) Even 400 ms short presaturation destroys iminos b) Watergate does not saturate water before excitation c) Watergate and flip-back give increased signal, even for s relaxation delay

9 Problem with T 1 relaxation - long T 1 of water (T 1 3s): water saturated during pulse sequence takes longer than normal recycle delays to recover and saturates biomolecule resonances via exchange and NOE during the recycle delay. This can be avoided by "water flip-back", which restores the water along z before acquisition. Then exchange and NOE help to restore biomolecule polarization more quickly (Bax, Pervushin). Gradient HSQC with flip-back, fig. 14C of review The water-selective pulse with phase -x brings the water to -z, the gradient purges any transverse components. The 180 pulse in t 1 returns it to z, and the remaining pulses together act like a 0 pulse on the water. So the water comes out of the pulse sequence along +z, and doesn t need time to relax. This also improves water suppression. Radiation damping: H O doesn t behave like weak signals, but restores itself along the z axis much faster than by T 1. The voltages that the water (weak signals) can induce in the receiver coil are comparable to (small compared to) a weak pulse. If the full water signal is excited, the voltage induced by in the receiver coil acts as a selective pulse at the water frequency, which brings the water back to z. The water signal appears broad. If the water is inverted, there typically will be a small transverse component, which again induces a voltages, which tips the magnetisation away from -z, which again increases the transverse component. This is visible as dispersive phase of the water signal in the inversion recovery. Finally, the water will return to +z much faster than by T 1. T 1 relaxation and radiation damping of HO (next page) Radiation damping restored the water in the watergate NOESY to z at the end of the mixing time. Radiation damping can be suppressed if the water signal dephased and rephased in a controlled way during t 1 (Otting, JMR B103 (1994), 88; Sklenar JMR A114 (1995), 13).

10 T and Radiation Damping of HO 1 a) After saturation, water relaxes with T > 1 s relaxation delay, s b) After inversion, water recovers after < 300 ms relaxation delay, s

11 Criteria for water suppression: - excitation profile: is even excitation possible / needed? - What is the duration of the water suppression element, compared to the T of the signals to be observed? - Does the water suppression element dephase or saturate the water signal? Must exchange of water and biomolecule protons be expected? Can radiation damping occur? Can "water flip-back" be incorporated? - Can resonances under the water signal (H α ) be observed at least in principle? The following table is anything but an exhaustive overview of water suppression techniques, but contains only the most common techniques. Water suppression - characteristics of techniques Technique Excitation profile duration a) Dephased / flip-back Presaturation Notch 0.1 ppm µs yes / no Jump-return Very uneven 100 µs no / yes Watergate Notch 1 ppm 3 ms Avoidable / yes Gradient echo Even 5 ms Avoidable /yes spin lock Even ms yes / no Purge gradients Even ms e) Avoidable / yes a) time added to a pulse sequence without water suppression element. Presaturation is still useful during shimming, and very weak for DO samples Jump-return can be useful to observe very broad iminos. Spin lock and purge gradients are not used much any more.

12 Cryo Probe - How the sensitivity is achieved; The signal is the same, the noise can be reduced. This one can achieve if one reduces the thermal noise, by cooling the receiver coil and the 1 H preamp to 50 K. You must make sure that you are really limited by thermal noise, so they built dedicated preamps with special bandpass filters for 13C and 15N. sketch of cryo probe hardware This is basically a sphisticated fridge, that gets He out of a bottle, then has a He circulation with a compressor and an expansion unit, that provides 0 K He. This He is circulated through the probe to cool the receiver coil and the 1 H preamp. Now the noise is smaller, but to get an advantage, you still have to digitise the noise, so you must operate with higher receiver gain. Presaturation on H O doesn t work well, because it is hard achieve high enough receiver gain. - Basically used like any other probe, but some special things: - The resonance circuits have a higher Q. Much more narrow tuning curve. Tuning has to be done more carefully, Less power needed for any pulse length. Jump-return doesn t work well, because the adjustment of the two pulses becomes too sensitive. - Examples A nice advantage of the lower practicable concentrations is that linewidths decrease, so that one actually gains resolution as well. HMBC on 0.3 mm RNA

13 Adiabatic pulses Shaped pulses, often, but not necessarily with a frequency sweep. Square and adiabatic inversion pulses square inversion pulse adiabatic inversion pulse z z M y M B eff y B eff x x square pulse, non-adiabatic shapes: magnetization rotates on a cone around effective field adiabatic pulse: a spin lock that starts far off-resonance and carries spinlocked magnetization into destination state. Magnetization is aligned along the time-dependent effective field. What is the problem? A 90 pulse of some field strength γb 1 excites a band width ± γb 1. A 1.5 µs C pulse has γb 1 = 0 khz and 40 khz bandwidth, which covers 00 ppm 13 C at 800 MHz. In contrast, a 1.5 µs 180 pulse inverts only 1 khz properly, corresponding to only 100 ppm at 500 MHz. So the problem is to have inversion pulses with large bandwidth.

14 Square versus adiabatic inversion pulse test in 1D HSQC: fid dec. 5 µ s, 0 khz square pulse 400 µ s, 10 khz chirp pulse offset, khz ppm -3

15 Adiabatic inversion pulses don t need careful calibration (but just a minimum amplitude, and are therefore also less sensitive to inhomogeneity of pulse over sample volume ( B1-inhomogeneity ). Adiabatic inversion pulses can be used to create decoupling sequences for broader decoupling with less power, but more side band problems.

16 Water suppression: criteria - excitation profile: is even excitation possible / needed? - What is the duration of the water suppression element, compared to the T of the signals to be observed? - Does the water suppression element dephase or saturate the water signal? Must exchange of water and biomolecule protons be expected? Can radiation damping occur? Can "water flip-back" be incorporated? - Can resonances under the water signal (Hα) be observed at least in principle?