Connections through space So far we have analyzed experiments that work because the system has scalar couplings. COSY, HETCOR, HOMO2DJ, MQF-COSY, and INADEQUATE tell us about the chemical structure of the system, but nothing about the conformation or stereochemistry (not entirely true, but bear with me here). We had seen at the very beginning that if we saturate a proton in the sample, it will relax by either zero- or double- quantum processes, giving energy (enhancing) the signals of protons dipolarly coupled to it (protons close by…). This was the nuclear Overhauser effect (NOE): We had seen that relaxation by either W2IS or W0IS will occur depending on the size of the molecule, actually its rate of tumbling, or correlation time, tc. bb (*) W1I W1S W2IS (***) ab ba (*) W0IS W1S W1I aa (***)

The two-spin system case We had established a relationship between the different rates or probabilities and the magnitude of the NOE. If we limit everything to 1H-1H (gI / gS = 1), we have: fI{S} is the NOE enhancement of nucleus I when we saturate nucleus S. You may also find this equation written in another way in some textbooks. We define the numerator as the cross-relaxation rate constant, sIS, and the denominator as the dipolar longitudinal relaxation, rIS. We already saw that the sign of the NOE will depend on the tumbling rate (size) of the molecule. There were two distinct cases, one in which w * tc << 1, or extreme narrowing limit, and w * tc >> 1, or diffusion limit. W2IS - W0IS hI = = fI{S} 2W1S + W2IS + W0IS sIS = W2IS - W0IS rIS = 2W1S + W2IS + W0IS hI = sIS / rIS= fI{S}

Correlation functions and spectral density We had mentioned before that the pathways for the system had to release energy to the lattice depended on the frequencies of different processes the system can undergo. In solution, this means rotation of the molecule (tc). The spins stay aligned with the external Bo, while the molecule turns, and this generates magnetic fields (fluctuating dipoles) at the frequency of the rotation that allow spins to release energy: We need a way of analyzing the way a molecule tumbles in solution. We define the correlation function of a system as the average of the molecular orientation at a certain time, and a little while (t) after that: It basically (cor)relates the orientation of the molecule at two different times. g(0) = 1, and g(t) decays exponentially as a function of t/ tc, tc being the correlation time of the molecule. g(t) = R(t) * R(t + t)

Spectral density As we now know, once we have a function of time, we can check the frequencies of the processes involved by doing an FT. The FT of g(t) is called the spectral density function, J(w), and since g(t) is a decaying exponential, J(w) is a Lorentzian curve: Depending on tc, the sluggishness of the molecule, its movement will be composed by frequencies slower, comparable, or faster than wo, the Larmor frequency of the system: 2 tc J(w) = 1 + w2tc2 J(w) wo wo * tc >> 1 wo * tc ≈ 1 wo * tc << 1 log(w)

The whole enchilada Since the probability of a transition depends on the different frequencies that the system has (the spectral density), the W terms are proportional the J(w). Also, since we need two magnetic dipoles to have dipolar coupling, the NOE depends on the strength of the two dipoles involved. The strength of a dipole is proportional to rIS-3, and the Ws will depend on rIS-6: The relationship is to the inverse sixth power of rIS, which means that the NOE decays very fast as we pull the two nuclei away from each other. For protons, this means that we can see things which are at most 5 to 6 Å apart in the molecule (under ideal conditions…). W0IS  gI2 gS2 rIS-6 tc / [ 1 + (wI - wS)2tc2] W2IS  gI2 gS2 rIS-6 tc / [ 1 + (wI + wS)2tc2] W1S  gI2 gS2 rIS-6 tc / [ 1 + wS2tc2] W1I  gI2 gS2 rIS-6 tc / [ 1 + wI2tc2]

Steady-state NOE In small, rigid molecules the following relationships are valid: Basically, they move really fast. There is a big simplification of the probability equations (the W stuff), and we end with a simple dependency for the NOE enhancement: hI = 0.5. Bummer. We had the rIS-6 dependency there and now it’s gone. The problem is that we have only two spins, and this basically means no ‘geometry’. This is normally the case for 1H-13C. Fortunately, when we look at the proton enhancements in a molecule, we are always looking at more than one proton. This means that if we irradiate a proton, it will be dipolarly coupled to several protons at the same time. (wI - wS) * tc<< 1 wS * tc<< 1 wS * tc << 1 (wI + wS) * tc<< 1 b a c rba rac

Steady-state NOE (continued) What happens is that we have competing relaxation mechan- isms for the proton we are saturating (two or more protons in the surroundings). Now the rates of the relaxation with the different protons becomes important (the respective Ws). The equations get really complicated, but if we are still in the extreme narrowing limit, we can simplify things quite a bit. In the end, we can establish a ‘simple’ relationship between the NOE enhancement and the internuclear distances: In order to estimate distances between protons in a molecule we can saturate one nuclei and analyze the relative enhancements of other protons. This is know as steady-state NOE. We take two spectra. The first spectrum is taken without any irradiation, and the second with. The two are subtracted, and the difference gives us the enhancement from which we can estimate distances... rIS-6 - SX fX{S} * rIX-6 fI{S} = hmax * rIS-6 + SX rIX-6

NOE difference spectroscopy  If our molecule has three protons, two of them at a fixed distance (a CH2), we have: Since we have a reference pair of protons (Ha and Hb) for which we know the distance, we can establish a reference NOE effect for it, and then calculate the distance between the other protons (Ha and Hc in this example): The sizes of the peaks are measured by integration. Since the effect is not symmetric, we usually do the I S and then the S I enhancement and take an average value. Hb C Ha Hb Hc Ha Hc _ = hab hac hab  rab-6 rac = rab * ( hab / hac ) -1/6 hac  rac-6

Transient NOE One of the problems of steady-state NOE is that we are continuously giving power to the system (saturation). This works well for small molecules, because W2 processes (double-quantum) are dominant and we have few protons. However, as the size and tc increase, other processes are more important (normal single-quantum spin-spin relaxation and zero-quantum transitions). Additionally, there are more protons in the surroundings of a larger molecule, and we have to start considering a process called spin diffusion: Basically, the energy transferred from I to S then diffuses to other nuclei in the molecule. We can see an enhancement of a certain proton even if it is really far away from the center we are irradiating, which would give us ambiguous results. Therefore, we need to control the amount of time we saturate the system. The longer we irradiate, the more spin diffusion… I S

Transient NOE (continued) There are also some technical difficulties if we try to do this by selective saturation. Since what we need is to see how a system returns to equilibrium through cross-relaxation, we can selectively invert one transition and then see how the NOE grows with time: A pulse sequence to do this could be the following: The last p / 2 pulse is usually called a read pulse, and its only job is to convert whatever magnetization is in <z> after tm into <xy> magnetization (detectable). All the equations are the same, but the NOE will also depend on the mixing time, tm. bb (**) W1I W1S W2IS (****) ab ba () W0IS W1S W1I aa (**) 180s 90 tm selective inversion

Transient NOE (…) If we do it for different tm values, we get NOE build-up curves, which in the case of two isolated protons and ideal conditions are exponentials that grow until they reach hmax. If we also take into account the T1/T2 relaxation, the NOE grows and then falls to zero (all the magnetization returns to the <z> axis…): hmax tm hmax tm

Transient NOE and NOESY If we had more spins and spin diffusion, the curve (for three spins, I, S, and X) would look like this: If we want to measure distances more or less accurately, we have to find a compromise between the mixing time tm that we use and the spin-diffusion we may have. Doing one proton at a time takes forever, so we obviously use a 2D experiment in which we study all protons at the same time. The sequence is called NOESY (NOE SpectroscopY), and it’s shown below: hmax Diffusion to X tm 90 90 90 t1 tm “inversion”

NOESY (continued) The first two pulses (with the t1 variable time) are basically an inversion for all protons, in which we label everything with chemical shifts and couplings (no re-focusing). What ends up in the <-z> axis evolves during the tm mixing time, and dipolarly coupled spins will suffer NOE cross- relaxation according to all the equations we’ve seen. The 2D spectrum after 2D FT has chemical shifts in both f1 and f2 (t2 has all the chemical shifts also) and cross-peaks for systems that are dipolarly coupled: The size of the cross-peak will depend on the internuclear distances. These are measured relative to a cross-peak for which we know a distance, using the cross-peak volumes. d (f1) Hc Hb C Ha Ha Hc Hb d (f2)

NOESY (…) Although this pulse sequence looks like the MQF-COSY pulse sequence, proper phase cycling removes COSY-type cross-peaks. One more thing we have to consider is the sign of the cross- peaks. Thye will depend on the correlation time tc, in the same way that the NOE enhancement does. We therefore have positive cross-peaks for w * tc << 1 and negative cross- peaks for w * tc << 1. As we said earlier, there is a range of w * tc values for which the enhancement can be zero (guess which looser had this happening to him?). If we express the NOE enhancement as a function of w * tc we get a sigmoid function, and we solve it for hI = 0: The options are to change tc (kind of hard), w (the magnet is pretty hard to change…), or trick the system to think that we are working at a very low field, where w * tc << 1 always. The experiment that does this is called ROESY, and it is based on spin-locking, which we will see next class. 5 + w2tc2 - 4w4tc4 hI = hI = 0 for w * tc = 1.12 10 + 23w2tc2 + 4w4tc4

Summary The NOE enhancement is, among other things, proportional to r-6, so it allows us to make determination of internuclear distances: stereochemistry and conformation. Depending on the size/rigidity (tc) of the molecule, steady- state or transient NOE experiments will give more accurate results: Small organic compounds: Steady-state NOE. Proteins, nucleic acids, saccharides: Transient NOE, build-up curves. We can do everything at once using NOESY (transient NOE). Next time Spin-locking - NMR at ‘lower’ fields… Uses in through-space correlation spectroscopy (ROESY) and J-correlation spectroscopy (TOCSY).