What is the mechanism of spins "dropping down" from the ft state to the a state and "fanning out" around the two cones, and what determines the rates (R1 = 1/T1 and R2 = 1/T2)ofNMR relaxation? These processes are intimately tied to the motion of molecules as they tumble ("reorient") in solution in their rapid Brownian motion, and measurement of the NMR relaxation parameters T1 and T2 can even give us detailed information about molecular dynamics (motion) from the point of view of each spin in the molecule. A simplified model of these physical processes and their consequences for NMR will help you to understand the effect of molecular size on relaxation and the NOE.
The nucleus cannot simply transfer energy to molecular motions (vibration, rotation, translation, etc.) by collisions because the nucleus is not really "attached" to the rest of the molecule. The tiny nucleus sits in a vacuum very far from the bonding electrons that hold the molecule together, and its only mechanism for interaction with the outside world is through its magnetic properties. Thus only magnetic fields can affect the nucleus and induce transitions from the 5 to the a state (longitudinal relaxation) or create random differences in the rate of precession (transverse relaxation). For T1 relaxation, a magnetic field must be oscillating at the frequency of the transition (the Larmor frequency, vo) in order to induce a transition from the 5 to the a state. Molecular motion ("tumbling") can generate these oscillating magnetic fields if the molecule is tumbling at the right frequency. There are several ways this can happen; we will focus on the major one, which is called the dipole-dipole interaction, and later mention one other mechanism.
Consider a nucleus such as a 13C nucleus within a molecule. If the carbon in question is a methine (CH) carbon, then it has a hydrogen atom (1H) rigidly attached to it at a distance of 1.1 A(1 A = 10 10 m). As the molecule tumbles, from the point of view of the 13C nucleus, the 1H nucleus is rotating around it in a circle of radius 1.1 A at the tumbling rate of the molecule. The 1H nucleus retains its orientation with respect to the external (Bo) magnetic field as the molecule tumbles because the nucleus is not "attached" to the molecule in any way. Precession of the 1H nucleus is not important for this phenomenon, so we can view the proton as a rigid magnet oriented along +z (a state) or — z (5 state). As the molecule tumbles, the tiny magnetic field of the 1H nucleus is "felt" by the 13C nucleus at its location in space (Fig. 5.12). When the 1H is above or below the13C, its field adds to the Bo field at the
location of the 13 C nucleus, but when it is to the left or right of the 13 C nucleus its magnetic field subtracts a slight amount from the Bo field at the position of the 13 C nucleus. Thus the effective magnetic field experienced by the 13C nucleus (Beff) is modulated by a sinusoidal variation whose frequency is the rate of tumbling of the molecule and whose amplitude is proportional to the 1H "magnet strength" (yh) and to the inverse third power of the distance (rCH = 1.1 A) between the 1H and the 13C. If the frequency of this perturbation is exactly equal to the Larmor frequency for the 13C nucleus (yCBo/2n), then 13C spins in the upper energy state will be stimulated to drop down to the lower energy state in a process similar to stimulated emission (Chapter 1, Section 1.4.7). Unlike stimulated emission, which occurs when we add a radio frequency signal to the sample, there are no photons absorbed or emitted, and the energy is coupled to the rotational motion of the molecule and released as heat. Because the process of relaxation involves interaction of the 13C nuclear magnet's field with the 1H nuclear magnet's field, the rate of stimulated transitions is proportional to both yC/r3 and to yH/r3, so it depends on the inverse sixth power (1/r6) of the distance between the two nuclei. This dipole-dipole interaction is central to the NOE as well, which also exhibits a 1/r6 dependence.
All of this requires that the molecule be tumbling at exactly the Larmor frequency of the nucleus that is undergoing relaxation in order to stimulate a transition from j state to a state. In fact, only a very small fraction of molecules is tumbling at this frequency at any one time. We can look at the distribution of tumbling rates as a histogram with tumbling frequency v (in hertz) on the horizontal axis and number of molecules tumbling at that frequency on the vertical axis. This function, which is a property of molecular size and shape as well as the viscosity of the solvent, is called the spectral density function or J(v). A simplified logarithmic plot is shown in Figure 5.13 for five different molecules ranging in molecular weight from 10 to 100,000 Da (100 kD). A typical "organic" molecule (a "small" molecule to the NMR spectroscopist) would be on the order of 100 Da, whereas a peptide, glycopeptide, or oligosaccharide might be in the range of 1000 Da ("medium-sized") and proteins and nucleic acids (DNA and RNA) would range from 10 to 100 kD ("large") in molecular weight. Each molecule can also be characterized by its average tumbling time tc (formally known as the rotational correlation time). This is essentially the average time it takes for the molecule to change its orientation with respect to the Bo field (z axis); it is the reciprocal of the average tumbling frequency. Small molecules tumble rapidly and have a short tc, whereas large molecules tumble slowly and have a long tc. As you can see from the histogram (Fig. 5.13), the distribution of tumbling frequencies for any size molecule is flat over a wide range of frequencies up to a maximum or cutoff value. Above this frequency the number of molecules drops rapidly to zero. The cutoff frequency is very high for small molecules (10-100 Da), lower for the "medium-sized" molecules (1 kD) and quite low for large molecules (10-100 kD). The graph is based on the tumbling of spherical molecules in water at 27 °C, with molecular density typical of proteins.
Because in each case we are dealing with the same number of molecules, as we squeeze the population of molecules into a smaller range of frequencies, the number of molecules at any one frequency within the range increases. This explains the higher level at the left side of the histogram as the molecular size increases. Because of the logarithmic horizontal scale, the differences are much larger than those shown in the plots—note the increase in vertical scaling at lower molecular weights. At 500 MHz, which is the Larmor frequency
for protons in a magnetic field of 11.74 T, the dotted line (vsq = vo) shows that the number of molecules tumbling at this rate is greatest for the 1 kD ("medium-sized") molecules (the 100 Da curve is actually only 15% of the height shown, in relation to the 1 kD curve). This would give the fastest T1 relaxation because a larger proportion of molecules of this size are tumbling at exactly the Larmor frequency. For smaller molecules, the much wider range of tumbling rates means that it is unlikely to find a molecule tumbling at exactly the Larmor frequency, and for larger molecules the Larmor frequency is higher than the "cutoff"; so very, very few molecules tumble this fast. This is consistent with experimental results: As molecular size increases from 10 to 1000 Da, the relaxation rate (R1 = 1/T1) increases (T1 decreases), and above this critical size (1000 Da) the relaxation rate falls off (T1 increases). This relationship is shown in Figure 5.14 for a pair of protons separated in a rigid molecule by a distance of 1.8 A at a field strength corresponding to a Larmor frequency of 500 MHz. Again, the molecular weight scale is based on a spherical molecule of density 1.42 g/L (typical for a protein) in water at 27 0 C. The critical molecular weight, where the reciprocal of the tumbling time (1/rc) is close to the Larmor frequency in radians s-1 (^o), depends on molecular shape as well as solvent viscosity. This "crossover" condition is usually written as: Tc ~ 1. We will see that molecules with this "medium" size are a problem for NOE experiments because the theoretical NOE falls to zero. In the "small molecule" regime (&o tc ^ 1) an increase in molecular weight increases the relaxation rate (decreases T1) and in the large molecule regime (^o tc ^ 1) an increase in molecular weight decreases the relaxation rate (increases T1).
Although T1 increases with molecular size for large molecules, note that the spin-spin or transverse relaxation rate T2 continues to decrease as molecular size increases (Fig. 5.14). This is harder to explain, but we can rationalize this effect by considering that the "fanning
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