Let's explore the spectral density!

The model-free formalism of Lipari and Szabo is a way to convert experimental NMR data into a limited number of generalized parameters describing the internal dynamics of a protein. However, the relaxation rates that are typically measured by NMR — the R₁, the R₂, and the steady-state nuclear Overhauser effect (nOe) — do not themselves appear in the model-free formulas. Instead we see a term, J(ω), and this constitutes the interface between the data and the model. This term refers to the spectral density, which is a measure of the power available to relax spins at a given angular frequency. The relaxation rates measured by NMR spectroscopists interrogate this density at known frequencies, which means that we can use those rates to assess general information about the shape of the spectral density function and thus constrain the model-free parameters.

In biomolecular NMR, these rates are most frequently measured on the nitrogen of a backbone amide group, in which case they fundamentally depend on the spectral density at three frequencies: 0, the Larmor frequency of nitrogen (ω_N), and the larmor frequency of the proton (ω_H). The precise relationships are as follows:

R₁ = D [3J(ω_N) + 6J(ω_N+ω_H) + J(ω_N-ω_H)] + C [3J(ω_N)]
R₂ = D/2 [4J(0) + 3J(ω_N) + 6J(ω_H)+ 6J(ω_N+ω_H) + J(ω_N-ω_H)] + C/6[J(0) + 3J(ω_N)]
steady-state nOe = 1 + R_NOE γ_H / R₁ γ_N
R_NOE = D [6J(ω_N+ω_H) - J(ω_N-ω_H)]
D = μ₀²ℏ²γ_N²γ_H²/64π²r_NH⁶
C = Δσ²ω_N²/3

where γ_H and γ_N are the gyromagnetic ratios of these nuclei, ℏ is the reduced Planck constant, and μ₀ is the magnetic constant (or vacuum permeability, if you prefer), and Δσ is the chemical shift anisotropy of the ¹⁵N nucleus (typically -160 - -170ppm).

I'm not going to cover precisely why they have these relationships today; instead I want to focus on how these relationships connect certain dynamic behaviors to particular observations about relaxation rates. The key to this is to think about how the spectral density looks. At right I have a simplified spectral density calculated for a rigid protein of reasonable NMR size (I only show the positive side of the function, the negative is a mirror image). While the particular shape of the spectral density function will depend strongly on the internal dynamics and overall size, certain general features will be the same for most proteins. It should be immediately evident, for instance, that J(0) >> J(ω_N) >> J(ω_H) (shown on the figure for a 500 MHz magnet). This implies that each relaxation rate reports on just one spot in the spectral density. R₂ should be proportional to J(0), R₁ to J(ω_N), and R_NOE to J(ω_H), keeping in mind that ω_H >> ω_N.

The shape of this curve derives in a fairly obvious way from the Lorentzian used to calculate it, in this case the Lipari-Szabo formalism, which if you'll recall is:


Where τ_m is the time it takes the protein to tumble through one radian in solution, S² is the order parameter for the bond in question, and τ_e is the correlation time of internal motions. The Lipari-Szabo model is not the only model of the spectral density, but most of the alternatives just add more Lorentzians or scaling factors. These models differ in the fine structure of the spectral density, but the overall shape (and the features I'm about to describe) is generally not affected.

It should be clear from examining this (and given that τ_m >> τ_e) that the point where ωτ_m = 1 divides the spectral density into two regions. Where ωτ_m <= 1, the first term dominates, and the spectral density is determined by S² and τ_m. Where ωτ_m >> 1, the second term dominates and the spectral density is essentially dependent on (1-S²) and τ_e. This being the case, you would expect highly flexible moieties (low S²) to have inefficient R₂ and R₁ relaxation and highly efficient NOE relaxation, and this we generally find to be the case.

Similarly, you would predict that increasing τ_m would cause R₂ to increase. The graph at right simulates relaxation rates for a typical, rigid backbone amide nitrogen (at 500 MHz) as the τ_m increases (note log scale on x). As you can see, the R₂ (red) does in fact get continuously higher as τ_m is increased; this is one of the reasons NMR spectroscopy of very large molecules is so difficult. Also note that R₁ (blue) goes through a maximum and then declines. This is because as τ_m increases, the point where ωτ_m = 1 shifts to lower and lower frequency. When |ω_N| > 1/τ_m, the spectral density at ω_N starts to fall off, reducing R₁. This might sound advantageous, but in fact it is another reason that spectroscopy on large molecules is difficult — their inefficient R₁ relaxation means that additional time must be scheduled after each transient to create a sufficiently sensitive steady state. Because even a simple spectrum can have 2048 transients, adding just a few fractions of a second per transient can rapidly amount to a significant increase in experiment time.

It's obvious that it would be questionable to map the spectral density based on just three relaxation rates, if for no other reason than that we have four unknowns and three pieces of data. This is typically addressed in three ways, which are often used in combination. The first is to reduce the spectral density, by making some general assumptions about the nature of the spectral density around ω_H and collapsing the J(ω_N +/- ω_H) terms into 0.87*J(ω_H). Another approach is to increase the number of relaxation rates measured, by incorporating R_1zz or other measurements, but many of these rates incorporate additional factors (such as ρ_HH) that must also be fit, so that their ability to reduce the dimensionality of the problem is sometimes limited.

The third approach is to take data at several fields. The Larmor frequencies ω_H and ω_N depend on the strength of the magnetic field in the spectrometer, while J(0) is obviously field-independent. As a result, each additional field of data taken improves the ratio between data and unknowns. This improvement is valuable even when the relaxation is being fit to a simplified representation such as the model-free formalism, and therefore dynamics experiments should always include measurements at more than one field if at all possible. Moreover, the field-dependence of relaxation rates can be very informative, in general terms, about the dynamics of the system.

In the simplified view it might seem that R₂ should be essentially independent of field strength, but observations show this not to be the case. R₂ increases at high fields primarily because of the chemical shift anisotropy contribution, which has a square field dependence and therefore increases with field to a greater degree than ω_N declines. As a result, R₂ has a sort of chevron appearance as you vary the field, with differences in dynamics primarily affecting the magnitude rather than the shape. This means that for R₂ the field-dependence is not particularly informative about the dynamics. However, if a residue has anomalous R₂ field-dependence with respect to the rest of the protein, this can be an indicator of a chemical exchange process on the μs - ms timescale.

Because relaxation due to chemical shift anisotropy makes a lesser contribution to R₁ (and depends entirely on J(ω_N) for this rate) the behavior of R₁ with respect to field is generally much simpler — for proteins, the R₁ almost always decreases as field increases. The degree to which this occurs, however, can be quite different depending on the dynamics behavior that is going on. The reason for that can be seen in the sample spectral densities to the left, calculated for a typical backbone amide (blue) and a flexible one (red). As you can see, the more flexible residue has a lower J(0) and a smaller slope between the flat portions of the spectral density than the rigid one. This means that the R₁ will be lower at high field and higher at low field, decreasing the field-dependence of the residue's relaxation. The exact magnetic field where this crossover occurs depends on the correlation times of the internal motion and global tumbling.

The gyromagnetic ratios of the hydrogen and nitrogen nuclei have opposite signs, so the heteronuclear NOE measured for these nuclei should be less than one. How much less depends on the relative ratio between J(ω_H) and J(ω_N). For flexible residues, the spectral density at large ω will be high (and that at lower ω will be low), this ratio will be large, and a low value will be measured in the hetNOE experiment. R_NOE typically has a steep field-dependence for flexible residues, and because this rate dominates the ratio, one tends to see greater field-dependence of the hetNOE for flexible residues. However, the situation for the hetNOE is more complex than for the other two rates because the spectral density around ω_H defines the relaxation. As a result, the internal correlation time (particularly if it's on the order of 100 ps - 1 ns) starts to dictate the shape of the spectral density, and hence the magnetic field-dependence of relaxation. For certain τ_e, the hetNOE will have no apparent field dependence, whether the residue is flexible or not.

Actually parameterizing the dynamics of a given group requires numerical fitting of the relaxation data, but for many questions a qualitative estimate will suffice. In these cases just examining the field-dependence of one or two relaxation rates (especially R₁ or NOE) can provide valuable insight into the heterogeneous dynamics of a given protein. In the next post I'll describe an example of a case in which this turns out to be true.