An Improved Picture of Methyl Dynamics in Proteins from SRLS Analysis of ²H Spin Relaxation

Abstract

Protein dynamics is intimately related to biological function. Core dynamics is usually studied with ²H spin relaxation of the ¹³CDH₂ group, analyzed traditionally with the model-free (MF) approach. We showed recently that MF is oversimplified in several respects. This includes the assumption that the local motion of the dynamic probe and the global motion of the protein are decoupled, the local geometry is simple, and the local ordering has axial symmetry. Because of these simplifications MF has yielded a puzzling picture where the methyl rotation axis is moving rapidly with amplitudes ranging from nearly complete disorder to nearly complete order in tightly packed protein cores. Our conclusions emerged from applying to methyl dynamics in proteins the slowly relaxing local structure (SRLS) approach of Polimeno and Freed (J. Phys. Chem. 1995, 99, 1099), which can be considered the generalization of MF, with all the simplifications mentioned above removed. The SRLS picture derived here for the B1 immunoglobulin binding domain of peptostreptococcal protein L, studied over the temperature range of 15 − 45 °C, is fundamentally different from the MF picture. Thus, methyl dynamics is characterized structurally by rhombic local potentials with varying symmetries, and dynamically by tenfold slower rates of local motion. On average potential rhombicity decreases, mode-coupling increases and the rate of local motion increases with increasing temperature. The average activation energy for local motion is 2.0 ± 0.2 kcal/mol. Mode-coupling affects the analysis even at 15 ^oC. The accuracy of the results is improved by including in the experimental data set relaxation rates associated with rank 2 coherences.

I. Introduction

NMR spin relaxation is a powerful method for studying protein dynamics.^1-9 The traditional probe for investigating backbone motion is the ¹⁵N-¹H bond and the common probe for studying side chain motion is the uniformly ¹³C-labeled and fractionally deuterated methyl group, ¹³CH₂D.^5,6,10-12 In this study we focus on the latter. Methyl dynamics in proteins is analyzed typically with the model-free (MF) approach,^13-15 that assumes that the global and local motions of the probe are decoupled due to the former being much slower than the latter. This is an approximation, and so are the high symmetries assigned implicitly to the diffusion, ordering and magnetic tensors involved, and the coincidence of their frames, which simplifies the local geometry. By virtue of these simplifications an analytical formula is obtained for the measurable spectral density,¹³ specific values of which enter the expressions for the experimental relaxation rates. The original MF spectral density¹³ is determined by an effective correlation time for local motion, τ_e, a squared generalized order parameter, S², representing the spatial restrictions at the site of the motion of the probe, and the global motion correlation time, τ_m. The latter is usually determined independently.

For methyl dynamics MF considers two local motions including rotation about the C-CH₃ axis and fluctuations of the C-CH³ axis.¹⁰ Moreover, the methyl rotation axis C-CH₃ (to be denoted M_z, with M representing the local ordering/local diffusion frame) is tilted at β_MQ = 110.5° from the magnetic quadrupolar frame, Q, which lies along the C-D bond (110.5^o is the tetrahedral angle taking r_CH = r_CD = 1.115 Å)¹². Yet, as pointed out above, the original MF formula,¹³ typically used in methyl dynamics analyses, features only one mode of local motion and has no provision for a “diffusion tilt”. These features entail further approximations (see below).

We have shown recently that the MF approach is oversimplified.^16-23 This has been accomplished by applying to NMR spin relaxation in proteins¹⁶ the Slowly Relaxing Local Structure (SRLS) approach of Freed and co-workers.^24-26 SRLS can be considered the generalization of MF, yielding the latter in asymptotic limits.^16,20,21,24 Unlike MF the SRLS model takes into account rigorously the dynamical coupling between the global motion of the protein and the local motion of the dynamic probe, brought into effect by a rhombic coupling potential. It features explicitly local motional modes parallel and perpendicular to the methyl rotation axis.^16-23,25,26 and accounts rigorously for the tilt between a rhombic local ordering frame, M, and the magnetic frame, Q. The Euler angles Ω_C'M, which relate the M frame to a local director frame, C’ (e.g., the equilibrium C-CH₃ orientation), are governed by the coupling/ordering potential. “Dynamical coupling” means that through the time dependence of Ω_C'M the locally reorienting dynamic probe follows the slower motion of the protein.

The MF simplifications have far-fetched implications. For a M frame tilted relative to the Q frame the spectral density, J^QQ(ω) (QQ denotes quadrupolar auto-correlated relaxation), comprises three generic spectral density functions, j_K(ω), with K = 0, 1, 2 (refs. ^{16, 25, 26}), by analogy with J(ω) for axial global diffusion of a rigid protein comprising three Lorentzian functions with K = 0, 1, 2. Yet, the MF spectral density consists of a single function which represents the K = 0 contribution. Various parameterizations of its form have been attempted to overcome the flaw of omission of the K = 1 and K = 2 contributions. If the local ordering frame, M, is rhombic, as we found it to be,²³ cross-term functions j_KK’(ω) will also enter the expression for J^QQ(ω). This renders the parameterized MF spectral density to be very different from the actual spectral density. However, in many cases the experimental data can be reproduced by force-fitting, with the statistical criteria fulfilled, but the best-fit parameters (S² and τ_e) highly inaccurate.²³

The form of J^QQ(ω) is parameterized in MF as follows. To accommodate two local motional modes S² is factored into the product [P₂(cos 110.5°)]² × S_axis² = 0.1 × S_axis² . The term [P₂(cos 110.5°)]² = 0.1 represents the squared order parameter for methyl rotation about _C−CH₃, and S_axis² the axial squared order parameter for motion of the C−CH₃ axis.^10,27a As outlined below in detail, factoring S² as shown, with the meaning of the constituents as indicated, is only valid when τ_e is in the extreme motional narrowing limit.^23,27b,c Yet, in practice finite values of τ_e are required in MF analyses to fit the experimental data.

Practical implications of the MF simplifications have been investigated recently²³ using the B1 immunoglobulin binding domain of peptostreptococcal protein L (to be called below “protein L”)¹² and ubiquitin²⁸ as test cases. The respective data were subjected to SRLS analysis²³ and the emerging dynamic pictures were compared with the corresponding previously obtained MF pictures.^12,28

We found that rhombic local potential/local ordering is required to analyze methyl dynamics consistently and insightfully.²³ MF analyses yield unduly large distributions in the value of S_axis² ranging from nearly complete disorder (S_axis² ~ 0.1) to nearly complete order (S _axis² ~ 1), often exhibiting three distinct maxima.^6,28-31 The (pervasive) low S_axis² values imply large-amplitude excursions of the C-CH₃ axis in tightly packed protein cores.⁶ Interpretation in terms of limited excursions using the 1D and 3D Gaussian Axial Fluctuations (GAF) models^32-34 is incompatible with axial symmetry around C−CH₃, inherent in the definition of S_axis² (ref. ^27a). Contrary to the problematic MF picture, SRLS interprets the variations in the experimental data as variations in the symmetry, and to some extent the magnitude, of the local ordering potential (or local ordering tensor).²³ The three categories of S_axis² values correspond to different forms (symmetries) of the rhombic local potential.²³ This is physically tenable, provides new and interesting site-specific structural information, and agrees with NMR J-coupling and reduced dipolar coupling,^35,36 molecular-dynamics^37,38 and molecular mechanics³⁹ studies. All of these investigations have shown that local structural asymmetry prevails at methyl sites in proteins, contrary to the axial S_axis² based MF picture.

The present paper is an extension of our previous study²³ which was based on ²H T₁ and T₂ data acquired for protein L¹² and ubiquitin²⁸ at ambient temperature and magnetic fields of 11.7 and 14.1 T. Kay and co-workers developed pulse sequences for measuring relaxation rates associated with double-quantum, two-spin-order and antiphase rank 2 coherences,¹¹ in addition to ²H T₁ and T₂.¹⁰ For protein L the Kay group acquired all five ²H relaxation rates at 5, 15, 25, 35 and 45 °C at a magnetic field of 11.7 T. At 25 (5) °C additional data were acquired at magnetic fields of 9.4, 14.1 and 18.8 (14.1) T. This is among the most extensive and robust data sets of autocorrelated ²H relaxation rates currently available. In the present study we used these data, kindly provided by Prof. L. E. Kay, to explore temperature, magnetic field and rank 2 coherence dependence, and treat several important aspects of methyl dynamics. The issue of relatively large uncertainties in the best-fit parameters, implied by the relatively narrow portion of J^QQ(ω) sampled by the experimental data, is addressed. Note that unlike the case of proton-bound heteronuclei, where ¹H contributes high-frequency J(ω) values through the NOE, for ²H relaxation only the ω = 0, ω_D and 2ω_D values, with ω_D denoting the ²H resonance frequency, are sampled at any given magnetic field. In this context the benefit of using up to four-field data sets, including rank 2 coherences, is explored herein.

We find that the protein L methyl sites exhibit rhombic potentials of different forms in the SRLS scenario instead of amplitudes of C-CH₃ motion of different extents in the MF scenario. The local motional modes are tenfold slower in the SRLS scenario. Mode-coupling is important even at 15 °C. On average potential rhombicity decreases, mode-coupling increases and the rate of local motion increases with increasing temperature. The average activation energy for local motion is 2.0 ± 0.2 kcal/mol. The accuracy of the results is improved by including in the experimental data set relaxation rates associated with rank 2 coherences.

The Theoretical Background appears in section II. The various topics mentioned above are treated under Results and Discussion in section III. Our conclusions appear in section IV.

II. Theoretical Background

The Theoretical Background relevant for this paper appears in ref. 23. For convenience a brief summary is presented below.

1. The Model-free (MF) approach

The original MF spectral density, J(ω), based on τ_e << τ_m (i.e., an effective local motion, τ_e, much faster than the global motion, τ_m), is given by:¹³

$$J\left(\omega \right)=S^2{\tau }_m∕(1+{{\tau }_m}^2{\omega }^2)+(1-S^2){\tau }_e’∕(1+{\tau }_e{’}^2{\omega }^2),$$ (1)

where 1/τ_e’ = 1/τ_m + 1/τ_e ~ 1/τ_e.

This equation has been adapted to methyl dynamics where two restricted local motions around and of the methyl averaging axis are considered^10,27a by setting S² equal to [P₂(cos 110.5°)]²× _Saxis² = 0.1×S_axis². The term 0.1 represents the squared order parameter associated with the motion around the C-CH₃ axis, and S_axis² the axial squared order parameter associated with motion of the C-CH₃ axis. The effective correlation time for local motion, τ_e, has been associated with both local motional modes. This yields the spectral density:^10,27a

$$J^{QQ}\left(\omega \right)={S_{axis}}^{2}0.1{\tau }_m∕(1+{\omega }^2{{\tau }_m}^2)+(1-{S_{axis}}^{2}0.1){\tau }_e’∕(1+{\omega }^2{\tau }_e{’}^2).$$ (2)

2. The Slowly Relaxing Local Structure (SRLS) model

The fundamentals of the stochastic coupled rotator slowly relaxing local structure (SRLS) theory^24,25 as applied to biomolecular dynamics²⁶ have been developed recently for NMR spin relaxation in proteins.^16-23 Two rotators, representing the global motion of the protein, R^C, and the local motion of the probe (C-D bond in this case), R^L, are treated. The motions of the protein and the probe are coupled by a local potential, U(Ω_C'M), where C’ denotes the local director fixed in the protein, and M the local ordering/local diffusion frame fixed in the probe. The Euler angles Ω_C'M are modulated by the local motion whereas the Euler angles Ω_LC', with L denoting the fixed laboratory frame, are modulated by the overall tumbling of the protein.²¹ If the protein is considered axially symmetric then a global diffusion frame C tilted relative to the C’ frame will be defined. The site-specific angles, β_CC’, are fixed in the protein. The various frames entering the SRLS/model and the magnetic quadrupoalr frame are shown in Figure 1.

Figure 1.

(a) Various reference frames which define the SRLS model: L – laboratory frame, C – global diffusion frame associated with protein shape, C’ – local director frame fixed in the protein, M – local ordering/local diffusion frame fixed in the C-D bond, Q – quadrupolar tensor frame along the C-D bond. (b) Application to methyl dynamics. The simple case of motion about the rotation axis of the methyl group is illustrated. The equilibrium orientation of the CH₂D−C bond (C^α−C^β for alanine, C^β−C^γ1 and C^β−C^γ2 for valine, etc.) is taken as the local director, C’. The local ordering frame, M, is assumed in this illustration for simplicity to be axially symmetric. Z_M orients preferentially parallel to C’. It also represents the methyl rotation axis, and it is tilted relative to Z_Q, i.e., the C−D bond, at β_MQ = 110.5° (when r_CH = r_CD is set equal to 1.115Å − ref. 23). The angle β_C'M is the stochastic angle between the instantaneous orientation of the M frame, Z_M, and its equilibrium orientation, C’.

Formally the diffusion equation for the coupled system is given by:

$$\frac{\partial }{\partial t}P(X,t)=-\widehat{\Gamma }P(X,t),$$ (3)

where X is a set of coordinates completely describing the system.

$$\begin{array}{cc}\hfill X& =({\Omega }_{C^{\prime }M},{\Omega }_{L{C}^{\prime }})\hfill \\ \hfill \widehat{\Gamma }& =\widehat{J}\left({\Omega }_{C^{\prime }M}\right){\mathbf{R}}^L{P}_{eq}\widehat{J}\left({\Omega }_{C^{\prime }M}\right)P_{eq}^{-1}+[\widehat{J}\left({\Omega }_{C^{\prime }M}\right)-\widehat{J}\left({\Omega }_{L{C}^{\prime }}\right)]{\mathbf{R}}^c{P}_{\mathrm{e}q}[\widehat{J}\left({\Omega }_{C^{\prime }M}\right)-\widehat{J}\left({\Omega }_{L{C}^{\prime }}\right)]P_{\mathrm{e}q}^{-1}\hfill \end{array}$$ (4)

where Ĵ(Ω_C'M) and Ĵ(Ω_LC’) are the angular momentum operators for the probe and the protein, respectively.

The Boltzmann distribution P_eq = exp [−u (Ω_C'M)]/ 〈exp [−u(Ω_C'M)]〉 is defined with respect to the (scaled) probe-cage interaction potential given by:

$$u\left({\Omega }_{C^{\prime }M}\right)=\frac{U\left({\Omega }_{C^{\prime }M}\right)}{k_{B}T}=-c_0^2{D}_{0,0}^2\left({\Omega }_{C^{\prime }M}\right)-c_2^2[D_{0,2}^2\left({\Omega }_{C^{\prime }M}\right)+D_{0,-2}^2\left({\Omega }_{C^{\prime }M}\right)].$$ (5)

This represents the expansion in the full basis set of Wigner rotation matrix elements, D^L_KM(Ω_C'M), with only lowest order, i.e. L = 2, terms being preserved.^21,23,25,26 The coefficient $c_0^2$ (given in units of _kBT) is a measure of the orientational ordering of the C-D bond with respect to the local director, C’, whereas $c_2^2$ measures the asymmetry of the ordering around the director. Time correlation functions are calculated as:

$$C_{M,K{K}^{\prime }}^J\left(t\right)=\langle {D_{M,K}^J}^{\ast }\left({\Omega }_{LM}\right)\mid \text{exp}(-\widehat{\Gamma }t)\mid D_{M,K^{\prime }}^J\left({\Omega }_{LM}\right)P_{eq}\rangle .$$ (6)

Their Fourier-Laplace transforms yield the spectral densities j^J_MKK’(ω).

In the case of zero potential, $c_0^2=c_2^2=0$, the solution of the diffusion operator associated to the time evolution operator features three distinct eigenvalues for the probe motion:

$$1∕{\tau }_K=6{R^L}_{\perp }+K^2({R^L}_{\parallel }-{R^L}_{\perp })\text{for}K=0,1,2,$$ (7)

where R^L_∥ = 1/(6τ_∥) and R^L_⊥ = 1/(6τ_⊥) = 1/(6τ₀). Only the diagonal terms, j_K(ω) (the functions j_KK’ denote the real part of j²_M,KK’ – see ref. 25), are non-zero and they can be calculated analytically as Lorentzian spectral densities, each defined by width 1/τ_K . When the ordering potential is axially symmetric, $c_0^2\ne 0$, $c_2^2=0$ again only diagonal terms persist, but they are given by infinite sums _of Lorentzian spectral densities which are defined in terms of eigenvalues 1/τ_i of the diffusion operator, and weighing factors c_K,i, such that:

$$j_K\left(\omega \right)=\sum _i\frac{c_{K,i}{\tau }_i}{1+{\omega }^2{\tau }_i^2}.$$ (8)

The eigevalues 1/τ_i represent modes of motion of the system, in accordance with the parameter range considered. Note that although in principle the number of terms in eq 8 is infinite in practice a finite number of terms is sufficient for numerical convergence of the solution.

Finally, when the local ordering potential is rhombic, $c_0^2\ne 0$, $c_2^2\ne 0$ both diagonal j_K(ω) and non-diagonal j_KK'(ω) terms are different from zero and need to be evaluated explicitly according to expressions analogous to eq 8.

The spectral densities j_KK’(ω) are defined in the M frame. If the M frame and the magnetic frame are tilted a Wigner rotation will be required to obtain the measurable auto-correlated spectral density, J^QQ(ω), from the j_K(ω) and j_KK’(ω) spectral densities.⁴⁰ Due to the additional symmetry j_M,K,K′ = j_{M,−K,−K′} only the diagonal terms, j_K(ω), with K = 0, 1, 2 and the non-diagonal terms, j_KK’(ω), with KK’ = (−2,2), (−1,1), (−1,2), (0,1), (0,2), (1,2), need to be considered.

For an axial magnetic frame, Q, one has the explicit expression:

$$J^{QQ}\left(\omega \right)={d^2}_{00}{\left({\beta }_{MQ}\right)}^2{j}_{00}\left(\omega \right)+2{d^2}_{10}{\left({\beta }_{MQ}\right)}^2{j}_{11}\left(\omega \right)+2{d^2}_{20}{\left({\beta }_{MQ}\right)}^2{j}_{22}\left(\omega \right)+4{d^2}_{00}\left({\beta }_{MQ}\right){d^2}_{20}\left({\beta }_{MQ}\right)j_{02}\left(\omega \right)+2{d^2}_{-10}\left({\beta }_{MQ}\right){d^2}_{10}\left({\beta }_{MQ}\right)j_{-11}\left(\omega \right)+2{d^2}_{-20}\left({\beta }_{MQ}\right){d^2}_{20}\left({\beta }_{MQ}\right)j_{-22}\left(\omega \right).$$ (9)

with only the diagonal terms, j_K(ω), with K = 0, 1, 2, and the non-diagonal terms, j_KK’(ω), with KK’ = (0, 2) , (−1,1) and, (−2, 2) contributing.

A convenient measure of the orientational ordering of the C−D bond is provided by the order parameters, $S_0^2=\langle D_{00}^2\left({\Omega }_{C’M}\right)\rangle$ and $S_2^2=\langle D_{02}^2\left({\Omega }_{C’M}\right)+D_{0-2}^2\left({\Omega }_{C’M}\right)\rangle$, which are related to the orienting potential (eq 5), hence $c_0^2$ and $c_2^2$, via the ensemble averages:

$$\langle D_{0n}^2\left({\Omega }_{C^{\prime }M}\right)\rangle =\int d{\Omega }_{C^{\prime }M}{D}_{0n}^2\left({\Omega }_{C^{\prime }M}\right)\text{exp}[-u\left({\Omega }_{C^{\prime }M}\right)]∕\int d{\Omega }_{C^{\prime }M}\text{exp}[-u\left({\Omega }_{C^{\prime }M}\right)]$$ (10)

One may convert to Cartesian ordering tensor components according to $S_{zz}=S_0^2$, $S_{xx}=\left(\sqrt{3∕2}{S}_2^2-S_0^2\right)∕2$, $S_{yy}=-\left(\sqrt{3∕2}{S}_2^2-S_0^2\right)∕2$. Note that S_xx + S_yy + S_zz = 0.

For ²H relaxation the measurable quantities are J^QQ(0), J^QQ(ω_D) and J^QQ(2ω_D). Together with the squared magnetic quadrupole interaction they determine the experimentally measured relaxation rates according to standard expressions for NMR spin relaxation.^41,42

In the present study we allowed for at most four fitting parameters including the potential coefficients $c_0^2$ and $c_2^2$, R^C defined in units of R^L (hence representing the time scale separation between the global and local motions) and the “diffusion tilt” β_MQ. We used R^C = 1/6τ_m with τ_m as determined in refs. ¹² and ⁴³ based on ¹⁵N T₁/T₂ ratios.⁴⁴ When β_MQ is set equal to zero then the SRLS spectral density is formally analogous with the original MF spectral density (eq 1).

The functions j_K(ω) (eq 8) and j_KK’(ω) (equations analogous to eq 8) are calculated during data fitting on the fly. In the methyl dynamics application the local potential (the equivalent of S² in MF) is low, with $\mid c_0^2\mid$ and $\mid c_2^2\mid$ on the order of 1 − 2 (in units of k_BT). The time scale separation, R^C, is also not too large. The computational effort was found to be very reasonable in this case. Thus, it took about 40 minutes on a 3.2 GHz Pentium 4 processor to fit the relaxation data of a given methyl group of protein L. Our SRLS-based fitting program is similar to the MF fitting programs. The only extra requirement on the part of the user is to determine a truncation parameter, which determines the size of the matrix representation required for convergence of the solution (given by eq 8 or similar equations). Several trial and error calculations carried out for typical parameter sets suffice. Our current software is available upon request. The “Theoretical background” sections of this paper and of references 21 and 23 comprise the information required for ab initio programming.

3. MF as SRLS asymptote

Equation 1, from which eq 2 has been derived, represents the SRLS solution in the Born-Oppenheimer (BO) limit where τ_m >> τ_e.^45,46a Equation 1 was obtained in early work⁴⁵ as a perturbational expansion of SRLS in the limit of τ_m >> τ_e for axial local ordering, isotropic global (R^C) and local (R^L) diffusion, and collinear magnetic and ordering tensors.⁴⁵ In this limit S² represents ${\left(S_0^2\right)}^2$ (eqs 5 and 10 with $c_2^2=0$) and τ_e is given by τ₀ of eq 7. When the coupling potential is very high then the phenomenon called renormalization^46b,c becomes important. The renormalized correlation time, τ_ren, is given approximately by $2{\tau }_0∕c_0^2$ where $c_0^2$ represents the potential coefficient.^46c _We found that in this limit τ_e agrees with τ_ren and S² agrees with ${\left(S_0^2\right)}^2$ (references ²⁰ and ²¹). Outside of the BO limit eq 1 is not valid for diffusive motion (or wobblein-a-cone, if the latter is associated with a cosine squared potential), and will consequently lead to force-fitting.

As already noted, eq 2 features two dynamic modes associated with the axial order parameters, [P₂(cos β_MQ)]² = [P₂(cos 110.5°)]² = 0.1 and S_axis², for motion around and of the CCH₃ axis, respectively, and a common correlation time, τ_e. A major inconsistency in eq 2 is having the local ordering/local diffusion axis, M, tilted at 110.5° from the magnetic axis, Q, but ignoring the K = 1 and K = 2 contributions (eq 8). Independent of the model assumed (e.g., see reference ^27d where the Woessner model is being developed), these terms enter the calculation of the exact J^QQ(ω) function as 3 sin² (110.5°) cos² (110.5°) j₁(ω) = 0.323 j₁(ω), and 0.75 sin⁴ (110.5°) j₂(ω) = 0.577 j₂(ω). Yet, J^QQ(ω) of eq 2 comprises only the K = 0 term. This is only valid in the extreme motional narrowing limit²³ where R^L_⊥ = R^C and R^L_∥ → ∞ (eq 7, ref. 26; see also eq 31 of ref. 27b and pertinent discussion). The condition that R^L_∥ → ∞ (equivalent to τ_e → 0 in MF) renders the functions j₁(ω) and j₂(ω) so small that the K = 1 and K = 2 terms can be ignored. When τ_e → 0 the second term of eq 2 can be ignored to obtain J^QQ(ω) = (1.5 cos² 110.5° – 0.5)² j₀(ω) = 0.1 j₀(ω), where j₀(ω) = S_axis² τ_m (1 + ω² τ_m²). It can be seen that in this limit the effect of the local motional modes consists of reducing the quadrupole interaction (featured by the relaxation rate expressions) consecutively by 0.1 and S_axis².

In practice combined ²H T₁ and T₂ auto-correlated relaxation rates cannot be fit from a statistical point of view with τ_e set equal to zero in eq 2 because the extreme motional narrowing limit has not been attained. Technically the data can often be fit with τ_e ≠ 0. However, in this case the quantities S _axis² and τ_e have only vague physical meaning and, as shown below, they provide a distorted picture of the actual situation if taken seriously.

III. Results and discussion

In principle one should first consider axial local potentials in the SRLS fitting process. We showed previously²³ that this leads to a physically problematic picture and implies inconsistencies between ²H auto-correlation in ¹³CDH₂ (ref. ¹²) and HC-HH cross-correlation in ¹³CH₃ (ref. ⁴³). The problems mentioned have been resolved by allowing for rhombic potentials.²³ Therefore in this study we allow for rhombic potentials from the start.

The form of J^QQ(ω) for methyl dynamics

The SRLS model yields the generic spectral densities, j_KK’(ω). The measurable spectral density, J^QQ(ω), is given by linear combinations of the relevant j_KK’(ω) functions. The symmetry of the local ordering (M frame) determines which KK’ quantum numbers are non-zero, and, together with the other physical parameters, the form of the j_KK’(ω) functions. The orientation of the M frame with respect to the magnetic quadrupolar frame determines the coefficients in the linear combination yielding J^QQ(ω) (eq 9). For a rhombic M frame and β_MQ = 110.5° J^QQ(ω) features large contributions from j₀₀(ω), j₂₂(ω) and j₂₀(ω), and smaller contributions from j₁₁(ω), j_2-2(ω) and j_1-1(ω) (eq 9). J^QQ(0), J^QQ(ω_D) and J^QQ(2ω_D) enter the expressions for the ²H spin relaxation rates.

The appropriate representation of methyl dynamics by J^QQ(ω) makes possible the determination of the physical parameters (in general, R^L, R^C, β_MQ, $c_0^2$ and $c_2^2$) despite the fact that relatively few values of J^QQ(ω) are available at any given magnetic field. As pointed out above, unlike heteronuclear ¹⁵N-¹H and ¹³C-¹H spin relaxation ²H spin relaxation does not feature high-frequency values of J(ω), with ω on the order of ω_H. Choy and Kay⁴⁷ have shown that even with twenty synthetic data points (²H T₁, T₂ and three relaxation rates associated with rank 2 coherences generated at 9.4, 11.7, 14.1 and 18.8 T) the results of fitting these data with a model-free spectral density were unacceptable. Only further parameterization of this function yielded statistically acceptable results by force fitting.

We illustrate below typical SRLS spectral densities used in methyl dynamics analysis (Figures 2–4). Figure 2 shows the j_KK’(ω) functions calculated using a typical parameter set (obtained by analyzing the data acquired for methyl T23 of protein L at magnetic fields of 9.4, 11.7, 14.1 and 18.8 T, 25 °C) featuring $c_0^2=1.82$, $c_2^2=-0.67$ and R^C = 0.017. Note that R^C is given in units of R^L. Hence 0.017 represents the ratio between the global and local motional rates. Since the global motional rate is known independently, the parameter R^C actually determines the local motional rate. The inset shows a compressed ω-range extending from 0 to 4000 MHz. Clearly the portion of the j_KK’(ω) functions sampled consist of a restricted region outside of which these functions are not defined experimentally. Note that the magnetic field range scanned is almost as large as feasible with currently available technology. Figure 3 shows the j_KK’(ω) functions calculated for $c_0^2=1.5$, $c_2^2=-0.5$ and R^C = 0.05, and Figure 4 shows the j_KK’(ω) functions of Figure 3 assembled into J^QQ(ω) for β_MQ = 69.5° (the complement of 110.5°; the time correlation functions (eq 6) are the same for β_MQ and (180° − β_MQ)). While SRLS fitting needs to account for the various j_KK’(ω) functions (e.g., Figs. 3 and 4) and their coefficients, model-free only needs to reproduce specific values of the J^QQ(ω) function (e.g., Fig. 4) because the MF spectral density is a parameterized version of the actual spectral density. We found that SRLS and MF spectral densities, which are formally equivalent, yield best-fit parameters with significantly smaller uncertainties in the SRLS scenario.

Figure 2.

j_KK’(ω) functions with KK’ = (0,0) – black, (1,1) – red, (2,2) – green, (2,0) – blue, (2,−2) – yellow, and (1,−1) – brown calculated with eq 8 and an analogous equation appropriate for cross-terms, using $c_0^2=1.82$, $c_2^2=-0.67$ and R^C = 0.017, which are the best-fit SRLS parameters obtained for methyl T23 at 25°C. The inset shows a compressed ω-range extending from 0 to 4000 MHz. j_KK’(ω) is given in units of 1/R^L and ω is given in units of R^L.

Figure 4.

J^QQ(ω) function assembled from the j_KK’(ω) functions shown in Figure 3, using β_MQ = 69.5°. J(ω) is given in units of 1/R^L and ω is given in units of R^L.

Figure 3.

j_KK’(ω) functions with KK’ = (0,0) – black, (1,1) – red, (2,2) – green, (2,0) – blue, (2,−2) – yellow, and (1,−1) – brown calculated with eq 8 and an analogous equation appropriate for cross-terms, using $c_0^2=1.5$, $c_2^2=-0.5$ and R^C = 0.05. j_KK’(ω) is given in units of 1/R^L and ω is given in units of R^L.

Fitting strategy and error estimation

Exhaustive grid searches are impractical with SRLS. To ascertain that the global minimum of the Least Squares Sum (LSS) “target” function has been reached in a given fitting process we tested various strategies. It was found effective to carry out a coarse grid search, where $c_0^2$, $c_2^2$, R^C and β_MQ were allowed to vary, with the starting value of β_MQ in the vicinity of the tetrahedral angle, followed by a finer grid search. This strategy was superior to one where β_MQ was fixed at the tetrahedral angle value. In general the minimization converged for numerical reasons to (180° – β_MQ) (which, as indicated above, yields the same time correlation functions as β_MQ). To enable meaningful comparison of potential coefficients among methyl groups we fixed β_MQ at 69.5° in the final calculation of each methyl group. An alternative (more tedious) strategy, which yielded very similar but not identical results, consisted of allowing β_MQ to vary freely and accepting only those sets of temperature-dependent fits where β_MQ was within half a degree of 69.5°. Clearly it is necessary to devise an effective automated fitting protocol based on both statistical and physical criteria. This effort is underway.

Both procedures outlined above comprise error estimation capabilities, which can be used in different ways. The Monte Carlo-based error estimation methods used in MF-based fitting,⁴⁸ which would involve hundreds of calculations of J^QQ(ω), are not practical with SRLS. In reference ¹² a strategy where part of the experimental data was eliminated systematically was used to evaluate uncertainties. Ultimately we used a combination of the various methods mentioned to estimate the errors in the best-fit parameters, found to be typically on the order of 10%.

Importance of the rank 2 coherences

Including the rank 2 coherences into the experimental data set increases the accuracy of the results, obviously with a higher but still acceptable reduced χ² value. This is illustrated in Table 1, using for simplicity axial potentials. It can be seen that practically the same results are obtained independent of the starting values with 16 data points, 8 of which are relaxation rates associated with the rank 2 coherences (rows 1−6). Using 8 data points comprising only the ²H T and T relaxation rates yields ${\left(S_0^2\right)}^2$ values lower by 4.8% (rows 7−12). The discrepancies are parameter-range dependent (not shown).

Table 1.

Best-fit parameters, listed under "output", obtained with combined fitting of 16 relaxation rates (8 relaxation rates including ²H T₁ and T₂ acquired at 9.4, 11.7, 14.1 and 18.8 T T, and 6 relaxation rates associated with the three rank 2 coherences acquired at 11.7 and 14.1 T) measured at 25 °C for methyl A50, using the input parameters shown under "input" (rows 1–6). Same as above, using only ²H T₁ and T₂ data (rows 7–12). The τ_m value used was 4.05 ns.¹² χ²_red values are also shown. The local motional rate is given by R^C × τ_m.

	Input		Output
	$c_0^2$	RC	$c_0^2$	RC	χ 2red
1	0.5	0.0035	0.88	0.0038	7
2	0.89	0.0035	0.88	0.0037	7
3	1.2	0.0035	0.88	0.0038	7
4	0.5	0.008	0.88	0.0038	7
5	0.89	0.008	0.89	0.0037	7
6	1.2	0.008	0.88	0.0038	7
7	0.5	0.0035	0.84	0.0038	0.1
8	0.89	0.0035	0.84	0.0039	0.2
9	1.2	0.0035	0.84	0.0038	0.1
10	0.5	0.008	0.85	0.0038	0.1
11	0.89	0.008	0.85	0.0038	0.1
12	1.2	0.008	0.85	0.0038	0.1

A difference of 4.8% in S_axis² implies differences in the potential coefficient, $c_0^2$, exceeding 20%, due to the shape of the squared order parameter versus $c_0^2$ function for high ${\left(S_0^2\right)}^2$ values. This has been discussed in detail in ref. 21. Note that Table 1 features an illustrative example. In general the differences between corresponding best-fit parameters determined with rank 2 coherences included or excluded might be larger.

Qualitative MF-based information

We checked whether adequate qualitative information could be obtained with MF. The parameter used in MF to estimate the strength of the local spatial restrictions is S_axis². The SRLS parameter, which serves the same purpose, is the potential coefficient, $c_0^2$. Table 2 shows groups of methyl moieties with very similar S_axis² values and the corresponding best-fit SRLS parameters. Ten data points (²H T₁, T₂ and the three relaxation rates associated with the rank 2 coherences acquired at 11.7 and 14.1 T) measured at 25 °C have been used for the SRLS calculations. The MF data shown in Table 2 were taken from ref. 12. We also show $c_0^2\left(\mathrm{MF}\right)$ derived from S² = 0.1×S_axis² using the axial versions of eqs 5 and 10. The penultimate and ultimate columns on the right show $R\left(c_0^2\right)=c_0^2\left(\mathrm{SRLS}\right)∕c_0^2\left(\mathrm{MF}\right)$ and R(τ) = τ₀(SRLS)/τ_e(MF), respectively. It can be seen that these ratios are larger than unity and in many cases vary considerably within a given group of similar S_axis² values, indicating that the variations in S_axis² (MF) and $c_0^2\left(\mathrm{SRLS}\right)$ are likely to differ qualitatively.

Table 2.

Combined fitting of 10 relaxation rates (²H T₁, T₂ and the three relaxation rates associated with the rank 2 coherences) acquired at 11.7 and 14.1 T, 25 °C for the depicted methyl groups. The data under "MF" were taken from ref. 12. The penultimate and ultimate columns on the right show $R\left(c_0^2\right)=c_0^2\left(\mathrm{SRLS}\right)∕c_0^2\left(\mathrm{MF}\right)$ and R(τ) = τ₀(SRLS)/τ_e(MF), respectively. The residues marked in bold face letters required an extended MF formula for data analysis.¹²

Methyl	MF				SRLS
	Saxis2	$c_0^2$	τe, ps	RC	$c_0^2$	$c_2^2$	τ0, ps	RC	$R\left(c_0^2\right)$	R(τ)
V2γ1	0.73	1.22	54	0.013	1.77	-0.82	101	0.025	1.5	1.9
T37	0.74	1.23	50	0.012	1.89	-0.92	97	0.024	1.5	1.9
T55	0.98	1.41	51	0.012	1.83	-0.95	113	0.028	1.3	2.2
T17	0.97	1.42	45	0.011	1.56	-0.82	117	0.029
I9δ	0.38	0.89	24	0.006	1.57	-0.50	28	0.007	1.8	1.2
L8δ1	0.30	0.79	35	0.009	-0.29	-0.50	53	0.013
L8δ2	0.30	0.79	41	0.010	-0.35	-0.50	57	0.014
T15	0.57	1.09	69	0.017	1.79	-0.95	105	0.026	1.6	1.5
L38δ1	0.56	1.08	34	0.008	1.68	-0.74	61	0.015	1.6	1.8
V47γ1	0.57	1.09	55	0.014	1.51	-0.58	93	0.023	1.4	1.7
I58δ	0.58	1.10	17	0.004	1.86	-0.56	32	0.008	1.7	1.9
V49γ2	0.62	1.13	40	0.010	1.68	-0.82	61	0.015	1.5	1.5
L56δ1	0.61	1.12	70	0.017	1.89	-1.09	117	0.029	1.7	1.7
L56δ2	0.61	1.12	38	0.009	1.60	-0.68	65	0.016	1.4	1.7
A61	0.60	1.11	46	0.011	1.68	-0.73	77	0.019	1.5	1.7
T3	0.88	1.33	39	0.010	1.78	-0.89	85	0.021	1.3	2.2
T28	0.88	1.33	41	0.010	1.82	-1.02	85	0.021	1.4	2.1
I4γ	0.87	1.32	24	0.006	1.84	-0.78	73	0.018	1.4	3.1
A33/A11	0.89	1.34	37	0.009	1.87	-0.91	101	0.025	1.4	2.7
A11/A33	0.82	1.29	49	0.012	1.98	-1.02	134	0.033	1.5	2.7
A18	0.81	1.28	57	0.014	1.90	-1.00	109	0.027	1.5	1.9
I58γ	0.82	1.29	27	0.007	1.79	-0.96	53	0.013	1.4	2.0
T46	0.69	1.20	63	0.016	1.89	-1.00	105	0.026	1.6	1.7
V49γ1	0.68	1.18	34	0.008	1.72	-0.68	69	0.017	1.5	2.0
T23	0.84	1.31	39	0.010	1.84	-0.94	81	0.020	1.4	2.1
A50	0.84	1.31	24	0.006	1.81	-0.93	53	0.013	1.4	2.2
A31	0.83	1.30	77	0.019	1.94	-1.15	134	0.033	1.5	1.7

Profiles of $c_0^2\left(\mathrm{SRLS}\right)$ (based on the best-fit parameters obtained with SRLS at 5 and 25 °C for the methyl groups of Tables 3 and 5) and the corresponding S_axis² (MF) values (taken from ref. 12) are shown in Figure 5. For clarity the methyl groups have been classified as follows. SRLS categories 1, 2 and 3 correspond to $c_0^2>1.65$, $1.49\le c_0^2\le 1.65$ and $c_0^2<1.49$, ($c_0^2>1.80$, $1.60\le c_0^2\le 1.80$, and $c_0^2<1.60$) at 5 (25) °C. MF categories 1, 2 and 3 correspond to S_axis² > 0.85, 0.6 ≤ S_axis² ≤ 0.85 and S_axis² < 0.6 (S_axis² > 0.89, 0.6 ≤ S_axis² ≤ 0.89 and S_axis² < 0.6) at 5 (25) ° C. Clearly the $c_0^2$ S_axis² profiles differ significantly, often exhibiting opposite trends at the same temperature, and different temperature dependences.

Table 3.

Best-fit parameters obtained with combined fitting of 5 relaxation rates (²H T₁, T₂ and the three relaxation rates associated with the rank 2 coherences) acquired at 11.7 T, 5 °C, for the depicted methyl groups. The global motion correlation time used was τ_m = 8.01 ns.¹²,⁴³ The quadrupole interaction was 167 kHz, and the r_CH = r_CD distance 1.115Å. Note that all the $c_2^2∕c_0^2$ values are negative.

Methyl	$c_0^2$	$c_2^2$	RC	τ0, ps	$\mid c_2^2∕c_0^2\mid$
L8δ1	1.49	–0.20	0.014	112	0.13
L38δ1	1.51	–0.33	0.012	96	0.22
L56δ2	1.54	–0.22	0.015	120	0.14
T55	1.19	–2.66	0.004	32	2.24
T23	1.53	–0.28	0.016	128	0.18
A18	1.37	–1.66	0.013	104	1.21
A50	1.51	–0.77	0.010	80	0.51
L38δ2	1.50	–0.51	0.014	112	0.34
T37	1.51	–0.65	0.027	216	0.43
T17	1.92	–1.69	0.019	152	0.88
V49γ2	1.62	–0.85	0.012	96	0.59
A6	1.55	–0.92	0.020	160	0.59
A61	1.51	–0.95	0.013	104	0.63
V47γ1	1.51	–0.61	0.016	128	0.40
V2γ2	1.56	–0.82	0.012	100	0.53
I9γ	1.56	–0.80	0.012	96	0.50

Table 5.

Best-fit parameters obtained with combined fitting of 5 relaxation rates (²H T₁, T₂ and the three relaxation rates associated with the rank 2 coherences) acquired at 11.7 T, 25 °C, for the depicted methyl groups. The global motion correlation time used was τ_m = 4.05 ns.¹²,⁴³ Note that all the $c_2^2∕c_0^2$ values are negative except for $c_2^2∕c_0^2$ of L8δ₁, marked with an asterisk.

Methyl	$c_0^2$	$c_2^2$	RC	τ0, ps	$\mid c_2^2∕c_0^2\mid$
L8δ1	–0.29	–0.50	0.013	53	1.72*
L38δ1	1.68	–0.74	0.015	61	0.44
L56δ2	1.60	–0.68	0.016	65	0.43
T55	1.83	–0.95	0.028	113	0.52
T23	1.84	–0.94	0.020	81	0.51
A18	1.90	–1.0	0.027	109	0.53
A50	1.81	–0.93	0.013	53	0.51
L38δ2	1.44	–0.46	0.017	70	0.32
T37	1.89	–0.92	0.024	97	0.49
T17	1.56	–0.82	0.029	118	0.53
T3	1.78	–0.89	0.021	113	0.50
V49γ2	1.68	–0.82	0.015	61	0.52
A6	1.98	–1.02	0.033	134	0.52
A61	1.68	–0.73	0.019	77	0.43
V47γ1	1.51	–0.58	0.023	93	0.38
V2γ2	1.80	–0.79	0.018	73	0.44
I9γ	1.69	–0.78	0.015	61	0.46
A31	1.94	–1.15	0.033	134	0.59

Figure 5.

Schematic representing trends in $c_0^2$ SRLS and S_axis² MF at 5 °C (a) and 25 °C (b). The $c_0^2$ and S_axis² values have been classified into three groups according to their magnitude, as outlined in the text. These categories are denoted as 1, 2 and 3 on the ordinate. As pointed out above, the average error in c²₀ has been estimated at 10%. A conservative estimate of the average error in S_axis², based on ref. 12, is 4%. These figures translate into an average error margin of ±1.5 times the black symbol size, and ±1.0 times the red symbol size.

Hence, care is to be exerted in MF analyses in interpreting squared order parameters and local motion correlation times in terms of physical or biological properties. S_axis² has been used extensively to derive residual configurational entropy and heat capacity, with far-fetched implications.³⁰ Recently a new term called, “polar dynamics”, based on relative S_axis² values, was set forth.⁴⁹ Small differences in S_axis² and τ_e (which is actually a composite depending on both S² and a bare rate of local motion) have been used to elucidate communication pathways in proteins.⁵⁰ Such inferences require accurate best-fit parameters.

Data fitting

Five relaxation rates (²H T₁ and T₂, and relaxation rates associated with two-quantum, two-spin order and antiphase rank 2 coherences) acquired at 5, 15, 25, 35 and 45°C, 11.7 T, have been measured for the methyl groups L8δ₁, L38δ₁, L56δ₂, T55, T23, A18, A50, L38δ₂, T37, T17, V49γ , A6, A61, V47γ₁, V2γ₂, and I9γ. These data were fit with SRLS allowing $c_0^2$, $c_2^2$ and R^C to vary while keeping β_MQ fixed at 69.5°. The best-fit parameters are shown in Tables 3–7.

Table 7.

Best-fit parameters obtained with combined fitting of 5 relaxation rates (²H T₁, T₂ and the three relaxation rates associated with the rank 2 coherences) acquired at 11.7 T, 45 °C, for the depicted methyl groups. The global motion correlation time used was τ_m = 1.74 ns.¹²,⁴³ Note that all the $c_2^2∕c_0^2$ values are negative.

Methyl	$c_0^2$	$c_2^2$	RC	τ0, ps	$\mid c_2^2∕c_0^2\mid$
L8δ1	0.87	–0.32	0.018	31	0.37
L38δ1	1.39	–0.50	0.020	35	0.36
L56δ2	1.93	–0.58	0.007	12	0.30
T55	3.94	–1.18	0.063	110	0.30
T23	4.10	–0.66	0.067	117	0.16
A18	2.68	–0.70	0.052	91	0.26
A50	3.05	–0.54	0.034	59	0.18
L38δ2	1.49	–0.50	0.015	26	0.34
T37	2.51	–0.57	0.044	77	0.23
T17	2.15	–0.48	0.022	38	0.22
T3	3.06	–0.59	0.049	125	0.19
V49γ2	2.03	–0.49	0.024	42	0.24
A6	2.48	–0.74	0.058	101	0.30
A61	2.36	–0.44	0.028	49	0.19
V47γ1	1.96	–0.49	0.034	59	0.25
V2γ2	2.20	–0.47	0.030	52	0.21
I9γ	2.42	–0.74	0.061	106	0.31
A31	2.03	–1.00	0.050	87	0.49

The results of fitting the data acquired at 5 °C, shown in Table 3, feature best-fit R^C values of 0.01-0.02 (with the exception of methyl T55). The strength of the local potential, given by $c_0^2$, is approximately 1.5 in units of k_BT. The highest potential asymmetry (rhombicity), as given by $\mid c_2^2∕c_0^2\mid$, is on the order of 0.6 (with the exception of methyl groups T55, A18 and T17). Inspection of the data shown in Tables 4-7, obtained at the higher temperatures, points to a diversified picture. We present in Table 8 results obtained by averaging over the methyl groups analyzed at any given temperature. It can be seen that on average R^C increases, τ₀ decreases, $c_0^2$ increases, $c_2^2$ decreases and $\mid c_2^2∕c_0^2\mid$ decreases with increasing temperature. Similar trends were observed with the alternative fitting strategy where β_MQ was also allowed to vary.

Table 4.

Best-fit parameters obtained with combined fitting of 5 relaxation rates (²H T₁, T₂ and the three relaxation rates associated with the rank 2 coherences) acquired at 11.7 T, 15 °C, for the depicted methyl groups. The global motion correlation time used was τ_m = 5.36 ns.¹²,⁴³ Note that the $c_2^2∕c_0^2$ values are negative, except for those marked with asterisks.

Methyl	$c_0^2$	$c_2^2$	RC	τ0, ps	$\mid c_2^2∕c_0^2\mid$
L8δ1	1.63	–0.67	0.012	64	0.41
L38δ1	–0.11	–0.51	0.011	59	4.64*
L56δ2	1.58	–0.80	0.015	80	0.51
T55	1.66	–0.77	0.024	129	0.46
T23	1.65	–0.77	0.024	129	0.47
A18	1.68	–0.82	0.018	97	0.49
A50	1.64	–0.80	0.013	70	0.49
L38δ2	1.44	+0.35	0.016	86	0.24*
T37	1.77	–0.88	0.022	118	0.50
T17	1.66	–0.80	0.019	102	0.48
T3	1.58	–0.78	0.025	200	0.49
V49γ2	1.61	–0.71	0.018	97	0.44
A6	1.64	–0.81	0.019	102	0.49
A61	1.64	–0.81	0.013	70	0.49
V47γ1	1.54	–0.68	0.016	85	0.44
V2γ2	0.99	–0.99	0.024	129	1.00
I9γ	1.54	–0.61	0.015	80	0.40

Table 8.

Average best-fit $c_0^2$ values, rhombicity ratios $\mid c_2^2∕c_0^2\mid$, local motion correlation times, τ₀, and time scale separations, R^C, obtained for the methyl groups in Tables 3 – 7. The global motion correlation times, τ_m, are also given.

t, °C	$<c_0^2>$	$<\mid c_2^2∕c_0^2\mid >$	<τ0>, ps	<RC>	τm, ns12,43
5	1.52	0.60	114	0.014	8.01
15	1.68	0.46	101	0.019	5.36
25	1.74	0.48	88	0.022	4.05
35	1.83	0.36	84	0.033	2.55
45	2.37	0.27	71	0.041	1.74

Decrease in the local motional correlation time, τ₀, with increasing temperature is expected. An activation energy of 2 ± 0.2 kcal/mol has been derived from the data of Table 8 based on the Arrhenius relation for the rate 1/6τ₀. Large site-specific variations in local motional correlation times of methyl groups in proteins have been predicted theoretically.^27a,51 The value of 2 kcal/mol pertains to the theoretically predicted range, and the tenfold lower SRLS rates are in significantly better agreement with the theoretical predictions than the MF rates.^27a,51

Based on the Arrhenius relation for the rate 1/6τ_m (with τ_m determined previously^12,43) we obtained an activation energy of 6.72 ± 0.36 kcal/mol for the global motion of protein L, in agreement with similar values obtained for other proteins in aqueous solution (see ref. 22 and relevant papers cited therein).

Mode-coupling, as expressed by the parameter <R^C> = <τ₀/τ_m>, increases with increasing temperature, in accordance with the activation energies for global and local motion. This is an interesting result further documented below by outlining the mode-composition at various temperatures.

The asymmetry of the local potential, as expressed by $<\mid c_2^2∕c_0^2\mid >$, decreases with increasing temperature. This is also interesting new information indicating that the local spatial restrictions at the site of the motion of the methyl group become more axially symmetric as the temperature is raised.

When the local potential is axially symmetric and β_MQ = 110.5° (with r_CH = r_CD = 1.115 Å) the local spatial restrictions at the methyl site are characterized solely by c²₀, which evaluates the potential strength. Clearly c²₀ is expected to decrease with increasing temperature. When the potential is rhombic and β_MQ is also allowed to vary, as done (in a controlled manner – see above) in the present study, the set of parameters c²₀, c²₂ and β_MQ, rather than c²₀ alone, evaluates the local restrictions. Changes in c²₀ with temperature represent in this case apparent changes in potential strength.

Inter-site comparison of the local restrictions based on parameter sets is not straightforward. We have been looking for simplified models, which evaluate this important property in a more direct manner. The combination where c²₀ is fixed at its Woessner-model-compatible value and β_MQ at 110.5°, with c²₂ and the local motion correlation time, τ, allowed to vary, was found to be appropriate. It yields results, which are similar at lower temperatures, and very close at higher temperatures, to those of the complete model. The parameter τ decreases consistently with increasing temperature, as required by physical viability, and |c²₂| decreases with increasing temperature in most cases. The typical decrease (exceptional increase) in structural rhombicity at most (specific) methyl sites with increasing temperature constitutes interesting new information. These developments will be reported shortly elsewhere.

Temperature-dependent mode composition

We illustrate the mode-coupling concept inherent to the SRLS model. The “pure”, i.e., unrestricted by a potential local motional mode has an eigenvalue of 6, and the “pure” global motion mode has an eigenvalue of 6×R^C, both in units of R^L.²⁴ In the original MF formula (eq 1), which is a limiting case of the two coupled rotator model,^20,21,24,45 the weighting factors of the global and local motional modes correspond to ${\left(S_0^2\right)}^2$ and $(1-{\left(S_0^2\right)}^2)$, respectively.

When the mode-decoupling limit is exceeded a larger number of modes contribute to the spectral density (eq 8). We show in Table 9 the dynamic modes associated with methyl I9γ at 5, 25 and 45 °C which contribute to the time correlation functions C₀(t), C₁(t) and C₂(t) (the labels 0, 1 and 2 are abridged versions of KK’ = (0,0), (1,1) = (–1,–1) and (2,2) = (–2,–2), eq 6). We focus first on C₀(t). For the largest time scale separations of R^C = 0.012, and a rhombic potential with $c_0^2=1.56$ and $c_2^2=-0.80$ obtained at 5 °C, three major local motional modes with eigenvalues in the vicinity of 6 make a fractional contribution of 0.778 to C₀(t). The eigenvalue of the global motion mode (shown in bold face letters) is equal to the “pure” eigenvalue of 0.072 = 6×0.012, indicating that the overall tumbling mode of the protein is, as expected, distinct from the local motional modes. The fractional contribution of the global motion mode is 0.092. Additional modes with eigenvalues ranging from 4.87 to 9.17, with various individual weights, contribute 0.098. The rest (0.052) is contributed by a large number of mixed modes with small individual weights (not shown). In can be seen that the two-mode limit is exceeded even though the time scale separation is relatively low (R^C = 0.012) and the potential relatively weak ($c_0^2=1.56$ in units of k_BT).

Table 9.

Eigenvalues, 1/τ_i (in units of R^L), and weighting factors, c_K.i, of the SRLS solution for C₀(t), C₁(t) and C₂(t) obtained using (a) $c_0^2=2.42,c_2^2=-0.74$ and R^C = 0.012, (b) $c_0^2=1.69,c_2^2=-0.78$ and R^C = 0.015, and (c) $c_0^2=2.42,c_2^2=-0.74$ and R^C = 0.061. These values represent the best-fit parameters obtained with rhombic potential fitting using the data acquired at a magnetic field of 11.7 T, and at 5, 25 and 45 °C, respectively for methyl I9γ. In all the cases shown β_MQ = 69.5°. The global motion mode contribution is marked in bold face letters.

C0(t)
a		b		c
1/τi	cK.i	1/τi	cK,i	1/τi	cK,i
5.30	0.297	5.40	0.309	6.29	0.396
5.94	0.264	6.06	0.270	9.46	0.130
7.92	0.194	8.08	0.188	7.41	0.123
8.08	0.057	8.23	0.050	19.83	0.023
4.87	0.031	4.83	0.024	9.70	0.021
5.94	0.023	21.7	0.009	23.03	0.016
9.17	0.010	9.37	0.008	14.84	0.006
0.072	0.092	0.090	0.121	0.355	0.263

C1(t)
a		b		c
1/τi	cK.i	1/τi	cK,i	1/τi	cK,i
6.14	0.321	6.09	0.299	6.41	0.223
5.92	0.309	6.49	0.290	1.69	0.221
6.43	0.313	6.85	0.265	1.51	0.141
1.77	0.051	1.61	0.130	7.80	0.140
19.75	0.003	19.80	0.009	9.17	0.111
				9.00	0.068
				7.55	0.048
				19.85	0.013
				22.06	0.010

C2(t)
a		b		c
1/τi	cK.i	1/τi	cK,i	1/τi	cK,i
5.42	0.506	5.11	0.573	4.70	0.365
6.45	0.380	6.89	0.335	4.65	0.242
6.80	0.111	7.47	0.086	4.82	0.149
20.02	0.002	20.26	0.005	9.62	0.097
				9.70	0.080
				6.29	0.026
				7.41	0.008

For R^C = 0.015, $c_0^2=1.69$ and $c_2^2=-0.78$, obtained for I9γ at 25 °C, three local motional modes with eigenvalues close to 6 make a combined fractional contribution of 0.767. The global motion eigenvalue is given by 0.090 = 6×0.016, which is again equal to the “pure” eigenvalue. Its contribution has increased to 0.121. Additional modes with eigenvalues ranging from 4.83 to 9.37 contribute 0.083. The rest (0.05) is contributed by a large number of mixed modes with small individual weighting factors. Note the presence of a mixed mode with an eigenvalue of 1.77, which contributes 0.051.

At 45 °C, where R^C = 0.061, $c_0^2=2.42$ and $c_2^2=-0.74$ was determined for I9γ, only two modes with eigenvalues relatively close to 6 (6.29 and 7.41), with a combined contribution of 0.519, are present. The global motion eigenvalue is given by the “pure” eigenvalue of 0.366 = 6×0.061, and its contribution is 0.263. Mixed or coupled modes contribute 0.218 to C₀(t). Mode-coupling is clearly important at 45 °C.

Dynamic modes with eigenvalues close to 6 contribute to C₁(t) (C₂(t)) 0.943, 0.854 and 0.411 (0.997, 0.994 and 0.791) at 5, 25 and 45 °C, respectively. For C₁(t) mode-coupling is significant at 35 °C and dominant at 45 °C. For C₂(t) mode-coupling is important at 45 °C.

The trends in the various parameters as a function of temperature have been discussed above.

Residual configurational entropy

Side-chain S_axis² values, which exhibit significantly larger variations than backbone S² values, have been used extensively in recent years to calculate residual configurational entropy.^30,52-54 This requires the equilibrium probability distribution function, P_eq(Ω_C'M), of the M frame relative to the local director, C’. P_eq(Ω_C'M) is calculated automatically in SRLS using a potential form as general as justified by the quality of the experimental data. The coefficients of this potential, $c_0^2$ and $c_2^2$, are determined by fitting the experimental ²H relaxation data. Model-free does not feature potential energy functions (hence equilibrium probability functions) explicitly. Equation 1 features a single order parameter consistent with axial potential/axial ordering. Hence one has to adopt a simple form of the potential after fitting, assuming that it is axially symmetric, and then use S², which is typically inaccurate because of force-fitting, to calculate the coefficient of this (axial) potential. Hence the residual configurational entropy derived from S² is often inaccurate.

Determining the form of the local potential compatible with data integrity, and accounting for potential rhombicity, are clear advantages of SRLS over MF. Currently the orientation of the spin-bearing bond vectors does not depend explicitly on the other degrees of freedom implying over-estimation of the partition function.⁵² Significant improvement on this important aspect is expected to be achieved within the scope of the “integrated approach” discussed below.

Side-chain rotamer inter-conversion

SRLS is a many-body mode-coupling approach.²⁴ _In principle it can handle any number of local motions coupled to one another and to the (asymmetric) global diffusion. The local potential is expanded in the complete basis set of the Wigner rotation matrix elements. The number of terms one may preserve is determined by the nature of the experimental data. We found that the sensitivity of the ²H relaxation data set (including in the current paper rank two coherences and relaxation rates acquired at four magnetic fields) does not justify preserving terms beyond the axial and (indispensable) rhombic L = 2 components. The local diffusion tensor is axially symmetric, accounting for diffusion about and of the C-CH₃ axis.²³

This scenario captures many of the major features of methyl dynamics as they emerged from early^55-57 and recent (ref. ²³ and the current paper) studies. The asymmetry of the local spatial restrictions is represented by the rhombicity of the SRSL potential. The dynamical coupling between the local and global motions (which may occur with arbitrary rates) is intrinsic to the SRLS model. General features of local geometry (e.g., the tilt between the magnetic and local ordering/local diffusion frames) are allowed for automatically by the SRLS formalism. All of the relevant physical quantities can be determined as best-fit parameters.

With regard to rotamer jumps − the SRLS model can include potential minima involving motion within the latter, with less frequent jumps between the minima. This is illustrated in Figure 4 of reference ²⁵ and discussed at length in that paper. However, terms of higher rank, L, and order, K, are required to generate such potentials. As pointed out above, data sensitivity does not justify including these terms in the SRLS potential.

Fast rotamer jumps and local librations can be treated separately and combined with SRLS. We have used this strategy (using the Stochastic Liouville Equation approach) in the context of ESR of a nitroxide label tethered to a helix mimicking a protein environment.^58,59 In a most general manner one can combine SRLS with molecular dynamics (MD) simulations, which account for any local motion including rotamer jumps in significantly populated conformations. Moreover, quantum chemical calculations can be used to determine magnetic tensors, and hydrodynamic methods to determine the global diffusion tensor.

One of us currently promoting such an “integrated approach”, applied so far to small molecules.^60,61 Our present and recently published²³ SRLS-based methyl dynamics papers constitute a critical step toward the application of the “integrated approach” to bio-macromolecules. This is currently probably the most advanced attempt to treat methyl dynamics in proteins. The recent study of Hu et al.³⁹ uses model-free (MF) (in particular, the squared generalized order parameter, S²) combined with molecular mechanics. Unlike SRLS, the MF method does not account for local structural asymmetry, mode-coupling and general features of local geometry.²¹ Only the torsional potential term associated with the C-C bond preceding the CCH₃ axis is considered in reference ³⁹. Conformational multiplicity and additional possible local motions are not accounted for.

Finally, let us point out that methyl dynamics is currently the leading method for studying with NMR mega-Dalton protein systems.^62,63 Therefore efforts to improve the analysis of the experimental data are timely and important.

IV. Conclusions

By applying SRLS to an extensive set of ²H spin relaxation data we have shown that appropriate analysis of methyl dynamics requires rhombic local potentials/local ordering. The model-free S_axis² -based “amplitude of motion” picture, implying extensive excursions of the CCH₃ axis in tightly packed protein cores, has been replaced with site-specific potential rhombicity derived with SRLS. The form of the local potential is an important structural property not determined so far with NMR spin relaxation. Potential rhombicity was found to decrease with increasing temperature. The rates for local motions increase on average with increasing temperature. They are approximately 10 times lower than their MF counterparts. For the first time activation energies for methyl motion in proteins are estimated with NMR spin relaxation at 2 ± 0.2 kcal/mol. These findings are consistent with theoretical predictions derived with molecular dynamics and molecular mechanics methods. The dynamical coupling between the global and the local motions increases with increasing temperature. The two-mode approximation is not borne out by our results. Rather, methyl dynamics is given by the superposition of quite a few modes. The intrinsic ill-definition of the measurable spectral density is reduced considerably using SRLS. The accuracy of the results can be improved by including in the experimental data set rank 2 coherences. Research prospects include elucidation of highly accurate site-specific information, the calculation of residual configurational entropy from experimentally determined rhombic potentials, and enhancements of the model to include rotamer jumps.

Table 6.

Best-fit parameters obtained with combined fitting of 5 relaxation rates (²H T₁, T₂ and the three relaxation rates associated with the rank 2 coherences) acquired at 11.7 T, 35 °C, for the depicted methyl groups. The global motion correlation time used was τ_m = 2.55 ns.¹²,⁴³ Note that all the $c_2^2∕c_0^2$ values are negative except for $c_2^2∕c_0^2$ of L38δ₁, marked with an asterisk.

Methyl	$c_0^2$	$c_2^2$	RC	τ0, ps	$\mid c_2^2∕c_0^2\mid$
L8δ1	1.49	–0.50	0.017	43	0.36
L38δ1	–0.54	–0.56	0.015	38	1.04*
L56δ2	1.46	–0.50	0.020	51	0.34
T55	1.99	–0.66	0.037	94	0.33
T23	2.17	–0.63	0.029	71	0.29
A18	2.09	–0.74	0.039	100	0.35
A50	2.41	–0.55	0.026	66	0.23
L38δ2	1.19	–0.52	0.016	41	0.44
T37	1.89	–0.58	0.032	82	0.31
T17	1.61	–0.54	0.040	102	0.30
T3	2.12	–0.62	0.030	122	0.29
V49γ2	1.83	–0.51	0.021	54	0.28
A6	2.06	–0.80	0.042	107	0.39
A61	1.67	–0.51	0.024	61	0.31
V47γ1	1.59	–0.51	0.029	74	0.32
V2γ2	1.75	–0.94	0.058	147	0.54
I9g	1.88	–0.90	0.044	112	0.41
A31	1.91	–0.92	0.044	112	0.48