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- I. Introduction
- II. Theoretical Background
- III. Results and discussion
- IV. Conclusions
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J Phys Chem B. Author manuscript; available in PMC 2010 June 15.

Published in final edited form as:

Published online 2007 October 17. doi: 10.1021/jp072156s

PMCID: PMC2885794

NIHMSID: NIHMS62771

Corresponding authors: Email: li.ca.uib.sl.1igsrmn@ave; Email: ude.llenroc.rmcc@fhj; Email: ti.dpinu@onemilop.oninotna

The publisher's final edited version of this article is available at J Phys Chem B

See other articles in PMC that cite the published article.

Protein dynamics is intimately related to biological function. Core dynamics is usually studied with ^{2}H spin relaxation of the ^{13}CDH_{2} 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 ^{o}C. The accuracy of the results is improved by including in the experimental data set relaxation rates associated with rank 2 coherences.

NMR spin relaxation is a powerful method for studying protein dynamics.^{1}^{-}^{9} The traditional probe for investigating backbone motion is the ^{15}N-^{1}H bond and the common probe for studying side chain motion is the uniformly ^{13}C-labeled and fractionally deuterated methyl group, ^{13}CH_{2}D.^{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 density^{13} is determined by an effective correlation time for local motion, *τ _{e}*, a squared generalized order parameter,

For methyl dynamics MF considers two local motions including rotation *about* the C-CH_{3} axis and fluctuations *of* the C-CH^{3} axis.^{10} Moreover, the methyl rotation axis C-CH_{3} (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

We have shown recently that the MF approach is oversimplified.^{16}^{-}^{23} This has been accomplished by applying to NMR spin relaxation in proteins^{16} 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-CH

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,

The form of *J ^{QQ}*(ω) is parameterized in MF as follows. To accommodate two local motional modes

Practical implications of the MF simplifications have been investigated recently^{23} using the B1 immunoglobulin binding domain of peptostreptococcal protein L (to be called below “protein L”)^{12} and ubiquitin^{28} as test cases. The respective data were subjected to SRLS analysis^{23} 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*_{axis}^{2} ranging from nearly complete disorder (*S*_{axis}^{2} ~ 0.1) to nearly complete order (*S *_{axis}^{2} ~ 1), often exhibiting three distinct maxima.^{6}^{,}^{28}^{-}^{31} The (pervasive) low *S*_{axis}^{2} values imply large-amplitude excursions of the C-CH_{3} 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−CH_{3}, inherent in the definition of *S*_{axis}^{2} (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*_{axis}^{2} 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 mechanics^{39} studies. All of these investigations have shown that local structural asymmetry prevails at methyl sites in proteins, contrary to the axial *S*_{axis}^{2} based MF picture.

The present paper is an extension of our previous study^{23} which was based on ^{2}H *T*_{1} and *T*_{2} data acquired for protein L^{12} and ubiquitin^{28} 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 ^{2}H *T*_{1} and *T*_{2}.^{10} For protein L the Kay group acquired all five ^{2}H 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 ^{2}H 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

We find that the protein L methyl sites exhibit rhombic potentials of different forms in the SRLS scenario instead of amplitudes of C-CH_{3} 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.

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

The original MF spectral density, *J*(ω), based on *τ _{e}* <<

$$J\left(\omega \right)={S}^{2}\phantom{\rule{thickmathspace}{0ex}}{\tau}_{m}\u2215(1+{{\tau}_{m}}^{2}\phantom{\rule{thickmathspace}{0ex}}{\omega}^{2})+(1-{S}^{2})\phantom{\rule{thickmathspace}{0ex}}{\tau}_{e}\u2019\u2215(1+{\tau}_{e}{\u2019}^{2}\phantom{\rule{thickmathspace}{0ex}}{\omega}^{2}),$$

(1)

where 1/*τ _{e}*’ = 1/

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 *S*^{2} equal to [*P*_{2}(cos 110.5°)]^{2}× _{S}_{axis}^{2} = 0.1×*S*_{axis}^{2}. The term 0.1 represents the squared order parameter associated with the motion around the C-CH_{3} axis, and *S*_{axis}^{2} the axial squared order parameter associated with motion of the C-CH_{3} axis. The effective correlation time for local motion, *τ _{e}*, has been associated with

$${J}^{QQ}\left(\omega \right)={{S}_{axis}}^{2}\phantom{\rule{thickmathspace}{0ex}}0.1\phantom{\rule{thickmathspace}{0ex}}{\tau}_{m}\u2215(1+{\omega}^{2}\phantom{\rule{thickmathspace}{0ex}}{{\tau}_{m}}^{2})+(1-{{S}_{axis}}^{2}\phantom{\rule{thickmathspace}{0ex}}0.1)\phantom{\rule{thickmathspace}{0ex}}{\tau}_{e}\u2019\u2215(1+{\omega}^{2}\phantom{\rule{thickmathspace}{0ex}}{\tau}_{e}{\u2019}^{2}).$$

(2)

The fundamentals of the stochastic coupled rotator slowly relaxing local structure (SRLS) theory^{24}^{,}^{25} as applied to biomolecular dynamics^{26} 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),

(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 **...**

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

The Boltzmann distribution *P _{eq}* = exp [−

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

(5)

This represents the expansion in the full basis set of Wigner rotation matrix elements, *D ^{L}_{KM}*(Ω

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

(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:

$$1\u2215{\tau}_{K}=6\phantom{\rule{thickmathspace}{0ex}}{{R}^{L}}_{\perp}+{K}^{2}\phantom{\rule{thickmathspace}{0ex}}({{R}^{L}}_{\parallel}-{{R}^{L}}_{\perp})\phantom{\rule{thickmathspace}{0ex}}\text{for}\phantom{\rule{thickmathspace}{0ex}}K=0,1,2,$$

(7)

where *R ^{L}*

$${j}_{K}\left(\omega \right)=\sum _{i}\frac{{c}_{K,i}{\tau}_{i}}{1+{\omega}^{2}{\tau}_{i}^{2}}.$$

(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

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,

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,

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\u2019M}\right)\rangle $ and ${S}_{2}^{2}=\langle {D}_{02}^{2}\left({\Omega}_{C\u2019M}\right)+{D}_{0-2}^{2}\left({\Omega}_{C\u2019M}\right)\rangle $, which are related to the orienting potential (eq 5), hence ${c}_{0}^{2}$ and ${c}_{2}^{2}$, via the ensemble averages:

$$\langle {D}_{0n}^{2}\left({\Omega}_{{C}^{\prime}M}\right)\rangle =\int d{\Omega}_{{C}^{\prime}M}{D}_{0n}^{2}\left({\Omega}_{{C}^{\prime}M}\right)\text{exp}[-u\left({\Omega}_{{C}^{\prime}M}\right)]\u2215\int d{\Omega}_{{C}^{\prime}M}\phantom{\rule{thinmathspace}{0ex}}\text{exp}[-u\left({\Omega}_{{C}^{\prime}M}\right)]$$

(10)

One may convert to Cartesian ordering tensor components according to ${S}_{zz}={S}_{0}^{2}$, ${S}_{xx}=\left(\sqrt{3\u22152}{S}_{2}^{2}-{S}_{0}^{2}\right)\u22152$, ${S}_{yy}=-\left(\sqrt{3\u22152}{S}_{2}^{2}-{S}_{0}^{2}\right)\u22152$. Note that *S _{xx}* +

For ^{2}H relaxation the measurable quantities are *J ^{QQ}*(0),

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

The functions *j _{K}*(ω) (eq 8) and

Equation 1, from which eq 2 has been derived, represents the SRLS solution in the Born-Oppenheimer (BO) limit where *τ _{m}* >>

As already noted, eq 2 features two dynamic modes associated with the axial order parameters, [*P*_{2}(cos *β _{MQ}*)]

In practice combined ^{2}H *T*_{1} and *T*_{2} 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

In principle one should first consider axial local potentials in the SRLS fitting process. We showed previously^{23} that this leads to a physically problematic picture and implies inconsistencies between ^{2}H auto-correlation in ^{13}CDH_{2} (ref. ^{12}) and HC-HH cross-correlation in ^{13}CH_{3} (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 SRLS model yields the generic spectral densities, *j _{KK’}*(ω). The measurable spectral density,

The appropriate representation of methyl dynamics by *J ^{QQ}*(ω) makes possible the determination of the physical parameters (in general,

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

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

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

Including the rank 2 coherences into the experimental data set increases the accuracy of the results, obviously with a higher but still acceptable reduced χ^{2} 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 ^{2}H *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).

Best-fit parameters, listed under "output", obtained with combined fitting of 16 relaxation rates (8 relaxation rates including ^{2}H *T*_{1} and *T*_{2} 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 **...**

A difference of 4.8% in *S*_{axis}^{2} 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.

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*_{axis}^{2}. 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*_{axis}^{2} values and the corresponding best-fit SRLS parameters. Ten data points (^{2}H *T*_{1}, *T*_{2} 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 *S*^{2} = 0.1×*S*_{axis}^{2} using the axial versions of eqs 5 and 10. The penultimate and ultimate columns on the right show $R\left({c}_{0}^{2}\right)={c}_{0}^{2}\left(\mathrm{SRLS}\right)\u2215\phantom{\rule{thickmathspace}{0ex}}{c}_{0}^{2}\left(\mathrm{MF}\right)$ and *R*(*τ*) = *τ*_{0}(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

Combined fitting of 10 relaxation rates (^{2}H *T*_{1}, *T*_{2} 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 **...**

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 and5)5) and the corresponding *S*_{axis}^{2} (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*_{axis}^{2} > 0.85, 0.6 ≤ *S*_{axis}^{2} ≤ 0.85 and *S*_{axis}^{2} < 0.6 (*S*_{axis}^{2} > 0.89, 0.6 ≤ *S*_{axis}^{2} ≤ 0.89 and *S*_{axis}^{2} < 0.6) at 5 (25) ° C. *Clearly **the* ${c}_{0}^{2}$ S_{axis}^{2} profiles differ significantly, often exhibiting opposite trends at the same temperature, and different temperature dependences.

Schematic representing trends in ${c}_{0}^{2}$ SRLS and *S*_{axis}^{2} MF at 5 °C (a) and 25 °C (b). The ${c}_{0}^{2}$ and *S*_{axis}^{2} values have been classified into three groups according to their magnitude, as outlined in the text. These categories are denoted as **...**

Best-fit parameters obtained with combined fitting of 5 relaxation rates (^{2}H *T*_{1}, *T*_{2} 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 **...**

Best-fit parameters obtained with combined fitting of 5 relaxation rates (^{2}H *T*_{1}, *T*_{2} 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 **...**

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*_{axis}^{2} 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*_{axis}^{2} values, was set forth.^{49} Small differences in *S*_{axis}^{2} and *τ _{e}* (which is actually a composite depending on both

Five relaxation rates (^{2}H *T*_{1} and *T*_{2}, 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 L8δ_{1}, L38δ_{1}, L56δ_{2}, T55, T23, A18, A50, L38δ_{2}, T37, T17, V49γ , A6, A61, V47γ_{1}, V2γ_{2}, and I9γ. These data were fit with SRLS allowing ${c}_{0}^{2}$, ${c}_{2}^{2}$ and *R ^{C}* to vary while keeping

Best-fit parameters obtained with combined fitting of 5 relaxation rates (^{2}H *T*_{1}, *T*_{2} 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 **...**

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

Best-fit parameters obtained with combined fitting of 5 relaxation rates (^{2}H *T*_{1}, *T*_{2} 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 **...**

Decrease in the local motional correlation time, *τ*_{0}, 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 1/6*τ*_{0}. 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 1/6*τ _{m}* (with

Mode-coupling, as expressed by the parameter <*R ^{C}*> = <

The asymmetry of the local potential, as expressed by $<\mid {c}_{2}^{2}\u2215{c}_{0}^{2}\mid >$, 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

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 *c*^{2}_{0} is fixed at its Woessner-model-compatible value and *β _{MQ}* at 110.5°, with

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

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 *C*_{0}(t), *C*_{1}(t) and *C*_{2}(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 *C*_{0}(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

Eigenvalues, 1/τ_{i} (in units of *R*^{L}), and weighting factors, *c*_{K.i}, of the SRLS solution for *C*_{0}(t), *C*_{1}(t) and *C*_{2}(t) obtained using (a) ${c}_{0}^{2}=2.42,\phantom{\rule{thickmathspace}{0ex}}{c}_{2}^{2}=-0.74$ and *R*^{C} = 0.012, (b) ${c}_{0}^{2}=1.69,\phantom{\rule{thickmathspace}{0ex}}{c}_{2}^{2}=-0.78$ and *R*^{C} = 0.015, and (c) ${c}_{0}^{2}=2.42,\phantom{\rule{thickmathspace}{0ex}}{c}_{2}^{2}$ **...**

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

Dynamic modes with eigenvalues close to 6 contribute to *C*_{1}(t) (*C*_{2}(t)) 0.943, 0.854 and 0.411 (0.997, 0.994 and 0.791) at 5, 25 and 45 °C, respectively. For *C*_{1}(t) mode-coupling is significant at 35 °C and dominant at 45 °C. For *C*_{2}(t) mode-coupling is important at 45 °C.

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

Side-chain *S*_{axis}^{2} values, which exhibit significantly larger variations than backbone *S*^{2} 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

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.

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 ^{2}H 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-CH_{3} 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 published^{23} 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, *S*^{2}) 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 CCH_{3} 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.

By applying SRLS to an extensive set of ^{2}H spin relaxation data we have shown that appropriate analysis of methyl dynamics requires *rhombic* local potentials/local ordering. The model-free *S*_{axis}^{2} -based “amplitude of motion” picture, implying extensive excursions of the CCH_{3} 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.

This work was supported by the Israel Science Foundation (Grant No. 279/03 to E.M.), the Binational Science Foundation (Grant No. 2000399 to E.M. and J.H.F.), and the Damadian Center for Magnetic Resonance at Bar-Ilan University, Israel. This work was also supported by the National Center for Research Resources of the National Institutes of Health (Grant No. P41-RR016292 to J.H.F.). A.P. acknowledges support of the Italian Ministry for Universities and Scientific and Technological Research projects FIRB and PRIN ex-40%.

We thank Prof. Lewis E. Kay of the University of Toronto for providing the experimental data used in this study.

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