Protein Dynamics in the Solid-State from ²H NMR Lineshape Analysis: a Consistent Perspective

Abstract

Deuterium lineshape analysis of CD₃ groups has emerged as a particularly useful tool for studying μs - ms protein motions in the solid-state. The models devised so far consist of several independently conceived simple jump-type motions. They are comprised of physical quantities encoded in their simplest form; improvements are only possible by adding yet another simple motion, thereby changing the model. The various treatments developed are case-specific; hence comparison amongst the different systems is not possible. Here we develop a new methodology for ²H NMR lineshape analysis free of these limitations. It is based on the microscopic-order-macroscopic-disorder (MOMD) approach. In MOMD motions are described by diffusion tensors, spatial restrictions by potentials/ordering tensors, and geometric features by relative tensor orientations. Jump-type motions are recovered in the limit of large orientational potentials. Model-improvement is accomplished by monitoring the magnitude, symmetry and orientation of the various tensors. The generality of MOMD makes possible comparison amongst different scenarios. CD₃ lineshapes from the Chicken Villin Headpiece Subdomain, and the Streptomyces Subtilisin Inhibitor, are used as experimental examples. All of these spectra are reproduced by using rhombic local potentials constrained for simplicity to be given by the L = 2 spherical harmonics, and axial diffusion tensors. Potential strength and rhombicity are found to be ca. 2 − 3 [k_BT]. The diffusion tensor is tilted at 120° from the C−CD₃ axis. The perpendicular (parallel) correlation times for local motion are 0.1 − 1.0 ms (3.3 − 30 μs). Activation energies in the 1.1 − 8.0 kcal/mol range are estimated. Future prospects include extension to the ²H relaxation limit, application to the ¹⁵N and ¹³C NMR nuclei, and accounting for collective motions and anisotropic media.

1. Introduction

Solid-state NMR spectroscopy applied to proteins is currently at the cutting-edge of bio-structural NMR. In parallel with advances in structure determination (e.g., Ref 1 and articles cited therein) there is a surge in the development of relaxation and conformational exchange methods for characterizing internal protein mobility, using mainly ¹⁵N and ¹³C as NMR probes (e.g., Refs 2 – 5 and articles cited therein). The relaxation limit relates to stochastic motions that are much faster than the relevant magnetic interactions. Conformational exchange is a deterministic process that relates to jumps among two or more symmetry-related sites. Together these methods cover the ps - ms range, with the protein structure mapped out effectively through uniform isotope-labeling with the heteronucleus observed.

These are very attractive features. However, the experimental information consists of few parameters − T₁ and T_1ρ for relaxation and T_1ρ for exchange − usually acquired at a given magnetic field and temperature.^2-5 Because of paucity of data one has to resort to severe simplifications in analyzing the experimental data. By contrast, for the ¹⁵N and ¹³C nuclei the sensitive slow-motional μs - ms range, which is amenable to investigation with NMR lineshape analysis, has been explored so far only qualitatively only in a few cases (e.g., Ref 6).

Unlike the ¹⁵N and ¹³C nuclei, the ²H nucleus bonded to sp3-hybridized carbon emerged as a particularly useful probe over the entire dynamic range.^7-28 It has been used extensively in the relaxation limit, which applies to motions with rates exceeding approximately 10⁷ s⁻¹. The typical relaxation parameters studied are ²H T_1Z and T_1Q.^{12,13,19,20,25} For example, methyl rotation in a small protein, investigated recently with multiple echo acquisition of T_1Z,¹⁹ has been interpreted in terms of jumps among three potential wells, with independent diffusive motion within the wells.²⁰

The slow-motional ²H range extends from approximately 10 μs to 1 ms (e.g., cf. Ref 23). Lineshape analyses provided insightful information on dynamic processes in hydrophobic protein cores,^17,18 membrane proteins,^29-33 collagen fibrils,⁸ peptides and proteins adsorbed onto mineral surfaces,^21-24,26,27 etc. Relation to protein stability, protein folding, the assembly of collagen molecules in fibrils, the packing of secondary structure elements in globular proteins, physisorption, etc., has been established. Lineshape analysis requires selective isotope labeling; effective protocols for carrying out such procedures are available.^34,35

Deuterium NMR was used in early work to study dynamics in the solid-state.^7-13,15,16 Torchia and co-workers pioneered the utilization of CD₂ and CD₃ probes to study μs − ms motions in small alanine-, methionine-, and leucine-containing peptides, and collagen fibrils (e.g., Refs 7 and 8). References 11, 12, 17 − 28 and 36 − 38 may be considered illustrative of the progress made since.

Theoretical developments have devised physically plausible but rather intricate and diverse models. Traditionally internal motions in proteins have been considered to be jump-type/barrier-crossing processes (e.g., Ref 39). Early on it was found that a single motion of this nature does not reproduce typical experimental spectra.⁴⁰ Rather, it is necessary to combine several simple motional modes, assuming they are independent.^23,39 The key feature of asymmetry, realized in early work,⁴⁰ has been modeled in terms of unequally populated (exchange-involved) sites,^17,18,21-24 motion on arcs of variable length,^17,18,23 small-angle jumps on cones centered at tilted bond axes,^21,23 and/or motion on distorted cones.⁴¹ Different authors conceived different combinations of simple motions. In such scenarios, where the physical quantities are encoded in their simplest form, model improvements are only possible by including in the analysis yet another simple motion, i.e., changing the overall model. Comparison among different systems featuring the same dynamic probe is not possible in view of model diversity. In some cases experimental ²H lineshapes could not be reproduced even with a combination of simple motions.⁴¹ Or, to accomplish this, quite intricate²³ or specific⁴² models had to be invoked.

No model can be proven to be unique. However, a general and comprehensive approach to the treatment of restricted motions has been developed within the scope of the Stochastic Liouville Equation (SLE) by Freed and co-workers. The SLE applies to restricted overall^43-48 as well as internal⁴⁹ motions. The major underlying factors − type of motion, spatial restrictions, and local geometry − are treated generally for the entire motional range, within their rigorous three-dimensional tensorial requirements.^43-49

The motion per-se is represented by a second rank diffusion tensor, allowed to be tilted from the magnetic tensor(s). The local spatial restrictions are represented by a potential, u, expanded in the full basis set of the generalized spherical harmonics. Typically only the lowest, L = 2, terms are used partly for convenience, but additional terms in the expansion can be used if and when they are needed.^44,45 The L = 2, K = 0 term represents the strength of the potential; the L = 2, K = 2 term represents its rhombicity. The key element of asymmetry is thus substantiated by the (rhombic) form of the local potential. A second rank ordering tensor may be defined in terms of u. The local geometry is given by the relative orientation of the various tensors involved. Energy barriers can be realized with L = 2 potentials (see below). Jump-type motions can be actualized in the limit of large barriers. By including L = 4, K = 0, 2 (or higher order) terms with large rhombic components in the expression for the local potential, a range of symmetries of the potential wells can be generated.⁴⁹

The SLE described above has been used extensively for ESR applications^46,48 and in some cases for NMR applications^14-16. Particularly relevant in the present context is the extension of the SLE called microscopic-order-macroscopic-disorder (MOMD) approach, where the director of the local restricting environment is distributed randomly over the sample.⁵⁰ MOMD/ESR was applied to nitroxide probes in liposomes,⁵⁰ and later to nitroxide-labeled proteins and DNA fragments in “frozen” solutions (i.e., scenarios where the global motion of the (typically large) molecule may be considered “frozen” on the ESR timescale).^51,52 The nitroxide label in “frozen” solutions is formally analogous to the ²H label in polycrystalline proteins. In this study we develop MOMD for the analysis of dynamic ²H NMR lineshapes. Substantial changes in MOMD/ESR, associated with the ²H spin Hamiltonian and the quadrupole echo excitation scheme, have been made to adapt it to ²H NMR.

In MOMD one can devise analysis scenarios differing in complexity by monitoring tensor magnitude, symmetry and orientation. This enables a continuous range of scenarios, and thus comparison amongst different scenarios, within the scope of the same general model.

Different methyl environments within the energy landscape have been revealed.^19,20 Modification of potential energy surfaces of protein side chains by proximal surfaces has been suggested.²⁴ Undoubtedly, it is of interest to determine the form of the potential energy surface associated with probe dynamics. We show below potentials and associated equilibrium distributions functions emerging from MOMD/²H NMR analysis. Substantial sensitivity to the symmetry of the local spatial restrictions has been detected. This is a new perspective of the local structure at methyl sites in peptides and proteins.

Vold and co-workers developed a comprehensive SLE-based computer program called EXPRESS, which treats jump-type motions in various contexts.³⁹ SLE-based methods for treating diffusive motions at different levels of complexity have also been developed. Thus, Wittebort & Szabo¹¹ treated protein side-chains experiencing multiple bond rotations. The spatial constrains are accounted for by restricting the amplitudes of the internal rotations. Small-amplitude motions, associated with diffusion, were found to be ineffective in causing relaxation (of the ¹³C label) whereas large-amplitude motions, associated with jumps, were found to be effective. Based on this it was concluded that jump models are appropriate. Within their scope excluded volume effects have been accounted for by considering only sterically allowed conformations. Concerted dynamics was accounted for by considering solely motions where the position of only a small number of atoms is altered.

The groups of Nordio (e.g., Ref 53), Moro (e.g., Ref 54), and Freed (e.g., Ref 55) also developed methods for treating chain flexibility. In Ref 55 torsional profiles of the chain bonds have been calculated ab initio. A limited number of allowed conformers that undergo torsional oscillations and conformational jumps have been identified. Similar results were obtained by Nordio, Moro and co-workers.^53,54 Thus, more elaborate analyses have shown that chain dynamics is determined by combined diffusive and jump-type processes. Vugmeyster et al. reaches similar conclusions with regard to methyl-group dynamics.²⁰

A method for simulating dynamic NMR spectra in rotating solids has been developed by Heaton.⁵⁶ While Magic Angle Spinning is handled effectively the treatment of restricted motions is rather limited. Diffusive and jump-type processes are treated alike. The motional rates are given by the elements k_ij of an exchange matrix, and the local restrictions by (p_i/p_j)^1/2×k_ij, where p_i and p_j are the populations of the states i and j. Typically a single rate constant is used in a given modeling procedure. Application to phenyl ring dynamics illustrates jump-type motion whereas application to lateral diffusion across curved membranes illustrates diffusive motion.⁵⁶ This method may be considered a simple limit of MOMD.

Drobny and co-workers have studied extensively furanose ring dynamics in DNA and RNA. Their most sophisticated computer program is called KIDMAS.⁴² In general, the low-energy conformations of the furanose ring cluster within two distinct energy minima identified with the pseudorotation states C2’-endo and C3’-endo. This scenario has been modeled in terms of arc-like displacements of the relevant bonds, and twisting of the rings.⁵⁷ In the GCGC methyltransferase DNA recognition site the barrier between the two pseudorotation states was found to be very small. In this case furanose ring dynamics has been modeled as reorientational diffusion in a harmonic potential. Thus, Drobny et al. developed SLE-based methods featuring specific restricted diffusive motions appropriate to treat furanose ring dynamics in DNA and RNA.^42,57

The following comment is in order. MOMD carefully considers the non-commutability of rotations. So do the programs EXPRESS³⁹ and KLDMAS⁴². However, the following important feature distinguishes MOMD from EXPRESS/KLDMAS. In MOMD the orientational diffusion is represented by a 3D second-rank tensor. One may use the effective 1D (isotropic diffusion tensor) version, improve the analysis by using the effective 2D (axial diffusion tensor) version, or else use the fully 3D (rhombic diffusion tensor) version. Which version is used depends on the nature of the experimental data. Likewise, the local potential may simply be described by the L = 2 and K = 0 term; this yields an axial second-rank local ordering tensor. Or, one may improve the analysis by also including the L = 2 and K = 2 term; this yields a rhombic second-rank local ordering tensor. Higher L-value and K = 0, K ≠ 0 terms may also be included; this will yield higher-rank ordering tensors. Various physically relevant features of local geometry can be implemented by varying the relative orientation of the local diffusion, local ordering and (rhombic) magnetic tensors. This is accomplished without changing the essence of the physical picture. Again we mention that in the limit of strong orientational potentials jump-type motions of various symmetries can be implemented.

EXPRESS has been designed to treat a collection of jump-type motions. The simplest model includes a single motional mode with its specific geometric features. To improve the analysis a different motional mode with independent geometric features has to be added, etc. The essence of the physical picture (in a geometric sense and otherwise) is changed with each addition.

KLDMAS has modeled so far furanose ring dynamics within the scope of a single motional mode in terms of a one-dimensional Brownian diffuser and an axial local potential.⁴² Model-improvement in a geometric sense and otherwise, likely to be necessary in future work, requires generalization of this description.

MOMD can be adapted to the calculation of relaxation parameters, and enhanced to include the effect of Magic Angle Spinning^23,25. It represents a limit of a more general two-body (protein and probe) coupled-rotator model known as the slowly relaxing local structure (SRLS) approach,^49,58,59 where the slow rotator (the protein) is so slow as to be considered “frozen”, yielding random orientations of the local director, i.e., MOMD. In recent years we applied the full SRLS approach to proteins in aqueous solution.^60-62 Thus, the same general physically relevant theoretical/computational tool is now available for studying protein dynamics in the solid and liquid states. Moreover, by implementing for protein NMR the complete SRLS theory, which also applies to the slow-motional range and to anisotropic solvents,⁴⁹ it will be possible to treat membrane proteins^29-33 which also execute slow (ms) global motion, and polycrystalline proteins which execute slow (ms) collective motions⁶³ in addition to the currently treated localized motions.

MOMD is applicable to ¹⁵N or ¹³C lineshapes. Currently few slow-motional spectra generated by these nuclei are available. A recent method, whereby internal motions are slowed down by coating the protein with trehalose without detrimental implications,⁶⁴ is expected to aid in acquiring informative slow-motional spectra.

The following comments are in order. (1) As pointed out above, MOMD and SRLS potentials can be tailored to feature potential wells⁴⁹. Within the scope of SRLS the probe executes infrequent jumps among potential wells, and fast but constrained motion within the wells.⁴⁹ This scenario is more general than just discrete jumps among wells. Currently available methods ignore mode-coupling, i.e., one body moving relative to another moving body. Therefore, even if both processes are treated, statistical independence has to be assumed − cf. Ref 20. (2) The local potential (usually defined by two coefficients), the local diffusion tensor (usually defined by two principal values and a tilt angle), and their orientational relationships to the body-fixed magnetic tensors, are the only physical quantities entering the MOMD analysis. In the recent “multi-simple-mode-treatments” each variable represents a different physical quantity.¹⁸

In this study we apply MOMD/²H NMR to deuterated leucine and valine methyl groups of the 36-residue Chicken Villin Headpiece Subdomain (HP36)¹⁸ and a deuterated methionine methyl group of the Streptomyces Subtilisin Inhibitor (SSI)⁴¹. All of these experimental spectra are reproduced with a rhombic potential and an axial diffusion tensor. We determine the strength and rhombicity of the potential for every deuterated methyl site. Typical potential forms, and associated equilibrium distribution functions, are illustrated. The local diffusion tensor is conveniently found to be axially symmetric and tilted from the quadrupole tensor. We determine the principal values and orientation of this tensor for every deuterated methyl site. Associated activation energies are estimated. HP36 and SSI have been analyzed previously with different multi-simple-mode models.

A brief summary of the main aspects of the Stochastic Liouville Equation underlying MOMD is presented in Section 2. Our results and their discussion appear in Section 3 and our Conclusions are summarized in Section 4.

2. Theoretical background

The theory for analyzing slow-motional magnetic resonance lineshapes within the scope of the stochastic Liouville equation (SLE) is given in Refs 43 − 46. A brief summary is presented below.

The SLE for the spin density matrix is given by:

$$\left(\frac{\partial }{\partial t}\right)\rho (\Omega ,t)=[-iH{\left(\Omega \right)}^X-{\Gamma }_{\Omega }]\rho (\Omega ,t),\mathrm{with}{\Gamma }_{\Omega }{P}_0\left(\Omega \right)=0$$ (1)

$H{\left(\Omega \right)}^X$ is the superoperator for the orientation-dependent spin Hamiltonian. ${\Gamma }_{\Omega }$ is a Markovian operator for the rotational reorientation of the spin-bearing moiety, with the Euler angles Ω → (α, β, γ) representing the orientational angles. P₀ (Ω) is the unique equilibrium probability distribution of ${\Gamma }_{\Omega }$.

A simple form of the diffusion operator, ${\Gamma }_{\Omega }$ is:⁴³

$$-{\Gamma }_{\Omega }=R{\nabla }_{\Omega }^{2}P(\Omega ,t)-(R∕k_{B}T){\left(\sin \beta \right)}^{-1}\partial ∕\partial \beta \left[\sin \beta TP(\Omega ,t)\right]$$ (2)

where R is the isotropic rotational diffusion coefficient, ${\nabla }_{\Omega }^2$ is the rotational diffusion operator in the Euler angles, Ω, and T is the restoring torque. The latter is equal to $\partial u∕\partial \beta$ for an axial restoring potential, e.g., $u\cong -3∕2c_0^2{\cos \beta }^2$ ($c_0^2$ is given in units of k_BT). The general expressions for rhombic diffusion tensor, R, are given in Ref 44. A comprehensive treatise of the SLE appears in Ref 46.

In this study the probe is for convenience taken to be an axial rotator, associated in the absence of a restricting potential with three decay rates, ${\tau }_K^{-1}=6R_{\perp }+K^2(R_{\mid \mid }-R_{\perp })$ where K = 0, 1, 2 (K is the order of the rank 2 diffusion tensor; $R_{\mid \mid }$ and $R_{\perp }$ are the principal values R). One may also define ${\tau }_{\mid \mid }=1∕\left(6R_{\mid \mid }\right)\mathrm{and}{\tau }_{\perp }=1∕\left(6R_{\perp }\right)$.

The general form of the local potential, $u\left({\Omega }_{\mathrm{CM}}\right)$, is:⁴³

$$u\left({\Omega }_{\mathrm{CM}}\right)=\sum _{L,K}{C}_K^L{D}_K^L\left({\Omega }_{\mathrm{CM}}\right)$$ (3)

C denotes the local director, given in this case by the equilibrium orientation of the C−CD₃ bond, and M denotes the local ordering frame (Figure 1a). If only the lowest, L = 2, terms are preserved, one obtains the rhombic potential:^44,45

Figure 1.

MOMD frames: L − lab frame, C − local director, M − local ordering/local diffusion frame, Q − frame/principal axis of (axial) quadrupole tensor <Q> (a). Stereo-chemical context for (leucine) C^γ−CD₃ motion (b). MOMD axes associated with C^γ−CD₃ motion: Q − principal axis of <Q>, M − principal ordering/diffusion axis. Here β_MQ = 110.5° (corresponding to motion around the C^β−C^γ bond for sp₃-hybridized carbon atoms) (c).

$$\begin{array}{cc}\hfill & u\left({\Omega }_{CM}\right)=U\left({\Omega }_{\mathrm{CM}}\right)∕k_{B}T\hfill \\ \hfill & -c_0^2{D}_{0,0}^2\left({\Omega }_{\mathrm{CM}}\right)-c_2^2[D_{0,2}^2\left({\Omega }_{\mathrm{CM}}\right)+D_{0,-2}^2\left({\Omega }_{\mathrm{CM}}\right)]\hfill \end{array}$$ (4)

The coefficient $c_0^2$ evaluates the strength of u(Ω_CM) and $c_2^2$ evaluates its non-axiality (both in units of k_BT).

Local order parameters are defined as:^44,45

$$\begin{array}{cc}\hfill & \left[D_{0,K}^2\left({\Omega }_{\mathrm{CM}}\right)\right]=\hfill \\ \hfill & =\int d{\Omega }_{\mathrm{CM}}{D}_{0,K}^2\left({\Omega }_{\mathrm{CM}}\right)\exp [-u\left({\Omega }_{\mathrm{CM}}\right)]∕\int d{\Omega }_{\mathrm{CM}}\exp [-u\left({\Omega }_{\mathrm{CM}}\right)],K=0,2\hfill \end{array}$$ (5)

For at least three-fold symmetry around the local director, C, and at least two-fold symmetry around Z_M, the principal axis of the local ordering tensor, only $S_0^2\equiv \left[D_{0,0}^2\right]\left({\Omega }_{\mathrm{CM}}\right)$ and $S_2^2\equiv \left[D_{0,2}^2\left({\Omega }_{CM}\right)\right]$ survive.⁴⁴ The Saupe scheme order parameters relate to $S_0^2$ and $S_2^2$ as $S{\sigma }_{\mathrm{xx}}=(\sqrt{3∕2S_2^2}-S_0^2)∕2$, $S_{\mathrm{yy}}=-(\sqrt{3∕2S_2^2}+S_0^2)∕2$ and $S_{\mathrm{zz}}=S_0^2$.

Figure 1a shows the MOMD frame structure for C−CD₃ bond dynamics. L denotes the laboratory frame. C (local director frame) and M (local ordering/local diffusion frame taken as coincident for simplicity) have been defined above. Q denoted the principal axis system (PAS) of the effective quadrupole tensor, <Q>. The Euler angles Ω_CM are time-dependent. The Euler angles Ω_MQ are time-independent. Since there is no “macroscopic order”, one has to calculate ²H spectra for every Ω_LC, and convolute the corresponding lineshapes according to a random distribution.⁵⁰

3. Results and discussion

3.1. The model

Fast methyl rotation in the C−CD₃ moiety reduces the C−D quadrupole constant from 167 kHz (e.g., Ref 60) to 167×[1.5 cos²(110.5°) − 0.5] = 52.8 kHz (the tetrahedral angle of 110.5° corresponds to sp3-hybridized carbon). Figure 1b shows the leucine side-chain stereo-chemistry. In the multi-simple-mode paradigm each component is consistent with a specific feature of Figure 1b. The model used to treat C−CD₃ dynamics in HP36 (Ref 18) combines (1) fast methyl rotation accounted for by a partially averaged quadrupole constant of 53.3 kHz; (2) restricted diffusion of the methyl axis on an arc approximated by small nearest-neighbor jumps; and (3) rotameric interconversion corresponding to jumps between four out of the nine non-equivalent configurations of the leucine side chain. The parameters varied include the length of the arc, the jump-rate on the arc, the rate of the rotameric jumps, and the populations of the sites among which the rotameric jumps occur (one major conformer and three equally populated minor conformers). The quadrupolar frame, Q, of the <Q> tensor is implicitly parallel to the C^γ−C^δ bond.

MOMD treats C^γ−CD₃ leucine dynamics with a conceptually simple mesoscopic tensorial approach. A single model-related frame, M, a single NMR-related frame, Q, and their relative orientation (β_MQ), are featured (Figure 1c). Relation to Figure 1b is implicit in the values of the MOMD parameters. Based on this frame structure underlying the theory presented in section 2, we have developed a theoretical/computational tool for analyzing ²H NMR limeshapes from polycrystalline proteins. The parameters varied include, in general, the strength and rhombicity of the local potential, u ($c_0^2$ and $c_2^2$ in Eq 4), the principal values of the local diffusion tensor, ($R_{\mid \mid }$ and $R_{\perp }$, and the relative orientation of the local ordering/local diffusion and quadrupolar tensors (β_MQ).

3.2. Key parameters; sensitivity and versatility

The time-range amenable to investigation is determined by the partially averaged quadrupole constant, <Q>. The inverse of <Q> (more generally, of a given magnetic interaction, Δ) in appropriate units is a typical correlation time, τ_<Q>, yielding a dynamic lineshape. For <Q> = 52.8 kHz one has τ_<Q> = 0.11 ms.

3.2.1. Simple parameter combination

Axial local potential, isotropic local diffusion, and collinear local ordering/local diffusion and quadrupolar frames constitute the simplest parameter combination. We check whether this scenario reproduces the main features of the ²H lineshapes from HP36 (depicted as ribbon diagram in Figure 2). The temperature-dependent ²H spectra from one of the two methyl groups of the leucine residues L75, L61 and L69, and the methyl groups of the valine residue V50, are represented by the red traces in Figure 3 (taken from Ref 18).

Figure 2.

Ribbon diagram of the protein HP36, with the selectively-deuterated methyl groups depicted.

Figure 3.

Experimental (red) and modeled in Ref 18 (blue) ²H lineshapes obtained from the selectively deuterated methyl groups L75, L61, L69 and V50 of HP36.

The potential coefficient, $c_0^2$, was varied in the 1 − 15 range, while the correlation time for local diffusion, τ = 1/(6R), was varied in the 1 μs − 5 ms range; the angle β_MQ was set equal to zero. Qualitative agreement with the low-temperature spectra in Figure 3 was achieved with <Q> = 52.8 kHz, an intrinsic linewidth ${\left(T_2^{\ast }\right)}^{-1}$, of 1 kHz, $c_0^2$ on the order of 2 (in units of k_BT), and local motional correlation times of 1 − 2 ms. Weak local potentials are consistent with previous SRLS analyses of methyl dynamics in protein dissolved in aqueous solution.⁶⁵ They are also consistent with relatively small squared generalized order parameters of methyl groups attached to long side-chains of proteins in solution⁶⁶. Figure 4a shows ²H MOMD spectra scanning the entire dynamic range, which for $c_0^2=2$ extends from τ ~ 2 ms (rigid-limit) to τ ~ 20 μs (extreme motional narrowing limit). In Figure 4b we show analogous lineshapes obtained for $c_0^2=4$. The agreement with the low-temperature spectra of Figure 3 is worse for $c_0^2=4$ as compared to $c_0^2=2$.

Figure 4.

²H MOMD lineshapes calculated using $c_0^2=2$ (a) and $c_0^2=4$ (b), motional correlation times as depicted, and β_MQ = 110.5°. Additional parameters used include <Q> = 52.8 kHz, and an intrinsic linewidth, ${\left(T_2^{\ast }\right)}^{-1}$, of 1 kHz.

The $c_0^2=2$ spectra (Figure 4a) are quite sensitive to τ both in the “shoulder” region (corresponding to the parallel orientation), and in the “horn” region (corresponding to the perpendicular orientation). The sensitivity of the analysis to parameter variations is substantially more limited for $c_0^2=4$ (Figure 4b). In both cases the central “depression” decreases in intensity with decreasing τ for τ > τ_<Q>, while the opposite trend is observed for τ < τ_<Q>. The overall spectral width decreases as the local motion becomes faster for both $c_0^2=2$ and $c_0^2=4$. The experimental spectra of L69 have a decreasing central depression with increasing temperature, and a largely temperature-invariant overall width, in disagreement with Figure 4. Thus, an important factor appears to be missing in these simulations, which do not reproduce even the relatively simple lineshape evolution of L69.

A straightforward improvement is to allow the angle β_MQ to vary. Since the local ordering tensor is axially symmetric, the angle β_MQ between the principal axes of the M and Q frames should affect the predicted spectrum. We found that within the scope of isotropic local diffusion the effect of β_MQ being different from zero is small. The multi-simple-mode approaches feature scalar parameters (formally analogous to the rate or correlation time for isotropic diffusion in MOMD) as descriptors of the local motions. That a single motional mode of this nature cannot reproduce experimental ²H lineshapes with characteristic geometric features is consistent with the finding delineated above. That appropriate tensorial description actualizes a single generally applicable motional mode is shown in the forthcoming.

3.2.2. Axial potentials and axial diffusion

We proceed by allowing for axial local diffusion. To test sensitivity we set $c_0^2=2$, ${\tau }_{\perp }$ equal to τ_<Q> = 0.11 ms, and vary ${\tau }_{\mid \mid }$ as shown in Figure 5 for β_MQ = 50° and 90° (note that in a mesoscopic approach preferential ordering does not have to – although it certainly may - point along a chemical bond). It can be seen that corresponding lineshapes in parts a and b differ to a very large extent, bearing out the substantially enhanced sensitivity of the analysis to the local geometry within the scope of axial diffusion.

Figure 5.

MOMD ²H lineshapes for β_MQ = 50° (a), 90° (b), and local motional correlation times as depicted. Additional parameters used include <Q> = 52.8 kHz and ${\left(T_2^{\ast }\right)}^{-1}$ = 1 kHz.

Although we allowed $c_0^2$, β_MQ, ${\tau }_{\perp }$ and ${\tau }_{\mid \mid }$ to vary freely, the lineshape evolutions depicted in Figure 3 have not been reproduced satisfactorily by our MOMD calculations. We conclude that an important factor is still missing in our simulations.

3.2.3. Rhombic potentials

A key feature that emerged from our previous SRLS-based studies of NMR relaxation in aqueous protein solutions,^{61,62,65,,67,68} and from previous studies of ²H lineshapes in polycrystalline peptides and proteins,^17-27,41 is the need to account for the asymmetry of the local spatial restrictions. Within the scope of MOMD this property is expressed in terms of rhombic local potentials/local ordering. We find that the main features of the Figure 3 lineshapes are reproduced with rhombic potentials with $c_0^2$ and $c_2^2$ on the order of 2 − 3. Our best-fit lineshapes (based on visual comparison) are shown in Figure 6, together with the respective best-fit parameters. For easy comparison with their experimental counterparts shown in Figure 3, we also specify temperatures in blue Figure 6.

Figure 6.

²H MOMD lineshapes calculated for rhombic potentials with $c_0^2$ and $c_2^2$ as depicted, axial local diffusion with ${\tau }_{\perp }$ and ${\tau }_{\mid \mid }$ as depicted, and tilt angle β_MQ = 120° (except for L69 − denoted with an asterisk − where β_MQ = 110.5°).

Detailed discussion of our results appears below. Note that we do not impose Arrhenius-type temperature-dependence on the motional rates. However, we do not accept results where the temperature-dependent lineshape evolution yields rates that vary in an un-physical manner.

3.3. MOMD applied to the Chicken Villin Headpiece Subdomain

All of the calculations presented below used <Q> = 52.8 kHz and an intrinsic linewidth of 1 kHz. . Initially the angle β_MQ was set equal to 110.5°. For leucine (valine) residues this is consistent with two-site exchange around the C^β−C^γ (C^α−C^β) bond (e.g., Ref 8). Except for the rather insensitive L69 lineshapes, the spectra shown in Figure 3 could not be reproduced satisfactorily with β_MQ = 110.5°. On the other hand, β_MQ ≅ 120° yielded satisfactory results.

The description of the local geometry in previous studies is very diverse. The angle between exchanging C−D or C−CD₃ bonds, considered as the main dynamic mode, was ascribed quite different values in different multi-mode analyses. It was determined as 108° − 112° in Ref 8, 32° − 90° in Ref 28, 90° in Ref 7, 60° − 80° in Ref 41, 120° in Ref 36, etc. As indicated above, these are effective angles. It is of interest, though, that the value of 120° determined by our MOMD analysis is the angular minimum for trans-gauche isomerization³⁶. We estimate the error in β_MQ as ±3°.

With β_MQ fixed at 120°, we proceeded varying $c_0^2$, $c_2^2$, ${\tau }_{\perp }=1∕\left(6R_{\perp }\right)$ and $t_{\mid \mid }=1∕\left(6R_{\mid \mid }\right)$. The lineshapes of L69 vary to a relatively small extent. The nearly constant overall width, and the slight reduction in the central depression with increasing temperature, have been reproduced with rhombic potential given by $c_0^2=2.0$ and $c_2^2=2.0$. The lower-limit of ${\tau }_{\perp }$, which is too slow to affect the analysis, has been set at 1.04 ms. ${\tau }_{\mid \mid }$ ranges from 29.8 μs at 254 K to 8.8 μs at 298 K, with an activation energy of E_a = 8.0 kcal/mol. The latter is substantially larger than the activation energies associated with all of the other methyl groups investigated (see below). This is consistent with L69 being effectively secluded in the protein core. Thus, unlike all of the other deuterated methyl groups of HP36, the deuterated methyl group of L69 yields at 298 K virtually identical spectra in hydrated and dry polycrystalline powder samples.¹⁸ By contrast, the authors of Ref 18 found that the activation energy for L69 is the smallest among all of the activation energies determined.

The errors in $c_0^2$, $c_2^2$, ${\tau }_{\perp }$ and ${\tau }_{\mid \mid }$ are estimated at 10%. The errors in the activation energies associated with $R_{\perp }$ and $R_{\mid \mid }$ given below are estimated at 15%. These evaluations apply to all of the MOMD calculations shown and discussed below.

The lineshapes of V50 feature a cusp at zero frequency at, and above, 263 K. The authors of Ref 23 pointed out difficulties in reproducing this feature. As shown below, MOMD reproduces it satisfactorily. The temperature-dependent lineshapes of V50 can be reproduced with a potential strength given by $c_0^2=2.1$, rhombicity increasing slightly from $c_2^2=2.9$ at 247 to $c_2^2=3.0$ at 298 K, and a decrease in ${\tau }_{\perp }$ from 0.42 ms (16.7 μs) at 247 K to 0.21 ms (7.6 μs) at 298 K. The activation energies are 1.9 kcal/mol for ${\tau }_{\perp }$ and 2.3 kcal/mol for ${\tau }_{\mid \mid }$.

The L61 linesape evolution with temperature has been reproduced with $c_0^2=2.1$, and $c_2^2$ increasing from 2.7 at 254 K to 3.2 at 298 K. ${\tau }_{\perp }$ changes from 0.42 ms at 254 K to 0.17 ms at 298 K with an activation energy of 3.3 kcal/mol. ${\tau }_{\mid \mid }$ changes from 12.8 ms at 254 K to 6.7 ms at 298 with an activation energy of 2.5 kcal/mol. The differences between corresponding parameters of V50 and L61 are relatively small, except for the activation energies, which are substantially larger for L61. This is likely to be due to factors associated with methyl-type and/or residue location (V50 is located in helix 1 and L61 at the interface of helix 2 and the loop connecting helices 2 and 1).

The lineshapes of L75 exhibit the largest variation as a function of temperature. The rhombicity of the local potential increases substantially from $c_2^2$ at 252 K to 3.4 at 298 K. The correlation time ${\tau }_{\perp }$ is too slow to affect the analysis. The correlation time ${\tau }_{\mid \mid }$ (10.4 μs at 252 K and 3.3 μs at 298 K) is associated with the very small activation energy of 1.1 kcal/mol (in Ref 18 L75 is associated with very large activation energy). That smaller energy barriers have to be surmounted in the unstructured C-terminal segment housing L75, as compared to secondary-structure elements housing L61, L69 and V50, is expected.

For virtually all of the methyl groups studied we find that the rhombicity of the local potential increases with increasing temperature. At first glance this is rather surprising. However, if two (or more) different sources of temperature-dependent structural changes are contributing to this potential, it would not be surprising. For example, the local ordering tensors (defined in terms of the respective potentials) associated with these sources could be mostly axial but tilted with respect to each other. When both sources are operative, say at higher temperatures, the effective potential becomes rhombic. When one nearly freezes out at lower temperatures, the rhombicity of the effective potential becomes reduced substantially.

We associate the persistent source with proximal restraints and the source prone to freeze out at lower temperatures with distal restraints. For HP36 the latter play a particularly important role in stabilizing the tertiary structure in view of the exceptionally small area/volume ratio of this protein, which renders the sequestering of the hydrophobic residues more difficult.

In this context we show in Figure 7 local potentials, u(β_CM,γ_CM), and the associated equilibrium probability distribution functions, P_eq(β_CM,γ_CM). The latter are depicted as X_{C =} Rsin(β_CM)cos(γ_CM), Y_C = Rsin(β_CM)sin(γ_CM) and Z_C = Rcos(β_CM), where R = exp(−u). We show data for L75 at 252 K, where $c_0^2=2.2$ and $c_2^2=2.0$, and at 298 K, where $c_0^2=2.2$ and $c_2^2=3.4$. At 252 K preferential ordering occurs along Z_C, indicating parallel ordering. This corresponds to a nearly axial/slightly rhombic scenario, in agreement with the distal restrains being nearly frozen. At 298 K preferential ordering occurs along X_C, indicating perpendicular ordering. We analyzed previously with SRLS ²H relaxation from methyl groups in proteins dissolved in aqueous solution. In some cases the rhombicity of the local potential was found to be very large and the tilt between the main ordering axis and the principal axis of the <Q> tensor nearly 110.5°. Xc ordering is consistent with this geometric scenario.

Figure 7.

Equilibrium probability distribution function, P_eq(β_CM,γ_CM), depicted as X_{C =} Rsin(β_CM)cos(γ_CM), Y_{C =} Rsin(β_CM)sin(γ_CM), and Z_{C =} Rcos(β_CM), where R = exp(−u) (part A), corresponding to the MOMD potential, u(β_CM,γ_CM), given by $c_0^2=2.2$ and $c_2^2=1.8$ (part B). Equilibrium probability distribution function, P_eq(β_CM,γ_CM) (part C), corresponding to the MOMD potential, u(β_CM,γ_CM), given by $c_0^2=2.2$ and $c_2^2=3.4$ (part D).

An interesting future prospect is to correlate the mesoscopic local potentials and associated equilibrium distribution functions obtained with MOMD with their atomistic counterparts calculated with molecular dynamics (MD) simulations.

Figure 8 shows experimental spectra from the surface-exposed residue L63 (left), and a free peptide comprising helix h3 of HP36, labeled at the position of L69 (right). Figure 9 shows the respective best-fit MOMD lineshapes. Corresponding parameters are quite similar, except for the significant increase in potential rhombicity with increasing temperature for L63 as compared to constant rhombicity for L69 in the free peptide. This can be rationalized as follows. For L63 the tilt between the proximal/distal ordering frames is virtually independent of temperature. For the free peptide it changes with temperature such that the effective potential becomes more rhombic at higher temperatures.

Figure 8.

Experimental (red) and modeled in Ref 18 (blue) ²H lineshapes from the deuterium-labeled-methyl group of the surface residue L63 and the peptide h3-L69 peptide.

Figure 9.

Best-fit MOMD lineshapes for the experimental spectra shown in Figure 8, calculated using <Q> = 52.8 kHz, ${\left(T_2^{\ast }\right)}^{-1}=1$ kHz, β_MQ = 120°, and the parameters depicted on the Figure.

Figure 10 shows MOMD spectra for L69 in the protein core and L69 in the free peptide at similar temperatures. In the spirit of the interpretation delineated in the preceding paragraph, the relative orientation of the proximal/distal ordering frames differs in these two scenarios such that the rhombicity of the effective potential is larger in the free peptide.

Figure 10.

MOMD ²H spectra of L69 for 254, 265 and 298 K reproduced from Figure 3 (left), and MOMD ²H spectra of the h3-L69 peptide (for 252, 267 and 298 K) reproduced from Figure 8B (right). The MOMD calculations were carried out using <Q> = 52.8 kHz, ${\left(T_2^{\ast }\right)}^{-1}=1$ kHz, β_MQ = 120°, and the parameters depicted on the Figure

The proximal/distal restraint proposal

Several independent studies of HP36 have provided information related to our proximal/distal restraint proposal. A brief summary of the findings relevant to the matter in point follows.

A large number of studies associated with HP36 focus on the folding aspect. The consensus, based primarily on molecular dynamics (MD) simulations, appears to be fast consolidation of the secondary structure and much slower consolidation of the tertiary structure. This may be associated with difficulty in the formation of distal restraints. Temperature-dependent backbone dynamics of HP36 in solution was studied previously using model-free as method of analysis.⁶⁹ The key parameter derived is the squared generalized order parameter, S². The results indicate that the temperature dependence of S is not described accurately by axial model potential functions for bond vector reorientation used previously in this context for larger proteins. This is consistent with the rhombic local potentials determined in the solid-state by the MOMD analysis.

Several published studied make possible a more direct comparison of their results with the proximal/distal restraint concept. For example, Herges and Wenzel⁷⁰ investigated the free-energy landscape of HP36 and the 40-residue HIV Accessory Protein (HAP) with molecular dynamics (MD) simulations. While HAP exhibits a distinct global minimum, HP36 exhibits quite a few metastable states with free-energy comparable to that of the global minimum. Reaching the global minimum took ten times longer for HP36 as compared to HAP. These observations are consistent with the very small protein HP36, but not the common protein HAP, requiring extra global-minimum-stabilizing distal restraints to fold properly.

Usually X-ray crystallography and NMR structures of single-domain proteins do not differ considerably. In the case of HP36 substantial differences have been detected.⁷¹ To reveal their origin MD simulations were carried out starting with the NMR structure. After 50 ns HP36 spontaneously adopted conformations more consistent with the crystal structure. Unlike the solution structure, these conformations featured a stable distance between F47 and R55, suggesting the formation of a cation−π interaction. Experimental double mutant cycles confirmed that the F47−R55 pair has a relatively large energetic coupling. In light of these observations the authors of Ref 71 consider the X-ray crystal structure an appropriate representation of HP36 in solution at neutral pH.

This may be rationalized as follows. The interaction between F47 and R55 is a distal restraint in the global minimum crystal structure. It is not stable in the solution structure which is an average comprising among others the global minimum. Recovering the latter requires extensive conformational search.

A recent study reported on the response of HP36-coated gold nanoparticles (AuNP)s) to ultrafast laser heating.⁷² Laser pulse radiation for maximal heating altered substantially the structure of proteins used previously as AuNP-coating agents. Surprisingly, it left intact the 3D structure of HP36. This may be rationalized as follows. In the coating layer the HP36 molecule is flanked by thiolated polyethylene glycol (briefly PEG). The potentially available distal interactions are utilized by PEG to stabilize the structure of HP36 in the coating layer. Such potential interactions are not available in other proteins.

Thus, independent studies provide information consistent with our proximal/distal restraint proposal.

Comparison with the multi-simple-mode analysis of HP36. The authors of Ref 18 specify the room-temperature (298 K) rates for L75 and L69. The values are 1.04×10⁴ 1/s (given as 6.5×10⁴ rad/s in the article) for L75 and 0.37×10⁴ 1/s (given as 2.3×10⁴ rad/s) for L69. The analogous MOMD data are 5.05×10⁴ 1/s (3.3 μs) for L75 and 1.9×10⁴ 1/s (8.8 μs) for L69. The MOMD rate constants are five times faster than the rate constants for the dominant motional mode of the simple-multi-mode scenario. In comparing them one must appreciate that the effective MOMD rates also include the effect of the potential barriers, which slow down the diffusive motion.^49,53

3.4. MOMD applied to the Streptomyces Subtilisin inhibitor

This protein has been ²H-labeled selectively at the methyl groups of residues Met 103, Met 73 and Met 70.⁴¹ Here S−CD₃ dynamics is investigated. ²H lineshapes were acquired from polycrystalline samples of the respective mutants as a function of hydration. Simulations based on discrete jumps around axes tilted from the S−CD₃ bond to various extents, and among sites with various relative populations, were carried out. Some spectra were reproduced satisfactorily. Several important spectra could not be reproduced following this approach. Figure 11C shows such a spectrum obtained from the methyl group of Met 73 in a sample containing 0.6 gr H₂O per gram protein-water. This scenario is typical of the native protein in the solid-state;⁴¹ hence, it is particularly important.

Figure 11.

Experimental ²H lineshape from the selectively deuterated methyl group of Met 73 of the Streptomyces Subtilisin inhibitor (C). Corresponding best-fit spectrum obtained in Ref 41 (C’). MOMD spectrum obtained for <Q> = 52.8 kHz, ${\left(T_2^{\ast }\right)}^{-1}=1$ kHz, β_MQ = 120°, $c_0^2=2.2$, $c_2^2=1.6$, ${\tau }_{\perp }=1.04$ ms and ${\tau }_{\mid \mid }=3.3\mu s$ (C”).

The Figure 11C spectrum features dominant intensity at zero frequency (ignoring the spike associated with residual HDO) and gradual descent to the baseline. Figure 11C’ shows the best-fit spectrum which the authors of Ref 41 could obtain. The model used consists of random jumps among four equally-populated sites on a distorted cone, with a rate of 10 MHz, 0.5 kHz Lorentzian linebroadening and 10 kHz Gaussian linebroadening. The shape of the spectrum around zero frequency, which is a dominant feature, has not been reproduced properly.

Figure 11C’’ shows the MOMD lineshape obtained for a rhombic local potential with $c_0^2=2.2$, $c_2^2=1.6$, ${\tau }_{\perp }=1.04\mu s$, and β_MQ = 120°. The agreement with the experimental spectrum of Figure 11C has been improved substantially. At least in this case, MOMD-based analysis yields better agreement than the multi-simply-mode approach.

4. Conclusions

The MOMD approach interprets the main features of ²H lineshapes representing μs − ms C−CD₃ and S−CD₃ bond dynamics in proteins in terms of a single (effective) dynamic mode. The latter features a local potential and a local diffusion tensor as qualifiers of structural dynamics. The local potential is typically rhombic; its strength and deviation from axiality are approximately 2 [k_BT]. The local diffusion tensor may be taken as axially symmetric, with principal axis tilted at 120° (the angular minimum for trans-gauche isomerization) from the C−CD₃ bond. The perpendicular (to the principal axis of the diffusion tensor) correlation time is in the 0.1 − 1.0 ms range. The parallel correlation time is in the 3.3 − 30 microsecond range. The analysis leads to Arrhenius temperature behavior with respective activation energies in the 1.1 − 8.0 kcal/mol range. The error in all of the parameters determined is approximately 10%.

The rhombicity of the local potential typically increases as a function of temperature. This is rationalized in terms of two different sources of temperature-dependent structural changes which we identify with persisting proximal restraints and distal restraints prone to freeze out at lower temperatures. This interpretation is consistent with other studies of HP36.

Previous approaches combined several independent restricted motional modes in a case-specific manner. Typically four quantities, differing in their physical nature, are used as descriptors. MOMD has the benefit of generality, simplicity of the dynamic picture, versatility and physical rigor. The SRLS/MOMD/NMR scheme makes possible analyzing internal protein mobility in the liquid and solid states with the same physically appropriate model.

Future prospects include extension to the ²H relaxation limit, application to ¹⁵N and ¹³C, inclusion of collective motions, and accounting for anisotropic (membrane-like) environments. Although we have not considered them in this study, more detailed MOMD analysis, and further comparison with experiment, could include the effects of a non-axial diffusion tensor, separate ordering and diffusion tensor frames, and additional terms in the expansion of the potential (Eq. 3). Even without such refinements, useful results were obtained, providing a consistent physical picture.