Protein Dynamics in the Solid State from (2)H NMR Line Shape Analysis. II. MOMD Applied to C-D and C-CD3 Probes.

Deuterium line shape analysis from mobile C-D and C-CD3 groups has emerged as a particularly useful tool for studying dynamics in the solid state. The theoretical models devised so far consist typically of sets of independent dynamic modes. Each such mode is simple and usually case-specific. In this scenario, model improvement entails adding yet another mode (thereby changing the overall model), comparison of different cases is difficult, and ambiguity is unavoidable. We recently developed the microscopic order macroscopic disorder (MOMD) approach as a single-mode alternative. In MOMD, the local spatial restrictions are expressed by an anisotropic potential, the local motion by a diffusion tensor, and the local molecular geometry by relative (magnetic and model-related) tensor orientations, all of adjustable symmetry. This approach provides a consistent method of analysis, thus resolving the issues above. In this study, we apply MOMD to PS-adsorbed LKα14 peptide and dimethylammonium tetraphenylborate (C-CD3 and N-CD3 dynamics, respectively), as well as HhaI methyltransferase target DNA and phase III of benzene-6-hexanoate (C-D dynamics). The success with fitting these four disparate cases, as well as the two cases in the previous report, demonstrates the generality of this MOMD-based approach. In this study, C-D and C-CD3 are both found to execute axial diffusion (rates R⊥ and R∥) in the presence of a rhombic potential given by the L = 2 spherical harmonics (coefficients c02 and c22). R⊥ (R∥) is in the 102-103 (104-105) s-1 range, and c02 and c22 are on the order of 2-3 kBT. Specific parameter values are determined for each mobile site. The diffusion and quadrupolar tensors are tilted at either 120° (consistent with trans-gauche isomerization) or nearly 110.5° (consistent with methyl exchange). Future prospects include extension of the MOMD formalism to include MAS, and application to 15N and 13C nuclei.

Introduction

2H NMR line shape analysis is a powerful method for studying structural dynamics in the solid state.[1,2] The [isopropyl-2H7]-leucine probe has been used in early work to investigate the dynamics of collagen fibrils.[3] In recent years, this probe has been used to study physisorption of the Ac-LKKLLKLLKKLLKL-OH (LKα14) peptide onto polystyrene (PS) and PS-COOH surfaces and carboxyl-functionalized gold nanoparticles.[4−6] The d3-alanine and d3-leucine probes have been used to investigate the dynamics of phospholamban embedded in phospholipid bilayers,[7,8] as well as partitioning and orientation of the lung surfactant peptide KL4 in phospholipid bilayers.[9] The probes d3-leucine and d8-valine have been used recently to study side-chain dynamics in the core of the Chicken Villin Headpiece Subdomain (HP36).[10,11] Extensive work associated with lipids, peptides, and proteins has been carried out by the groups of Seelig (e.g., ref (12)), Davis (e.g., ref (13)), Griffin (e.g., ref (14)), Oldfield (e.g., ref (15)), Brown (e.g., ref (16)), etc. Deuterated tyrosine side chains have served as a probe for studying peptide dynamics (e.g., ref (17)). 2H-Labeled methylene groups have been used to study polymer dynamics (e.g., ref (18)). Deuterium NMR has been used extensively to study liquid crystals.[19−21] Work focusing on NMR line shape and relaxation analysis in solids, where key aspects have been treated, is described in refs (22−25). We also contributed in the past to this field of research.[17,26−28]

Traditionally, dynamics in the solid-state have been envisioned as jump-type motions associated with barrier-crossing and hops among energy wells (e.g., ref (29)). Only when the activation energy turned out to be very small were simple diffusive models developed and utilized (e.g., ref (30)). However, a single (typically jump-type) simple motion often does not reproduce satisfactorily dynamic 2H line shapes in the solid state. Rather, several independent simple motions have to be combined (e.g., refs (4−6), (10), and (11)). These multi-simple-mode models are typically case-specific; hence, comparison among different systems is difficult. In this scenario, the agreement between calculated and experimental spectra can only be improved by adding yet another simple mode; this alters the overall model. Finally, the multi-simple-mode concept is inherently ambiguous, as many different combinations of simple motions are possible.

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[31−35] as well as internal[36−38] motions. The major underlying factors−type of motion, spatial restrictions, and geometry−are treated generally for the entire motional range, within their rigorous three-dimensional tensorial requirements.

Particularly relevant in the present context is the extension of the SLE called microscopic-order-macroscopic-disorder (MOMD) approach, where the director of the restricting environment is distributed randomly over the sample.[39] MOMD/ESR was applied to nitroxide probes in liposomes,[40] and later to internal motions in 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 time scale).[41,42] The ESR nitroxide label in “frozen” solutions is formally analogous to the NMR 2H label in polycrystalline proteins. In MOMD, parameter combinations differing in complexity can be devised by monitoring tensor magnitude, symmetry, and orientation.[39] This enables a continuous range of scenarios, and thus comparison among different systems, within the scope of the same general model. Thereby, the problematic issues associated with previous 2H line shape analyses (see above) are expected to be largely resolved.

On the basis of this reasoning, we recently developed MOMD[39] for the analysis of dynamic 2H NMR line shapes.[43] Our approach was applied to C–CD3 dynamics in HP36 and an example of S–CD3 dynamics in the Streptomyces Subtilisin Inhibitor (SSI).[43] The emerging physical picture depicts the spin-bearing probe executing axially symmetric diffusion (with rate constants R⊥ and R∥) in the presence of a rhombic local potential given by the L = 2 and K = 0, 2 spherical harmonic functions (with coefficients c02 and c22). The axial diffusion frame, M, is tilted from the axial quadrupolar tensor frame, Q, at 120° or nearly 110.5° (angle βMQ). The physical processes consistent with these geometric features are trans–gauche isomerization and methyl exchange, respectively. The uncertainty of the results has been estimated at 10% for c02 and c22, 15% for the motional rates, and 3% for the angle βMQ.[43]

It is important to apply MOMD to additional systems differing in nature to test its generality. Also, one should determine whether the parameter combination delineated above is appropriate in all cases. If it is not, adequate parameter combinations need to be devised. It is, in addition, insightful to compare MOMD analyses of different systems. Finally, it is of interest to compare the MOMD-based pictures with the corresponding multi-simple-mode-based pictures.

These are the objectives of the present study. Along these lines, we apply MOMD to C–CD3 dynamics in the PS-adsorbed LKα14 peptide[4] and in dimethylammonium tetraphenylborate.[44] In addition, we apply MOMD to C–D dynamics in the furanose ring of HhaI methyltransferse target DNA[30] and benzene-6-hexanoate in phase III.[45] The parameter combination found suitable to describe 2H dynamics in HP36 and SSI is also found suitable for describing 2H dynamics in the systems investigated in this work. The quantitative analysis reveals interesting new system-specific information.

The six different cases treated so far (four in this work and two in the study where the MOMD/NMR was developed[43]) demonstrate the benefit and power of model generality. Future prospects of this work include incorporation of magic angle spinning in the MOMD formalism, application to 15N and 13C nuclei, and further generalization of the MOMD model.

MOMD[39,43] represents the limit of the two-body coupled-rotator slowly relaxing local structure (SRLS) approach[36−38] where the slow rotator is “frozen”, yielding random orientations of the local director. In recent years, we applied SRLS to NMR relaxation from proteins in aqueous solution.[46−50] Thus, the same general physically relevant theoretical/computational tool is now available for studying protein dynamics in the solid and liquid states.

A theoretical summary is provided in section . Results and discussion are summarized in section . Our conclusions appear in section .

Theoretical Summary

The MOMD theory as applied to NMR has been delineated in ref (43). A brief summary is given below. Figure shows the MOMD frame scheme for a deuterium nucleus. L is the space-fixed laboratory frame. C is the local director frame fixed in the molecule. M denotes the principal axes system (PAS) of the local ordering tensor, S, taken the same (for simplicity) as the PAS of the local diffusion tensor, R. Q denotes the PAS of the quadrupolar tensor. The M and Q frames are fixed in the probe.

Figure 1

MOMD frames: L - lab frame, C - local director frame, M - PAS of the local ordering/local diffusion tensor, Q - PAS of the quadupolar tensor, Q, or the partially averaged-by-methyl-rotation quadrupolar tensor, ⟨Q⟩. The Euler angles ΩMQ are fixed. The Euler angles ΩCM are time-dependent. 2H NMR spectra are calculated for every ΩLC and convoluted according to a random distribution.[43]

The Euler angles ΩCM (associated with the orientation and diffusion of the probe relative to the local director) are time-dependent. The Euler angles ΩMQ = (αMQ, βMQ, γMQ) are time-independent. Given that the Q frame is axially symmetric, one has γMQ = 0. For simplicity, the angle αMQ is set equal to zero. Thus, the orientation of ZM (main ordering/diffusion axis) relative to ZQ (the known principal axis of the quadrupolar tensor, or the partially averaged-by-methyl-rotation quadrupolar tensor) is given by the polar angle, βMQ. Since there is no “macroscopic order”, one has to calculate 2H spectra for every ΩLC, and convolute the corresponding line shapes according to a random distribution.[43]

The stochastic Liouville equation (SLE) for the spin density matrix is given by[31−43] is the superoperator for the orientation-dependent spin Hamiltonian. ΓΩ is a Markovian operator for the rotational reorientation of the spin-bearing moiety (probe), with the Euler angles Ω → (α, β, γ) (ΩCM in the notation above) representing the orientational angles. P0(Ω) is the unique equilibrium probability distribution of ΓΩ.

A simple form of the diffusion operator, ΓΩ, is[31]where R is the isotropic rotational diffusion rate, ∇Ω2 is the rotational diffusion operator in the Euler angles, Ω, and T is the restoring torque. The latter is equal to ∂u/∂β for an axial restoring potential, e.g., u ≅ −3/2c02(cos β)2 (the coefficient c02 is given in units of kBT). The expression of ΓΩ for rhombic diffusion tensor and rhombic potential is given in ref (32).

In this study, we are using an axial diffusion tensor, R, associated in the absence of a restricting potential with three decay rates, τ–1 = 6R⊥ + K2(R∥ – R⊥), where K = 0, 1, and 2 (K is the order of the rank 2 diffusion tensor). R∥ and R⊥ are the principal values of R; one may also define τ∥ = 1/(6R∥) and τ⊥ = 1/(6R⊥).

For a uniaxial local director, one may expand the potential in the complete basis set of the Wigner rotation matrix elements, D0,(0, θ, φ), which are proportional to the spherical harmonics. One has[32]with u(θ, φ) and c being dimensionless. If only the lowest, L = 2, terms are preserved, one obtains the real potential:[32,39]with c02 evaluating potential strength and c22 potential rhombicity. This form of u(ΩCM) is used herein.[43]

Local order parameters are defined as[32]

For at least 3-fold symmetry around the local director, C, and at least 2-fold symmetry around the principal axis of the local ordering tensor, ZM, only S02 ≡ ⟨D0,02(ΩCM)⟩ and S22 ≡ ⟨D0,22(ΩCM) + D0,–22(ΩCM)⟩ survive.[32] The Saupe scheme order parameters relate to irreducible tensor components, S02 and S22, as S = ((3/2)1/2S22 – S02)/2, S = −((3/2)1/2S22 + S02)/2, and S = S02.

The diffusion operator of eq comprises the rate constant R (principal value of the isotropic diffusion tensor) and the parameter c02 (coefficient of the axial potential). The (enhanced) diffusion operator used in this study comprises the rate constants R⊥ and R∥ (principal values of the axial diffusion tensor) and the parameters c02 and c22 (coefficients of the rhombic potential). The diffusion operator ΓΩ is given in the M frame; the superoperator is given in the Q frame. Therefore, in solving the SLE (eq ), one has to convert the M frame into the Q frame. This is accomplished by the Wigner rotation based on the angle βMQ.

The solution of the SLE (eq ), with the appropriate forms of ΓΩ and P0(Ω), and the specified values of Q (quadrupole constant), c02, c22, R⊥, R∥, and βMQ, represents the calculated MOMD spectrum.

Results and Discussion

The parameters c02, c22, R∥, and R⊥ are allowed to vary in the data-fitting process; the angle βMQ is fixed as delineated below.

General Assessments

Local Spatial Restrictions

In our recent application of MOMD to C–CD3 dynamics of leucine residues of HP36 and S–CD3 dynamics of a methionine residue of SSI, we determined c02 and c22 to be approximately 2–3 kBT.[43]2H line shapes from C2′–2H2 in HhaI methyltransferase target DNA have been used in previous work to study furanose ring dynamics.[30] The local spatial restrictions have been expressed in terms of an axial potential, the coefficient of which has been determined to be approximately 5 kBT.

The dominant simple mode associated with C–D and C–CD3 dynamics is two-site exchange.[3−6,10,11,14,44,45] In this process, the population ratio, p1/p2, represents implicitly the spatial restrictions (favoring one conformer over the other). p1/p2 was found to range from 3/7 to 4/6 in ref (4), from 2/8 to 3.8/6.2 in ref (5), and from 2/8 to 1 in ref (45). In the extreme motional narrowing limit, one has 1/T2 = δ2 × (p1 × p2)/kex (δ, separation between chemical shifts; kex, exchange rate). For a rigid particle reorienting diffusively in a strong potential given by u = −c02D0,02, one has in the extreme motional narrowing limit 1/T2 = Δ2 × (S02)2 × τ (Δ, anisotropic magnetic interaction; S02 = ⟨D0,02⟩, axial order parameter; τ, diffusive correlation time). In this limit, p1 × p2 and (S02)2 may be considered to be formally analogous. The p1/p2 values depicted above correspond to S02 = 0.45–0.63, which in turn correspond[46] to c02 = 2–3 kBT.

Thus, for C–D and C–CD3 dynamics in the system investigated in this work, the coefficients c02 and c22 of the MOMD potentials are most likely on the order of 2–3 kBT.

Local Motion

The multimode models yielded exchange rates on the order of 5 × 102 to 5 × 106 (103–106) s–1 for C–D (C–CD3 with spinning methyl group) dynamics. This is expected on the basis of Q ∼ 170 kHz for C–D and ⟨Q⟩ ∼ 50 kHz (∼[1.5 cos2(110.5°) – 0.5] × 170, where 110.5° is the tetrahedral angle corresponding to rCH = rCD = 1.115 Å)[43] for C–CD3. In our previous application of MOMD to HP36 and SSI, we found that R⊥ is quite close to its rigid limit, whereas R∥ is moderate to fast on the dynamic C–CD3 time scale.[43] Whether R⊥ and R∥ of other cases of C–CD3 dynamics, and instances of C–D dynamics, exhibit similar patterns is to be determined.

Note that R⊥ and R∥ are geometrically related principal values of an axial diffusion tensor, R, the orientation of which is given by βMQ. If necessary, the symmetry of R can be lowered (i.e., R ≠ R), and its orientation specified by αMQ and βMQ. The form of the local potential (eq ), hence the complexity (in terms of magnitude, symmetry, and orientation) of the local ordering tensor, can also be enhanced. The single-mode description will still be in effect. Within such a broad scope, implied by the tensorial representation of the physical quantities, it is not necessary to resort to multimode descriptions (which, in principle, can be implemented in MOMD).

Local Geometry

The geometric feature of particular interest is the orientation of ZM relative to ZQ, given by βMQ. MOMD describes axial diffusive motion in the presence of a rhombic potential, rather than simple (jump-type) motions.[43] However, from a geometric point of view, one may relate βMQ to geometric features associated with relevant simple motions, and view MOMD as enhanced versions of those motions.

Figure A illustrates trans (left)–gauche (right) isomerization in a polymethylene chain segment.[51] It can be seen that the deuterium atom D1 in the gauche configuration assumes the orientation originally occupied by the deuterium atom D2 in the trans configuration. The configurational projection along the C–C bond (Figure B) illustrates the angular minimum of 120° for (jump-type) trans–gauche isomerization. From the perspective of MOMD, the blue axes denote ZQ, the red axis denotes ZM, and βMQ describes their relative orientation. Thus, βMQ = 120° is a physically plausible option for C–D dynamics.

Figure 2

Trans (left)–gauche (right) isomerization illustrated for a polymethylene chain segment. D1 and D2 denote the interconverting deuterium atoms (part A). Projection along the C–C bond of the trans and gauche configurations, illustrating the angular minimum of 120° for the isomerization process. From the perspective of MOMD, the blue axes denote ZQ, the red axis denotes ZM, and βMQ is equal to 120° (part B).

Figure illustrates methyl exchange between two out of the four corners of the carbon tetrahedron.[4] Within the scope of this process, ZM (red) is tilted at 110.5° from ZQ (blue). Thus, βMQ = 110.5° is a physically plausible option for C–CD3 dynamics.

Figure 3

Methyl jumps between two out of the four corners of the carbon tetrahedron, involving an angular change of 110.5°. From the perspective of MOMD, the blue axes denote ZQ, the red axis denotes ZM, and βMQ is equal to 110.5°.

In fitting the experimental data, we set first βMQ equal to 110.5 or 120°. If both values yield physically relevant and statistically appropriate (i.e., acceptable) results, the alternative with the better statistics will be selected. If only one βMQ value yields acceptable results, that value is selected. If none of these values yields acceptable results, βMQ is allowed to vary in the 90–130° range (the angle we call βMQ has been found to be 108–112° in ref (3), 32–90° in ref (45), and 60–80° in ref (52)). The digressions from βMQ equal to 120 or 110.5° have been found in this study to be relatively small (see below).

Error Estimates

The errors in the best-fit MOMD parameters determined in this work are as follows: 3% in βMQ, 10% in c02 and c22, and 15% in R∥ and R⊥.

Illustrative Simulations

Figure A shows 2H line shapes obtained with c02 = 2.0, c22 = 3.5, βMQ = 110.5°, and motional rates as depicted. Dynamic powder patterns similar to the red (R⊥ = 0.79 × 104 s–1 and R∥ = 6.31 × 104 s–1) and black (R⊥ = 0.79 × 104 s–1 and R∥ = 63.1 × 104 s–1) spectra are encountered in quite a few cases. These parameters belong to the predicted ranges. Dynamic powder patterns similar to the green and blue spectra, where R⊥ = 3.16 × 104 s–1 is quite fast, are encountered rarely.

Figure 4

2H MOMD spectra obtained for c02 = 2.0, c22 = 3.5, βMQ = 110.5°, and R⊥ and R∥ as depicted. The diffusion rates are given in units of 104 s–1 (part A). 2H MOMD spectra obtained for c02 = 2.2, R⊥ = 0.016 and (a) c22 = 1.8 and R∥ = 2.0 in units of 104 s–1; (b) c02 = 2.6 and R∥ = 3.16 in units of 104 s–1; and (c) c02 = 2.8 and R∥ in units of 104 s–1 (part B). In each panel, the various colors designate the value of βMQ as follows: 90° (black), 100° (red), 110° (green), 120° (blue), and 130° (violet). Additional parameters used include ⟨Q⟩ = 52.8 kHz and an intrinsic line width of 1 kHz.

Figure B shows a set of dynamic 2H powder patterns with c02 = 2.2, R⊥ = 1.6 × 102 s–1, c22, and R∥ as depicted, and βMQ varying from 90 to 130° in steps of 10° (see figure captions). Clearly, the 2H line shapes are very sensitive to the rhombicity of the potential (c22), the parallel diffusion rate (R∥), and the local geometry (βMQ).

The illustrative simulations presented in this section apply to deuterium nuclei executing axial local diffusion in a rhombic local potential. Specifics of the local motion, e.g., the relation of the local ordering/local diffusion frame to the local stereochemistry, are determined by the system investigated (see below).

MOMD Analysis of PS-Adsorbed LKα14 Peptide; C–CD3 Dynamics

As pointed out above, 2H line shapes from the peptide Ac-LKKLLKLLKKLLKL-OH (LKα14), deuterated selectively at various leucine positions with [isopropyl-2H7], have been used extensively to study physisorption onto biomineral surfaces (refs (4−6)). The samples L5, L8, L11, and L14 adsorbed onto the apolar polystyrene (PS) surface have been investigated in ref (4) (the numbers denote the position of the isotope-labeled Leu residue in the peptide sequence). L8 has also been studied in its unbound (lyophilized) form, as well as adsorbed onto the polar surface of carboxyl-functionalized gold nanoparticles. The respective 2H spectra are shown by the black traces in Figure A (see figure captions for specific designations). The spikes in the center of Figure Ab and Ac are associated with residual HDO; they should be ignored.

Figure 5

Experimental (black) and calculated (red) 2H NMR line shapes of unbound LKα14 labeled with d7-Leu at position 8 (L8) (a); PS-adsorbed L5 (b); PS-adsorbed L8 (c); PS-adsorbed L11 (d); and PS-adsorbed L14 (e). L8 adsorbed onto carboxyl-functionalized gold nanoparticles (f). Reproduced with permission from ref (4) (part A). 2H MOMD spectra that reproduce the experimental line shapes of part A, obtained for βMQ = 120° and the parameters depicted in the figure. The diffusion rates are given in units of 104 s–1. Additional parameters used include ⟨Q⟩ = 52.8 kHz and an intrinsic line width of 1 kHz (part B, a–f).

The red traces in Figure A represent the best-fit calculated spectra obtained in ref (4) with the following model. (1) The effective quadrupole constant, ⟨Q⟩, is allowed to vary. (2) Two-site methyl exchange with an angular separation of approximately 110° occurs. Its rate, kex, and the population ratio, p1/p2, are allowed to vary. (3) The instantaneous Cα–Cβ bond executes equidistant jumps around its equilibrium orientation on the surface of a cone with a half-angle of 10°. The rate of this motion, kcone, is allowed to vary. The contribution of the γ-deuteron is ignored in view of low signal intensity.

The experimental 2H spectrum of the unbound peptide (Figure Aa) could not be reproduced satisfactorily with this model even when the cone half-angle was also allowed to vary, and the contribution of the γ-deuteron was accounted for.[5] The experimental 2H spectra shown in Figure Ab–Af (black traces) have been reproduced very well (red traces).

The MOMD spectra that constitute best-fits to the experimental spectra of ref (5) (black traces in Figure A) are shown in Figure B. Spectra 5Bb and Be (L5 and L14 adsorbed onto PS) on the one hand, and spectra 5Bc and Bd (L8 and L11 adsorbed onto PS) on the other hand, are considered to be the same. MOMD reproduces well the experimental spectra, including the line shape shown in Figure Aa. The 2H spectra of L5 and L14 have been reproduced with βMQ = 120°, c02 = 2.2, c22 = 2.6, R⊥ = 1.6 × 102 s–1, and R∥ = 2.0 × 104 s–1. The 2H spectra of L8 and L11 feature reduced potential strength (c02 = 2.0 instead of 2.2), reduced potential rhombicity (c22 = 2.2 instead of 2.6), and enhanced R∥ (2.5 × 104 s–1 instead of 2.0 × 104 s–1). βMQ and R⊥ are the same. The end-on view of the helical structure of LKα14 depicted in ref (4) shows that L5 and L14 are relatively close to the PS surface whereas L8 and L11 are farther away from it. The MOMD description points to reduced spatial restrictions and enhanced motional rate, R∥, for Leu side chains located farther from the PS surface.

Unbound L8 (Figure Ba) differs from PS-adsorbed L8 (Figure Bc). The potential form/symmetry changes upon adsorption, and the local diffusion becomes more axial (i.e., R∥/R⊥ becomes larger). As indicated above, the 2H spectrum of unbound L8 (Figure Aa) could not be reproduced with the multimode models of refs (4) and (5). Allowing the local potential to be rhombic, and the local motion to be represented by a tensorial quantity, appears to be very important in this case.

The 2H spectrum of L8 adsorbed onto carboxyl-functionalized gold nanoparticles (Figure Bf) was reproduced with βMQ = 110.5°. It exhibits a potential form/symmetry that differs substantially from both its unbound form (Figure Ba) and its PS-adsorbed form (Figure Bc). The axiality of the local diffusion (R∥/R⊥) is similar for L8 adsorbed onto carboxyl-functionalized gold nanoparticles and L8 adsorbed onto PS. However, the absolute values differ, with both rates being twice slower for L8 adsorbed onto carboxyl-functionalized gold nanoparticles. The polarity, and in general the chemical nature, of the surface, is clearly an important factor in determining the structural dynamics of the Leu side chains. R∥ is quite slow (1.3 × 104 s–1) for both unbound L8 and L8 adsorbed onto carboxyl-functionalized gold nanoparticles; R⊥ is substantially slower in the latter sample (8.0 × 102 s–1 versus 2.0 × 103 s–1).

Thus, MOMD reproduces all of the experimental spectra shown by the black traces in Figure A, quantifying structural dynamics in terms of the form of the local potential, and the principal values and orientation of the local diffusion tensor. The three different forms of L8 are compared in terms of these qualifiers.

The black traces in Figure represent experimental 2H spectra acquired in ref (5) at room temperature (RT) from L1, L4, L5, L7, L8, L11, L12, and L14 adsorbed onto PS and from unbound L8 (the red traces represent the best-fit simulations from ref (5)). We consider the spectra of L1, L4, L7, L8, and L11 on the one hand, and L5 and L14 on the other hand, to be the same.

Figure 6

Experimental 2H line shapes from L1, L4, L5, L7, L8, L11, L12, and L14 adsorbed onto a polystyrene surface and unbound L8 (black traces; the red traces represent the best-fit simulations from ref (5)). Reproduced from ref (5). Copyright 2010 PNAS.

The spectrum of L12 is similar to the spectrum of unbound L8. The authors of ref (5) also report sum frequency generation (SFG) experiments which yielded the orientation of the leucine side chains relative to the surface normal. Tilt angles of 30–48° for L4, L7, L8, and L11, 70° for L1, 68–74° for L5 and L14, and 82° for L12 have been determined with SFG. Except for L1 (the N-terminal residue of the peptide, which might act from the perspective of Leu side-chain dynamics more like residues farther away from the surface), the MOMD results are consistent with the SFG results. That is, small and large SFG-derived tilt angles correspond to different line shapes, and L12 is exceptional. All three groups are quantified in terms of potential form and diffusion tensor. Reference (5) has singled out only L12 as exceptional based on its outstandingly large cone half-angle (allowed to vary in ref (5)) agreeing with a very large SFG-derived tilt angle.

The parameter combination used in ref (5) is different from the parameter combination used in ref (4). In ref (4), the parameters ⟨Q⟩, kex, p1/p2, and kcone have been allowed to vary; the cone half-angle, θ, has been set equal to 10°, and the contribution of the γ-deuteron has been ignored. In ref (5), the parameters p1/p2, ⟨Q⟩, θ, and the percent contribution of the γ-deuteron have been allowed to vary; kex has been fixed at 6 × 106 s–1 and kcone at 600 s–1. In yet another study, L8 adsorbed onto PS and PS-COOH surfaces has been investigated as a function of temperature in the −40 to +60 °C range.[6] In that study, the parameters ⟨Q⟩, kcone, and p1/p2 have been allowed to vary; kex has been fixed at 3 × 106 s–1, the angle θ at 35°, and the contribution of the γ-deuteron at 16%. Within the scope of the analysis of ref (6), p1 and p2 exhibit different trends as a function of temperature for the PS-adsorbed and PS-COOH-adsorbed peptide. In particular, above 20 °C, p1 and p2 diverge for PS-adsorbed L8 and converge for PS-COOH-adsorbed L8. This has been interpreted in terms of p1/p2 being consistent with the Boltzmann relation p1/p2 = exp(−(E2 – E1)/kBT) (where E2 – E1 is the energy difference between the site with population p2 and the site with population p1) for PS-COOH-adsorbed L8 but inconsistent for PS-adsorbed L8.

The analyses in refs (4), (5), and (6) differ in the identity of the fixed and allowed-to-vary parameters and the values of the fixed parameters. Taken separately, each scenario is plausible; taken together, ambiguities are encountered. Table shows the results obtained for L8 adsorbed onto PS at 20 °C. Within the scope of the various analyses, kex varies by an order of magnitude, kcone by a factor of 5, θ by a factor of 3.5, and the contribution of the γ-deuteron is zero in ref (4) and 16% in refs (5) and (6).

Table 1

Best-Fit Parameters Obtained for L8 Adsorbed onto the PS Surface at 20 °C in refs (4−6)a

ref	⟨Q⟩ (kHz)	kex (s–1)	kcone (s–1)	p1/p2	θ°	γ-deuteron
(4)	49	6 × 105	2 × 103	4/6	10*
(5)	47	6 × 106 *	6 × 102 *	3.6/6.4	11	16%
(6)	46	3 × 106 *	6 × 102	3.6/6.4	35*	16%*

Items designated by asterisks have been fixed in the respective calculations. The data of ref (6) have been obtained from 2H spectra obtained under magic angle spinning (MAS).

MOMD and the multimode models of refs (4−6) can only be compared in general terms. The MOMD model consists of a single dynamic mode comprised of a rhombic potential and an axial diffusion tensor. The multimode models of refs (4−6) are different versions of three independent simple dynamic modes comprising high-symmetry (scalar) parameters.

MOMD Analysis of Furanose Dynamics in HhaI Methyltransferase Target DNA; C–D Dynamics

Furanose dynamics in HhaI methyltransferase target DNA 2H-labeled at the C2′ position have been studied with 2H line shape analysis.[30] The experimental 2H spectra of cytosines C6 and C8 in the DNA sequence are shown by the black traces in Figure Aa and Ab, respectively. The model utilized takes into account the global motion of the DNA and the (independent) local motion of the furanose ring. The global motion consists of jumps with rate kex on the surface of a cone with a half-angle of 20° centered at the helix axis.

Figure 7

Experimental 2H line shapes from the 2H′ deuterons of the nucleotides C6 (a) and C8 (b) of the HhaI methyltransferase target DNA at a hydration level of 10% and 300 K. The calculated line shapes have been obtained as described in the text. Reproduced with permission from ref (30) (part A). 2H MOMD spectra that reproduce the experimental line shapes of part A, obtained for βMQ = 120° and the parameters depicted in the figure. The diffusion rates are given in units of 104 s–1. Additional parameters used include Q = 170 kHz and an intrinsic line width of 1 kHz (part B).

The local motion is represented by C2′–2H2 moving with rate D rad2/s in the presence of an axial potential with coefficient κ kBT. The experimental 2H spectra (black traces in Figure A) have been reproduced in ref (30) with kex = 1.0 × 104 s–1, D = 1.8 × 107 rad2/s, and κ = 5 kBT (blue traces in Figure A).

Calculated MOMD spectra which constitute good fits to the experimental spectra of Figure A, and which reproduce the trend observed in going from C6 to C8, are shown in Figure Ba and Bb. For the C6 nucleotide, MOMD analysis yielded c02 = 2.1, c22 = 3.0, R⊥ = 8.0 × 102 s–1, R∥ = 2.0 × 104 s–1, and βMQ = 120°. For C8, all the parameters except for c02, which is equal to 2.2, are the same.

The experimental spectrum shown in Figure Ab differs from the experimental spectrum shown in Figure Aa in the intensity of the central “peak” relative to the intensity of the “horns”. This feature could be reproduced by changing slightly the shape of the MOMD potential. Note that the spectrum shown in Figure Aa, which could not be reproduced with the models of refs (4) and (5), is similar to the spectrum shown in Figure Aa. Once again, our results highlight the fact that allowing the local potential to be rhombic is an important factor in rendering MOMD a general approach.

MOMD Analysis of Dimethyl-d6 Ammonium Tetraphenylborate; N–CD3 Dynamics

2H line shape (and relaxation) analysis has been used to study the dynamics of the ammonium cation, and a series of methylammonium cations, in their polycrystalline salts with tetraphenylborate.[44] These compounds are particularly interesting in view of the peculiar size and shape of the delocalized tetraphenylborate anion. Here we focus on (CD3)2NH2B(C6H5)4, for which temperature-dependent line shapes are given in ref (44). Figure A shows experimental spectra acquired at 245 (B), 270 (D), 275 (C), and 320 (A) K. Spectrum 8AB has been reproduced in ref (44) as a rigid-limit powder pattern with ⟨Q⟩ = 50 kHz. Spectrum 8AA has been interpreted to result from methyl exchange between two sites with an angular change of 118.4°, and ⟨Q⟩ = 45.8 kHz. Spectra 8AC and 8AD (and all of the spectra acquired between 250 and 300 K) have been interpreted as a superposition of spectra 8AA and 8AB, with temperature-dependent populations. It was concluded that there are temperature-dependent distributions of cations which do, and cations which do not, experience methyl exchange.

Figure 8

Experimental 2H line shapes from dimethyl-d6 ammonium tetraphenylborate acquired at 320 K (A), 245 K (B), 275 K (C), and 270 K (D). Spectrum B was simulated as a rigid-limit powder spectrum with ⟨Q⟩ = 50 kHz. Spectrum A was simulated with ⟨Q⟩ = 45.8 kHz and fast methyl exchange between two sites with an angular separation of 118.4°. Reproduced with permission from ref (44) (part A) Copyright 1990 Walter de Gruyter GmbH. 2H MOMD spectra that reproduce the experimental line shapes of part a, obtained for βMQ = 109.5° and the parameters depicted in the figure. The diffusion rates are given in units of 104 s–1. Additional parameters used include ⟨Q⟩ = 50 kHz and an intrinsic line width of 1 kHz. For convenience, the experimental temperatures are also depicted (part B).

The compound (CH3)2ND2B(C6H5)4 has also been studied in ref (44). The 2H spectrum obtained at 325 K has been reproduced with two-site exchange with an angular change of 103° (not shown).

MOMD reproduces the main features of the spectral evolution 8AB → 8AD → 8AC → 8AA within the scope of a single mode. The respective simulations are shown in Figure B. All of the line shapes in this figure have been obtained with βMQ = 109.5°, which corresponds to methyl exchange between two corners of the positively charged nitrogen ion tetrahedron. For the 270 K spectrum, MOMD yielded a potential given by c02 = 1.9 and c22 = 2.9 and local diffusion given by R⊥ = 1.3 × 102 s–1 and R∥ = 3.16 × 104 s–1. Increasing the temperature to 275 K implied a potential of a somewhat different form, and faster R⊥ and R∥. For the 320 K spectrum, MOMD yielded a further modified potential form, the same R⊥ and substantially faster R∥.

By plotting ln R∥ as a function of 1/T, we obtained an apparent activation energy of 8 ± 0.2 kcal/mol. Taking the ratio c22/c02 as a measure of rhombicity, one obtains 1.5, 1.5, and 1.7 at 270, 275, and 320 K, respectively. Larger potential rhombicity at higher temperature was encountered in our previous MOMD-based study of Leu side-chain dynamics in HP36.[43] We interpreted this finding in terms of two axial structural components contributing to the effective rhombic ordering tensor defined in terms of c02 and c22. At lower temperatures, one of these components partly “freezes out”, rendering the effective ordering more axially symmetric. We offer this interpretation here as well.

The 2H spectrum of (CH3)2ND2B(C6H5)4 at 325 K has been reproduced with MOMD employing the same parameters as those used to obtain Figure AA, except that R∥ has been found to be equal to 1.0 × 105 s–1 instead of 3.2 × 105 s–1 (not shown).

MOMD Analysis of 2H Line Shapes from Benzenehexa-6-hexanoate; C–D Dynamics

Benzene-n-alkanoates (where n represents the number of carbon atoms in the alkanoate chain) are compounds that exhibit extensive polymorphism in the solid state. The n = 6, 7, and 8 homologues have been studied previously with 13C and 2H line shape analysis, with the isotope labels applied at different molecular positions.[45] The various experimental spectra have been reproduced by superposing two-to-four rhombic powder patterns. Each rhombic spectrum has been interpreted as a motionally averaged spectrum ensuing from fast exchange between two side chain conformers. Exchange between rapidly exchanging pairs was considered to be slow. Temperature dependence has been interpreted to reflect changes in the relative population of the superposed rhombic spectra.

While most of the 2H line shapes presented in ref (45) exhibit features typical of multicomponent spectra (e.g., multiple pairs of “shoulders”, representing multiple “parallel” orientations in powder distributions), the phase III spectra of BHA6 2H-labeled at the β-carbon position (BHA6-βd2) exhibit features consistent with a single dynamic mode (spectrum shape, in general, and dominant spectral features, in particular, that depend on the pulse separation in the quadrupole echo sequence[45] might emerge both from a single spectrum in the slow-motional regime, as well as from two superposed fast-motional spectra).

Figure A, left column, shows experimental spectra from BHA6-βd2 acquired in phase IV at −46 °C and in phase III at −25, −12, and 13 °C. The phase IV spectrum has been reproduced as an isotropic powder pattern with 3/4×Q = 119 kHz. The phase III spectra have been interpreted in terms of two superposed rhombic powder patterns with temperature-dependent relative populations (Figure A, right column).

Figure 9

Experimental 2H line shapes from BHA6-βd2 in phase IV (−46 °C) and in phase III (−25, −12, and 13 °C) (upper left column). Calculated spectra obtained in ref (45) by superposing two rhombic powder patterns with temperature-dependent populations as delineated in the text (upper right column). Reproduced with permission from ref (45) (part A). 2H MOMD spectra reproducing the experimental line shapes shown in part A, left column, obtained for the parameters depicted in the figure. They include the best-fit values of the potential coefficients c02 and c22; the best-fit values of the diffusion rates, R⊥ and R∥ given in units of 104 s–1; and the fixed angles βMQ. Additional parameters used include 3/4×Q = 119 kHz and an intrinsic line width of 1 kHz. For convenience, the experimental temperatures (in °C) are also depicted (part B).

The experimental spectra shown in Figure A, left column, have been reproduced with MOMD in terms of a single dynamic mode characterized by rhombic local potential and axial local diffusion (Figure B). 3/4×Q = 119 kHz from the −46 °C phase IV spectrum has been used. The strength of the local potential, estimated by c02, decreases with increasing temperature. The rhombicity of the potential, as evaluated by the ratio c22/c02, is 1.4, 1.4, and 1.7 at −25, −12, and 13 °C, respectively. We offered above a general qualitative interpretation to increasing potential rhombicity with increasing temperature: two axial contributions to the local structural restrictions at higher temperatures with one of them partly “freezing out” at lower temperatures. We offer this interpretation here as well.

The perpendicular motional rate, R⊥, is relatively slow, however, with a sizable effect on the analysis. As expected, it increases with increasing temperature, ranging from 1.6 × 102 s–1 at −25 °C to 2.5 × 102 s–1 at 13 °C. An Arrhenius-type activation energy of 1.70 ± 0.24 kcal/mol has been calculated for it. R∥ increases from 2.0 × 104 s–1 at −25 °C to 7.9 × 104 s–1 at 13 °C, and an Arrhenius-type activation energy of 5.05 ± 0.71 kcal/mol has been calculated for this component. The angle βMQ was found to be 100.5, 112.5, and 120.5° at −25, −12, and 13 °C. This is the only example where the angle βMQ had to be adjusted as a function of temperature. This might reflect changes in the nature of the axial contributions to the local spatial restrictions (see above), in addition to their temperature-dependence.

Thus, we found with MOMD analysis that the temperature-dependent spectra of BHA6-βd2 in phase III can be interpreted in terms of a single dynamic mode (with small variations in the angle βMQ). Motional rates, associated activation energies, and the strength and rhombicity of the local potential at the 2H-labeled sites were determined. This picture is to be compared with two independent exchange processes in their extreme motional narrowing limit, having temperature-dependent relative populations.

The following comments are in order. (1) We have treated in this study C–D, C–CD3, and N–CD3 dynamics. Within a good approximation (maximum deviation 9.5°), the main diffusion axes may be identified with relevant preceding chemical bonds. (2) The MOMD-based reproduction of some of the experimental spectra considered is not perfect. However, our objective has been to determine whether key features of dynamic 2H line shapes from very different systems and experimental conditions can by reproduced reasonably well with the same set of general and physically relevant parameters. In general, this goal has been reached. Work aimed at improving the agreement between theory and experiment without overfitting the experimental data is in progress.

Conclusions

Quite a few systems (six in total) differing in nature have been subjected to MOMD analysis. In all of these cases, the local potential, given in units of kBT, was found to be weak. Typical values of the axial coefficient, c02, are 2.1–2.2. A particularly weak potential is associated with BHA6-βd2 in phase III, where c02 = 1.8 at 13 °C. This might indicate intermolecular interactions that are weaker than typically encountered in solids. The homologues BHA7 and BHA8 form a discotic phase at higher temperatures; our results suggest an inherent tendency of BHA6 to also do so.

For the C–CD3 Leu probe, the axial potential coefficient, c02, varies at room temperature from 2.0 to 2.2 for both HP36, and LKα14 adsorbed onto a PS surface. Thus, the strength of the local potential is similar for these two systems. On the other hand, the rhombic potential coefficient, c22, varies from 2.0 to 3.4 for HP36 and from 2.2 to 2.6 for LKα14 adsorbed onto a PS surface. This is pointing to greater uniformity in the shape of the local potential at the Leu side chains of the surface-adsorbed peptide (note that we compare sets of four cases associated with HP36 with four cases associated with LKα14).

The furanose ring of HhaI methyltransferase target DNA, the methylammonium and dimethylammonium cations, and BHA6-βd2 in phase III, exhibit relatively larger deviations of the local potential at the 2H site from axiality as compared to HP36 and LKα14.

The diffusion rate, R⊥, is generally small but not negligible. For the furanose ring of HhaI methyltransferase target DNA, R⊥ is quite large (7.9 × 102 s–1), in agreement with this outstandingly mobile structural element (cf. ref (30)).

The diffusion rate R∥ ranges from 1.3 × 104 to 7.9 × 104 s–1. The methylammonium and dimethylammonium cations and BHA6-βd2 in phase III exhibit relatively large R∥ values. They also exhibit relatively large apparent activation energies: 8 ± 0.2 kcal/mol for the DMA cation and 5.05 ± 0.7 kcal/mol for BHA6-βd2 in phase III.

Two experimental line shapes−the 2H spectrum of Met73 in Streptomyces Subtilisin Inhibitor[41] and the 2H spectrum of lyophilized LKα14 labeled at position 8 (see above)−could not be reproduced with the multi-simple-mode paradigm. They have been reproduced with MOMD.

The agreement between corresponding experimental and MOMD-derived 2H line shapes can be improved in several ways. One could vary the angle αMQ in addition to the angle βMQ, include the L = 1 and L = 3 spherical harmonics in the expression for the local potential, and/or devise additional potential forms. All of these enhancements are associated with lowering the symmetry of the physical quantities entering the model. However, to fit such detailed models would require extensive sets of temperature-dependent 2H spectra.

Prospects include the extension of MOMD so that it applies to 2H spectra recorded under MAS (in progress), and developing MOMD for 15N and 13C nuclei. Concerted SRLS analysis of NMR relaxation in solution and MOMD analysis of NMR line shapes in the solid state for the same system is an important objective.