Side chain and methyl dynamics

Similar to backbone motions (section 4.1.4.1, page 172), analytical functions for the interpreta-tion of the model-independent generalized order parameterS²of side chains and methyls were introduced. In the Woessner model [Woessner,1962a], the H,C bond vector of a methyl group freely diffuses about a fixed axis, which yields an order parameter

S_Woessner² = [P₂(cosβ)]², (4.6)

withβ being the angle between the bond vector and the rotation axis andP₂(cosβ)the second Legendre polynomial ¹₂ 3 cos²β−1

[Lipari and Szabo, 1982a,b]. The same equation was derived, assuming rotameric jumps of the methyl group among three equivalent sites.

For an ideal tetrahedral geometry, β =180^◦−109.5^◦ =70.5^◦, yielding S²_Woessner =0.111.

Therefore, the generalized order parameter for a methyl group is given by

S²=S²_axis×S²_Woessner=0.111×S²_axis, (4.7)

whereS²_axis is the order parameter for the motion of the rotation axis of the methyl group. As before, the generalized order parameter S² was determined as the ratio of the experimentally obtained dipolar coupling anisotropy to the rigid-limit value. The here employed rigid-limit

1H,¹³C bond length for methyl groups was 1.115 Å [Henry and Szabo, 1985, Ishima et al., 2001].

Furthermore, S²_axis in equation 4.7 can be linked to a motional model, as the diffusion-in-the-cone model, described in the previous section (cf. Figure 4.5, page 176). Alternatively, the restricted diffusion model for side chain motions can be employed [London and Avitabile, 1978,Wittebort and Szabo,1978,Lipari and Szabo,1982b,Nicholson et al.,1992,Engelke and Ruterjans,1998,Daley and Sykes,2004], as rotations about the preceeding axis also reorientates the methyl rotation axis. For example, as depicted by the structural model in Figure 4.6D for valine, three different rotameric states are illustrated, which are yielded byχ1rotations about the Cα,Cβbond vector. However, the sterical hindrance in context of a globular protein reduces the possible motional range, which is especially pronounced for longer chain amino acids [Hanssum and Ruterjans,1983], as isoleucine and methionine. The order parameter for restricted diffusion, S²_rest, was derived to be [Lipari and Szabo,1982b]

S²_rest= [P2(cosβ0)]²+3 sin²β0sin²γ0

whereβ₀ is the angle between the restricted reorientation axis and the methyl rotation axis, in the case of valine, between the Cα,Cβand the Cβ,Cγ1/Cγ2 bond vector, respectively. ±γ₀ defines the angular range of rotation about the restricted reorientation axis (Cα,Cβfor valine) and is the measure for side chain motion.β₀, however, is a fixed angle and equal to 70.5^◦for all methyl-bearing amino acids, except for methionine, whereβ₀is 80^◦. For both cases,β₀=70.5^◦ (black) andβ₀=80^◦(red),S²_restwas plotted as a function ofγ₀(Figure 4.6A).

Figure4.6:Motionalmodelsforsidechainandmethyldynamics.(A)Thesquaredorderparameter,assumingtherestricteddiffusionmodel,wasplottedasafunction oftheangularreorientationγ0,usingequation4.8(page182).(B)Theorderparameterforafreerotationaboutthesidechainχ1,χ2,χ3,χ4dihedralanglesofalysine moleculeisshown(equation4.9,page184).βwassetto70.5◦,assumingidealtetrahedralgeometry.Theredcurvedepictstheexperimentallydeterminedorder parameterforLys60ofα-spectrinSH3,usinga15%RAPsample.(C)Two-and(D)three-sitejumpmodel,whichlinkstheorderparameterandtheasymmetry rotamericpopulations[Schandaetal.,2011a].Forstructuralillustration,threerotamericstatesofvalineareshown.For(D)theconditionsp1≥p2≥p3and∑ipi= wereassumed.TheanalyticalexpressionsforbothmodelswerekindlyprovidedbyDr.PaulSchanda(IBS,Grenoble).Thefiguresin(D)wereadaptedfromSchanda etal.[2011a],however,here,thesquaredorderparameterwasplotted.

Free rotation corresponds to an angular range of γ₀=±180^◦, therefore, yielding the min-imum value of the order parameter, S²_rest,min= [P₂(cosβ₀)]² (0.111 and 0.207 for β₀ equal to 70.5^◦and 80^◦). As pointed out byLipari and Szabo[1982b], this theoretical minimum ofS²_rest can be used as the definition of a lower limit of the generalized order parameter for restricted diffusion. That way, if the experimentally determined generalized order parameterS²is smaller thanS_rest,min² ×S²_Woessner= [P2(cosβ0)]²×P₂(cosβ), then the restricted diffusion model is not sufficient to describe the present side chain dynamics.

In principle, if the reorientation about the successive side chain rotation axes are not sterically hindered, then the order parameterS²_nfor thenth carbon can be described by [Woessner,1962a, Wallach,1967,Lipari and Szabo,1982b]

S²_n= [P₂(cosβ)]²ⁿ. (4.9)

Figure 4.6B depicts the order parameterS²_nas a function of the side chain positionn, expected for a lysine residue, wheren=0,1,2,3,4 means theα,β,γ,δandposition, respectively. The order parameter decreases while moving along the side chain from the backbone to the terminal carbon. This trend was also observed in MD simulations [Levy et al.,1981b,Best et al.,2005].

In the present thesis, side chain and methyl motions were investigated by REDOR dephasing experiments, employing the RAP labeling scheme. A representative section of experimentally derived side chain and methyl order parameters and asymmetries is shown in Figure 4.7, next to the backbone values, where available. All employed values are given in the appendix (Table 4, page 220, Table 5, page 221, Table 6, page 223, Table 7, page 224).

The here depicted backbone and side chain dynamics parameters (Figure 4.7) were obtained, using a 15% RAP sample ofα-spectrin SH3. For the methyl parameters two different samples were employed. For valine and leucine methyl parameters, a selectively Val/Leu methyl-labeled

13CD2H sample ofα-spectrin SH3 was used, due to the high sensitivity, while the alanine and threonine methyl parameters shown in Figure 4.7 were determined, employing a 5% RAP sam-ple. It should be noted, that, as compared to selectively methyl labeled samples, the RAP label-ing enables the access to all aliphatic sites, includlabel-ing all types of methyl groups (Alaβ, Ileγ2, Ileδ1, Leuδ1/2, Met, Thrγ2, Valγ1/2, cf. Table 7, page 224).

In principle, the methyl proton concentration for the CD2H sample was expected to be higher than for the 5% RAP sample, especially within the hydrophobic protein core, which typi-cally shows the highest methyl density. In the¹³CD2H sample either one of the methyl groups of valine and leucine residues is protonated, yielding an average proton concentration of about 17%

for any Valγor Leuδmoiety (according to equation 3.6, page 37), however, in a 5% RAP labeled sample, the methyl concentration was shown to be on the order of 1-2% (cf. Table 3.2, page 38).

As reported recently [Schanda et al.,2011a], the REDOR sequence is particularly sensitive to remote protons (vide supra). Therefore, we compared order parameters and asymmetries ob-tained from the¹³CD2H sample to the 5% RAP sample ofα-spectrin SH3, and yielded the same values within the experimental error. In Figure 4.7, instead of the generalized order parameter, the squared order parameter of the methyl symmetry axis,S²_axis, is plotted, according to equation 4.7 (page 182).

Alanine residues are particularly interesting in terms of dynamics, as the methyl group directly relates to backbone motions, since it is directly bonded to the Cαatom [Henry et al.,1986]. In this manner, the spectroscopic advantages of methyl groups, as high sensitivity and resolution, can be exploited to probe the backbone dynamics with improved experimental accuracy and precision. As shown in Figure 4.7, the order parameter for the backbone Ala-¹³Cα and the adjacent ¹³Cβ are, as expected, rather similar, for all alanine residues. The slightly smaller order parameter for the¹Hβ,¹³Cβdipole tensor of alanine is indicative of additional librational motion of the ¹³Cα,¹³Cβmethyl rotation axis. Except for Val23, Val46 and Leu31, all methyl groups in Figure 4.7 show small motional amplitudes. This is also hinted by small asymmetry parameters.

The asymmetry parameter contains information about the anisotropy of the side chain motion on the sub-microsecond timescale. Analytical equations to link the methyl order and asymmetry parameter to a motional model were given bySchanda et al.[2011a]. To determine an expres-sion for rotameric populations as a function of dipole anisotropy and asymmetry, two (three) dipolar tensors were weighted by p₁andp₂=1−p₁(p1, p₂andp₃=1−p₁−p₂) and rotated by 120^◦ to each other to simulate a two-site (three-site) jump scenario. Prior to rotation, all tensors were tilted by the tetrahedral angle 109.5^◦. The principal components (eigenvalues) of

Figure 4.7:Experimentally derived¹H,¹³C order parameters and asymmetries for backbone, side chain and methyl resonances ofα-spectrin SH3. For the backbone and side chain dynamics parameters, a 15% RAP sample was employed. The parameters for valine and leucine methyl groups were obtained from a selectively Val/Leu methyl-labeled¹³CD₂H sample ofα-spectrin SH3, while alanine and threonine parameters were obtained, using a 5% RAP sample, respectively. Valγ1/2, Leuδ1/2, Lysβ1/2, Lysγ1/2S² andηvalues were averaged. The error bar for the average value ¯xwas determined as∆x¯=σx+¹₂p

(∆x1)²+ (∆x2)², whereσxis the standard deviation of the values x1andx2and∆x1and∆x2are the experimental uncertainties. For methyl resonances, the squared order parameter of the methyl rotation axis,S²_axis, is plotted, according to equation 4.7 (page 182). A¹H,¹³C methyl bond length of 1.115 Å [Henry and Szabo,1985,Ishima et al.,2001] was assumed. All employed values are given in the appendix