Equations describing relaxation rates of 15N nuclei relaxing by dipolar and chemical shift anisotropy mechanisms in terms of spectral density functions are given as (Abragam 1989; Korzhnev et al. 2001):
$${R_1}=\frac{1}{4}{d^2}\left[ {J\left( {{\omega _H} - {\omega _N}} \right)+3J\left( {{\omega _N}} \right)+6J\left( {{\omega _H}+{\omega _N}} \right)} \right]+\frac{1}{3}{c^2}J\left( {{\omega _N}} \right)$$
(1)
$${R_2}=\frac{1}{8}{d^2}\left[ {4J\left( 0 \right)+J\left( {{\omega _H} - {\omega _N}} \right)+3J\left( {{\omega _N}} \right)+6J\left( {{\omega _H}} \right)+6J\left( {{\omega _H}+{\omega _N}} \right)} \right]+\frac{1}{{18}}{c^2}\left[ {4J(0)+3J({\omega _N})} \right]+{R_{ex}}$$
(2)
where \(d=\frac{{{\mu _0}}}{{8{\pi ^2}}}\frac{{{\gamma _N}{\gamma _H}h}}{{\left\langle {r_{{NH}}^{3}} \right\rangle }},\quad c={\omega _N}\Delta \sigma\) and other symbols have their usual meaning. It has to be mentioned that in all calculations the vibrationally averaged N–H distance, rNH = 1.04 Å (Ottiger and Bax 1998) and the chemical shift anisotropy of the 15N chemical shift tensor Δδ = − 170 ppm (Yao et al. 2010) were used.
The conformational exchange contribution to the transverse relaxation rate, R
ex
, is proportional to the square of the 15N Larmor frequency, ω
N
. This term can be written as \({R_{ex}}=\Phi \omega _{N}^{2}\) (Peng and Wagner 1995). The proportionality factor Φ represents the effectiveness of conformational exchange processes and is independent on magnetic field strength facilitating direct comparison of chemical exchange terms determined at different magnetic field strengths for different proteins.
Model-free approach spectral density function takes the form (Lipari and Szabo 1982):
$$J\left( \omega \right)=\frac{2}{5}\left[ {\frac{{{S^2}{\tau _R}}}{{1+{{\left( {\omega {\tau _R}} \right)}^2}}}+\frac{{\left( {1 - {S^2}} \right)\tau }}{{1+{{\left( {\omega \tau } \right)}^2}}}} \right]$$
(3)
where \({\tau ^{ - 1}}=\tau _{R}^{{ - 1}}+\tau _{{\operatorname{int} }}^{{ - 1}}\). Performing the complete MFA analysis of relaxation data for N-residue protein one has to determine 3N local parameters, S2, τint, and R
ex
for each residue. Additionally one global parameter τ
R
or six parameters characterizing either isotropic or fully anisotropic overall tumbling, respectively, have to be determined. Extension of the spectral density function for isotropic motion (Eq. 3) to the anisotropic one, based on the formalism developed by Woessner (1962), was implemented to the protein relaxation studies (Tjandra et al. 1995; Baber et al. 2001). Allowing for the positive degree of freedom of a computational task it means that besides R1, R2, and 15N-{1H} NOE at a single magnetic field, at least one additional set of relaxation parameters has to be measured. It happens, however, that the number of available relaxation data is insufficient, due to sample instability or lack of experimental time and additional data processing has to be applied (Jaremko et al. 2014). Often only R1 and R2 relaxation rates at a single magnetic field are at one’s disposal.
A joined analysis of Q = R2/R1 and D = 2R2 − R1 or P = R1R2 values allows obtaining semi-quantitative insight into the protein dynamics owing to the different relations of these quantities to the MFA parameters and, therefore, untangling these parameters from experimental data. The Q parameter is quasi-insensitive to both local MFA parameters, S2 and τint, in a reasonably broad range of their values (Fig. 1A, B). Therefore, it is well suited for the evaluation of the overall tumbling correlation times of proteins comprising residues with diverse local mobility. On the other hand, the P values are quasi-insensitive to τint but decrease considerably with the increased amplitude of local motions, as manifested at smaller values of the S2 order parameter (Fig. 1B). The D parameter is even less sensitive to τint changes than Q and P, but it displays a modest sensitivity to S2 changes. All three quantities, Q, D, and P, are sensitive to the chemical exchange term and increase with the R
ex
enlargement (Fig. 1C). Simultaneous effect of S2 and τint, changes on Q, P and D is shown in Figs. S1–S3 (Supporting Information). One has to be aware of the opposite effects of fast (ps–ns) and slow (µs–ms) motions on the P values. Both these effects can compensate one another leaving the P value unchanged and, thus, hiding chemical exchange effect. The D values are also sensitive to such compensation. They are, however, less sensitive to fast motions and more sensitive to slow ones than P values and, therefore, should retain at least partially the ability of detection of R
ex
terms.
Use of Q, D, and P values in the analysis of a backbone protein dynamics requires several simplifying assumptions bearing a number of consequences. It has been noticed (Peng and Wagner 1992) that the spectral density functions at three highest frequencies J(ω
H
+ ω
N
), J(ω
H
), and J(ω
H
− ω
N
) are only a small fraction of two other component J(0) and J(ω
N
) and can be neglected in Eqs. (1) and (2) describing D values (Habazettl and Wagner 1995) or P values (Kneller et al. 2002). As a result following expressions can be written:
$$Q=\frac{2}{3}\frac{{J(0)}}{{J({\omega _N})}}+\frac{1}{2}$$
(4a)
$$P={A_2}J(0)J({\omega _N})+{A_3}{[J({\omega _N})]^2}$$
(4c)
$${\text{where }}{A_1}=\left( {{d^2}+\frac{4}{9}{c^2}} \right);{\text{ }}{A_2}=6{\left( {\frac{{{d^2}}}{4}+\frac{{{c^2}}}{9}} \right)^2};{\text{ }}{A_3}=\frac{1}{2}{\left( {\frac{3}{4}{d^2}+\frac{1}{3}{c^2}} \right)^2}$$
In the approach utilizing the Q ratio for the estimation of global correlation time, the assumption τint = 0.0 is made resulting in a simplified spectral density function (Kay et al. 1989).
$$J(\omega )=\frac{2}{5}\left[ {\frac{{{S^2}{\tau _R}}}{{1+{{(\omega {\tau _R})}^2}}}} \right]$$
(5)
In order to obtain so estimated global correlation time, τ
R
(Q), one has to compute it from Eq. (8) given by Kay et al. (1989). Additionally, neglecting second term in Eq. (4c) and assuming \({({\omega _N}{\tau _R})^2}>>1\) one obtains:
$$Q=\frac{2}{3}{\left( {{\omega _N}{\tau _R}} \right)^2}+\frac{1}{2}$$
(6a)
$$D={A_1}{S^2}{\tau _R}$$
(6b)
$$P={A_2}{\left( {\frac{{{S^2}}}{{{\omega _N}}}} \right)^2}$$
(6c)
Use of the Q values in the evaluation of an overall correlation time results in the τR underestimation provided the overall tumbling is isotropic (Korzhnev et al. 1997). Influence of the input τR and magnetic field strength values on the value of the apparent τR is demonstrated in Fig. 2. In the utmost situations (parts of plots below the dashed line in the Fig. 2 corresponding to intense internal motion: S2 = 0.7, τint = 100 ps, slow overall tumbling: τR = 32 ns and very high magnetic field: 23.5 T) the τR evaluation derived from the Q values breaks down; relative errors exceed 25%. Use of Q values retains the sense only if correlation time of internal motion, τint, is short and its amplitude small (Fig. S4).
In the case of anisotropic tumbling, the determined value of the orientation averaged overall correlation time τR = 0.5/(D1 + D2 + D3), can be either larger or smaller than the τR value estimated from Q values, depending on the orientation of the N–H vector. Appropriate comparison is presented in Table 1 and Fig. 3.
Table 1 Anisotropic tumbling visibly influences on the Q and D values, while its effect is strongly attenuated regarding P values, with variability ranges 25, 19, and 3%, respectively
Estimation of the average generalized order parameters \(S_{{av}}^{2}\) from the experimentally observed P values was proposed by Kneller et al. (2002) using formula:
$$S_{{av}}^{2}=\sqrt {\frac{{\left\langle P \right\rangle }}{{{P_{\hbox{max} }}}}} ,$$
where \(\left\langle P \right\rangle\) is experimentally observed 10% trimmed mean value and Pmax is determined from the relaxation parameters calculated for a rigid molecule (S2 = 1.0, R
ex
= 0.0) which reorients with τR(Q). This formula results directly from Eq. (6c). Use of medians is superior to the trimmed mean values since the distributions of Q, D, and P data are most commonly non gaussian (Table S1) and robust statistics has to be used in their description (Maronna et al. 2006). Medians allow not only avoiding influence of outliers but also eliminating residues from the unstructured segments of protein characterized by inherently small Q values and resulting in extremely skewed Q distributions. Robust statistics also facilitates identifying residues undergoing chemical exchange as outliers in Q, D, and P sets. In the following text Q, D, and P medians are solely used and denoted as \(\tilde {Q}\), \(\tilde {D}\), and \(\tilde {P}\). Therefore, the Kneller et al. formula is rewritten as
$$S_{{av}}^{2}=\sqrt {\frac{{\tilde {P}}}{{{P_{\hbox{max} }}}}}$$
(7)
This approach is also extended to the estimation of site specific generalized order parameters, \(S_{i}^{2}\) utilizing either D or P values:
$$S_{i}^{2}=\frac{{{D_i}}}{{{D_{\hbox{max} }}}}$$
(8a)
$$S_{i}^{2}=\sqrt {\frac{{{P_i}}}{{{P_{\hbox{max} }}}}}$$
(8b)
One has to be aware of possible systematic deviations of so estimated \(S_{i}^{2}\) values. As it was shown earlier, the Q-derived τR values are underestimated in isotropically tumbling molecules. Accordingly, the Pmax is underestimated as well, resulting in the overestimation of \(S_{i}^{2}\) values (Fig. 4). This effect is especially pronounced for slower internal motions (long τint) with large amplitudes (small S2) at high magnetic fields. A similar effect was reported for the relaxation data in ATPase α-domain (Gu et al. 2016).
It is stated that the Q values do not distinguish between the effects of motional anisotropy and chemical exchange (Kneller et al. 2002), while the analysis of P data significantly attenuates the effects of motional anisotropy (c.f. Table 1) permitting rapid identification of residues undergoing chemical exchange, R
ex
. It will be shown in the next section, however, that the attenuation of motional anisotropy is not sufficient to identify unequivocally the R
ex
influenced residues. The elevated P values not always allow identifying residues affected by chemical exchange. In fact, only simultaneous outlying Q, D, and P values point out unequivocally to the chemical exchange.