Analysis of NMR Spin-Relaxation Data Using an Inverse Gaussian Distribution Function

Abstract

Spin relaxation in solution-state NMR spectroscopy is a powerful approach to explore the conformational dynamics of biological macromolecules. Probability distribution functions for overall or internal correlation times have been used previously to model spectral density functions central to spin-relaxation theory. Applications to biological macromolecules rely on transverse relaxation rate constants, and when studying nanosecond timescale motions, sampling at ultralow frequencies is often necessary. Consequently, appropriate distribution functions necessitate spectral density functions that are accurate and convergent as frequencies approach zero. In this work, the inverse Gaussian probability distribution function is derived from general properties of spectral density functions at low and high frequencies for macromolecules in solution, using the principle of maximal entropy. This normalized distribution function is first used to calculate the correlation function, followed by the spectral density function. The resulting model-free spectral density functions are finite at a frequency of zero and can be used to describe distributions of either overall or internal correlation times using the model-free ansatz. To validate the approach, ¹⁵N spin-relaxation data for the bZip transcription factor domain of the Saccharomyces cerevisiae protein GCN4, in the absence of cognate DNA, were analyzed using the inverse Gaussian probability distribution for intramolecular correlation times. The results extend previous models for the conformational dynamics of the intrinsically disordered, DNA-binding region of the bZip transcription factor domain.

Introduction

Conformational dynamics of proteins and other biological macromolecules play substantial roles in functions such as allostery, catalysis, molecular recognition, regulation, and signaling (1, 2, 3). NMR spectroscopy is often employed to experimentally explore the dynamic and kinetic properties of biological macromolecules. Spin-relaxation experiments probe motions with atomic resolution over a broad range of timescales (4, 5). In addition, multiple methods and applications for ¹H, ²H, ¹³C, and ¹⁵N nuclei in proteins and nucleic acids currently exist that make such explorations possible (5, 6, 7). As systems with increasingly complex dynamical properties and as experimental methods become more powerful, correspondingly, more sophisticated approaches are necessary for analysis of data and linkage to theoretical or computational motional models (7, 8, 9).

Probability distributions for correlation times have long been used to model autocorrelation functions, C(t), and spectral density functions, J(ω), for NMR spin relaxation experiments in solution, particularly in flexible polymers (10, 11, 12). As examples of applications to biological macromolecules, the Cole-Cole distribution was employed by Buevich and c-oworkers in an analysis of relaxation data for the highly disordered S-peptide (13), and the Mittag-Leffler distribution, arising from a fractional diffusion equation, was employed by Abergel and co-workers to model intramolecular motions in proteins (14, 15, 16, 17). However, many of the aforementioned distribution functions, such as the Fuoss-Kirkwood, Cole-Cole, and Mittag-Leffler models, yield spectral density functions that diverge as the frequency ω approaches zero (12). Such functions are not straightforward to apply to experimental data sets that include transverse (spin-spin) relaxation or other measurements, such as relaxometry (18), that depend on J(ω) at frequencies equal or near zero, unless another exponentially decaying process is assumed that reduces the associated autocorrelation function to zero sufficiently rapidly. For example, Abergel and co-workers were able to model intramolecular dynamics for folded proteins using the Mittag-Leffler distribution function because the truncation from overall rotational diffusion ensured convergence at J(0). In contrast, Buevich and co-workers, in a more disordered system, could not rely on overall rotational diffusion for truncation, and thus J(0) ≈ J(ω = 1 s⁻¹) was chosen arbitrarily to avoid divergence.

Alternatively, autocorrelation functions can be modeled as sums of exponential decays, essentially modeling the probability distributions as sums of δ functions and yielding spectral density functions that are sums of Lorentzian terms. In this vein, Modig and Poulsen estimated the moments of the underlying distribution of correlation times after fitting a sum of exponential functions to experimental data (19), whereas Ferrage and co-workers fitted a sum of exponential functions, fixing the spacing of time constants to reduce the number of optimizable parameters (20). Increasingly accurate molecular dynamics simulations have also been used to develop and validate models for both distributions and sums of exponential processes (8, 15, 21, 22, 23, 24).

This work derives the inverse Gaussian probability distribution from general properties of spectral density functions at low and high frequencies for macromolecules, in conjunction with the principle of maximal entropy (25). Importantly, the maximal entropy approach constrains the spectral density function to be convergent at ω = 0, facilitating the application to biological macromolecular systems. Schrödinger originally derived the inverse Gaussian distribution to characterize first-passage times in Brownian motion (26). The theory and application of the inverse Gaussian distribution have been reviewed by Folks and Chhikara (27).

The inverse Gaussian spectral density formalism is applied to a set of backbone ¹⁵N relaxation data acquired for the bZip (DNA-binding) domain of the Saccharomyces cerevisiae protein GCN4 under the assumption that the inverse Gaussian probability distribution represents the distribution of intramolecular correlation times for the reorientation of the backbone amide N-H bond vector in a local molecular frame of reference. The yeast transcription factor GCN4 (28), which has homologs in mammals, is an example of a protein with extensive intrinsically disordered regions (IDRs) under certain conditions (29, 30, 31). The bZip domain, when attached to DNA substrate, contains a stable, highly basic, N-terminal helical region that inserts into the DNA major groove and a C-terminal region that forms a leucine zipper dimer (32, 33). In the absence of DNA, the N-terminal region becomes partially disordered with significant residual helicity, whereas the C-terminal leucine zipper remains ordered (34, 35, 36, 37). Results obtained by the application of the inverse Gaussian probability distribution provide additional insights into the flexibility of the IDR in GCN4.

Methods

The autocorrelation function for a continuous distribution of exponential terms is given by (38, 39)

$$C\left(t\right)={\int }_0^{\infty }p\left(\tau \right)e^{-\frac{t}{\tau }}d\tau ,$$ (1)

in which p(τ) is the probability density function for correlation times τ. The spectral density function important in NMR spin relaxation theory is given by (38, 39)

$$\begin{array}{c}J\left(\omega \right)=\frac{2}{5}Re\{{\int }_0^{\infty }{e}^{-i\omega t}{\int }_0^{\infty }p\left(\tau \right)e^{-\frac{t}{\tau }}d\tau dt\}\\ =\frac{2}{5}Re\{{\int }_0^{\infty }p\left(\tau \right){\int }_0^{\infty }{e}^{-i\omega t}{e}^{-\frac{t}{\tau }}dtd\tau \}.\\ =\frac{2}{5}{\int }_0^{\infty }p\left(\tau \right)\frac{\tau }{1+{\omega }^2{\tau }^2}d\tau \\ =\frac{2}{5}Re\{\underset{0}{\overset{\infty }{\int }}p\left(\tau \right)\underset{0}{\overset{\infty }{\int }}{e}^{-i\omega t}{e}^{-\frac{t}{\tau }}dtd\tau \}\end{array}$$ (2)

The second step is obtained by exchanging the order of integration. When ωτ ≪ 1,

$$J\left(\omega \right)\approx \frac{2}{5}{\int }_0^{\infty }p\left(\tau \right)\tau d\tau =\frac{2}{5}\langle \tau \rangle ,$$ (3)

and when ωτ ≫ 1,

$$J\left(\omega \right)\approx \frac{2}{5}{\int }_0^{\infty }p\left(\tau \right)\frac{1}{{\omega }^2\tau }d\tau =\frac{2}{5{\omega }^2}\langle \frac{1}{\tau }\rangle .$$ (4)

These two limits are encountered most frequently in spin-relaxation measurements of biological macromolecules performed at high static magnetic fields. For example, investigations of ¹⁵N relaxation in amide moieties in proteins or imino moieties in nucleic acids include measurements of the ¹⁵N R₁ and R₂ autorelaxation rate constants and the ¹H-¹⁵N heteronuclear cross-relaxation rate constant σ_NH. These relaxation rate constants depend on the ¹H-¹⁵N dipole-dipole interaction with directly attached ¹H spins and on the ¹⁵N chemical shift anisotropy. They are also dependent on the values of the spectral density function at frequencies ω = 0, ω_N, ω_H, and ω_H ± ω_N (40, 41, 42). Consequently, ω_Nτ > 1 and ω_Hτ > 1 for τ > 0.26 ns at a static magnetic field of 14.1 T (600 MHz).

The original model-free formalism of Lipari and Szabo parameterizes the model spectral density function using the mean correlation time <τ> (38, 39). This choice gives a one-parameter exponential Padé approximation of the correlation function (43, 44). The above considerations suggest that a two-parameter approximation should be based on both <τ> and <1/τ>. This idea can be made more rigorous by defining p(τ) using the maximal entropy principle with the constraints <τ> and <1/τ> codified as Lagrange multipliers, the result of which gives (25)

$$p\left(\tau \right)=c{e}^{{\lambda }_1\tau +\frac{{\lambda }_2}{\tau }}.$$ (5)

c, λ₁, and λ₂ are constants determined by the following requirements:

$$\begin{array}{c}1={\int }_0^{\infty }p\left(\tau \right)d\tau \\ \tau ={\int }_0^{\infty }p\left(\tau \right)\tau d\tau .\\ \frac{1}{\tau }={\int }_0^{\infty }\frac{p\left(\tau \right)}{\tau }d\tau \end{array}$$ (6)

The family of generalized inverse Gaussian probability distributions satisfies the functional normalized form of Eq. 5:

$$p_n\left(\tau \right)=\frac{{\mu }^{-n}{\tau }^{n-1}}{2K_n\left(\lambda /\mu \right)}{e}^{-\frac{\lambda \left({\tau }^2+{\mu }^2\right)}{2{\mu }^2\tau }},$$ (7)

in which λ₁ = −λ/(2μ²), λ₂ = −λ/2, which can be identified as the coefficients of τ and 1/τ in the exponential function, and K_n(x) is the modified Bessel function of the second kind (Basset or Macdonald function) (45, 46). The prefactor to the exponential arises from the constraint of a normalized distribution. These distributions have similarly skewed shapes in which μ is a location parameter and λ is a scale parameter. The choice of n = −1/2 is made herein for three reasons: 1) in this case, μ = <τ> and is thus independent of the scale parameter, 2) the distribution decays more rapidly at large values of τ rather than other simple choices such as n = 0 or 1, and 3) the resulting distribution function is more tractable mathematically because K_−1/2(x) = K_1/2(x) = (π/(2x))^1/2e^−x. The normalized probability distribution resulting from evaluating Eq. 7 for when n = −1/2 is called the inverse Gaussian probability density function:

$$p\left(\tau \right)={\left(\frac{\lambda }{2\pi {\tau }^3}\right)}^{1/2}{e}^{-\frac{\lambda {\left(\tau -\mu \right)}^2}{2{\mu }^2\tau }}={\left(\frac{{\mu }^3}{2\pi {\sigma }^2{\tau }^3}\right)}^{1/2}{e}^{-\frac{\mu {\left(\tau -\mu \right)}^2}{2{\sigma }^2\tau }},$$ (8)

in which, once again, μ = <τ> is the mean correlation time and the variance of the distribution is σ² = μ³/λ. Note that <1/τ> <τ> = <μ/τ> = 1 + μ/λ = 1 + σ²/μ² ≥ 1 with equality being approached as σ → 0, and the distribution becomes a δ function centered around μ (yielding a single exponential correlation function). Thus, <1/τ> <τ> > 1 indicates that the correlation function is multiexponential and is used in some cases herein as a convenient representation of the breadth of the distribution (rather than using σ itself). The mode of the distribution is given by

$${\tau }_{mode}=\frac{1}{2\mu }\left[-3{\sigma }^2+{\left(9{\sigma }^4+4{\mu }^4\right)}^{\frac{1}{2}}\right]$$ (9)

and approaches μ as <1/τ> <τ> → 1 (or σ/μ → 0) or 0 as <1/τ> <τ> → ∞ (or σ/μ → ∞). Examples of the inverse Gaussian probability distribution are shown in Fig. 1 a.

Figure 1.

Inverse Gaussian. For all figures, the lines are all dependent on the parameter <1/τ><τ>, which is equal to 5 (dashed, purple), 2 (dashed-dotted, green), and 1.1 (dotted, orange). (a) Normalized inverse Gaussian probability distribution functions p(τ). (b) Normalized correlation functions C_IG(τ). The black line gives the single exponential correlation function (the limiting case as <1/τ> <τ> → 1). (c) Spectral density functions J_IG(ω). The black line gives the single Lorentzian spectral density function (the limiting case as <1/τ><τ> → 1). To see this figure in color, go online.

The autocorrelation function is obtained by integrating over the probability distribution to yield

$$C_{IG}\left(t\right)=\underset{0}{\overset{\infty }{\int }}p\left(\tau \right)e^{-\frac{t}{\tau }}d\tau =e^{-\frac{\lambda X}{\mu }}/\left(X+1\right)=e^{-\frac{{\mu }^{2}X}{{\sigma }^2}}/\left(X+1\right),$$ (10)

in which X = (1 + 2t/λ)^1/2 − 1 = (1 + 2tσ²/μ³)^1/2 − 1. Additionally, the integrated area of C_IG(t) is μ, and the first derivative of C_IG(t) at t = 0 is <1/τ>. The spectral density function is given by the real part of the Fourier transform of the autocorrelation function:

$$J_{IG}\left(\omega \right)=\frac{2}{5}Re\underset{0}{\overset{\infty }{\int }}{e}^{-i\omega t}{C}_{IG}\left(t\right)dt=\frac{1}{5}Re\left\{{\left(\frac{2\pi \lambda }{\omega }\right)}^{1/2}{e}^{-\frac{i\pi }{4}}w\left(Z\right)\right\}=\frac{1}{5}Re\left\{{\left(\frac{2\pi {\mu }^3}{\omega {\sigma }^2}\right)}^{1/2}{e}^{-\frac{i\pi }{4}}w\left(Z\right)\right\},$$ (11)

in which

$$Z=\frac{-\left(1-\omega \mu \right)+i\left(1+\omega \mu \right)}{2\mu {\left(\omega /\lambda \right)}^{1/2}}=\frac{-\left(1-\omega \mu \right)+i\left(1+\omega \mu \right)}{2\sigma {\left(\omega /\mu \right)}^{1/2}}.$$ (12)

The Faddeeva function is w(Z) = exp(−Z²)erfc(−iZ) (47), and erfc(x) is the complementary error function. Examples of the autocorrelation and spectral density functions obtained from Eqs. 10 to 11 are shown in Fig. 1, b and c, respectively.

The model-free ansatz gives C(t) = C_O(t){S² + (1 − S²)C_I(t)} as the total autocorrelation function for a macromolecule, in which S² is the square of the generalized order parameter from Lipari and Szabo and C_o(t) and C_I(t) are the autocorrelation functions for overall and intramolecular motions, respectively (38, 39). If the inverse Gaussian correlation and spectral density functions of Eqs. 10 and 11 represent the overall rotational dynamics of a nuclear spin in a molecule and the intramolecular motions, assumed to be independent of overall motions, are described by a single exponential decay with an effective correlation time τ_e, then the spectral density function becomes

$$J\left(\omega \right)=S^2{J}_{IG}\left(\omega \right)+\left(1-S^2\right)J_{IG}\left(\omega -i/{\tau }_e\right).$$ (13)

The second term of Eq. 13 is derived by extending the Fourier transform into the complex domain and is an efficient way of representing the Fourier transform of C_IG(t)exp(−t/τ_e). Alternatively, if the correlation and spectral density functions of Eqs. 10 and 11 represent the internal dynamics of a nuclear spin in a biomolecule with an exponential overall rotational diffusion correlation function of correlation time τ_m, then the spectral density function analogously becomes

$$J\left(\omega \right)=\frac{2}{5}\frac{S^2{\tau }_m}{1+{\omega }^2{\tau }_m^2}+\left(1-S^2\right)J_{IG}\left(\omega -i/{\tau }_m\right).$$ (14)

In this case, the second term is the Fourier transform of C_IG(t)exp(−t/τ_m). In summary, Eqs. 10, 11, 13, and 14 are the main theoretical results of this work.

For comparison to Eq. 14, the original Lipari-Szabo model-free formalism is given by (38, 39)

$$J\left(\omega \right)=\frac{2}{5}\left\{\frac{S^2{\tau }_m}{1+{\omega }^2{\tau }_m^2}+\frac{\left(1-S^2\right)\tau }{1+{\omega }^2{\tau }^2}\right\},$$ (15)

in which 1/τ = 1/τ_m + 1/τ_e and τ_e is an effective internal correlation time. τ_m is defined as above. The extended model-free formalism is given by (38, 39)

$$J\left(\omega \right)=\frac{2}{5}\left\{\frac{S^2{\tau }_m}{1+{\omega }^2{\tau }_m^2}+\frac{\left(S_f^2-S^2\right)\tau }{1+{\omega }^2{\tau }^2}+\frac{\left(1-S_f^2\right){\tau }^{\prime }}{1+{\omega }^2{\tau }^{\prime 2}}\right\},$$ (16)

in which 1/τ = 1/τ_m + 1/τ_s, 1/τ′ = 1/τ_m + 1/τ_f, τ_f and τ_s are internal correlation times for fast and slow timescales, respectively, and S_f² is the square of the generalized order parameter for motions on the fast timescale.

Results

The above formalism was applied to a set of backbone ¹⁵N relaxation data acquired for the bZip domain of the yeast transcription factor GCN4. These data were acquired at four static magnetic fields, corresponding to ¹H Larmor frequencies 600, 700, 800, and 900 MHz (29). The data were processed by standard methods to give reduced spectral density values. Fig. 2 a shows data for J(0), and Fig. 2 b shows data for J(0.87ω_H) with 0.87ω_H/2π = 522 and 783 MHz, referring to spectral densities at the lowest and highest frequencies sampled; these illustrate the regions of the bZip domain affected by slower and faster motions, respectively. In the original report by Gill and co-workers (29), the resulting spectral density data were analyzed using the model-free (two-exponential, Eq. 15) and extended model-free (three-exponential, Eq. 16) formalisms. The fitted parameters for the residues in the coiled coil already incorporate the effects of rotational diffusion anisotropy through the assumption of a local isotropic τ_m. The previous analysis found the diffusion tensor anisotropy to be modest and thus unlikely to affect the results in this scenario (48).

Figure 2.

Spectral density values at (a) ω = 0 and (b) ω = 0.87ω_H/(2π) = 522 (open circles) and 783 (solid circles) MHz frequencies, obtained from results reported by Gill et al. (29). Error bars represent the standard deviation in the experimental data values.

The spectral density data for residues 5–28 in the N-terminal basic region and the first turn of the coiled-coil α-helix and residues 56–58 in the C-terminus of the bZip domain of GCN4 were analyzed using Eq. 14. Representative sample fits of Eq. 14 to the experimental data are illustrated in Fig. 3 for amino acid residues 11 and 22. The resulting probability distributions p(τ) are also shown in Fig. 3. For these mobile residues, the simple model-free formalism of Eq. 15 fits the data poorly, and thus more complex motional models such as Eq. 14 are necessary. The qualities of fits to Eq. 14, as measured by the reduced χ² or Akaike information criterion statistics (49), have a median improvement of 6.9% when compared to the extended three-exponential model-free formalism, even though Eq. 14 has four adjustable parameters and the extended model-free formalism has five.

Figure 3.

Relaxation data for residues 11 (a–c) and 22 (d–f) of the bZip domain of GCN4. (a, b, d, and e) The circles are the experimental data from Gill and co-workers (29). The green solid line is the fitted result using Eq. 14, the orange dotted line is the result when fitted with the extended model-free (three-exponential) formalism (Eq. 16), and the purple dashed line is the result when fitted with the model-free formalism (Eq. 15). (b) and (e) show the zoomed-in views of the low field regime of the data shown in (a) and (d). (c and f) The probability distributions resulting from the fitted data in (a) and (c) are shown. The vertical green dashed lines indicate the values of μ. For residue 11, fitted values using Eq. 14 are S² = 0.46 ± 0.04, μ = 0.89 ± 0.01 ns, <τ><1/τ> = 1.9 ± 0.5, and τ_m = 8.9 ± 0.9 ns. For residue 22, fitted values are S² = 0.75 ± 0.02, μ = 0.92 ± 0.02 ns, <τ><1/τ> = 2.3 ± 0.6, and τ_m = 14.5 ± 0.6 ns. Data analyses were performed using an in-house software written in Python; optimizations of model parameters were performed using the “curve_fit” algorithm and implemented using the trust region reflective method (63) for minimization of χ² and a three-point numerical method for the calculation of the Jacobian matrix. Uncertainties in fitted parameters were obtained from Monte Carlo simulations. To see this figure in color, go online.

Fitted values of S², τ_m, μ, and <τ><1/τ> for the bZip DNA-binding domain of GCN4 are shown in Fig. 4. As noted above, data for only residues 5–28 and 56–58 were fitted with the inverse Gaussian spectral density function, Eq. 14. Data for the highly ordered coiled-coil residues 29–55 were fitted with the Lipari-Szabo model-free formalism, Eq. 15.

Figure 4.

Distribution function values of (a) μ, (b) <τ><1/τ>, (c) S², and (d) local τ_m for the bZip domain of GCN4 fitted to Eq. 14 or the model-free formalism (Eq. 15), depending on the particular region. Regions of the protein are colored identically to Gill et al. (29): region 1 of bZip (residues 3–12), pink diamonds; region 2 of bZip (residues 13–25), green squares; coiled coil (residues 26–55), black circles; disordered C-terminus (residues 56–58), orange triangles. Eq. 15 was used to fit data for residues 29–55 and is the limit of Eq. 14 as <τ><1/τ> → 1, meaning that μ = τ_e. In (c), the solid line drawn between residues 5 and 15 is calculated using Eq. 18 with <S²> = 0.93, S₀² = 0.665, and N₀ = 15, and the solid line between residues 56 and 58 is calculated using Eq. 18 with <S²> = 0.76, S₀² = 0.86, and N₀ = 56. Uncertainties in fitted parameters were obtained from Monte Carlo simulations. To see this figure in color, go online.

Discussion

NMR spin-relaxation methods are powerful approaches for characterizing intramolecular dynamics because relaxation rate constants are functions of the spectral density function, J(ω), for stochastic processes that modulate nuclear spin Hamiltonians. Model spectral density functions based on finite sums of terms or on continuous distribution functions have been applied to various investigations of biological macromolecules. However, historically, the most commonly used distribution functions diverge as ω approaches zero, a disadvantage because J(0) is critical for interpreting various relaxation measurements performed on biomolecules. This work derives a functional form of the spectral density function based on the inverse Gaussian probability distribution. The resulting spectral density function is convergent as ω approaches zero and can represent the overall rotational dynamics or intramolecular stochastic dynamics of a nuclear spin interaction using the model-free ansatz.

To demonstrate the application of this theoretical approach, backbone ¹⁵N relaxation data for the bZip domain of GCN4 were fitted using the latter assumption, Eq. 14. The results were compared to the two-exponential (Eq. 15) and three-exponential (Eq. 16) model-free formalisms. Specifically, the inverse Gaussian model-free formalism proved advantageous in fitting the partially disordered, N-terminal basic-region domain of GCN4. The qualities of fits, as measured by the reduced χ² or Akaike information criterion statistics (49), were comparable to or better than for the extended model-free formalism. The smaller number of free parameters in the inverse Gaussian spectral density function (four) compared to the three-exponential model-free formalism (five) is advantageous particularly because the number of relaxation rate constants measured per spin is frequently small, limiting statistical power. In addition, the inverse Gaussian spectral density function is based on fundamental aspects of spectral density functions for overdamped processes as distinguished from other distributions, such as the Mittag-Leffler function, derived for specific physical models for molecular motions.

As discussed earlier, the extended three-exponential model-free formalism can be regarded as a δ function representation of a continuous probability distribution. In support of this interpretation, the values of τ_s and τ_f obtained using Eq. 16 are often similar in magnitude to the mean and mode, respectively, of the fitted inverse Gaussian probability distribution (data not shown). The optimal choice of a given continuous probability distribution or a multiexponential model in the interpretation of NMR spin-relaxation data cannot be established from fitting the available data alone. The model independence and the small number of adjustable parameters of the inverse Gaussian spectral density function are attractive features, but the approach or combination of approaches that will ultimately prove optimal is currently unknown (19). Additional experimental data over a broader frequency range (from relaxometry) and advances in computer simulations are likely to be helpful in addressing these issues.

The intramolecular dynamics of the bZip domain and other IDRs have been discussed extensively elsewhere (9, 29, 34, 35, 50, 51, 52, 53, 54, 55, 56). The following discussion focuses on the results for the residues analyzed with the inverse Gaussian spectral density function: residues 5–28 and 56–58. The values of the parameters in common with the extended model-free spectral density function, S² and τ_m, are similar, with correlation coefficients R = 0.96 and 0.95, respectively. The inverse Gaussian spectral density function additionally characterizes the timescales of intramolecular motions by the parameters μ and <τ><1/τ>. The average values of these parameters are relatively constant for residues 5–23 in the basic region, where μ = 0.916 ± 0.010 ns and <τ><1/τ> = 1.99 ± 0.33. The average most probable value is τ_mode = 0.25 ± 0.02 ns.

The transformational properties of probability distributions can be used to convert p(τ) (Eq. 8) into a distribution for the parameters of a physical model via τ. For example, if τ = τ₀exp[E_a/(k_BT)], then

$$p\left(E_a\right)=\frac{1}{k_{B}T}{e}^{-\frac{\lambda {\left(\tau -\mu \right)}^2}{2{\mu }^2\tau }}\sqrt{\frac{\lambda }{2\pi \tau }},$$ (17)

in which k_B is the Boltzmann constant and T is the temperature. The resulting distribution for the average value of <τ><1/τ> = 1.99 is shown in Fig. 5. The mean, mode, and SD of the distribution are −0.46 k_BT, −0.64 k_BT, and +0.94 k_BT, respectively. The distribution is slightly skewed to the right when compared to a Gaussian distribution with the same mean and SD.

Figure 5.

Distribution of activation energies (solid) for <τ><1/τ> = 1.99 compared to a (dashed) Gaussian distribution with mean = −0.46 k_BT and SD of 0.94 k_BT.

As discussed previously, residues ∼13–19 in the basic region substantially populate transient helical conformations, resulting in a break in the general decrease in S² toward the N-terminus from the well-ordered, coiled-coil, leucine zipper region (9). In Fig. 4, the most N-terminal region (pink) and the C-terminal region (yellow) display relatively smooth profiles for S². As shown by the solid black lines in Fig. 4, the order parameters for these residues can be fitted with a simple model:

(18)

in which N is a given residue number and <S²> is a mean multiplicative decrease in S² per residue relative to a reference residue N₀ with order parameter S₀². A similar approach has been used by Schneider, Dellwo, and Wand to characterize the C-terminal region of ubiquitin (57).

The above results provide an enhanced description of the intramolecular dynamics of the bZip domain in the absence of DNA, as cartooned in Fig. 6. The coiled-coil region is characterized as a relatively rigid rod, with an average S² = 0.92 ± 0.01, indicating restricted local motions. The most N-terminal residues of the coiled coil exhibit reduced order parameters, which appears to suggest fraying, followed by a significant drop in the order parameters occurring between residues 28 and 20. The order parameters for residues 14–19 are relatively uniform, consistent with previously identified, transient, helical conformations. Another sharp drop in S² at residue 13 leads to the next set of residues in the most N-terminal region, residues 5–13. In this portion of the molecule, the fit to Eq. 18 suggests that individual peptide units have limited local motions on timescales smaller than 5 ns, but the cumulative effect from residue to residue results in a decrease in S² toward the N-terminus. Local interactions must still restrict these motions because the mean reduction in S² per residue is smaller than observed for the more disordered C-terminal residues. This model suggests that each peptide unit constitutes a local persistence length, like the Kuhn length in polymers, with restricted multidirectional joints in between each unit (58, 59, 60). In this region, compared to the values observed in the coiled-coil domain, the reduced values of the local, effective, overall correlation time, τ_m, may reflect segmental motions on timescales similar to overall rotation. Thus, in these data, the effects of segmental motions are not distinguishable from the effects of overall tumbling. The spatial extents and timescales of intramolecular and overall motions in IDRs have recently been discussed by Blackledge and co-workers based on extensive spin-relaxation (61) and molecular dynamics simulations (8, 62). Our results for the IDR of the bZip domain of GCN4 share qualitative features with this earlier work. In addition, they also agree with previous investigations suggesting that partially ordered bZip conformations of GCN4 form initial-encounter complexes with DNA and then rapidly rearrange to the high-affinity state with fully formed, basic-region recognition helices (9, 29). The restriction of conformational space in the basic region for the apo bZip domain then reduces the entropic penalty of binding to target DNA molecules (34).

Figure 6.

A cartoon of the bZip domain of GCN4 in which the ribbons are color coded as in Fig. 4. This cartoon illustrates a model of the bZip domain of GCN4 with the DNA substrate removed after some time (simulated). The coiled-coil region is modeled as a pair of rigid rods, and the bZip region is modeled by multiple rigid rods of some persistence length unit (in this cartoon, the persistence length is one residue), all connected by multidirectional joints with restricted flexibility.

Conclusion

In summary, using the principle of maximal entropy, we have obtained the inverse Gaussian probability density as a distribution function for correlation times that mimic the low and high frequency behavior of the spectral density function, central to the theory of NMR spin relaxation. The resulting spectral density function is well behaved as the frequency approaches zero and can be used in analyzing the relaxation rate constants that depend on J(0), such as the transverse relaxation rate R₂, as well as relaxation at very low magnetic fields. The inverse Gaussian probability distribution function and its associated spectral density function are particularly suited for the analysis of highly flexible loops or IDRs of proteins, such as the basic region of the bZip transcription factor domain of GCN4.

Author Contributions

A.H., F.F., and A.G.P. performed theoretical calculations and wrote the manuscript. A.H. and A.G.P performed numerical calculations and data fitting. A.G.P. and F.F. designed the research project.

Editor: Michael Sattler.