DNA base order parameter determination without influence of chemical exchange

Abstract

Nuclear magnetic resonance (NMR) spectroscopy is a versatile tool used to investigate the dynamic properties of biological macromolecules and their complexes. NMR relaxation data can provide order parameters S², which represent the mobility of bond vectors reorienting within a molecular frame. Determination of S² parameters typically involves the use of transverse NMR relaxation rates. However, the accuracy in S² determination can be diminished by elevation of the transverse relaxation rates through conformational or chemical exchange involving protonation/deprotonation or non-Watson-Crick base-pair states of nucleic acids. Here, we propose an approach for determination of S² parameters without the influence of exchange processes. This approach utilizes transverse and longitudinal ¹³C chemical shift anisotropy (CSA) – dipole-dipole (DD) cross-correlation rates instead of ¹³C transverse relaxation rates. Anisotropy in rotational diffusion is taken into consideration. An application of this approach to nucleotide base CH groups of a uniformly ¹³C/¹⁵N-labeled DNA duplex is demonstrated.

1. Introduction

DNA and RNA are inherently dynamic. As polyelectrolytes with titratable groups, nucleic acids undergo chemical exchange involving protonation/deprotonation and electrostatic interactions with counterions [1, 2]. Nucleic acids also undergo substantial conformational exchange, as shown by nuclear magnetic resonance (NMR) and single-molecule studies [3–5]. Even B-form DNA, which may appear to be rigid and static, undergoes dynamic processes involving low-population states such as Hoogsteen base pairs [6–9], base-pair opening [10], and extrahelical base flipping [11]. The transitions between the major and minor conformational states occur, typically on a micro- to milli-second timescale.

While the slow dynamics involving the minor states are functionally important, the molecular properties of the major state are thermodynamically important, for example when the contribution of conformational entropy to a binding free energy is considered [12, 13]. So-called generalized order parameters S² are useful for assessment of the mobility of particular moieties in biomolecules [14]. S² parameters have been determined for DNA and RNA as well (e.g., Refs. [15–23]). Changes in S² parameters are often used to estimate changes in local conformational entropy [24–26].

NMR relaxation data allows for determination of order parameters [27]. Typical methods for order parameter determination involve the use of transverse relaxation rates along with other relaxation data. However, transverse relaxation rates may significantly elevate due to an exchange process occurring on a micro- to millisecond timescale [27]. The R_1ρ and CPMG methods can in principle cancel the exchange contribution (R_ex) to the apparent transverse relaxation rates if the ¹³C spin-lock RF strength or the CPMG frequency is large enough compared to the exchange rate. However, there is a practical limit for the cancelation. R_ex can be significant if the exchange is sufficiently fast and accompanies a large chemical shift difference between the ground and excited states. This may adversely impact the accuracy in determining the order parameters for the major state. For CH groups, homonuclear ¹³C-¹³C scalar couplings (J_CC) may also significantly increase apparent R₂ relaxation rates measured for uniformly ¹³C-,¹⁵N-labeled biomolecules, depending on chemical shifts and experimental conditions. These effects can diminish the accuracy of determining the order parameter S².

In this article, we present an approach to determine order parameters for DNA base ¹³C-¹H bond vectors without influence of the exchange or J_CC modulation. ¹³C NMR relaxation of these aromatic CH groups occurs via ¹³C-¹H dipole-dipole (DD) and ¹³C CSA mechanisms. In our current approach, instead of ¹³C R₂ or R_1ρ relaxation rates, we utilize rates for cross correlation between ¹³C chemical shift anisotropy (CSA) relaxation and ¹³C-¹H dipole-dipole (DD) relaxation mechanisms. Importantly, the exchange processes do not affect the cross-correlation rates [28, 29]. Therefore, this approach allows for determination of order parameters without influence of chemical exchange. Recently, we used a similar approach to determine order parameters for histidine side chains that undergo chemical exchange [30]. The quantitative use of the ¹³C CSA-DD cross-correlation rates requires accurate information of ¹³C CSA. We take advantage of the ¹³C CSA values available for nucleic acid base CH groups, which have been extensively investigated by several groups [31–35]. This approach allows us to determine the S² order parameters without influences of conformational or chemical exchange processes.

2. Materials and Methods

2.1. Sample preparation

The ¹³C, ¹⁵N labeled 15-bp DNA of the sequence shown in Figure 1 was produced by the endonuclease sensitive repeat amplification (ESRA) method [37] and purified through anion-exchange chromatography using a Resource-Q column (GE Healthcare), as previously described [39]. The NMR sample used in this study was a 0.5 ml solution sealed in a Norell 5-mm NMR tube containing 0.3 mM ¹³C,¹⁵N-labeled 15-bp DNA dissolved in a buffer of 10 mM potassium phosphate (pH 7.4), 100 mM NaCl, and 5% D₂O.

Figure 1.

15-bp DNA duplex used in the current study and ¹H-¹³C correlation spectra for DNA base CH groups. The DNA duplex was designed to have the sequence of PDB 9ANT [36] in the middle but the terminal regions are different to incorporate HaeIII restriction enzyme sites (5’-GG|CC-3’) for the ESRA method [37]. The residue numbering is based on Fernandez et al.[38]. The ¹H and ¹³C resonances were previously assigned [39, 40]. ¹³C relaxation and ¹³C CSA-DD cross-correlation data for C8 of adenine and guanine bases, C6 of cytosine and thymine bases, and C2 of adenine bases were analyzed in the current study.

2.2. NMR experiments

All NMR experiments were performed at 25°C with a Bruker Avance III spectrometer operated at the ¹H frequency of 600 MHz. A cryogenic ¹H/¹³C/¹⁵N/³¹P QCI probe with a z gradient coil was used for NMR detection. The ¹³C relaxation and ¹³C CSA-DD cross-correlation data were recorded for adenine C2 and C8, cytosine C6, guanine C8, and thymine C6 atoms (see Figure 1 for their positions and resonances). The resonances were assigned in our previous work [39, 40]. In each experiment, the ¹H and ¹³C acquisition times were 54 ms and 48 ms, respectively. At least 32 scans per free induction decay (FID) were recorded for each relaxation or cross-correlation measurement. The NMR data were processed with the NMR-Pipe program [41], and the spectra were analyzed with the NMRFAM-SPARKY program [42].

Our pulse sequences used to measure ¹³C CSA-DD cross-correlation rates for DNA/RNA are shown in Figure 2A–B. Compared to the pulse sequences of Kojima et al. [43], which were designed for random fractionally ¹³C-labeled nucleic acids, ours are different in several points. Our pulse sequences are designed for uniformly ¹³C-/¹⁵N-labeled nucleic acids and implements the symmetrical reconversion approach [29, 44]. As Pelupessy et al. demonstrated, the symmetrical reconversion approach remarkably improves accuracy in measurements of cross-correlation rates [29, 44]. The approach involves 4 sub-experiments with assorted combinations of Schemes A, B1, and B2 for the path selections 1 and 2 (Figure 2C–E). The pulse sequence implements selective pulses for ¹³C nuclei of adenine C2/C8, cytosine C6, guanine C8, and thymine C6 atoms. The ¹³C dimension of these pulse sequences implements homonuclear ¹³C-¹³C decoupling, which suppresses splitting due to one-bond ¹³C-¹³C scalar couplings. The delay Δ (see Figure 2) was varied, and the cross-correlation rates were determined through fitting with:

$$\sqrt{\left|\frac{A_{II}{A}_{III}}{A_I{A}_{IV}}\right|}=\text{tanh}(2\eta \Delta ),$$ (1)

where A_I, A_II, A_III, and A_IV are the signal amplitudes in the sub-experiments I, II, III, and IV, respectively. In the symmetrical reconversion approach, the adverse impact of ¹³C-¹³C scalar couplings on ¹³C CSA-DD cross-correlation are canceled in Eq. 1 [29, 44]. The following values of delay Δ were used: 6, 10, 15, and 20 ms for measuring the transverse ¹³C CSA-DD cross-correlation rates; and 100, 150, 200, and 250 ms for measuring the longitudinal ¹³C CSA-DD cross-correlation rates. Other experimental details about measuring the cross-correlation rates are given in the caption of Figure 2.

Figure 2.

Measurement of ¹³C CSA-DD cross-correlation rates for nucleic acid base purine C8, pyrimidine C6, and adenine C2 atoms of uniformly ¹³C/¹⁵N-labeled nucleic acids. (A) NMR pulse sequence to measure transverse ¹³C CSA-DD cross-correlation rates η_xy. Black thin and bold bars represent hard rectangular 90° and 180° pulses, respectively. Short bold bars represent soft rectangular 90° (1 ms) pulses selective to water ¹H magnetization. For ¹³C and ¹³C’, black bell shapes represent Q3-shaped 180° pulses (1 ms) [45], and the gray and purple bell shapes represent a Q5-shaped 90° pulse [45] and its time-reversal, respectively. ¹⁵N 180° pulses were adiabatic broadband pulses (1 ms). Pulse phases are x unless indicated otherwise. The delay τ_a was set to 1 ms, which takes the ¹³C Q3 pulse width into account. The delay δ_a was set to the length of the Q3 pulse plus the initial value of the evolution time t₁. The delay Δ is varied to determine the ¹³C CSA-DD cross-correlation rates through nonlinear least-squares fitting calculations with Eq. 1. To cancel cross-correlation involving ¹⁵N or ¹³C’ nuclei during each Δ period, a ¹⁵N 180° pulse was applied at the middle position and ¹³C’ 180° pulses were applied at the first and third quarter positions. For the ¹H dimension, the carrier position was at the H₂O resonance and the spectral width was 24 ppm. For the ¹³C dimension, the carrier position was 146 ppm and the spectral width was 20 ppm. Phase cycles are as follows. ϕ₁ = (y, −y); ϕ₂ = [2x, 2(−x)]; ϕ₃ = [8x, 8(−x)]; and receiver = [(x, −x, −x, x), 2(−x, x, x, −x), (x,-x, −x, x), (−x, x, x, −x), 2(x, −x, −x, x), (−x, x, x, −x)]. ζ = −y for sub-experiments I and II, whereas ζ = x for sub-experiments III and IV. φ_A = [16x, 16(−x)], φ_B = [16(−y), 16y], and φ_C = y in the path selection 1, whereas φ_A = [4x, 4(−x)], φ_B = −y, and φ_C = [4x, 4(−x)] in the path selection 2. (B) NMR pulse sequence to measure longitudinal ¹³C CSA-DD cross-correlation rates η_z. The meanings of individual symbols for delays are identical to those in Panel A. The delay τ_a was set to 1 ms, which takes the ¹³C Q3 pulse width into account. Phase cycles are as follows. ϕ₁ = (y, −y); ϕ₂ = y; ϕ₃ = [2x, 2(−x)]; ζ = [16x, 16(−x)]; and receiver = [(x, −x, −x, x), 2(−x, x, x, −x), (x,-x, −x, x), (−x, x, x, −x), 2(x, −x, −x, x), (−x, x, x, −x)]. φ_A = [8x, 8(−x)], φ_B = [8(−y), 8y], and φ_C = y in the path selection 1, whereas φ_A = [4x, 4(−x)], φ_B = −y, and φ_C = [4x, 4(−x)] in the path selection 2. (C, D) Schemes for selecting either 2C_zH_z or C_z term. (E) Four sub-experiments with assorted combinations of Schemes A and B for the path selections 1 and 2. The sub-experiments are conducted in an interleaved manner.

¹³C R₁ and R_1ρ relaxation rates and heteronuclear NOE for DNA base CH were measured as previously described [16, 46]. The R₁ and R_1ρ rates were determined through fitting to 8 time-point data using a mono-exponential function. The experiments were conducted using two different ¹³C spin-lock RF strengths (2.67 kHz and 1.41 kHz), which were calibrated as described [47]. The R₂ relaxation rates were determined from the R₁ and R_1ρ relaxation rates along with the effective field calculated with the offset from the ¹³C carrier position and ¹³C spin-lock RF strength [48].

2.3. Determination of order parameters and correlation times

Order parameters and correlation times for DNA base CH groups were determined through nonlinear least-squares fitting to ¹³C R₁ relaxation rates, [¹H−] ¹³C heteronuclear NOEs, transverse ¹³C CSA-DD cross-correlation rates, and longitudinal ¹³C CSA-DD cross correlation rates. An ideal B-form structure generated using the Xplor-NIH software [49] was used to compute parameters involving vector orientations. The fitting calculations were performed using the MATLAB software. Uncertainties were estimated using a Monte Carlo approach. Theoretical and computational details are described in the next section.

3. Theory / Calculation

3.1. ¹³C longitudinal auto-relaxation and cross-relaxation rates

The longitudinal (R₁) ¹³C relaxation rates for nucleotide base CH groups involve both ¹³C-¹H dipole-dipole (DD) and ¹³C CSA mechanisms and are given by [27, 32]:

$$R_1={\sum }_i{R}_1^{DD,i}+R_1^{CSA}$$ (2)

where $R_1^{DD,i}$ is a rate for longitudinal relaxation through DD interaction with a neighbor atom i; and $R_1^{CSA}$ is a rate for longitudinal relaxation through CSA mechanism. These rates are expressed as a linear combination of the reduced spectral density functions J_auto(w):

$$R_{1,i}^{DD}=\frac{d_i^2}{4}\left[3J\left({\omega }_C\right)+J\left({\omega }_{X,i}-{\omega }_C\right)+6J\left({\omega }_{X,i}+{\omega }_C\right)\right]$$ (3)

$$R_1^{CSA}=c^{2}J\left({\omega }_C\right)$$ (4)

In these equations, w_C and w_X,i are the Larmor frequencies multiplied by 2π for ¹³C and the nucleus of a neighbor atom i, respectively. The parameter d_i for each DD term is:

$$d_i=\left(\frac{{\mu }_0}{4\pi }\right){\gamma }_C{\gamma }_{X,i}ħr_i^{-3},$$ (5)

where μ₀ is the vacuum permeability; γ_C and γ_H are the nuclear gyromagnetic ratios for ¹³C and ¹H nuclei, respectively; ℏ is the Planck constant divided by 2π; and r_i is the distance between the ¹³C nucleus and the nucleus of a neighbor atom i.

The influence of ¹³C-¹⁵N DD interactions on ¹³C relaxation can be safely neglected because of the small gyromagnetic ratio of ¹⁵N. However, ¹³C-¹³C DD interactions can make significant contribution to the ¹³C R₁ relaxation rates because J_auto(w_X,i – w_C) ≈ J_auto(0) for homonuclear DD interactions and this is much larger than J_auto(w_C) involved in ¹³C-¹H DD terms for ¹³C R₁ relaxation.

As shown in Figure 1, adenine C2/C8 and guanine C8 atoms are not bonded to any carbon atoms, whereas cytosine C6 and thymine C6 atoms are bonded to a C5 carbon atom. In principle, R₁^DD for ¹³C-¹³C can be predicted with Eqs. 2–5. However, the order parameters S² for ¹³C-¹³C and ¹³C-¹H bonds may be significantly different, depending on how the base moves. For example, a χ-angle rotation about the glycosyl N1-C1’ bond alters the orientation of the C6-H6 bond but does not affect the orientation of the C6-C5 bond of pyrimidine. S² parameters must be defined separately for ¹³C-¹³C and ¹³C-¹H bonds if ¹³C R₁ rates for cytosine and thymine C6 atoms are used. However, precise determination of S² parameters for ¹³C-¹³C bond is difficult because only ¹³C R₁ rates are significantly impacted by ¹³C-¹³C interactions. Therefore, for the determination of S² parameters, we include ¹³C R₁ data for purine CH groups at C2 and C8 positions, but not for cytosine and thymine CH groups at C6 positions.

The parameter c in Eq. 4 is given by:

$$c=\Delta {\sigma }_{\mathit{\text{auto}}}{\omega }_C/\sqrt{3}.$$ (6)

Δσ_auto represents the effective anisotropy for auto-relaxation through the CSA mechanism [32, 50]:

$$\Delta {\sigma }_{\mathit{\text{auto}}}={\left[{\left({\sigma }_{zz}-{\sigma }_{xx}\right)}^2+{\left({\sigma }_{zz}-{\sigma }_{yy}\right)}^2-\left({\sigma }_{zz}-{\sigma }_{xx}\right)\left({\sigma }_{zz}-{\sigma }_{yy}\right)\right]}^{1/2},$$ (7)

where σ_xx, σ_yy, and σ_zz, are the principal components of the CSA tensor. When rotational diffusion is isotropic, the reduced spectral density function for auto relaxation is given by [14]:

$$J(\omega )=\frac{2}{5}\left[\frac{S^2{\tau }_r}{1+{\left(\omega {\tau }_r\right)}^2}+\frac{\left(1-S^2\right){\tau }_i^{\prime }}{1+{\left(\omega {\tau }_i^{\prime }\right)}^2}\right],$$ (8)

where τ_r is the molecular rotational correlation time; τ_i is the internal motion correlation time; and ${\tau }_{i^{\prime }}={\left({\tau }_r^{-1}+{\tau }_i^{-1}\right)}^{-1}$. However, typically, rotational diffusion of nucleic acid molecules is not isotropic because their molecular shape is elongated. For anisotropic rotational diffusion, the reduced spectral density function for the CH DD vector is given as follows:

$$J(\omega )=\frac{2}{5}{\sum }_n{A}_n\left[\frac{S^2{\tau }_n}{1+{\left(\omega {\tau }_n\right)}^2}+\frac{\left(1-S^2\right){\tau }_{i,n}^{\prime }}{1+{\left(\omega {\tau }_{i,n}^{\prime }\right)}^2}\right]$$ (9)

When the anisotropy is axially symmetric, three terms are involved (i.e, n = 1, 2, 3) and the correlation times τ₁, τ₂, and τ₃ are related to the axially symmetric diffusion tensor components D_∥ and D_⊥ as follows [51]:

$${\tau }_1={\left(6D_{\perp }\right)}^{-1}$$ (10)

$${\tau }_2={\left(5D_{\perp }+D_{\parallel }\right)}^{-1}$$ (11)

$${\tau }_3={\left(2D_{\perp }+4D_{\parallel }\right)}^{-1},$$ (12)

where D_∥ and D_⊥ are the diffusion coefficients for rotations about the symmetry axis and about an axis perpendicular to the symmetry axis, respectively. The amplitudes for the Lorentzian functions in the J(w) functions are [32, 51, 52]:

$$A_1=\frac{1}{4}{\left(3{\text{cos}}^2\phi -1\right)}^2$$ (13)

$$A_2=\frac{3}{4}{\text{sin}}^{2}2\phi$$ (14)

$$A_3=\frac{3}{4}{\text{sin}}^4\phi$$ (15)

where φ is the angle between the spin interaction vector and the main principal axis of the diffusion tensor. Strictly speaking, the S² parameters for DD and CSA interactions are not necessarily identical. However, considering that the angle between a DNA base CH bond and ¹³C CSA xx principal axis is only up to 32 degrees [35], the order parameters for DD and CSA spectral density functions were assumed to be identical, as typically assumed for ¹⁵N CSA and NH bond of protein backbone NH groups.

The ¹³C-¹H cross relaxation rate σ_CH through ¹³C-¹H DD interactions is given by [27]:

$${\sigma }_{CH}=\frac{d^2}{4}\left[6J\left({\omega }_H+{\omega }_C\right)-J\left({\omega }_H-{\omega }_C\right)\right].$$ (16)

The parameter d is given by Eq. 5 with r_i and γ_X,i being the CH bond length and the ¹H nuclear gyromagnetic ratio, respectively. The steady-state heteronuclear [¹H−] ¹³C NOE is:

$$NOE=1+\left({\gamma }_H/{\gamma }_C\right)\left({\sigma }_{CH}/R_1\right).$$ (17)

Therefore, ¹³C-¹H cross relaxation rates σ_CH can be determined from the heteronuclear NOE data and ¹³C R₁ rates. We should note that exchange could impact the apparent cross-relaxation rates if the mobility is substantially different between the ground state and the excited state. However, this impact is not problematic in determining the order parameters S² because the J(0) term has a dominant influence on S² determination but the σ_CH rate does not involve any J(0) term.

3.2. ¹³C transverse relaxation rates

The transverse (R₂) ¹³C relaxation rates for nucleotide base CH groups involve both ¹³C-¹H dipole-dipole (DD) and ¹³C CSA mechanisms and are given by [27, 32]:

$$R_2={\sum }_i{R}_2^{DD,i}+R_2^{CSA}+R_{ex}$$ (18)

$$R_{2,i}^{DD}=\frac{d_i^2}{8}\left[4J(0)+3J\left({\omega }_C\right)+J\left({\omega }_{X,i}-{\omega }_C\right)+6J\left({\omega }_{X,i}\right)+6J\left({\omega }_{X,i}+{\omega }_C\right)\right]$$ (19)

$$R_2^{CSA}=\frac{c^2}{6}\left[4J(0)+3J\left({\omega }_C\right)\right]$$ (20)

Unlike the case for ¹³C R₁ rates, the influence of ¹³C-¹³C DD interactions on the ¹³C R₂ relaxation rates is negligible because the sum of the J(0) terms for ¹³C-¹H DD and ¹³C CSA mechanisms is far larger than the J(0) term for the ¹³C-¹³C DD mechanism.

In Eq. 18, the term R_ex represents the exchange contribution arising from dynamics on a micro- to milli-second timescale. R_ex can virtually be zero if the ¹³C spin-lock RF strength is sufficiently greater than the exchange rate. However, when the exchange rate is relatively large (e.g., > 10⁴ s⁻¹), the R_ex term can remain sizable and increase R₂. Typically, R₂ rates are measured through CPMG or R_1ρ experiments. When these experiments are conducted for uniformly labeled ¹³C,¹⁵N-labeled DNA or RNA, homonuclear ¹³C-¹³C scalar couplings may also cause undesired modulations. C6 atoms of cytosine, thymine, and uracil are bonded to C5. Although adenine C2/C8 and guanine C8 atoms are not bonded to any carbon, ²J_CC couplings in aromatic groups can be sizable. The ¹³C-¹³C scalar couplings and the exchange contribution can elevate apparent R₂ rates and may diminish the accuracy in determination of CH order parameters.

3.3. ¹³C CSA-DD cross-correlation rates

The transverse (η_xy) and longitudinal (η_z) ¹³C CSA-DD cross-correlation rates for nucleotide base CH groups are given as follows [32, 53]:

$${\eta }_{xy}=-(1/6)d\Delta {\sigma }_{\mathit{\text{cross}}}{\omega }_C\left[4J_{\mathit{\text{cross}}}(0)+3J_{\mathit{\text{cross}}}\left({\omega }_C\right)\right]$$ (21)

$${\eta }_z=-d\Delta {\sigma }_{\mathit{\text{cross}}}{\omega }_C{J}_{\mathit{\text{cross}}}\left({\omega }_C\right)$$ (22)

where J_cross(w) is a reduced spectral density function for ¹³C CSA-DD cross correlation (see Section 3.4). Δσ_cross is an effective anisotropy for CSA-DD cross correlation [54, 55]:

$$\Delta {\sigma }_{\mathit{\text{cross}}}=P_2\left(\text{cos\hspace{0.17em}}{\theta }_{DD,CSA}\right)\left({\sigma }_{xx}-{\sigma }_{zz}\right)+P_2\left(\text{cos\hspace{0.17em}}{\theta }_{DD,CSA}^{\prime }\right)\left({\sigma }_{yy}-{\sigma }_{zz}\right)$$ (23)

P₂(x) is the second Legendre polynomial, ½ (3ײ − 1); θ_DD,CSA is the angle between the DD vector and the CSA xx principal axis; and θ’_DD,CSA is the angle between the DD vector and the CSA yy principal axis. When rotational diffusion is anisotropic and axially symmetric, the reduced spectral density functions are given by [52, 54, 56]:

$$J_{\mathit{\text{cross}}}(\omega )=\frac{2}{5}{\sum }_n{A}_n\left[\frac{S^2{\tau }_n}{1+{\left(\omega {\tau }_n\right)}^2}+\frac{\left(P_2\left(\text{cos\hspace{0.17em}}{\theta }_{DD,CSA}\right)-S^2\right){\tau }_{i,n}^{\prime }}{1+{\left(\omega {\tau }_{i,n}^{\prime }\right)}^2}\right].$$ (24)

Unlike R₂ relaxation rates, the transverse CSA-DD cross-correlation rates η_xy are unaffected by exchange [28]. Both R₂ and η_xy are dominated by J(0). The predominant influence of J(0) without influence of exchange is clearly advantageous for accurate determination of order parameters. Another advantage is that the η_xy cross-correlation rates measured by the pulse sequence (Figure 2A) implementing the symmetrical reconversion approach [44] are not impacted by ¹³C-¹³C scalar couplings. Although selective 180° ¹³C pulses are used, ¹³C-¹³C scalar couplings during the Scheme B (Figure 2D) may cause undesired attenuation. However, the attenuation is canceled when the ratio of signal intensities is calculated using Eq. 1.

3.4. Fitting to determine order parameters and correlation times

Several research groups investigated ¹³C CSA parameters for nucleotide base CH groups of DNA and RNA using solid NMR, solution NMR, and quantum chemical calculations [31–35]. In our current study, we use the values of the CSA parameters from Table I of Ying et al. [35], as these parameters were rigorously determined from multiple types of solution NMR data and well accepted in the literature.

To determine order parameters and correlation times, the following weighted sum of squared differences between the observed and calculated relaxation data was minimized:

$${\chi }^2={\sum }_m{\sum }_j\frac{{\left(Y_{obs,j,m}-Y_{calj,m}\right)}^2}{{\sigma }_{Y_{obs,j,m}}^2}$$ (25)

The index j is for the type of data: namely, 1) transverse ¹³C CSA-DD cross-correlation rates η_xy; 2) longitudinal ¹³C CSA-DD cross-correlation rates η_z; 3) ¹³C-¹H cross-relaxation rates σ_CH; and 4) ¹³C R₁ relaxation rates (for adenine and guanine CH only; see Section 3.1). The index m is for CH groups. The cross-relaxation rates σ_CH determined from the experimental NOE and R₁ data with Eq. 17 were used instead of heteronuclear NOE, because heteronuclear NOE is also impacted by ¹³C-¹³C DD interactions through ¹³C R₁ relaxation. ${\sigma }_{Y_{obs,j,m}}$ in the denominator of each term in Eq. 25 is the uncertainty for each observed value. The total number of data was 4n_AG +3n_CT, where n_AG is the number of analyzed CH groups in adenine and guanine bases; and n_CT is the number of analyzed CH groups in cytosine and thymine bases. During minimization of χ², the effective rotational correlation time τ_r,eff [= (2D_∥ + 4D_⊥)⁻¹], the rotational anisotropy r_anis (= D_∥/D_⊥), two polar angles for the main principal axis of the diffusion tensor with respect to the structure atomic coordinate system were optimized as global parameters, and S² and τ_i were optimized as individual parameters defined for each CH group. Thus, the total number of fitting parameters was 2n_AG + 2n_CT + 4. The effective correlation time τ_r,eff and the anisotropy r_anis were converted into D_⊥ [= τ_r,eff ⁻¹(2r_anis + 4)⁻¹] and D_∥ [= τ_r,eff ⁻¹r_anis(2r_anis + 4)⁻¹] for the use of Eqs. 10–12.

4. Results and Discussion

4.1. ¹³C CSA-DD cross-correlation rates

Using the pulse sequences shown in Figure 2, we measured transverse (η_xy) and longitudinal (η_z) ¹³C CSA-DD cross-correlation rates for CH groups nucleotide bases of the 15-bp ¹³C/¹⁵N-labeled DNA duplex at 25°C. Some examples of time-course data for the signal intensity ratios in these measurements are shown in Figure 3A together with best-fit curves obtained with Eq. 1. The measured η_xy and η_z rates for individual CH groups are shown in Figure 3B and 3C, respectively.

Figure 3.

¹³C CSA-DD cross-correlation data for the 15-bp DNA duplex at 25°C. The NMR data were obtained at the ¹H frequency of 600 MHz. (A) Examples of build-up of signals arising from transverse or longitudinal ¹³C CSA-DD cross-correlation. (B) Transverse ¹³C CSA-DD cross-correlation rates measured for the individual base CH groups. (C) Longitudinal ¹³C CSA-DD cross-correlation rates measured for the individual base CH groups. (D) Apparent molecular rotational correlation times τ_r,app estimated from η_xy / η_z using Eq. 26. In each panel, data are colored as follows: A C8, red; A C2, magenta; C C6, blue; G C8, black; and T C6, green. Note that despite a large variation in the ¹³C CSA-DD cross-correlation rates among different CH groups, the variation in τ_r,app is remarkably smaller.

The measured cross-correlation rates clearly differ for distinct types of base CH groups. For example, for both η_xy and η_z rates, CH groups at A C2 positions exhibited significantly larger cross-correlation rates than CH groups at A C8 positions of the same nucleotide residues. Compared to CH groups at pyrimidine (i.e., C or T) C6 positions, CH groups at purine (i.e, A or G) C8 positions tended to exhibit smaller cross-correlation rates. This strong dependence of ¹³C CSA-DD cross-correlation rates can be explained by Eqs 21–23. The variation in ¹³C CSA parameters directly impacts η_xy and η_z rates through Δσ_cross but has smaller impact on ¹³C R₁ and R₂ relaxation rates because ¹³C CSA relaxation makes only ~25–30% contribution at the ¹H frequency of 600 MHz. The ¹³C CSA parameters for nucleotide bases significantly depend on types of CH groups. The large variation in the ¹³C CSA-DD cross-correlation rates clearly reflects the dependence of ¹³C CSA parameters on CH types.

In fact, the ratio η_xy/η_z, which cancels Δσ_cross (see Eqs. 25–26), exhibited far smaller variations compared to variations of η_xy and η_z rates. The ratio η_xy/η_z conveniently allows us to estimate the apparent molecular rotational correlation time τ_r,app. Assuming that the internal motion correlation time is far smaller than the rotational correlation time, τ_r,app for an isotropic diffusion model can be approximated by:

$${\tau }_{r,app}\approx \frac{1}{2{\omega }_c}\sqrt{6\left(\frac{{\eta }_{xy}}{{\eta }_z}\right)-7}$$ (26)

which is derived from Eqs. 21–22 and the first term of Eq. 8. This is convenient because it requires neither fitting calculations nor CSA parameters. Figure 3D shows the apparent correlation time τ_r,app estimated from the ratio η_xy / η_z for each residue. The τ_r,app calculated individually for each CH group exhibited a small degree of variation, most likely due to significant anisotropy in rotational diffusion for the cylindrical shape of the 15-bp DNA duplex, as shown below. Nonetheless, the variation in τ_r,app among different CH groups was obviously far smaller than the variations in ¹³C CSA-DD cross-correlation rates.

4.2. ¹³C relaxation rates

Dependence of ¹³C relaxation data on the base CH types was not as clear as that of ¹³C CSA-DD cross-correlation data. Figure 4 shows the transverse ¹³C transverse relaxation rates R₂, the ¹³C longitudinal relaxation rates R₁, and the heteronuclear [¹H−]¹³C NOE data for the 15-bp DNA duplex at 25°C. The ¹³C R₂ rates were determined from the R₁ rates and the R_1ρ rates. Some CH groups exhibited significantly higher R₂ rates at the ¹³C spin-lock RF strength of 1.41 kHz than at 2.67 kHz (open and closed circles, respectively, in Figure 4A), suggesting the presence of an exchange contribution (R_ex) to ¹³C transverse relaxation. Interestingly, dependence on the ¹³C spin-lock RF strength was particularly significant for some adenine C2 nuclei. Further investigations are required to find out the origin of the exchange contribution. Nonetheless, the exchange contribution influences the subsequent determination of order parameters.

Figure 4.

¹³C relaxation data for DNA base CH groups for the 15-bp DNA duplex at 25°C. The NMR data were obtained at the ¹H frequency of 600 MHz. (A) ¹³C transverse relaxation rates R₂ determined from the R_1ρ and R₁ relaxation rates and the effective field for the ¹³C spin lock. Two data with different ¹³C spin-lock RF strengths are shown (open circles, 1.4 kHz; closed circles, 2.6 kHz). (B) ¹³C longitudinal relaxation rates R₁. (C) Heteronuclear [¹H−]¹³C NOE. (D) ¹³C-¹H cross relaxation rates σ_CH determined from the heteronuclear NOE and ¹³C R₁ data.

¹³C R₁ relaxation rates of pyrimidine C6 were generally larger than those of purine C8 and C2 (Figure 4B). Most likely, this trend is due to a homonuclear ¹³C-¹³C DD relaxation mechanism for pyrimidine C6-C5 pairs. Due to its J(0) term in Eq. 3, the homonuclear DD relaxation mechanism makes significant contrition to the ¹³C R₁ rates. This effect is specific to pyrimidine C6 atoms because they are bonded to C5, but purine C8 and C2 atoms are not bonded to any other carbon atoms (see Figure 1).

We also determined ¹³C-¹H cross-relaxation rates σ_CH from the heteronuclear [¹H−] ¹³C NOE and ¹³C R₁ data. Whereas the heteronuclear NOE values were nearly uniform (Figure 4C), the cross-relaxation rates σ_CH were not (Figure 4D). To determine order parameters, we used σ_CH rates instead of heteronuclear NOE data because the heteronuclear NOE for pyrimidine C6 is affected by adjacent ¹³C nuclei through R₁ relaxation (see Eq. 17) but σ_CH rates are unaffected for both pyrimidine and purine CH groups.

4.3. Order parameters and correlation times for internal motions

Using the experimental data shown in Figures 3 and 4, we analyzed the mobility of CH groups for the individual DNA bases. Through minimization of the χ² function defined by Eq. 25, we determined S² parameters and internal motion correlation times τ_i. Fitting calculations were conducted using the ¹³C CSA-DD cross-correlation rates η_xy and η_z, ¹³C-¹H cross relaxation rates σ_CH, and ¹³C R₁ rates, as described in Section 3.4. Figure 5A–B shows the order parameters S² and the internal motion correlation times τ_i for individual CH groups. The order parameters S² were between 0.8 and 1.0 for the DNA base CH groups. The internal motion correlation times τ_i were determined to be 50 – 150 ps. The effective molecular rotational correlation time (2D_∥ + 4D_⊥)⁻¹ and the anisotropy (D_∥/D_⊥) of rotational diffusion were determined to 5.93 ± 0.04 ns and 2.48, respectively.

Figure 5.

Order parameters S² and internal motion correlation time τ_i determined through nonlinear least-squares fitting to the data of ¹³C CSA-DD cross-correlation rates η_xy and η_z, ¹³C-¹H cross-relaxation rates σ_CH, and ¹³C R₁ rates. (A) Order parameters S². (B) Internal motion correlation times τ_i. (C) Comparison of S² parameters determined by conventional methods and those determined by the current approach. The global fitting optimizing the rotational diffusion parameters as well as S² and τ_i were used for each. (D) The same as C except that the rotational diffusion parameters were fixed to (2D_∥ + 4D_⊥)⁻¹ = 5.93 ns and D_∥/D_⊥ = 2.48.

We also conducted fitting calculations using the ¹³C R₂ rates, ¹³C R₁ rates, and ¹³C-¹H cross-relation rates σ_CH. This corresponds to a conventional order parameter determination. We used the ¹³C R₂ rates determined through the R_1ρ experiments with a ¹³C spin-lock RF strength of 2.67 kHz. Through this conventional fitting, the effective molecular rotational correlation time was calculated to be 6.28 ± 0.01 ns, which is significantly larger than the value obtained with the other fitting using ¹³C CSA-DD cross-correlation rates. An interpretation of the larger rotational correlation time from the fitting using ¹³C R₂ rates is that the presence of R_ex in the R₂ rates cause an overestimate of the effective molecular rotational correlation time. Such an overestimate of the effective molecular rotational correlation time can lead to smaller order parameters in the fitting calculation. In fact, as shown in Figure 5C, the order parameters determined through this conventional approach tended to be smaller than those determined with the ¹³C CSA-DD cross-correlation rates.

For further examination, we conducted the fitting using the same data set of ¹³C R₂, R₁, and σ_CH rates along with the rotational diffusion parameters obtained from the fitting with the ¹³C CSA-DD cross-correlation data. In Figure 5D, the order parameters S² from this fitting are compared with those from the fitting with the ¹³C CSA-DD cross-correlation data. These two sets of S² parameters were in better agreement when the same rotational diffusion parameters were used. These results suggest that even relatively small R_ex terms in R₂ rates may adversely impact the accuracy in determination of rotational diffusion parameters and order parameters if many residues undergo conformational or chemical exchange.

5. Conclusions

In this work, we have demonstrated that the use of ¹³C CSA-DD cross-correlation rates together with ¹³C longitudinal auto- and cross-relaxation data allows for determination of DNA base order parameters without influences of exchange. This approach can provide accurate order parameters because it does not rely on ¹³C R₂ or R_1ρ rates, which can be affected by the R_ex term or J_CC modulation. This approach can be particularly useful for protein-DNA or protein-RNA complexes because many nucleotide residues can undergo exchange, particularly around their binding interfaces [57, 58]. Accurate order parameters would be invaluable for evaluating the role of conformational entropy of nucleic acids in the process of molecular association with proteins.

Highlights:

• DNA base order parameters can be determined without any impact of chemical exchange.

• ¹³C-¹³C interactions do not affect this method.

• ¹³C NMR relaxation and cross-correlation rates are combined for the analysis.

• Rotational diffusion anisotropy is taken into consideration.

• The method is also applicable to RNA.