NMR Methods for Characterizing the Basic Side Chains of Proteins: Electrostatic Interactions, Hydrogen Bonds, and Conformational Dynamics

Abstract

NMR spectroscopy is a powerful tool for studying protein dynamics. Conventionally, NMR studies on protein dynamics have probed motions of protein backbone NH, side-chain aromatic, and CH₃ groups. Recently, there has been remarkable progress in NMR methodologies that can characterize motions of cationic groups in protein side chains. These NMR methods allow investigations of the dynamics of positively charged lysine (Lys) and arginine (Arg) side chains and their hydrogen bonds as well as their electrostatic interactions important for protein function. Here, describing various practical aspects, we provide an overview of the NMR methods for dynamics studies of Lys and Arg side chains. Some example data on protein–DNA complexes are shown. We will also explain how molecular dynamics (MD) simulations can facilitate the interpretation of the NMR data on these basic side chains. Studies combining NMR and MD have revealed the highly dynamic nature of short-range electrostatic interactions via ion pairs, especially those involving Lys side chains.

1. INTRODUCTION

To elucidate mechanisms underlying protein functions, it is essential to understand the atomic-level behavior of individual side chains at functionally important sites on proteins. Charged side chains and their electrostatic interactions are of fundamental importance for molecular recognition, association, and catalysis by proteins (Honig & Nicholls, 1995; Sheinerman, Norel, & Honig, 2000). The mobility of these side chains should affect electrostatic interactions and hydrogen bonds crucial for the functional activities of proteins. However, the dynamic aspects of these fundamental interactions involving the charged moieties of protein side chains are not well understood from experimentally obtained data.

Solution NMR spectroscopy is a powerful tool in experimental research on protein dynamics and has provided insight into dynamics–function relationship of proteins (Boehr, Dyson, & Wright, 2006; Clore & Iwahara, 2009; Loria, Berlow, & Watt, 2008; Mittermaier & Kay, 2009; Palmer, 2014). However, when protein dynamics are studied, the vast majority of NMR studies focus on the motions of either protein backbone or side-chain CH₃ groups. NMR investigations of polar or charged side chains remain rare. More investigations into the dynamic behavior of these side chains would deepen our knowledge about the functional roles of protein dynamics.

Our goal in this chapter is to provide other researchers with a practical overview of the NMR methods for dynamics investigations of lysine (Lys) and arginine (Arg) side chains. Hydrogen bonds and electrostatic interactions involving Lys side-chain NH₃⁺ groups and Arg N_εH groups are of particular interest. Over the past decade, we have conducted NMR studies of basic side chains, primarily to obtain a better understanding of protein–DNA interactions. Lys and Arg side chains are crucial for electrostatic interactions with DNA phosphates and for recognition of DNA bases (Luscombe, Laskowski, & Thornton, 2001; Privalov, Dragan, & Crane-Robinson, 2011; Rohs et al., 2010). Due to our research background, the majority of the examples shown here are for protein–DNA complexes. However, the NMR methodology for Lys and Arg side chains introduced in this chapter is applicable to other protein systems as well. Furthermore, we explain how molecular dynamics (MD) simulations can facilitate interpretation of NMR data regarding the behavior of basic side chains.

2. NMR OF Lys AND Arg SIDE CHAINS

2.1. ¹H, ¹³C, and ¹⁵N Resonances of Lys/Arg Side Chains

2.1.1. Chemical Shifts and Protonation States

Typical ¹H, ¹³C, and ¹⁵N chemical shifts of Lys and Arg side chains of proteins are shown in Table 1. Side-chain methylene ¹H and ¹³C resonances can be assigned using standard three-dimensional (3D) NMR spectra, such as HCCH-TOCSY, HCCH-COSY, H(CCO)NH, and C(CO)NH spectra (Clore & Gronenborn, 1998). ¹H and ¹⁵N resonances from the NH/NH₂/NH₃⁺ groups of Lys and Arg side chains are located in unique regions in ¹H–¹⁵N heteronuclear spectra (e.g., see Fig. 1A), and additional experiments are required to assign these ¹H and ⁵N resonances, as explained in Section 2.4. The NMR-based determination of the pK_a of protein side chains has been reviewed recently (Hass & Mulder, 2015; Platzer, Okon, & McIntosh, 2014). Because the typical pK_a of Lys side chains is 10.5, Lys side-chain amino groups are usually protonated in the form of NH₃⁺ in neutral and acidic pH conditions. Lys side-chain ¹⁵N_ζ chemical shifts are typically 31–35 ppm in the protonated state (i.e., NH₃⁺) (Andre, Linse, & Mulder, 2007; Iwahara et al., 2007; Poon et al., 2006). NH₃⁺ groups that form a hydrogen bond with other moieties tend to exhibit a relatively large ¹⁵N chemical shift (Blaum et al., 2010; Esadze, Li, Wang, Brüschweiler, & Iwahara, 2011; Sepuru, Iwahara, & Rajarathnam, 2018). The deprotonated state (i.e., NH₂) of a Lys side-chain amino group exhibits ¹⁵N chemical shift of 22–26 ppm in the deprotonated state (Andre et al., 2007; Kougentakis et al., 2018; Poon et al., 2006; Takayama et al., 2008). The typical pK_a of Arg side chains has recently been updated to 13.8 (Fitch, Platzer, Okon, Garcia-Moreno, & McIntosh, 2015). Due to their typical pK_a values, the deprotonated states of Lys and Arg side chains are rare at physiological pH, especially for Arg. In this chapter, we focus on the positively charged, protonated states of Lys and Arg side chains.

Table 1.

Typical ¹H, ¹³C, and ¹⁵N Chemical Shifts (in ppm) of Lys and Arg Residues^a

	Lys		Arg
1H	HN	7.5–8.8	HN	7.3–9.2
	Hα	3.8–4.7	Hα	3.8–4.8
	Hβ2/β3	1.5–2.0	Hβ2/β3	1.4–2.1
	Hγ2/γ3	1.1–1.6	Hγ2/γ3	1.2–1.8
	Hδ2/δ3	1.0–2.3	Hδ2/δ3	2.8–3.4
	Hε2/ε3	2.7–3.1	Hεb	7.2-9.2
	Hζ1/ζ2/ζ3 (NH3+)c	7.2–8.2	Hη11/η12/ η21/η22d	6.7–8.7
	Hζ1/ζ2 (NH2)e	0.8–1.2
13C	C═O	171–183	C═O	173–180
	Cα	54–60	Cα	53–60
	Cβ	30–36	Cα	28–33
	Cγ	22–28	Cγ	24–30
	Cδ	26–32	Cδ	40–46
	Cε	39–45	Cζ	154–167
15N	N (backbone)	116–126	N (backbone)	116–126
	Nζ (NH3+)c	30–35	Nεb	78–90
	Nζ (NH2)f	22–26	Nη1/η2d	68–74

^aUnless indicated otherwise, data are from the Biological Magnetic Resonance Bank (BMRB). Each range corresponds to average ± standard deviation in the BMRB chemical shift statistics table.

^bFrom data in the literature (e.g., Esadze et al., 2016; Nguyen, Hoffpauir, & Iwahara, 2017; Yamazaki, Pascal, Singer, Formankay, & Kay, 1995).

^cFrom data in the literature (e.g., Iwahara, Jung, & Clore, 2007; Poon et al., 2006).

^dFrom data in the literature (e.g., Nieto et al., 1997; Yamazaki et al., 1995).

^eObserved only for those buried in a hydrophobic core (Kougentakis et al., 2018; Takayama, Castaneda, Chimenti, Garcia-Moreno, & Iwahara, 2008).

^fFrom data in the literature (Andre et al., 2007; Kougentakis et al., 2018; Poon et al., 2006; Takayama et al., 2008).

Fig. 1.

(A) ¹H and ¹⁵N resonances of the Lys and Arg side chains of the HoxD9 homeodomain bound to 24-bp specific DNA at 35°C. (B) Effect of D₂O in the sample buffer. D₂O for NMR lock should separately be sealed to avoid exchange between isotopically different species (i.e., NH₃⁺, NDH₂⁺, ND₂H⁺). (C) Two types of coaxial NMR tubes for separating the sample and D₂O.

2.1.2. Degenerate ¹H Resonances of Lys NH₃⁺

The resonances of the three ¹H nuclei within a Lys NH₃⁺ group are degenerate due to rapid rotations around the C_ε–N_ζ bond. This degeneracy occurs even in the presence of hydrogen bonds with other moieties. For example, in ubiquitin, the Lys27 NH₃⁺ group forms three hydrogen bonds with other moieties, but the bond rotations of this NH₃⁺ occur on a nanosecond timescale, making the ¹H resonances of its three ¹H nuclei completely degenerate (Esadze et al., 2011). As described in Section 5, the hydrogen bonding of Lys side chains is dynamic, allowing rapid C_ε–H_ζ bond rotations. The intrinsic ¹⁵N transverse relaxation of Lys NH₃⁺ is very slow owing to the rapid C_ε–N_ζ bond rotations. Although rapid hydrogen exchange with water causes some adverse effects on NMR experiments (see Section 2.3), heteronuclear ¹H–¹⁵N NMR of Lys NH₃⁺ groups is useful for investigating the conformational dynamics, hydrogen-bond dynamics, and ion-pair dynamics of proteins.

2.1.3. Hindered 180° Rotations of Arg Guanidinium C_ζ–N Bonds

Arg side-chain guanidinium moieties involve five hydrogen (H_ε, H_η11, H_η12, H_η21, and H_η22) and three nitrogen (N_ε, N_η1, and N_η2) atoms. In general, it is relatively easy to analyze the ¹H and ¹⁵N resonances of N_εH groups but not those of N_ηH₂ groups. The so-called hindered 180° rotations along the N_ε–C_ζ, C_ζ–N_η1, and C_ζ–N_η2 bonds within each planar guanidinium moiety cause interconversions between the two N_ηH₂ groups and between the two hydrogen atoms of each N_ηH₂ group. These interconversions tend to occur in the intermediate exchange regime on the ¹H and ¹⁵N chemical shift timescale at physiological temperature (Yamazaki et al., 1995). Because these interconversions cause severe broadening of ¹H_η resonances, the four ¹H_η resonances of Arg N_ηH₂ groups are rarely isolated. This broadening effect makes the NMR investigations of Arg N_ηH₂ groups challenging, though quantitative kinetic analysis of the C_ζ–N bond rotationsis possible in favorable cases (Gerecht, Figueiredo, & Hansen, 2017; Nieto et al., 1997). In contrast, the ¹H and ¹⁵N resonances of Arg N_εH groups are unaffected by the hindered 180° bond rotations and exhibit relatively sharp NMR signals when the hydrogen exchange is slow enough. Thus, N_εH groups are most convenient for ¹H–¹⁵N NMR investigations of Arg side-chain guanidinium moieties.

2.2. Impact of Hydrogen Exchange on Lys/Arg Side-Chain NMR

2.2.1. Direct Influence of Hydrogen Exchange on ¹H Line Shapes

Hydrogen exchange with water broadens the ¹H line shapes of signals from the NH/NH₂/NH₃⁺ groups of Arg and Lys side chains. When direct ¹H observation of a particular NH, NH₂, or NH₃⁺ group is possible, its hydrogen exchange rate, k_ex,w, is typically <200 s⁻¹. This corresponds to a slow exchange regime on the NMR chemical shift timescale because the ¹H chemical shift difference (Δδ) between NH/NH₂/NH₃⁺ and water resonances is typically >1000 Hz (i.e., k_ex,w ≪ 2πΔδ). This slow exchange increases the apparent ¹H transverse relaxation rate by the first-order rate constant for the transition from NH/NH₂/NH₃⁺ to water. Note that this rate constant is virtually identical to the hydrogen exchange rate k_ex,w (sum of the rate constants for the forward and backward transitions), since the population of water is far larger in this exchange. Therefore, the line width of the ¹H signal from NH/NH₂/NH₃⁺ is broadened by k_ex,w/πHz due to hydrogen exchange.

2.2.2. Indirect Influence of Hydrogen Exchange on ¹⁵N Line Shapes

Hydrogen exchange impacts the ¹⁵N dimension unless in-phase single-quantum coherence is maintained throughout the ¹⁵N evolution period. This broadening effect on the ¹⁵N dimension is easier to understand for HMQC because multiple-quantum terms such as 2N_xH_y should be impacted by hydrogen exchange in the manner explained above. Actually, hydrogen exchange broadens ¹⁵N line shapes not only in HMQC spectra but also in HSQC spectra. The impact of hydrogen exchange on ¹⁵N line shape in HSQC spectra occurs through scalar relaxation of antiphase single-quantum terms (Iwahara et al., 2007). In the first half of the ¹⁵N evolution period (t₁/2) before a ¹H 180° pulse, the 2N_yH_z term generated through the first INEPT scheme produces 2N_xH_z, N_x, and N_y terms for NH; 2N_xH_z, N_x, N_y, 4N_xH_zH_z, and 4N_yH_zH_z for NH₂; and 2N_xH_z, N_x, N_y, 4N_xH_zH_z, 4N_yH_zH_z, 8N_xH_zH_zH_z, and 8N_xH_zH_zH_z for NH₃⁺ groups. For antiphase single-quantum ¹⁵N transverse terms with respect to ¹H, scalar relaxation arises from autorelaxation of the coupled ¹H nuclei (Abragam, 1961; Bax, Ikura, Kay, Torchia, & Tschudin, 1990) and substantially increases the relaxation rates of the 2N_x(_y)H_z, 4N_x(_y)H_zH_z, and 8N_x(_y)H_zH_zH_z terms, compared to the relaxation rates of the in-phase single-quantum terms N_x(_y). The scalar relaxation rate R_sc for each ¹H nucleus is given by R_sc = ρ_HH + k_ex,w, where ρ_HH is the rate of autorelaxation arising from dipole–dipole (DD) interactions with external ¹H nuclei (Iwahara et al., 2007). Scalar relaxation rates for the N₊, 2N₊H_z, 4N_+HzHz, and 8_N+H_zH_zH_z terms are 0, R_sc, 2R_sc, and 3R_sc, respectively (Esadze et al., 2011). The hydrogen exchange rate k_ex,w is typically faster than ρ_HH rates and intrinsic ¹⁵N relaxation rates, and therefore governs ¹⁵N line shapes in HSQC spectra for NH₃⁺ groups (Anderson et al., 2013; Esadze et al., 2011; Iwahara et al., 2007; Zandarashvili, Esadze, & Iwahara, 2013). In HSQC, hydrogen exchange broadens ¹⁵N line width by 1.5k_ex,w/π Hz for Lys NH₃⁺ groups and by 0.5k_ex,w/πHz for Arg N_εH groups. The impacts of scalar relaxation and hydrogen exchange on the ¹⁵N dimension of heteronuclear ¹H–¹⁵N correlation experiments are summarized in Fig. 2A and B.

Fig. 2.

The importance of maintaining the in-phase single-quantum term during the ¹⁵N evolution period in ¹H–¹⁵N NMR experiments for Lys and Arg side chains. (A) Impact of scalar relaxation on different single-quantum terms of ¹⁵N coherence. (B) Broadening of ¹⁵N line shapes due to hydrogen exchange in the ¹⁵N dimensions of HSQC, HMQC, and HISQC spectra for Lys NH₃⁺ and Arg N_εH groups. The increases in ¹⁵N half-height widths are given in Hz units. (C) The pulse sequence of the water-flip-back HISQC experiment for Lys NH₃⁺ or Arg N_εH groups. Thin and bold bars in black represent hard rectangular 90° and 180° pulses, respectively. Water-selective half-Gaussian (2.1 ms) and soft-rectangular (1.2 ms) 90° pulses are represented by half-bell and short-bold shapes, respectively. Unless indicated otherwise, pulse phases are along x, and the carrier position for ¹H was set to the position of the water resonance. The ¹⁵N carrier position is set to ~33 ppm for Lys NH₃⁺ groups and to ~85 ppm for Arg N_εH groups. A gray bell-shape for ¹⁵N represents a 1.0-ms rSNOB 180° pulse (Kupče, Boyd, & Campbell, 1995) selective to Lys ¹⁵N_ζ or Arg ¹⁵N_ε nuclei. The delays τ_a and τ_b are set as follows: τ_a = 2.7 ms and τ_b = 1.3 ms for Lys NH₃⁺ groups; τ_a = 2.3 ms and τ_b = 2.7 ms for Arg N_εH groups. This setting for Arg side chains suppresses signals from N_ηH₂ groups; τ_b = 1.3 ms can be used to avoid this suppression of N_ηH₂ signals. Phase cycles: φ₁ = (y, −y); φ₂ = (2(y), 2(−y)); φ₃ = (4x, 4(−x));receiver = (x, 2(−x), x, −x, 2x, −x). Quadrature detection in the t₁ domain was achieved using States-TPPI, incrementing the phase φ₁. The third ¹H 90° pulse along −x (before the gradient g₄) serves two purposes: it purges unnecessary ¹⁵N antiphase magnetization by generating multiple quantum coherence and it flips the water magnetization back to +z. In the HISQC experiment, ¹H WALTZ-16 decoupling pulses maintain ¹⁵N in-phase single-quantum coherence during the ¹⁵N evolution period t₁ and thereby suppress the broadening of ¹⁵N line shapes due to rapid scalar relaxation arising from hydrogen exchange. (D) HISQC drastically improves ¹H–¹⁵N correlation spectra for Lys NH₃⁺ groups through suppression of scalar relaxation during the ¹⁵N evolution period. This panel compares the HMQC, HSQC, and HISQC spectra recorded for the Lys NH₃⁺ groups of the ¹⁵N-labeled Egr-1 zinc-finger–DNA complex at pH 5.8 and 10°C. For each spectrum, the data were acquired using the same number of scans and the same digital resolutions and were processed identically. F₁-slices at the position indicated by an arrow are shown to indicate ¹⁵N line shapes.

2.2.3. Self-decoupling Due to Hydrogen Exchange

Due to the self-decoupling effect (Cavanagh, Fairbrother, Palmer, Rance, & Skelton, 2007; Harbison, 1993), the apparent magnitude of J-coupling between the ¹⁵N and ¹H nuclei of NH/NH₂/NH₃⁺ becomes smaller than the intrinsic value when the hydrogen exchange rate is comparable to or faster than π¹J_NH s⁻¹. Theoretical consideration of the self-decoupling effect through hydrogen exchange is presented in the supporting information of Takayama et al. (2008). Because the hydrogen exchange rates for Lys NH₃⁺ groups at pH > 7 are typically far greater than π¹J_NH (Segawa, Kateb, Duma, Bodenhausen, & Pelupessy, 2008), self-decoupling of ¹J_NH can easily occur for Lys side chains. Under such conditions, H2(C)N spectra recorded without ¹H decoupling in the ¹⁵N dimension exhibit singlet (rather than quartet) ¹⁵N resonances of Lys NH₃⁺ groups due to the self-decoupling effect.

2.3. Importance of Maintaining In-Phase Single-Quantum ¹⁵N Coherence

2.3.1. HISQC Drastically Improves Lys NH₃^{+ 15}N Line Shapes

To avoid the severe broadening of ¹⁵N line shapes due to rapid scalar relaxation arising from hydrogen exchange, Iwahara et al. developed NH₃⁺-selective heteronuclear in-phase single-quantum coherence (HISQC) and its derivatives (Iwahara et al., 2007). The pulse sequence of the HISQC experiment is shown in Fig. 2C. In this ¹H–¹⁵N heteronuclear correlation experiment, the in-phase single-quantum term N_x or N_y is created at the beginning of the ¹⁵N evolution period, and the in-phase single-quantum coherence N₊ (=N_x + iN_y) is maintained via the ¹H WALTZ decoupling scheme throughout the evolution period. By maintaining the in-phase single-quantum terms N_x and N_y and thereby removing the scalar relaxation from the t₁ time domain for the ¹⁵N dimension, the HISQC experiment drastically improved the sensitivity and resolution in detection of signals from NH₃⁺ groups, compared to the corresponding HMQC and HSQC experiments (Fig. 2D). Although the typical range of the ¹⁵N chemical shifts of Lys NH₃⁺ groups is relatively narrow, the signal dispersion in HISQC spectra is typically excellent owing to the intrinsically slow ¹⁵N transverse relaxation of NH₃⁺ groups. Many NMR pulse sequences for NH₃⁺ groups have implemented the principle of HISQC and minimized the adverse impacts of scalar relaxation of antiphase terms with respect to ¹H nuclei (Anderson et al., 2013; Esadze et al., 2011; Esadze, Zandarashvili, & Iwahara, 2014; Iwahara et al., 2007; Zandarashvili et al., 2013; Zandarashvili, Li, Wang, Brüschweiler, & Iwahara, 2011).

2.3.2. HISQC Can Also Improve Arg ¹⁵N_ε Line Shapes

The hydrogen exchange rates k_ex,w of Arg N_εH groups are generally slower than those of Lys NH₃⁺ group. Furthermore, while hydrogen exchange of Lys NH₃⁺ groups broadens the Lys ¹⁵N line width by 1.5k_ex,w/π in HSQC, hydrogen exchange of Arg N_εH groups broadens the Arg ¹⁵N line width only by 0.5k_ex,w/π (see Fig. 2B). Because of the smaller impact of scalar relaxation arising from hydrogen exchange, the advantage of ¹H–¹⁵N HISQC for Arg N_εH groups is not as great as that for Lys NH₃⁺groups. However, when hydrogen exchange is rapid (e.g., due to a relatively high pH), the HISQC spectra of Arg N_εH groups can be significantly better than the HSQC spectra (Yuwen & Skrynnikov, 2014a). To maximize the sensitivity for Arg N_εH groups, the delay ω_b in the HISQC pulse sequence in Fig. 2D should be set to 2.7 ms. This setting also suppresses signals from Arg N_ηH₂ groups and simplifies the spectrum, allowing the use of a narrower spectral width for the ¹⁵N dimension. TROSY for Arg N_εH groups is not as advantageous as that for backbone NH groups because the ¹⁵N chemical shift anisotropy (CSA) of Arg N_ε (−114 ppm) (Trbovic et al., 2009) is significantly smaller than that of backbone N (−162 to −173 ppm) (Yao, Grishaev, Cornilescu, & Bax, 2010). The ¹⁵N dimension of TROSY also suffers from scalar relaxation arising from hydrogen exchange.

2.4. Resonance Assignment for Lys/Arg Side Chains

2.4.1. Resonance Assignment for Lys Side-Chain NH₃⁺ Groups

Several ¹H, ³C, ⁵N triple-resonance experiments for resonance assignment of Lys side-chain amino groups have been developed (Andre et al., 2007; Esadze et al., 2014; Iwahara et al., 2007). Resonance assignment for Lys side-chain NH₃⁺ groups requires long-range correlation spectra. Short-range correlations between NH₃⁺ and ¹H_ε/¹³C_ε resonances are usually insufficient for unambiguous assignment due to poor dispersion of their chemical shifts. Due to high mobility, few ¹H–¹H NOE cross-peaks are observed for Lys NH₃⁺ groups, rendering NOE-based resonance assignment difficult. We found that the combined use of (H2C)N(CC) H-TOCSY, H3NCECD, H3NCG, H2(C)N, and HISQC spectra can greatly facilitate assignment of ¹H and ¹⁵N resonances for Lys side-chain NH₃⁺ groups (Esadze et al., 2014). Fig. 3 shows an example of this approach. H2(C)N (Andre et al., 2007) and (H2C)N(CC)H-TOCSY (Esadze et al., 2014) experiments allow observation of the ¹⁵N resonances of Lys NH₃⁺ groups even if direct ¹H detection of the NH₃⁺ groups is impossible due to their rapid hydrogen exchange. Because multiple side-chain ¹H resonances are observed at high resolution along the ¹H direct-detection dimension with excellent separation by sharp ¹⁵N_ζ resonances, the (H2C)N(CC)H-TOCSY spectrum greatly facilitates ¹⁵N_ζ resonance assignment for Lys NH₃⁺ groups. The temperature dependence of the H2(C)N spectra allows us to track the change in ¹⁵N_ζ resonances as a function of temperature. These data together with the 3D H3NCG and H3NCECD spectra allowed us to unambiguously assign ¹H–¹⁵N HISQC signals from the Lys NH₃⁺ groups of some protein–DNA complexes at relatively low temperature.

Fig. 3.

Assignment of the ¹H and ¹⁵N resonances of Lys NH₃⁺ groups using 2D ¹³C-decoupled HISQC, 2D H2(C)N (Andre et al., 2007), 2D (H2C)N (CC)H-TOCSY (Esadze et al., 2014), 3D H3NCG (Esadze et al., 2014), and 3D H3NCECD (Esadze et al., 2014) spectra. The spectra were recorded for the ¹³C/¹⁵N-labeled Egr-1 zinc-finger–DNA complex at indicated temperatures and pH 5.8.

2.4.2. Resonance Assignment for Arg Guanidinium Moieties

Some ¹H, ³C, ⁵N triple-resonance experiments on Arg side chains have been proposed (Andre et al., 2007; Iwahara & Clore, 2006; Yamazaki et al., 1995). Compared to the guanidinium N_ηH₂ moieties, Arg ¹⁵N_ε and ¹H_ε resonances are relatively easy to assign. When broadband ¹⁵N 180° pulses are used, 3D HNCACB for backbone amide groups also gives signals from Arg side chains, which correlate Arg side-chain ¹H_ε, ¹⁵N_ε, and ¹³C_γ/¹³C_δ resonances (Iwahara & Clore, 2006). Alternatively, the same correlation signals can be observed selectively for Arg side chains by using ¹⁵N_ε-selective pulses (see Section 2.4.3) in 3D HNCACB. NOE cross-peaks of ¹H_ε nuclei in 3D ¹⁵N-edited NOESY also facilitate resonance assignment for Arg N_εH groups. Fig. 4 shows an example of Arg ¹⁵N_ε and ¹H_ε resonance assignment based on ¹³C_γ/¹³C_δ resonances and NOE data. Some double- or triple-resonance experiments were developed for resonance assignment of Arg side-chain N_ηH₂ groups (Gerecht et al., 2017; Löhr & Rüterjans, 1998; Yamazaki et al., 1995), though this is more challenging due to the above-mentioned hindered 180° rotations that tend to occur in the intermediate exchange regime at physiological temperature.

Fig. 4.

(A) ¹H–¹⁵N HISQC spectra recorded for the Arg N_εH groups of the Antp homeodomain–DNA complex at pH 5.8 and 25°C (Nguyen, Hoffpauir, et al., 2017). (B) Strips of 3D ¹⁵N-edited NOESY and broadband HNCACB (Iwahara & Clore, 2006) spectra for some Arg side chains of the Antp homeodomain–DNA complex. The broadband HNCACA spectrum shows correlation of Arg side-chain ¹H_ε, ¹³C_γ, ¹³C_δ, and ¹⁵N_ε resonances.

2.4.3. Selective ¹⁵N Pulses for Lys and Arg Side Chains

The ¹⁵N resonances of Lys (~25–35 ppm) and Arg (~65–90 ppm) side chains are significantly different from those of protein backbone (~105–135 ppm). Largely because the practically available RF strength for the ¹⁵N channel is relatively weak (up to ~7 kHz), standard ¹H–¹⁵N NMR experiments optimized for the backbone give significantly compromised signals from Lys and Arg side chains (Iwahara & Clore, 2006; Iwahara et al., 2007). The pulse imperfection for off-resonance ¹⁵N nuclei adversely impacts quantitative ¹H–¹⁵N heteronuclear NMR experiments, particularly measurements of ¹⁵N transverse relaxation rates. The simultaneous observation with a reasonably high digital resolution for ¹⁵N requires aliasing in the ¹⁵N dimension, which can cause accidental overlaps of nonaliased and aliased signals. For these reasons, we typically conduct ¹H–¹⁵N NMR experiments for the backbone, Arg side chains, and Lys NH₃⁺ groups separately. For selective observation of Lys or Arg side-chain signals, we use 1-ms rSNOB pulses (Kupče et al., 1995) as ¹⁵N 180° pulses selective for Lys NH₃⁺ or Arg N_εH ¹⁵N nuclei. The rSNOB pulse works as an inversion pulse for N_z and as a refocusing pulse for N_x and N_y magnetizations. The bandwidth of a 1-ms rSNOB pulse is approximately ±1.1 kHz and is well suited to selectively observe either Arg ¹⁵N_ε (75–90 ppm) nuclei or Lys ¹⁵N_ζ (30–35 ppm; NH₃⁺) nuclei. We incorporate ¹⁵N rSNOB pulses into many Lys- or Arg-selective NMR experiments.

2.4.4. Chemical Approach

A chemical approach can be used for resonance assignment of basic side chains that form intermolecular ion pairs with DNA or RNA (Anderson et al., 2015). In this approach, ¹H–¹⁵N correlation spectra for the basic side chains of the protein–DNA (or RNA) complexes are compared with and without oxygen-to-sulfur substitution in a backbone phosphate group. This chemical modification is relatively mild because it retains the tetrahedral geometry and overall charge of the phosphate group. Thiophosphoramidites, which are commercially available from Glen Research Co., allow synthesis of DNA containing phosphorodithioate using standard DNA synthesizers. Site-specific oxygen-to-sulfur substitution in a backbone phosphate significantly perturbs the NMR chemical shifts of the interfacial cationic group at the modification site and allows one to identify the NMR signal from the interfacial Lys or Arg groups. This approach was demonstrated initially with dithioate derivatives of phosphate (Anderson et al., 2015) and later with monothioate derivatives of phosphate (Nguyen, Zandarashvili, White, & Iwahara, 2016). Although synthesis of DNA/RNA containing a monothioate derivative produces a mixture of R_P and S_P diastereomers, chromatographic separation of these two forms is possible (Frederiksen & Piccirilli, 2009; Nguyen et al., 2016).

2.5. Sample Conditions for NMR Studies of Lys/Arg Cations

2.5.1. Two Critical Factors: pH and Temperature

A major challenge for NMR studies of Lys NH₃⁺ and Arg N_εH groups is their rapid hydrogen exchange with water. For example, the hydrogen exchange rates for Lys NH₃⁺ groups can exceed 1000s⁻¹ at pH 7.5 and 27°C (Segawa et al., 2008). Under such conditions, direct observation of the ¹H signals from Lys NH₃⁺ groups is virtually impossible. However, direct ¹H detection can become possible if hydrogen exchange is substantially slowed down due to (1) lower pH, (2) lower temperature, or (3) structural effects such as hydrogen bonding and steric hindrance (Esadze et al., 2011; Iwahara et al., 2007; Poon et al., 2006; Takayama et al., 2008; Tomlinson, Ullah, Hansen, & Williamson, 2009). For the feasibility of heteronuclear ¹H–¹⁵N NMR experiments for Lys and Arg side chains, temperature and pH are two critical factors that should be set carefully.

The dependence of hydrogen exchange rates on pH is particularly strong. In general, a decrease in pH by 1 unit reduces the hydrogen exchange rate by a factor of 10 (Englander, Downer, & Teitelbaum, 1972; Liepinsh & Otting, 1996). For example, if the pH is decreased from 7.5 to 5.8, the hydrogen exchange rates become 50-fold slower. It should be noted that hydrogen exchange rates depend on the buffer used (e.g., slower in a succinate buffer than in a phosphate buffer at the same pH). The temperature dependence of hydrogen exchange rates is also strong. Upon a decrease in temperature by 10°C, hydrogen exchange rates become ~two- to fourfold slower for Lys NH₃⁺ and Arg N_εH groups, depending on the activation energy. The temperature dependence of hydrogen exchange seems to be stronger for Arg N_εH groups than for Lys NH₃⁺ groups (Segawa et al., 2008). We typically conduct NMR experiments on Lys and Arg side chains at pH 5–6 and 2–35°C.

2.5.2. Coaxial NMR Tubes to Avoid Partial Deuteration by D₂O

For quantitative NMR experiments on Lys NH₃⁺ and Arg N_εH groups, a D₂O for NMR lock should not be included in the sample buffer. In general, the ¹⁵N chemical shifts of NH/NH₂/NH₃⁺ moieties are significantly different when they are partially deuterated in the form of NHD, NH₂D, or NHD₂ by D₂O. The partial deuteration causes not only additional signals (see Fig. 1B) (Iwahara et al., 2007) but also additional broadening effects on the ¹⁵N dimension due to the exchange between ¹H-attached and ²H-attached states for which ¹⁵N resonances differ (Yuwen & Skrynnikov, 2014a). We use coaxial NMR tubes to resolve the problems arising from deuterated NH/NH₂/NH₃⁺ moieties by separating D₂O from the sample solution. The sample solution (typically, 0.4–1.0 mM protein) does not contain D₂O, and therefore, deuteration of NH/NH₂/NH₃⁺ moieties does not occur. Depending on the applications, we use two different configurations of NMR coaxial tubes (Fig. 1C). In one type of NMR coaxial tube (e.g., Shigemi cat. # SC-002), the protein solution is in the inner tube, and D₂O is in the outer tube. For this type, we use a sample solution in the inner tube (270 μL) and D₂O in the outer tube (~100 μL). In the other type of NMR coaxial tube (e.g., Norell cat. # NI5CCI-B), we use a sample solution in the outer tube (370 μL) and D₂O in the inner stem (~ 100 μL). In general, this type of configuration with a thin inner stem gives a higher sensitivity for the same solution, mainly because the sample molarity in the detected region of the tube is higher due to a larger volume. However, a practical problem about this coaxial tube is that shimming is more difficult. Shimming for coaxial tube samples should not rely on deuterium signals because D₂O and the sample solution are located in different compartments. In our experience, the gradient shimming method based on H₂O imaging (van Zijl, Sukumar, Johnson, Webb, & Hurd, 1994) works fine in the one-dimensional mode but not in the three-dimensional mode to achieve magnetic-field homogeneity of solution samples in coaxial NMR tubes. When a high-salt concentration is required, the configuration with the sample in the inner tube is more suitable for the reason described in some papers on NMR studies of high ionic-strength samples (Takeda et al., 2011; Voehler, Collier, Young, Stone, & Germann, 2006).

2.5.3. Dealkalization of NMR Tubes

Due to the intrinsically rapid hydrogen exchange, the ¹H signals from Lys side-chain NH₃⁺ groups are sensitive to an increase in pH. Previously, we often found that a gradual increase in pH occurred over a few weeks in NMR tubes even in the presence of buffer. This increase was noticeable in NMR experiments because the signals from Lys NH₃⁺ groups became significantly weaker or even disappeared due to the more rapid hydrogen exchange. In fact, the original spectra of such samples were restored through buffer exchange. We speculate that the increase in pH might be due to gradual contamination of sodium ions leached from the glass surface through the Na⁺–H⁺ exchange mechanism (Bunker, 1994). We found that dealkalization treatment of NMR tubes can prevent the gradual increase in pH. In this treatment, we put 0.5 M nitric acid into coaxial NMR tubes, incubate the tubes in a hot water bath (60–80°C) for 2–3 h, extensively wash the tubes with pure water, and dry them in a heat dryer. We routinely perform this dealkalization for new coaxial tubes before their use for NMR experiments on Lys NH₃⁺ and Arg N_εH groups.

3. CONFORMATIONAL DYNAMICS OF Lys/Arg SIDE CHAINS

The ¹⁵N NMR relaxation and scalar coupling data of Lys NH₃⁺ and Arg N_εH groups can provide information on the mobility of the cationic moieties of these basic side chains. Here, we describe practical aspects of the analysis of the conformational dynamics of Lys and Arg side chains.

3.1. ¹⁵N Relaxation Analysis for Lys NH₃⁺ Groups

The theoretical details of the ¹⁵N relaxation of Lys side-chain NH₃⁺ groups are somewhat complicated due to cross-correlation within the AX₃ spin system and are beyond the scope of this chapter. Readers interested in the theoretical issues regarding the ¹⁵N relaxation of Lys NH₃⁺ groups should refer to our previous publication (Esadze et al., 2011). General descriptions of the cross-correlation effects on the NMR relaxation of AX₃ spin systems can be found in other articles (Kay & Torchia, 1991; Kumar, Grace, & Madhu, 2000; Ollerenshaw, Tugarinov, & Kay, 2003).

3.1.1. Removal of Adverse Effects of Multispin Terms

Fig. 5 shows the pulse sequences used to measure ¹⁵N relaxation for Lys side-chain NH₃⁺ groups. The first step for measuring ¹⁵N longitudinal (R₁) and transverse (R₂) relaxation rates is to create the ¹⁵N in-phase single-quantum term via coherence transfer from ¹H to ¹⁵N nuclei through a refocused INEPT scheme (Ernst, Bodenhausen, & Wokaun, 1987). The product operator terms N_x, 2N_yH_z, 4N_xH_zH_z, and 8N_yH_zH_zH_z are generated in the period of 2τ_b in the first refocused INEPT scheme of our pulse sequence for ¹⁵N R₁ and R₂ measurements. The 2N_yH_z and 8N_yH_zH_zH_z terms are eliminated by the pulsed field gradient (PFG) g₄ after the ¹H 90°(−x) and ¹⁵N 90°(γ) pulses at the end of the refocused INEPT scheme, but the 4N_xH_zH_z term is not eliminated because a PFG alone cannot destroy the homonuclear zero-quantum coherence 4N_zH_yH_y (van de Ven, 1995). The coefficients of the transfers from 2N_yH_z to N_x and to 4N_xH_zH_z are given by cos²θsinθ and (3cos²θ − 1)sinθ, respectively, in which θ = 2π ¹J_NH τ_b and ¹J_NH represents the one-bond ¹H–¹⁵N scalar coupling constant (Ernst et al., 1987). The use of the time τ_b satisfying 3cos²θ − 1 = 0 thus eliminates the 4N_xH_zH_z term but retains the N_x term, and this condition is achieved by ρ_b = 2.1 ms in the original pulse sequences (Esadze et al., 2011). This approach was also used for ¹³C R₁ and R₂ relaxation measurements for protein CH₃ groups (Kay et al., 1992; Palmer, Wright, & Rance, 1991). A practical problem of this approach is that it reduces the efficiency of the transfer from 2N_yH_z to N_x (i.e., cos²θsinθ) and weakens the signals from NH₃⁺ groups. Recently, we resolved this problem by implementing schemes that eliminate any contributions arising from 4N_xH_zH_z (Nguyen, Lokesh, et al., 2017). The pulse sequences shown in Fig. 5 implement such schemes and allow the use of τ_b = 1.3 ms to maximize cos²θsinθ, improving the sensitivity by a factor of 1.8 without compromising the accuracy in ¹⁵N relaxation measurements (Nguyen, Lokesh, et al., 2017).

Fig. 5.

¹⁵N relaxation measurements for Lys NH₃⁺ and Arg N_εH groups (Esadze et al., 2016, 2011; Nguyen, Lokesh, Volk, & Iwahara, 2017). Symbols representing pulses are the same as those used in Fig. 2C. The delays τ_a = 2.7 ms and τ_b = 1.3 ms are used for Lys NH₃⁺ groups, whereas τ_a = 2.3 ms and τ_b = 2.7 ms are used for Arg N_εH groups. The ¹⁵N carrier position is set to ~33 pm for Lys NH₃⁺ groups and to ~85 ppm for Arg N_εH groups. The ¹H carrier position is set to the H₂O resonance. Quadrature detection in the t₁ domain was achieved using States-TPPI, incrementing the phase φ₁. Although the use of τ_b = 1.3 ms for Lys NH₃⁺ groups retains 4N_xH_zH_z terms, the impact of these terms is eliminated in other parts of these pulse sequences (Nguyen, Lokesh, et al., 2017). To maintain the in-phase single-quantum ¹⁵N coherence, the ¹H WALTZ composite pulses are used during the t₁ period. (A) ¹⁵N longitudinal relaxation measurement. A ¹H 180° pulse that does not affect H₂O resonance is applied every 10 ms during the delay T_r for longitudinal relaxation. A recycle delay between scans is set to 2 s in this experiment. Phase cycles: φ₁ = (2y, 2(−y)), φ₂ = (y, −y), φ₃ = (4x, 4(−x)), φ₄ = (8y, 8(−y)), and receiver = (x, −x, −x, x, 2(−x, x, x, −x), x, −x, −x, x). (B) ¹⁵N transverse relaxation measurement. The ¹H carrier position is shifted to 7.8 ppm at the position of the first vertical arrow (after the PFG g₄) and set back to the position of water resonance at the position of the second arrow. Typically we use the CPMG frequency ν_CPMG of 417 Hz. For this, the RF strength ω_CW/2π of ¹H CW during the CPMG is set to 5 kHz, satisfying ω_CW/2π = 2kν_CPMG (k, integer) (Hansen, Vallurupalli, & Kay, 2008). However, when a lower CPMG frequency is used, a stronger RF strength is required for ¹H CW to maintain the in-phase single-quantum ¹⁵N coherence during the CPMG scheme. For Arg N_εH groups, the CPMG frequency, the RF strength, and ¹⁵N carrier position should be set to avoid the recoupling conditions for modulation by ²J_NN couplings (Nguyen & Iwahara, 2018). The delays ξ₁ and ξ₂ are for alignment of ¹H magnetization and given by ξ₁ = 1/ω_CW = (4/π)τ_90H and ξ₂ = τ_90N − (2/π)τ_90H (Hansen et al., 2008; Hansen & Kay, 2007), in which τ₉₀ represents the relevant 90° pulse length. A recycle delay of 2.9 s is used. Phase cycles: φ₁ = (4y, 4(−y)), φ₂ = (8y, 8(−y)), φ₃ = x, φ₄ = (x, −x), φ₅ = (2y, 2 (−y)), φ₆ = (2x, 2(−x)), φ₇ = (2(−y), 2y), and receiver = (x, −x, x, −x, 2(−x, x, −x, x), x, −x, x, −x). (C) Heteronuclear ¹H–¹⁵N NOE measurement. Measurement with ¹H saturation (5 s) is performed with a train of 180°x and 180°(−x) pulses (RF strength, 11 kHz) at an interval of 10 ms. The reference spectrum is measured without the scheme in the bracket. For measurements of heteronuclear NOE, we set a recycle delay (including the saturation period) to 12s for a 600-MHz spectrometer and to 18s for 750 or 800 MHz spectrometer. Phase cycles: φ₁ = (y, −y), φ₂ = (4x, 4y, 4(−x), 4(−y)), φ₃ = (2x, 2(−x)), and receiver = (x, −x, −x, x, −x, x, x, −x).

3.1.2. ¹⁵N R₁ Relaxation Measurements for Lys NH₃⁺ Groups

The ¹⁵N R₁ relaxation rates of Lys side-chain NH₃⁺ groups are typically 0.2–1.5 s⁻¹ at the ¹H frequencies of 600–800 MHz. To measure the Lys ¹⁵N R₁ rates with the pulse sequence shown in Fig. 5A, we typically record eight spectra in an interleaved manner with different lengths for the relaxation period T_r (=0.1, 0.2, 0.4, 0.6, 0.9, 1.2, 1.6, and 2.1 s). The minimum delay, T_r = 0.1 s, is long enough to let the 4N_zH_yH_y term completely decay through its rapid relaxation (Nguyen, Lokesh, et al., 2017). We record the eight datasets in an interleaved manner, typically with 16 scans per free induction decay (FID) using a cryogenic probe. The total time for recording is ~20–30 h. The ¹⁵N R₁ rates are determined through nonlinear least-squares fitting with a single exponential function.

3.1.3. ¹⁵N R₂ Relaxation Measurements for Lys NH₃⁺ Groups

Owing to rapid C_ε–N_ζ bond rotations, the Lys side-chain NH₃⁺ groups exhibit very slow ¹⁵N transverse relaxation. For the protein–DNA complexes we studied (17–30 kDa), while ¹⁵N T₂ relaxation times for the protein backbone NH groups were 50–80 ms, those for the Lys NH₃⁺ groups were typically >300 ms even for the Lys side chains that form ion pairs with DNA. Fig. 5B shows the pulse sequence for the ¹⁵N transverse relaxation measurement, in which the CW-CPMG scheme (Hansen et al., 2008) is implemented to maintain the in-phase single-quantum coherence N₊. This scheme contains two CPMG blocks (Carr & Purcell, 1954; Meiboom & Gill, 1958) that sandwich a refocusing pulse, and the ¹H continuous wave (CW) is applied to eliminate scalar relaxation during each CPMG blocks. As is known for ¹³C transverse relaxation for CH₃ groups, the ¹⁵N transverse relaxation for NH₃⁺ groups should intrinsically be biexponential due to the strong effects of DD–DD cross-correlation between three ¹⁵N–¹H DD interactions (Esadze et al., 2011). However, the biexponential decay and single-exponential decay according to exp(−R_2,iniT) are virtually indistinguishable for the first 30% of decay, for which the use of single-exponential fitting for determination of the initial transverse relaxation rate R_2,ini is valid (Esadze et al., 2011). The theoretical expression of the R_2,ini rate is formulated in Esadze et al. (2011), and its practical use is described below (Section 3.3). Due to the use of only <30% decay, measurements of R_2,ini rates for Lys NH₃⁺ groups are typically not as precise as those of R₁ rates in terms of error percentage. To compensate for this problem, we measure the ¹⁵N R_2,ini rates using a larger number of scans (32 scans per FID) than that for ¹⁵N R₁ measurements. We typically record nine spectra in an interleaved manner with different lengths for the transverse relaxation period (up to 120 ms). The total time for recording ¹⁵N transverse relaxation data for Lys NH₃⁺ groups is ~35–50 h.

The pulse sequence shown in Fig. 5B can also be used to detect microsecond to millisecond timescale motions of NH₃⁺ groups (Esadze et al., 2011) because the CW-CPMG scheme is well suited for relaxation dispersion experiments (Baldwin, Hansen, Vallurupalli, & Kay, 2009; Hansen et al., 2008). In fact, the relaxation dispersion data recorded with various CPMG frequencies in this pulse sequence were used to study the slow dynamics of Lys NH₃⁺ groups (Esadze et al., 2016, 2011).

3.1.4. Heteronuclear ¹⁵N NOE for Lys NH₃⁺ Groups

Due to rapid C_ε–N_ζ bond rotations, heteronuclear NOEs for Lys side-chain NH₃⁺ groups are negative and typically ranges from −1.8 to −3.3. The pulse sequence to record heteronuclear ¹⁵N NOE data for Lys NH₃⁺ groups is shown in Fig. 5C (Esadze et al., 2011; Nguyen, Lokesh, et al., 2017). The effects of DD/DD cross-correlation, which causes transitions between N_z and 4N_zH_zH_z terms and between H_z and 4N_zH_zH_z terms, are simplified by using 180° pulses instead of 120° pulses in the ¹H saturation period (Esadze et al., 2011). Incomplete recovery of water ¹H magnetization can introduce substantial errors in the heteronuclear NOEs and this problem is particularly serious when hydrogen exchange is rapid (Grzesiek & Bax, 1993; Idiyatullin, Daragan, & Mayo, 2001; Li & Montelione, 1994; Skelton et al., 1993). In the pulse sequence shown in Fig. 5C, this problem is avoided by implementation of the water-flip-back scheme (Grzesiek & Bax, 1993) and the use of a very long recycle delay (Renner, Schleicher, Moroder, & Holak, 2002). Our recycle delays in the heteronuclear NOE experiments for Lys NH₃⁺ groups are 12 and 18 s at the ¹H frequencies of 600 and 800 MHz, respectively. If a shorter time is used, the magnitude of heteronuclear NOE may appear to be larger due to weaker signal intensity in the reference spectrum. With 48 scans per FID and 100 complex points for each subspectrum, the total recording time is typically ~55–80h.

3.2. ¹⁵N Relaxation Analysis for Arg N_εH Groups

3.2.1. NMR Relaxation Measurements for Arg ¹⁵N_ε Nuclei

The pulse sequences shown in Fig. 5 can be used to measure ¹⁵N R₁, R₂, and heteronuclear NOE for Arg side-chain N_εH_ε groups as well, though changes in some parameters are required due to the differences between Lys NH₃⁺ and Arg N_εH groups. The caption of Fig. 5 shows the required changes, many of which are related to the different spin systems (AX₃ vs AX) and ¹J_NH coupling constants (74 vs 92 Hz) of Lys NH₃⁺ and Arg N_εH groups. In the ¹⁵N relaxation measurements for Arg N_εH groups, we typically use a ¹⁵N carrier position at ~84ppm and a ¹⁵N spectral width of ~6–12ppm. These settings together with the use of ¹⁵N rSNOB 180° pulses in the INEPT schemes allow ¹⁵N relaxation measurements for Arg side-chain ¹⁵N_ε nuclei only. Although simultaneous detection of Arg side-chain ¹⁵N_ε and backbone ¹⁵N nuclei is possible, separate measurements have the following advantages. First, selective observation of Arg ¹⁵N_ε allows the use of a narrower ¹⁵N spectral width for a high resolution without any concerns about undesired overlaps with folded signals from backbone NH or Lys NH₃⁺ groups. Second, R₂ measurements with the ¹⁵N carrier position set to the Arg N_ε resonance allow application of on-resonance 180° pulses in the CPMG scheme. Simultaneous R₂ measurements of backbone and Arg side-chain ¹⁵N nuclei require the use of a ¹⁵N carrier position that introduces significant off-resonance effects to the CPMG scheme (Korzhnev, Tischenko, & Arseniev, 2000) and decrease the data quality for either type of ¹⁵N nuclei. Furthermore, selective observation of Arg N_εH groups allows higher sensitivity because the conditions optimized for backbone NH groups considerably weaken the signals from Arg N_εH groups unless broadband ¹⁵N 180° pulses are used (Iwahara & Clore, 2006). For these reasons, we conduct the ¹⁵N relaxation experiments for backbone NH groups, Lys NH₃⁺ groups, and Arg N_εH group separately.

3.2.2. Impact of ¹⁵N–¹⁵N Couplings on Arg ¹⁵N_ε R₂ Measurements

The presence of two geminal couplings to ¹⁵N_η1 and ¹⁵N_η2 nuclei can in principle affect measurements of transverse relaxation of Arg ¹⁵N_ε nuclei. However, our recent study shows that the impact of the ²J_NN couplings on CPMG-based transverse relaxation measurements for Arg ¹⁵N_ε nuclei is typically negligible (Nguyen & Iwahara, 2018). This result is due to the relatively small magnitudes of the ²J_NN couplings (1.1–1.3 Hz) and also due to the so-called SITCOM mechanism (Dittmer & Bodenhausen, 2006) that efficiently quenches homonuclear J modulation. The use of selective refocusing pulses (e.g., REBURP) in the CPMG scheme was previously proposed to avoid modulation through the homonuclear ¹⁵N–¹⁵N scalar couplings (Yuwen & Skrynnikov, 2014b). However, the use of long shaped pulses during the CPMG scheme allows only low ν_CPMG frequencies, which may not completely cancel the exchange contribution to the measured R₂ relaxation rates. Such use of ¹⁵N_ε-selective shaped pulses during the CPMG scheme is unnecessary because the SITCOM effect effectively quenches the evolution of ²J_NN couplings of Arg side chains during the CPMG scheme with typical rectangular ¹⁵N 180° pulses. However, the ¹⁵N_ε R₂ rates may be overestimated for very mobile Arg side chains, if the following condition (recoupling condition) (Aeby & Bodenhausen, 2008) is accidentally met:

$${\Omega }_S/\left(2\pi \right)=4n{\nu }_1{\nu }_{\mathit{CPMG}}/\left(2{\nu }_1+{\nu }_{\mathit{CPMG}}\right)$$ (1)

where n is an integer; Ω_S/(2π), the offset in Hz for off-resonance spin S; ν₁, the RF strength in Hz for refocusing pulses; ν_CPMG, the CPMG field frequency in Hz. When ν₁ ≫ ν_CPMG, Eq. (1) becomes Ω_S/(2π) ≈ 2nν_CPMG. When the difference between Ω_S/(2π) and the nearest recoupling condition is less than 10% of ν_CPMG, significant amplitude modulation occurs and the apparent R₂ rate becomes larger than the actual R₂ relaxation rate. Not surprisingly, this adverse effect is weaker for a smaller magnitude of the homonuclear J coupling. When Ω_S/(2π) is more significantly different from the recoupling conditions, effectively complete quenching of J-modulation occurs, allowing accurate measurements of the R₂ rates. Recoupling conditions can be readily avoided through appropriate settings of the CPMG frequency ν_CPMG, the RF strength ν₁, and the ¹⁵N carrier position.

3.2.3. ¹⁵N_ε Relaxation Analysis Using Direct ¹³C_ζ Detection

Hansen and coworkers analyzed Arg side-chain ¹⁵N_ε relaxation through direct detection of ¹³C_ζ nuclei (Werbeck, Kirkpatrick, & Hansen, 2013). A remarkable advantage of this approach is that it can be used at any pH because rapid hydrogen exchange does not impact ¹³C_ζ detection. In fact, this method was successfully applied to a 42-kDa protein at pH 8.2. ¹H detection of Arg side-chain N_εH groups at this pH is virtually impossible due to the rapid hydrogen exchange with water. Because the ¹³C_ζ resonances are unaffected by hydrogen exchange, ¹⁵N_ε relaxation analysis through direct ¹³C_ζ detection is possible. A drawback of this method is that it provides relaxation rates for 2C_zN_x and 2C_zN_z terms but not those for N_x and N_z terms. Werbeck et al. demonstrated that the order parameters determined using this method agree well with those determined using the standard ¹⁵N relaxation method.

3.3. Determination of Lys/Arg Side-Chain Order Parameters

3.3.1. Overall Procedures

The ¹⁵N relaxation data for Lys and Arg side chains at multiple magnetic fields are used to calculate order parameters for relevant bonds. The overall procedures for this calculation are shown in Fig. 6A. To determine the order parameter and the correlation times for internal motions, the following function is minimized for each side chain:

$${\chi }^2=\sum _{\mathit{Type}}\sum _{\mathit{Field}}\frac{{\left(Y_{\mathit{obs}}-Y_{\mathit{cal}}\right)}^2}{w^2},$$ (2)

where Y_obs and Y_cal stand for the observed and calculated quantities. The squared difference terms are summed for different types of relaxation parameters (i.e., R₁, R₂, and NOE) and for different magnetic fields. The use of multiple magnetic fields is important because up to three parameters for internal motions are optimized. Based on availability of our NMR spectrometers, we typically combine ¹⁵N relaxation data at 600 and 800 MHz for Lys NH₃⁺ groups, and combine those at 600 and 750 MHz for Arg N_εH groups. In Eq. (2), 1/w² corresponds to the weight of each squared difference (Y_obs − Y_cal)². For this kind of fitting calculation, two different weights have been used: w = Y_obs (e.g., Dellwo & Wand, 1989; Nicholson et al., 1992) or w = σ, where σ represents the uncertainty in Y_obs (e.g., Mandel, Akke, & Palmer, 1995; Palmer, Rance, & Wright, 1991). Although the latter approach may seem to be more reasonable, it is not necessarily the case when there is bias causing a large systematic error in Y_obs that deviates from the true value. An example of such bias is on NOE data impacted by the saturation transfer effect (Grzesiek & Bax, 1993; Li & Montelione, 1994). The approach using w = Y_obs is safer if some systematic errors may exist in the experimental data. Calculation of Y_cal requires the use of an appropriate spectral density function relevant to the ¹⁵N relaxation parameters. The parameters involved in the spectral density function are optimized through minimization of the sum of the squared differences represented by Eq. (2).

Fig. 6.

(A) Procedures for determination of the order parameters S² for Lys and Arg side chains. (B) Motion model for Lys side-chain NH₃⁺ groups.

3.3.2. Determination of Molecular Rotational Correlation Time

Calculation of Y_cal in Eq. (2) requires the molecular rotational correlation time. However, the ¹⁵N relaxation data of Lys and Arg side chains are not suitable for determination of the molecular rotational correlation time because many Lys and Arg side chains are highly flexible and internal motions govern the ¹⁵N relaxation parameters. Therefore, the molecular rotational correlation time is determined using backbone ¹⁵N R₁ and R₂ relaxation data for rigid regions of the same protein under the identical conditions. For calculations of rotational diffusion anisotropy, the atomic coordinates of the 3D structure are also required. In many cases, the axially symmetric diffusion model (Woessner, 1962) is suitable for the rotational diffusion calculation. Through nonlinear least-squares fitting, rotational diffusion parameters (D_∥, D_⊥, and two polar angles for the main principal axis) for the axially symmetric diffusion model are determined from the R₁/R₂ ratio data for backbone ¹⁵N nuclei (Tjandra, Feller, Pastor, & Bax, 1995). For this calculation, we use a C program (Iwahara, Peterson, & Clubb, 2005). The effective rotational correlation time τ_r,eff and the anisotropy of the rotational diffusion r are given by (2D_∥ + 4D_⊥)⁻¹ and D_∥/D_⊥, respectively (Woessner, 1962). The effective rotational correlation time τ_r,eff is used as a fixed-value parameter to calculate Y_cal when Lys/Arg side-chain order parameters are determined through minimization of Eq. (2).

3.3.3. Determination of Lys NH₃⁺ Order Parameters

The spectral density function for dipole–dipole (DD) autorelaxation of the ¹⁵N nucleus in an NH₃⁺ group is given by (Esadze et al., 2011):

$$j_{\mathit{DD}}\left(\omega \right)=\frac{{\zeta }^2}{4\pi }\left[\frac{\left(1/9\right)S_{\mathit{axis}}^2{\tau }_m}{1+{\omega }^2{\tau }_m^2}+\frac{\left(8/9\right)S_{\mathit{axis}}^2{\tau }_1}{1+{\omega }^2{\tau }_1^2}+\frac{\left(1/9\right)\left(1-S_{\mathit{axis}}^2\right){\tau }_2}{1+{\omega }^2{\tau }_2^2}+\frac{\left(8/9\right)\left(1-S_{\mathit{axis}}^2\right){\tau }_3}{1+{\omega }^2{\tau }_3^2}\right]$$ (3)

where $\zeta =\sqrt{(6\pi /5)}({\mu }_0/4\pi )(h/2\pi ){\gamma }_H{\gamma }_N{r}_{\mathit{NH}}^{-3}$; μ₀, the vacuum permeability; h, Planck’s constant; γ, the nuclear gyromagnetic ratio; r_ij, the N–H distance; $S_{\mathit{axis}}^2$, the generalized order parameter for the symmetry axis of the NH₃⁺ group; τ_m, the overall molecular rotational correlation time; ${\tau }_1^{-1}-={\tau }_m^{-1}+{\tau }_f^{-1}$; ${\tau }_{2-1}={\tau }_m^{-1}+{\tau }_i^{-1}$; ${\tau }_{3-1}={\tau }_m^{-1}+{\tau }_f^{-1}+{\tau }_i^{-1}$; τ_i, the correlation time for reorientation of the symmetry axis; and τ_f, the correlation time for bond rotation around the symmetry axis. Fig. 8B depicts the physical meanings of the parameters $S_{\mathit{axis}}^2$, τ_f, and τ_i. The spectral density function of Eq. (3) is a modification of Eq. (15) of Kay and Torchia (1991) for CH₃ groups, and is derived from the product of correlation functions for overall motions, internal motions of symmetry axis, and bond rotation (Esadze et al., 2011). The total number of parameters involved is identical for these equations and if τ_i ≫ τ_f, the equations become the same. Eq. (3) is more general because it is applicable even when the inequality τ_i ≫ τ_f does not hold. This aspect of Eq. (3) is important because hydrogen bonds may slow down the bond rotation and make τ_i < τ_f. The ¹⁵N relaxation through CSA is negligible because |σ_∥ – σ_⊥| is as small as 15 ppm for Lys NH₃⁺ groups (Sarkar et al., 1987). With the spectral density function j_DD(ω), the ¹⁵N relaxation parameters of the NH₃⁺ group are given by:

$$R_1=3j_{\mathit{DD}}\left({\omega }_N\right)+6j_{\mathit{DD}}\left({\omega }_H+{\omega }_N\right)+j_{\mathit{DD}}\left({\omega }_H-{\omega }_N\right)$$ (4)

$$R_{2,\mathit{ini}}=2j_{\mathit{DD}}\left(0\right)+\left(3/2\right)j_{\mathit{DD}}\left({\omega }_N\right)+3j_{\mathit{DD}}\left({\omega }_H+{\omega }_N\right)+3j_{\mathit{DD}}\left({\omega }_H\right)+\left(1/2\right)j_{\mathit{DD}}\left({\omega }_H-{\omega }_N\right)$$ (5)

$$\mathit{NOE}=1+\left({\gamma }_H/{\gamma }_N\right)\left(6j_{\mathit{DD}}\left({\omega }_H+{\omega }_N\right)-j_{\mathit{DD}}\left({\omega }_H-{\omega }_N\right)\right)/R_1,$$ (6)

Fig. 8.

NMR experiments for observing hydrogen-bond scalar couplings of Lys NH₃⁺ groups (Anderson et al., 2013; Zandarashvili et al., 2011). (A, B) 2D heteronuclear correlation experiments to observe signals arising from hydrogen-bond scalar couplings. Resonances of the coupling partner nuclei (¹³C═O for (A) and ³¹P for (B)) are directly observed in these experiments. Symbols representing pulses are the same as those used in Fig. 2C. Shaped ¹³C pulses represent IBURP-2 180° pulses (1.1 ms). Delays: τ_a = 2.7 ms; δ = 2.6 ms, T = 212 ms; and T_d = 206 ms. Carrier positions: ¹H, the position of the water resonance; ¹⁵N, 33 ppm; ¹³C_ε, 45 ppm; ¹³C═O, 177 ppm; and ³¹P, −3ppm (³¹P referencing to trimethyl phosphate). Phase cycles: φ₁ = (2x, 2(−x)), φ₂ = (4x, 4(−x)), φ₃ = (x, −x), φ₄ = (8x, 8(−x)), φ₅ = (8x, 8y, 8(−x), 8(−y)), and receiver = (x, −x,− x, x, 2(−x, x, x, −x), x, −x, −x, x). Quadrature detection in the t₁ domain was achieved using States-TPPI for φ₁. (C, D) 2D spin-echo J-modulation constant-time HISQC experiments to measure the magnitude of ^h3J_NC (C) or ^h3J_NP (D) couplings. Phase cycles: ψ₁ = (2x, 2 (−x)), ψ₂ = (4x, 4(−x)), ψ₃ = (y, −y), ψ₄ = (8x, 8y, 8(−x), 8(−y)), and receiver = (x, −x, −x, x, 2(−x, x, x, −x), x, −x, −x, x). Quadrature detection in the t₁ domain was achieved using States-TPPI for ψ₁. (E) Observation of ^h3J_NC couplings for the Lys NH₃⁺ groups of ubiquitin (Zandarashvili et al., 2011). (F) Observation of intermolecular ^h3J_NP couplings between Lys side-chain ¹⁵N and DNA ³¹P nuclei (Anderson et al., 2013). For the spin-echo J-modulation data, the difference spectra (i.e., subspectrum b minus subspectrum a) are shown to demonstrate the presence of the hydrogen-bond scalar couplings, although the magnitudes of these couplings are determined using the signal intensities in the individual subspectra, as described in Section 4.1.

For calculations of the order parameters and the internal motion correlation times for Lys NH₃⁺ groups, the complete source code of the Mathematica® script is available in the supporting information of Esadze et al. (2011). This script does not take the anisotropy of molecular rotational diffusion into consideration because the rotational anisotropy exerts only very minor effects on ¹⁵NH₃⁺ relaxation, as is evident from Eq. (6) of Tjandra et al. (1995) and the small overall order parameters (i.e., $S_{\mathit{axis}}^2/9$) for NH₃⁺ groups.

3.3.4. Determination of Arg N_εH Order Parameters

Since each Arg side-chain ${ }^{15}{\text{N}}_{\epsilon }^1\text{H}$ group can be regarded as an AX spin system, the relaxation parameters of Arg ¹⁵N_ε nuclei can be analyzed using the same equations as those for backbone ¹⁵N nuclei. Therefore, computer programs to calculate the order parameters and the correlation times for the internal motions of backbone NH groups [e.g., the ModelFree program by Palmer (Mandel et al., 1995); the Mathematica® script by Spyracopoulos (Spyracopoulos, 2006)] can be used to compute the corresponding model-free parameters for Arg N_εH groups. However, it should be noted that different settings are required for the ¹⁵N CSA values and the ¹⁵N–¹H distance r_NH of Arg side-chain N_εH groups: |σ_∥ − σ_⊥| = 114 ppm and r_NH = 1.04 Å should be used for Arg side-chain ¹⁵N_ε nuclei (Trbovic et al., 2009), whereas |σ_∥ − σ_⊥| = 170 ppm and r_NH = 1.02 Å are typically used for backbone ¹⁵N nuclei (Tjandra, Szabo, & Bax, 1996).

3.3.5. Different Dynamic Properties of Lys and Arg Side Chains

To show typical ranges of Lys/Arg side-chain order parameters, the data for the Lys and Arg side chains of the Antp homeodomain and the Egr-1 zinc-finger protein in the free and DNA-bound states are shown in Fig. 7. Note that the order parameters $S_{\mathit{axis}}^2$ of Lys NH₃⁺ groups are defined for the C₃ symmetry axis, which corresponds to the C_ε–N_ζ bond. Although the same number of rotatable bonds (i.e., four covalent bonds) is involved between Lys C_α and C_ε atoms and between Arg C_α and N_ε atoms, the ranges of the measured order parameters were remarkably different for Lys C_ε–N_ε and Arg N_ε–H_ε bonds. Compared to Lys C_ε–N_ε bonds, Arg N_ε–H_ε bonds exhibited a wider range of order parameters as well as larger changes in order parameters upon binding to DNA. The mobility of Arg side chains seems to be more sensitive to the surrounding environment, compared to Lys side chains.

Fig. 7.

Lys and Arg side-chain order parameters determined for the Egr-1 zinc-finger protein and the Antp homeodomain in the free form and in the complexes with specific DNA (Esadze et al., 2016; Nguyen, Hoffpauir, et al., 2017).

3.4. ³J_CN Measurements to Analyze Side-Chain Dynamics

Three-bond J-coupling data reflect the dynamics of torsion angles (Chou, Case, & Bax, 2003; Perez, Löhr, Rüterjans, & Schmidt, 2001; Zandarashvili et al., 2011). Those between ¹⁵N and ¹³C nuclei are relatively easy to measure using quantitative J-modulation experiments. The spin-echo J-modulation constant-time HSQC or HNCG experiments (Hu & Bax, 1997) can be used to measure the ³J_NC couplings between backbone ¹⁵N and side-chain ¹³C_γ nuclei relevant to χ₁ angles of Lys and Arg side chains and other side chains. For ³J_CN couplings between Lys ¹³C_γ and ¹⁵N_ζ nuclei relevant to χ₄ angles can be measured through the spin-echo ³J_CN-modulation constant-time HISQC experiment selective for NH₃⁺ groups (Zandarashvili et al., 2011). The ³J_CN coupling constants are typically less than 0.5–3 Hz. The precision of the measured coupling constants ³J_CN relevant to χ₄ angles is high owing to the very slow ¹⁵N relaxation of the Lys NH₃⁺ groups allowing the use of a long period for J-modulation. ³J_CN data can be interpreted by means of the Karplus equation (Karplus, 1959) together with empirical parameters (see Section 5.1). Due to the conformational fluctuation, the experimental ³J_NC data agree better with those predicted from MD trajectories than with those calculated from crystal structures (see Section 5.1).

4. ION PAIRS AND HYDROGEN BONDS INVOLVING Lys/Arg SIDE CHAINS

NMR studies of Lys and Arg side chains provide unique information on electrostatic interactions, hydrogen bonds, and their dynamics. Recent NMR studies illuminated the dynamic nature of short-range electrostatic interactions via ion pairs (also known as salt bridges).

4.1. Hydrogen-Bond Scalar Couplings

4.1.1. Heteronuclear Correlation via Hydrogen-Bond Scalar Couplings

Hydrogen-bond scalar couplings reflect the orbital overlaps in hydrogen bonds and provide unique information about hydrogen bonding (Grzesiek, Cordier, Jaravine, & Barfield, 2004). Hydrogen-bond scalar couplings provide direct evidence of hydrogen bonds and, in fact, are included in IUPAC’s recently updated criteria for hydrogen bonds (Arunan et al., 2011). Hydrogen-bond scalar couplings can also provide information about dynamics because these couplings are influenced by the transient distortion or breakage of the hydrogen bonds as well (Jaravine, Alexandrescu, & Grzesiek, 2001; Markwick, Sprangers, & Sattler, 2003; Zandarashvili et al., 2016, 2011).

Owing to the very slow ¹⁵N transverse relaxation of NH₃⁺ groups, which permits the use of a long period for J-modulation, hydrogen-bond scalar coupling constants in the range of 0.1–1.0 Hz can readily be measured for Lys side chains of midsize (~30kDa) systems studied by NMR. This allows direct detection and characterizations of hydrogen bonds involving Lys NH₃⁺ by NMR. Fig. 8A and B shows the NMR pulse sequences for observing ¹⁵N–¹³C or ¹⁵N–³¹P scalar couplings across hydrogen bonds involving Lys NH₃⁺ groups (Anderson et al., 2013; Zandarashvili et al., 2011). The F₂ dimension corresponds to ¹Hζ resonances of the NH₃⁺ groups, and the F₁ dimension corresponds to ¹³C or ³¹P resonances of nuclei coupled to Lys ¹⁵N_ζ. Some examples are shown in Fig. 8E and F. The ^h3J_NP couplings with DNA phosphate ³¹P nuclei were observed for the interfacial Lys side chains of several protein–DNA complexes (Anderson et al., 2013, 2015; Chen et al., 2015; Esadze et al., 2016; Zandarashvili, Esadze, et al., 2015; Zandarashvili, Nguyen, et al., 2015).

4.1.2. Determination of Hydrogen-Bond Scalar Coupling Constants

Although the abovementioned heteronuclear correlation spectra via hydrogen-bond scalar couplings are useful to identify the acceptor groups for the hydrogen bonds of Lys NH₃⁺ groups, these spectra do not provide quantitative information on the hydrogen-bond scalar couplings. The values of |^h3J_NC| or |^h3J_NP| can be determined using the spin-echo ^h3J-modulation constant-time HISQC experiments selective for NH₃⁺ groups (Fig. 8C and D) (Anderson et al., 2013; Zandarashvili et al., 2011). In these experiments, two subspectra are recorded in an interleaved manner: one with the ¹³C or ³¹P 180° pulses at position a (subspectrum a) and the other at position b (subspectrum b). The values of |^h3J| can be calculated from I_a/I_b = cos2πJ(T_d + δ), in which I_a and I_b represent signal intensities in subspectra a and b, respectively. For ^h3J_NP coupling, we use a cryogenic ¹H/¹³C/¹⁵N/³¹P QCI probe (Bruker) operated at the ¹H frequency of600 MHz. This probe is well suited to study hydrogen-bond scalar couplings involving DNA/RNA ³¹P nuclei. Using this probe, we were also able to observe ^h3J_NP and ^h2J_HP couplings between zinc-coordinating His side-chain ¹H/¹⁵N and DNA phosphate ³¹P nuclei in the Egr-1 zinc-finger–DNA complex as well (Chattopadhyay, Esadze, Roy, & Iwahara, 2016).

4.2. Dynamic Ion Pairs and Hydrogen Bonds

4.2.1. Dynamic Hydrogen Bonding and Ion Pairing

Although one may expect that side chains forming hydrogen bonds and/or ion pairs could be immobilized, recent NMR data show that this situation is not always the case for basic side chains (Anderson et al., 2013; Chen et al., 2015; Esadze et al., 2016; Nguyen, Hoffpauir, et al., 2017; Trbovic et al., 2009; Zandarashvili et al., 2011). Hydrogen bonds involving basic side chains can rapidly break and form, depending on the energy barriers involved. If the breakage and formation of hydrogen bonds occur on a picosecond–nanosecond timescale, the NMR order parameters and internal motion correlation times reflect the hydrogen-bond dynamics. A recent NMR study on the temperature dependence of the Lys side-chain ¹⁵N relaxation suggested that the free energy of activation (ΔG^‡) is as small as 2–5 kcal/mol for breaking hydrogen bonds of Lys NH₃⁺ groups, though the enthalpy of activation (ΔH^‡) is as large as 5–14kcal/mol (Zandarashvili & Iwahara, 2015). Recent NMR studies showed that the Lys NH₃⁺ groups interacting with DNA phosphate groups exhibit small order parameters $(S_{\mathit{axis}}^2<0.5)$ and sizable ^h3J_NP couplings, indicating the coexistence of high mobility and hydrogen bonding (Anderson et al., 2013; Chen et al., 2015; Esadze et al., 2016; Nguyen, Hoflpauir, et al., 2017). This somewhat puzzling behavior can be understood in terms of the dynamic equilibrium between the contact ion-pair (CIP) and solvent-separated ion-pair (SIP) states (Iwahara, Esadze, & Zandarashvili, 2015), as described in Section 5.2. It should be noted that the CIP–SIP transitions do not decouple (but do attenuate) the ^h3J_NP couplings between the Lys ¹⁵N and DNA ³¹P nuclei as long as the |α〉 or |β〉 spin states of the coupled ³¹P nuclei are largely maintained during the ^h3J_NP evolution period for the ¹⁵N coherence. Small order parameters of the interfacial Lys NH₃⁺ groups can also be explained by the CIP–SIP transitions occurring on a pico-to-nanosecond timescale, as suggested by MD simulations (Chen et al., 2015; Esadze et al., 2016).

4.2.2. Self-decoupling Due to Molecular Dissociation

Hydrogen-bond scalar couplings are very useful data that represent direct evidence of hydrogen bonds. However, it should be noted that the intermolecular hydrogen-bond scalar couplings between two molecules can be observed only when the residence time of the complex is sufficiently long. When the residence time is shorter than or comparable to 1/(π^hJ), the self-decoupling effect makes the apparent coupling constant smaller than ^hJ. We recently studied the self-decoupling effect theoretically and experimentally (Zandarashvili et al., 2016). Self-decoupling is easy to qualitatively understand. Consider a protein side-chain ¹⁵N nuclei coupled to DNA phosphate ³¹P nuclei for example. In solution, the protein can dissociate from a DNA molecule and associate with another DNA molecule of the same kind. When this happens, the spin state of the interacting ³¹P nuclei can change from |α〉 to |β〉 or vice versa. Therefore, rapid translocation of the protein effectively mixes the |α〉 to |β〉 spin states of the coupled ³¹P nuclei and causes decoupling of ^h3J_NP for protein ¹⁵N nuclei. A Liouville equation description allows the self-decoupling effect of intermolecular hydrogen-bond scalar coupling to be quantitatively analyzed and gain information about the residence time of macromolecular complexes. This approach was demonstrated for intermolecular hydrogen-bond scalar couplings between Lys side-chain ¹⁵N and DNA ³¹P nuclei measured for protein–DNA complexes at various ionic strengths (Zandarashvili et al., 2016). The self-decoupling-based method is unique in that it does not require different chemical shifts for the interconverting states, while such conditions are crucial for other NMR methods for kinetic investigations of protein–DNA interactions (Iwahara, Zandarashvili, Kemme, & Esadze, 2018). Although self-decoupling is typically regarded as a nuisance in NMR scalar coupling measurements (Cavanagh et al., 2007; Harbison, 1993; Kuboniwa, Grzesiek, Delaglio, & Bax, 1994), the quantitative analysis of self-decoupling presents a useful tool for kinetic investigations of processes that accompany breakage of intermolecular hydrogen bonds.

5. DATA INTERPRETATION FACILITATED BY MOLECULAR DYNAMICS SIMULATIONS

NMR spectroscopy can provide atomic-level information about the structures and dynamics of proteins and their complexes, but experimental data are available only for particular analyzable atoms of proteins. In contrast, MD simulations can give information on all atoms of the system including solvent molecules and free ions, but the computational studies are based on empirical models that need experimental validation, especially for electrostatic interactions. NMR and MD are complementary in this sense. MD simulations have greatly facilitated the interpretation of NMR data on basic side chains and their electrostatic interactions (Chen et al., 2015; Esadze et al., 2016, 2011; Trbovic et al., 2009; Zandarashvili et al., 2011). In what follows, we describe how MD simulations can facilitate the interpretation of NMR data on basic side chains.

5.1. NMR vs MD for Dynamics of Basic Side Chains

5.1.1. Lys/Arg Side-Chain Order Parameters From MD Trajectories

The calculation of order parameters from MD trajectories and comparison of experimental and computational order parameters have been conducted since 1980s (Chatfield, Szabo, & Brooks, 1998; Levy et al., 1985; Lipari, Szabo, & Levy, 1982). However, it is only recently that NMR- and MD-derived order parameters were compared for cationic moieties of basic side chains (Chen et al., 2015; Esadze et al., 2016, 2011; Trbovic et al., 2009). For such comparison, the behavior of Lys C_ε–N_ζ and Arg N_ε–H_ε bonds in the MD trajectories should be analyzed. Lys C_ε–N_ζ bonds are used because the NMR order parameter for each Lys side-chain NH₃⁺ is defined for its symmetry axis.

The dynamic behavior of a bond vector from MD trajectories can be studied by calculating the autocorrelation function for internal motions C_I(t). This function is given by (Case, 2002; Lipari & Szabo, 1982):

$$C_I\left(t\right)=\langle P_2\left(\mu \left(t_0\right)\mu \left(t_0+t\right)\right)\rangle ,$$ (7)

where μ(t₀)μ(t₀+t) is the projection of a unit vector pointing along a bond vector at time t₀ onto itself at time t₀ + t; P₂(x) = (3x² − 1)/2 is the second Legendre polynomial; and the brackets denote a time average over the trajectory. The trajectory frames are superimposed onto an optimal (in the least squares sense) common reference frame to remove the effects of overall tumbling and translation. Then, the time dependence of the autocorrelation for the reorientational motion is better described in Clore’s extended model-free formulism (Clore et al., 1990):

$$C_I\left(t\right)=S^2+\left(1-{S_f}^2\right)\exp \left(-t/{\tau }_f\right)+\left({S_f}^2-S^2\right)\exp \left(-t/{\tau }_i\right),$$ (8)

where $S_f^2$ and τ_f correspond to the amplitude and correlation time due to fast librational motion; and S² and τ_i are the order parameter and correlation time of the reorientational motion of a bond vector. Using a single exponential in the case of methyl group dynamics is often satisfactory (Frederick, Sharp, Warischalk, & Wand, 2008), while for the more chemically complex basic side chains, we found that method of analysis gave poor results. The implication is that these side chains have a rich dynamical range of motion and are strongly coupled to their surroundings with varied time scales. Some C_I(t) functions calculated for the Lys C_ε–N_ζ and Arg N_ε–H_ε bonds of the Egr-1 zinc-finger protein bound to its target DNA are shown in Fig. 9A. Use of sufficiently long (on μs order) MD trajectories is important for accurate computation of the order parameters (Bowman, 2016). This seems to be particularly important for the analysis of Arg side chains because we noticed that the C_I(t) functions for Arg N_ε–H_ε bonds tend to exhibit slow convergence or decay.

Fig. 9.

Comparison of NMR-derived vs MD-derived order parameters for the Lys and Arg side chains of the Egr-1 zinc-finger protein bound to its target DNA. (A) Examples of autocorrelation functions calculated for Lys C_ε–N_ζ and Arg N_εH bonds from 0.6-μs MD trajectories. (B) Correlation between the NMR-derived and MD-derived order parameters for the Lys NH₃⁺ and Arg N_εH groups of the Egr-1–DNA complex.

The correlation between the NMR and MD order parameters for the Lys and Arg side chains of the Egr-1 zinc-finger protein bound to its target DNA is shown in Fig. 9B. In general, the order parameters calculated from MD trajectories tend to be smaller than the NMR-derived order parameters, as pointed out by Brüschweiler and coworkers (Gu, Li, & Brüschweiler, 2014). This is because the impact of C_I(t) on the overall correlation function relevant to NMR relaxation is scaled by exp(−t/τ_r) (Lipari & Szabo, 1982), thereby making NMR data insensitive to motions on a 10⁻⁸–10⁻⁶s timescale. Since τ_r is typically 5–50ns for most proteins analyzed by NMR, an order parameter as a plateau of C_I(i) beyond 50 ns has only little impact on NMR relaxation. In addition, depending on the details of the system, some find consistent estimates of S² from different force fields (Bowman, 2016), while the details of methyl dynamics can display more variations (O’Brien et al., 2016). Nevertheless, significant correlations have been found between the NMR and MD order parameters for Arg/Lys side chains (Chen et al., 2015; Esadze et al., 2016, 2011; Trbovic et al., 2009). These correlations with the experimental data validate MD-based investigations into the properties of basic side chains that are difficult to assess by NMR (see Sections 5.2 and 5.3).

5.1.2. Three-Bond Scalar Couplings From MD Trajectories

The dihedral angles χ of long side chains can dynamically change through conformational fluctuations. Three-bond ³J_NC coupling constants relevant to Lys and Arg side-chain χ angles can be measured through spin-echo ³J_NC modulation experiments, as described in Section 3.4. The three-bond coupling constants can directly be compared with those predicted from MD trajectories. This analysis utilizes the Karplus equation (Karplus, 1959) for the relationship between the ³J_NC coupling constant and the dihedral angle χ between the coupled ¹⁵N and ¹³C nuclei:

$${ }^3{J}_{\mathit{NC}}=A{\cos }^2\chi +B\cos \chi +C$$ (9)

The parameters A, B, and C are empirically determined, for example, A = 1.29, B = −0.49, and C = 0.37 for χ₁ angles (Perez et al., 2001) and A = 1.29, B = −0.49, and C = 0.63 for Lys χ₄ angles (Huang & MacKerell, 2013). Ensemble averages, 〈³J_NC〉, are calculated from χ angles in numerous snapshots sampled from MD trajectories. Interestingly, experimental ³J_NC data tend to exhibit far better correlation with the 〈³J_NC〉 data predicted from the MD ensemble than with those predicted from the crystal structure (Chen et al., 2015; Zandarashvili et al., 2011), as shown in Fig. 10. These results suggest that the χ torsion angles of these side chains are actually as dynamic as seen in the MD simulation. Thus, ³J_NC data can also illuminate the dynamic nature of basic side chains.

Fig. 10.

³J_NC coupling data indicating dynamic alterations of side-chain torsion angles. (A) Correlations between the observed and predicted χ₄-relevant ³J_NζCγ couplings for the Lys side chains of ubiquitin (Zandarashvili et al., 2011), the Antp homeodomain–DNA complex, and the Egr-1–DNA complex (Chen et al., 2015). The ³J_NζCγ couplings were predicted from crystal structures (PDB 1UBQ, 9ANT, and 1AAY) or MD ensembles. (B) Corresponding data for the ³J_NCγ couplings between backbone ¹⁵N and side-chain ¹³C_γ nuclei related to the χ₁ angles of ubiquitin side chains. (C) Distribution of the χ₁ and χ₄ angles of the K11 side chain of ubiquitin in the MD trajectory (Zandarashvili et al., 2011). Arrows indicate the angles in the crystal structure.

5.2. Ion-Pair Dynamics in MD Simulations

Ion pairs with charged side chains can involve both strong short-range electrostatic interactions and hydrogen bonds. Although the NMR data of hydrogen-bond scalar couplings (^h3J_NP) between protein ¹⁵N and DNA ³¹P nuclei indicate the presence of hydrogen bonds in the intermolecular ion pairs, the NMR S² and ³J_NC data clearly indicate that the Lys side chains interacting with DNA phosphates are highly mobile (Anderson et al., 2013; Chen et al., 2015). How can the Lys side-chain NH₃⁺ groups exhibit high mobility despite the simultaneous presence of hydrogen bonds and strong short-range attractive electrostatic interactions? While the answer to this question was not immediately obvious from the NMR investigations alone, MD simulations provided useful insight into this question and facilitated interpretation of the NMR data.

5.2.1. Analysis of CIP–SIP Transitions

The dynamic behavior of biomolecular ion pairs becomes evident through analysis of the distances between cationic and anionic moieties in MD trajectories. Fig. 11A shows the N…O distances from the interfacial Lys side-chain NH₃⁺ group to the closest DNA phosphate in the 0.6-μs MD trajectories for the Antp homeodomain–DNA complex and the Egr-1 zinc-finger–DNA complex (Chen et al., 2015). These N…O distances dynamically fluctuated between two ranges: one between 2.5 and 3.2 Å, corresponding to the CIP states, and the other between 3.8 and 6.0 Å, corresponding to the SIP states. The transitions between the CIP and SIP states occurred on a pico-to-nanosecond timescale. This dynamic behavior of biomolecular ion pairs is reasonable in light of the fundamental concepts that were established for small ions over the last several decades (Fennell, Bizjak, Vlachy, & Dill, 2009; Marcus & Hefter, 2006; Masnovi & Kochi, 1985; Masunov & Lazaridis, 2003; Peters & Li, 1994; Pettitt & Rossky, 1986; Szwarc, 1972). The order parameters $S_{\mathit{axis}}^2$ and ³J_NζCγ coupling constants predicted from the MD trajectories were consistent with experimental data from NMR experiments. The CIP–SIP transitions make the interfacial Lys side chains highly dynamic, making the order parameters small, whereas the major presence of CIP causes sizable hydrogen-bond scalar couplings ^h3J_NP.

Fig. 11.

Dynamic transitions between the CIP and SIP states of the intermolecular ion pairs of Lys side-chain NH₃⁺ and DNA phosphate groups observed in the 0.6-μs MD simulations for the Antp–DNA and Egr-1–DNA complexes. (A) The trajectories of the distances from the Lys N_ζ atoms to the closest DNA phosphate oxygen atoms are shown for the intermolecular ion pairs for which the presence of CIP was experimentally confirmed. (B) Probability distribution of the CIP and SIP states as a function of the distance from the Lys N_ζ atom to the closest DNA phosphate oxygen atom. (C) PMFs for the intermolecular ion pairs of the protein side-chain NH₃⁺ and DNA phosphate groups in the Antp homeodomain–DNA complex and the Egr-1 zinc-finger–DNA complex.

5.2.2. Free-Energy Analysis of the CIP–SIP Equilibria

MD trajectories enable calculations of the potentials of mean force (PMFs) for these ion pairs. PMFs represent the effective potential or free-energy landscapes given as a function of the interionic distance. Fig. 11C shows the PMFs as a function the N…O distance (r_NO) for the ion pairs of the Lys side-chain NH₃⁺ and DNA phosphate groups (Chen et al., 2015). These PMFs were obtained though analysis of the probability distribution p(r_NO) (Fig. 11B), which is based on a histogram of the r_NO values in the MD trajectory. The free energy is calculated as G = −RT ln p(r_NO), where R is the gas constant; T is the standard temperature (i.e., T = 298.15 K). The two minima in PMFs correspond to the CIP and SIP states. The peak between these states corresponds to the effective free energy barrier for the CIP–SIP transition, provided that the PMFs represent the reaction coordinates. With a higher energy barrier, the CIP–SIP transitions should be slower. For the protein–DNA ion pairs whose CIP states were experimentally detected, the free energy differences between CIP and SIP states were determined to be 0.8–1.6 kcal/mol. The energy barriers for CIP → SIP transitions were determined to be 2.2–3.2kcal/mol, which are qualitatively consistent with the mean lifetimes of the CIP states. The relatively small free energy difference between CIP and SIP as well as the low-energy barrier between these states can explain the highly dynamic nature of the interfacial ion pairs involving Lys side chains.

5.3. Conformational Entropy of Basic Side Chains

Changes in the conformational entropies of protein side chains can make significant contribution to the overall change in free energy upon molecular association (Wand, 2013; Wand & Sharp, 2018). Because basic side chains are important constituents of electrostatic interactions in molecular recognition by proteins, the conformational entropies of these side chains are of particular interest. For CH₃-containing side chains, the order parameters $S_{\mathit{axis}}^2$ for CH₃ groups are used as “meters” of side-chain conformational entropy (Kasinath, Sharp, & Wand, 2013; Li & Brüschweiler, 2009). For Lys and Arg side chains, however, the order parameters of their cationic groups do not necessarily serve as good meters of the conformational entropy of the entire side chain because CH₂ moieties could remain flexible even if the cationic moieties are restricted by hydrogen bonds (Li & Brüschweiler, 2009; Trbovic et al., 2009). Lys and Arg side-chain conformational entropies can be calculated from the distributions of the dihedral angles sampled during an MD simulation (Karplus & Kushick, 1981; Trbovic et al., 2009):

$$S=-R\int P\left(\overrightarrow{\chi }\right)\ln P\left(\overrightarrow{\chi }\right)\text{d}\overrightarrow{\chi },$$ (10)

where R is the gas constant and $P(\overrightarrow{\chi })$ is the probability density as a function of the dihedral angles χ₁, χ₂, χ₃, and χ₄ of each Arg or Lys side chain. We tested the dependence of the entropies on the bin size of the integral mesh and chose 5° (Esadze et al., 2016). The bin size of the integral mesh affects the absolute entropy but, given reasonable resolution, does not affect the change in conformational entropy. Fig. 12A displays the change in conformational entropy of each Arg/Lys side chain of the Egr-1 zinc-finger protein upon binding to its target DNA, which was calculated from the MD trajectories using Eq. (10) (Esadze et al., 2016). Fig. 12B shows the correlation between the changes in the NMR-derived order parameters of the Lys NH₃⁺ and Arg N_εH groups and the changes in the MD-derived conformational entropy of the same side chains upon complex formation.

Fig. 12.

Changes in side-chain conformational entropy for the Lys and Arg side chains of the Egr-1 zinc-finger protein upon binding to the target DNA. (A) Changes in the side-chain conformational entropy of Lys and Arg side chains calculated from 0.6-μs MD trajectories for the free protein and the complex. (B) Correlation between changes in the NMR order parameters and changes in the MD conformational entropy upon complex formation.

6. CONCLUDING REMARKS

NMR experiments on Lys and Arg side chains can provide unique insights into their conformational dynamics, hydrogen bonds, and electrostatic interactions. Functionally important Lys and Arg side chains tend to show relatively slow hydrogen exchange due to hydrogen bonding, which facilitates NMR-based characterizations of these side chains. For the NMR experiments to work, optimization of pH and temperature can be necessary so that hydrogen exchange becomes slow enough to detect ¹H signals from the cationic groups. The intrinsic ¹⁵N transverse relaxation of Lys NH₃⁺ groups is very slow, which makes it possible to precisely measure small scalar couplings, including those across hydrogen bonds. The pulse programs and parameter sets for the pulse sequences introduced in this chapter are available upon request at our laboratory website (https://scsb.utmb.edu/labgroups/iwahara/software/).