Measurement of ¹⁵N longitudinal relaxation rates in ¹⁵NH₄⁺ spin systems to characterise rotational correlation times and chemical exchange

Graphical abstract

Highlights

• •A pulse sequence is derived to select ¹⁵N coherences within the ¹⁵NH₄⁺ spin-system.
• •A sequence for measurement of ¹⁵N spin-relaxation rates in ¹⁵NH₄⁺ is presented.
• •The effective correlation time of ¹⁵NH₄⁺ bound to a 41 kDa protein is characterised.

Abstract

Many chemical and biological processes rely on the movement of monovalent cations and an understanding of such processes can therefore only be achieved by characterising the dynamics of the involved ions. It has recently been shown that ¹⁵N-ammonium can be used as a proxy for potassium to probe potassium binding in bio-molecules such as DNA quadruplexes and enzymes. Moreover, equations have been derived to describe the time-evolution of ¹⁵N-based spin density operator elements of ¹⁵NH₄⁺ spin systems. Herein NMR pulse sequences are derived to select specific spin density matrix elements of the ¹⁵NH₄⁺ spin system and to measure their longitudinal relaxation in order to characterise the rotational correlation time of the ¹⁵NH₄⁺ ion as well as report on chemical exchange events of the ¹⁵NH₄⁺ ion. Applications to ¹⁵NH₄⁺ in acidic aqueous solutions are used to cross-validate the developed pulse sequence while measurements of spin-relaxation rates of ¹⁵NH₄⁺ bound to a 41 kDa domain of the bacterial Hsp70 homologue DnaK are presented to show the general applicability of the derived pulse sequence. The rotational correlation time obtained for ¹⁵N-ammonium bound to DnaK is similar to the correlation time that describes the rotation about the threefold axis of a methyl group. The methodology presented here provides, together with the previous theoretical framework, an important step towards characterising the motional properties of cations in macromolecular systems.

1. Introduction

Monovalent cations such as potassium and sodium regulate many enzymes via binding to either active sites or to allosteric sites [1], [2], [3]. The dynamics and movements of these ions are therefore crucial factors to understand the regulation of enzymes by monovalent cations and experimental insight into the dynamics of cations therefore becomes important in order to characterise many biological processes. Solution NMR spectroscopy is a powerful technique to probe the dynamics of nuclear spin, where in particular nuclear spin-relaxation rates have been used to report on the dynamics of small ions [4], [5], [6], [7] to large macromolecular complexes [8], [9], [10], [11]. The obtained nuclear spin-relaxation rates often report on both the rotational correlation time of the nuclear spin as well as on chemical exchange between different magnetic environments and a separation of the different contributions to the observed nuclear spin-relaxation rate is thus important. Several strategies have therefore previously been employed to separate the different contributions to spin-relaxation rates, including, measurements at several magnetic field strengths [12], measurements of cross-correlated relaxation rates [13], [14], and measurements of the relaxation rate of related anti-phase coherences [15], [16].

Recently it was shown that ¹⁵NH₄⁺ can be used as a proxy for potassium to probe potassium-binding sites in nucleic acids and enzymes [17], [18], [19], [20]. This method relies on several characteristics of the ammonium ion: (i) The ionic radius of the ammonium ion is similar to the ionic radius of potassium, 1.44 Å versus 1.33 Å [21], [22], such that ammonium generally binds to potassium binding-sites in macromolecules. (ii) Under physiological conditions the chemical exchange of the ammonium protons with the bulk solvent is so fast that free ammonium is not observed in NMR correlation spectra, however, the protection of the ammonium ion, for example by a protein environment, slows the exchange of the protons with the bulk solvent to such an extend that these are observed in two-dimensional ¹⁵N-¹H correlation spectra [19]. (iii) Finally, protein-bound ammonium ions appear to have a fast internal correlation time such that the line broadening due to the ¹⁵N-¹H dipolar-dipole interactions is limited.

The recently developed theory for ¹⁵N spin relaxation in ¹⁵NH₄⁺ spin systems together with the possibility of obtaining ¹⁵N-¹H correlation maps of protein-bound ¹⁵NH₄⁺ open up the possibility of quantifying the dynamics of ammonium ions, even within potassium binding-sites of large proteins, and thus for correlating cation dynamics with macromolecular function. Given the current development of techniques to probe ammonium ions in proteins and nucleic acids it is therefore of interest to derive methods to experimentally measure nuclear spin-relaxation rates to report on the rotational correlation time and chemical exchange of ammonium ions. The advantage of the ¹⁵NH₄⁺ spin system is the availability of a wealth of coherences and spin density elements, whose relaxation rates each report differently on the rotational correlation time and chemical exchange. Herein NMR pulse sequences are developed firstly to select different spin density matrix elements and secondly to measure ¹⁵N-based longitudinal relaxation rates of the ¹⁵NH₄⁺ spin system. Applications to ¹⁵N-ammonium in an acidic aqueous solution and ¹⁵N-ammonium bound in a potassium binding-site of a ∼41 kDa domain of the protein DnaK are presented to illustrate the general utility of the derived pulse sequences.

2. Theory

Time-evolution of the ¹⁵NH₄⁺ spin systems: The time evolution of a spin system is generally given by the Liouville-von-Neumann equation [23], [24], [25]:

$$\frac{d\sigma (t)}{\mathit{dt}}=-i[{\widehat{\mathcal{H}}}_0\text{,}\sigma (t)]-\mathrm{\Gamma }(\sigma (t)-{\sigma }_{\text{eq}})$$ (1)

where ${\widehat{\mathcal{H}}}_0$ is the time independent part of the spin-Hamiltonian, σ_eq is the equilibrium density operator, and $\mathrm{\Gamma }$ is the spin-relaxation super-operator, which was derived previously for the ¹⁵N-ammonium spin-system [20]. The time-independent part of the Hamiltonian is here given by:

$${\widehat{\mathcal{H}}}_0={\widehat{\mathcal{H}}}_Z+{\widehat{\mathcal{H}}}_J$$ (2)

where the total Zeeman Hamiltonian is, ${\widehat{\mathcal{H}}}_Z=(H_{z1}+H_{z2}+H_{z3}+H_{z4}){\omega }_{\text{H}}+N_z{\omega }_{\text{N}}$, and the ¹⁵N-¹H scalar-coupling Hamiltonian is given by ${\widehat{\mathcal{H}}}_J=\pi J(2H_{z1}{N}_z+2H_{z2}{N}_z+2H_{z3}{N}_z+2H_{z4}{N}_z)$, and J is the nitrogen-proton one-bond scalar coupling constant.

As shown previously, there are nine single quantum ¹⁵N transitions within the tetrahedral ¹⁵NH₄⁺ spin system, however, due to the tetrahedral symmetry and degeneracy of the ¹⁵NH₄⁺ spin system there are only five characteristic frequencies of the single quantum ¹⁵N coherences: −4πJ + Ω, −2πJ + Ω, Ω, 2πJ + Ω, 4πJ + Ω, where Ω is the offset from the RF carrier. The frequency of a ¹⁵N transition depends on the spin states of the four ammonium protons, for example, when all the protons are in the α state, the frequency is 4πJ + Ω. In turn, the spin state of the four protons can be described using different basis sets; for example a Zeeman basis given by the eigenfunctions to the Zeeman Hamiltonian or a Cartesian basis. In the Zeeman basis set the transitions fall in three spin manifolds, A₁, T₂, and E according to the T_d symmetry of the ammonium ion as discussed previously [20]. For the development of the pulse sequence below the spin densities of the four ammonium protons are most conveniently described using the product operator formalism [26] and the Cartesian basis set.

In the product operator formalism the equilibrium density operator of the ammonium ion is given by σ_eq ∝ γ_H(H_z1 + H_z2 + H_z3 + H_z4) + γ_NN_z, where γ_H and γ_N are the gyromagnetic ratio of the proton and the nitrogen nuclear spin, respectively, and H_zi (i = 1, 2, 3, 4) and N_z are the Cartesian product operator describing the longitudinal magnetisation of the four protons and the nitrogen spin, respectively. The transverse nitrogen magnetisation is written as N₊ = N_x + i N_y, where i is the imaginary unit. Thus, using the product operator formalism the nine single quantum nitrogen transitions are described by ${\mathcal{B}}_{\text{xyz}}$ ={N₊, 2N₊H_z, 4N₊H_zH_z, 8N₊H_zH_zH_z, 16N₊H_zH_zH_zH_z, N₊H₊H₋, 2N₊H₊H₋H_z, 4N₊H₊H₋H_zH_z, N₊H₊H₋H₊H₋} in the Cartesian basis set, where the following notation has been used: H_z = H_z1 + H_z2 + H_z3 + H_z4; H_zH_z = H_z1H_z2 + H_z1H_z3 + H_z1H_z4 + H_z2H_z3 + H_z2H_z4 + H_z3H_z4; H_zH_zH_z = H_z1H_z2H_z3 + H_z1H_z2H_z4 + H_z1H_z3H_z4 + H_z2H_z3H_z4; H_zH_zH_zH_z = H_z1H_z2H_z3H_z4; H₊H₋ = ${\sum }_{i\ne j}{H}_{+\text{,}i}{H}_{-\text{,}j}$; H₊H₋H_z = ${\sum }_{i\ne j\ne k}{H}_{+\text{,}i}{H}_{-\text{,}j}{H}_{z\text{,}k}$; H₊H₋H_zH_z = ${\sum }_{i\ne j\ne k\ne l}{H}_{+\text{,}i}{H}_{-\text{,}j}{H}_{z\text{,}k}{H}_{z\text{,}l}$; H₊H₋H₊H₋ = ${\sum }_{i\ne j\ne k\ne l}{H}_{+\text{,}i}{H}_{-\text{,}j}{H}_{+\text{,}k}{H}_{-\text{,}l}$, The advantage of using the Cartesian basis set here is that spin density elements present during the pulse sequence are more easily identified and the relaxationoftheproton spins by external sources is conveniently implemented as additions to the auto-relaxation rates as described previously [20].

The evolution of spin density matrix elements under the scalar coupling Hamiltonian forms the basis for a separation of different nitrogen single quantum coherences. It is therefore of interest to characterise how the spin density elements of the basis $\mathcal{B}$_xyz evolve under the scalar coupling Hamiltonian. Only considering scalar coupling and ignoring relaxation, it was shown previously that [20]

Firstly it is noted that, due to the symmetry of the ammonium ion, the scalar coupling Hamiltonian is only mixing coherences within three groups: {N₊, 2N₊H_z, 4N₊H_zH_z, 8N₊H_zH_zH_z, 16N₊H_zH_zH_zH_z,}, {N₊H₊H₋, 2N₊H₊H₋H_z, 4N₊H₊H₋H_zH_z}, and N₊H₊H₋H₊H₋ (N₊H₊H₋H₊H₋ commutes with the scalar coupling Hamiltonian). Thus, starting from in-phase transverse nitrogen magnetisation, N_x, gives:

$$N_x\stackrel{\tau 2\pi J{\mathit{H}}_z{N}_z}{\to }\frac{1}{8}(3+4\cos 2\pi J\tau +\cos 4\pi J\tau )N_x+\frac{1}{8}(2\sin 2\pi J\tau +\sin 4\pi J\tau )2N_y{\mathit{H}}_z-\frac{1}{8}(1-\cos 4\pi J\tau )4N_x{\mathit{H}}_z{\mathit{H}}_z-\frac{1}{8}(2\sin 2\pi J\tau -\sin 4\pi J\tau )8N_y{\mathit{H}}_z{\mathit{H}}_z{\mathit{H}}_z+\frac{1}{8}(3-4\cos 2\pi J\tau +\cos 4\pi J\tau )16N_x{\mathit{H}}_z{\mathit{H}}_z{\mathit{H}}_z{\mathit{H}}_z$$ (4)

Eq. (4) implies that evolution of N_x for a period of 1/(2J) generates the quadruple anti-phase coherence 16N_xH_zH_zH_zH_z. Moreover, because the single anti-phase coherence, 2N_x,yH_z, is easily generated from the equilibrium density operator it’s time evolution is also of particular interest here:

$$2N_x{\mathit{H}}_z\stackrel{\tau 2\pi J{\mathit{H}}_z{N}_z}{\to }\frac{1}{2}(2\sin 2\pi J\tau +\sin 4\pi J\tau )N_y+\frac{1}{2}(\cos 2\pi J\tau +\cos 4\pi J\tau )2N_x{\mathit{H}}_z+\frac{1}{2}(\sin 4\pi J\tau )4N_y{\mathit{H}}_z{\mathit{H}}_z-\frac{1}{2}(\cos 2\pi J\tau -\cos 4\pi J\tau )8N_x{\mathit{H}}_z{\mathit{H}}_z{\mathit{H}}_z-\frac{1}{2}(2\sin 2\pi J\tau -\sin 4\pi J\tau )16N_y{\mathit{H}}_z{\mathit{H}}_z{\mathit{H}}_z{\mathit{H}}_z$$ (5)

Eqs. (4), (5) imply that by choosing appropriate values of τ in a pulse sequence a series of different coherences can be generated by evolving either in-phase transverse N_x magnetisation or the single anti-phase coherence 2N_xH_z.

It should be noted that inclusion of proton spin-flip, R₁(H_z), in the evolution described in Eq. (3) causes differential relaxation of the ¹⁵N single quantum Cartesian density elements and can effectively be viewed as a five-site exchange between the transitions of the ¹⁵NH₄⁺ multipliet. Thus, the signal with a frequency of −4πJ + Ω exchanges magnetisation with the signal at −2πJ + Ω, the signal with a frequency of −2πJ + Ω exchanges with the signals at −4πJ + Ω and at Ω, and so forth. The consequence of proton spin-flip can therefore be a change in the characteristic frequencies, linewidths and intensities, which in turn can be calculated from the eigenvalues and eigenvectors of a Liouvillian that includes both the scalar coupling and proton spin-flip, R₁(H_z). It can be shown that inclusion of R₁(H_z) in the Liouvillian, and assuming that R₁(H_z) < 2π|J|, changes the characteristic frequencies to $\left\{-4\pi J{[1-{\zeta }^2]}^{1/2}+\mathrm{\Omega }\text{,}-2\pi J{[1-{\zeta }^2]}^{1/2}+\mathrm{\Omega }\text{,}\mathrm{\Omega }\text{,}2\pi J{[1-{\zeta }^2]}^{1/2}+\mathrm{\Omega }\text{,}4\pi J{[1-{\zeta }^2]}^{1/2}+\mathrm{\Omega }\right\}$, where $\zeta =R_1(H_z)/(2\pi J)\text{.}$ Moreover, when the single anti-phase coherence 2N₊H_z is excited and detected, for example in a standard coupled ¹⁵N-¹H correlation spectrum, the intensity ratio of the five signals is 1:r:0:r:1, with $r=1-5\zeta \left(\zeta -\frac{4i}{5}\sqrt{1-{\zeta }^2}\right)$. For experimental spectra of ¹⁵NH₄⁺ the effects on the frequencies and the intensity ratios are generally small. For example, when J = −70 Hz and R₁(H_z) = 35 s⁻¹, which is well within the range of the data shown below, the characteristic frequencies are {−139.56 Hz, −69.79 Hz, 0 Hz, 69.79 Hz, 139.56 Hz} and the absolute intensity ratio is 1:1.02:0:1.02:1. It should be noted that since r is a complex number, the proton spin-flip also causes a relative phase-shift of the signals within the multiplet, as also observed previously for two-site chemical exchange [27]; this phase-shift is approximately 10° for the example above.

3. Results

Pulse sequence for measuring the longitudinal relaxation rates of ¹⁵NH₄⁺: ¹⁵N-ammonium ions in an acid aqueous solution or bound to proteins or nucleic acids can be characterised by ¹⁵N-¹H correlations spectra [17], [18], [19], [20]. Such two-dimensional spectra can be obtained using standard ¹⁵N-¹H correlation experiments, where after an initial INEPT, 90_x(¹H) – τ – 180_x(¹H), 180_x(¹⁵N) – τ – 90_y(¹H) with τ = 1/(4J), a spin density operator proportional to 2N_zH_z is generated. Subsequently a 90_x(¹⁵N) pulse generates −2N_yH_z for chemical shift evolution and a final INEPT transfers anti-phase 2N_x,yH_z magnetisation to transverse proton magnetisation for detection. When the ¹⁵N-¹H scalar coupling is allowed to evolve during the 2N_x,yH_z chemical shift evolution period signals at the five ¹⁵N characteristic frequencies −4πJ + Ω, −2πJ + Ω, Ω, 2πJ + Ω, and 4πJ + Ω (Eq. (5)) are observed with an intensity ratio of 1:1:0:1:1 [20] in the limit where R₁(¹H) $\lesssim$ 2π|J|(see Section 2). The pulse sequence used here for generating subspectra of the ¹⁵NH₄⁺ quintet and for measuring relaxation rates of the longitudinal density operator elements is shown in Fig. 1 and is based on the ¹⁵N-¹H correlation experiment described above.

Fig. 1.

Pulse sequence to measure longitudinal relaxation rates of individual spin density matrix elements of the ¹⁵NH₄⁺ spin-system. The ¹H carrier is placed on the water and the ¹⁵N carrier is placed in the middle of the ¹⁵NH₄⁺ region (∼20 ppm). All pulses are applied at the highest possible power levels, with the exception of the ¹H water selective pulses (open bell-shaped pulses) and the ¹⁵N decoupling, where a ∼350 Hz field strength and a 1.5 kHz WALTZ-16 decoupling [28] are employed, respectively. The element immediately following the initial INEPT (grey box) is used to select different coherences of the ¹⁵NH₄⁺ spin system, as detailed in the text. The delay used are: τ_a = 3.47 ms and τ_b varied as described in the main text. Pulses without annotation are applied with x-phase. The phase cycle is: φ₁ = x, −x, φ₂ = 2(x), 2(−x), φ₃ = 4(x), 4(−x), φ_rec = x, −x, −x, x. The phase of φ₄ is chosen as a part of the coherence selection element and detailed in the text and in Fig 3. Quadrature detection in the indirect ¹⁵N dimension is achieved by altering ϕ₂ and ϕ_rec in the States-TPPI manner. Gradients are used to remove artifacts and are applied using a sine bell-shaped profile for 0.5 ms with maximum strengths of, g1: 24 G/cm, g2: 7.3 G/cm, g3: 35 G/cm, g4: 15 G/cm, g5: 8.6 G/cm, g6: 11.2 G/cm, g7: 8.6 G/cm.

The sequence in Fig. 1 differs from a standard ¹⁵N-¹H correlation experiment by the insertion of a selection element between a and b and a relaxation delay prior to the ¹⁵N chemical shift evolution. The final INEPT between c and acquisition selects for the two spin-order longitudinal spin density element, 2N_zH_z, and transfers it to H_y magnetisation for detection while a pair of ¹⁵N 90° pulses, with the first pulse phase-cycled x, −x, is placed immediately prior to acquisition to remove anti-phase 2N_zH_xy coherences.

Selecting subspectra of the ¹⁵NH₄⁺ quintet: The initial 90_ϕ1 ¹⁵N pulse of the selection element between a and b in Fig. 1 generates the ±2N_yH_z anti-phase spin density element. Free precession of 2N_yH_z during the coherence selection element can be characterised schematically using a vector model, where each of the four observed signals of the ¹⁵NH₄⁺ quintet are represented by a vector (Fig. 2). A ¹⁵N 180° pulse is applied in the middle of the coherence selection element such that chemical shift evolution of ¹⁵N is refocused. Relaxation during the selection element is disregarded initially and therefore only the evolution under the ¹⁵N-¹H scalar coupling Hamiltonian is considered immediately below.

Fig. 2.

(A) Vector diagram showing the evolution of transverse ¹⁵N magnetisations of the ¹⁵NH₄⁺ spin-system under the ¹⁵N-¹H scalar coupling Hamiltonian. The vectors correspond to the four transitions observed in the coupled ¹⁵N spectrum that is obtained by exciting and detecting the anti-phase coherence 2N_xyH_z. Specifically, the red vector corresponds to the transition where all the ammonium protons are in the α-state, green vector corresponds to the transitions where three of the protons are in the α-state, blue vector corresponds to the transitions where one of the protons is in the α-state, and black vector corresponds to the situation where all the protons are in the β-state. (B) Evolution of the anti-phase coherence −2N_yH_z under the scalar coupling Hamiltonian for a time of |1/4J| (J < 0 for ¹⁵N-¹H scalar couplings). Selection along the y-axis generates an intensity ratio of 0:−1:0:1:0 of the five ¹⁵N characteristic frequencies and a spin density element proportional to N_z−16N_zH_zH_zH_zH_z, while selection along the x-axis generates an intensity ratio of 1:0:0:0:1 of the five ¹⁵N characteristic frequencies and a spin density element proportional to 2N_zH_z + 8N_zH_zH_zH_z. (C) Evolution of the anti-phase coherence −2N_yH_z under the scalar coupling Hamiltonian for a time of |1/6J|. Selection along the y-axis generates an intensity ratio of 1:1:0:−1:−1 of the five ¹⁵N characteristic frequencies, while selection along the x-axis generates an intensity ratio of 1:−1:0:−1:1 of the five ¹⁵N characteristic frequencies.

The selection element shown in the pulse sequence of Fig. 1 relies on both a variation of the delay τ_b and on the phase ϕ₄. In the limiting case when τ_b = τ_a the evolution under the scalar coupling Hamiltonian is refocused, while when τ_b = 0 the scalar coupling evolves for 1/|2J| and a spin density element proportional to 8N_zH_zH_zH_z is obtained (Eq. (5)). For other values of τ_b the evolution of the scalar coupling can conveniently be followed using a vector diagram as shown in Fig.2A. Specifically, Fig.2B and C show the selection of four different spin density elements of the ¹⁵NH₄⁺ spin-system. It should be noted that differential relaxation of the density elements {N₊, 2N₊H_z, …, 16N₊H_zH_zH_zH_z} during the selection element becomes significant for application to ¹⁵NH₄⁺ systems with large proton spin-flip rates, R₁(H_z). As discussed below the effect of differential relaxations can be taken into account in the analysis of relaxation decay curves by introducing model parameters that describe the state immediately following the selection element.

The pulse sequence in Fig. 1 and the selection element were initially verified using ¹⁵NH₄⁺ in an acidic aqueous solution. Under acidic conditions (pH = 2.82) the exchange rate of the ammonium protons with the water is slowed to an extent where ¹⁵N-¹H correlation spectra can be obtained and where the pulse sequence in Fig. 1 can be verified. Fig. 3 shows the generation of four different spectra obtained by varying τ_b and ϕ₄. The obtained spectra are in excellent agreement with the predicted ratio of the intensities of the multiplet structure.

Fig. 3.

Selection of spin density matrix elements of ¹⁵NH₄⁺ dissolved in an acidic aqueous solution (pH = 2.82). The spectra were obtained using the sequence shown in Fig 1, with T_relax = 0 and τ_b and ϕ₄ as shown above each of the panels. (A) The selection of N_x − 16N_xH_zH_zH_zH_z coherences; corresponding to Fig2B with selection along x. (B) The selection of 2N_xH_z + 8N_xH_zH_zH_z coherences; corresponding to Fig2B with selection along the y-axis. (C) and (D) Selection shown in Fig2C with selection phase along the x- and y-axis, respectively.

Linear combinations of the spectra in Fig. 3 can be used to generate spectra of four of the individual lines of the ¹⁵NH₄⁺ quintet, where for example, the spectra in Fig.3A–C can be used to generate a spectrum only consisting of the αααα line. It should be noted however that although spectra of the four lines can be generated, pure spectra of all of the five Cartesian ¹⁵N single quantum spin density elements, N₊, 2N₊H_z, …, 16N₊H_zH_zH_zH_z, cannot be generated using the described sequence. A spectrum of pure 2N₊H_z and pure 8N₊H_zH_zH_z can be generated from Fig.3B and D, or from spectra with τ_b = τ_a and τ_b = 0 s, respectively. A further consequence is that the intensity of the four lines, αααα, αααβ, αβββ, and ββββ can be used to derive the relative intensity of 2N₊H_z and 8N₊H_zH_zH_z. On the contrary, the pure spectra corresponding to the three spin density elements N₊, 4N₊H_zH_z, and 16N₊H_zH_zH_zH_z are correlated such that their individual intensities cannot be separated using the selection elements in Fig. 1. Such a correlation is the result of an underdetermined system where four lines are observed in the coupled ¹⁵N-¹H NMR spectra, however originating from five spin density elements. On the other hand, when the contributions from the five possible spin density elements are know, the intensities of the four lines can be calculated.

Measuring longitudinal relaxation rates: In general, the relaxation rate of the five longitudinal spin density elements, N_z, 2N_zH_z, 4N_zH_zH_z, 8N_zH_zH_zH_z and 16N_zH_zH_zH_zH_z are different, since they have different contributions from the spectral density function J(ω) and different contributions from the proton spin-flip rate, R₁(H_z) [20]. When the rotational diffusion of the ¹⁵NH₄⁺ ion is described accurately by one correlation time, τ_C, the five relaxation rates, R₁(N_z), R₁(2N_zH_z), R₁(4N_zH_zH_z), R₁(8N_zH_zH_zH_z), and R₁(16N_zH_zH_zH_zH_z), as well as cross-correlated relaxations between the longitudinal spin density elements, only depend on two parameters, that is, the rotational correlation time τ_C and the proton spin-flip rate, R₁(H_z). Thus, as described below using the Liouvillian in Table 4 of Werbeck and Hansen [20], which includes both auto- and cross-correlated relaxations, the two parameters τ_C,eff and R₁(H_z) can be obtained from the intensities of the four lines observed in coupled ¹⁵NH₄⁺ spectra recorded for different values T_relax.

A relaxation delay, T_relax, is inserted after the selection element and before the ¹⁵N chemical shift evolution in the sequence in Fig. 1, which allows the relaxation decay of the four lines {αααα, αααβ, αβββ, ββββ} to be obtained in order to characterise the dynamics of the ammonium ion. Initially ¹⁵NH₄⁺ in acidic aqueous solutions (pH 2.82) was used as a model system. For such a system the proton spin-flip rate R₁(H_z) is given by the off-rate of the ammonium protons with the bulk solvent: ${}^{15}{\text{NH}}_4^{+}+{\text{H}}_2^{\ast }\text{O}\stackrel{k_{\text{ex}}}{\to }{}^{15}{\text{NH}}^{\ast }{\text{H}}_3^{+}+{\text{H}}^{\ast }\text{OH}$.

Although there are many possible combinations of τ_b and ϕ₄, each allowing for a different selection of spin density elements, focus below is on selecting two initial states, that is (i) 2N_zH_z + 8N_zH_zH_zH_z and (ii) N_z −16N_zH_zH_zH_zH_z. The decay curves obtained for ¹⁵NH₄⁺ in an acidic aqueous solution following these two different selections is shown in Fig. 4. It is noted that in the logarithmic plots the decay curves show a double sigmoidal shape reporting on the relaxation of the different spin density elements present during the relaxation delay. For example, for the selection of an element proportional to N_z −16N_zH_zH_zH_zH_z the faster relaxation of 16N_zH_zH_zH_zH_z is seen by the first sigmoidal shape and thereafter the relative ratio of the intensity of the four observed lines approaches the 1:2:0:−2:−1 ratio that is characteristic for the in-phase magnetisation. Similarly, for selection of an initial spin density element proportional to 2N_zH_z + 8N_zH_zH_zH_z the ratio of the intensity of the four signals approaches 1:1:0:1:1, which is characteristic of the single anti-phase coherence 2N₊H_z.

Fig. 4.

Decay curves of the four lines observed in the coupled ¹⁵NH₄⁺ spectrum of ¹⁵NH₄⁺ in an acidic aqueous solution; pH 2.82 at 278 K, and recorded at a static magnetic field of 11.74 T. The solid lines are obtained from best-fits of all the data shown to an evolution of the Liouvillian describing the decay of the five Cartesian longitudinal spin density elements (see text). (A) Decay curves obtained after selecting for 2N_zH_z + 8N_zH_zH_zH_z and using the pulse sequence of Fig 1. (B) Decay curves obtained after selecting for N_z −16N_zH_zH_zH_zH_z. The double sigmoidal shape of the decay curves in the logarithmic plot shows first the faster relaxation of the quadruple anti-phase spin density element 16N_zH_zH_zH_zH_z followed by a substantially slower relaxation of the in-phase longitudinal magnetisation N_z.

The decay curves were analysed by evolution of the Liouvillian in the Cartesian basis, Γ, during the relaxation delay T_relax. Here the Liouvillian is a function of the two parameters τ_C and R₁(H_z) and describes the auto- and cross-correlated relaxation rates within the basis set {E/2, H_z, 2H_zH_z, 4H_zH_zH_z, 8H_zH_zH_zH_z, N_z, 2N_zH_z, 4N_zH_zH_z, 8N_zH_zH_zH_z, 16N_zH_zH_zH_zH_z}. Using a larger basis set that includes also the zero-quantum density elements such as N_zH₊H₋ and 2N_zH₊H₋H_z, did not change the quality of the fit nor did it change the obtained parameters. Briefly, a vector describing the initial state v₀ = {0, 0, 0, 0, 0, I₀(N_z), I₀(2N_zH_z), I₀(4N_zH_zH_z), I₀(8N_zH_zH_zH_z), I₀(16N_zH_zH_zH_zH_z)} was evolved for a time of T_relax. The intensity of the density elements, E/2, H_z, 2H_zH_z, 4H_zH_zH_z, and 8H_zH_zH_zH_z were assumed to vanish, because phase ϕ₁ in the pulse sequence of Fig. 1 is phase-cycled {x,−x} with no concomitant change in the receiver phase, which eliminates contributions from density elements not proportional to N_z or N_y. After evolving the initial state v₀ for a time of T_relax the intensities of v(T_relax) were converted to intensities of the four lines observed, {I_αααα, I_αααβ, I_αβββ, I_ββββ}. The best-fit model parameters, τ_C, R₁(H_z), and v₀, were subsequently obtained by a Levenberg-Marquardt least-squared fitting procedure (see Section 5).

Analysis of the decay curves in Fig. 4 gives a correlation time of τ_C = 1.63 ps ± 0.03 ps and R₁(H_z) = 4.34 s⁻¹ ± 0.02 s⁻¹. Moreover, the intensities obtained for the initial states were v₀ = {0, 0, 0, 0, 0, 0.017 ± 0.001, 0.552 ± 0.001, 0.002 ± 0.001, 0.480 ± 0.001, 0.002 ± 0.001} for the data shown in Fig.4A and v₀ = {0, 0, 0, 0, 0, −1.106 ± 0.002, 0.003 ± 0.001, 0.037 ± 0.002, 0.002 ± 0.001, 0.991 ± 0.005} for the data shown in Fig.4B, respectively. It is seen that the obtained selection is very similar to the predicted selection, with I₀(2N_zH_z)/I₀(8N_zH_zH_zH_z) = 1.15 (predicted value = 1) and I₀(N_z)/I₀(16N_zH_zH_zH_zH_z) = −1.12 (predicted value = −1). The small deviation from the predicted values could be a result of relaxation during the selection element and if τ_a is slightly different from 1/|4J|.

In order to verify the obtained correlation time and proton spin-flip rate a second set of data was obtained at a static magnetic field strength of 16.44 T (700 MHz proton frequency). Here, the same two experiments were recorded as above, that is (i) selecting for an initial state proportional to 2N_zH_z + 8N_zH_zH_zH_z and (ii) selecting for an initial state proportional to N_z – 16N_zH_zH_zH_zH_z. The quality of the data obtained at 16.44 T are similar to those obtained at 11.74 T and simultaneous analysis of the two data set at 16.44 T gives τ_C = 1.47 ps ± 0.04 ps and R₁(H_z) = 4.32 s⁻¹ ± 0.02 s⁻¹. The spin-flip rate obtained at 16.44 T agrees extremely well with the spin-flip rate obtained at 11.74 T. Contributions from the ¹⁵N chemical shift anisotropy (CSA) relaxation mechanism to the relaxation rates are expected to vanish for the ammonium ion due to its tetrahedral symmetry. The slightly shorter correlation time obtained at 16.44 T compared to 11.74 T confirms that contributions from ¹⁵N CSA to the relaxation is negligible, since an ¹⁵N CSA contribution to the longitudinal relaxation rates would lead to larger relaxation rates at higher fields and thus an artificially too long correlation times being obtained at higher fields. Moreover, the correlation time obtained here is in good agreement with the rotational correlation time obtained previously from measurement of in-phase longitudinal R₁(N_z) relaxation rates of ¹⁵N-ammonium in water and at 276.5 K; τ_C = 1.41 ps [5].

Application to ¹⁵N-ammonium bound to the 41kDa ATP binding domain of DnaK: The pulse sequence of Fig. 1 together with the equations for the auto- and cross-correlated relaxation rates within the ¹⁵NH₄⁺ spin system provide the basis to characterise the local dynamics and chemical exchange properties of ammonium ions in various environments. The applications above to ¹⁵N-ammonium in an acidic aqueous solution provide the correlation time of the ammonium ion and thus provide a validation of the pulse sequence shown in Fig. 1 for the measurement of longitudinal relaxation rates of ¹⁵NH₄⁺ longitudinal spin density elements. The correlation time for ammonium ions in various solvents have been characterised [6], however the correlation time of ammonium ions within specific monovalent cation binding-sites in proteins have not been characterised previously. Previous applications [19], [20] have shown that ¹⁵N-ammonium within potassium binding-sites in medium-large proteins can be probed using ¹⁵N-¹H correlation spectra and the pulse sequence presented above in Fig. 1 therefore opens up for the possibility of measuring longitudinal relaxation rates of ¹⁵N-ammonium within potassium-binding sites in medium-large proteins.

The activity of the bacterial Hsp70 homologue DnaK relies on the binding of two potassium ions, where the two potassium ions in the ATP binding domain have been shown to be crucial for the ATP cycle [22]. Of interest here is that potassium can be substituted by ammonium with the enzyme retaining more than half of its activity [22]. Previous applications have shown that ¹⁵N-ammonium within the two potassium binding sites of a 41 kDa domain of DnaK can be probed using ¹⁵N-¹H correlation spectra, when ADP and inorganic phosphate are added to create an environment that protects the ammonium ion from the bulk solvent. An initial application below to ¹⁵N-ammonium bound to one of the two potassium binding-sites in DnaK will illustrate the applicability of the method of measuring longitudinal relaxation rates of ¹⁵N-ammonium in medium-large proteins.

Selection of the initial state and measurements of longitudinal relaxation rates of ¹⁵N-ammonium in DnaK at 18.79 T (800 MHz proton frequency) and at 278 K is shown in Fig. 5.

Fig. 5.

Characterising ¹⁵N longitudinal relaxation of ¹⁵NH₄⁺ bond to a ∼41 kDa domain of DnaK at a field of 18.79 T and a temperature of 278 K. (A) ¹⁵N-¹H correlation spectrum obtained after selecting for 2N₊H_z + 8N₊H_zH_zH_z. (B) ¹⁵N-¹H correlation spectrum obtained after selecting for N₊ − 16N₊H_zH_zH_zH_z using the pulse sequence of Fig 1. (C) Relaxation decay curve obtained following the selection in A. (D) Relaxation decay curve obtained following the selection in B. Shown in solid lines are the global fit of all the data shown in C and D.

As for the application to ¹⁵N-ammonium in acidic aqueous solutions shown above two relaxation experiments were carried out, that is, selecting for an initial state proportional to 2N_zH_z + 8N_zH_zH_zH_z (Fig.5A) and selecting for an initial state proportional to N_z – 16N_zH_zH_zH_zH_z (Fig.5B). A simultaneous analysis of the two set of data and neglecting chemical exchange provides a proton spin-flip rate of R₁(H_z) = 19.7 s⁻¹ ± 1.4 s⁻¹ and an effective correlation time of τ_C,eff = 66.3 ± 3.5 ps. These model parameters correspond to auto-relaxation rates of R₁(N_z) = 1.30 ± 0.07 s⁻¹, R₁(2N_zH_z) = 26.6 ± 1.5 s⁻¹, R₁(4N_zH_zH_z) = 48.6 ± 2.9 s⁻¹, R₁(8N_zH_zH_zH_z) = 68.0 ± 4.3 s⁻¹, and R₁(16N_zH_zH_zH_zH_z) = 85.5 ± 5.7 s⁻¹. Moreover, the intensities obtained for the initial states are v₀ = {0, 0, 0, 0, 0, 0.028 ± 0.027, 0.632 ± 0.017, −0.018 ± 0.024, 0.432 ± 0.021, 0.011 ± 0.077} for the data shown in Fig.5C and v₀ = {0, 0, 0, 0, 0, −1.693 ± 0.040, 0.021 ± 0.016, 0.091 ± 0.026, −0.002 ± 0.020, 0.276 ± 0.086} for the data shown in Fig.5D, respectively. The large proton spin-flip rate causes relaxation during the selection element, such that the selections shown in Fig.5A and B are not exactly proportional to 2N_zH_z + 8N_zH_zH_zH_z and N_z – 16N_zH_zH_zH_zH_z, respectively. This becomes particularly apparent for the selection of N_z – 16N_zH_zH_zH_zH_z (Fig.5B and D), where the fast relaxation of the quadruple anti-phase element leads to a ratio of I₀(16N_zH_zH_zH_zH_z)/I₀(N_z)∼−0.16 instead of −1. It should be stressed that a determination of the initial state is included in the least-squared analysis via the model parameter v₀ and as such the different selections serve merely as a means of providing initial states that are different enough to allow for an accurate determination of both R₁(H_z) and τ_C,eff.

It is interesting to note that the effective rotational correlation time obtained here for the ammonium ion is very similar to the rotational correlation time of a methyl group within a protein environment, which has been found to be between 25 ps $\lesssim$ τ_Me $\lesssim$ 125 ps, depending on residue type and temperature [29], [30], [31]. A previous investigation of the dynamics of lysine side chains has shown that the correlation time for the rotation about the threefold axis of the lysine —NH₃⁺ group reports on hydrogen bonding [32]. The potassium binding sites in DnaK are lined with negative charges from aspartic acid side chains and phosphate groups and hydrogen-bonding and/or salt-bridging is therefore possible and could explain the ca. 40 times longer correlation time for DnaK-bound ammonium compared to free ammonium.

It was assumed above that the rotational correlation function for ¹⁵NH₄⁺ bound to DnaK can be described accurately with one effective correlation time and chemical exchange events, for example dissociation of ¹⁵NH₄⁺ from the binding site, were neglected. The dependence of the ¹⁵NH₄⁺ longitudinal relaxation rates on temperature and the dependence of the rates on the static magnetic field strength are still to be explored. Such future explorations will, for example, give information about whether the correlation function for the rotational diffusion of ammonium ions bound to proteins is accurately described by one correlation time, τ_C,eff, or if more elaborate models for the correlation function need to be employed. The temperature dependence of the derived relaxation rates will aid to elucidate possible chemical exchange events such as dissociation of the ammonium ion.

4. Conclusions

In summary, NMR pulse schemes have been developed to both select different longitudinal spin density matrix elements of ¹⁵N-ammonium and also to obtain the longitudinal ¹⁵N relaxation rates of these longitudinal spin density elements. An initial application to ¹⁵NH₄⁺ in an acidic aqueous solution was used to validate the pulse scheme and the new method to derive the effective correlation time of the ¹⁵NH₄⁺ spin system.

An application of the derived pulse scheme to probe the dynamics of enzyme-bound ammonium ions was subsequently described, where in particular the possibility of characterising ¹⁵N-ammonium bound to a 41 kDa domain of DnaK at 278 K shows the very general applicability of the method. For such a system, the protein itself is expected to have a rotational correlation time of approximately 36 ns at 278 K, thus confirming that the derived method is applicable to characterise potassium binding-site in medium-large proteins.

The pulse scheme and method presented here provides an avenue for further investigations of protein-bound ammonium ions to elucidate the properties of potassium-binding sites in large proteins and also characterise the kinetic aspects of monovalent cation binding in such systems.

5. Material and methods

Sample preparations: A sample of ¹⁵NH₄⁺ in an acidic aqueous solution was prepared by dissolving 21 mg of ¹⁵NH₄Cl in 2 ml of 100% H₂O containing 50 mM sodium phosphate, 2 mM EDTA, 25 mM NaCl, 2 mM NaN₃. The pH was subsequently adjusted to 2.82 to slow the chemical exchange of the ammonium protons with the H₂O. The NMR sample of the ATP binding domain of DnaK from Thermus thermophilus was prepared as explained previously[19]. The protein concentration was ∼100 μM in 100% H₂O containing 150 mM ¹⁵NH₄Cl, 0.5 mM ADP, 50 mM (NH₄)H₂PO₄, 5 mM MgCl₂, 1 mM DTT, 1 mM NaN₃ and 75 mM Tris, pH 7.5.

NMR experiments: The NMR experiments were performed on a Bruker Avance III 500 MHz (11.7 T) spectrometer using an HCN Prodigy probe, a Bruker Avance III 700 MHz (16.4 T) spectrometer using a TCI cryogenic inverse triple-resonance probe, and a Bruker Avance III HD 800 MHz (18.8 T) spectrometer equipped with a cryogenic inverse triple-resonance TCI probe. All spectra were measured with an external D₂O reference insert (Wilmad coaxial insert Z278513 (Sigma–Aldrich)), such that no D₂O was added to the sample buffer.

The longitudinal relaxation rates of free ¹⁵N-ammonium at pH 2.82 were measured at a static magnetic field of 11.7 T and 16.5 T using the pulse scheme shown in Fig. 1. For the spectra showing the initial states in Fig. 3, an inter-scan delay of 1 s was used, T_relax = 4 ms, and 32 complex point were acquired in the indirect ¹⁵N frequency dimension. Relaxation delays T_relax of {0.004 s, 0.010 s, 0.020 s, 0.020 s, 0.030 s, 0.040 s, 0.050 s, 0.060 s, 0.080 s, 0.100 s, 0.120 s, 0.140 s, 0.160 s, 0.200 s, 0.300 s, 0.400 s, 0.500 s, 0.600 s, 0.800 s, 0.800 s, 1.000 s, 2.000 s, 5.000 s, 10.00 s, 15.00 s} were used for all relaxation experiments. Eight scans were acquired for each FID leading to a net acquisition time of 10 h for each initial state {τ_b,ϕ₄}.

Longitudinal relaxation rates of ¹⁵N-ammonium bound to DnaK were measured at a static magnetic field strength 18.8 T. A total of 32 complex points were acquired in the indirect ¹⁵N frequency dimension and 32 T_relax relaxation delays were used: {0.002 s, 0.004 s, 0.004 s, 0.008 s, 0.016 s, 0.032 s, 0.048 s, 0.064 s, 0.080 s, 0.096 s, 0.128 s, 0.128 s, 0.160 s, 0.192 s, 0.224 s, 0.256 s, 0.288 s, 0.320 s, 0.352 s, 0.384 s, 0.416 s, 0.448 s, 0.480 s, 0.512 s, 0.512 s, 0.600 s, 0.700 s, 0.800 s, 0.900 s, 1.000 s, 1.250 s, 1.500 s} for each initial state {τ_b,ϕ₄}. An inter-scan delay of 1 s was used and 40 scans were obtained for each FID, leading to total acquisition time of 34 h for each initial state.

Data analysis: All spectra were processed using nmrPipe [33] and signal intensities were quantified using the program FuDA [34] by assuming a common line shape for a given cross-peak during a relaxation series as described previously [15].

Relaxation decay curves for the four lines, I_αααα(T_relax), I_αααβ(T_relax), I_αβββ(T_relax), and I_ββββ(T_relax) as a function of T_relax were analysed using the propagation of the full Liouvillian and the best-fit model parameters were obtained by minimisation of the target function:

$${\chi }^2({\tau }_C\text{,}{R}_1({\mathit{H}}_z)\text{,}{\mathbf{v}}_0)={\displaystyle \sum _{\lambda =\{\alpha \alpha \alpha \alpha \text{,}\dots \text{,}\beta \beta \beta \beta \}}}{\displaystyle \sum _{T_{\mathit{relax}}}}{(I_{\lambda }^{\mathit{calc}}(T_{\mathit{relax}})-I_{\lambda }^{\mathit{exp}}(T_{\mathit{relax}}))}^2/{\sigma }_{\mathit{exp}}^2$$ (6)

where v₀ is the initial state described in the main text. The first sum is over the four lines, αααα, αααβ, αβββ, and ββββ, and the second sum is over the different relaxation delays T_relax. Moreover, $I_{\lambda }^{\mathit{exp}}(T_{\mathit{relax}})$ and ${\sigma }_{\mathit{exp}}$ are experimental intensities of the four lines observed in ¹⁵N-¹H correlation spectra and their uncertainties, respectively, $I_{\lambda }^{\mathit{calc}}(T_{\mathit{relax}})$ are calculated intensities obtained by numerical propagation of the initial state using the Liouvillian, Γ.

Calculated intensities, $I_{\lambda }^{\mathit{calc}}(T_{\mathit{relax}})$ were obtained by first calculating a vector v(T_relax), which describes the time-dependence of the population of the longitudinal coherences, {I(N_z), … I(16N_zH_zH_zH_zH_z)}:

$$\mathbf{v}(T_{\mathit{relax}})=\exp (-\mathbf{\Gamma }({\mathrm{\tau }}_{\text{C}}\text{,}{R}_1({\mathit{H}}_{\text{z}}))T_{\mathit{relax}}){\mathbf{v}}_0$$ (7)

Subsequently, the intensities of the four lines observed in the ¹⁵N-¹H correlation spectra were calculated as described previously [20]:

$$\left(\begin{array}{c}\hfill I_{\alpha \alpha \alpha \alpha }\hfill \\ \hfill I_{\alpha \alpha \alpha \beta }\hfill \\ \hfill I_{\alpha \beta \beta \beta }\hfill \\ \hfill I_{\beta \beta \beta \beta }\hfill \end{array}\right)=\left(\begin{array}{ccccc}\hfill -1/4\hfill & \hfill 1\hfill & \hfill -3/2\hfill & \hfill 1\hfill & \hfill -1/4\hfill \\ \hfill -1/2\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 1/2\hfill \\ \hfill 1/2\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill -1/2\hfill \\ \hfill 1/4\hfill & \hfill 1\hfill & \hfill 3/2\hfill & \hfill 1\hfill & \hfill 1/4\hfill \end{array}\right)\left(\begin{array}{c}\hfill I(N_z)\hfill \\ \hfill I(2N_z{\mathit{H}}_{\text{z}})\hfill \\ \hfill I(4N_z{\mathit{H}}_{\text{z}}{\mathit{H}}_{\text{z}})\hfill \\ \hfill I(8N_z{\mathit{H}}_{\text{z}}{\mathit{H}}_{\text{z}}{\mathit{H}}_{\text{z}})\hfill \\ \hfill I(16N_z{\mathit{H}}_{\text{z}}{\mathit{H}}_{\text{z}}{\mathit{H}}_{\text{z}}{\mathit{H}}_{\text{z}})\hfill \end{array}\right)$$ (8)

Finally best-fit model parameters were determined by minimising the target function χ² in Eq. (6) using in-house written software based on the LMFIT python library [35].