Microsecond Time Scale Proton Rotating-Frame Relaxation under

Article pubs.acs.org/JPCB

Microsecond Time Scale Proton Rotating-Frame Relaxation under Magic Angle Spinning Published as part of The Journal of Physical Chemistry virtual special issue “Recent Advances in Connecting Structure, Dynamics, and Function of Biomolecules by NMR”. Petra Rovó and Rasmus Linser* Department Chemie und Pharmazie, Ludwig-Maximilians-Universität München, 81377 München, Germany ABSTRACT: This paper deals with the theoretical foundation of proton magic angle spinning rotating-frame relaxation (R1ρ) and establishes the range of validity and accuracy of the presented approach to describe lowamplitude microsecond time scale motion in the solid state. Beside heteronuclear dipolar and chemical shift anisotropy interactions, a major source of relaxation for protons is the homonuclear dipolar interaction. For this latter relaxation process, no general analytical equation has been published until now, which would describe the R1ρ relaxation at any spinning speed, spin-lock field, or tilt angle. To validate the derived equations, we compared the analytical relaxation rates, obtained by solving the master equation within the framework of Redfield theory, with numerically simulated relaxation rates. We found that for small opening angles (∼10°), the relaxation rates obtained with stochastic Liouville simulations agree well with the analytical Redfield relaxation rates for a large range of motional correlation times. However, deviations around the rotary-resonance conditions highlight the fact that Redfield treatment of the solid-state relaxation rates can only provide qualitative insights into the microsecond time scale motion.

■

INTRODUCTION Solid-state nuclear magnetic resonance (NMR) relaxation data contain information about both coherent (direct spin interactions) and incoherent (stochastic motion) processes that lead to rapid signal intensity loss in relaxation rate measurements. The usual approach to extracting motionrelated dynamic information involves suppression of coherent contributions by applying high deuteration and fast magic angle spinning (MAS) speeds.1−5 While such precautions are necessary, the interpretation of incoherent solid-state proton rotating-frame relaxation in the presence of radio frequency (rf) fields (R1ρ) is still a challenge for three major reasons.4,5 (i) Redfield relaxation theory gradually fails when the correlation time of the stochastic motion (τc) approaches the time scale of 1 the relaxation mechanism (T1ρ = R ).6 (ii) The signal

contributions outside of the rotary-resonance conditions are sufficiently suppressed to a first approximation. The incoherent relaxation of the amide proton is determined by stochastic fluctuation of the atomic positions of the interacting nuclei with respect to the external magnetic field and to each other. The three mechanisms by which an amide proton relaxes are the 1 H−1H homonuclear dipolar, the 1H−15N heteronuclear dipolar, and the 1H chemical shift anisotropy (CSA) relaxation. General spin−lattice relaxation in periodically perturbed systemswith perturbations due to phase-modulated rf fields or MASwere first described by Haeberlen and Waugh,8 and their concept was applied to derive the relaxation equations for rotating-frame heteronuclear dipolar and CSA relaxation mechanisms under MAS by Kurbanov et al.9 The applicability and limitations of these Redfield relaxation equations to describe ps−ms motions are discussed by Schanda et al.4 and Smith et al.5 Here, we focus on homonuclear dipolar relaxation and derive the corresponding solid-state rotating-frame relaxation rate equations within the framework of Redfield theory and only list the analogue equations for heteronuclear dipolar and CSA relaxations. To establish the range of validity and accuracy of the derived Redfield relaxation equations, we analyze a large number of simulated relaxation profiles that were generated by

1ρ

oscillations due to recoupling of anisotropic interactions near and at the rotary-resonance conditions (where the effective rf field, ωe, equals 1/2, 1, or 2 times the spinning speed, ωr) interfere with the relaxation mechanisms and prevent accurate extraction of relaxation rates from the signal intensity decay.3,7 (iii) Protons can rarely be treated as isolated spins; they relax in densely coupled relaxation equation systems where both autoand cross-relaxation processes contribute to the signal intensity decays. In order to formulate the problem, let us consider a hypothetical amide proton rotating-frame relaxation study of a fully deuterated and labile proton back-exchanged protein at high spinning speeds (40 kHz), where the coherent © 2017 American Chemical Society

Received: April 8, 2017 Revised: May 17, 2017 Published: May 23, 2017 6117

DOI: 10.1021/acs.jpcb.7b03333 J. Phys. Chem. B 2017, 121, 6117−6130

Article

The Journal of Physical Chemistry B

relaxation rates for solid-state rotating-frame spin−lattice relaxation (R1ρ). The general concepts and the theory of nuclear spin relaxation were formulated more than 50 years ago by Solomon,14 Wangsness, Bloch,15 and Redfield6 and further discussed by Abragam.16 A detailed derivation of the solutionstate equations can be found in refs 17−19, rotating-frame spin relaxations under rf irradiation are discussed in refs 20−23, and the relevant concepts for solid-state relaxation equations are given in refs 4, 8, 9, 24, and 25. Here, we consider in detail only the case of homonuclear dipolar relaxation and note that in the case of axially symmetric anisotropic interactions the correlation functions and the spectral densities are the same for dipolar, CSA, and quadrupolar relaxations. We do not discuss cross-correlated relaxations here. The theoretical foundation of solid-state CSA/DD cross-correlated relaxation can be found in refs 13 and 26. The total, time-dependent spin Hamiltonian, /̂ (t ), is given as a sum of the deterministic Hamiltonian (/̂ 0) and stochastic Hamiltonian (/̂ 1), where /̂ 0 defines the reference interaction frame in which only the /̂ 1 is active. Stochastic perturbations are driven by the time-dependent fluctuations of the anisotropic interactions that arise between two dipoles (dipolar interaction) or between a dipole and the external magnetic field (CSA interaction) or for a quadrupole with itself (quadrupolar interaction). For the sake of simplicity, we disregard now the time-dependent fluctuation of the isotropic interactions, for example, the change of the isotropic chemical shift due to the slow conformational exchange processes, and consider only the anisotropic fluctuation of the interaction tensors. To benefit from the advantageous properties of the rotational transformations, it is advisible to express the anisotropic coupling between the interacting spins in their irreducible spherical tensor representations. In the principal axis frame (PAS), a traceless symmetric anisotropic coupling tensor (A) given in its eigenbase with eigenvalues of {axx, ayy, azz} acts on the I ⃗ and S⃗ spins as follows

using stochastic Liouville space simulations with a two-site jump model. The good agreement between the analytic and simulated relaxation rate constants would corroborate the use of Redfield relaxation equations in the analysis of experimental solid-state relaxation data.

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METHODS

Numerical Simulations. Stochastic Liouville simulations were performed using the Spinach spin dynamics simulation software (version 1.9).10 To describe the equation of motion, Spinach uses the Fokker−Planck formalism.11,12 The Fokker− Planck formalism greatly simplifies the simulations because it treats the sample spinning as a time-independent, static interaction term in the Liouvillian, and also, it performs powder averaging in a single run so that no averaging grid integration is required. In the present simulations, the Wigner D-function rank of the rotor coordinate expansion was set to 4; no improvement was seen with higher rotor harmonic ranks. Powder averaging was performed with 100 different orientations using a two-angle spherical grid. Stochastic Liouville simulations were performed in the rotating frame with highfield truncated Hamiltonians. Such a truncation prevents accurate simulation of motions at the time scale of the Larmor frequency but gives accurate results for μs−ms motions.5 All simulations were performed with a magnetic field strength of 14.1 T (600 MHz 1H Larmor frequency) and 40 kHz MAS. The pulse sequence for proton rotating-frame relaxation measurements was a π/2 excitation pulse on the 1H channel followed by a continuous-wave on-resonance spin-lock of increasing duration and an acquisition of 125 ms. No 15N decoupling was applied during the spin-lock period. The evolution of the 1H+ operator was followed for 100 ms in 1 ms steps, and for the decays displayed in Figure 7, it was followed in 100 ns steps. The simulated spectra used Lorentzian line broadening of 10 Hz. Simulations were performed with a nonselective damping rate of 10.0 Hz for all states to ensure that the spin system relaxes back to thermal equilibrium. The signal intensities in the time-domain spectra were fitted with mono- and biexponential decay functions in the form of I = I0 exp(−R1ρt) and I = Ia0 exp(−Ra1ρt) + Ib0 exp(−Rb1ρt). The first time point (t = 0 ms) was omitted in the fits. The simulated relaxation rates in Figures 7 and 8 are the results of monoexponential fits. The simulations involved a 1Ha−1Hb spin pair, which exchanged within two different subsystems differing only in the relative orientations of their interaction tensors; the isotropic chemical shifts were not varied during the simulations. The isotropic chemical shifts of the different spins were 0.0 and 3.0 ppm for the 1Ha and 1Hb spins. The distance between 1Ha and 1Hb was set to 3.00 Å, resulting in a dipolar coupling constant of 27951 rad·s−1. During simulation of homonuclear dipolar relaxations, an isotropic chemical shift tensor was assumed for both 1H spins, thereby eliminating the need for suppression of CSA/DD cross-correlated relaxation.13 A series of simulations were performed by varying the onresonance spin-lock rf field strength (ωe), the exchange rate (kex), and the exchange jump angle (θ). The populations of the two substates were kept constant at a 0.5:0.5 ratio.

I ⃗· A· S ⃗ = axxIxSx + ayyIySy + azzIzSz =

1 1 (3azz − Tr{A})T20 + (axx − ayy)[T22 + T2 − 2] 2 6 (1)

where 3azz − Tr{A} and axx − ayy are the axiality and rhombicity of the interaction tensor and T2m are the secondrank irreducible spherical tensor representations of the spin operator defined as T20 = T2 ± 2

1 (3IzSz − I S⃗ ⃗) 6 1 = I ±S ± 2

1 T2 ± 1 = ± (IzS ± + I ±Sz) 2 (2)

The isotropic, zero-rank coupling tensor component (A00) is invariant to rotation and does not contribute to relaxation. The antisymmetric first-rank components (A10, A1±1) are not directly observable in high-field NMR spectra, but they can contribute to CSA relaxation.27 However, here we ignore this relaxation contribution for now. The dipolar interaction tensor is traceless and axially symmetric, so that axx = ayy and azz = −2axx, and thus, the second term on the right-hand side of eq 1 vanishes. The magnitude of the anisotropy is determined by the nature and the distance of the interacting nuclei, so that

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THEORETICAL BACKGROUNDS Stochastic Hamiltonians. In this section, we briefly summarize the well-known equations that help us derive the 6118


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The Journal of Physical Chemistry B azz ≡ δ D = −2

μ0 γIγSℏ 4π r

+1

3

T1̃ p =

(3)

x =−1

where δD is the dipolar coupling, μ0 is the vacuum permeability, γI and γS are the gyromagnetic ratios of I and S spins, ℏ is the Planck constant, and r is the distance between the two interacting nuclei. In the following, instead of the dipolar coupling δD, we use the dipolar coupling constant d, which is half of δD. Two rotational transformations and the corresponding Euler angles can define the relative position of the interacting spin pairs with respect to the rotor and laboratory frames. For the dipolar interaction Hamiltonian (/̂ D) in the laboratory frame under MAS, it has the form of

+1

T1̃ q =

∑

×

×

q =−1 p =−1 1

=

+1

⟨11pq|2p + q⟩+ 20(p + q)(Ω PL)

ωIt )+1yq(ωeSt , βeS , ωSt )T1xT1y

(9)

2

∑

+ 20m ′(α(t ), β(t ), γ(t ))+ m2 ′ m(ωR t , θM , 0)

m ′=−2

(10)

Correlation Functions and Spectral Densities. The functions given by the consecutive Wigner transformations from the PAS to the interaction frame, + 20m(Ω PI(t )), are called “spatial functions” and they involve all spatial and timedependent information about the interaction. It is common to write the interaction Hamiltonian as a product of these timedependent spatial functions and time-independent spin operators (T2m) multiplied by a scalar (k) that defines the strength of the interaction (e.g., k = − 6 d for dipolar relaxation). (For nonaxially symmetric interactions, the strength of the interaction depends also on the rhombicity.) ̃ /̂ 1(t ) = k ∑ + 20m(Ω PI(t ))T2m = k ∑ + 20m* (Ω PI(t ))T2†m m

m

(11)

(5)

In isotropic samples (such as liquid or solid powders), the random spatial functions are assumed to be statistically stationary, so that their temporal change can be expressed by an autocorrelation function in the form of Cm(τ ) = ⟨+ 20m(t )+ 20m* (t + τ )⟩

(12)

where ⟨...⟩ symbols indicate ensemble averaging and the asterisk denotes a complex conjugate. The real part of the Fourier transform of the correlation function yields the power spectral density function

⎛1 1 ⎞ 2 5⎜ ⎟T1pT1q ⎝ p q −(p + q)⎠

∞

1

∑ ∑

+1

∑ ∑

+1xp(ωeIt , βeI ,

+ 20m(Ω PL) =

T2m = T2p + q 1

(8)

where

For heteronuclear dipolar and CSA interactions, it is advisible to convert the rank-2 spin operators into the sum of two rank-1 operators to allow locking of only one of the interacting spins during irradiation. This transformation can be done using the rule of summation of irreducible tensor operators

1

2 θM , 0)+ mn (ωet , βe , ωIt )T2m

p , q =−1 x , y =−1

+2

∑ ∑

+ m2 ′ m(ωR t ,

̃ /̂ D(t ) = − 6 d

ωe = ω12 + ΔΩ2 is the effective rf field strength, derived from the irradiation field strength, ω1, and from the chemical shift offset from the carrier, ΔΩ, βe = arctan(ω1/ΔΩ) is the effective tilt angle, and ωI is the Larmor frequency of I. Homonuclear spin pairs experiencing the same spin-lock irradiation frequency transform simultaneously; thus, a secondrank Wigner rotation can transform them from the laboratory frame to the interaction frame

= ( −1) p + q

+ 20m ′(α(t ), β(t ), γ(t ))

For the case of a heteronuclear spin pair

where + lm ′ m(α , β , γ ) are the elements of the (2l + 1)dimensional Wigner rotation matrix, α, β, and γ are the timedependent Euler angles defining the relative orientation of the PAS and the rotor-fixed frames, and ωR and θM = arctan(√2) are the spinning speed and the magic angle. To describe relaxation under spin-lock irradiation, we shall consider the evolution of the magnetization components in the interaction representation (denoted with a tilde). This requires transformation of the Hamiltonian into the doubly rotating frame. This can be achieved by another rotational transformation with angles of (ω e t, β e , ω I t), where

m =−2

∑ m ′ , m , n =−2

+ 20m ′(α(t ), β(t ), γ(t ))+ m2 ′ m(ωR t , θM , 0)T2m

2 + mn (ωet , βe , ωIt )T2m

+2

̃ /̂ D(t ) = − 6 d

(4)

∑

(7)

The dipolar Hamiltionan of a homonuclear spin pair in the interaction representation under spin-lock and MAS conditions will thus become

m , m ′=−2

T2̃ n =

+1yp(ωeSt , βeS , ωSt )T1y

∑ y =−1

+2

/̂ D(t ) = − 6 d

+1xp(ωeIt , βeI , ωIt )T1x

∑

Jm (ω) =

⟨11pq|2p + q⟩T1pT1q

∫−∞ Cm(τ)e−iωτ dτ = 2 ∫0

∞

Cm(τ )e−iωτ dτ

(6)

(13)

1 1 2 where p q −(p + q) are the Wigner j symbols and ⟨11pq|2p + q⟩ are the Clebsch−Gordan coefficients. Now, the two rank-1 spin operators (T1p and T1q) can be rotated separately using rank-1 Wigner rotational matrices

For solid samples, the correlation function depends on the index m;25,28,29 however, for isotropic solid samples, this difference is negligible and one can approximate Cm with a single function irrespective of the rank of the Wigner functions.9 This approximation enables the use of the widely applied “liquid-state” model-free formalism30 to describe

q =−1 p =−1

(

)

6119


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field strength, and spinning speed. The terms where m ≠ n are the so-called nonsecular terms, for which the spatial functions contain exponentials with different frequencies, such as exp(i(ωm − ωn)t). The cross relaxation between such terms is averaged to zero because their interaction frames rotate with different frequencies; thus, only the secular m = n terms remain in eq 18.

restricted motion in the solid state. For motions on a single time scale, the spectral density function has the form of J(ω) =

⎞ τc 2⎛ ⎜(1 − S2) ⎟ 5⎝ 1 + (ωτc)2 ⎠

(14)

where the order parameter S2 defines the spatial restriction of the motion and the correlation time τc characterizes the time scale of motion. The value of S2 ranges between 0 and 1 for unrestricted and fully restricted motion, respectively. Note that there is no overall tumbling in the solid state, and thus, all motions must be restricted to some extent. Redfield Relaxation Theory. The detailed description of spin−lattice relaxation within the framework of Redfield theory has first been described by Redfield6 and was later extensively discussed by Abragam,16 Fischer,17 and Goldman.18 In this section, we just give an overview of the main equations and highlight the important assumptions that are relevant for solidstate relaxation theory. The equation of motion for the spin density operator in the presence of relaxation is governed by the stochastic Hamiltonian, and it can be given by the Liouville−von Neumann equation in the interaction representation (σ̃) as dσ (̃ t ) ̃ = −i[/̂ 1(t ), σ (̃ t )] dt

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Homonuclear Dipolar Relaxation. Here, we consider the homonuclear dipolar relaxation of two protons (Iz and Iz′), which could be either magnetically equivalent (AA spin system) or nonequivalent (AB spin system). From a theoretical perspective, the distinction between these two types of mechanisms is arbitrary because the derived relaxation rate equations are, in principle, the same. The relaxation rates of a magnetically equivalent spin pair can be given as the sum of the auto- and cross-relaxation rates of a magnetically nonequivalent spin pair (vide infra). To facilitate the relaxation rate calculations, it is advisible to perform a basis transformation from Iz and Iz′ to Iz + Iz′ and Iz − Iz′. For distinguishable protons, both the Iz + Iz′ and Iz − Iz′ spin states contribute to the relaxation, while for indistinguishable protons, the second component is not observable. The calculation of the relaxation rate constant is performed in this new basis with the sum magnetization. Solving the master equation (eq 18) for the solid-state homonuclear dipolar relaxation rate in the doubly rotating frame gives

(15)

After some modifications, such as formal integration from time zero to t, ensemble averaging, and using the condition that ̃ the ensemble average of /̂ (t ) is zero, we obtain 1

d⟨σ (̃ t )⟩ =− dt

∫0

t

̃ ̃ ⟨[/̂ (t ), [/̂ (t + τ ), σ (̃ t + τ )]]⟩ dτ (16)

If we assume that the time scale of the evolution of σ̃ is slow in comparison to the time scale of the random fluctuation (i.e., t ≫ τ), then we can extend the integration to infinity and also replace σ̃(t + τ) by σ̃(t). This approximation, regarding the relative ratio of the time scale of the signal intensity decay and the correlation time of the motion, sets the slow time scale limit of the Redfield relaxation theory and serves as the major restriction for the quantification of μs−ms time scale motion using Redfield theory.4,5 To account for the finite lattice temperature, we can replace σ̃(t) by σ̃(t) − σ̃eq, where σ̃eq is the thermal equilibrium density operator. With these modifications, eq 16 becomes dσ (̃ t ) =− dt

∫0

∞

2

R1IIρ = 3dII2

μ γI 2ℏ

where dII = − 4π0

rII 3

(19)

is the homonuclear dipolar coupling

constants of two protons at a distance of rII and dij2(β) are the second-rank reduced Wigner rotation matrix elements. X and Y represent the spin states, where X ≡ Y for autorelaxation and X ≠ Y for cross-relaxation. Note that there is no cross-relaxation between magnetically equivalent sites. In the following equations, we use the approximation that |ωI| ≫ | ± ωe ± ωr| and thus neglect that the high-frequency terms are also modulated by the spinning and spin-lock frequencies. These equations simplify to the well-known solution-state rotating-frame relaxation rate equations in the limit of ωr = 0.19,22,31 If both ωr and ωe equal zero, they give back equations of the solution-state transverse relaxation rates (R2). AB Spin System. For general off-resonance autorelaxation of an AB spinsytem, eq 19 yields

̃ ̃ ⟨[/̂ (t ), [/̂ (t + τ ), σ (̃ t ) − σeq̃ ]]⟩ dτ

dσ (̃ t ) † = −k 2 ∑ [T2̃ m , [T2̃ n , σ (̃ t ) − σeq̃ ]] dt m,n ∞

Tr{[X , T2m]† [Y , T2m]}[dln2(θM)]2

2 × [dmn (βe)]2 J(lωr + mωe + nωI)

With the definition of the interaction Hamiltonian as it appears in eq 11, we can then rewrite eq 17 as

∫0

∑ l , m , n =−2

(17)

×

RESULTS

⟨+ 20m(Ω PI(t ))+ 02n*(Ω PI(t + τ ))⟩ dτ (18)

Equation 18 is the master equation of spin density matrix evolution as it appears in the interaction representation. Here, + 20m(Ω PI(t + τ )) and + 20n*(Ω PI(t + τ )) contain all of the oscillatory terms related to the Larmor frequencies, spin-lock 6120


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The Journal of Physical Chemistry B R1AB ρ (βe) =

1 2⎧ 1 dII⎨ (1 + 3 cos2 2βe)J(ωr) 4 ⎩ 24

Now we consider the condition where the NOE and ROE cross-relaxation contributions cancel each other out. This corresponds to the tilt angle of arctan(1/√2) ≈ 35.3°.32,33 From eqs 20 and 23, we get

1 (1 + 3 cos2 2βe)J(2ωr) 48 ⎡ 3 1 + sin 4 βe⎢J(2ωe + ωr) + J(2ωe + 2ωr) ⎣ 4 2 ⎤ 1 + J(2ωe − 2ωr) + J(2ωe − ωr)⎥ ⎦ 2 ⎡ 3 1 + sin 2 2βe⎢J(ωe + ωr) + J(ωe + 2ωr) ⎣ 8 2 ⎤ 1 3 + J(ωe − 2ωr) + J(ωe − ωr)⎥ + (5 − cos 2βe) ⎦ 2 4 ⎫ 3 J(ωI) + (3 + cos 2βe)J(2ωI)⎬ ⎭ 2 (20) +

1 2 4 2 dII J(ωr) + J(2ωr) 32 3 3 4 8 8 + J(ωe − 2ωr) + J(ωe − ωr) + J(ωe + ωr) 3 3 3 4 1 2 + J(ωe + 2ωr) + J(2ωe − 2ωr) + J(2ωe − ωr) 3 3 3 2 1 + J(2ωe + ωr) + J(2ωe + 2ωr) +28J(ωI) 3 3 + 40J(2ωI)

1 1 1 9 = dII2 J(ωr) + J(2ωr) + J(ωI) 4 6 12 2 3 3 + 3J(2ωI) + J(2ωe − 2ωr) + J(2ωe − ωr) 8 4 3 3 + J(2ωe + ωr) + J(2ωe + 2ωr) 4 8

μ1AB (35.3°) = ρ

{

}

(21)

1 2 1 2 dII J(2ωr) + J(ωr) + 3J(ωI) + 6J(2ωI) 4 3 3

{

}

(22)

The general off-resonance relaxation rate for homonuclear cross-relaxation (ROE) gives

R1AA ρ (βe) =

⎛ 1 1 ⎧ (βe) = dII2⎨(1 + 3 cos2 2βe)⎜ − J(ωr) μ1AB ρ ⎝ 24 4 ⎩ ⎡3 ⎞ 1 − J(2ωr)⎟ + sin 4 βe⎢ J(2ωe − 2ωr) ⎣8 ⎠ 48 ⎤ 3 3 3 + J(2ωe − ωr) + J(2ωe + ωr) + J(2ωe + 2ωr)⎥ ⎦ 4 4 8 ⎫ + 3 sin 2 βeJ(ωI) + 6 cos2 βeJ(2ωI)⎬ ⎭ (23)

(27)

3 2 1 dII sin 2 2βe[J(ωr − ωe) + J(ωr + ωe)] 4 8

{

1 sin 4 βe[J(ωr − 2ωe) + J(ωr + 2ωe)] 2 1 sin 2 2βe[J(2ωr − ωe) + J(2ωr + ωe)] + 16 1 + sin 4 βe[J(2ωr − 2ωe) + J(2ωe + 2ωr)] 4 1 1 + (7 − 3 cos 2βe)J(ωI) + (5 + 3 cos 2βe)J(2ωI) 4 2

}

(28)

(Formally this is the relaxation rate equation of solid-state rotating-frame quadrupolar relaxation, and the strength of the

1 2 1 1 dII − J(ωr) − J(2ωr) 4 6 12 3 3 3 + J(2ωe − 2ωr) + J(2ωe − ωr) + J(2ωe + ωr) 8 4 4 3 + J(2ωe + 2ωr) + 3J(ωI) (24) 8

{

e 2qQ 2

3

interaction ( 4 dII ) should be replaced by 3 4ℏ , where e is the charge on the electron, Q is the nuclear quadrupole moment, and eq is the principal value of the electric field gradient.) For on-resonance spin-locks, βe = π/2, eq 28 simplifies to

}

R1AA ρ (π /2) =

At the βe = 0 condition, we get the relaxation equation for solid-state homonuclear NOE μ1AB (0) ≡ σ II = ρ

1 2 dII{J(ωI) + 4J(2ωI)} 4

+

which simplifies to the solid-state equivalent of on-resonance ROE cross-relaxation μ1AB (π /2) = ρ

(26)

for the residual cross-relaxation. Indeed, from eq 27, it is apparent that the cross-relaxation at this tilt angle is negligible in comparison to the autorelaxation rate as it depends on spectral densities evaluated at ωI and 2ωI, and no lowfrequency terms affect the cross-relaxation rate constant. AA Spin System. Equations 20−27 assumed that the chemical shifts of the interacting spins are different so that the spins can cross-relax with each other. For protons at magnetically equivalent sites (AA spin system), no crossrelaxation occurs and their autorelaxation can be given by taking the double commutator in eq 19 with a coupled spin state, Iz + Iz′ at both sites. This yields

and gives the longitudinal relaxation rate when βe = 0 AB R1AB ρ (0) ≡ R1 =

}

for autorelaxation and

which simplifies for on-resonance (βe = π/2) conditions to R1AB ρ (π /2)

{

R1AB ρ (35.3°) =

3 2 dII{J(2ωe − 2ωr) + 2J(2ωe − ωr) 16

+ 2J(2ωe + ωr) + J(2ωe + 2ωr) + 10J(ωI) + 4J(2ωI)}

⎤ 1 2 ⎡ 2 1 dAB⎢ − J(ωr) − J(2ωr) + 6J(2ωI)⎥ ⎣ ⎦ 4 3 3

(29)

For βe = 0, we get the longitudinal relaxation of the AA spin system.

(25) 6121


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The Journal of Physical Chemistry B AA R1AA ρ (0) ≡ R1 =

3 2 dII{J(ωI) + 4J(2ωI)} 4

R1ISρ(0) ≡ R1IS =

(30)

(35)

It is interesting to note that at any tilt angle the sum of the homonuclear dipolar auto- and cross-relaxation rates (eqs 20 and 23) gives the homonuclear dipolar relaxation rate of an AA spin system (eq 28). AB AA R1AB ρ (βe) + μ1ρ (βe) = R1ρ (βe)

The equations describing the heteronuclear cross-relaxation rates between the dipolar coupled spins have the same form in solid and in solution states22 if high-frequency modulations by the spinning speed and the effective irradiation field is neglected.

(31)

Hetero- and Homonuclear Dipolar Relaxation of AM Spin Systems. In the late 1960s, Haeberlen and Wough laid down the foundation of the nuclear spin relaxation in the presence of periodical time perturbation such as rf field irradiation or sample spinning.8 This concept was later further developed by Mansfield34 and Vega.21,35 The relaxation equations relevant to the present study (heteronuclear dipolar and CSA relaxation under both rf irradiation and sample spinning) were given by Kurbanov et al.9 Here we give a more general equation for heteronuclear dipolar relaxation that could also describe the homonuclear dipolar relaxation of an AM spin system, (e.g., 13Cα−13CO), where the irradiation field locks the two interacting nuclei at different effective tilt angles (at βeI and βeS) with different effective field strengths (ωeI and ωeS). The master equation (eq 18) should now include the heteronuclear dipolar Hamiltonian as it appears in eq 9. 2

R1ISρ = 3dIS2

1

μ1AX (βe) = ρ

μ1AB (0) ≡ σ AX = ρ

R1CSA ρ =

× (− 1)

R1CSA ρ (βe) = (32)

1

Tr{[X , T1p]† [Y , T1p]}⟨11p0|2p⟩2

l =−2 p , x =−1

1 [dlp2(θM)]2 [dxp (βe)]2 Jp (lωr

+ xωe + pωI , η)

(38)

ωI 2Δσ 2 ⎧ 1 ⎨ sin 2 βe[J(ωe − 2ωr) 3 ⎩9

+ 2J(ωe − ωr) + 2J(ωe + ωr) + J(ωe + 2ωr)] ⎫ 1 + (3 + cos 2βe)J(ωI)⎬ 4 ⎭

Here, dIS is the heteronuclear dipolar coupling constants of two nuclei at a distance of rIS, X and Y are Iz and Iz for autorelaxation, Iz and Sz are for cross-relaxation, and d1ij(β) and d2ij(β) are the first- and second-rank reduced Wigner rotation matrix elements, respectively. Under the condition of I = 1H and S = 15N and spin-lock applied on the 15N channel, eq 32 gives back the general offresonance heteronuclear dipolar relaxation equation (cf. eqs 8− 10 in ref 9). For 1H rotating-frame heteronuclear dipolar relaxation, the I and S indexes have to be swapped so that they become

(39)

At on-resonance spin locks, we get ωI 2Δσ 2 2 4 J(ωe − 2ωr) + J(ωe − ωr) 18 3 3 4 2 + J(ωe + ωr) + J(ωe + 2ωr) + 3J(ωI) (40) 3 3

R1CSA ρ (π /2) =

{

}

1 2 1 dIS sin 2 βe[J(ωe − 2ωr) + 2J(ωe − ωr) 4 3

{

and for the longitudinal relaxation CSA R1CSA = ρ (0) ≡ R1

+ 2J(ωe + ωr) + J(ωe + 2ωr) + 9J(ωS)] 1 + (3 + cos 2βe)[3J(ωI) + 6J(ωI + ωS) 4

ωI 2Δσ 2 J (ω I ) 3

(41)

Equations 40 and 41 are equivalent to eqs 18−20 in ref 9. Applications. From the above list of relaxation equations, it is apparent that all solid-state rotating-frame spin−lattice relaxation mechanisms depend on spectral densities sampled at nωe ± mωe, lωI ± nωe ± mωe or ωI ± ωS ± nωe ± mωe frequencies, where l, m, and n are 0, 1, or 2. These are the spinning and rf field sidebands of the usual sampling frequencies common in solution-state expressions. As the Larmor frequencies are orders of magnitude larger than ωe or ωr, the higher-order term modulations were neglected in the above equations. In general, rotating-frame relaxation rates (R1ρ) are sensitive to motions in the microsecond regime, while longitudinal relaxation rates (R1) depend rather on stochastic perturbations

(33)

For on-resonance rotating-frame relaxation, it is 1 2 2 4 dIS J(ωe − 2ωr) + J(ωe − ωr) 8 3 3 4 2 + J(ωe + ωr) + J(ωe + 2ωr) + J(ωI − ωS) 3 3

{

+ 3J(ωI) + 6J(ωS) + 6J(ωI + ωS)}

2

∑ ∑

where the notations are same as in eq 32, Δσ is the reduced CSA (Δσ = σzz − 1/2(σxx + σyy)), and η is the shielding asymmetry (η = 3(σyy − σxx)/2Δσ) of the chemical shielding tensor. With the high-frequency approximation, eq 38 yields

× [d1yq(βeS)]2 Jp + q (lωr + xωeI + yωeS + pωI + qωS)

R1ISρ(π /2) =

ωI 2Δσ 2 3 x−p

1 × ⟨11pq|2p + q⟩2 (− 1)x − p + y − q [dl2p + q(θM)]2 [dxp (βeI)]2

}

1 2 dIS[6J(ωI + ωS) − J(ωI − ωS)] 4

For on-resonance irradiation (βe = π/2), the heteronuclear dipolar cross-relaxation vanishes. Chemical Shift Anisotropy Relaxation. The rotatingframe spin−lattice relaxation of CSA interaction is

Tr{[X , T2p + q]† [Y , T2p + q]}

+ J(ωI − ωS)]

(36)

(37)

l =−2 p , q =−1 x , y =−1

R1ISρ(βe) =

1 2 dIS cos βe[6J(ωI + ωS) − J(ωI − ωS)] 4

The heteronuclear NOE can be given as

1

∑ ∑ ∑

1 2 dIS{3J(ωI) + 6J(ωI + ωS) + J(ωI − ωS)} 4

(34)

and the longitudinal relaxation rate is observed with βe = 0 6122


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Figure 1. Dependence of the 1H R01ρ rate constant (first row) and 1H R1 rate constants (second row) on the order parameter (S2) and correlation time (τ) for homonoculear dipolar relaxation (a,d), for heteronuclear dipolar relaxation (b,e), and for CSA relaxation (c,f) obtained from eqs 29, 30, 34, 35, 40, and 41, using a simple single time scale model-free spectral density function as it appears in eq 14. In all cases, the magnetic field strength was 14.1 T (600 MHz 1H Larmor frequency), rHH = 3.00 Å, rHN = 1.02 Å, ΔσH = 8 ppm, the spinning speed was 40 kHz, and ωe was 0 Hz (R2 limit).

Figure 2. 1H R1ρ relaxation rate constants describing the homonoculear dipolar relaxation (a,d), heteronuclear dipolar relaxation (b,e), and CSA relaxation (c,f) mechanisms. The plots display the dependencies of the R1ρ(π/2) on the correlation time at different spinning speeds (a−c) and at different rf field strengths (d−f). The selected field strengths are near the half- and full-rotary-resonance conditions at 40 kHz spinning speed. The curves were obtained using eqs 29, 34, 40, and 14 assuming spectral density function with a single correlation time and order parameter of 0.90. In all cases, the magnetic field strength was set to 14.1 T (600 MHz 1H Larmor frequency), rHH = 3.00 Å, rHN = 1.02 Å, and ΔσH = 8 ppm.

that occur in the ∼100 ps regime. This correlation time dependence of the 1H−1H homonuclear, 1H−15N heteronuclear, and 1H CSA relaxation mechanisms is demonstrated in Figure 1. Note that the 1H CSA relaxation is almost negligible in comparison to the dipolar relaxation mechanisms. Also, the relaxation of a 1H spin due to the dipolar interaction with the directly attached 15N is expected to be much larger than its relaxation due to dipolar interaction with a nearby proton. Here, we consider a proton-diluted system where the effective 1 H−1H distance is relatively large; in the calculations, we set rHH to be 3.00 Å. In solid-state rotating-frame relaxation measurements, the maxima of the R1ρ rates are dependent not only on the extent of motion (with larger amplitude motion, R1ρ increases gradually)

but also on the spinning speed and the irradiation frequency, as demonstrated in Figure 2. At higher spinning speeds, one can expect a decrease of the R1ρ rates irrespective of the relaxation mechanism. This is in line with the observation that coherent contributions to the solid-state line width can also be suppressed with application of higher spinning speeds. The underlying principles of the coherent and incoherent mechanisms are fundamentally the same.26 The irradiation field dependencies of the different relaxation processes show distinct differences. In Figure 2d−f, we display the correlation time dependence of R1ρ at a 40 kHz spinning speed at different irradiation frequencies; ωe = 0 is the R2 limit, and 19, 39, and 79 kHz correspond to near-rotary-resonance conditions. As expected from eqs 29, 34, and 40, the homonuclear dipolar 6123


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Figure 3. 1H R1ρ relaxation rates as a function of the irradiation rf field strength, ωe/2π, calculated with eqs 29, 33, and 40 with different correlation times (τc) and order parameters using a simple single time scale model-free spectral density function, as appears in eq 14. The magnetic field strength was 14.1 T (600 MHz 1H Larmor frequency), rHH = 3.00 Å, rHN = 1.02 Å, ΔσH = 8 ppm, and the spinning speed was 40 kHz.

Figure 4. 1H R1ρ relaxation rates as a function of the spinning speed, ωr/2π, calculated with eqs 29, 33, and 40 with different correlation times (τc) and order parameters using a simple single time scale model-free spectral density function, as appears in eq 14. The magnetic field strength was 14.1 T (600 MHz 1H Larmor frequency), rHH = 3.00 Å, rHN = 1.02 Å, ΔσH = 8 ppm, and the irradiation rf field strength was 10 kHz.

resonance condition becomes too narrow to be observable unambiguously. Also, coherent anisotropic recoupling close to the resonance conditions would prevent the quantification of R1ρ rates (vide infra). It is also informative to analyze the spinning speed dependencies of the 1H relaxation mechanisms. In Figure 4, we plot the same R1ρ rates as in Figure 3 but now as a function of ωr; the irradiation field is set to 10 kHz. Again, we see two resonant R1ρ peaks, now with intensity ratios of 1:2. Clearly, a strong R1ρ(ωr) dependence reflects μs motion; otherwise, the R1ρ vs ωr profile would be flat.36 From a practical perspective, measuring the spinning speed dependence of the R1ρ rates has limited applicability because the achievable range of stable spinning conditions with a solidstate NMR probe is relatively narrow. Besides, sample heating due to increasing inner friction at higher MAS speeds interferes with the dynamic processes and hampers accurate characterization of motional parameters. It is more straightforward to measure R1ρ rates as a function of ωe as in regular solution-state

relaxation is sensitive to on-resonance spin-lock irradiation close to the half- and full-rotary-resonance conditions (where ωe ≈ 0.5ωr or ωr), while heteronuclear dipolar and CSA relaxations show dependence on the full-rotary and twice the rotary-resonance conditions (where ωe ≈ ωr or 2ωr). Accordingly, we see higher R1ρ rates (with maxima at slower correlation times) when ωe is near the corresponding resonance conditions. The dependence of the on-resonance R1ρ rates on the effective spin-lock field strength is more noticeable in Figure 3. Here, we see two resonance conditions (dubbed as the resonant R1ρ peak), with intensity ratios of 2:1. The R1ρ rates increase substantially in cases where the correlation time of the stochastic motion is in the tens of microseconds regime. The time scale of motion defines the line width of the appearing resonant R1ρ peaks, so that faster motion results in broader conditions. For τc ≪ (ωr)−1, no resonant R1ρ peak is expected, and nanosecond time scale motions only slightly elevate the baseline relaxation rates. In the limit of τc ≫ (ωr)−1, the 6124


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Figure 5. Time dependence of the diagonal (blue), cross-peak (green), and sum of the diagonal and cross-peak (yellow) intensities at different effective spin-lock tilt angles (β). (a) NOESY, where formally the tilt angle is zero; (b) on-resonance ROESY with β = 90°; (c) off-resonance ROESY with β = 35.3°. Relaxation rate constants are listed in the figure. The curves were calculated with (a) eqs 25 and 22; (b) eqs 21 and 24; and (c) eqs 26 and 27 assuming a single time scale spectral density function with an order parameter of 0.95 and correlation time of 10 ps in (a) and 1 μs in (b) and (c). In the calculations, the magnetic field strength was 14.1 T (600 MHz 1H Larmor frequency), rHH = 3.00 Å, the spinning speed was 40 kHz, and in (b) and (c) the effective irradiation field strength was 12 kHz.

Figure 6. Spinning speed and irradiation spin-lock field strength dependencies of the homonuclear dipolar cross-relaxation plotted as a function of the correlation time of the motion. (a) 1H−1H NOE, (b) hypothetical 1H−1H ROE with ωe = 0 Hz, and (c) on-resonance 1H−1H ROE. The curves were calculated with (a) eq 25 and (b,c) eq 24 assuming a single time scale spectral density function with an order parameter of 0.90, a 14.1 T magnetic field strength, and rHH = 3.00 Å.

⎡ R i μ ji ⎤⎡ i ⎤ ⎡ i⎤ d ⎢ Iz ⎥ ⎢ 1ρ 1ρ ⎥ ⎢ I z ⎥ = −⎢ ij j ⎥⎢ j ⎥ dt ⎢⎣ Izj ⎥⎦ ⎣ μ1ρ R1ρ ⎦⎣ Iz ⎦ d or in brief Iz(t ) = −RIz(t ) dt

relaxation dispersion measurements.37 Uneven sample heating due to increasingly higher irradiation frequencies could be compensated with suitable preparation blocks in a pulse sequence.38,39 Nevertheless, spinning speed-dependent studies can test the efficiency of suppression of coherent effects outside of the resonance conditions;1 furthermore, they can provide insights into hundreds of nanosecond to few microsecond time scale protein dynamics, as demonstrated for a microcrystalline ubiquitin, where the sample was spun at 60, 90, and 110 kHz spinning speeds.39 1 H Auto- and Cross-Relaxation in the Rotating Frame. 1H rotating-frame relaxation as it appears in eq 28 describes the decay of the sum of the magnetization in a homonuclear twospin system where the two spins are equivalent. However, if the peaks are separated in the spectra, then the magnetization decay is expected to be biexponential, where both the auto- (eq 20) and the cross-relaxation (eq 23) influence the time evolution of the spins. In a homonuclear 2D spectrum (e.g., NOESY or ROESY) auto- and cross-relaxation will manifest in the peak intensities at the diagonal and at the cross-peaks, respectively. The time evolution of the spin-locked magnetization of a two-spin system can be described by a coupled differential equation system

(42)

where R is the relaxation matrix and Iz is the vector containing Iiz, Ijz, and Ri1ρ, Rj1ρ and μij1ρ, μji1ρ are the auto- and cross-relaxation rate constants of Iiz and Ijz. R could be defined at any spin-lock tilt angle, so that if βe = 0, we get back the Solomon equations for a two-spin system.14,40 The formal solution of eq 42 gives Iz(t ) = Iz(0) exp( −Rt )

(43)

where the diagonalization of R gives the relaxation rates of the diagonal and cross-peaks. For an isolated homonuclear spin pair, Ri1ρ = Rj1ρ = R1ρ and μij1ρ = μji1ρ = μ1ρ; hence, the intensity evolution of the auto- and cross-peaks will be 1 [1 + exp( −2μ1ρ t )] exp[−(R1ρ − μ1ρ )t ] 2 1 aij = aji = − [1 − exp( −2μ1ρ t )] exp[−(R1ρ − μ1ρ )t ] 2 aii = ajj =

(44) 1

In H rotating-frame relaxation measurements, the second homonuclear dimension is not recorded, and thus, the observed peak intensity appears as the sum of the diagonal and the cross6125


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Figure 7. Selected time domain data of the numerical simulations of the 1H R1ρ decay curves of a 1H−1H spin pair at a distance of 3.0 Å. Simulation conditions were the following: 14.1 T magnetic field strength; ωr = 40 kHz; exchange rate of 0 (first row) and 25 kHz (second row) with an opening semiangle of 5°. The irradiation field strength is indicated in the figures, and 20 kHz corresponds to the half-rotary-resonance condition. A isotropic chemical shielding tensor was assumed for both protons with an isotropic value of 0.0 and 3.0 ppm. The chemical shift was not changed upon exchange. Simulations were performed with an overall 10 Hz damping rate. Solid yellow and green lines indicate the mono- and biexponential fits. The fitted monoexponential rates are indicated in the figure. No fitting was possible for the data points at the rotary-resonance condition. Insets show the expansion of the decays in the range of 0−10 ms. Note the oscillatory behavior of the signal intensity close to and at the half-rotaryresonance condition.

However, if the correlation time of the motion is in the ∼10−5 s regime, both NOE and ROE cross-peaks are expected, which show large spinning speed dependencies. Increased spinning speed suppresses the homonuclear cross-relaxation and shifts the sensitive regime to faster motions. At a given spinning speed, ROE cross-peak intensities will depend on the applied irradiation field strength in a similar manner as seen for the homonuclear dipolar autorelaxation (cf. Figures 2d and 6c). Simulations. To check the validity and accuracy of the derived solid-state rotating-frame relaxation equations, we performed rotating-frame Liouville space simulations using the spin dynamics simulation software Spinach.10 The basic concepts and application of the stochastic Liouville formalism for Markovian-type jump models are discussed in refs 35, 37, 41, 42, and 43. The simulations were performed with an equal-population two-site jump model. The exchange reaction between the two states can be described as

peak that would be detected in a homonuclear 2D spectrum. This gives aii + aij = exp[−(R1ρ + μ1ρ )t ]

(45)

Using the observation made in eq 31, we get

ai = exp[−R1AA ρ t]

(46)

that is, the peak intensity decay for a homonuclear two-spin system is monoexponential and the relaxation is governed by the autorelaxation rate equation of an AA spin system. Figure 5 illustrates the above observations. Independent of the spin-lock tilt angle, the time evolution of dipolarly coupled 1 H spins in a two-spin system follows a monoexponential decay (yellow curves) with a relaxation rate that is the sum of the auto- and cross-relaxation rates. At the tilt angle of 35.3°, the cross-relaxation is negligible and the autorelaxation is equal to the apparent relaxation rate. As long as the homonuclear dipolar relaxation of a proton is governed primarily by the dipolar interaction with its nearest proton neighbor, the two-spin approximation holds, and thus, the apparent relaxation rate will be close to monoexponetial 1 1 with a rate of RAA 1ρ . However, if multiple H− H interactions contribute to the homonuclear relaxation process, then Ri1ρ ≠ Rj1ρ and μij1ρ ≠ μji1ρ, which results in an apparent biexponential decay curve. The determination of the accurate rate constants for such processes would require analysis of the full relaxation rate matrix, whose analysis is highly demanding if not impossible. However, monoexponential decay would still occur at a spin-lock tilt angle of 35.3°. Therefore, for densely coupled 1H−1H systems, off-resonance (βe = 35.3°) R1ρ measurements are highly recommended.33 The equations describing the homonuclear dipolar crossrelaxation rates reveal some interesting characteristics of NOE and ROE cross-peak intensities (Figure 6). For solid-state samples with ps to hundreds of ns time scale motion, no NOE or ROE cross-peak is expected in 2D homonuclear spectra. This is why, instead of NOESY or ROESY experiments, direct dipolar recouplings such as RFDR or R-symmetry sequences are in use to gain information about spacial 1H−1H proximities.

k12

A1 ⇄ A 2 k 21

(47)

where k12 and k21 are the elementary rate constants for the forward and reverse reactions and the equilibrium populations are p1 = k21/(k12 + k21) and p2 = k12/(k12 + k21). The exchange rate is defined as the sum of the forward and backward rates, kex = k12 + k21, and the correlation time of the exchange is τex = 1/ kex. The differential equation system for a two-site exchange process can be written in a matrix form as d ⎡ A1 ⎤ ⎡−k12 k 21 ⎤⎡ A1 ⎤ ⎥⎢ ⎥ ⎢ ⎥=⎢ dt ⎣ A 2 ⎦ ⎣⎢ k12 −k 21⎥⎦⎣ A 2 ⎦

(48)

If we assume that the two conformations have the same energy and hence the same forward and reverse rates, then the p1 and p2 populations are equal. The order parameter of such an exchange process can be defined as S2 = 6126

1 (3 cos2 θ + 1) 4

(49) DOI: 10.1021/acs.jpcb.7b03333 J. Phys. Chem. B 2017, 121, 6117−6130

Article


Figure 8. Comparison between analytical (solid lines) and simulated (dots) solid-state homoculear dipolar on-resonance R1ρ relaxation rates as a function of the irradiation field strength at 14.1 T magnetic field strength, 40 kHz spinning speed, and 3.0 Å1H−1H distance. Relaxation rates simulated at 19, 20, 21, 39, 40, and 41 kHz spin-lock field strengths were excluded from the plots. (a) Exchange time scale dependence at an opening angle of 9.4° (S2 = 0.98). (b) Order parameter dependence at an exchange rate of 25 kHz (τex = 40 μs). (c) R1ρ relaxation rates calculated with Redfield theory (eq 29) as a function of the correlation time and order parameter at ωe = 12 kHz. (d) R1ρ relaxation rates obtained by stochastic Liouville simulations as a function of correlation time and order parameter at ωe = 12 kHz. Note that the scale in (c) and (d) is logarithmic. (e) Ratio of the analytical (RR1ρ) and simulated rates (RS1ρ). For a large range of correlation times, there is a good agreement between the two rates; however, for motions around 10 μs, the Redfield relaxation rates overestimate the simulated rates by a factor of 1.2−2.0.

Simulated 1H−1H Homonuclear Dipolar Relaxation. Next, we investigated the agreement between the analytical Redfield relaxation rates and the rates obtained with the stochastic Liouville simulations. Figure 8 demonstrates the acceptable agreement between the analytical and simulated solid-state homonuclear dipolar rotating-frame relaxation rates of a dipolarly coupled 1H spin pair. The coincidence is better at relatively fast (τex < 1 μs) or for relatively slow (τex > 200 μs) motions and worse at the intermediate regime and close to the resonance conditions; also, the agreement is better at smaller opening angles. We found that even far from the resonance conditions (e.g., at ωe = 12 kHz at 40 kHz spinning speed) the Redfield relaxation rate constants overstate the actual relaxation rate by a factor of 1.2−2.0 (Figure 8e). The reason for the deviations between the Redfield and the simulated relaxation rates stems from different sources: (i) The validity of Redfield relaxation for microsecond time scale motion is debatable;4,5 (ii) homonuclear dipolar recoupling close to the resonance condition superimposes with the relaxation process, resulting in oscillatory signal intensity decay where the fits might be biased (cf. Figure 7); (iii) the decays are fitted with a monoexponential function, albeit the expected decay is multiexponential.29 The Redfield relaxation rate equations with a spectral density function, as appears in the model-free formalism, are expected to agree with the rates of the stochastic two-site jump model only if the initial part, and not the whole multiexponential decay curve, is fitted with a monoexponential function. These problems are expected to influence analysis of the experimental relaxation rates; therefore, we advise interpreting the fitted motional parameters only at a qualitative level and also avoiding the vicinity of the rotary-resonance conditions (within 2−4 kHz) because the experimental disentanglement of

To avoid CSA/DD cross-correlated relaxations, the three major anisotropic interactions of an amide proton (homo- and heteronuclear DD and CSA interactions) were simulated separately, so that the signal intensity decay was governed by only one particular relaxation mechanism. The stochastic Liouville simulations of heteronculear DD and CSA relaxation mechanisms have been already reported;4,5,9,38 therefore, here we focus on the simulation of the homonuclear dipolar relaxation. Figure 7 displays the simulated relaxation decays of a dipolarly coupled 1H−1H spin pair with (bottom) and without (up) microsecond time scale (τex = 40 μs) motion at different irradiation frequencies (ω1/2π = 10, 18, 19, and 20 kHz), at a 40 kHz spinning speed. In the absence of slow motion, the signal intensity decays with the preset 10 Hz damping rate. In the presence of slow motion, the relaxation rate is increased by the exchange contribution, which gets larger in the vicinity of the (half-)rotary-resonance conditions (ω1/2π ≈ 20 kHz). The coherent contribution is manifested in the oscillation of the signal intensity decays. These oscillations get enhanced, and the effective, initial signal intensity drops as the irradiation frequency approaches the (half-)rotary-resonance condition. The relaxation rate at the rotary-resonance condition cannot be obtained. The presence and extent of oscillation is independent of the slow time scale motion; thus, it contains no direct information about the molecular dynamics. However, the overall relaxation decay, on which this oscillation is superimposed, reports about the slow time scale motion. Although the decay is expected to be multiexponential, we found it adequate to fit the simulated time domain data with a single monoexponential decay function. Solid green lines in Figure 7 represent the biexponential fits, which do not differ significantly from the monoexponential fits. 6127


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spin-lock frequencies (up to 100 kHz), (ii) shorter required spin-lock durations, and (iii) appearance of resonance conditions at lower irradiation field strengths. However, quantification of the motional parameters based on 1H relaxation rates is more complicated than analysis of 15N rates due to the (i) larger influence of residual coherent contributions to the relaxation decays, (ii) NOESY- and ROESY-type cross-relaxation between dipolarly coupled homonuclear spins, and (iii) multispin interactions where both the distance and the relative angle of the interacting pairs change simultaneously. Coherent effects can be sufficiently suppressed by using higher overall deuteration levels and higher spinning speeds;1,2,4 cross-relaxation can be identified from cross-peaks in 15N-edited ROSEY or NOESY spectra, and it can be avoided by applying off-resonance (35.3°) R1ρ spin-locks. The angle and distance dependence of the homonuclear dipolar relaxation in a dipolarly coupled proton network cannot be accurately described with a single order parameter and correlation time; only stochastic Liouville simulations including multiple spin interactions can shed light on the exact relaxation processes for such densely coupled spin systems.

the coherent and incoherent processes at these conditions seems to be impossible.

■

DISCUSSION After laying down the theoretical foundation of the solid-state 1 H rotating-frame relaxation, the question arises whether it could be used to gain new insights into a protein’s motion and whether it has any practical benefit over the well-established and understood 15N relaxation. Despite the fact that proton relaxation methodology is only in its infancy and needs further investigations, we are absolutely convinced that the answer to these questions is definitely yes. For highly deuterated proteins, where the effective 1H−1H distance is >2.5 Å, the amide proton relaxation is dominated by the 1H−15N heteronuclear relaxation. For those cases, one would expect a correlation between the measured 1H and 15N R1ρ rates as they reflect the same local motion via the same interaction type. This correlation has been recently demonstrated for a perdeuterated and partially labile proton backexchanged wild-type ubiquitin sample38 where the amide 1H and 15N R1ρ ratesmeasured far from the resonance conditionsshowed good agreement with each other. This implies that for perdeuterated proteins 1H relaxation can be used to confirm the findings of the 15N relaxation measurements. Moreover, heteronuclear 1H−15N dipolar relaxation is sensitive only to rotational motion of the amide bond vector because the length of the covalent bond does not change on the ps−ms time scale. In contrast, homonuclear 1H−1H dipolar interaction can sense both rotational as well as translational motions through the alteration of the apparent effective 1H−1H distances. Therefore, deviation in the 15N−1H R1ρ correlation plot, where 1H R1ρ ≫ 15N R1ρ, can indicate translational local motion. Also, heteronuclear 1H−15N dipolar relaxation is sensitive only to local motion. For high-amplitude, microsecond time scale amide bond reorientational motions, the 15N and the 1H relaxation rates can be so high that the corresponding 1H−15N correlation peak broadens beyond the detection level. In this case, the 15N rates of the surrounding residues close in space would not indicate any motion because their 1H−15N bond angle is not changing. However, the nearby amide protons can sensitively report on the high-amplitude motion of the “invisible” neighbor because all of the involved 1H−1H interactions would change dramatically. Similarly, the neighboring amide protons could be sensitive reporters of a 180° peptide bond flip where, in principle, no apparent change is expected for the involved 1H−15N heteronuclear dipolar interaction (which applies for close-to-180° flips). In general, we believe that the 1H and 15N R1ρ experiments may be used as complementary techniques and the combination of the two probes could give more comprehensive information about the backbone motion than does either one alone. Furthermore, if some protons are reintroduced selectively into certain carbon-bound positions of amino acid side chains (e.g., into methyl groups as CHD2), then 1H R1ρ relaxation can even report about the dynamics of the hydrophobic core of the protein. Such approaches are already in use in solution-state relaxation-dispersion-based protein NMR techniques,44,45 and the labeling schemes and pulse sequences can be readily tailored for solid-state NMR needs. From a practical perspective, the measurement of 1H R1ρ is beneficial over 15N R1ρ due to the (i) larger range of accessible

■

CONCLUSIONS Here, we have derived analytical equations for general offresonance solid-state homonuclear dipolar rotating-frame relaxation (R1ρ(βe)) for AA and AB spin systems, which gives back the standard longitudinal (R1) and on-resonance rotatingframe relaxation for βe = 0 and 90°, respectively. We solved the master equation within the framework of Redfield theory taking into consideration the signal modulation due to the Larmor frequency, spin-lock irradiation field, and spinning speed and analyzed the derived equations using the model-free approach. We found that the relaxation rate of a two-spin system with nonidentical spins (AB spin system, where both auto- and cross-relaxation occur between the spins) is equivalent to the autorelaxation rate of an identical spin pair (AA spin system, where only autorelaxation occurs) at any effective spin-lock tilt angle. We analyzed the spin-lock field strength and spinning speed dependencies of the three major relaxation mechanismshomonuclear dipolar, heteronuclear dipolar, and CSA relaxationthat occur for an amide proton and established the conditions where microsecond time scale motion affects the relaxation rates. To validate our observations, we performed a large number of stochastic Liouville space simulations using the spin dynamics software Spinach. For most conditions, we found excellent agreement between the analytical and simulated rates; however, around the rotary-resonance conditions for motions at a ∼10−100 μs time scale, the rates obtained from two-site jump simulations deviate slightly from the ones obtained analytically, therefore hindering the quantitative Redfield treatment of experimental data. Nonetheless, elevated R1ρ rates around the resonance conditions can give qualitative insights into microsecond time scale exchange processes of solid-state samples.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +49 (0)89 2180-77652. Fax: +49 (0)89 2180-77646. ORCID

Petra Rovó: 0000-0001-8729-7326 Rasmus Linser: 0000-0001-8983-2935 6128


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The Journal of Physical Chemistry B Notes

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Ilya Kuprov, Matthias Ernst, and Paul Schanda for helpful discussions. The authors acknowledge support from the Deutsche Forschungsgemeinschaft (SFBs 803 and 749, as well as the Emmy Noether program), the Verband der Chemischen Industrie (VCI) in terms of a Liebig junior group fellowship, the Excellence Cluster CIPSM, and the Max Planck Society.

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Microsecond Time Scale Proton Rotating-Frame Relaxation under

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