Site-based Description of R_1ρ Relaxation in Local Reference Frames

Abstract

Nuclear magnetic spin relaxation in the presence of an applied radiofrequency field depends critically on chemical exchange processes that transfer nuclear spins between chemical or conformational environments with distinct resonance frequencies. Characterization of chemical exchange processes in R_1ρ relaxation dispersion, CEST, and DEST experiments provides powerful insights into chemical and conformational kinetics of biological macromolecules. The present work reformulates expressions for magnetization evolution and the R_1ρ relaxation rate constant by focussing on the orientations of the tilted rotating frames of reference for magnetization components in different sites, by treating the spin-locking field strength as a perturbation to free-precession evolution, and by applying the Homotopy Analysis and Laplace transform methods to approximate solutions to the Bloch-McConnell equations. The results provide an expression for R_1ρ that is invariant to the topology of the kinetic scheme, an approximate equation for evolution of spin-locked z-magnetization, and an approach for effective simplification of chemical exchange topologies for 2- and N-site chemical exchange processes. The theoretical approach also provides an accurate approximation for relaxation during a constant-amplitude radiofrequency field in the absence of exchange.

Graphical Abstract

Introduction

The family of techniques based on nuclear spin relaxation in the presence of a spin-locking radiofrequency (rf) field, including R_1ρ, CEST, and DEST methods, enable highly detailed characterization of chemical exchange kinetic processes in nuclear magnetic resonance (NMR) spectroscopy [1, 2, 3]. When chemical exchange is fast on the chemical shift time scale, Bloch-Wangsness-Redfield (BWR) theory can be used to obtain analytical expressions for the R_1ρ relaxation rate constant for arbitrary numbers of exchanging sites N ≥ 2 (in the following, states or sites are used interchangeably) [4, 5, 6]. Expressions for R_1ρ applicable to all exchange time scales were first obtained for 2-site exchange by Trott and Palmer [7]. The initial results subsequently were extended to higher accuracy [8, 9], more general conditions [10], and N-site (N > 2) kinetic schemes [11, 12, 13]. As evident in the first 2-site [7] and N-site results [11], the R_1ρ relaxation rate constant is uniquely sensitive to the resonance frequency offsets of minor (sparsely populated) chemical or conformational states that cannot be directly observed in first-order NMR spectra. This property of the relaxation of spin-locked magnetization enables characterization of structure and dynamics of sparsely populated states of biological macromolecules using R_1ρ, CEST, and DEST experiments [14, 2, 15, 16, 17, 18, 19, 20, 21].

The present work is concerned primarily with the temporal evolution of the spin-locked z-magnetization in the tilted rotating frame subject to N-site chemical exchange, rather than only the R_1ρ relaxation rate constant, which is usually defined as the least negative real eigenvalue of the Bloch-McConnell evolution matrix [7]. This analysis focuses on the role of the differences in tilt angles for spin-locked magnetization components in different chemical or conformational states [2]. The Homotopy Analysis Method (HAM) [22] is used to obtain approximate solutions for the Bloch-McConnell equations that are highly accurate when chemical exchange is intermediate-to-slow. The Laplace transform approach developed by Abergel and coworkers [8, 5, 6] also is used to obtain an expression for the long-time evolution of the spin-locked z-magnetization that only requires operations on N × N matrices.

Theory

The following discussion focuses on 2-site exchange; however, the extension to N > 2 sites is conceptually straightforward. Indeed, the following analysis leads to expressions for N-site exchange that have the same formal structure as expressions for 2-site exchange, in constrast to earlier derivations [11, 12, 13]. Initially, the relaxation of isolated spin-1/2 magnetization is treated in a spherical basis in the tilted rotating frame of reference. This treatment also leads to new perspectives for magnetization in the laboratory Cartesian frame of reference.

The Bloch-McConnell equation in the Cartesian rotating frame of reference is given by:

$$\frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}\hfill {\overrightarrow{M}}_A(t)\hfill \\ \hfill {\overrightarrow{M}}_B(t)\hfill \end{array}\right]=\left\{\left[\begin{array}{cc}\hfill {\mathbf{L}}_A\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathbf{L}}_B\hfill \end{array}\right]+\mathbf{K}\right\}\left[\begin{array}{c}\hfill {\overrightarrow{M}}_A(t)\hfill \\ \hfill {\overrightarrow{M}}_B(t)\hfill \end{array}\right]+\left[\begin{array}{c}\hfill R_{1A}{\overrightarrow{M}}_A^0\hfill \\ \hfill R_{1B}{\overrightarrow{M}}_B^0\hfill \end{array}\right]$$ (1)

In this equation,

$${\mathbf{L}}_X=\left[\begin{array}{ccc}\hfill -R_{2X}\hfill & \hfill -{\Omega }_X\hfill & \hfill 0\hfill \\ \hfill {\Omega }_X\hfill & \hfill -R_{2X}\hfill & \hfill -{\omega }_1\hfill \\ \hfill 0\hfill & \hfill {\omega }_1\hfill & \hfill -R_{1X}\hfill \end{array}\right]$$ (2)

is the Bloch evolution matrix, ${\overrightarrow{M}}_X(t)=[M_{xX}(t),M_{yX}(t),M_{zX}(t){]}^T$ is the Cartesian magnetization vector, ${\overrightarrow{M}}_X^0=[0,0,M_{zX}^0{]}^T$ is the corresponding equilbrium magnetization vector, R_1X and R_2X are the relaxation rate constants in the absence of exchange, and Ω_X is the resonance offset frequency for a spin in site X = {A, B}. The radio-frequency (rf) field has frequency ω_rf, amplitude (Rabi frequency) ω₁, and is assumed to have x-phase. The kinetic rate matrix is given by:

$$\mathbf{K}=\left[\begin{array}{cc}\hfill -k_{12}\hfill & \hfill k_{21}\hfill \\ \hfill k_{12}\hfill & \hfill -k_{21}\hfill \end{array}\right]\otimes \mathbf{I}$$ (3)

in which the forward and reverse kinetic rate constants are k₁₂ and k₂₁, respectively. Herein, 0 and I are the empty and identity matrices with conformal dimensions, respectively. In the following, the above equation is modified to:

$$\frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}\hfill \Delta {\overrightarrow{M}}_A(t)\hfill \\ \hfill \Delta {\overrightarrow{M}}_B(t)\hfill \end{array}\right]=\left\{\left[\begin{array}{cc}\hfill {\mathbf{L}}_A\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathbf{L}}_B\hfill \end{array}\right]+\mathbf{K}\right\}\left[\begin{array}{c}\hfill \Delta {\overrightarrow{M}}_A(t)\hfill \\ \hfill \Delta {\overrightarrow{M}}_B(t)\hfill \end{array}\right]$$ (4)

by defining, $\Delta {\overrightarrow{M}}_X(t)={\overrightarrow{M}}_X(t)-{\overrightarrow{M}}_X^{ss}$ in which:

$$\left[\begin{array}{c}\hfill {\overrightarrow{M}}_A^{ss}\hfill \\ \hfill {\overrightarrow{M}}_B^{ss}\hfill \end{array}\right]=-{\left\{\left[\begin{array}{cc}\hfill {\mathbf{L}}_A\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathbf{L}}_B\hfill \end{array}\right]+\mathbf{K}\right\}}^{-1}\left[\begin{array}{c}\hfill R_{1A}{\overrightarrow{M}}_A^0\hfill \\ \hfill R_{1B}{\overrightarrow{M}}_B^0\hfill \end{array}\right]$$ (5)

are the steady-state values of the magnetization components.

The Bloch-McConnell equation is transformed in three steps. The first transformation symmetrizes the kinetic matrix using the following matrix:

$$\mathbf{S}=\left[\begin{array}{cc}\hfill \sqrt{p_A}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \sqrt{p_B}\hfill \end{array}\right]\otimes \mathbf{I}$$ (6)

in which p_A = k₂₁/k_ex is the equilibrium population of site A, p_B = k₁₂/k_ex is the equilibrium population of site B, and k_ex = k₁₂ + k₂₁. The kinetic matrix is transformed to:

$${\mathbf{S}}^{-1}\mathbf{KS}=\left[\begin{array}{cc}\hfill -k_{12}\hfill & \hfill \sqrt{k_{12}{k}_{21}}\hfill \\ \hfill \sqrt{k_{12}{k}_{21}}\hfill & \hfill -k_{21}\hfill \end{array}\right]\otimes \mathbf{I}$$ (7)

but the Bloch evolution matrices are unaffected.

The second transformation rotates the Bloch evolution matrix for each site to the local tilted rotating frame using the matrix:

$$\mathbf{R}=\left[\begin{array}{cc}\hfill {\mathbf{R}}_y({\theta }_A)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathbf{R}}_y({\theta }_B)\hfill \end{array}\right]$$ (8)

in which:

$${\mathbf{R}}_y(\theta )=\left[\begin{array}{ccc}\hfill \cos \theta \hfill & \hfill 0\hfill & \hfill \sin \theta \hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill -\sin \theta \hfill & \hfill 0\hfill & \hfill \cos \theta \hfill \end{array}\right]$$ (9)

The transformed Bloch evolution matrix is given by:

$$\begin{gathered} {\mathbf{R}}_y^{-1}({\theta }_X){\mathbf{L}}_X{\mathbf{R}}_y({\theta }_X)= \\ \left[\begin{array}{ccc}\hfill -(R_{2X}{\cos }^2{\theta }_X+R_{1X}{\sin }^2{\theta }_X)\hfill & \hfill -{\omega }_X\hfill & \hfill -\frac{\Delta R_X}{2}\sin (2{\theta }_X)\hfill \\ \hfill {\omega }_X\hfill & \hfill -R_{2X}\hfill & \hfill 0\hfill \\ \hfill -\frac{\Delta R_X}{2}\sin (2{\theta }_X)\hfill & \hfill 0\hfill & \hfill -(R_{2X}{\sin }^2{\theta }_X+R_{1X}{\cos }^2{\theta }_X)\hfill \end{array}\right] \end{gathered}$$ (10)

in which θ_X = tan⁻¹(ω₁/Ω_X) is the tilt angle, ${\omega }_X=({\omega }_1^2+{\Omega }_X^2{)}^{1∕2}$ is the effective field in the rotating frame, and ΔR_X = R_2X − R_1X for magnetization in site X. The kinetic matrix becomes:

$${\mathbf{R}}^{-1}{\mathbf{S}}^{-1}\mathbf{KSR}=\left[\begin{array}{cc}\hfill -k_{12}\mathbf{I}\hfill & \hfill \sqrt{k_{12}{k}_{21}}{\mathbf{R}}_y(\Delta \theta )\hfill \\ \hfill \sqrt{k_{12}{k}_{21}}{\mathbf{R}}_y(-\Delta \theta )\hfill & \hfill -k_{21}\mathbf{I}\hfill \end{array}\right]$$ (11)

in which Δθ = θ_B − θ_A.

The last transformation converts the Cartesian basis into the spherical basis using the matrix:

$$\mathbf{U}=\left[\begin{array}{cc}\hfill \mathbf{u}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \mathbf{u}\hfill \end{array}\right]$$ (12)

in which:

$$\mathbf{u}=\frac{1}{\sqrt{2}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill i\hfill & \hfill -i\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{2}\hfill \end{array}\right]$$ (13)

The Bloch evolution matrix becomes:

$${\overline{\mathbf{L}}}_X={\mathbf{u}}^{-1}{\mathbf{R}}_y^{-1}({\theta }_X){\mathbf{L}}_X{\mathbf{R}}_y({\theta }_X)\mathbf{u}={\mathbf{L}}_X^0+{\mathbf{D}}_X$$ (14)

in which:

$$\begin{gathered} {\mathbf{L}}_X^0=\left[\begin{array}{ccc}\hfill -R_{2\rho X}-i{\omega }_X\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -R_{2\rho X}+i{\omega }_X\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -R_{1\rho X}\hfill \end{array}\right] \\ {\mathbf{D}}_X=\frac{\Delta R_X}{2}\left[\begin{array}{ccc}\hfill 0\hfill & \hfill {\sin }^2{\theta }_X\hfill & \hfill -\frac{1}{\sqrt{2}}\sin (2{\theta }_X)\hfill \\ \hfill {\sin }^2{\theta }_X\hfill & \hfill 0\hfill & \hfill -\frac{1}{\sqrt{2}}\sin (2{\theta }_X)\hfill \\ \hfill -\frac{1}{\sqrt{2}}\sin (2{\theta }_X)\hfill & \hfill -\frac{1}{\sqrt{2}}\sin (2{\theta }_X)\hfill & \hfill 0\hfill \end{array}\right] \end{gathered}$$ (15)

and the rotating frame relaxation rate constants are defined as:

$$\begin{gathered} R_{2\rho X}=\frac{1}{2}[R_{2X}(1+{\cos }^2{\theta }_X)+R_{1X}{\sin }^2{\theta }_X] \\ {R}_{1\rho X}=R_{2X}{\sin }^2{\theta }_X+R_{1X}{\cos }^2{\theta }_X \end{gathered}$$ (16)

The kinetic matrix becomes:

$$\overline{\mathbf{K}}={\mathbf{U}}^{-1}{\mathbf{R}}^{-1}{\mathbf{S}}^{-1}\mathbf{KSRU}=\left[\begin{array}{cc}\hfill -k_{12}\mathbf{I}\hfill & \hfill \sqrt{k_{12}{k}_{21}}{\overline{\mathbf{R}}}_y(\Delta \theta )\hfill \\ \hfill \sqrt{k_{12}{k}_{21}}{\overline{\mathbf{R}}}_y(-\Delta \theta )\hfill & \hfill -k_{21}\mathbf{I}\hfill \end{array}\right]$$ (17)

in which:

$${\overline{\mathbf{R}}}_y(\Delta \theta )=\left[\begin{array}{ccc}\hfill \frac{1}{2}(\cos \Delta \theta +1)\hfill & \hfill \frac{1}{2}(\cos \Delta \theta -1)\hfill & \hfill \frac{1}{\sqrt{2}}\sin \Delta \theta \hfill \\ \hfill \frac{1}{2}(\cos \Delta \theta -1)\hfill & \hfill \frac{1}{2}(\cos \Delta \theta +1)\hfill & \hfill \frac{1}{\sqrt{2}}\sin \Delta \theta \hfill \\ \hfill -\frac{1}{\sqrt{2}}\sin \Delta \theta \hfill & \hfill -\frac{1}{\sqrt{2}}\sin \Delta \theta \hfill & \hfill \cos \Delta \theta \hfill \end{array}\right]$$ (18)

For completeness, the transformed magnetization is given by:

$$\left[\begin{array}{c}\hfill \Delta {\overline{M}}_A(t)\hfill \\ \hfill \Delta {\overline{M}}_B(t)\hfill \end{array}\right]={\mathbf{U}}^{-1}{\mathbf{R}}^{-1}{\mathbf{S}}^{-1}\left[\begin{array}{c}\hfill \Delta {\overrightarrow{M}}_A(t)\hfill \\ \hfill \Delta {\overrightarrow{M}}_B(t)\hfill \end{array}\right]$$ (19)

with a similar expression for the equilibrium and steady-state magnetizations.

With these transformations, the Bloch-McConnell equation is written as:

$$\frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}\hfill \Delta {\overline{M}}_A(t)\hfill \\ \hfill \Delta {\overline{M}}_B(t)\hfill \end{array}\right]=\{\Lambda +\Gamma \}\left[\begin{array}{c}\hfill \Delta {\overline{M}}_A(t)\hfill \\ \hfill \Delta {\overline{M}}_B(t)\hfill \end{array}\right]$$ (20)

in which:

$$\begin{gathered} \Lambda =\left[\begin{array}{cc}\hfill {\mathbf{L}}_A^0-k_{12}\mathbf{I}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathbf{L}}_B^0-k_{21}\mathbf{I}\hfill \end{array}\right] \\ \Gamma =\left[\begin{array}{cc}\hfill {\mathbf{D}}_A\hfill & \hfill \sqrt{k_{12}{k}_{21}}{\overline{\mathbf{R}}}_y(\Delta \theta )\hfill \\ \hfill \sqrt{k_{12}{k}_{21}}{\overline{\mathbf{R}}}_y(-\Delta \theta )\hfill & \hfill {\mathbf{D}}_B\hfill \end{array}\right] \end{gathered}$$ (21)

The matrix Λ is diagonal by construction and the matrix Γ is empty (all diagonal elements are zero). Importantly for the following discussion, the only frequency (imaginary) terms, ±ω_A and ±ω_B, appear on the diagonal of Λ.

Once a solution to Eq. 20 is found or approximated, the solution can be transformed back into the laboratory frame basis by using:

$$\left[\begin{array}{c}\hfill {\overrightarrow{M}}_A(t)\hfill \\ \hfill {\overrightarrow{M}}_B(t)\hfill \end{array}\right]=\mathbf{SRU}\left[\begin{array}{c}\hfill \Delta {\overline{M}}_A(t)+{\overline{M}}_A^{ss}\hfill \\ \hfill \Delta {\overline{M}}_B(t)+{\overline{M}}_B^{ss}\hfill \end{array}\right]$$ (22)

Alternatively, the matrix S commutes with R and U, so the total magnetization in the tilted rotating frame, in the Cartesian basis, is obtained as:

$$\left[\begin{array}{c}\hfill {\tilde{M}}_A(t)\hfill \\ \hfill {\tilde{M}}_B(t)\hfill \end{array}\right]=\mathbf{SU}\left[\begin{array}{c}\hfill \Delta {\overline{M}}_A(t)+{\overline{M}}_A^{ss}\hfill \\ \hfill \Delta {\overline{M}}_B(t)+{\overline{M}}_B^{ss}\hfill \end{array}\right]$$ (23)

The former transformation would be used in analyses of CEST or DEST experiments and the latter in analyses of R_1ρ experiments.

Initial insights from formulation of the Bloch-McConnell equation in the tilted rotating frame of reference are obtained by considering the limit in which ω_A ≫ {(k₁₂k₂₁)^1/2, ΔR_A} and ω_B ≫ {(k₁₂k₂₁)^1/2, ΔR_B}. This limit always is reached if ω₁ ≫ {(k₁₂k₂₁)^1/2, ΔR_A, ΔR_B}. In this limit, off-diagonal elements of D_X and ${\overline{\mathbf{R}}}_y(\pm \Delta \theta )$ are non-secular and can be set to zero. The Bloch-McConnell equation becomes:

$$\frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}\hfill \Delta {\overline{M}}_A(t)\hfill \\ \hfill \Delta {\overline{M}}_B(t)\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill {\mathbf{L}}_A^0-k_{12}\mathbf{I}\hfill & \hfill \sqrt{k_{12}{k}_{21}}\mathbf{C}\hfill \\ \hfill \sqrt{k_{12}{k}_{21}}\mathbf{C}\hfill & \hfill {\mathbf{L}}_B^0-k_{21}\mathbf{I}\hfill \end{array}\right]\left[\begin{array}{c}\hfill \Delta {\overline{M}}_A(t)\hfill \\ \hfill \Delta {\overline{M}}_B(t)\hfill \end{array}\right]$$ (24)

in which:

$$\mathbf{C}=\left[\begin{array}{ccc}\hfill \frac{1}{2}(\cos \Delta \theta +1)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2}(\cos \Delta \theta +1)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \cos \Delta \theta \hfill \end{array}\right]$$ (25)

Each secular pair of magnetization components now relaxes independently of other magnetization components. This result shows the importance of the ratio (k₁₂k₂₁)^1/2/ω₁ = (p_Ap_B)^1/2k_ex/ω₁: if this ratio is much less than unity, the six-dimensional coupled evolution of the Bloch-McConnell equation simplifies to three independent two-dimensional coupled evolution equations. For example, the resulting equation for evolution of z-magnetization in the tilted frame of reference becomes:

$$\frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}\hfill \Delta {\overline{M}}_{zA}(t)\hfill \\ \hfill \Delta {\overline{M}}_{zB}(t)\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill -R_{1\rho A}-k_{12}\hfill & \hfill \sqrt{k_{12}{k}_{21}}\cos \Delta \theta \hfill \\ \hfill \sqrt{k_{12}{k}_{21}}\cos \Delta \theta \hfill & \hfill -R_{1\rho B}-k_{21}\hfill \end{array}\right]\left[\begin{array}{c}\hfill \Delta {\overline{M}}_{zA}(t)\hfill \\ \hfill \Delta {\overline{M}}_{zB}(t)\hfill \end{array}\right]$$ (26)

This equation shows the fundamental importance for the R_1ρ experiment of the difference in tilt angles, Δθ, for spin magnetization in sites A and B, as highlighted by Al-Hashimi and coworkers [2]. To illustrate this point, the radiofrequency field is assumed to be resonant with the spin in site A. If the difference in resonance frequencies Δω = ∣Ω_B − Ω_A∣ becomes very large , then Δθ → ∣π/2∣ and the z-magnetization components in the two sites relax independently. This is the slow exchange limit in the rotating frame. In the other limit, ω₁ → ∞ and Δθ → 0. Assuming for simplicity that ΔR ≪ k_ex, the eigenvalues of the above matrix become −R₂ and −(R₂ + k_ex) in which R₂ = p_AR_2A + p_BR_2B. In this limit, chemical exchange is fully suppressed for one (the more highly populated site) of the z-magnetization components, as expected.

As shown by Koss and coworkers [12], the R_1ρ relaxation rate constant can be estimated from the negative reciprocal of the trace of the inverse of the Bloch-McConnell evolution matrix. Thus, using Eq. 26:

$$\begin{gathered} R_{1\rho }\approx -{\text{Tr}}^{-1}\left({\left[\begin{array}{cc}\hfill -R_{1\rho A}-k_{12}\hfill & \hfill \sqrt{k_{12}{k}_{21}}\cos \Delta \theta \hfill \\ \hfill \sqrt{k_{12}{k}_{21}}\cos \Delta \theta \hfill & \hfill -R_{1\rho B}-k_{21}\hfill \end{array}\right]}^{-1}\right) \\ \approx -\frac{(R_{1\rho A}+k_{12})(R_{1\rho B}+k_{21})-k_{12}{k}_{21}{\cos }^2\Delta \theta }{R_{1\rho A}+R_{1\rho B}+k_{12}+k_{21}} \\ \approx \frac{R_{1\rho A}{R}_{1\rho B}+k_{21}{R}_{1\rho A}+k_{12}{R}_{1\rho B}+k_{12}{k}_{21}{\sin }^2\Delta \theta }{R_{1\rho A}+R_{1\rho B}+k_{ex}} \end{gathered}$$ (27)

Assuming k_ex ≫ R_1ρA + R_1ρB gives:

$$\begin{gathered} R_{1\rho }\approx p_A{R}_{1\rho A}+p_B{R}_{1\rho B}+p_A{p}_B{\sin }^2\Delta \theta k_{ex} \\ \approx {\overline{R}}_{1\rho }+p_A{p}_B{k}_{ex}[\sin {\theta }_B\cos {\theta }_A-\cos {\theta }_B{\sin }_A{]}^2 \\ \approx {\overline{R}}_{1\rho }+p_A{p}_B{k}_{ex}\frac{{\omega }_1^2\Delta {\omega }^2}{{\omega }_A^2{\omega }_B^2} \\ \approx {\overline{R}}_{1\rho }+p_A{p}_B{k}_{ex}\frac{{\sin }^2\overline{\theta }\Delta {\omega }^2}{{\omega }_A^2{\omega }_B^2∕{\omega }_e^2} \end{gathered}$$ (28)

in which $\overline{\theta }={\tan }^{-1}({\omega }_1∕\overline{\Omega })$, $\overline{\Omega }=p_A{\Omega }_A+p_B{\Omega }_B$, and ${\omega }_e=({\omega }_1^2+{\overline{\Omega }}^2{)}^{1∕2}$ are the tilt angle, resonance frquency offset, and effective frequency for the population-average magnetization in the rotating frame. The last line is equivalent to the expression originally derived by Trott and Palmer:

$$\begin{gathered} R_{1\rho }={\overline{R}}_{1\rho }+{\sin }^2\overline{\theta }{R}_{ex} \\ ={\overline{R}}_{1\rho }+\frac{{\sin }^2\overline{\theta }{p}_A{p}_B{k}_{ex}\Delta {\omega }^2}{{\omega }_A^2{\omega }_B^2∕{\omega }_e^2+k_{ex}^2} \end{gathered}$$ (29)

in the limit that ${\omega }_A^2{\omega }_B^2∕{\omega }_e^2\gg k_{ex}^2$, except that the definition ${\overline{R}}_{1\rho }=p_A{R}_{1\rho A}+p_B{R}_{1\rho B}$ differs from the definitions used originally [7]. The new definition is more accurate when ω₁ is small and the rf field is near resonance with $\overline{\Omega }$.

An additional result is obtained from Eq. 24 if the radiofrequency carrier is positioned exactly between the two resonances and ω₁ = Δω/2. In this case, cos Δθ = 0 and the z-magnetizations in sites A and B relax independently of each other with decay constants R_2ρA + k₁₂ and R_2ρB + k₂₁, respectively. Essentially, the two magnetizations relax as if exchange were infinitely slow. Miloushev and Palmer showed that the R_1ρ relaxation rate constant can be calculated analytically when the radiofrequency carrier is positioned midway between the two exchanging resonances or when p_A = p_B [9]. Al-Hashimi and coworkers noted the maximum in R_1ρ is obtained when ω₁ = Δω/2 [2]. The decoupling of evolution of the two z-magnetization components is a more general phenomenon and occurs because the effective fields for sites A and B are orthogonal and cos Δθ = 0, whenever ${\Omega }_A{\Omega }_B+{\omega }_1^2=0$.

The basis of HAM has been described by Liao [22] and used to describe evolution of magnetization during shaped rf pulses by Crawley and Palmer [23].

The concept of HAM is to transform the solution of a solvable linear differential operator ℒ[] into the desired solution of a second, possibly non-linear, differential operator 𝒩[] by solving an iterative series of linear ‘deformation equations’. The transformed Bloch-McConnell equations provide the following definitions of the operators necessary for HAM:

$$ℒ[\overrightarrow{g}(t)]=\left\{\frac{\mathrm{d}}{\mathrm{d}t}-\Lambda \right\}\overrightarrow{g}(t)$$ (30)

$$𝒩[\overrightarrow{g}(t)]=\left\{\frac{\mathrm{d}}{\mathrm{d}t}-\Lambda -\Gamma \right\}\overrightarrow{g}(t)$$ (31)

in which $\overrightarrow{g}(t)$ is a vector function. The approximate solution to the Bloch-McConnell equation is given by:

$$\left[\begin{array}{c}\hfill \Delta {\overline{M}}_A(t)\hfill \\ \hfill \Delta {\overline{M}}_B(t)\hfill \end{array}\right]=\sum _{m=0}^M\left[\begin{array}{c}\hfill \Delta {\overline{M}}_{Am}(t)\hfill \\ \hfill \Delta {\overline{M}}_{Bm}(t)\hfill \end{array}\right]$$ (32)

in which M is the order of approximation.

The 0th-order approximation conveniently is chosen as the solution to Eq. 30:

$$\left[\begin{array}{c}\hfill \Delta {\overline{M}}_{A0}(t)\hfill \\ \hfill \Delta {\overline{M}}_{B0}(t)\hfill \end{array}\right]={\mathrm{e}}^{\Lambda t}\left[\begin{array}{c}\hfill \Delta {\overline{M}}_A(0)\hfill \\ \hfill \Delta {\overline{M}}_B(0)\hfill \end{array}\right]$$ (33)

The 1st-order HAM deformation equation is:

$$\begin{gathered} ℒ\left[\begin{array}{c}\hfill \Delta {\overline{M}}_{A1}(t)\hfill \\ \hfill \Delta {\overline{M}}_{B1}(t)\hfill \end{array}\right]=c_0𝒩\left[\begin{array}{c}\hfill \Delta {\overline{M}}_{A0}(t)\hfill \\ \hfill \Delta {\overline{M}}_{B0}(t)\hfill \end{array}\right] \\ \Rightarrow \left\{\frac{\mathrm{d}}{\mathrm{d}t}-\Lambda \right\}\left[\begin{array}{c}\hfill \Delta {\overline{M}}_{A1}(t)\hfill \\ \hfill \Delta {\overline{M}}_{B1}(t)\hfill \end{array}\right]=-c_0\Gamma \left[\begin{array}{c}\hfill \Delta {\overline{M}}_{A0}(t)\hfill \\ \hfill \Delta {\overline{M}}_{B0}(t)\hfill \end{array}\right] \end{gathered}$$ (34)

in which c₀ is called the convergence control parameter (vide infra). The solution to this equation is:

$$\begin{gathered} \left[\begin{array}{c}\hfill \Delta {\overline{M}}_{A1}(t)\hfill \\ \hfill \Delta {\overline{M}}_{B1}(t)\hfill \end{array}\right]=-c_0{\mathrm{e}}^{\Lambda t}{\int }_0^{t}d{t}^{\prime }{\mathrm{e}}^{-\Lambda t^{\prime }}\Gamma \left[\begin{array}{c}\hfill \Delta {\overline{M}}_{A0}(t^{\prime })\hfill \\ \hfill \Delta {\overline{M}}_{B0}(t^{\prime })\hfill \end{array}\right] \\ =-c_0{\mathrm{e}}^{\Lambda t}{\int }_0^{t}d{t}^{\prime }{\mathrm{e}}^{-\Lambda t^{\prime }}\Gamma {\mathrm{e}}^{\Lambda t^{\prime }}\left[\begin{array}{c}\hfill \Delta {\overline{M}}_A(0)\hfill \\ \hfill \Delta {\overline{M}}_B(0)\hfill \end{array}\right] \end{gathered}$$ (35)

The mth-order deformation equation is:

$$\begin{gathered} ℒ\left[\begin{array}{c}\hfill \Delta {\overline{M}}_{Am}(t)\hfill \\ \hfill \Delta {\overline{M}}_{Bm}(t)\hfill \end{array}\right]=\{ℒ+c_0𝒩\}\left[\begin{array}{c}\hfill \Delta {\overline{M}}_{A(m-1)}(t)\hfill \\ \hfill \Delta {\overline{M}}_{B(m-1)}(t)\hfill \end{array}\right] \\ \Rightarrow \left\{\frac{\mathrm{d}}{\mathrm{d}t}-\Lambda \right\}\left[\begin{array}{c}\hfill \Delta {\overline{M}}_{Am}(t)\hfill \\ \hfill \Delta {\overline{M}}_{Bm}(t)\hfill \end{array}\right]=(1+c_0)\left\{\frac{\mathrm{d}}{\mathrm{d}t}-\Lambda \right\}\left[\begin{array}{c}\hfill \Delta {\overline{M}}_{A(m-1)}(t)\hfill \\ \hfill \Delta {\overline{M}}_{B(m-1)}(t)\hfill \end{array}\right] \\ -c_0\Gamma \left[\begin{array}{c}\hfill \Delta {\overline{M}}_{A(m-1)}(t)\hfill \\ \hfill \Delta {\overline{M}}_{B(m-1)}(t)\hfill \end{array}\right] \end{gathered}$$ (36)

The solution to this equation is:

$$\begin{gathered} \left[\begin{array}{c}\hfill \Delta {\overline{M}}_{Am}(t)\hfill \\ \hfill \Delta {\overline{M}}_{Bm}(t)\hfill \end{array}\right]=(1+c_0)\left[\begin{array}{c}\hfill \Delta {\overline{M}}_{A(m-1)}(t)\hfill \\ \hfill \Delta {\overline{M}}_{B(m-1)}(t)\hfill \end{array}\right] \\ -c_0{\mathrm{e}}^{\Lambda t}{\int }_0^{t}d{t}^{\prime }{\mathrm{e}}^{-\Lambda t^{\prime }}\Gamma \left[\begin{array}{c}\hfill \Delta {\overline{M}}_{A(m-1)}(t^{\prime })\hfill \\ \hfill \Delta {\overline{M}}_{B(m-1)}(t^{\prime })\hfill \end{array}\right] \end{gathered}$$ (37)

This series solution will be called Method 1 in the following. The exponential of a diagonal matrix is obtained by exponentiating each diagonal element. Consequently the matrix calculations in the above equations can be performed analytically. For example, the first-order expression becomes:

$$\left[\begin{array}{c}\hfill \Delta {\overline{M}}_{A1}(t)\hfill \\ \hfill \Delta {\overline{M}}_{B1}(t)\hfill \end{array}\right]=-c_0\mathbf{W}(t)\left[\begin{array}{c}\hfill \Delta {\overline{M}}_A(0)\hfill \\ \hfill \Delta {\overline{M}}_B(0)\hfill \end{array}\right]$$ (38)

in which the elements of W are given by:

$$W_{ij}(t)={\Gamma }_{ij}\frac{{\mathrm{e}}^{{\Lambda }_{jj}t}-{\mathrm{e}}^{{\Lambda }_{ii}t}}{{\Lambda }_{jj}-{\Lambda }_{ii}}$$ (39)

Note that W_ii(t) = 0 because Γ_ii = 0. Equation 38 is equivalent to the result of first-order perturbation theory, but no assumptions have been made that Γ depends on a small parameter. In addition, Eq. 37 provides a convenient generalization to higher-order terms in the series solution.

Equation 24 suggests another HAM series approximation to the Bloch-McConnell equation by defining: Λ′ = Λ + C and Γ′ = Γ − C. In this case, Λ′ is not diagonal, but can be diagonalized by diagnalizing three 2 × 2 submatrices, rather than a general 6 × 6 matrix, as illustrated by Eq. 26. Given the factorization ${\Lambda }^{\prime }=\mathbf{V}\widehat{\Lambda }{\mathbf{V}}^{-1}$, in which $\widehat{\Lambda }$ is diagonal, the Bloch-McConnell equation is transformed to:

$$\frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}\hfill \Delta {\widehat{M}}_A(t)\hfill \\ \hfill \Delta {\widehat{M}}_B(t)\hfill \end{array}\right]=\left\{\widehat{\Lambda }+\widehat{\Gamma }\right\}\left[\begin{array}{c}\hfill \Delta {\widehat{M}}_A(t)\hfill \\ \hfill \Delta {\widehat{M}}_B(t)\hfill \end{array}\right]$$ (40)

in which

$$\left[\begin{array}{c}\hfill \Delta {\widehat{M}}_A(t)\hfill \\ \hfill \Delta {\widehat{M}}_B(t)\hfill \end{array}\right]={\mathbf{V}}^{-1}\left[\begin{array}{c}\hfill \Delta {\overline{M}}_A(t)\hfill \\ \hfill \Delta {\overline{M}}_B(t)\hfill \end{array}\right]$$ (41)

and $\widehat{\Gamma }={\mathbf{V}}^{-1}{\Gamma }^{\prime }\mathbf{V}$. Equation 40 has the same structure as Eq. 20 and is solved exactly as shown by Eqs. 33-37 using

$$\begin{gathered} ℒ[\overrightarrow{g}(t)]=\left\{\frac{\mathrm{d}}{\mathrm{d}t}-\widehat{\Lambda }\right\}\overrightarrow{g}(t) \\ 𝒩[\overrightarrow{g}(t)]=\left\{\frac{\mathrm{d}}{\mathrm{d}t}-\widehat{\Lambda }-\widehat{\Gamma }\right\}\overrightarrow{g}(t) \end{gathered}$$ (42)

The results of the HAM approximation then are transformed back to the original variables. This approach will be called Method 2 in the following.

The usual approaches to theoretical analysis of R_1ρ relaxation focuses on exchange between states (as in Method 1), but Method 2 suggests that additional insights are obtained by focusing instead on exchange between magnetization components. Thus, the symmetrized Bloch-McConnell equations in the laboratory frame spherical basis (i.e. appying the unitary transformations S and U, but not R) can be reordered to give:

$$\frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}\hfill \Delta {\check{M}}^{-}(t)\hfill \\ \hfill \Delta {\check{M}}^{+}(t)\hfill \\ \hfill \Delta {\check{M}}_z(t)\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill {\Gamma }_{-}\hfill & \hfill 0\hfill & \hfill i\frac{{\omega }_1}{\sqrt{2}}\mathbf{I}\hfill \\ \hfill 0\hfill & \hfill {\Gamma }_{+}\hfill & \hfill -i\frac{{\omega }_1}{\sqrt{2}}\mathbf{I}\hfill \\ \hfill i\frac{{\omega }_1}{\sqrt{2}}\mathbf{I}\hfill & \hfill -i\frac{{\omega }_1}{\sqrt{2}}\mathbf{I}\hfill & \hfill {\Gamma }_z\hfill \end{array}\right]\left[\begin{array}{c}\hfill \Delta {\check{M}}^{-}(t)\hfill \\ \hfill \Delta {\check{M}}^{+}(t)\hfill \\ \hfill \Delta {\check{M}}_z(t)\hfill \end{array}\right]$$ (43)

In this expression,

$$\left[\begin{array}{c}\hfill \Delta {\check{M}}_A(t)\hfill \\ \hfill \Delta {\check{M}}_B(t)\hfill \end{array}\right]={\mathbf{U}}^{-1}{\mathbf{S}}^{-1}\left[\begin{array}{c}\hfill \Delta M_A(t)\hfill \\ \hfill \Delta M_B(t)\hfill \end{array}\right]$$ (44)

is reordered so that $\Delta {\check{M}}^{-}(t)=[\Delta {\check{M}}_A^{-}(t)\Delta {\check{M}}_B^{-}(t){]}^T$ and:

$${\Gamma }_{-}=\left[\begin{array}{cc}\hfill -i{\Omega }_A-R_{2A}-k_{12}\hfill & \hfill \sqrt{k_{12}{k}_{21}}\hfill \\ \hfill \sqrt{k_{12}{k}_{21}}\hfill & \hfill -i{\Omega }_B-R_{2B}-k_{21}\hfill \end{array}\right]$$ (45)

with corresponding expressions for the other terms. Notably, Γ₋, Γ₊, Γ_z in general have the dimension N × N and describe the evolution of magnetization in the absence of an applied rf field. Although not pursued herein, HAM applied to the Eq. 43 leads to a power series solution in ω₁.

The Bloch-McConnell matrix in Eq. 43 is inverted by use of the Schur complement to give the first-order estimate of R_1ρ as [13]:

$$\begin{gathered} R_{1\rho }=-T{r}^1\{{\Gamma }_{-}^{-1}+{\Gamma }_{+}^{-1}\phantom{\}} \\ \phantom{\{}+{\Gamma }_z^{-1}{\left[\mathbf{I}+\frac{{\omega }_1^2}{2}({\Gamma }_{-}^{-1}{\Gamma }_z^{-1}+{\Gamma }_{+}^{-1}{\Gamma }_z^{-1})\right]}^{-1}\left[\mathbf{I}-\frac{{\omega }_1^2}{2}({\Gamma }_{-}^{-2}+{\Gamma }_{+}^{-2})\right]\} \end{gathered}$$ (46)

in which only matrices of size N × N must be inverted. In this expression, the details of the kinetic topology are contained in the individual Γ₋, Γ₊, Γ_z matrices, so that the above formal expression does not change for different kinetic schemes. This property contrasts with the use of the Schur complement acting on matrix blocks for each state, in which case different topologies generate distinct and sometimes very complex expressions [13]. Equations 43 and 46 also clearly show the effect of the rf field as a perturbation to the free-precession evolution of magnetization. When ω₁ → ∞, $R_{1\rho }\to -1∕Tr\{({\Gamma }_{+}^{-1}+{\Gamma }_{-}^{-1})\}$, which yields the expected limiting value R_1ρ = p_AR_2A + p_BR_2B for k_ex ≫ {R_2A, R_2B}.

The focus on magnetization components also provides an additional perspective when applied to the evolution of magnetization in the tilted rotating reference frame. Reordering Eq. 20 gives:

$$\frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}\hfill \Delta {\overline{M}}^{-}(t)\hfill \\ \hfill \Delta {\overline{M}}^{+}(t)\hfill \\ \hfill \Delta {\overline{M}}_z(t)\hfill \end{array}\right]=\Gamma \left[\begin{array}{c}\hfill \Delta {\overline{M}}^{-}(t)\hfill \\ \hfill \Delta {\overline{M}}^{+}(t)\hfill \\ \hfill \Delta {\overline{M}}_z(t)\hfill \end{array}\right]$$ (47)

in which,

$$\Gamma =\left[\begin{array}{ccc}\hfill {\Gamma }_{11}\hfill & \hfill {\Gamma }_{12}\hfill & \hfill {\Gamma }_{13}\hfill \\ \hfill {\Gamma }_{21}\hfill & \hfill {\Gamma }_{22}\hfill & \hfill {\Gamma }_{23}\hfill \\ \hfill {\Gamma }_{31}\hfill & \hfill {\Gamma }_{32}\hfill & \hfill {\Gamma }_{33}\hfill \end{array}\right]$$ (48)

the on-diagonal blocks are given by:

$$\begin{gathered} {\Gamma }_{11}={\Gamma }_{22}^{\ast }=\left[\begin{array}{cc}\hfill -i{\omega }_A-R_{2\rho A}-k_{12}\hfill & \hfill \sqrt{k_{12}{k}_{21}}(\cos \Delta \theta +1)∕2\hfill \\ \hfill \sqrt{k_{12}{k}_{21}}(\cos \Delta \theta +1)∕2\hfill & \hfill -i{\omega }_B-R_{2\rho B}-k_{21}\hfill \end{array}\right] \\ {\Gamma }_{33}=\left[\begin{array}{cc}\hfill -R_{1\rho A}-k_{12}\hfill & \hfill \sqrt{k_{12}{k}_{21}}\cos \Delta \theta \hfill \\ \hfill \sqrt{k_{12}{k}_{21}}\cos \Delta \theta \hfill & \hfill -R_{2\rho B}-k_{21}\hfill \end{array}\right] \end{gathered}$$ (49)

and the off-diagonal blocks are given by:

$$\begin{gathered} {\Gamma }_{12}={\Gamma }_{21}=\left[\begin{array}{cc}\hfill \frac{\Delta R_A}{2}{\sin }^2{\theta }_A\hfill & \hfill \sqrt{k_{12}{k}_{21}}(\cos \Delta \theta -1)∕2\hfill \\ \hfill \sqrt{k_{12}{k}_{21}}(\cos \Delta \theta -1)∕2\hfill & \hfill \frac{\Delta R_B}{2}{\sin }^2{\theta }_B\hfill \end{array}\right] \\ {\Gamma }_{13}={\Gamma }_{23}={\Gamma }_{31}^T={\Gamma }_{32}^T=\left[\begin{array}{cc}\hfill -\frac{\Delta R_A}{2\sqrt{2}}\sin (2{\theta }_A)\hfill & \hfill \frac{\sqrt{k_{12}{k}_{21}}}{\sqrt{2}}\sin \Delta \theta \hfill \\ \hfill -\frac{\sqrt{k_{12}{k}_{21}}}{\sqrt{2}}\sin \Delta \theta \hfill & \hfill -\frac{\Delta R_B}{2\sqrt{2}}\sin (2{\theta }_B)\hfill \end{array}\right] \end{gathered}$$ (50)

In many experiments, the primary interest is in the evolution of the two spin-locked z-magnetizations. The Bloch-McConnell equation for the six magnetization components can be transformed into a matrix differential equation for evolution of the z-magnetization components. First, note that

$$\frac{d^n}{d{t}^n}\Delta {\overline{M}}_z(t)=[{\Gamma }_{31}{\Gamma }_{32}{\Gamma }_{33}]{\Gamma }^{n-1}\left[\begin{array}{c}\hfill \Delta {\overline{M}}^{-}(t)\hfill \\ \hfill \Delta {\overline{M}}^{+}(t)\hfill \\ \hfill \Delta {\overline{M}}_z(t)\hfill \end{array}\right]$$ (51)

The pair of equations for n = 1 and n = 2 is solved for $\Delta {\overline{M}}^{-}(t)$ and $\Delta {\overline{M}}^{+}(t)$, which are substituted into the equation for n = 3. After some algebraic simplification, the resulting third-order differential equation for $\Delta {\overline{M}}_z(t)$ is:

$$\frac{d^3}{d{t}^3}\Delta {\overline{M}}_z(t)={\Theta }_2\frac{d^2}{d{t}^2}\Delta {\overline{M}}_z(t)+{\Theta }_1\frac{d}{dt}\Delta {\overline{M}}_z(t)+{\Theta }_0\Delta {\overline{M}}_z(t)$$ (52)

in which,

$${\Theta }_j=[{\Gamma }_{31}{\Gamma }_{32}{\Gamma }_{33}]{\Gamma }^2{\Psi }_j$$ (53)

$$\begin{gathered} {\Psi }_0=\left[\begin{array}{c}\hfill -(\alpha -\beta {\Gamma }_{32}^{-1}{\Gamma }_{31}{)}^{-1}(\gamma -\beta {\Gamma }_{32}^{-1}{\Gamma }_{33})\hfill \\ \hfill -(\beta -\alpha {\Gamma }_{31}^{-1}{\Gamma }_{32}{)}^{-1}(\gamma -\alpha {\Gamma }_{31}^{-1}{\Gamma }_{33})\hfill \\ \hfill \mathbf{I}\hfill \end{array}\right] \\ {\Psi }_1=\left[\begin{array}{c}\hfill -(\alpha -\beta {\Gamma }_{32}^{-1}{\Gamma }_{31}{)}^{-1}\beta {\Gamma }_{32}^{-1}\hfill \\ \hfill -(\beta -\alpha {\Gamma }_{31}^{-1}{\Gamma }_{32}{)}^{-1}\alpha {\Gamma }_{31}^{-1}\hfill \\ \hfill 0\hfill \end{array}\right] \\ {\Psi }_2=\left[\begin{array}{c}\hfill (\alpha -\beta {\Gamma }_{32}^{-1}{\Gamma }_{31}{)}^{-1}\hfill \\ \hfill (\beta -\alpha {\Gamma }_{31}^{-1}{\Gamma }_{32}{)}^{-1}\hfill \\ \hfill 0\hfill \end{array}\right] \end{gathered}$$ (54)

$$\begin{gathered} \alpha ={\Gamma }_{31}{\Gamma }_{11}+{\Gamma }_{32}{\Gamma }_{21}+{\Gamma }_{33}{\Gamma }_{31} \\ \beta ={\Gamma }_{31}{\Gamma }_{12}+{\Gamma }_{32}{\Gamma }_{22}+{\Gamma }_{33}{\Gamma }_{32} \\ \gamma ={\Gamma }_{31}{\Gamma }_{13}+{\Gamma }_{32}{\Gamma }_{23}+{\Gamma }_{33}{\Gamma }_{33} \end{gathered}$$ (55)

The above expressions are simplified for 2-site exchange because Γ₁₃ = Γ₃₂ (Eq. 50). In essence, a first-order 6 × 6 matrix differential equation has been replaced by a third-order 2 × 2 differential equation. For N-site exchange, the first-order 3N × 3N matrix differential equation is reformulated as a third-order N × N differential equation.

The Laplace transform of Eq. 52 has the solution:

$$\begin{gathered} \Delta {\overline{M}}_z(s)=[s^3\mathbf{I}-{\Theta }_2{s}^2-{\Theta }_{1}s+{\Theta }_0{]}^{-1}[s^2-\{{\Theta }_2\Delta {\overline{M}}_z(0)-\Delta {\overline{M}}_z^{\prime }(0)\}s \\ -{\Theta }_1\Delta {\overline{M}}_z(0)-{\Theta }_2\Delta {\overline{M}}_z^{\prime }(0)+\Delta {\overline{M}}_z^{″}(0)] \end{gathered}$$ (56)

in which:

$$\begin{gathered} \Delta {\overline{M}}_z^{\prime }(0)=\frac{d}{dt}\Delta {\overline{M}}_z(t){\mid }_{t=0}={\Gamma }_{33}\Delta {\overline{M}}_z(0) \\ \Delta {\overline{M}}_z^{″}(0)=\frac{d^2}{d{t}^2}\Delta {\overline{M}}_z(t){\mid }_{t=0}=\gamma \Delta {\overline{M}}_z(0) \end{gathered}$$ (57)

and the second equality in each expression above is obtained assuming that the initial magnetization is aligned along the tilted frames of reference for the A and B sites.

A [0,1] Padé approximation is obtained from the s → 0 limit of Eq. 56 using Eq. 57 [8, 5]:

$$\Delta {\overline{M}}_z(s)=[s\mathbf{I}+\mathbf{B}{\Theta }_1^{-1}{\Theta }_0{]}^{-1}\mathbf{B}(1+{\Theta }_1^{-1}{\Theta }_2{\Gamma }_{33}-{\Theta }_1^{-1}\gamma )\Delta {\overline{M}}_z(0)$$ (58)

in which:

$$\mathbf{B}=[\mathbf{I}+{\Theta }_1^{-1}\{{\Theta }_2-{\Gamma }_{33}\}\{{\Theta }_1+{\Theta }_2{\Gamma }_{33}-\gamma {\}}^{-1}{\Theta }_0{]}^{-1}$$ (59)

The inverse Laplace transform of Eqn. 58 yields:

$$\Delta {\overline{M}}_z(t)=\text{exp}[-\mathbf{B}{\Theta }_1^{-1}{\Theta }_{0}t]\mathbf{B}(\mathbf{I}+{\Theta }_1^{-1}{\Theta }_2{\Gamma }_{33}-{\Theta }_1^{-1}\gamma )\Delta {\overline{M}}_z(0)$$ (60)

as an approximate solution for the long-time decay components of the z-magnetization.

Methods

All calculations were performed using Python 3.6. Numerical solutions of Eq. 4 were obtained using the SciPy odeint routine. Open-source code for all calculations is available at Github (https://github.com/hanskoss/HAMr1r).

Results and Discussion

The above theoretical analyses have yielded a number of novel results for evolution of spin-locked magnetization and for approximate expressions for the R_1ρ relaxation rate constant. In the following, the accuracy of the analytical approximations are compared to numerical results. In addition, insights obtained by covarying ω₁ and ω_rf are presented for 2- and N-site chemical exchange topologies.

If c₀ = −1, the HAM solutions are identical to an expansion of exp[(Λ + Γ)t] in a series of directional derivatives [24]. A similar series has been used in the analysis of Carr-Purcell-Meiboom-Gill relaxation dispersion by Koss and coworkers [25, 26]. As in the earlier work, such expansions are are accurate at low order when exchange is relatively slow. In contrast to other applications of HAM [23], variation of c₀ did not appreciably affect convergence of the series approximations to the Bloch-McConnell equations. Consequently, all calculations performed herin used c₀ = −1.

When ω₁ is sufficiently large, only secular components in the local tilted frames of reference exchange magnetization through the kinetic process, as shown by Eq. 24. This result is illustrated in Figure 1. Initial magnetization was aligned along the local tilted rotating reference frames, as would be obtained experimentally using an adiabatic sweep [27], so that the initial magnetization is given by [0, 0, p_A, 0, 0, p_B]^T. The x- and y- components of the magnetization remain small in magnitude, because these components are isolated from z-magnetization in Eq. 24. The x- and y-magnetizations are not identically zero at all times because ω₁ is not infinitely greater than (k₁₂k₂₁)^1/2. The zeroth-order result for method 1 does not couple the z-magnetization for sites A and B and fails to reproduce the numerical results. The zeroth-order result using method 2 is identical to the solution of Eq. 24 and exactly matches the numerical results for z-magnetization, while predicting the ideal results that x- and y-magnetizations are zero. The first-order result using method 2 is nearly exact for all magnetization components. The single exponential approximation to the evolution of $\Delta {\tilde{M}}_{zA}(t)$ magnetization using Eq. 29 is highly accurate.

Figure 1:

HAM approximations for R_1ρ relaxation in limit of large ω₁. (a) M_x(t), (b) M_y(t), and (c) M_z(t) in local tilted frame for magnetization in site A. (d) M_x(t), (e) M_y(t), and (f) M_z(t) in local tilted frame for magnetization in site B. (black, solid line) Numerical solution of the Bloch-McConnell equation (Eq. 4). (green, dot-dashed line) Zeroth-order HAM approximation using method 1. (orange, dashed line) Zeroth-order HAM approximation using method 2. (reddish-purple, dotted line) First-order HAM approximation using method 2. (c) (blue, dotted line). Single exponential approximation using Eq. 29. Calculations use Ω_A/2π = −200 Hz, Ω_B/2π = 0 Hz, Δω/(2π) = 200 Hz, ω₁ = Δω, k_ex = 0.1Δω, p_A = 0.8, p_B = 0.2, R_1A = R_1B = 1 s⁻¹, R_2A = R_2B = 20 s⁻¹, and c₀ = −1. Initial magnetization is aligned along the local tilted rotating frames with amplitudes p_A and p_B, respectively.

When ω₁ = Δω/2 is sufficiently large and the radiofrequency carrier is positioned midway between Ω_A and Ω_B, cos Δθ = 0, and the z-magnetization in sites A and B become decoupled and relax independently, as shown by Eq. 24. This result is illustrated in Figure 2. Initial magnetization was aligned along the local tilted rotating reference frames. As in Fig. 1, the x- and y- components of the magnetization remain small in magnitude. The zeroth-order results using method 1 and method 2 now match the numerical results for z-magnetization, while predicting the ideal results that x- and y-magnetizations are zero. The first-order result using method 2 again is nearly exact for all magnetization components. The single exponential approximation to the evolution of $\Delta {\tilde{M}}_{zA}(t)$ magnetization using Eq. 29 is highly accurate.

Figure 2:

HAM approximations for R_1ρ relaxation in limit of large ω₁ and Ω_A = −Ω_B. (a) M_x(t), (b) M_y(t), and (c) M_z(t) in local tilted frame for magnetization in site A. (d) M_x(t), (e) M_y(t), and (f) M_z(t) in local tilted frame for magnetization in site B. (black, solid line) Numerical solution of the Bloch-McConnell equation (Eq. 4). (green, dot-dashed line) Zeroth-order HAM approximation using method 1. (orange, dashed line) Zeroth-order HAM approximation using method 2. (reddish-purple, dotted line) First-order HAM approximation using method 2. (c) (blue, dotted line) Single exponential approximation using Eq. 29. Calculations use Ω_A/2π = −100 Hz, Ω_B/2π = 100 Hz, Δω/(2π) = 200 Hz, ω₁ = Δω/2, k_ex = 0.1Δω, p_A = 0.8, p_B = 0.2, R_1A = R_1B = 1 s⁻¹, R_2A = R_2B = 20 s⁻¹, and c₀ = −1. Initial magnetization is aligned along the local tilted rotating frames with amplitudes p_A and p_B, respectively.

The results using method 2 are accurate for relatively low orders of approximation when chemical exchange is slow on the chemical shift time scale or when (k₁₂k₂₁)^1/2/ω₁ << 1. In practice, results are highly accurate using the second-order (M = 2) expansion if this ratio is < 0.2 and using the fifth-order (M = 5) approximation if this ratio is < 0.5. Even-order terms in the expansion primarily contribute to the accuracy of the evolution of z-magnetization and odd-order terms primarily contribute to the accuracy of the evolution of transverse magnetization.

Approximate expressions for R_1ρ relaxation rate constants derived by estimation of the largest (least negative) eigenvalue of the Bloch-McConnell rate matrix generally are accurate when p_B << p_A [7, 9, 12]. The dependence of the difference in site populations is reduced in higher-order approximations, for example, comparing the results obtained by Trott and Palmer (Eq. 29) to those obtained by Miloushev and Palmer [9]:

$$\begin{gathered} \frac{\mathrm{d}}{\mathrm{d}t}{R}_{1\rho }={\overline{R}}_{1\rho }+{\sin }^2\overline{\theta }{R}_{ex} \\ ={\overline{R}}_{1\rho }+\frac{{\sin }^2\overline{\theta }{p}_A{p}_B{k}_{ex}\Delta {\omega }^2}{{\omega }_A^2{\omega }_B^2∕{\omega }_e^2+k_{ex}^2-{\sin }^2\overline{\theta }{p}_A{p}_B\Delta \omega \left(1+\frac{2k_{ex}^2(p_A{\omega }_A^2+p_B{\omega }_B^2)}{{\omega }_A^2{\omega }_B^2+{\omega }_e^2{k}_{ex}^2}\right)} \end{gathered}$$ (61)

In contrast, the HAM method is less dependent on the values of the site populations provided the above limits on (k₁₂k₂₁)^1/2/ω₁ are observed. As shown in Fig. 3, the effective relaxation rate constant obtained by integration of the estimated M_zA(t) magnetization in an R_1ρ experiment agrees closely with the results of Eq. 61, although the magnetization decay is clearly multi-exponential.

Figure 3:

HAM approximations for R_1ρ relaxation high and low populations of state A. (a) M_zA(t) in the local tilted frame for magnetization in site A with site population p_A = 0.9. (b) M_zA(t) in the local tilted frame for magnetization in site A with site population p_A = 0.6. (black, solid line) Numerical solution of the Bloch-McConnell equation (Eq. 4). (reddish-purple, dashed line) fourth-order HAM approximation using method 2 with c₀ = −1. (orange, dash-dotted line) Exponential decay with decay constant determined from the integral of the HAM approximation. (blue, dash-dot-dotted line) Exponential decay with rate constant determined from Eq. 29. (green, dotted line) Exponential decay with rate constant determined from Eq. 61. Calculations use Ω_A/2π = −200 Hz, Ω_B/2π = 0 Hz, Δω/(2π) = 200 Hz, ω₁ = 0.25Δω, k_ex = 0.25Δω, R_1A = R_1B = 1 s⁻¹, R_2A = R_2B = 20 s⁻¹. Initial magnetization is aligned along the local tilted rotating frames with amplitudes p_A and p_B, respectively.

The evolution of z-magnetization can be approximated using Eq. 60 for cases in which the transverse magnetization components, in the tilted rotating frame of reference, are not of interest. Equation 60 is an accurate description of the evolution of magnetization at long times, after decay of initial transients. An example of this approach is shown in Figure 4. The graphs show the evolution of z-magnetization in sites A and B for three different rf carrier positions: on-resonance with spins in site A, at the midpoint between resonances for sites A and B, and on-resonance with spins in site B, as might be sampled as part of a relaxation dispersion measurement. The exchange process is in the intermediate regime with (k₁₂k₂₁)^1/2/ω₁ = 0.8, giving rise to complicated evolution profiles that are nonetheless well approximated by Eq. 60. The decay rate for magnetization in site A obtained from the least negative eigenvalue of the argument to the exponential in Eq. 60 agrees well with results calculated from Eq. 61. If the evolution of magnetization becomes highly oscillatory, then the factor B is not accurate, leading to divergent evolution profiles. In such cases, fixing B = I yields convergent results comparable to the first-order accuracy of Eq. 29.

Figure 4:

Approximations for R_1ρ relaxation of states A and B. (a,c,e) M_zA(t) in the local tilted frame for magnetization in site A. (b,d,f) M_zB(t) in the local tilted frame for magnetization in site B. (black, solid line) Numerical solution of the Bloch-McConnell equation (Eq. 4); (reddish-purple, dashed line) calculation from Eq. 60; and (blue, dotted line) calculation from Eq. 60 with B = I. The rf carrier positions were (a, b) equal to the resonance frequency for site A, (c, d) midway between the resonance frequencies for sites A and B, and (e,f) equal to the resonance frequency for site B. Other parameters are p_A = 0.8, Ω_A/2π = 0 Hz, Ω_B/2π = 200 Hz, Δω/(2π) = 200 Hz, ω₁ = 0.5Δω, k_ex = Δω, R_1A = R_1B = 1 s⁻¹, R_2A = R_2B = 20 s⁻¹. Initial magnetization is aligned along the local tilted rotating frames with amplitudes p_A and p_B, respectively.

When chemical exchange is not extremely fast, the position of the minor (state B) resonance is identified as a local maximum of $R_{ex}=(R_{1\rho }-{\overline{R}}_{1\rho })∕{\sin }^2\overline{\theta }$ as a function of resonance offset (or local minimum in resonance intensity in CEST experiments) [3]. Equation 24 suggests another strategy for determining Δω, and hence Ω_B. If resonance offset is varied while maintaining a fixed tilt angle θ_A, a local maximum is obtained when cos Δθ = 0. The simplest approach is to fix θ_A = 45° or 135°, depending on whether Ω_A is upfield or downfield of Ω_B, respectively; in these cases, the local maximum occurs for ω₁ = Δω/2 [2]. Also as indicated by Eq. 26, at this critical point, evolution of magnetization in state A is decoupled from evolution of magnetization is state B (Fig. 2). In practice, resonance offsets may be measured from $\overline{\Omega }$, rather than Ω_A. If ω₁ and ω_rf are shifted to maintain Δθ = 90°, then:

$$\begin{gathered} 0={\Omega }_A{\Omega }_B+{\omega }_1^2 \\ =-(p_B\Delta \omega +{\omega }_1)(p_A\Delta \omega -{\omega }_1)+{\omega }_1^2 \\ =2{\omega }_1^2-{\omega }_1\Delta \omega (p_A-p_B)-p_A{p}_B\Delta {\omega }^2 \end{gathered}$$ (62)

in which Ω_A = sgn(Δω)(−p_BΔω − ω₁) and Ω_B = sgn(Δω)(p_AΔω − ω₁). The solution to this quadratic equation is:

$$\begin{gathered} {\omega }_1=\frac{\Delta \omega }{4}(p_A-p_B\pm \sqrt{1+4p_A{p}_B}) \\ =\frac{\Delta \omega }{4}(1-2p_B\pm \sqrt{1+4p_B-4p_B^2}) \end{gathered}$$ (63)

When p_A ≫ p_B:

$${\omega }_1\approx \{\Delta \omega ∕2,-p_B\Delta \omega \}$$ (64)

The first solution is the desired result and is identical to that obtained if resonance offset is measured relative to Ω_A. The second solution trivially yields a second maximum in R_1ρ at the resonance position of the major state A. These results are exact when ω₁ ≫ {(k₁₂k₂₁)^1/2, ΔR_A, ΔR_B} and approximate otherwise. Once Δω is determined, the desired value of Ω_B is obtained using the known value of Ω_A because Δω = Ω_B − Ω_A.

Examples of the two approaches are shown in Figure 5 for (k₁₂k₂₁)^1/2/ω₁ = 0.03 and 0.075. In Fig. 5a, local maxima in R_1ρ are observed at ω₁/2π = 100 and ω₁/2π = 98 Hz for the two different values of k_ex when resonance offset is measured from Ω_A = −ω₁. Both maxima are observed at ω₁/2π = 98 Hz when resonance offset is measured from $\overline{\Omega }=-{\omega }_1$. These results are in excellent agreement with with expected value of ω₁ = Δω/2 for Δω/2π = 200 Hz. At the maxima, the tilt angles are θ_A = 135° and θ_B = 45° when resonance offset is measured from Ω_A. The tilt angles are θ_A = 140° and θ_B = 50° when resonance offset is measured from $\overline{\Omega }$. In both cases, ∣θ_B − θ_A∣ = 90° and cos Δθ = 0. The maximum values of R_1ρ are (green solid line) 24.8, (orange dotted line), 22.7, (blue, dashed line) 41.1, and (reddish-purple dashed-dotted line) 38.4 1/s, compared with limiting theoretical values of $R_{1\rho }={\overline{R}}_{1\rho }+k_{12}=25.1,23.1,44.0$, and 41.9 1/s, respectively. In Fig. 5b, R_ex is shown as a function of Ω_A in the established approach for identifying the resonance frequency of the sparsely populated state. Local maxima are observed at −Ω_A/2π = (black solid) 199 and (reddish-purple, dashed) 198 Hz, compared to the expected value of Δω = 200 Hz. Note that the approach illustrated in Fig. 5a does not require knowledge of ${\overline{R}}_{1\rho }$ for calculation of R_ex as in Fig. 5b.

Figure 5:

Alternate methods of determining Δω. (a) R_1ρ calculated as a function of ω₁ with the constraint (green, solid line and blue dashed line) Ω_A = −ω₁ or (orange, dotted line and reddish-purple dashed-dotted line) Ω_A = −p_BΔω − ω₁, giving a constant θ_A or $\overline{\theta }={135}^{\circ }$, respectively. Values of Ω_B = Δω − ω₁ or Ω_B = p_AΔω − ω₁, respectively. Values of k_ex were (green, solid line and orange dotted line) 0.1Δω or (blue dashed line and reddish-purple dashed-dotted line) 0.25Δω. (b) R_ex calcuated as a function of Ω_A with ω₁ = Δω/2. Values of (green, solid line) k_ex = 0.1Δω and (blue, dashed line) k_ex = 0.25Δω. Other parameters are Δω/(2π) = 200 Hz, p_A = 0.9, p_B = 0.1, R_1A = R_1B = 1 s⁻¹, and R_2A = R_2B = 20 s⁻¹. R_1ρ relaxation rates are calculated numerically and agree closely with appproximations using Eq. 61.

Many of the above results generalize in straightforward fashion to N-site chemical exchange, in particular Eq. 46 and Eqs. 52-60, simply by expanding the dimensionality of the relevant sub-matrix blocks. The observation that ∣Δθ∣ = 90° for specific combinations of ω₁ and resonance offsets remains true for N-sites. For any two sites X and Y, ∣Δθ_XY∣ = ∣θ_X − θ_Y∣ = 90° when ${\Omega }_X{\Omega }_Y+{\omega }_1^2=0$. If at the same time, ω₁ is sufficiently large that non-secular interactions can be neglected (vide supra), then the z-magnetization components in sites X and Y relax independently of each other, while remaining connected to the other sites. A triangular A-B-C system is considered as an example. If ∣Δθ_BC∣ = 90°, then the triangular system becomes equivalent to a pseudo-linear C-A-B system (while cross-exchange between states B and C is eliminated, the on-diagonal matrix elements for sites B and C remain unchanged). In contrast, if ∣Δθ_AC∣ = 90°, then then the triangular system becomes equivalent to a pseudo-linear A-B-C system. These effects are illustrated in Figure 6. The system has Ω_A < Ω_B < Ω_C. The value of ω₁ is varied from 0 to Δω_CB while setting Ω_B = −ω₁. The values of R_1ρ are plotted as a function of ω₁ for the triangular A-B-C system and the two pseudo-linear systems C-A-B and A-B-C. As shown in Fig. 6, Δθ_BC = 90° when ω₁/2π = 100 Hz; at this point Ω_B = 135° and Ω_C = 45°. As predicted, R_1ρ for the triangular and C-A-B linear systems coincide. In addition, Δθ_AC = 90° when ω₁/2π = 128.1 Hz; at this point Ω_A = 150.7° and Ω_C = 60.7°. As predicted, R_1ρ for the triangular and A-B-C linear systems coincide. The effective decoupling and linearization illustrated in this example is a potential test for the identification of the topology of an N-site chemical exchange scheme.

Figure 6:

Decoupling and linearziation of triangular A-B-C kinetic scheme with Ω_A < Ω_B < Ω_C. (a) R_1ρ calculated as a function of ω₁ with the constraint Ω_B = −ω₁, giving Ω_A = −Δω_BA−ω₁, and Ω_C = Δω_CB−ω₁. (black, solid line) numerical eigenvalue of the triangular system; (reddish-purple, dashed line) numerical eigenvalue of the pseudo-linear C-A-B system; and (orange, dashed-dotted line) numerical eigenvalue of the pseudo-linear A-B-C system. Vertical blue dotted lines are drawn at ω₁/2π = 100 and 128.1 Hz. (b) Absolute values of (reddish-purple, dashed line) Δθ_BC and (orange, dashed-dotted line) Δθ_AC. Vertical blue dotted lines are drawn at ω₁/2π = 100 and 128.1 Hz. The horizontal black line is drawn at 90°. Site populations are p_A = 0.85, p_B = 0.05, p_C = 0.10. Resonance frequency differences were Δω_BA/2π = 100 Hz and Δω_CB/2π = 200 Hz. Relaxation rate constants are R_1A = R_1B = R_1C = 1 s⁻¹, and R_2A = R_2B = R_2C = 20 s⁻¹. Kinetic parameters are k₁₂ + k₂₁ = 150, k₁₃ + k₃₁ = 100, and k₂₃ + k₃₂ = 226 s⁻¹.

Furthermore, the relationship ${\Omega }_X{\Omega }_Y+{\omega }_1^2=0$ can be solved to obtain values of resonance offsets and 0 ≤ ω₁ ≤ Δω_XY/2 that maintain ∣θ_X − θ_Y∣ = 90°. Figure 7 illustrates linearization of the A-B-C triangular system to the pseudo-linear C-A-B and A-B-C systems. The agreement between the calculated values of R_1ρ for the triangular and linear systems is excellent except when ω₁ is not sufficiently large.

Figure 7:

Band decoupling and linearziation of triangular A-B-C kinetic scheme. Unless noted, parameters are identical to those in Fig. 6 (a) R_1ρ calculated as a function of ω₁ and Ω_B. (black, solid line) numerical eigenvalue of the triangular system; (reddish-purple, dashed line) numerical eigenvalue of the pseudo-linear C-A-B system; vertical blue dotted line is drawn at Δω_CB/2π. (b) Value of ω₁ obtained from ${\Omega }_B{\Omega }_C+{\omega }_1^2=0$ used in calculations shown in (a). (a,b) Ω_A = −Δω_BA + Ω_B, and Ω_C = Δω_CB + Ω_B. (c) R_1ρ calculated as a function of ω₁ and Ω_A. (black, solid line) numerical eigenvalue of the triangular system; (reddish-purple, dashed line) numerical eigenvalue of the pseudo-linear A-B-C system; vertical blue dotted line is drawn at Δω_CA/2π. (d) Value of ω₁ obtained from ${\Omega }_A{\Omega }_C+{\omega }_1^2=0$ used in calculations shown in (c). (c,d) Ω_B = Δω_BA + Ω_A and Ω_C = Δω_CA + Ω_A.

The orthogonality of effective fields is expected to change, and in many cases amplify, the sensitivity of R_1ρ to variations of exchange parameters. In conventional off-resonance R_1ρ or CEST experiments, the informed choice for a (typically) constant ω₁ is not trivial. A hypothetical experiment based on known or predicted site resonance freqencies has the advantage of having defined ω₁ by choosing two sites for which the effective fields are set to be orthogonal. Following a previously published example of three-site triangular kinetics with mixed slow- and fast-intermediate exchange [28], the impact of various exchange parameters on R_1ρ were simulated for hypothetical experiments with ∣θ_X − θ_Y∣ = 90° (Fig. 8). Conventional R_1ρ or CEST experiments only offer similar distinguishability with scenario-specific optimized ω₁; however, they only offer superior distinguishability between R_1ρ rate constants when exchange becomes fast and requires larger values of ω₁. Setting ω₁ to achieve orthogonality of effective fields might emerge as a reasonable experimental choice, especially when minor site chemical shifts are known or need to be validated.

Figure 8:

Distinguishing R_1ρ or CEST exchange parameters using orthogonal effective rf fields in a triangular A − B − C kinetic scheme. The −Ω_A scale in the hypothetical experiments covers the sites according to the order Ω_A < Ω_B < Ω_C. (a-f) R_1ρ rates were obtained using orthogonal effective fields, with the constraint ${\Omega }_X{\Omega }_Y+{\omega }_1^2=0$. In panels (a-c), X = A and Y = C; in panels (d-f), X = A, Y = B for −Ω_A < Ω_B; and X = B, Y = C for −Ω_A > Ω_B. (g-i) R_1ρ rates were obtained for conventional off-resonance R_1ρ and CEST type experiments, using constant rf fields ω₁/2π ≕ (g) 80 Hz, (h) 152 Hz, and (i) 83 Hz. For all panels, the reference case (black, solid line) has the following parameters: Ω_B/2π = 69 Hz; Ω_C/2π = 276.5 Hz; p_B = 1.52; p_C = 1.565; k₁₂ + k₂₁ = 3170 s⁻¹; k₃₁ + k₃₁ = 695 s⁻¹; k₂₃ + k₃₂ = 1250 s⁻¹; R_2A = R_2B = R_2C = 10 s⁻¹; R_1A = R_1B = R_1C = 1 s⁻¹. This reference case has been adapted from a kinetic scenario described for residue Ile38 in Cadherin-11 [28]. All ω₁ settings were set based on this reference case; all other simulated R_1ρ curves to simulate other exchange parameters were based on this reference ω₁ set. The left panels (a,d,g) illustrate z-fold changes of p_B, (reddish-purple) z > 1, and (blue) z < 1; ∣z − 1∣ within each color group is increasing with dashed – dash-dot-dotted – dash-dotted – dotted; z factors are 0.05, 0.2, 0.5, 0.75, 1.25, 1.5, 2, 3. The center panels (b,e,h) illustrate z-fold changes in k₁₂ + k₂₁. Color schemes and line styles are defined as in the left panels; z factors are 0.00, 0.05, 0.2, 0.5, 2, 5, 10, 20. The right panels (c,f,i) illustrate z-fold changes in Δω_CA, (reddish-purple) z >= 1, (orange) 0 < z < 1, and (blue) z <= 0. Line styles are as defined as in the left panels; z factors are −5, −0.5, 0, 0.1, 0.3, 0.75, 0.9, 1.25, 1.5, 2, 5. The magnitudes of ω₁ in panels (g-i) were determined by grid searches varying ω₁ and T to maximize the average difference between the reference case and other simulated exchange parameters for each panel.

In addition, in the absence of chemical exchange, the first-order HAM approximation yields extremely accurate resuts for evolution during a rectangular pulse. The results shown in Figure 9, for initial z-magnetization, are nearly exact and should be compared to Figure 5 of Crawley and Palmer [23]. The improvement illustrates a major strength of HAM: different choices for the linear operator, initial approximation, and c₀ can favorably impact the accuracy of the resulting approximate solutions.

Figure 9:

HAM approximations for rectangular (a,b,c) 90°, (d,e,f) 180°, (g,h,i) 270°, and (j,k,l) 360° pulses applied to initial z-magnetization. Values of (a,d,g,j) M_x(Ω), (b,e,h,k), M_y(Ω), and (c,f,i,l) M_z(Ω) are shown as functions of resonance offset Ω. Magnetization components in absence of relaxation (black dotted line), first-order HAM approximation of the Bloch equations (reddish-purple dashed line), and exact solution of the Bloch equations (blue solid line). Calculations used ω₁/(2π) = 250 Hz, R₁ = 2 s⁻¹, R₂ = 200 s⁻¹, and c₀ = −1.

Conclusions

Nuclear magnetic spin relaxation in the presence of a spin-locking rf field, variously implemented in R_1ρ, CEST, and DEST experiments, has emerged as an extremely powerful probe of biological macromolecules, enabling detailed descriptions of chemical and conformational states whose populations or kinetics are modified during biological processes. The present work, by examining the form of the Bloch-McConnell equation for 2-site chemical exchange in local frames of reference, emphasizes the importance of the ratio (k₁₂k₂₁)^1/2/ω₁ and the difference in local tilt angles θ_B − θ_A in determining the qualitative properties of coupled magnetization evolution. The results also identify a special property of the R_1ρ relaxation rate constant that allows determination of Δω by varying the rf carrier frequency and the spin-locking field strength in tandem. This same approach also can lead to effective decoupling and simplification of exchange topologies for N-site kinetic schemes in favorable circumstances. A compact trace expression for R_1ρ is derived that has the same functional form for arbitrary number of sites N. Finally an approximate equation is obtained for evolution of coupled z-magnetization in the tilted rotating frame of reference. These results advance qualitative understanding of rotating-frame relaxation and may suggest fruitful experimental approaches.

Highlights.

Dependence of R_1ρ relaxation on local tilt angles for pairs of sites.

Equation for R_1ρ rate constant for N-site exchange.

Long-time evolution of z-magnetization for N-site exchange.

Simplification of R_1ρ relaxation for N-site exchange.