General Expressions for Carr—Purcell—Meiboom—Gill Relaxation Dispersion for N-Site Chemical Exchange

Abstract

The Carr—Purcell—Meiboom—Gill (CPMG) nuclear magnetic resonance experiment is widely used to characterize chemical exchange phenomena in biological macromolecules. Theoretical expressions for the nuclear spin relaxation rate constant for two-site chemical exchange during CPMG pulse trains valid for all time scales are well-known as are descriptions of N-site exchange in the fast limit. We have obtained theoretical expressions for N-site exchange outside of the fast limit by using approximations to an average Liouvillian describing the decay of magnetization during a CPMG pulse train. We obtain general expressions for CPMG experiments for any N-site scheme and all experimentally accessible time scales. For sufficiently slow chemical exchange, we obtain closed-form expressions for the relaxation rate constant and a general characteristic polynomial for arbitrary kinetic schemes. Furthermore, we highlight features that qualitatively characterize CPMG curves obtained for various N-site kinetic topologies, quantitatively characterize CPMG curves obtained from systems in various N-site exchange situations, and test distinguishability of kinetic models.

NMR spectroscopy provides tools to study chemical exchange and kinetic processes in proteins and nucleic acids in a site-specific manner.^1–3 Relaxation dispersion experiments, including R_1ρ and Carr—Purcell—Meiboom—Gill (CPMG) methods, target chemical exchange processes in the microsecond-to-millisecond time range. R_1ρ and CPMG experiments measure nuclear spin relaxation rate constants, while quenching the effects of exchange processes of interest to varying degrees. The resulting relaxation rate dispersion curves are analyzed to extract kinetic information and probe chemical and conformational states with populations as low as ~0.5%. R_1ρ and the related CEST/DEST experiments utilize spin—lock irradiation with variable effective field strengths, while CPMG experiments apply trains of refocusing pulses with variable repetition frequencies. CPMG experiments have been applied to, for example, interactions in a fuzzy complex between an intrinsically disordered MKK4 domain and MAPK p38alpha,⁴ formation of E-Cadherin dimerization intermediate,⁵ dynamic ligand-induced allostery in imidazol glycerol phosphate synthase,⁶ transient formation of dG-dT Watson—Crick like base pairs in RNA,⁷ superoxide dismutase dimerization,⁸ and helix—coil transitions in allosteric processes.⁹(For reviews, see refs 1–3, 9, and 10.)

Although the theoretical bases of R_1ρ and CPMG experiments are closely related,² closed-form expressions for CPMG relaxation rates are more difficult to obtain because evolution of magnetization is more complex in the presence of a refocusing pulse train than during spin locking. For two-site exchange, general expressions have been obtained by Carver et al.,¹¹ Jen,¹² and Baldwin;¹³ compact results have been obtained for fast exchange¹⁴ and for very slow exchange.¹⁵ Expressions for N-site (N > 2) fast-limit exchange have been obtained from an eigenanalysis¹⁶ and applied in a linear three-site case.¹⁷ However, most analyses of linear three-site exchange in CPMG experiments have relied on numerical integration of the Bloch—McConnell equations.^18–20

Analytical approximations for N-site chemical exchange relaxation dispersion in CPMG experiments enable both more efficient data analysis and extension to additional chemical exchange topologies. For example, a triangular kinetic scheme has been discovered for E-Cadherin, which can proceed to the dimer form either directly or via an intermediate that has been termed X-dimer.⁵ Another potentially large group of N-site, particularly four-site, processes arise through coupled equilibria for allosteric processes.²¹ A theoretical study pointed out the difficulties arising when trying to fit synthetic data to an (exact) four-state model.²¹ Experimental strategies to enhance the data basis for N-site data fitting have also been described.²⁷

We recently have obtained general approximations for N-site chemical exchange in R_1ρ experiments.²² In the present paper, we obtain general expressions for CPMG experiments for arbitrary N-site kinetic schemes and all experimentally accessible time scales. The most comprehensive expressions are algebraically complex but can be used in efficient curve-fitting algorithms. For sufficiently slow chemical exchange, we obtain closed-form algebraic expressions. These approximations are compact for linear two-site and three-site exchange (linear and triangular). Furthermore, we highlight features that qualitatively characterize CPMG curves obtained from systems in various N-site exchange situations.

THEORY

Definition of R_cpmg(τ_cp) in Terms of the Average Evolution Matrix G(τ_cp).

The CPMG experiment consists of m repetitions of the pulse sequence element:

$${\tau }_{cp}-{180}^{°}-2{\tau }_{cp}-{180}^{°}-{\tau }_{cp}$$ (1)

in which τ_cp is a delay period and 180° is a radiofrequency pulse with a 180° rotation angle, applied to transverse magnetization. Assuming that the 180° pulses are ideal, only initial magnetization proportional to the raising operator needs to be considered in the following. Evolution of transverse magnetization during a single free-precession period t is described by

$$\mathbf{M}(t)=\exp \mathbf{\{}(i\mathrm{\Omega }-\mathbf{R}+\mathbf{K})t\}\mathbf{M}(0)=\exp \left\{\mathbf{L}t\right\}\mathbf{M}(0)$$ (2)

in which Ω is a diagonal matrix with elements Ω_ii = Ω_i giving the resonance offset for a nuclear spin in the i^th site, R is a diagonal matrix with elements R_ii = R_2i giving the transverse relaxation rate constant in the absence of the exchange process for a nuclear spin in the i^th site, the elements of the N × N kinetic rate matrix K are K_ij = k_ji, the (pseudo) first-order rate constant for transitions from state j to state i (j ≠ i) and

$$K_{jj}=-\sum _{\begin{array}{c}i=1\\ i\ne j\end{array}}^N{k}_{ji}$$ (3)

The effect of the 180° pulses is to invert the signs of Ω_i. Thus, evolution of magnetization through the CPMG sequence becomes

$$\begin{gathered} \mathbf{M}(T)=\mathbf{M}\left(4m{\tau }_{cp}\right) \\ ={\left[\exp \left\{\mathbf{L}{\tau }_{cp}\right\}\exp \left\{2{\mathbf{L}}^{*}{\tau }_{cp}\right\}\exp \left\{\mathbf{L}{\tau }_{cp}\right\}\right]}^m\mathbf{M}(0) \end{gathered}$$ (4)

in which the asterisk indicates complex conjugation.

Following Allerhand and Theile,¹⁶ a similarity trans-formation is applied to symmetrize the matrix L:

$$\begin{gathered} \widehat{\mathbf{M}}(t)=\mathbf{U}\mathbf{M}(t) \\ \widehat{\mathbf{L}}=\mathbf{U}\mathbf{L}{\mathbf{U}}^{-1}=\mathrm{\Omega }+\mathbf{R}+\mathbf{U}\mathbf{K}{\mathbf{U}}^{-1}=\mathrm{\Omega }+\mathbf{R}+\widehat{\mathbf{K}} \end{gathered}$$ (5)

in which U_ij = d_ij p_i^−1/2 and p_i is the population of the i^th state. The definitions

$$\begin{gathered} \widehat{\widehat{\mathbf{L}}}=\widehat{\mathbf{L}}-\left\{i{\mathrm{\Omega }}_1-R_{21}\right\}\mathbf{E} \\ \widehat{\widehat{\mathbf{M}}}(t)=\widehat{\mathbf{M}}(t){\mathrm{e}}^{R_{21}t} \end{gathered}$$ (6)

are useful, although not essential. Using these results, the evolution of magnetization through the CPMG sequence becomes

$$\widehat{\widehat{\mathbf{M}}}(T)={\left[\exp \left\{\widehat{\widehat{\mathbf{L}}}{\tau }_{cp}\right\}\exp \left\{2\widehat{\widehat{{\mathbf{L}}^{\mathrm{*}}}}{\tau }_{cp}\right\}\exp \left\{\widehat{\widehat{\mathbf{L}}}{\tau }_{cp}\right\}\right]}^m\widehat{\widehat{\mathbf{M}}}(0)$$ (7)

For convenience in the following, the average evolution matrix is defined as

$$\exp \left\{\mathbf{G}({\tau }_{cp})\right\}=\exp \left\{\widehat{\widehat{\mathbf{L}}}{\tau }_{cp}\right\}\exp \left\{2\widehat{\widehat{{\mathbf{L}}^{\mathrm{*}}}}{\tau }_{cp}\right\}\exp \left\{\widehat{\widehat{\mathbf{L}}}{\tau }_{cp}\right\}$$ (8)

Consider the rate of change of magnetization between m and m + 1:

$$\begin{gathered} \frac{\widehat{\widehat{\mathbf{M}}}\left(4[m+1]{\tau }_{cp}\right)-\widehat{\widehat{\mathbf{M}}}\left(4m{\tau }_{cp}\right)}{4{\tau }_{cp}} \\ =\frac{1}{4{\tau }_{cp}}\left\{\exp \left\{\mathbf{G}({\tau }_{cp})\right\}-\mathbf{E}\right\}\widehat{\widehat{\mathbf{M}}}(4m{\tau }_{cp}) \\ =\frac{1}{4{\tau }_{cp}}\left\{\mathbf{G}({\tau }_{cp})+\frac{1}{2}{\left(\mathbf{G}({\tau }_{cp})\right)}^2+\frac{1}{6}{\left(\mathbf{G}({\tau }_{cp})\right)}^3+\cdots \right\}\widehat{\widehat{\mathbf{M}}}(4m{\tau }_{cp}) \end{gathered}$$ (9)

Similarly, the rate of change of magnetization between m − 1 and m is

$$\begin{gathered} \frac{\widehat{\widehat{\mathbf{M}}}\left(4m{\tau }_{cp}\right)-\widehat{\widehat{\mathbf{M}}}\left(4[m-1]{\tau }_{cp}\right)}{4{\tau }_{cp}} \\ =\frac{1}{4{\tau }_{cp}}\left\{\mathbf{E}-\exp \left\{-\mathbf{G}\left({\tau }_{cp}\right)\right\}\right\}\widehat{\widehat{\mathbf{M}}}\left(4m{\tau }_{cp}\right) \\ =\frac{1}{4{\tau }_{cp}}\left\{\mathbf{G}\left({\tau }_{cp}\right)-\frac{1}{2}{\left(\mathbf{G}\left({\tau }_{cp}\right)\right)}^2+\frac{1}{6}{\left(\mathbf{G}\left({\tau }_{cp}\right)\right)}^3+\cdots \right\}\widehat{\widehat{\mathbf{M}}}\left(4m{\tau }_{cp}\right) \end{gathered}$$ (10)

Averaging these two expressions gives the difference equation:

$$\begin{gathered} \frac{\widehat{\widehat{\mathbf{M}}}\left(4[m+1]{\tau }_{cp}\right)-\widehat{\widehat{\mathbf{M}}}\left(4[m-1]{\tau }_{cp}\right)}{8{\tau }_{cp}} \\ =\frac{1}{4{\tau }_{cp}}\left\{\mathbf{G}({\tau }_{cp})+\frac{1}{6}{\left(\mathbf{G}({\tau }_{cp})\right)}^3+\dots \right\}\widehat{\widehat{\mathbf{M}}}\left(4m{\tau }_{cp}\right) \\ =\frac{1}{4{\tau }_{cp}}\left\{\mathbf{G}\left({\tau }_{cp}\right)\right\}\left\{\mathbf{E}+\frac{1}{6}{\left(\mathbf{G}({\tau }_{cp})\right)}^2+\dots \right\}\widehat{\widehat{\mathbf{M}}}\left(4m{\tau }_{cp}\right) \end{gathered}$$ (11)

The above ansatz defines a “coarse-grained” differential equation:

$$\frac{d\widehat{\widehat{\mathbf{M}}}(T)}{dT}=\left(\frac{1}{4{\tau }_{cp}}\mathbf{G}\left({\tau }_{cp}\right)\right)\widehat{\widehat{\mathbf{M}}}(T)=-\mathrm{\Gamma }\widehat{\widehat{\mathbf{M}}}(T)$$ (12)

in which T = 4mτ_cp. The second bracketed term on the last line of eq 11 is assumed to be minimally di erent from the identity matrix; this assumption is tantamount to assuming that the fractional change in magnetization during a single CPMG block is small. By inspection, the chemical exchange rate matrix is Γ = —G(τ_cp)/(4τ_cp). Although not used herein, the pulsing rate ν_cpmg = 1/(4τ_cp) can be defined.

Definition of R_cpmg by a Two-Propagator Average Evolution Matrix and by H(τ_cp).

Eq 8 can be rewritten as

$$\begin{gathered} \exp \left\{\mathbf{G}\left({\tau }_{cp}\right)\right\}=\exp \left\{\widehat{\widehat{\mathbf{L}}}{\tau }_{cp}\right\}\exp \left\{2\widehat{\widehat{{\mathbf{L}}^{\mathrm{*}}}}{\tau }_{cp}\right\}\exp \left\{\widehat{\widehat{\mathbf{L}}}{\tau }_{cp}\right\} \\ =\exp \left\{\widehat{\widehat{\mathbf{L}}}{\tau }_{cp}\right\}\exp \left\{2\widehat{\widehat{{\mathbf{L}}^{\mathrm{*}}}}{\tau }_{cp}\right\}\exp \left\{2\widehat{\widehat{\mathbf{L}}}{\tau }_{cp}\right\}\exp \{-\widehat{\widehat{\mathbf{L}}}{\tau }_{cp}\} \\ =\exp \left\{{\mathrm{e}}^{\widehat{\widehat{\mathbf{L}}}{\tau }_{\mathit{cp}}}\mathbf{H}\left({\tau }_{cp}\right){\mathrm{e}}^{-\widehat{\widehat{\mathbf{L}}}{\tau }_{\mathit{cp}}}\right\} \end{gathered}$$ (13)

with the central two propagators defining an alternative average Liouvillian matrix H(τ_cp) with

$$\exp \left\{\mathbf{H}\left({\tau }_{cp}\right)\right\}=\exp \left\{2\widehat{\widehat{{\mathbf{L}}^{\mathrm{*}}}}{\tau }_{cp}\right\}\exp \left\{2\widehat{\widehat{\mathbf{L}}}{\tau }_{cp}\right\}$$ (14)

so that

$$\mathrm{\Gamma }=-\frac{1}{4{\tau }_{cp}}{\mathrm{e}}^{\widehat{\widehat{\mathbf{L}}}{\tau }_{cp}}\mathbf{H}\left({\tau }_{cp}\right){\mathrm{e}}^{-\widehat{\widehat{\mathbf{L}}}{\tau }_{cp}}$$ (15)

The CPMG relaxation rate constant R_cpmg(τ_cp) can be obtained from the largest (least negative) eigenvalue of Γ, or equivalently of the average Liouvillian H(τ_cp)/(4τ_cp), because the eigenvalues are unaffected by the transformation from H(τ_cp) to Γ:

$$R_{cpmg}\left({\tau }_{cp}\right)=-\max \left(\mathrm{eigenvalues}\left(\mathbf{H}\left({\tau }_{cp}\right)/4{\tau }_{cp}\right)\right)+R_{21}$$ (16)

and the term R₂₁ arises from eq 6. Computation of the exponential terms in H(τ_cp) is more efficient than the corresponding G(τ_cp), which is why this equation is preferable. In addition, for slow exchange, convenient approximations can be obtained from H(τ_cp) (vide infra, eqs 21–27). For completeness, relaxation of the average magnetization during a CPMG experiment, as defined by Allerhand and Thiele¹⁶ for fast exchange, is recapitulated in Supporting Information, section S2.

Approximation of the Evolution Matrix H(τ_cp).

In order to find approximations for R_cpmg(τ_cp), the evolution matrix H(τ_cp) must be defined. The right-hand side of eq 14 is a Hermitian positive-definite matrix and consequently has positive eigenvalues. This equation also has a maximum value of unity when τ_cp = 0 and approaches 0 as τ_cp approaches infinity. As a result

$$\begin{gathered} \mathbf{H}\left({\tau }_{cp}\right)=\log \left[\exp \left\{2\widehat{\widehat{{\mathbf{L}}^{\mathrm{*}}}}{\tau }_{cp}\right\}\exp \left\{2\widehat{\widehat{\mathbf{L}}}{\tau }_{cp}\right\}\right]=\log \left[\mathbf{Z}\left({\tau }_{cp}\right)\right] \\ =\log \left[\mathbf{E}-\left(\mathbf{E}-\mathbf{Z}\left({\tau }_{cp}\right)\right)\right] \\ =-\sum _{k=1}^{\infty }\frac{1}{k}{\left(\mathbf{E}-\mathbf{Z}\left({\tau }_{cp}\right)\right)}^k \end{gathered}$$ (17)

in which $\mathbf{Z}\left({\tau }_{cp}\right)=\exp \{2\widehat{\widehat{{\mathbf{L}}^{\mathrm{*}}}}{\tau }_{cp}\}\exp \{2\widehat{\widehat{\mathbf{L}}}{\tau }_{cp}\}$ has been introduced for compactness. The infinite series on the last line of eq 17 converges slowly and is practical in the present circumstance only if E — Z(τ_cp) is sufficiently small that the series can be truncated after the first term to give H(τ_cp) ≈ Z(τ_cp) — E.

Highly accurate Padé approximations are known for the matrix logarithm, allowing the series of eq 17 to be represented to arbitrary accuracy.²³ The following Padé approximations have increasing accuracies and are sufficient for the present applications:

$$\begin{gathered} {\mathbf{H}}_{10}\left({\tau }_{cp}\right)\approx -\mathbf{E}+\mathbf{Z}\left({\tau }_{cp}\right)=-\left(\mathbf{E}-\exp \left\{2\widehat{\widehat{{\mathbf{L}}^{\mathrm{*}}}}{\tau }_{cp}\right\}\exp \left\{2\widehat{\widehat{\mathbf{L}}}{\tau }_{cp}\right\}\right) \\ {\mathbf{H}}_{11}\left({\tau }_{cp}\right)\approx 2\left\{-\mathbf{E}+\mathbf{Z}\left({\tau }_{cp}\right)\right\}{\left\{\mathbf{E}+\mathbf{Z}\left({\tau }_{cp}\right)\right\}}^{-1} \\ {\mathbf{H}}_{22}\left({\tau }_{cp}\right)\approx 3\left\{-\mathbf{E}+{\mathbf{Z}}^2\left({\tau }_{cp}\right)\right\}{\left\{\mathbf{E}+4\mathbf{Z}\left({\tau }_{cp}\right)+{\mathbf{Z}}^2\left({\tau }_{cp}\right)\right\}}^{-1} \\ {\mathbf{H}}_{33}\left({\tau }_{cp}\right)\approx \frac{1}{3}\left\{-11\mathbf{E}-27\mathbf{Z}\left({\tau }_{cp}\right)+27{\mathbf{Z}}^2\left({\tau }_{cp}\right)+11{\mathbf{Z}}^3\left({\tau }_{cp}\right)\right\} \\ {\left\{\mathbf{E}+9\mathbf{Z}\left({\tau }_{cp}\right)+9{\mathbf{Z}}^2\left({\tau }_{cp}\right)+3{\mathbf{Z}}^3\left({\tau }_{cp}\right)\right\}}^{-1} \end{gathered}$$ (18)

in which the subscripts in the expressions H_jk(τ_cp) give the highest order power in the numerator and denominator polynomials in Z(τ_cp). H₁₀(τ_cp) is identical to the lowest order term in the expansion shown in eq 17. Estimates H_jk(τ_cp) with j = k are generally superior to approximations with j ≠ k. Values of $\exp \{2\widehat{\widehat{\mathbf{L}}}{\tau }_{cp}\}$ (and its complex conjugate) needed to obtain Z(τ_cp) and H(τ_cp) are calculated by standard analytical or numerical methods for the matrix exponential,²⁴ or with an approximation for the matrix exponential when exchange is sufficiently slow (k_ex < Δω/10) (vide infra).

Generalized R_cpmg(τ_cp) Eigenvalue Approximations.

The eigenvalues, λ, of H(τ_cp) are obtained as the roots of the characteristic polynomial:

$$f(\lambda )=|\lambda \mathbf{E}-\mathbf{H}({\tau }_{cp})|$$ (19)

in which |A| is the determinant of a matrix A. Exact eigenvalues can be calculated analytically for N ≤ 4 (and numerically for any N); however, in many cases of practical interest, simple approximations to the eigenvalue are accurate and generate simple algebraic expressions. The k^th order approximation to the largest (least negative) eigenvalue can be obtained iteratively from the Newton—Raphson algorithm as

$${\lambda }_k={\lambda }_{k-1}-\frac{f\left({\lambda }_{k-1}\right)}{f^{\prime }\left({\lambda }_{k-1}\right)}={\lambda }_{k-1}+\frac{1}{Tr\left\{{\left({\lambda }_{k-1}\mathbf{E}-\mathbf{H}({\tau }_{cp})\right)}^{-1}\right\}}$$ (20)

in which λ₀ = 0 and the second line is a matrix form of the equation. This algorithm converges quadratically to the eigenvalue; more rapidly converging approximations to the desired eigenvalue²⁵ are provided in the Supporting Information, section S4.

After obtaining Z(τ_cp), expressions for R_cpmg are obtained in three steps: (i) a Padé approximation of H(τ_cp) is obtained from eq 18, (ii) the least negative eigenvalue of H(τ_cp) is calculated exactly or by approximation with eq 20, and (iii) the eigenvalue is substituted into eq 16. Choices of level of approximations for H(τ_cp) and for the eigenvalue are discussed in Results. An alternative formulation based on the Cayley—Hamilton theorem is given in the Supporting Information, section S3.

Slow Exchange.

The above general approach can be simplified to yield compact algebraic expressions when exchange is slow. The matrix exponential has a series representation:²⁶

$$\exp \left\{2\widehat{\widehat{\mathbf{L}}}{\tau }_{cp}\right\}=\exp \left\{2{\tau }_{cp}(\widehat{\mathrm{\Omega }}+\mathbf{V})\right\}=\sum _{k=0}^{\infty }\frac{1}{k!}{D}_V^{[k]}\left(2{\tau }_{cp},\widehat{\mathrm{\Omega }}\right)$$ (21)

in which $\widehat{\mathrm{\Omega }}=\mathrm{\Omega }-i{\mathrm{\Omega }}_1\mathbf{E},\mathbf{V}=\hat{\mathbf{K}}-\left(\mathbf{R}-R_{21}\mathbf{E}\right)$

$$D_V^{[k]}\left(2{\tau }_{cp},\widehat{\mathrm{\Omega }}\right)=k{\mathrm{e}}^{2{\tau }_{cp}\widehat{\mathrm{\Omega }}}{\int }_0^{2{\tau }_{cp}}{\mathrm{e}}^{-t\widehat{\mathrm{\Omega }}}\mathbf{V}{D}_V^{[k-1]}(t,\widehat{\mathrm{\Omega }})dt$$ (22)

is the k^th directional derivative and $D_V^{[0]}(2{\tau }_{cp},\widehat{\mathrm{\Omega }})={\mathrm{e}}^{2{\tau }_{cp}}\widehat{\mathrm{\Omega }}$. The directional derivative describes the infinitesimal change in $\exp \{2{\widehat{\widehat{\mathbf{L}}}}_{cp}\}$ in the direction V. When exchange is slow, the series in eq 22 can be truncated at k = 1, with the result that

$$\begin{gathered} \exp \left\{\widehat{\mathbf{H}}\left({\tau }_{cp}\right)\right\}=\exp \left\{2\widehat{\widehat{{\mathbf{L}}^{\mathrm{*}}}}{\tau }_{cp}\right\}\exp \left\{2\widehat{\widehat{\mathbf{L}}}{\tau }_{cp}\right\} \\ \approx \mathbf{E}+2{\int }_0^{2{\tau }_{cp}}{\mathrm{e}}^{-t\widehat{\mathrm{\Omega }}}\mathbf{V}{\mathrm{e}}^{t\widehat{\mathrm{\Omega }}}dt \\ =\mathbf{E}+2\mathbf{V}\mathrm{o}{\int }_0^{2{\tau }_p}\psi (t)dt=\mathbf{E}+2\mathbf{V}\mathrm{o}\mathbf{\Psi }\left({\tau }_{cp}\right) \end{gathered}$$ (23)

in which ° denotes the Hadamard product, for which ${\left({\mathbf{A}}^{°}\mathbf{B}\right)}_{ij}=A_{ij}{B}_{ij},{\psi }_{ij}(t)=\exp \left[-i\mathrm{\Delta }{\omega }_{ij}t\right]$ and

$$\mathbf{\Psi }{\left({\tau }_{cp}\right)}_{ij}=\left\{\begin{array}{cc}2{\tau }_{cp}& i=j\\ \frac{-i\left(1-\exp \left[-i2\mathrm{\Delta }{\omega }_{ij}{\tau }_{cp}\right]\right)}{\mathrm{\Delta }{\omega }_{ij}}& i\ne j\end{array}\right\}$$ (24)

in which Δω_ij = Ω_i — Ω_j. Using eq 18 the simplest approximation to H(τ_cp) is

$${\mathbf{H}}_{10}\left({\tau }_{cp}\right)=2\mathbf{V}\circ \mathbf{\Psi }\left({\tau }_{cp}\right)$$ (25)

with

$$H_{10}{\left({\tau }_{cp}\right)}_{ij}/\left(4{\tau }_{cp}\right)=\left\{\begin{array}{cc}\sqrt{k_{ij}{k}_{ji}}\mathrm{sinc}\left[\mathrm{\Delta }{\omega }_{ij}{\tau }_{cp}\right]\exp \left[-i\mathrm{\Delta }{\omega }_{ij}{\tau }_{cp}\right]& i>j\\ K_{ii}-\mathrm{\Delta }{R}_{2i}& i=j\\ \sqrt{k_{ij}{k}_{ji}}\mathrm{sinc}\left[\mathrm{\Delta }{\omega }_{ij}{\tau }_{cp}\right]\exp \left[i\mathrm{\Delta }{\omega }_{ji}{\tau }_{cp}\right]& i<j\end{array}\right\}$$ (26)

Assuming ΔR_2i = 0, the general characteristic polynomial for eq 26 can be written as

$$f(\lambda )=\sum _{u=0}^N{\lambda }^u\sum _{s=0}^{N-u}({\delta }_{s,0}+\sum _{\begin{array}{l}{h}_v,j_v=1\\ 0<v\le s\\ h_0=0,j_0=0\\ h_{v-1}<h_v\\ h_v\ne j_v\end{array}}^N{(-1)}^{\varphi }\prod _{g=1}^s(\sqrt{k_{h_g{j}_g}{k}_{j_g{h}_g}}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\{\mathrm{\Delta }{\omega }_{h_g{j}_g}{\tau }_{cp}\}\sum _{b=1}^s{\delta }_{h_g,j_b}))({\delta }_{u+s,N}+\sum _{\begin{array}{l}{l}_r,m_r=1\\ 1<r\le N-u-s\\ l_0=0,m_0=0\\ l_{r-1}<l_r\ne m_r\\ l_r\notin \{h_1\dots h_s\}\\ l_r\notin \{j_1\cdots j_s\}\end{array}}^N\prod _{g=1}^{N-u-s}{k}_{l_g{m}_g})$$ (27)

with k_x0, k_0x = 0, δ_ij is the Kronecker δ, N is the number of sites, and ϕ is the number of closed kinetic structures within a given set of {kh1j1, kj1h1, kh2j2, kj2h2, ..., khnjn, kjnhn}. Details and examples on expanding this equation are provided in the Supporting Information, sections S5.1 and S5.2. Simple algebraic expressions for the largest negative eigenvalue λ can be obtained by using this characteristic polynomial (eq 27), and eqs 20 or S14, to calculate R_cpmg(τ_cp) from eq 16. The eigenvalue can equivalently be obtained from eqs 26 and 20 (second line) or S15–S16. Truncation of the series in eqs 21 and 22 beyond the first term or use of higher order Padé approximations from eq 18 do not give consistently more accurate expressions for R_cpmg(τ_cp) (see Results).

Two-State Exchange.

Expressions for R_cpmg(τ_cp) for two-site exchange are well-known. For such a system

$$\widehat{\mathbf{L}}=\left[\begin{array}{cc}-k_{12}& {\left(k_{12}{k}_{21}\right)}^{1/2}\\ {\left(k_{12}{k}_{21}\right)}^{1/2}& i\mathrm{\Delta }{\omega }_{21}-k_{21}-\mathrm{\Delta }{R}_{21}\end{array}\right]$$ (28)

in which Δω_k1 = Ω_k —Ω₁, ΔR_k1 = R_2k — R₂₁, and k_ex = k₁₂ + k₂₁. In slow exchange, the largest non-negative eigenvalue λ in eq 20 can be calculated to obtain R_cpmg(τ_cp), from the characteristic polynomial in eq 27. The first-order approximation λ₁ yields

$$R_{cpmg}\left({\tau }_{cp}\right)=R_{21}+\frac{p_1{p}_2{k}_{ex}^2\left[1-{\mathrm{sinc}}^2\left(\mathrm{\Delta }{\omega }_{21}{\tau }_{cp}\right)\right]+p_2{k}_{ex}\mathrm{\Delta }{R}_{21}}{k_{ex}+\mathrm{\Delta }{R}_{21}}$$ (29)

More accurate closed-form expressions are obtained by using higher order approximations or the exact solution for the eigenvalue with the characteristic polynomial from eqs 27 and S14 (see the Supporting Information, section S5.2).

Three-State and N-State Exchange.

For the linear three-site exchange, in which site 1 (species A) exchanges with sites 2 (species B) and 3 (species C), sites 2 and 3 do not exchange:

$$B\underset{k_{12}}{\overset{k_{21}}{\rightleftharpoons }}A\underset{k_{31}}{\overset{k_{13}}{\rightleftharpoons }}C$$

For such a system

$$\widehat{\widehat{\mathbf{L}}}=\left[\begin{array}{ccc}-k_{12}-k_{13}& {\left(k_{12}{k}_{21}\right)}^{1/2}& {\left(k_{13}{k}_{31}\right)}^{1/2}\\ {\left(k_{12}{k}_{21}\right)}^{1/2}& i\mathrm{\Delta }{\omega }_{21}-k_{21}-\mathrm{\Delta }{R}_{21}& 0\\ {\left(k_{13}{k}_{31}\right)}^{1/2}& 0& i\mathrm{\Delta }{\omega }_{31}-k_{31}-\mathrm{\Delta }{R}_{31}\end{array}\right]$$ (30)

For a slow exchange, we obtain the characteristic polynomial (eq 27) and the expression for λ₁ in eq 20:

$$\begin{gathered} R_{cpmg}\left({\tau }_{cp}\right)=R_{21}+\left[\left(B_0-k_{12}{k}_{21}\left(k_{31}+\mathrm{\Delta }{R}_{31}\right){\mathrm{sinc}}^2\left(\mathrm{\Delta }{\omega }_{21}{\tau }_{cp}\right) \\ -k_{13}{k}_{31}\left(k_{21}+\mathrm{\Delta }{R}_{21}\right){\mathrm{sinc}}^2\left(\mathrm{\Delta }{\omega }_{31}{\tau }_{cp}\right)\right) \\ /\left(B_1+k_{12}{k}_{21}{\mathrm{sinc}}^2\left(\mathrm{\Delta }{\omega }_{21}{\tau }_{cp}\right)+k_{13}{k}_{31}{\mathrm{sinc}}^2\left(\mathrm{\Delta }{\omega }_{31}{\tau }_{cp}\right)\right)\right] \end{gathered}$$ (31)

in which B0 = (k12 + k13)(k21 + ΔR21)(k31 + ΔR31) and B1 = (k21 + ΔR21)(k31 + ΔR31) + (k12 + k13) (k21 + k31 + ΔR21 + ΔR₃₁). More accurate closed-form expressions are obtained by using higher order approximations (eqs 20, S14), or the exact solution (Supporting Information, section S5.4), for the eigenvalue with the characteristic polynomial from eq 27 (Supporting Information, section S5.1, eq S18). Matrices $\widehat{\widehat{\mathbf{L}}}$ for other kinetic schemes are provided in the Supporting Information (Figure S1).

Sections S5.1 and S5.2 in the Supporting Information demonstrate how to expand eq 27 to obtain the characteristic polynomial for the triangular three-site exchange scheme under slow exchange conditions (eq S14). With eq 20, assuming ΔR₂₁ = ΔR₃₁ = 0, we obtain for the triangular scheme:

$$\begin{gathered} R_{cpmg}\left({\tau }_{cp}\right)=R_{21}-\frac{X_1+X_2}{Y_1+Y_2} \\ {X}_1=k_{12}{k}_{21}\left(k_{31}+k_{32}\right)\left(1-\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}{\left\{\mathrm{\Delta }{\omega }_{21}{\tau }_{cp}\right\}}^2\right) \\ +k_{13}{k}_{31}\left(k_{23}+k_{21}\right)\left(1-\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}{\left\{\mathrm{\Delta }{\omega }_{31}{\tau }_{cp}\right\}}^2\right) \\ +k_{23}{k}_{32}\left(k_{12}+k_{13}\right)\left(1-\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}{\left\{\mathrm{\Delta }{\omega }_{32}{\tau }_{cp}\right\}}^2\right) \\ {X}_2=2\sqrt{k_{13}{k}_{31}{k}_{23}{k}_{32}{k}_{12}{k}_{21}}\left(1-\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left\{\mathrm{\Delta }{\omega }_{21}{\tau }_{cp}\right\}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left\{\mathrm{\Delta }{\omega }_{31}{\tau }_{cp}\right\}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left\{\mathrm{\Delta }{\omega }_{32}{\tau }_p\right\}\right) \\ {Y}_1=k_{12}{k}_{21}\left(1-\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}{\left\{\mathrm{\Delta }{\omega }_{21}{\tau }_{cp}\right\}}^2\right)+k_{13}{k}_{31}\left(1-\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}{\left\{\mathrm{\Delta }{\omega }_{31}{\tau }_{cp}\right\}}^2\right) \\ +k_{23}{k}_{32}\left(1-\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}{\left\{\mathrm{\Delta }{\omega }_{32}{\tau }_{cp}\right\}}^2\right) \\ {Y}_2=k_{12}\left(k_{23}+k_{31}+k_{32}\right)+k_{13}\left(k_{21}+k_{23}+k_{31}\right)+k_{21}\left(k_{31}+k_{32}\right)+k_{23}{k}_{31} \end{gathered}$$ (32)

Examples of characteristic polynomials of four-site exchange schemes are provided in the Supporting Information, section S5.3.

RESULTS

Nomenclature.

In order to distinguish between the various approximations described above, we use the nomenclature Exp_aLog_nλ_k to describe CPMG relaxation rate constants. In this notation, “Exp_a” denotes the type of approximation used for calculating the exponential term in eq 14: “Exp∞” denotes any solution that is based on the exact solution, while “Exp_m” refers to an m^th order exponential series approximation, as described in eqs 21 and 22; Exp₁ refers to the first-order truncation from eqs 24 and 25. The notation “Log_n” denotes the diagonal Pade(´n,n) approximation used to calculate the logarithm of said exponential in eq 18 (Log₀, as an exception, referring to Padé(1,0), or H₁₀), and “λ_k” denotes the k^th order approximation for the eigenvalue, as described in eq 25 (if not otherwise specified, using the Halley’s method algorithm, eq S16, for k > 1).

Accuracy of Approximations.

Numerical calculations show that the accuracy of the approximations to R_cpmg(τ_cp) improve for Exp_∞Log_nλ_k as n and k are increased for all chemical exchange time scales. Closed-form expressions are obtained for Exp₁Log₀λ_k and become more accurate as k increases. These latter approximations perform well for sufficiently slow exchange phenomena, as judged by the exchange rates between the major site and any minor sites. Preferably, k_1x + k_x1 should be an order of magnitude smaller than |Δω_i1| for 1 < i ≤ N in order to use Exp₁Log₀λ_k. The accuracy of approximations to R_cpmg(τ_cp) also improves as the minor site populations decrease. Generally, Exp₁Log₀λ_k approximations are more robust with regard to a faster (but still slow) exchange between minor sites than to a faster exchange between the major and minor sites. For a two-site exchange, the approximation Exp_∞Log₂λ₂ is nearly indistinguishable from the Carver—Richards equation generally used to fit CPMG relaxation dispersion data.

We illustrate the accuracy of various approximations to R_cpmg(τ_cp) for the linear (C—A—B) three-state scheme in Figure 1. In the example presented here, exchange between major and the minor sites is (just) sufficiently slow and minor site populations are sufficiently small, so that Exp₁Log₀λ_k approximations can be used and all approximations with k > 1 give high accuracies. For Exp_∞Log_nλ_k approximations, even though higher order Padé approximations give more accurate results for the matrix logarithm, the most important contributor to accuracy is the order k of the eigenvalue approximation.

Figure 1.

Approximations for the three-site linear exchange of the B—A—C type. (Solid black) Exact solution; (dashed) Exp∞Log_nλ_k approximations with (blue) n = 1, k = 2; (orange) n = 2, k = 2; (red) n = 2, k = 3; (dotted) Exp₁Log₀λ_k approximations, with (blue) k = 1; (cyan) k = 2 (Newton); (orange) k = 2; (red) k = 3. Parameters: k₁₂ + k₂₁ = 200 s⁻¹; k₁₃ + k₃₁ = 300 s⁻¹; Δω_AB = 1800 s⁻¹; Δω_AC = −2600 s⁻¹; p_A = 0.85; p_B = 0.08; p_C = 0.07.

Figure 2 compares approximations to R_cpmg(τ_cp) for the four-site kite scheme (scheme shown in Figure 6, with sites denoted as A, B, C, and D), exemplifying a case in which exchange between the major site and minor sites B and D is in the slow—intermediate, rather than the very slow—slow, range. The Exp_∞Log_nλ_k approximations are superior to Exp₁Log₀λ_k. This reinforces the recommendation to use Exp₁Log₀λ_k approximations only if the major site exchanges sufficiently slow with other sites, with k_1x being an order of magnitude smaller than |Δω_i1|. Empirically, slow exchange is indicated by the difference R_cpmg(τ_cp →∞) — R_cpmg(τ_cp → 0) becoming independent of the static magnetic field strength(B₀).²⁷

Figure 2.

R_cpmg approximations for the four-site kite scheme. The exact solution (solid black) is approximated by Exp₁Log₀λ_k (dotted), Exp_∞Log₂λ_k (dashed), or Exp_∞Log₃λ_k (solid). Colors: (red) k = 3; (orange) k = 2; (blue) k = 1. Characteristically, Exp_∞Log_nλ_k is always smaller than the exact solution and improves with a larger n. The Exp₁Log₀λ_k approximation is simple, but not accurate in this case because exchange between the major site A and any minor site B or D is not sufficiently slow. Parameters: k₁₂ + k₂₁ = 200 s⁻¹, k₁₃ + k₃₁ = 300 s⁻¹, k34 + k43 = 700 s⁻¹, and k14 + k41 = 350 s⁻¹; Δω_AB = 810 s−1; Δω_AC = −2900 s⁻¹; Δω_AD = 1100 s⁻¹; p_A = 0.92; p_B = 0.06; p_C = 0.06; p_D = 0.06.

Figure 6.

CPMG curves can help to differentiate between various four-site schemes. Left: schematic illustration of four-site schemes. Right: exact solutions (solid), Exp_∞Log₂λ₂ (dotted), and Exp_∞Log₂λ₃ (dashed blue, star) for CPMG curves for a variety of four-site schemes. Colors: kite (cyan), star (blue), quadratic (red), and linear (black). Parameters: k₁₂ + k₂₁ = 200 s⁻¹; k₁₃ + k₃₁ = 20 s⁻¹; k₁₄ + k₄₁ = 70 s⁻¹; k₂₄ + k₄₂ = 400 s⁻¹; k₃₄ + k₄₃ = 150 s⁻¹; Δω_AB = 1000 s⁻¹; Δω_AC = −2900 s⁻¹; Δω_AD = 1100 s⁻¹; p_A = 0.88; p_B = 0.05; p_C = 0.03; p_D = 0.04.

Reliability of CPMG Rate Models Based on a Single Eigenvalue.

The solutions for R_cpmg(τ_cp) used in this paper are based on the least negative eigenvalue of the effective matrix, H(τ_cp) (eq 16). The evolution of magnetization for any of the sites in an N-site kinetic scheme consists of N components (see the Supporting Information). In practice, a single R_cpmg(τ_cp) normally is obtained by monoexponential fitting to experimental data. In most cases, this is possible because one of the decay components dominates (after initial rapid decay of components with small amplitudes) or because decay constants are too similar to be resolved.

When exchange between the major site A and any minor X site is in the fast limit, the observed magnetization of these sites is averaged. The resulting time decay of the average magnetization is practically identical to the decays obtained for the single-eigenvalue approximations used in this paper. More generally, the “exact” single-eigenvalue solution is not identical to the multieigenvalue solution only for a very limited range of conditions, and this limitation only applies to small values of 1/τ_cp: (a) when minor site populations are large and (b) when any minor site, especially one that is not connected to the major site, is “kinetically isolated”, meaning that the kinetic rate constants connecting the isolated site and the major site are much smaller than the rates connecting the major site with other sites. An example of such a situation is illustrated in Figure S1, in which the exchange between site C—D in a linear B—A—C—D scheme is much slower than exchange between A and B or A and C; the problem resolves as soon as site D is connected to the major site A via another path, for example in the kite scheme. Fitting algorithms using single-eigenvalue approximations could avoid such difficulties by setting appropriate parameter boundaries or, better, by using more robust models (three-site triangular, four-site kite, or four-site quadratic).

Distinguishing Various Exchange Models and CPMG Curve Features.

Already in the three-site case with one dominant (major) site, fundamentally different schemes have to be considered: triangular, linear B—A—C, and linear A—B—C. We have illustrated sample CPMG relaxation dispersion curves in Figure 3, together with a two-state CPMG dispersion curve for reference. In order to test whether these different kinetic schemes can be distinguished by CPMG experiments, we created synthetic data for B—A—C and triangular models, based on the parameters defined in the caption to Figure 3, and subsequently fit these synthetic data with the other kinetic models (two-site and three-site linear B—A—C for data from the triangular scheme). We varied the number of magnetic fields and the number of spins utilized in a global analysis (as would be performed in practice), while the resonance frequency offsets for each spin were randomly varied between synthetic data sets. Fitting results are reported as average root-mean-square (RMS) deviations between simulated and fitted dispersion curves calculated separately for small and large values of τ_cp. As found previously by others,²⁷ the results show that data should be recorded at two static magnetic field strengths; however, more fields are not beneficial in distinguishing different kinetic schemes. The number of resonances included in the data set is important: a larger number of spins always enhances distinguishability between kinetic schemes. Figure 4 only shows RMS deviations for 20 randomized spin sets; for practical applications, it is important to be able to distinguish data for most combinations of resonance offets. Accordingly, we provide a plot with lower cutoffs (Figure S3), revealing that in many cases at least 8 spins are required to distinguish between kinetic models.

Figure 3.

CPMG curves differentiate between various three-site schemes. Early sections (small 1/τ_cp) of the CPMG dispersion curve depend on exchange between the major and minor sites. At larger 1/ τ_cp, exchange between minor sites determines the shape of the curve. Top: exact solutions (solid) and Exp_∞Log₂λ₂ approximations (dotted, near perfect overlap) for CPMG curves for a variety of three-site schemes. Colors: triangular (black); B—A—C linear (cyan); A—B—C linear (orange); two-site (red, for reference). Parameters: k₁₂ + k₂₁ = 300 s⁻¹ (for two-site, 400 s⁻¹); k₁₃ + k₃₁ = 100 s⁻¹; k₂₃ + k₃₂ = 300 s⁻¹; Δω_AB = 1300 s⁻¹; Δω_AC = −1700 s⁻¹; p_A = 0.92; p_B = 0.05 (two-site p_B = 0.08); p_C = 0.03.

Figure 4.

Dependency of scheme distinguishability on magnetic field selection and spin number. CPMG curves for various schemes (triangular and linear B—A—C) were generated using parameters from Figure 1 (k₁₂ + k₂₁ = 300 s⁻¹; k₁₃ + k₃₁ = 100 s⁻¹; k₂₃ + k₃₂ = 300 s⁻¹; p_A = 0.92; p_B = 0.05; p_C = 0.03). Data were generated for m static magnetic fields (see labels in the figure) and n spins. Resonance frequencies for the n spins were randomly generated over the range Δω_AB (450—2500 s⁻¹) and Δω_AC (±450—2500 s⁻¹) at the lowest magnetic field and scaled linearly by the magnitude of the magnetic field. The synthetic data for m static magnetic fields and n spins were globally fit with other models (linear BAC and two-site) and the RMS deviation calculated for each resulting data set. For the RMS calculation, different parts of each of the resulting mn CPMG curves were selected (left panel, small 1/τ_cp; right panel, large 1/τ_cp). The largest RMS was selected to obtain a measure that represents the distinguishability between given schemes for a parameter set. This process was repeated 20 times to test for various Δω_AB/Δω_AC sets. The averages of the 20 resulting values are shown. (For a lower cutoff of distinguishability, see Figure S3.) Maximum RMS (yellow) for the left panel, 1.51; maximum RMS (yellow) for the right panel, 0.64.

Large values of 1/τ_cp are useful to distinguish triangular from B—A—C or two-state models, but not to distinguish linear from two-state schemes (Figure 4, right panel). Inspection of Figure 3 suggests that at large 1/τ_cp the shape of the dispersion curve depends primarily on exchange between minor sites, which is why the CPMG rate constants for the linear A—B—C scheme and the triangular scheme converge toward large 1/τ_cp. On the other hand, the linear B—A—C (no exchange between minor sites) and the A—B schemes also give similar (lower) CPMG rates for large 1/τ_cp.

For small 1/τ_cp, the linear B—A—C and the triangular scheme have very similar CPMG relaxation rates (Figure 3). The only difference between these two schemes is the presence of exchange between the minor sites in the triangular scheme. This observation suggests that minor site exchange does not influence the CPMG relaxation rates at very small 1/τ_cp. Rather, the CPMG relaxation rates at small 1/τ_cp depend primarily on exchange between the observed major site and any directly connected minor sites. Exceptions to this trend are observed for situations with relatively fast exchange between remote minor sites B and C and if B exchanges relatively fast with the major state A (as illustrated in Figure S2).

Having found that a sufficient number of spins and at least two magnetic fields are needed to distinguish kinetic schemes, we more fully explored parameter space, using a fixed number of spins (8) and magnetic fields (3). Because of the distinct features identified for small and large values of 1/τ_cp, we report the average RMS fitting errors separately for small and large 1/τ_cp in Figure 5. For small 1/τ_cp, we generally observe that larger kinetic exchange rates (as k₁₂ + k₂₁, for example) and minor site populations enhance distinguishability. However, if the population of a minor site is low, increasing exchange rates for that site hardly enhances distinguishability (and vice versa). For large 1/τ_cp (the R_cpmg(τ_cp) “tail”), larger minor site populations enhance distinguishability and larger values k₂₃ + k₃₂ also generally enhance distinguishability when other rate constants are small or when the minor site populations are not very large. Most importantly, in many cases, smaller exchange rates between major and minor sites, k₁₃ + k₃₁ and k₁₂ + k₂₁, improve distinguishability between the triangular and the linear scheme.

Figure 5.

Distinguishability of the triangular kinetic scheme from the BAC three-state model. Top left: fits of synthetic triangular data to a BAC three-state model (Exp₁Log₀λ_2,Newton); (solid) synthetic triangular scheme data; (dotted) best fits. In this global fit, 8 randomly generated Δω_AB (450—2500 s⁻¹) and Δω_AC (±450—2500 s⁻¹) spin sets were generated at the lowest magnetic field (black, 11.7 T) and scaled linearly to the other two fields (blue, 18.7 T; red, 22.2 T). Other parameters: k₁₂ + k₂₁ = 80 s⁻¹; k₁₃ + k₃₁ = 100 s⁻¹; k₂₃ + k₃₂ = 150 s⁻¹; p_A = 0.88; p_B = 0.04; p_C = 0.08. Only the fit with the largest deviation is shown, for which Δω_AB = 2084 s⁻¹; Δω_AC = −1675 s⁻¹. The RMS deviations for different characteristic sections of the curve are shown in the figure. Top right: this figure uncovers the basis for the difference between linear (red) and triangular (black and other colors), specifically in the “tail” region (small 1/τ_cp) of the CPMG curves. The Exp₁Log₀λ_2,Newton approximations (dashed black/red) are very similar to the exact solutions (solid). For some curves, only certain terms from the constant term of the characteristic polynomial (eq 27; naming of terms in eq 32) were selected to analyze their contributions to the full (red) approximation: X1, Y1, Y2 (orange); X2, Y1, Y2 (blue); X1, X2, Y2 (cyan). Parameters: k₁₂ + k₂₁ = 50 s⁻¹; k₁₃ + k₃₁ = 70 s⁻¹; k₂₃ + k₃₂ = 150 s⁻¹; Δω_AB = 1300 s⁻¹; Δω_AC = −1700 s⁻¹; p_A = 0.88; p_B = 0.04; p_C = 0.08. Bottom: General conditions for distinguishability between triangular and linear schemes. The graph is based on multiple calculations as in the top left panel. In each of the 16 panels, 25 points with different values of k₁₂/k₂₃ were analyzed. For each point, 20 RMS values from a set of 3 × 8 CPMG curves, with randomized Δω_AB (700—2000 s⁻¹) and Δω_AC (±700—2000 s⁻¹) were obtained. In order to get an estimate of a lower cutoff for distinguishability, the 10th percentile values are shown. For the bottom left panel, the large 1/(4τ_cp) (20—300 s⁻¹) region was selected for analysis (see also top left panel). For the bottom right panel, the small 1/(4τ_cp) “tail” region (1200—1500 s⁻¹) was selected for analysis. p_b was set to 0.04. To aid in convergence, some variables were constrained between upper and lower bounds during fitting: p_b between 0.005 and 0.25; k₁₂ + k₂₁ between 10 s⁻¹ and 400 s⁻¹; Δω_AB between 450 s⁻¹ and 2500 s⁻¹.

Because the characteristic polynomial for the triangular scheme is available (eqs 27 and S18), we can decompose the Exp₁Log₀λ₂ approximation and identify terms that explain the observation that the R_cpmg(τ_cp) dispersion curve “tail” is generally larger for triangular schemes than for the respective linear BAC scheme. Important terms are also listed in eq 32 for the Exp₁Log₀λ₁ approximation. We have found that is useful to inspect the distinct contributions to the numerator of Exp₁Log₀λ₁ and Exp₁Log₀λ₂; we show the contributions of the numerator to the CPMG curve in Figure 5 (top right panel). Even though component X1 from eq 3 vanishes at large τ_cp, component X2 represents a cubed sinc function, which grows faster than the squared sinc functions in X1. Generally, large exchange rates k_ij + k_ji, with as large as possible minor site populations, would therefore be expected to give a larger X1 and a larger numerator for large 1/τ_cp; the X1 term is indeed larger than X2 in the given example. The denominator is dominated by Y2 at large 1/τ_cp or more specifically large k₂₁k₃₁ + k₂₁k₃₂ + k₃₁k₂₃ in the given example. Due to the high population of the major site, the term k₂₁k₃₁ would be expected to yield a larger denominator, while the other two summands k₂₁k₃₂ and k₃₁k₂₃ each include only one rate constant that scales with the major site population. For that reason, a large numerator (with large kinetic rate constants) and a small denominator is best achieved with large k₂₃ + k₃₂, and some balanced constellation of k₁₂, k₁₃, k₂₁, and k₃₁. This explains why the ability to distinguish triangular from linear schemes does not simply increase with generally larger k_ij + k_ji. At smaller 1/τ_c, the Y1 term dominates the denominator. Y1 does not include any k₂₁k₃₁ factor, which would particularly “penalize” a combination of a large k₂₁ and k₃₁, which is why generally higher exchange rates k_ij + k_ji improve distinguishability at small 1/τ_c. The trends observed when comparing the distinguishability between triangular and three-state linear BAC models are generally augmented when trying to fit triangular scheme data to a two-state model (Figure S4), with a slightly higher complexity of the results.

We confirmed the observations made for three-site schemes by the consideration of different four-site kinetic topologies, specifically linear, kite, quadratic, and star schemes (Figure 6). The quadratic and kite schemes have the largest minor exchange contributions in the given example; therefore, the CPMG rates at large 1/τ_cp are largest for these schemes. The star scheme has no minor exchange contribution; therefore, CPMG rates at large 1/τ_cp are small. For small 1/τ_cp, we can focus on the exchange contributions between the major site and any minor site to qualitatively understand the magnitude of the CPMG rates. Specifically, linear and quadratic schemes on one hand (both with two minor sites bound to A), as well as kite and star schemes on the other hand (both with three minor sites bound to A), have very similar CPMG relaxation rates. We find that these observations are generally robust as long as minor site populations are not very large, and in particular for three-site schemes, as long as exchange between minor sites is not fast relative to the other exchange rates in the scheme.

DISCUSSION

N-Site chemical exchange processes are frequently encountered in structural biology, but more specific, quantitative applications will be needed to become a major driver of biomedical understanding and innovation. NMR spectroscopy provides experiments, for example, CPMG relaxation dispersion, to obtain quantitative information about N-site processes. However, extracting quantitative information from CPMG data is often challenging because of insufficient data to fit multiple parameters, or because numerical computation limits the easily accessible range of kinetic models. Closed-form approximations for CPMG data obtained for linear schemes with three fast-exchanging sites have been introduced by Gray et al.¹⁵ Herein, we introduce two sets of approximations for any kinetic scheme that also cover the entire exchange regime. We also feature general observations regarding the theoretically obtained exact CPMG rates for N-site schemes.

Previous N-site studies have been limited in the interpretation of CPMG data by the lack of appropriate models to fit slow exchange. For example, in the CPMG study of a Fyn3 SH3 domain mutant, chemical shifts obtained for Val55 could not be obtained with high precision because exchange fell into the slow range for this residue.²⁸ If sufficiently accurate approximations are not available, CPMG data can be fit numerically either directly²⁹ or using a geometric approach in a parameter space with precalculated solutions.³⁰ However, in N-site exchange, the size of the parameter space escalates quickly. In addition, multiple schemes have to be considered, especially due to the possibility that the various sites can be assigned to different positions in the kinetic schemes. For example, in the linear three-site case, A—B—C, A—C—B, and B—A—C schemes will have to be tested, as exemplified in the context of fast exchange by Neudecker et al.³¹ Recording sufficient data to fit to a larger parameter space is not necessarily a major problem because CPMG experiments are commonly recorded for multiple residues with distinct frequency differences between the analyzed states, and CPMG data can be recorded for different nuclei.^28,32 Importantly, the potentially large number of parameters, models, and local minima require efficiently converging fitting algorithms. Many efficient fitting routines, such as the Levenberg—Marquardt algorithm, perform best if closed-form derivatives of the target expression are available. The approximations presented herein can be used in these minimization procedures.

An approximation of the eigenvalue λ is an integral part of all Exp_aLog_nλ_k approximations we show herein. The simple λ₁ approximation has been introduced by us recently for R_1ρ experiments.²² We find that either the Newton—Raphson or the Halley (linearized Laguerre) method (eq 20, Supporting Information, section S4) can be used to iteratively obtain Exp_aLog_nλ_k approximations that are of higher order. The Halley method converges faster, but each iteration is computationally more expensive. We recommend to generally use the Halley approximation for k > 2. For k ≤ 2, we find, as expected, the following order of accuracies: Exp_aLog_nλ₁ < Exp_aLog_nλ₂ (Newton) < Exp_aLog_nλ₂ (Halley). When minor site populations are large or when exchange rates are very unequal, increasing k can improve the accuracy of many approximations arbitrarily, at the expense of more complex expressions. The closed-form Exp₁Log₀λ_k (k ≥ 1) approximations are accurate for slow enough exchange. Slow enough usually means that the rate constant sum for the exchange between the major and any minor site is an order of magnitude smaller than the corresponding chemical shift differences between linked sites; recording CPMG data at two different fields helps to determine whether the system of interest is in slow enough exchange. For all other cases, Exp_∞Log_nλ_k (n ≥ 1, k ≥ 1) approximations can be used. We find that even challenging problems, such as a four-site quadratic scheme with comparably large minor populations and very unequal exchange rates, can usually be fit well with Exp_∞Log_nλ_k (n = 2, k = 2), with the possibility to enhance the approximation by increasing the order of approximation for Log_n, or, as outlined above more importantly, λ_k. The expressions are not necessarily simple or closed-form algebraic expressions because of the Exp∞ element in the approximation. However, derivatives can be calculated from the matrix expressions and used in data analysis curve-fitting algorithms. We have shown in the Results section that Exp_∞Log_nλ_k approximations are limited in use for small values of 1/τ_cp when any minor site is kinetically isolated from the major site because the “exact” single-eigenvalue solution for R_cpmg(τ_cp) will be incorrect because the magnetization decays contain significant contributions from more than one eigenvalue component. Some models are more prone to this problem than others. For example, the triangular-site scheme might be preferred over the linear three-site scheme as kinetic isolation is less problematic in the triangular case.

The general nature of the new Exp_aLog_nλ_k expressions suggests that they might be useful in other contexts. Specifically, we tested how the iterative Halley approximations could be used for N-site R_1ρ approximations, expanding the results reported previously.²² We find that the use of second-order approximations for the eigenvalues (λ) yielding R_1ρ fit the numerical data excellently in most scenarios. Virtually perfect fits were obtained for the previously tested four-state kite case (Figure S5); in addition, we also tried more challenging conditions by assuming relatively large populations, giving excellent agreement with the exact solution.

Practical guidelines to distinguish between two- and three-site exchange in the fast exchange regime by inspecting the CPMG rate curves have been established previously,¹⁵ highlighting the ability to distinguish two-site and linear three-site schemes by obtaining data at small 1/τ_cp. We can now expand this observation to state that R_cpmg(τ_cp) at small 1/τ_cp is mostly influenced by direct exchange between the major and any attached minor site. This means that in most cases (exemplified for four-state), kite and star schemes on one hand and linear and quadratic schemes on the other hand will have very similar R_cpmg(τ_cp) at small 1/τ_cp. An exception to this rule occurs, here exemplified for the four-site linear scheme, if exchange processes connecting a “remote” minor site D (in B—A—C—D) to the major site A are much faster than exchange processes between another directly connected minor site and A. Furthermore, we find that at large 1/τ_cp, minor exchange dominates the magnitude of R_cpmg(τ_cp). In other words, the “tail” of the R_cpmg(τ_cp) curve can be used to qualitatively identify minor state exchange processes (for example, discriminating kite from star schemes, or triangular from linear three-site schemes). We corroborated these qualitative findings by simulating R_cpmg(τ_cp) curves for a large section of the parameter space for a given triangular or linear BAC model and then fitting those data with R_cpmg(τ_cp) expressions derived for different kinetic models. We tested the conditions under which the different kinetic schemes can be distinguished and also identified parameters that are particularly important for the difference between triangular and linear models at small 1/τ_cp. We were also able to rationalize the influence of minor exchange on shape of the CPMG dispersion curve by inspecting individual terms in the compact Exp₁Log₀λ₁ approximation for the triangular case (eq 32). This example illustrates that closed-form approximations provide qualitative insight into chemical exchange in CPMG experiments (or other relaxation experiments).

We provide general approximations for CPMG experiments that can be used in most N-site exchange scenarios. For sufficiently slow exchange, we provide compact closed-form expressions for Exp₁Log₀λ_k of any kinetic scheme. For other cases, arbitrarily accurate approximations Exp_∞Log_nλ_k were also found; however, these expressions can become very complex algebraically. In order to assess the large variety of accessible approximations, we provide an interactive python script to visualize these results together with the corresponding exact solutions and exponential decays (Supporting Information). This visualization tool has also helped us to gain qualitative insight into CPMG curve features that result from complex kinetic processes. Specifically, we find that the tail of the CPMG curve is useful to analyze exchange between minor sites, while exchange between the major site and attached minor site is observed at small 1/τ_cp.