Introduction

The structures spontaneously adopted by proteins dictate their function. The interactions that stabilize these structures are individually weak and at biologically relevant temperatures are easily rearranged by thermal motion. Consequently, biological molecules are inherently dynamic (Karplus and Kuriyan 2005) and are best understood as ensembles of inter-converting conformers rather than in terms of single static structures (Lange et al. 2008; Lindorff-Larsen et al. 2005). Moreover, it is increasingly apparent that conformational fluctuations play crucial roles in enabling the functions of protein molecules (Karplus and Kuriyan 2005). Observing and characterizing such motions however remains challenging.

Solution NMR spectroscopy is particularly well suited to characterizing such molecular dynamics over a broad spectrum of time-scales (Ishima and Torchia 2000; Mittermaier and Kay 2006; Palmer et al. 1996). Among the different methodologies, paramagnetic relaxation enhancement (Iwahara and Clore 2006), R 1ρ (Palmer and Massi 2006) and Carr-Purcell-Meiboom-Gill (CPMG) (Hansen et al. 2008; Korzhnev and Kay 2008; Palmer et al. 2001) experiments are particularly useful for the study of exchange processes involving the inter-conversion between dominant (ground) and sparsely populated (conformationally ‘excited’) states. Insight into biological processes as diverse as molecular recognition, ligand binding, enzyme catalysis and protein folding, has been achieved using these techniques (Boehr et al. 2006; Fraser et al. 2009; Henzler-Wildman and Kern 2007; Ishima et al. 1999; Korzhnev et al. 2004, 2010; Sugase et al. 2007; Tang et al. 2007).

A rich theoretical basis for understanding PRE, R 1ρ and CPMG-based experiments as they pertain to systems undergoing chemical exchange has emerged over many years. Notably, analytical expressions exist that describe the effective R 1ρ or R 2 relaxation rate as a function of the exchange parameters, radio frequency (RF) field strengths and shift offsets (Allerhand et al. 1966; Carver and Richards 1972; Deverell et al. 1970; Jen 1978; Luz and Meiboom 1963; Miloushev and Palmer 2005; Palmer et al. 2001, 2005; Palmer and Massi 2006; Trott and Palmer 2002). While explicit use of these equations in fits of experimental data has now given way to numerical approaches involving fast computers, analytical expressions remain useful for providing physical insight that various parameters have on the measured relaxation rates. Most recently, Palmer and coworkers and Ishima and Torchia have derived analytical expressions for CPMG and R 1ρ experiments that extend the relations that were originally presented many years ago (Ishima and Torchia 1999; Miloushev and Palmer 2005; Trott and Palmer 2002).

The majority of the derived formulae for transverse relaxation in the presence of CPMG or R 1ρ fields are based on the assumption that the spins in question exchange between states with the same intrinsic transverse relaxation rates, ΔR 2  = 0, although Allerhand and Thiele recognized very early on that large differences in transverse relaxation rates of exchanging spins can certainly influence the CPMG profile (Allerhand and Thiele 1966). More recently, Ishima and Torchia have explored through simulations how CPMG dispersion profiles are affected in the case where ΔR 2  ≠ 0 (Ishima and Torchia 2006). In addition, Miloushev and Palmer have presented an expression to account for relaxation differences in R 1ρ experiments that is valid in certain limits (Miloushev and Palmer 2005). There are clearly many cases where the assumption ΔR 2  = 0 is reasonable. However, recent advances in NMR methodology have opened the possibilities of studying biological molecules of ever-increasing size (Fiaux et al. 2002; Sprangers and Kay 2007), leading to examples where differential relaxation becomes an important issue. This is the case for the 560 kDa αB-crystallin oligomeric ensemble (Baldwin et al. 2011a, b), where we have recently characterized a disorder (ground state) to order (excited state) transition with ΔR 2  ~ 200 s−1 (Baldwin et al. 2012), illustrated schematically in Fig. 1. In an effort to understand how such differences affect measured R 1ρ relaxation rates we have derived an expression for R 1ρ in the limit ΔR 2  ≠ 0 that is valid over a large range of ΔR 2 values (see below). We show that differential transverse relaxation can significantly influence R 1ρ rates but that remarkably measurements of signs of chemical shift differences, based on comparison of R 1ρ values obtained with spinlock fields on either sides of the ground state correlation (Korzhnev et al. 2003; Trott and Palmer 2002), are much less affected by ΔR 2  ≠ 0.

Fig. 1
figure 1

Schematic of a two-site exchange process involving the interconversion between protein conformers whereby a fragment is rigidly bound or unattached to the core of the protein. In the case where the protein is small (molecular mass of 5–10 kDa, a) it may be that differential transverse relaxation in the two states can be neglected in the analysis of exchange data (although this must be rigorously established). In the case of a high molecular weight complex, such as αB crystallin (aggregate molecular mass of close to 600 kDa, illustrated schematically in b) ΔR 2 can be large (≈200 s−1), complicating analysis of dispersion data

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Results and discussion

We consider here the case of two-site conformational exchange involving the interconversion of a molecule between ground (G) and excited (E) states, \( G\underset{{k_{EG} }}{\overset{{k_{GE} }}{\rightleftarrows}}E \) and focus on a single spin reporter. In what follows, the fractional populations of the two states are given by p G  = k EG /k ex and p E  = 1 − p G  = k GE /k ex , the overall exchange rate is k ex  = k EG  + k GE , the spin of interest in each of the two states evolves with frequencies ω G and ω E (Δω = ω E −ω G , rad/s) in the absence of chemical exchange and the difference in intrinsic transverse spin relaxation rates is given by ΔR 2 . As has been described in the literature (Anet and Basus 1978; Skrynnikov et al. 2002), chemical exchange leads to a shift in the position of the observed resonance from ω G to ω OBS where in the limit that p G  ≫ p E ,

$$ \omega_{ex} = \omega_{OBS} - \omega_{G} = \frac{{k_{EG} k_{GE} \Updelta \omega }}{{\left( {k_{EG} + \Updelta R_{2} } \right)^{2} + \left( {\Updelta \omega } \right)^{2} }}. $$
(1)

In a typical R 1ρ experiment that is of interest here, magnetization initially along the z-axis is rotated into the x–z plane by a pulse along the y axis with flip angle \( \theta_{flip} = \tan^{ - 1} \left( {\omega_{1} /\delta_{OBS} } \right),\delta_{OBS} = \omega_{OBS} - \omega_{RF} \) where ω RF is the carrier frequency. The resulting spin-locked magnetization subsequently evolves for a defined period of time under the influence of an RF pulse of strength ω 1 applied along the x-axis at an offset of δ OBS from the observed ground state frequency. At the completion of this interval magnetization is subsequently restored to the z-axis by a y pulse with flip angle \( - \theta_{flip} \). The decay rate of the signal, R 1ρ can be analyzed to determine the properties of the underlying conformational exchange (Palmer and Massi 2006). The evolution of magnetization during the spin-lock period is given by a set of coupled linear differential equations (McConnell 1958),

$$ \frac{d}{dt}\left( {\begin{array}{*{20}c} {M_{x}^{G} } \\ {M_{y}^{G} } \\ {M_{z}^{G} } \\ {M_{x}^{E} } \\ {M_{y}^{E} } \\ {M_{z}^{E} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} { - R_{2}^{G} - k_{GE} } \hfill & { - \delta_{G} } \hfill & 0 \hfill & {k_{EG} } \hfill & 0 \hfill & 0 \hfill \\ {\delta_{G} } \hfill & { - R_{2}^{G} - k_{GE} } \hfill & { - \omega_{1} } \hfill & 0 \hfill & {k_{EG} } \hfill & 0 \hfill \\ 0 \hfill & {\omega_{1} } \hfill & { - R_{1}^{G} - k_{GE} } \hfill & 0 \hfill & 0 \hfill & {k_{EG} } \hfill \\ {k_{GE} } \hfill & 0 \hfill & 0 \hfill & { - R_{2}^{E} - k_{EG} } \hfill & { - \delta_{G} - \Updelta \omega } \hfill & 0 \hfill \\ 0 \hfill & {k_{GE} } \hfill & 0 \hfill & {\delta_{G} + \Updelta \omega } \hfill & { - R_{2}^{E} - k_{EG} } \hfill & { - \omega_{1} } \hfill \\ 0 \hfill & 0 \hfill & {k_{GE} } \hfill & 0 \hfill & {\omega_{1} } \hfill & { - R_{1}^{E} - k_{EG} } \hfill \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {M_{x}^{G} } \\ {M_{y}^{G} } \\ {M_{z}^{G} } \\ {M_{x}^{E} } \\ {M_{y}^{E} } \\ {M_{z}^{E} } \\ \end{array} } \right) + \overrightarrow {{M_{0} }} $$
(2)

where \( \vec{M}_{O} = \left( {0,0,R_{1}^{G} M_{O}^{G} ,0,0,R_{1}^{E} M_{O}^{E} } \right)^{T} \), T is ‘transpose’, R G/E 1 and R G/E 2 are longitudinal and transverse relaxation rates for the spin in states G, E and M G/E 0 are thermal equilibrium magnetization values, proportional to p G and p E . In what follows we assume \( R_{1}^{E} = R_{1}^{G} = R_{1} \) and \( R_{2}^{E} \ne R_{2}^{G} \). Note that the assumption of equivalent R 1 values is reasonable even if states G and E differ significantly in correlation time or in intrinsic dynamics since R 1 rates are in general small for the spin ½ probes of dynamics typically used in studies of biomolecules.

Following the lead of Trott and Palmer (2002) an expression for the R 1ρ rate when \( R_{2}^{E} \ne R_{2}^{G} \) is derived by evaluating,

$$ \det |\hat{R} - \lambda \hat{E}| = 0 $$
(3)

where \( \hat{E} \) is the identity matrix and \( \hat{R} \) is the 6 × 6 square matrix in Eq. (2). Of the six eigenvalues of \( \hat{R} \), four are complex, and of the two that are real only one is small in magnitude, corresponding to the R 1ρ value measured experimentally (Trott and Palmer 2002). Evaluating the characteristic polynomial that derives from Eq. (3), with the substitutions \( \Updelta R_{2} = R_{2}^{E} - R_{2}^{G} \) and \( \Updelta R = R_{2}^{G} - R_{1}^{{}} \) generates 230 non-zero terms of the form \( C(\Updelta \omega )^{i} \delta_{G}^{j} \omega_{1}^{k} k_{EG}^{l} k_{GE}^{m} (\Updelta R)^{n} (\Updelta R_{2} )^{o} \lambda^{p} \) where C is a number and the sum of all superscripts is equal to 6. In general it is not possible to obtain an analytical solution for the roots (eigenvalues) of a polynomial of order 6, however, as discussed by Trott and Palmer (2002) only the leading terms of the polynomial must be retained to calculate an accurate expression for R 1ρ . Because in many practical cases both the eigenvalue of interest (λ, corresponding to −R 1ρ ) and ΔR are small compared to the other elements in each of the 230 terms (Δω, δ G , ω 1 , k EG , k GE , ΔR 2 ) it is possible to include only those terms in the characteristic polynomial for which \( n + p \le 1 \) without unduly sacrificing accuracy. This follows directly from the fact that for higher powers of ΔR and λ the terms become small. The linear (in λ) equation that results, Aλ + B = 0, contains 76 terms, with λ = −B/A = \( - R_{1\rho } \).

After a lengthy derivation, and with the following definitions,

$$ \begin{aligned} \delta_{j} & = \omega_{j} - \omega_{RF} ,\quad j \in \{ G,E\} \hfill \\ \overline{\delta }& = p_{G} \delta_{G} + p_{E} \delta_{E} = \delta_{G} + p_{E} \Updelta \omega \hfill \\ \Upomega_{G}^{2} &= \omega_{1}^{2} + \delta_{G}^{2} \hfill \\ \Upomega_{E}^{2}& = \omega_{1}^{2} + \delta_{E}^{2} \hfill \\ \overline{\Upomega }_{{}}^{2}& = \omega_{1}^{2} + \overline{\delta }_{{}}^{2} \hfill \\ \theta &= \tan^{ - 1} \left( {\omega_{1} /\overline{\delta } } \right) \hfill \\ \end{aligned} $$
(4)

we arrive at the central result of the paper,

$$ R_{1\rho } = C_{R1} R_{1}^{{}} \cos^{2} \theta + (C_{R2} R_{2}^{G} + R_{2ex} )\sin^{2} \theta $$
(5)

where \( \sin^{2} \theta = \omega_{1}^{2} /\bar{\Upomega }^{2} \) and \( \cos^{2} \theta = \bar{\delta }^{2} /\bar{\Upomega }^{2} \). Note that \( \theta = \theta_{flip} \) only in the limit \( \omega_{G} + p_{E} \Updelta \omega \approx \omega_{OBS} \). The coefficients C R1 , C R2 and R 2ex include explicit contributions from ΔR 2 and are given by

$$ \begin{aligned} C_{R1} & = \frac{{F_{2}^{P} + \left( {F_{1}^{P} + \Updelta R_{2} \left( {F_{3} - F_{2} } \right)} \right)\tan^{2} \theta }}{{D_{{}}^{P} + \Updelta R_{2} F_{3} \sin^{2} \theta }} \\ C_{R2} & = \frac{{\left( {D_{{}}^{P} \csc^{2} \theta - F_{2}^{P} \cot^{2} \theta } \right) - F_{1}^{P} + \Updelta R_{2} F_{2} }}{{D_{{}}^{P} + \Updelta R_{2} F_{3} \sin^{2} \theta }} \\ R_{2ex} & = \frac{{F_{1}^{P} k_{ex} + \Updelta R_{2} F_{1} }}{{D_{{}}^{P} + \Updelta R_{2} F_{3} \sin^{2} \theta }} \\ \end{aligned} $$
(6)

These are, in turn, recast in terms of three expressions that are independent of ΔR 2 , present in the derivation by Trott and Palmer (2002),

$$ \begin{aligned} F_{1}^{P} &= p_{G} p_{E} \Updelta \omega^{2} \\ F_{2}^{P} &= k_{ex}^{2} + \omega_{1}^{2} + \frac{{\delta_{G}^{2} \delta_{E}^{2} }}{{\bar{\delta }^{2} }} \\ D_{{}}^{P}& = k_{ex}^{2} + \frac{{\Upomega_{G}^{2} \Upomega_{E}^{2} }}{{\overline{\Upomega }^{2} }} \\ \end{aligned} $$
(7)

and an additional three terms that are all multiplied by ΔR 2 in C R1 ,C R2 and R 2ex ,

$$ \begin{aligned} F_{1}^{{}} &= p_{E} \left( {\Upomega_{G}^{2} + k_{ex}^{2} + \Updelta R_{2} p_{G} k_{ex} } \right) \\ F_{2} &= 2k_{ex} + \frac{{\omega_{1}^{2} }}{{k_{ex} }} + \Updelta R_{2} p_{G}^{{}} \\ F_{3} &= 3p_{E} k_{ex} + \left( {2p_{G} k_{ex} + \frac{{\omega_{1}^{2} }}{{k_{ex} }} + \Updelta R_{2} + \frac{{\Updelta R_{2} p_{E}^{2} k_{ex}^{2} }}{{\Upomega_{G}^{2} }}} \right)\left( {\frac{{\Upomega_{G}^{2} }}{{\omega_{1}^{2} }}} \right) \\ \end{aligned} $$
(8)

Equation (5) is complicated and it is useful to consider certain limiting cases. Assuming for the moment that ΔR 2  = 0 it is straightforward to show that

$$ R_{1\rho } = R_{2} + \frac{{\left( {k_{ex}^{2} + \omega_{1}^{2} + \frac{{\delta_{G}^{2} \delta_{E}^{2} }}{{\bar{\delta }_{{}}^{2} }}} \right)\left( {R_{1} - R_{2} } \right)\cos^{2} \theta + p_{G} p_{E} \Updelta \omega^{2} \left( {k_{ex} + R_{1} - R_{2} } \right)\sin^{2} \theta }}{{k_{ex}^{2} + \frac{{\Upomega_{G}^{2} \Upomega_{E}^{2} }}{{\bar{\Upomega }_{{}}^{2} }}}} $$
(9)

(\( R_{2}^{E} = R_{2}^{G} = R_{2} \)) which is identical to Eq. (11) of Trott and Palmer (2002) (not withstanding a small typographical error in their expression). In the limit that ΔR/k ex  ≪ 1 and δ G  ≫ p E Δω it follows that

$$ R_{1\rho }^{P} = R_{1} \cos^{2} \theta + (R_{2} + R_{ex}^{P} )\sin^{2} \theta $$
(10)

where

$$ R_{ex}^{P} = \frac{{p_{G} p_{E} \Updelta \omega^{2} k_{ex} }}{{\frac{{\Upomega_{E}^{2} \Upomega_{G}^{2} }}{{\bar{\Upomega }_{{}}^{2} }} + k_{ex}^{2} }} \cong \frac{{p_{G} p_{E} \Updelta \omega^{2} k_{ex} }}{{\Upomega_{E}^{2} + k_{ex}^{2} }}, $$
(11)

that is Eq. (21) of Trott and Palmer (2002).

Equation (5) can be simplified in the limit when ΔR 2 is small by performing a Taylors series expansion about ΔR 2  = 0, retaining only the linear terms and enforcing δ G  ≫ p E Δω, p G  ≫ p E . It can be shown that under these conditions

$$ \begin{aligned} C_{R1}^{{\Updelta R_{2} \to 0}} &= 1 + \frac{{p_{E} \Updelta \omega^{2} }}{{k_{ex}^{2} + \Upomega_{E}^{2} }}\tan^{2} \theta + \frac{{\Updelta R_{2} \Upxi }}{{k_{ex}^{2} + \Upomega_{E}^{2} }}\tan^{2} \theta \\ C_{R2}^{{\Updelta R_{2} \to 0}} &= 1 - \frac{{p_{E} \Updelta \omega^{2} }}{{k_{ex}^{2} + \Upomega_{E}^{2} }} - \frac{{\Updelta R_{2} \Upxi }}{{k_{ex}^{2} + \Upomega_{E}^{2} }} \\ R_{2ex}^{{\Updelta R_{2} \to 0}} &= \frac{{p_{E} \Updelta \omega^{2} k_{ex} }}{{k_{ex}^{2} + \Upomega_{E}^{2} }}\left( {1 + \frac{{\Updelta R_{2} }}{{k_{ex} }}\left( {\frac{{k_{ex}^{2} + \Upomega_{G}^{2} }}{{\Updelta \omega^{2} }} - \frac{{2k_{ex}^{2} + \omega_{1}^{2} }}{{k_{ex}^{2} + \Upomega_{E}^{2} }}} \right)} \right) \\ \end{aligned} $$
(12)
$$ \Xi = p_{E} \left( {\left( {1 - 3\cos ^{2} \theta } \right)k_{{ex}} - \left( {\frac{{2k_{{ex}}^{2} + \omega _{1}^{2} }}{{k_{{ex}}^{2} + \Omega _{E}^{2} }}} \right)\left( {\frac{{\Delta \omega ^{2} }}{{k_{{ex}} }}} \right)} \right) $$

Substitution of the expressions in Eq. (12) into Eq. (5) with the requirement that ΔR/k ex  ≪ 1 and retaining only the leading terms gives

$$ R_{1\rho }^{{\Updelta R_{2} \to 0}} = R_{1} \cos^{2} \theta + \left( {R_{2}^{G} + R_{2ex}^{{\Updelta R_{2} \to 0}} } \right)\sin^{2} \theta . $$
(13)

From the expression for \( R_{2ex}^{{\Updelta R_{2} \to 0}} \)in Eq. (12) it is clear that so long as

$$ \frac{{\Delta R_{2} }}{{k_{{ex}} }}{\text{ }} < < \left| {\frac{{\Delta \omega ^{2} \left( {k_{{ex}}^{2} + \Omega _{E}^{2} } \right)}}{{\left( {k_{{ex}}^{2} + \Omega _{G}^{2} } \right)\left( {k_{{ex}}^{2} + \Omega _{E}^{2} } \right) - \left( {2k_{{ex}}^{2} + \omega _{1}^{2} } \right)\Delta \omega ^{2} }}} \right| $$
(14)

differential transverse relaxation makes little contribution to R 1ρ , while as k ex increases the effects of ΔR 2 become more pronounced (see below).

Values of R 1ρ as a function of ω 1 have been calculated with (i) Eq. (11) of Trott and Palmer (2002), modified as they describe with R 1 and R 2 in this expression replaced by population weighted averages, (ii) with Eq. (5) or (iii) numerically for exchange parameters indicated in the legend to Fig. 2a, b. When ΔR 2  = 0 (Fig. 2a) excellent agreement is obtained with the numerically computed values. The slight deviations observed for small ω 1 values are in keeping with the assumptions used in the derivation of analytical expressions based on linearization of the characteristic polynomial of the matrix in Eq. (2). Significant deviations are noted for the Trott and Palmer expression when ΔR 2 is large (\( R_{2}^{G} \) = 6 s−1, \( R_{2}^{E} \)= 200 s−1), as expected, while rates calculated with Eq. (5) are in good agreement with those obtained numerically, Fig. 2b. These differences decrease as ω 1 increases but do not go to zero in the present case as this would require in addition that ΔR 2 /k ex  ≪ 1, a condition that is not fulfilled in this example (see legend to Fig. 2). Of note, similar calculations, but as a function of k ex , show that the differences between the expressions do converge to the exchange free rate in the tilted frame in the limit that k ex  → ∞ with R 1 and R 2 values given by population weighted averages. We have also performed an extensive grid search in parameter space to ascertain the accuracy of Eq. (5) more generally. Numerical simulations for \( R_{2}^{G} \le R_{2}^{E} \le 500 {\text{s}}^{ - 1} \), 0.1 % ≤ p E  ≤ 10 %, 12.5 rad/s ≤ Δω ≤ 12,500 rad/s, 10 s−1≤k ex  ≤ 20,000 s−1, 25 rad/s ≤ ω 1  ≤ 100,000 rad/s, 0.001 rad/s ≤ δG ≤ 100,000 rad/s show that, so long as \( (\delta_{G}^{2} + \omega_{1}^{2} )^{0.5} > A(p_{E} \Updelta \omega )^{2} \) with A = 0.1, errors are less than 2 s−1. Figure 2c plots the deviations in R 1ρ values calculated from Eq. (5) or from the formula of Trott and Palmer for a number of \( R_{2}^{E} \) values as a function of A. As the value of A decreases in the expression above the range of possible δ G , ω 1 values increases, leading to cases where R 1ρ becomes sufficiently large that retaining only those terms up to linear in λ in the polynomial of Eq. (3) is no longer appropriate so that Eq. (5) becomes less accurate.

Fig. 2
figure 2

a, b Plots of R 1ρ (ω 1 ) calculated with (i) Eq. (11) of Trott and Palmer (2002), modified such that R 1 and R 2 in this expression are replaced by population weighted averages (red), (ii) with Eq. (5) of the present manuscript (blue) and (iii) by solving numerically for R 1ρ using Eq. (2) (black) for ΔR 2  = 0 (a) and \( R_{2}^{G} = 6\,{\text{s}}^{ - 1} \), \( R_{2}^{E} = 200 {\text{s}}^{ - 1} \) (b). Values of k ex  = 1,000 s−1, Δω = 1,000 rad/s, R 1  = 2.5 s−1, δ G  = 0 rad/s, p B  = 3 % have been used. In the limit \( \omega_{1} \to \infty \) Eq. (5) reduces to \( R_{1\rho } = R_{2}^{G} + \frac{{\Updelta R_{2} p_{E} }}{{1 + \Updelta R_{2} /k_{ex} }} \) that differs slightly from a population weighted average of rates, \( p_{G} R_{2}^{G} + p_{E} R_{2}^{E} \). Note that an identical expression is obtained in this limit for the transverse relaxation rate derived from the CPMG experiment as well. c Errors in R 1ρ values calculated from Eq. (5) [blue, R 1ρ (Eq. (5))] or Eq. (11) of Trott and Palmer [red, R 1ρ (Eq. (11), T&P)] for \( R_{2}^{E} = \left( {6, \, 100, \, 200, \, 500\, {\text{s}}^{ - 1} } \right) \) as a function of A, with \( (\delta_{G}^{2} + \omega_{1}^{2} )^{0.5} > A(p_{E} \Updelta \omega )^{2} \) and spanning the range \( R_{2}^{G} \le R_{2}^{E} \le 500\, {\text{s}}^{ - 1} \), 0.1 % ≤ p E  ≤ 10 %, 12.5 rad/s ≤ Δω ≤ 12,500 rad/s, 10 s−1≤k ex  ≤ 20,000 s−1, 25 rad/s ≤ ω 1  ≤ 100,000 rad/s, 0.001 rad/s ≤ δG ≤ 100,000 rad/s. For A = 0.1 errors are less than 2 s−1 using Eq. (5). Values of R 1ρ calculated by numerically computing the eigenvalues of the matrix in Eq. (3) are denoted by R 1ρ (N)

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As an interesting application of the work described here we consider the measurement of R 1ρ as a function of offset from the ground state peak to obtain the sign of Δω, and hence the value for the resonance position of the spin in the excited conformer. This information is not available from Carr-Purcell-Meiboom-Gill experiments that are typically recorded to characterize exchanging systems since relaxation dispersion profiles are only sensitive to |Δω|. In their original derivation of an expression for R 1ρ outside the fast exchange regime Trott and Palmer (2002) noted that R 1ρ values are particularly sensitive to the placement of the spin lock field relative to the resonance frequency in the ground state, with a maximum in R 2ex when the carrier is positioned on resonance with the peak derived from the excited state. Thus, R 1ρ (δ G  = Δω) < R 1ρ (δ G  = −Δω). Our laboratory has exploited this result by measuring R 1ρ values where the spin lock is placed equidistantly upfield and downfield of the major state correlation in a pair of experiments. Positions of the carrier and the magnitude of the spinlock field (ω 1 ) are optimized on a per-residue basis, taking into account the exchange parameters that have been measured previously, by maximizing the difference in R 1ρ values for a given spin lock duration, T relax . Using this approach we have developed a suite of off-resonance R 1ρ experiments with weak spinlock fields to measure the signs of chemical shift differences for 1Hα, 1HN, 15N, 13Cα and 13C-methyl nuclei in proteins that exchange between different conformations on the ms time-scale (Auer et al. 2009, 2010; Baldwin and Kay 2012; Korzhnev et al. 2003, 2005).

Equations (12) and (13) presented above can be used to derive an expression for the difference in R 1ρ relaxation rates recorded with the spinlock carrier placed downfield \( (R_{1\rho }^{ + } ) \) and upfield \( (R_{1\rho }^{ - } ) \) of the ground state resonance position, \( \Updelta R_{1\rho } = R_{1\rho }^{ + } - R_{1\rho }^{ - } \), that forms the basis for measurement of the signs of chemical shift differences. Under the same set of assumptions that were used in the derivation of Eq. (12) it can be shown that

$$ \begin{aligned} \Updelta R_{1\rho }^{{\Updelta R_{2} \to 0}} = \frac{{4p_{E} |\delta_{OPT} |\Updelta \omega^{3} k_{ex} \sin^{2} \theta }}{{\left( {k_{ex}^{2} + \Upomega_{OPT}^{2} + \Updelta \omega^{2} } \right)^{2} - 4\delta_{OPT}^{2} \Updelta \omega^{2}}} \hfill \\ \, \left( {1 - \frac{{\Updelta R_{2} }}{{k_{ex}}}\left( {\frac{{2\left( {k_{ex}^{2} + \Upomega_{OPT}^{2} + \Updelta \omega^{2} } \right)\left( {2k_{ex}^{2} + \omega_{1}^{2}} \right)}}{{\left( {k_{ex}^{2} + \Upomega_{OPT}^{2} + \Updelta\omega^{2} } \right)^{2} - 4\delta_{OPT}^{2} \Updelta \omega^{2}}} - \frac{{\left( {\Upomega_{OPT}^{2} + k_{ex}^{2} }\right)}}{{\Updelta \omega^{2} }}} \right)} \right) \hfill \\\end{aligned} $$
(15)

where

$$ \begin{aligned} \delta _{{OPT}} & = \omega _{{OBS}} - \omega _{{RF,OPT}} \approx \omega _{G} - \omega _{{RF,OPT}} \\ \Omega _{{OPT}}^{2} & = \omega _{{1,OPT}}^{2} + \delta _{{OPT}}^{2} \\ \end{aligned} $$
(16)

and ω RF,OPT , ω 1,OPT are the optimized position of the carrier and spinlock field strength, respectively.

From Eq. (15) it is clear that the sign of Δω determines the sign of ΔR 1ρ , as expected from the work of Trott and Palmer (2002); note that it can be proved that the sign of the denominator of the first term is >0 and extensive numerical simulations establish that even for ΔR 2 values as large as 500 s−1 the signs of ΔR 1ρ and Δω are the same. Thus, measurement of the sign of ΔR 1ρ allows the determination of the sign of Δω and for positive (negative) values of ΔR 1ρ the excited state peak is downfield (upfield) of the ground state correlation. It is not immediately transparent from Eq. (15) how ΔR 2  ≠ 0 influences this scenario; however because ΔR 2 is multiplied by an expression that is, in turn, given by the difference of two positive terms there is the possibility for attenuation of the effects of differential relaxation. Indeed, this turns out to be the case, as numerical simulations indicate. Figure 3ai–iii plot \( R_{1\rho }^{ + } \) (i), \( R_{1\rho }^{ - } \) (ii) and ΔR 1ρ (iii) calculated from Eq. (2) numerically, as a function of Δω and k ex for a spin exchanging between two states with ΔR 2  = 500 s−1 (blue) or 0 (red). Shown in Fig. 3aiv–vi are the differences between each of these values, Z(ΔR 2  = 500 s−1) − Z(ΔR 2  = 0 s−1), where Z \( \in \) {\( R_{1\rho }^{ + } \),\( R_{1\rho }^{ - } \),ΔR 1ρ }, as labelled along the vertical axis of each panel. Most notably, ΔR 1ρ is rather insensitive to ΔR 2 , even for very large values. Similar plots for the exchange induced frequency shift (ppm), that can also be used to establish the sign of Δω (Skrynnikov et al. 2002), are shown in Fig. 3bi–iii. Here ϖ ex (Δω,k ex ) values evaluated at 11.7 and 18.8T are shown, along with the difference, Δϖ ex  = ϖ ex (11.7T) − ϖ ex (18.8T) for both ΔR 2  = 500 s−1 (blue) and ΔR 2  = 0 s−1 (red). In addition absolute values of the differences between ϖ ex  = ϖ ex (ΔR 2  = 500 s−1) and ϖ ex (ΔR 2  = 0) are plotted in Fig. 3biv (11.7T) and 3Bv (18.8T), respectively, along with |Δϖ ex (ΔR 2  = 500 s−1) − Δϖ ex (ΔR 2  = 0)| in Fig. 3bvi. Values of Δϖ ex are significantly more sensitive to ΔR 2 than are ΔR 1ρ rates, as indicated in the Figure. Indeed, in a previous application involving studies of chemical exchange in the 560 kDa αB crystallin complex we noted that Δϖ ex values were much smaller than anticipated based on the measured exchange parameters, while large ΔR 1ρ values were still obtained from which signs of chemical shift differences could be established (Baldwin and Kay 2012).

Fig. 3
figure 3

a Plots of \( R_{1\rho }^{ + } \) (i), \( R_{1\rho }^{ - } \) (ii) and ΔR 1ρ (iii) calculated numerically, as a function of Δω and k ex , with ΔR 2  = 500 s−1 (blue) or 0 (red). (iv–vi) Corresponding differences, Z(ΔR 2  = 500 s−1) − Z(ΔR 2  = 0 s−1), where Z \( \in \) {\( R_{1\rho }^{ + } \),\( R_{1\rho }^{ - } \),ΔR 1ρ }, as labelled along the vertical axes. b (i–iii) Exchange induced frequency shifts (ppm), ϖ ex (Δω,k ex ), calculated for static magnetic fields of 11.7 and 18.8T along with Δϖ ex  = ϖ ex (11.7T) − ϖ ex (18.8T), ΔR 2  = 500 s−1 (blue) and ΔR 2  = 0 s−1 (red). (iv–vi) |ϖ ex (ΔR 2  = 500 s−1) − ϖ ex (ΔR 2  = 0 s−1)| at 11.7T (iv) and 18.8T (v), |Δϖ ex (ΔR 2  = 500 s−1) − Δϖ ex (ΔR 2  = 0 s−1)| (vi)

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In summary, we have derived an expression for R 1ρ for a spin exchanging between two sites with ΔR 2  ≠ 0. While analytical expressions of the sort presented here can be used for data fitting we much prefer to use (exact) numerical approaches that start from the Bloch-McConnell equations. Nevertheless, formula are valuable as a ‘teaching tool’ providing physical insight into how the range of exchange parameters can influence measured relaxation rates. The work described here adds to a growing body of theoretical analyses of CPMG and R 1ρ dispersion experiments and considers the important case of differential transverse relaxation that will become increasingly relevant as applications to high molecular weight biomolecules become more common.