Dynamical Models of Chemical Exchange in Nuclear Magnetic Resonance Spectroscopy

Abstract

Chemical exchange line-broadening is an important phenomenon in nuclear magnetic resonance (NMR) spectroscopy, in which a nuclear spin experiences more than one magnetic environment as a result of chemical or conformational changes of a molecule. The dynamic process of chemical exchange strongly affects the sensitivity and resolution of NMR experiments, and increasingly provides a powerful probe of the inter-conversion between chemical and conformational states of proteins, nucleic acids, and other biological macromolecules. A simple and often used theoretical description of chemical exchange in NMR spectroscopy is based on an idealized two-state jump model (the random-phase or telegraph signal). However, chemical exchange can also be represented as a barrier-crossing event that can be modeled using chemical reaction rate theory. The time scale of crossing is determined by the barrier height, the temperature, and the dissipation modeled as collisional or frictional damping. This tutorial explores the connection between the NMR theory of chemical exchange line-broadening and strong-collision models for chemical kinetics in statistical mechanics. Theoretical modeling and numerical simulation are used to map the rate of barrier-crossing dynamics of a particle on a potential energy surface to the chemical exchange relaxation rate constant. By developing explicit models for the exchange dynamics, the tutorial aims to elucidate the underlying dynamical processes that give rise to the rich phenomenology of chemical exchange observed in NMR spectroscopy. Software for generating and analyzing the numerical simulations is provided in the form of Python and Fortran source codes.

I. Introduction

Chemical exchange in NMR spectroscopy describes the behavior of nuclear spins subject to stochastic fluctuations in their magnetic environments arising from transitions between in molecular chemical or conformational states. Chemical exchange provides insight into essential biomolecular dynamics associated with critical events, such as protein and nucleic acid folding, ligand binding, allostery, and catalytic turnover (1-3). Developments in theoretical and experimental understanding of chemical exchange phenomena aids in the characterization of these biological events by providing insight into the conformational dynamics of the underlying processes.

A variety of theoretical models have been developed to interpret chemical exchange (2). In each case, some underlying dynamical model is adopted that leads to time-dependent changes in the resonance frequency of the affected nuclear spin. The most popular models assume that dynamics consist of instantaneous jumps between two or more discrete states (4-6), although Schurr and co-workers considered a model with continuous Gaussian fluctuations in resonance frequency (7). While the frequency associated with each state is unchanged by jumps, the time series is interrupted, leading to a decorrelation in time. This type of discontinuous jump-model, sometimes referred to as a random-phase model or telegraph signal, captures long timescale decorrelation, but fails to account for shorter timescale dynamical fluctuations that may involve state-to-state transitions.

Chemical exchange leads to changes in resonance frequencies and relaxation rate constants of affected nuclear spins, and hence to NMR resonance lineshapes. In the simple two-state telegraph model, a single nuclear spin exchanges between states with different resonance frequencies ω₁ and ω₂, with Δω = ω₂ − ω₁, according to the kinetic scheme:

$$A_1\underset{k_{-1}}{\overset{k_1}{\rightleftharpoons }}{A}_2$$

in which k_ex = k₁ + k₁ is the sum of the forward and reverse kinetic rate constants, the equilibrium populations of the two states are p₁ = k₁/k_ex and p₂ = k₁/k_ex, and p₁ + p₂ = 1. Stochastic changes in resonance frequencies, arising from transitions between the two states, lead to dephasing of components of the spin magnetization perpendicular to the static magnetic field. This results in shifts in resonance frequencies and additional contributions to transverse relaxation (with rate constant R₂ = 1/T₂). The NMR spectrum resulting from the chemical exchange process can be calculated from the Bloch-McConnell equations (1). Illustrative spectra for p₁ = 0.8, p₂ = 0.2, and Δω = 1000 s⁻¹ for different values of k_ex are shown in Figure 1.

Figure 1.

NMR spectra for two-site chemical exchange. Parameters were p₁ = 0.8, p₂ = 0.2, ω₁ = −200 s⁻¹, ω₂ = 800 s⁻¹, and Δω = 1000 s⁻¹, yielding the average resonance frequency p₁ω₁ + p₂ω₂ = 0 for convenience. The values of k_ex are (dash-dotted line) 250 s⁻¹, (dashed line) 1000 s⁻¹, (solid line) 2500 s⁻¹, and (dotted line) 4000 s⁻¹, corresponding to slow, intermediate, fast, and very fast exchange on the chemical shift time scale, respectively. The inset shows a vertical expansion of the region from 250-1000 s⁻¹. The transverse relaxation rate constants obtained for the major peak in each spectrum by fitting a Lorentzian lineshape function over the region −500 to 300 s⁻¹ were 49.7, 113.8, 61.6, and 39.5 s⁻¹, respectively. Spectra were calculated from the Bloch-McConnell equations (1).

In the absence of chemical exchange, individual resonance lines are observed at the frequencies ω₁ and ω₂ with integrated intensities proportional to p₁ and p₂ (not shown). When chemical exchange is slow on the NMR chemical shift time scale, k_ex < Δω, resolved resonance lines are still observed. However, the linewidths are increased because the transitions between states increase transverse relaxation rate constants. If the site populations are unequal with p₁ > p₂ then k₋₁ > k₁ and the resonance line for the minor population is preferentially broadened. In intermediate exchange or coalescence, k_ex ≈ Δω, and a single very broad and shifted resonance line results, in many cases broadened to be unobservable in practice. In fast exchange, k_ex > Δω, and a single resonance line is observed at the population-averaged resonance frequency with a linewidth that becomes smaller as k_ex becomes even larger. The reduction in linewidth, or transverse relaxation rate constant, as k_ex increases in the fast-exchange limit is called motional narrowing (vide infra). As illustrated by this simple example, the dramatic effects of the chemical kinetic process on the NMR lineshapes, or equivalently resonance frequencies and transverse relaxation rate constants, is the basis for investigating chemical or conformational kinetic processes in biological macromolecules.

As an intrinsic physical phenomenon, chemical exchange as evidenced in NMR spectroscopy must comply with physical kinetic theories of chemical reactions. In order to broaden the interdisciplinary understanding of chemical exchange, the present tutorial reviews statistical mechanical theories of chemical kinetics relevant to the modeling and interpretation of the phenomenon of chemical exchange in NMR spectroscopy. By mapping the chemical exchange phenomenon onto the dynamics of a particle moving stochastically in a classical biphasic potential (8,9), reaction rate theory and models of barrier crossing dynamics can be used to explore the microscopic exchange dynamics and interpret chemical exchange in terms of the underlying chemical dynamics.

In this tutorial, we employ state-of-the-art theories for chemical dynamics to generalize jump-like models of chemical exchange to barrier crossing dynamics of a particle along a reaction coordinate. We demonstrate that the introduction of a continuously varying coordinate variable, as opposed to a set of discrete states, can capture not only the long-time decorrelation associated with barrier crossing, but also the shorter time scale transient dynamics, associated with fluctuations within potential energy wells. Stochastic evolution is described by the strong-collision model. This model assumes that the state of the system is randomized upon each collision, in accord with the equilibrium distribution. Collisions occurs at a rate α, leading to an exponential distribution of collision times, and the average change in energy due to a collision is large compared to k_BT. The resulting detailed models are shown to be consistent with the predictions of standard discrete-state jump models, while adding a higher degree of realism in the underlying dynamics. This tutorial serves to elucidate the underlying phenomenon of exchange by reducing the complexity of detailed computational simulations of proteins and other macromolecules to the essential dynamics that dictate the rate of chemical exchange (10,11).

The tutorial is organized as follows. Theoretical models of chemical exchange are developed by proposing a mapping between changes in coordinates and changes in resonance frequencies. The resulting time-dependent frequencies are used to evaluate time correlation functions associated with NMR observables. The predictions of the theoretical expressions are compared with the results of numerical simulations for the dynamics of a continuously varying coordinate in a biphasic double-well potential. The dynamics are explored for a variety of parameterizations of the potential wells and as a function of the rate of collisions. In an extension of these results for chemical exchange, the models are used to simulate Carr-Purcell-Meiboom-Gill (CPMG) relaxation dispersion decay (2,10). Finally, suggestions are made for generalizations of this work to more detailed models of chemical.

II. Theory

We consider, as a minimal model of chemical exchange dynamics, a single nuclear spin whose state is defined by a coordinate variable q(t) that depends on time. Changes in the variable q(t) capture changes in the environment of the nucleus, reflecting changes in the chemical or conformational state of the system. The associated resonance frequency of the nucleus, ω(t), is a function of the state of the system so that ω(t)=ω[q(t)]. Without loss of generality, the NMR radio frequency carrier can be assumed to be on-resonance with the ensemble average of <ω(t)>, so that <ω(t)> = 0 by construction. As a model of the exchange dynamics, we adopt a standard model of chemical reaction dynamics, involving transitions between two mechanically stable states in a double-well (biphasic) potential. In terms of this simple model, the dynamics of the resonance frequency of the nuclear spin ω[q(t)] involves fluctuations within and between two basins. The rate of transition between states is a function of barrier height, temperature, and dissipation.

In the simplest case, q(t) is a two-state telegraph signal taking on values 0 and 1, so that q(t) = 1 if the spin is in state 1 with resonance frequency ω₁ and q(t) = 0 if the spin is in the state 2 with resonance frequency ω₂. Thus, the instantaneous resonance frequency is expressed as ω(t) = q(t)ω₁ + (1 − q(t))ω₂. This is the two-state jump model commonly encountered in the theory of NMR spectroscopy and illustrated in Fig. 1 (1). In the two-state jump model, every collision is a strong collision in which the system loses memory of its prior state. The effect of transitions between q(t) = 1 and q(t) = 0, representing frequency jumps between ω₁ and ω₂, can be determined by a variety of mathematical approaches (1,2,5). In the following, we adopt an alternative approach commonly employed in reaction rate theory in which q(t) is a continuous rather than discrete variable. This allows exploration of a wider variety of models and enables a more detailed analysis of the underlying system dynamics.

II.A. Strong-collision dynamics in the discrete state jump-model

As an introduction to the approach utilized in the tutorial, a connection is established between strong-collision models for chemical dynamics and transverse spin relaxation arising from chemical exchange. The complex-valued time-domain NMR signal (omitting a number of constants of proportionality) for a single stochastic realization of the two-state jump process (also called the telegraph process) is given by (5,12):

$$\begin{gathered} s^{+}(t)=\text{exp}\left\{i\underset{0}{\overset{t}{\int }}[q(t^{\prime }){\omega }_1+(1-q(t^{\prime })){\omega }_2]d{t}^{\prime }\right\} \\ =\text{exp}\left\{-i\Delta \omega \underset{0}{\overset{t}{\int }}\left[q(t^{\prime })-p_1\right]d{t}^{\prime }\right\} \\ =q(t)\text{exp}\left\{-i{p}_2\Delta \omega t\right\}+[1-q(t)]\text{exp}\left\{i{p}_1\Delta \omega t\right\} \end{gathered}$$ (1)

in which s^±(t) = s_x(t) ± is_y(t) and s_x(t) and s_y(t) are the quadrature components of the signal. The second equality is obtained by noting that p_i = <q(t)> is the equilibrium population of the ith state and recalling that Δω = ω₂ − ω₁ and <ω(t)> = p₁ω₁ + p₂ω₂ = 0. The third equality is obtained in the absence of state changes, so that q(t) is either 0 or 1 for a given member of the ensemble. The observable NMR signal is the average over the ensemble of realizations of the stochastic process, and is denoted <s⁺>(t). The ensemble average of the third equality reduces to <s⁺>(t) = p₁exp(iω₁t) + p₂exp(iω₂t), as expected in the absence of state changes. The usual theoretical analysis of spin relaxation in NMR spectroscopy would proceed essentially by calculation of the ensemble average of the second line of Eq. 1 to obtain the ensemble average NMR signal, including exchange broadening effects (vide infra) (1). Instead, herein, the autocorrelation function of the NMR signal is calculated as:

$$C(\tau )=\langle s^{+}(t)s^{-}(t+\tau )\rangle =\underset{T\to \infty }{\text{lim}}\frac{1}{T}\underset{0}{\overset{T}{\int }}{s}^{+}(t)s^{-}(t+\tau )dt$$ (2)

in which the second equality assumes the system dynamics are stationary and ergodic. The autocorrelation function of the NMR signal in the absence of state changes then is given by:

$$\begin{gathered} C(\tau )=\langle \left(q(t)e^{-i{p}_2\Delta \omega t}+[1-q(t)]e^{i{p}_1\Delta \omega t}\right)\phantom{\rangle } \\ \phantom{\langle }\times \left(q(t+\tau )e^{i{p}_2\Delta \omega (t+\tau )}+[1-q(t+\tau )]e^{-i{p}_1\Delta \omega (t+\tau )}\right)\rangle \\ =p_1{e}^{i{p}_2\Delta \omega \tau }+p_2{e}^{-i{p}_1\Delta \omega \tau } \end{gathered}$$ (3)

in which the third line of Eq. 1 has been substituted into Eq. 3. The final result in Eq. 3 is obtained by noting that q(t)² = q(t) and that in the absence of state changes, q(t) = q(t+τ). The Laplace transform of the resulting autocorrelation function is:

$$C(s)=\underset{0}{\overset{\infty }{\int }}C(\tau )e^{-s\tau }d\tau =\frac{p_1}{s-i{p}_2\Delta \omega }+\frac{p_2}{s+i{p}_1\Delta \omega }$$ (4)

To proceed, q(t) is assumed to be time-dependent because strong collisions cause state-to-state transitions. A theorem from statistical mechanics states that the Laplace transform of the autocorrelation function of a dynamical system in the presence of strong collisions $\tilde{C}(s)$ is a function of the autocorrelation function in the absence of collisions C(s) (8,9):

$$\tilde{C}(s)=\frac{C(s+\alpha )}{1-\alpha C(s+\alpha )}$$ (5)

in which α is the collision rate. Substituting Eq. 4 into Eq. 5 gives:

$$\tilde{C}(s)=\frac{s+\alpha -i\Delta \omega \Delta p}{s^2+\{\alpha -i\Delta \omega \Delta p\}s+p_1{p}_2\Delta {\omega }^2}$$ (6)

in which Δp = p₂ − p₁. Equation 6 is the Laplace transform of the two-state jump process in the strong-collision model. The inverse Laplace transform of Eq. 6 yields the corresponding autocorrelation function in the time domain:

$$\begin{gathered} \tilde{C}(t)=e^{-(\alpha -i\Delta \omega \Delta p)t∕2}\{\cosh \left(\frac{t}{2}\sqrt{{\alpha }^2-\Delta {\omega }^2-2i\alpha \Delta \omega \Delta p}\right)\phantom{\}} \\ +\phantom{\{}\frac{\alpha -i\Delta \omega \Delta p}{\sqrt{{\alpha }^2-\Delta {\omega }^2-2i\alpha \Delta \omega \Delta p}}\sinh \left(\frac{t}{2}\sqrt{{\alpha }^2-\Delta {\omega }^2-2i\alpha \Delta \omega \Delta p}\right)\} \end{gathered}$$ (7)

This equation is exact, but is also complicated. Fortunately, useful limiting results can be obtained as described below.

When transitions between states are sufficiently fast, the transverse relaxation rate constant is given by the decay of the autocorrelation function at long times (after any initial transients have decayed to zero) (4,6). This corresponds to the limit s → 0 in the Laplace domain. The usefulness of the analysis in the Laplace domain is that the long-time behavior is inferred from the small s limit, allowing relatively simple approximations to be applied.

To begin, Eq. 6 can be expressed as:

$$\tilde{C}(s)=\frac{1}{s+\frac{p_1{p}_2\Delta {\omega }^2}{s+\alpha -i\Delta \omega \Delta p}}=\frac{1}{s+D(s)}$$ (8)

D(s) can be expanded in a Taylor’s series as:

$$D(s)\approx \frac{p_1{p}_2\Delta {\omega }^2}{\alpha -i\Delta \omega \Delta p}-s\frac{p_1{p}_2\Delta {\omega }^2}{{\left(\alpha -i\Delta \omega \Delta p\right)}^2}+\dots$$ (9)

Keeping only the first two terms of the series, Eq. 8 becomes:

$$\begin{gathered} \tilde{C}(s)=\frac{1}{s\left(1-\frac{p_1{p}_2\Delta {\omega }^2}{{\left(\alpha -i\Delta \omega \Delta p\right)}^2}\right)+\frac{p_1{p}_2\Delta {\omega }^2}{\alpha -i\Delta \omega \Delta p}} \\ ={\left(1-\frac{p_1{p}_2\Delta {\omega }^2}{{\left(\alpha -i\Delta \omega \Delta p\right)}^2}\right)}^{-1}\frac{1}{s{\left(1-\frac{p_1{p}_2\Delta {\omega }^2}{(\alpha -i\Delta \omega \Delta p{)}^2}\right)}^{-1}\frac{p_1{p}_2\Delta {\omega }^2}{\alpha -i\Delta \omega \Delta p}} \end{gathered}$$ (10)

This equation is isomorphous to a result derived from the Bloch-McConnell equations by Abergel and Palmer (Eq. 23 in (13)). The last line of Eq. 10 is the Laplace transform of $\tilde{C}(t)=A{e}^{i\varphi }\text{exp}\left[(i{\Omega }_{sc}-R_{sc})t\right]$ in which:

$$\begin{gathered} A=\mid {\left(1-\frac{p_1{p}_2\Delta {\omega }^2}{(\alpha -i\Delta \omega \Delta p{)}^2}\right)}^{-1}\mid \\ \varphi =\text{Arg}\left[{\left(1-\frac{p_1{p}_2\Delta {\omega }^2}{(\alpha -i\Delta \omega \Delta p{)}^2}\right)}^{-1}\right] \\ {\Omega }_{sc}=\text{Im}\left[{\left(1-\frac{p_1{p}_2\Delta {\omega }^2}{(\alpha -i\Delta \omega \Delta p{)}^2}\right)}^{-1}\frac{p_1{p}_2\Delta {\omega }^2}{\alpha -i\Delta \omega \Delta p}\right] \\ {R}_{sc}=\text{Re}\left[{\left(1-\frac{p_1{p}_2\Delta {\omega }^2}{(\alpha -i\Delta \omega \Delta p{)}^2}\right)}^{-1}\frac{p_1{p}_2\Delta {\omega }^2}{\alpha -i\Delta \omega \Delta p}\right] \end{gathered}$$ (12)

A and ϕ are the amplitude and phase of the long-time component of the autocorrelation function, Ω_sc is the resonance frequency, and R_sc is the transverse relaxation rate constant in the two-state strong-collision model. Following the procedure of Abergel and Palmer yields an explicit form for R_sc (Eq. 31 in (13)):

$$R_{sc}=p_1{p}_2\Delta {\omega }^2\alpha \left[\frac{{\alpha }^2+\Delta {\omega }^2}{{\left({\alpha }^2+\Delta {\omega }^2\right)}^2-p_1{p}_2\left(5{\alpha }^2+\Delta {\omega }^2\right)\Delta {\omega }^2}\right]$$ (13)

(a related expression for Ω_sc is given by Eq. 32 in (13), but not needed in this tutorial).

If collisions are in the fast limit on the NMR chemical shift time scale, α >> Δω, then the bracketed term in Eq. 13 approaches α⁻² and:

$$R_{sc}=p_1{p}_2\Delta {\omega }^2∕\alpha$$ (14)

Equation 14 also can be obtained in straightforward fashion by retaining only the first term in the series in Eq. 9 and assuming α >> Δω. These results demonstrate that the autocorrelation function of the NMR signal in the fast-limit strong-collision two-state telegraph model is a single exponential decay with rate constant R_sc. For comparison, either the random-phase model or Redfield (BWR) theory applied to the ensemble average of the second line of Eq. 1 give the fast-limit expression for R_BWR as the integral of the autocorrelation function for the resonance frequency fluctuations C(τ) = <δω(t) δω(t + τ)> (1,5), in which δω(t) = ω(t) − <ω(t)> is the instantaneous resonance frequency fluctuation. For the two-state telegraph process, C(τ) = p₁p₂^Δω2 exp(−k_exτ), in which δω(t) = −Δω[q(t) − p₁], and C(0) = ⟨δω²(t)⟩ = p₁p₂Δω², yielding:

$$R_{BWR}=\underset{0}{\overset{\infty }{\int }}\langle \delta \omega (t)\delta \omega (t+\tau )\rangle dt=p_1{p}_2\Delta {\omega }^2∕k_{ex}$$ (15)

Equations 14 and 15 identify the collision rate α as being equal to the exchange rate k_ex. This casts the strong-collision relaxation rate constant in the familiar form used in NMR spectroscopy. In statistical mechanics, a strong collision imparts variation in particle energy in the order of k_BT. In contrast, in NMR spectroscopy, only collisions that lead to changes in resonance frequency contribute to relaxation properties of the affected nuclear spin. That is, only the fraction of “collisions” that lead to transitions between states with different resonance frequencies are meaningful for spin relaxation. For the two-state telegraph model, k_ex is the only non-zero eigenvalue of the kinetic transition matrix and consequently the only rate constant appearing in Eq. 15, and by extension, Eqs. 13 and 14. Equations 14 and 15 also provide an important identity between the (slow) long-time decay of the NMR signal and the autocorrelation function of the (fast) resonance frequency fluctuations in the fast (or BWR) limit.

Figure 2 compares the results for the strong-collision autocorrelation function calculated from Eq. 7 and the single-exponential approximations with rate constants given by Eqs. 13 and 14. When α < Δω, the imaginary component of the exact autocorrelation function is large and the fast-limit (or BWR) approximation of Eq. 14 is not accurate (Fig. 2a and 2b). As α increases so that α > Δω, the imaginary component of the autocorrelation function decreases in magnitude and the fast-limit approximation becomes increasingly accurate (Fig. 2c and 2d). When α > 4Δω, the exact autocorrelation function becomes nearly real and the fast-limit approximation is nearly exact (Fig. 2d). The extended formula for R_sc in Eq. 13 is accurate for intermediate (Fig. 2b) and slow exchange for times t longer than approximately (4-5)/α, after the initial fast decay of the autocorrelation function (Fig. 2a), as well as fast exchange (Figs. 2c and 2d). For the given illustrative parameters, Eq. 13 underestimates the relaxation rate constant obtained from lineshape fitting by a maximum of 7.5% for intermediate exchange and becomes highly accurate in the fast-exchange regime (see captions to Figs. 1 and 2). The remainder of the tutorial focuses for simplicity on the fast-limit regime for simplicity, but as indicated by Eq. 13 and the results in Fig. 2, the approaches developed are more generally applicable outside this limit.

Figure 2.

Comparison of exact autocorrelation functions for the strong-collision model and single-exponential approximations. Parameters were p₁ = 0.8, p₂ = 0.2, Δω = 1000 s⁻¹ and α varied between (a) 250 s⁻¹, (b) 1000 s⁻¹, (c) 2500 s⁻¹, and (d) 4000 s⁻¹ (identical as for Fig. 1). Plots show the magnitudes of the (solid, black lines) and (dotted, black lines) imaginary components of the exact complex autocorrelation function calculated from Eq. 7; (dashed, reddish-purple lines) the fast-limit single-exponential approximation of Eq. 14; and (dash-dot-dotted, orange lines) the single-exponential approximation of Eq. 13 with amplitude A given by Eq. 12. The relaxation rate constants calculated from Eq. 13 are 46.2, 105.3, 61.2, and 39.4, respectively (Fig. 2a-d) and are 7.0%, 7.5%, 0.6%, and 0.1% smaller than the values obtained from fitting the corresponding NMR spectra (Fig. 1).

Additional details regarding the Laplace transform approach used to determine theoretical expressions for NMR spin relaxation rate constants are provided elsewhere (4-6,13). A similar strong-collision model has been applied to model the free-induction decay, corresponding to <s⁺>(t), in NMR spectroscopy by Goldman (12,14).

II.B. Strong-collision dynamics in the continuous reaction coordinate model

In the two-state jump-model discussed above, the states of the system are discrete and the coordinate variable q(t) is restricted to values of 0 and 1. This is essentially a square-well model, in which each well is associated with a unique resonance frequency and the transit time across the barrier between wells is short compared to residence times within the wells. A more realistic model of the chemical dynamics employs a continuous positional variable q(t) moving in a double-well (biphasic) potential. The continuous dependence of the variable q(t) provides a more realistic model for classical reaction dynamics. We use the double-well potential function (8):

$$V(q)=\phantom{\{}\left\{\begin{array}{cc}\frac{1}{2}m{\omega }_A^2(q+q_A{)}^2\hfill & q<-a\hfill \\ Q-\frac{1}{2}m{\omega }_1^2{q}^2\hfill & -a\le q<b\hfill \\ V_B+\frac{1}{2}m{\omega }_B^2(q-q_B{)}^2\hfill & q\ge b\hfill \end{array}\phantom{\}}\right\}$$ (16)

in which Q is the energy barrier height between wells, V_B is the difference in energy between the well minima, q_A and q_B are the positions of the well minima, m is the particle mass, and ω₁ is the barrier transition frequency in units of time⁻¹ (note that the frequency parameters here are constants and are distinct from time-dependent NMR resonance frequency ω(t)). These parameters are specified to set the shape of the potential and the definitions:

$$\begin{gathered} a=2Q∕(m{\omega }_1^2{q}_A) \\ b=2(Q_{AB}-V_B)∕(m{\omega }_1^2{q}_B) \\ {\omega }_A={\omega }_1\sqrt{a∕(q_A-a)} \\ {\omega }_B={\omega }_1\sqrt{b∕(q_B-b)} \end{gathered}$$ (17)

ensure that V(q) and dV(q)/dq are continuous. The potential can be recast in dimensionless variables:

$$\widehat{V}(\widehat{q})=\phantom{\{}\left\{\begin{array}{cc}\frac{1}{2}{\widehat{\omega }}_A^2(\widehat{q}+{\widehat{q}}_A{)}^2\hfill & \widehat{q}<-\widehat{a}\hfill \\ \widehat{Q}-\frac{1}{2}{\widehat{q}}^2\hfill & -\widehat{a}\le \widehat{q}<\widehat{b}\hfill \\ {\widehat{V}}_B+\frac{1}{2}{\widehat{\omega }}_B^2(\widehat{q}-{\widehat{q}}_B{)}^2\hfill & \widehat{q}\ge \widehat{b}\hfill \end{array}\phantom{\}}\right\}$$ (18)

in which energy variables ${\widehat{V}}_B=V_B∕(k_{B}T)$ and $\widehat{Q}=Q∕(k_{B}T)$ are in units of k_BT, length variables are redefined as $\widehat{x}=[m{\omega }_1^2∕(k_{B}T){]}^{1∕2}$ for x = {q, q_A, q_B, a, b}, and frequency variables are redefined as ${\widehat{\omega }}_{\gamma }={\omega }_{\gamma }∕{\omega }_1$ for γ = {1, A, B}. Dimensionless time variables are defined as $\widehat{t}={\omega }_{1}t$ and the dimensionless collision rate is $\widehat{\alpha }=\alpha ∕{\omega }_1$ (vide infra). For convenience, the circumflex will be omitted in the following discussion, but all variables should be regarded as dimensionless, unless otherwise noted. Examples of the potential are shown in Figure 3. In the following, populations of site 1 and site 2 were obtained by integrating the canonical ensemble probability assuming q < 0 corresponds to site 1 and q > 0 corresponds to site 2.

Figure 3.

Biphasic potential calculated from Eqs. 17 and 18. (solid) Parameters are Q = 7, V_B = 2, and ω_A = ω_B = 0.5 yield q_A = 8.37, q_B = 7.07, a = 1.67, and b = 1.41. (dashed) Parameters Q = 7, V_B = 2, ω_A = 0.5, and ω_B = 0.25 yield q_A = 8.37, q_B = 13.04, a = 1.67, and b = 0.77. (dotted) Chemical shift profile ω(q) for the random-coil model adapted to (dashed) asymmetric potential. The Boltzmann-weighted average <ω(q)> = 0 for q > 0 All parameters are dimensionless as described in the text.

III. Methods

The stepwise procedure followed in modeling strong collision dynamics is provided alongside the algorithm used to conduct the simulations. The mapping of the simple dynamical model to actual physical systems is discussed, including the particular example of protein folding. The use of the dynamical model to describe signal decay in a CPMG experiment is discussed.

III.A. Simulations for strong collisions

The above theoretical discussion suggests a simple strategy for simulating chemical exchange dynamics using the biphasic potential model:

1. Generate a simulation of q(t) = q(t_n) = q(nΔt) = q_n for a particle in the biphasic potential in the absence of collisions by solving Newton’s equations of motion for a series of time steps n = {0,…, n_max − 1} with duration Δt. In the present work, the velocity Verlet algorithm was used for propagating q(t) and the velocity v(t).

2. Add strong collisions that occur in time with an exponential probability distribution. Each collision randomly reassigns the particle velocity from a Gaussian distribution centered at zero with variance <v²> = k_BT/m (= 1 in dimensionless units).

3. Map from q(t) to ω[q(t) = ω(t).

4. Calculate the autocorrelation function of δω(t) given by <δω(t)δω(t+τ)>.

5. Calculate the autocorrelation function for s^±(t) using Eq. 2 and the definition:

   $$s^{\pm }(t)=s^{\pm }(n\Delta t)=\text{exp}\left(\pm i\Delta t\sum _{k=0}^n{\omega }_k\right)$$

6. Calculate the ensemble-average NMR signal <s⁺>(t) by averaging s⁺(t) over multiple independent trajectories, q(t).

The finite-difference equations of motion for the coordinate variable q(t) and the velocity v(t) are defined by the velocity Verlet algorithm (in dimensionless units):

$$\begin{gathered} q(t+\Delta t)=q(t)+v(t)\Delta t+\frac{F(t)}{2}\Delta t^2 \\ v(t+\Delta t)=v(t)+\frac{F(t+\Delta t)+F(t)}{2}\Delta t \end{gathered}$$ (19)

in which the dimensionless instantaneous acceleration of a particle at a particular position q is given by:

$$F(q)=-\frac{dV(q)}{dq}=\phantom{\{}\left\{\begin{array}{cc}-{\omega }_A^2(q-q_A)\hfill & q<-a\hfill \\ q\hfill & -a\le q<b\hfill \\ -{\omega }_B^2(q-q_B)\hfill & q\ge b\hfill \end{array}\phantom{\}}\right\}$$ (20)

Python code illustrating the simulation algorithm is shown in Box 1. Autocorrelation functions can be calculated by approximations of the integral in Eq. 2 through a summation (substituting δω(t) for s⁺(t) as needed) or more efficiently by Fourier transformation.

Box 1. Algorithm for strong collisions.

III.B. Mapping generalized coordinates to resonance frequencies

The mapping of the coordinate variable q(t) to the resonance frequency ω(t) was performed in two ways. In the first approach, a resonance frequency of 1 was assigned if q < 0 and a resonance frequency of 0 was assigned if q ≥ 0. The values of the resonance frequencies were then shifted and scaled so that <ω(t)> = 0 and <ω(t)²>^1/2 τ_c = 0.1 in which τ_c is the long-time decay time for the autocorrelation function of q(t). The latter constraint ensures that the NMR fast-exchange limit is reached. This mapping reduces the simulated model to the two-state telegraph jump process analyzed theoretically above.

In the second approach, the chemical shifts were modeled initially using ω(q) = −erf(q − q₀) in which erf is the error function. The value of q₀ is chosen so that ω(q) ≈ 1 for q < 0 and ω(q) varies between +1 and −1 with <ω(q)> ≈ 0 for q ≥ 0, as illustrated in Fig. 3. This function was then shifted and scaled so that <ω(t)> = 0 and <ω(t)²>^1/2 τ_c = 0.1, as for the first approach. This second approach is a simple model of a major state 1 with a fixed resonance frequency (such as would occur in a folded state of a protein) and a minor state 2 in which local conformational fluctuations drive concomitant time-dependent variations in resonance frequencies. This mapping will be called the random-coil model below.

To illustrate the above models, a ¹³Cα spin has a secondary chemical shift, Δδ, of 2-4 ppm in an α-helical conformation of a protein. The secondary shift is defined as the measured chemical shift minus the chemical shift expected for an unstructured (random-coil) peptide. Thus, the telegraph model assigns a single frequency equal to Δδ in the folded state and a single frequency equal to 0 in the unfolded state. The random-coil model also assigns a single frequency equal to Δδ in the folded state. However, the unfolded state is described by a fluctuating distribution of frequencies from −Δδ to + Δδ, with an average of 0, representing in a simple manner the averaging of chemical shifts over the distributions of conformations sampled by a disordered peptide.

III.C. Carr-Purcell-Meiboom-Gill relaxation dispersion

Carr-Purcell-Meiboom-Gill (CPMG) and R_1ρ relaxation dispersion experiments have developed as powerful approaches for investigation of μs-ms time-scale dynamic processes in proteins and other macromolecules (2). In the CPMG experiment, relaxation is measured during a train of spin-echo sequences: (τ_op − 180° − τ_op)_n, in which “180°” is a radiofrequency refocusing pulse, τ_op is the spin-echo delay time, the total relaxation delay period is T = 2nτ_op, and n is the number of spin-echo units applied. In the ideal case, each 180° pulse merely acts to invert the sign of the evolution frequencies. As such, in the above protocols, the effect of a CPMG pulse train is modeled by multiplying ω(t) by a square wave varying between +1 and −1 with 50% duty cycle and period 4τ_op, prior to further analysis of ω(t). The autocorrelation function of s⁺(t) is calculated as above, or an ensemble average < s⁺>(t) is estimated by averaging multiple stochastic trajectories, to mimic the signal decay in an actual CPMG experiment.

The variation in the observed relaxation rate as a function of the pulse delay τ_op is called a CPMG relaxation dispersion curve. In the fast-exchange limit, analysis of the relaxation dispersion curve for a two-state model yields:

$$R_{CPMG}({\tau }_{cp})=p_1{p}_2\Delta {\omega }^2{\tau }_{ex}\left[1-\frac{{\tau }_{ex}}{{\tau }_{cp}}\tanh \left(\frac{{\tau }_{cp}}{{\tau }_{ex}}\right)\right]$$ (21)

in which the exchange time τ_ex = 1/k_ex. More complex expressions for R_CPMG(τ_op) valid for all chemical exchange time scales have been reported and widely used in the analysis of experimental data (2).

IV. Results

Strong-collision simulations were performed using in-house Python 3.6 or Fortran 77 programs; equivalent results were obtained with either programming language (but the Fortran 77 routine is much faster). Numerical and graphical analyses of simulation trajectories were performed using in-house Python 3.6 programs. The Python and Fortran programs used in this tutorial are provided as Supplementary Information.

IV.A. Strong-collision dynamics in the biphasic potential

Initial simulations were performed using parameters Q = 7, V_B =2, and ω_A = ω_B = 0.5 (solid curve in Fig. 3). The site populations are p₁ = 0.88 and p₂ = 0.12. Simulations were performed using values of collision rate α ranging between 10⁻³ to 10². Dynamics were underdamped for α < 0.3 and overdamped otherwise. The simulated time series for the coordinate variable q(t) and the autocorrelation function of q(t) for the simulation with α = 2.5 are shown in Figure 4.

Figure 4.

Simulation for the strong-collision dynamics of the coordinate q(t) in a biphasic double-well potential with Q = 7, V_B = 2, and equal well frequencies ω_A = ω_B = 0.5. (solid trace in Fig. 2). The simulation used a time step of 0.01 and consisted of 2³⁴ steps with a collision rate of α = 2.5; q(t) was stored every 2¹⁰ steps. (a) q(t) at every 1000 stored sample point during the simulation. (b) (black) Autocorrelation function C(τ) of q(t) fit with a (reddish-purple, dashed line) mono- and (blue, dotted line) bi-exponential decay function; 10 replicate simulations were averaged to produce the final autocorrelation function. The inset shows the fast initial decay of the autocorrelation function, which is well-described by the bi-exponential fit. The fitted parameters are amplitudes a₁ = 3.9 and a₂ = 25.2 and decay times τ₁ = 9.4 and τ₂ = 5110, respectively. All parameters are dimensionless as described in the text. The simulated dynamics in the potential are shown in SI movie 1.

In contrast to the two-state telegraph model, the coordinate varies continuously over a range of q. The particle is found to spend the majority of the time in the left well, which has coordinate q < 0 and has lower potential energy, making occasional transitions to the right well, which has coordinate q > 0. Thus, the particle has longer residence times in the left well when compared with the right well. The autocorrelation function is bi-exponential, with a short time constant, τ₁ = 9.4, arising from fluctuations within the potential energy wells and a longer time constant, τ₂ = 5110, arising from transitions between wells.

The simulation strategy was validated by comparing the rate constants obtained from the long-time decay of the autocorrelation function to rate constants for passage over the barrier obtained from simulations of the reactive flux (15) and from theoretical estimates of the transition rate (16). In these approaches, k = κk_TST, in which k is the sum of the forward and backward barrier-crossing rate constants, k_TST is transition-state theory estimate of the rate constant, and κ is the transmission coefficient. For compactness, the theoretical calculations are outlined in the Appendix.

For the parameters given in Fig. 4, k_TST = 6.09 × 10⁻⁴ and κ = 0.33; the transition rate is lower than that predicted by transition state theory due to slow spatial diffusion over the barrier. The resulting time constant τ = 1/k = 4980 agrees well with the fitted long-time decay of the autocorrelation function, τ₂ = 5110. Figure 5 shows a more extensive comparison between the transmission coefficients obtained from theoretical estimates of the rate constant, the reactive flux method, and the long-time decay of the simulated autocorrelation function.

Figure 5.

Barrier-crossing transmission coefficients for the strong-collision model. The simulations and calculations used Q = 7, V_B = 2, ω_A = ω_B = 0.5. (solid circles) Simulated transmission coefficients are obtained as 1/(τk_TST) in which τ is the long-time decay constant of the simulated autocorrelation function. (open circles) Simulated transmission coefficients are obtained from the reactive flux method (15). The solid curve shows the theoretical result and dashed lines show the limiting κ_EB and κ_SD transmission coefficients for the strong-collision model, as described in the Appendix.

The variation in the transmission coefficient displays the classic Kramers turnover. At low collision frequencies, the exchange rate increases in proportion to the collision frequency, while at high collision frequencies, the exchange rate decreases in proportion to the reciprocal of the collision frequency. At intermediate collision frequencies, the rate is a maximum and the transition state theory provides a reasonable estimate of the exchange rate. Segments of the simulations for the strong-collision model with α = 0.04 and 2.5 are animated in Supplementary Movies 1 and 2; these parameters correspond to the energy dissipation (ED) and spatial diffusion (SD) limits of the Kramers reaction rate theory discussed in the Appendix.

IV.B. Comparison of continuous and discrete state frequency mappings

The telegraph signal ω(t) resulting from the mapping of the coordinate variable q(t) to the frequency is shown in Figure 6a; the autocorrelation function is shown in Figure 6b. The telegraph signal only captures transitions between potential wells and consequently the autocorrelation function is mono-exponential. The long time constant from the autocorrelation function of q(t) and from the telegraph signal, τ_c = 5110, agree well as expected. The spin relaxation rate constant is obtained, using Eq. 15, as the integral of the autocorrelation function; from the mono-exponential fitting parameters, R_BWR = <δω(t)²>τ_c = 1.96 × 10⁻⁶. The real part of the complex NMR signal is shown in Figure 6c; the corresponding autocorrelation function is shown in Figure 6d. The relaxation rate constant obtained as the reciprocal of the decay time constant of the autocorrelation function is R_sc = 1.90 × 10⁻⁶, in good agreement with R_BWR. The agreement between these two results is consistent with the theoretical results above. Empirically, the simulated autocorrelation function of the frequency fluctuations converges more rapidly than the simulated ensemble average NMR signal, as evident in the residual noise in Fig. 6d. Thus, obtaining the relaxation rate constant from the autocorrelation function of the frequency fluctuations is more efficient in practice if the stochastic process is in the fast (BWR) limit.

Figure 6.

Two-state telegraph model for dynamics in the biphasic potential with equal well frequencies. (a) Sample of ω(t) at times during the simulation with parameters chosen so <δω(t)²>^1/2 = 1.96 × 10⁻⁵ (b) (black) Autocorrelation of ω(t) fit with a (reddish-purple, dashed line) mono-exponential function with amplitude <δω(t)²> = 3.83 × 10⁻¹⁰ and decay time τ_c = 5110; <δω(t)²>^1/2τ_c = 0.1 and an estimated value of R_BWR = <δω(t)²>τ_c = 1.96 × 10⁻⁶. The inset shows only a mono-exponential decay. (c) Sample of Re[s⁺(t)] at times during the simulation. (d) (black) Real part of the autocorrelation of s⁺(t) fit with a (reddish-purple, dashed line) mono-exponential function with initial amplitude fixed at 1.0 and decay time constant of 5.27 × 10⁵, yielding R_sc = 1.90 × 10⁻⁶ in good agreement with R_BWR. All parameters are dimensionless as described in the text.

A similar analysis was performed using the random-coil model (vide supra). The results are shown in Figure 7. The function ω(t), obtained from q(t), is shown in Figure 7a; the autocorrelation function is shown in Figure 7b. The autocorrelation function is bi-exponential, reflecting the fact that the model includes both fast fluctuations within potential wells and slower transitions between wells. The spin relaxation rate constant is obtained, using Eq. 15, as the integral of the autocorrelation function; for a bi-exponential autocorrelation function, R_BWR = a₁τ₁ + a₂τ₂, in which a_i and τ_i are the amplitude and time constant for the ith exponential term. From the bi-exponential fitting parameters, R_BWR = 1.10 × 10⁻⁶; the reduced relaxation rate constant, compared to the telegraph model, reflects the effect of the short time constant from the fast fluctuations with the potential well. The real part of the complex NMR signal is shown in Figure 7c; the corresponding autocorrelation function is shown in Figure 7d. The relaxation rate constant obtained as the reciprocal of the decay time constant of the autocorrelation function is R_sc = 1.04 × 10⁻⁶, in good agreement with R_BWR. Note that the autocorrelation of the NMR signal in Fig. 7d is mono-exponential even when the autocorrelation function of ω(t) in Fig. 7b is bi-exponential in the fast exchange limit as described above and illustrated in Fig. 2.

Figure 7.

Two-state random-coil model for the biphasic potential with equal well frequencies. (a) Sample of ω(t) at times during the simulation with parameters chosen so <ω(t)²>^1/2 = 1.96 × 10⁻⁵. (b) (black) Autocorrelation of ω(t) fit with (reddish-purple, dashed line) mono- and (blue, dotted line) bi-exponential functions. The inset shows the fast initial decay of the autocorrelation function, which is well-described by the bi-exponential fit. The fitted parameters are amplitudes a₁ = 1.67 × 10⁻¹⁰ and a₂ = 2.15 × 10⁻¹⁰ and decay times τ₁ = 8.5 and τ₂ = 5110; <δω(t)²>^1/2τ₂ = 0.1 and an estimated value of R_BWR = 1.10 × 10⁻⁶. (c) Sample of Re[s⁺(t)] times during the simulation. (d) (black) Real part of the autocorrelation of s⁺(t) fit with a (reddish-purple, dashed line) mono-exponential function with initial amplitude fixed at 1.0 and decay time constant of 9.60 × 10⁵, yielding R_sc = 1.04 × 10⁻⁶ in good agreement with R_BWR. All parameters are dimensionless as described in the text.

A second simulation was performed for the parameters given in Fig. 3 for the dashed curve: Q = 7, V_B = 2, ω_A = 0.5, ω_B = 0.25, and α = 2.5, giving k_TST = 3.41 × 10⁻⁴ and κ = 0.33; the resulting long-time constant τ = 1/k = 8890. Similar plots as for the above model for the case of equal well frequencies are shown in Figures 8, 9, and 10. The site populations are now p₁ = 0.79 and p₂ = 0.21. The long-time decay constant of the autocorrelation function for q(t), shown in Fig. 8, is now τ₂ = 10440 in agreement with the theoretical result. Similar levels of agreement are found between R_sc and R_BWR for (Fig. 9) telegraph and (Fig. 10) random-coil models for the resonance frequencies. Notably, the autocorrelation function for the frequency fluctuations ω(t) in the random-coil model (Fig. 10b) is distinctly bi-exponential, reflecting the effects of chemical shift averaging in the broader potential well for state 2. Again, the autocorrelation of the NMR signal in Fig. 10d is mono-exponential even when the autocorrelation function of ω(t) is bi-exponential in the fast exchange limit.

Figure 8.

Strong-collision model simulation of q(t) for the biphasic potential with unequal well frequencies. Parameters were Q = 7, V_B = 2, ω_A = 0.5, and ω_B = 0.25 (dashed trace in Fig. 2). The simulation used a time step of 0.01 and consisted of 2³⁴ steps with a collision rate of α = 2.5; q(t) was stored every 2¹⁰ steps. (a) q(t) at every 1000 stored sample point during the simulation. (b) (black, solid) Autocorrelation function of q(t) fit with a (reddish-purple, dashed line) mono- and (blue, dotted line) bi-exponential decay function; 10 replicate simulations were averaged to produce the final autocorrelation function. The inset shows the fast initial decay of the autocorrelation function, which is well-described by the bi-exponential fit. The fitted parameters are amplitudes a₁, = 6.0 and a₂ = 77.1 and decay times τ₁ = 30.7 and τ₂ = 10440. All parameters are dimensionless as described in the text. The simulated dynamics in the potential are shown in SI movie 2.

Figure 9.

Two-state telegraph model for the biphasic potential with unequal well frequencies. (a) Sample of ω(t) at times during the simulation with parameters chosen so <δω(t)²>^1/2 = 9.60 × 10⁻⁶. (b) (black, solid) Autocorrelation of ω(t) fit with a (reddish-purple, dashed line) mono-exponential function with amplitude <δω(t)²> = 9.17 × 10⁻¹¹ and decay time τ_c = 10440; <δω(t)²>^1/2τ_c = 0.1 and an estimated value of R_BWR = <δω(t)²>τ_c = 9.58 × 10⁻⁷. The inset shows only the mono-exponential decay. (c) Sample of Re[s⁺(t)] at times during the simulation. (d) (black, solid) Real part of the autocorrelation of s⁺(t) fit with a (reddish-purple, dashed line) mono-exponential function with initial amplitude fixed at 1.0 and decay time constant of 1.13 × 10⁶, yielding R_sc = 8.83 × 10⁻⁷ in good agreement with R_BWR. All parameters are dimensionless as described in the text.

Figure 10.

Two-state random-coil model for the biphasic potential with unequal well frequencies. (a) Sample of ω(t) at times during the simulation with parameters chosen so <ω(t)²>^1/2 = 9.60 × 10⁻⁶. (b) (black, solid) Autocorrelation of ω(t) fit with (reddish-purple, dashed line) mono- and (blue, dotted line) bi-exponential functions. The inset shows the fast initial decay of the autocorrelation function, which is well-described by the bi-exponential fit. The fitted parameters are amplitudes a₁ = 4.52 × 10⁻¹¹ and a₂ = 4.49 × 10⁻¹¹ and decay times τ₁ = 32.0 and τ₂ = 10420; <δω(t)²>^1/2τ₂ = 0.1 and an estimated value of R_BWR = 4.69 × 10⁻⁷. (c) Sample of Re[s⁺(t)] at times during the simulation. (d) (black) Real part of the autocorrelation of s⁺(t) fit with a (reddish-purple, dashed line) mono-exponential function with initial amplitude fixed at 1.0 and decay time constant of 2.47 × 10⁶, yielding R_sc = 4.05 × 10⁻⁷ in good agreement with R_BWR All parameters are dimensionless as described in the text.

In simulations of both model potentials using the random-coil model for ω(t), the effect of averaging the chemical shift within the rightmost well (with higher minimum potential energy and consequently reduced population compared to the leftmost well) reduces the transverse relaxation rate constant by approximately a factor of two. The result that fast time scale fluctuations of the resonance frequencies within the second well reduce, rather than increase, the transverse relaxation rate constant is a characteristic feature of the fast-exchange regime in NMR spectroscopy. As the rates of frequency fluctuations become much larger than the range of frequencies sampled, the NMR transverse relaxation rate constant approaches 0, a process called motional narrowing (1).

IV.C. CPMG relaxation dispersion for strong-collision dynamics

CPMG relaxation dispersion was estimated using the two-state telegraph mapping of resonance frequencies applied to simulations using the strong-collision model in the asymmetric biphasic double-well potential (the dashed line in Fig. 3) and τ_cp = 1.05 × 10⁴ and 2.62 × 10³. The ensemble signal <s⁺(t)> and the autocorrelation function of s⁺(t) are shown in Figure 11. The results show the effect of the CPMG pulse train as τ_cp is reduced. In the absence of the CPMG pulse train, the decay rate is ~ 9 × 10⁻⁷ and drops to ~ 2 × 10⁻⁷ for τ_cp = 1.05 × 10⁴ and to ~ 2 × 10⁻⁸ for τ_cp = 2.62 × 10³.

Figure 11.

CPMG relaxation dispersion for the strong-collision model, the biphasic potential with unequal well frequencies, and the telegraph signal mapping of resonance frequencies. Parameters were Q = 7, V_B = 2, ω_A = 0.5, and ω_B = 0.25 (dashed trace in Fig. 3). The simulations used a time step of 0.01 and a collision rate of α = 2.5; q(t) was stored every 2¹⁰ steps. Values of τ_cp were 1.05 × 10⁴ and 2.62 × 10³. (a) Values of s⁺(t) for 640 trajectories of length 2²⁹ steps were averaged to obtain < s⁺(t) >. (b) Autocorrelation functions of s⁺(t) were calculated as in Figure 8 for 20 individual trajectories of 2³⁴ steps and averaged. In each figure, black lines are simulated results. Fits with single exponential functions are shown for (blue, dotted line) absence of CPMG block, (green, dash-dotted line) CPMG block with τ_cp = 1.05 × 10⁴, and (reddish-purple, dashed line) CPMG block with τ_cp = 2.62 × 10³. The decay time constants in the absence of an applied CPMG pulse train agree well between the NMR signal and its autocorrelation function. The decay rates in the absence of the CPMG sequence are (a) 9.49 × 10⁻⁷ and (b) 9.16 × 10⁻⁷ (in good agreement with the results shown in Fig. 8), the decay rates are (a) 2.07 × 10⁻⁷ and (b) 2.29 × 10⁻⁷ for τ_cp = 1.05 × 10⁴, and the decay rates are (a) 1.97 × 10⁻⁸ and (b) 2.26 × 10⁻⁸ for τ_cp = 2.62 × 10³.

An important aspect of CPMG experiments is illustrated by comparing relaxation dispersion curves acquired for the two-state telegraph and random-coil mapping of resonance frequencies applied to simulations using the strong-collision model in the asymmetric biphasic double-well potential. The relaxation rate constants determined as in Fig. 11 are shown in Figure 12 for τ_cp values ranging from 327 to 4.19 × 10⁴. The simulated data were fit using Eq. 21 with an added constant offset. As can be seen in the inset to Fig. 12, the offset is zero for the data using the two-state model and positive for the random-coil model. The averaging of resonance frequencies in the well with q(t) > 0 gives rise to an initial fast decay of the autocorrelation function shown in Fig. 10b, with a time constant of 32. This dynamic process is faster than the fastest pulsing used in the CPMG simulations (τ_cp = 327) and consequently, the dephasing caused by this process cannot be refocused by the 180° pulses in the CPMG train. Instead, the plateau value represents the relaxation rate constant from dynamic processes faster than pulsing. Windowless CPMG and R_1ρ experiments overcome this limitation and allow faster dynamic processes to be characterized (17-19).

Figure 12.

CPMG relaxation dispersion for (filled circles, solid line) two-state and (open circles, dashed line) random-coil models. Relaxation rate constants shown as circles were obtained as described in Fig. 11. Simulated points were fit with Eq. 19 augmented by a constant offset parameter. Optimized values of τ_ex = 10330 and 10200, for the telegraph and random-coil models, respectively, in agreement with the results shown in Figs. 9 and 10. The optimized offset was 0 for the telegraph model and 1.36 × 10⁻⁹ for the random-coil model. The limiting relaxation rate constant for the random-coil model agrees well with the value of 1.44 × 10⁻⁹ obtained as the product of the amplitude and decay time for the fast component of the autocorrelation function shown in Fig. 10b, confirming that the apparent plateau represents the contribution from dynamics processes faster than the CPMG pulsing rates. Data have been normalized by the relaxation rate constant in the absence of pulsing, R_BWR, for display.

V. Discussion

Chemical exchange has emerged as one of the most powerful phenomena in NMR spectroscopy for investigating the conformational dynamics and chemical kinetics of biological macromolecules (2). The present tutorial is intended to introduce chemical exchange dynamics and illustrate the connections between theoretical approaches conventionally used in NMR spectroscopy to model chemical exchange and chemical reaction rate theory in statistical mechanics.

V.A. Strong-collision models for chemical exchange

A theoretical expression for the NMR signal was derived within the strong-collision model and shown to be similar to expressions derived from conventional NMR approaches for chemical exchange (e.g. Redfield theory or the random-phase model). Numerical simulations of the dynamics were performed for a particle moving stochastically on a classical one-dimensional biphasic double-well potential using the strong-collision model. Simulations of dynamics of simple models provide ready access to the kinetic rate constants on the chemical exchange timescale μs-ms) while capturing essential features of more complex molecular dynamics (e.g. macromolecules). Excellent agreement was obtained between the simulated rate constants for barrier crossing and the calculated rate constants based on the formal statistical mechanical theory of kinetic processes.

The theoretical derivation of the NMR spin relaxation rate constant for chemical exchange is commonly obtained using discrete-state jump models for the underlying chemical dynamics. Such approaches conform well to the theoretical strong-collision model, remembering that “collisions” have a somewhat different meaning in the two approaches. However, as is well-known in statistical mechanics, a range of kinetic rate constants frequently can be obtained from simulations in either the strong-collision model or the frictional “weak-collision” (Langevin) model, with the appropriate choice of parameters. In the weak-collision Langevin model, collisions are effectively continuous, each resulting in a change in the energy that is small compared to k_BT. Similar results to those presented for the strong-collision model have been obtained using Langevin dynamics with the algorithm of Grønbech-Jensen and coworkers (20,21) (not shown).

The simulations of a continuous variable model of dynamics in a biphasic double-well potential were mapped onto jump models for chemical exchange by defining discrete states for ranges of the position of the particle, for example, setting state 1 for q < 0 and state 2 for q > 0. The associated simulations were found to agree with theoretical results and can be used to explore the behavior of more complex models of chemical exchange, allowing for asymmetry between potential energy wells, as well as more complex mapping between the position of the particle and the NMR resonance frequency of a nuclear spin. These features mimic commonly occurring situations in which one “state” of a protein or other macromolecule is highly ordered, with limited stochastic variation in resonance frequencies, and the other “state” is highly disordered, with large stochastic variation in resonance frequencies.

The results of the theoretical derivations and numerical simulations have been presented in dimensionless units, and can be rescaled to natural units as desired by choice of ω₁ and dimensional analysis. For example, the dimensionless value of the NMR relaxation rate constant R_sc = 1.96 × 10⁻⁶ is obtained from the telegraph signal model for the strong-collision model with equal potential well frequencies (Fig. 6). A choice of ω₁ = 1.0 × 10⁷ s⁻¹ then gives R_sc = 19.6 s⁻¹. The simulation used <δω(t)²>^1/2τ_c = <δω(t)²>^1/2/k_ex = 0.1, so from Eq. 14, and recalling p₁ = 0.88 and p₂ = 0.12 for the given potential, k_ex = 1960 s⁻¹ and Δω = 603 s⁻¹; these are values in the range typical of conformational changes in biological macromolecules detected in actual NMR experiments (2).

The numerical simulations in the biphasic potential were performed for the most part using parameters consistent with fast-limit or BWR time scale chemical exchange for simplicity in comparing results to Eqs. 14 and 15. Extensive use was made of the autocorrelation functions of the resonance frequency fluctuations or of the NMR resonance frequency to obtain estimates of the transverse relaxation rate constant. As shown by Eq. 13 and results in Fig. 2, the long-time decay constant of the autocorrelation function is a good estimate of the relaxation rate constant outside of the fast-limit. The actual range of application of Eq. 13 and related expressions for R_1ρ and CPMG relaxation dispersion have been discussed elsewhere (2,4,5,13). The simulations can be performed for stochastic processes on any time scale and, outside of the range of application of the autocorrelation function approach, the ensemble-average free induction decay can be modeled by summation of multiple independent simulations, in a similar fashion to the simulations represented in Fig. 12a.

V.B. Further consideration of CMPG relaxation dispersion

The CPMG experiment is one of the most common and effective techniques used to assess exchange on the microsecond-to-millisecond timescale. The 180° pulses in the CPMG sequence have the effect of reversing the sense of precession of transverse magnetization and thus can be represented by inversion of the sign of the precession frequencies. In the present work, the effect of the CPMG pulse train was simulated by multiplying the NMR signal by a square wave function sq[πt/(2τ_cp)] with period 4τ_cp and varying between +1 and −1. As described in Figs. 11 and 12, the simulated relaxation dispersion data provides an excellent fit to Eq. 21. As an alternative approach, Skrynnikov and coworkers have shown that the effective relaxation rate constant in a CPMG experiment can be obtained directly from the autocorrelation function of the frequency fluctuations, in a generalization of Eq. 15, also by multiplying by a square wave:

$$\begin{gathered} R_{CPMG}({\tau }_{cp})=\underset{0}{\overset{\infty }{\int }}\langle \delta \omega (t)sq[\pi t∕(2{\tau }_{cp})]\delta \omega (t+\tau )sq[\pi (t+\tau )∕(2{\tau }_{cp})]\rangle dt \\ =\underset{0}{\overset{\infty }{\int }}\langle \delta \omega (t)\delta \omega (t+\tau )\rangle \langle sq[\pi t∕(2{\tau }_{cp})]sq(\pi (t+\tau )∕(2{\tau }_{cp})]\rangle dt \\ =\underset{0}{\overset{\infty }{\int }}\langle \delta \omega (t)\delta \omega (t+\tau )\rangle tri[\pi \tau ∕(2{\tau }_{cp})]dt \end{gathered}$$ (22)

in which tri(x) is a triangle-wave function that consists of linear segments connecting the extrema of the cosine function (10). The second line of Eq. 20 is obtained because the fluctuations δω(t) are uncorrelated with the time of applications of the 180° pulses and the third line is obtained because the autocorrelation of the square-wave function is the triangle-wave function. Integration of the last line of Eq. 22 for ⟨δω(t)δω(t + τ)⟩ = p₁p₂^{Δω2e−τ/τ_ex} yields Eq. 21. The integration can be performed by expanding tri(x) in a Fourier cosine series, integrating each term in the Fourier series, and summing the resulting series (22).

V.C. Generalization to dynamics in multiphasic many-well potentials

The two-state kinetic model explored in this work exhibits the principal features of chemical exchange, but also represents one of the most common scenarios encountered in experimental NMR spectroscopy. However, the simulation protocols developed above for a two-well biphasic potential are extendible to more complex kinetic schemes. As one example, a linear three-site potential function (C-A-B) can be designed in a similar fashion as for the biphasic potential, as follows:

$$V(q)=\phantom{\{}\left\{\begin{array}{cc}{V}_B+\frac{1}{2}m{\omega }_B^2(q-q_B{)}^2\hfill & q>b\hfill \\ Q_{AB}-\frac{1}{2}m{\omega }_1^2(q-q_{AB}{)}^2\hfill & b\ge q>a\hfill \\ \frac{1}{2}m{\omega }_A^2{q}^2\hfill & a\ge q>-c\hfill \\ Q_{AC}-\frac{1}{2}m{\omega }_1^2(q+q_{AC}{)}^2\hfill & -c\ge q>-d\hfill \\ V_C+\frac{1}{2}m{\omega }_C^2(q+q_C{)}^2\hfill & q\le -d\hfill \end{array}\phantom{\}}\right\}$$ (23)

which includes parameters describing the two energy barrier heights (Q_AC, Q_AB), the energy difference between the main state and the two minor states (V_B, V_C), the positions of the minor state minima (q_B, q_C), the locations of the barriers (q_AB, q_AC), barrier transition frequencies (ω₁), the well frequencies (ω_A, ω_B, ω_C), and the mass (m). Equation 23 is formulated with the position of the well A at q = 0. This potential also assumes for simplicity that the barrier frequencies are identical; the potential could be generalized to distinct barrier frequencies. The potential function is made continuous by the following definitions:

$$\begin{gathered} a=(m{\omega }_1^2{q}_{AB}^2-2Q_{AB})∕(m{\omega }_1^2{q}_{AB}) \\ b=\left\{2(Q_{AB}-V_B)-m{\omega }_1^2{q}_{AB}(q_{AB}-q_B)\right\}∕(m{\omega }_1^2(q_B-q_{AB}) \\ c=(m{\omega }_1^2{q}_{AC}^2-2Q_{AC})∕(m{\omega }_1^2{q}_{AC}) \\ d=\left\{2(Q_{AC}-V_C)-m{\omega }_1^2{q}_{AC}(q_{AC}-q_C)\right\}∕(m{\omega }_1^2(q_C-q_{AC}) \\ {\omega }_A={\omega }_1\sqrt{(q_{AB}-a)∕a} \\ {\omega }_B={\omega }_1\sqrt{(q_{AB}-b)∕(b-q_B)} \\ {\omega }_C={\omega }_1\sqrt{(q_{AC}-d)∕(d-q_C)} \end{gathered}$$ (24)

Additionally, the AC barrier location (q_AC) is constrained and is calculated using the following equation:

$$q_{AC}=q_{AB}\sqrt{Q_{AC}∕Q_{AB}}$$ (25)

Manipulation of the independent variables allows for the simulation of symmetric or asymmetric models with identical or distinct well frequencies. Strong-collision dynamic simulations can then be performed similarly as for the biphasic potential model described. Theoretical results for N-site (N > 2) chemical exchange always can be computed numerically from the Bloch-McConnell equations, although approximate analytical solutions also are available for comparison with simulations (2). Efforts to parameterize Markov state models from molecular dynamics simulations, together with calculations of NMR chemical shifts from molecular structures, extend beyond the idealized two- and three-state models given by Eqs. 16 and 23 (10,11).

V.D. Other possible mappings of coordinates to resonance frequencies

The models presented above can be modified or generalized by using other mappings between the particle position q(t) and the spin resonance frequencies. A common justification for the application of the two-site model in NMR spectroscopy relies on the assumption of rapid exchange between subsets of states. For example, if states 1 to M are in mutual fast exchange, and sites M + 1 to N are in mutual fast exchange, then the N-state system reduces to an effective two-state system with averaged site properties for the two sets of states. As noted by Trott and Palmer (23), convergence to this simplified two-state result depends on all the parameters of the spin system in complex fashion. The random-coil model used herein tacitly assumed barrierless averaging of shifts to a mean of zero in the q > 0 state. Simulations in the three-state potential described above would allow for exploration of the convergence to a two-state system by averaging of the states A and C as the AC barrier is reduced relative to the AB barrier.

VI. Conclusion

The theoretical analysis and simulations presented above are intended to illustrate the connection between the NMR phenomena of chemical exchange line-broadening and nuclear spin relaxation and statistical mechanical chemical reaction rate theories of barrier crossing dynamics. In this work, we have attempted to show the value of the natural connections between the two. The numerical simulations provide insights into how changes in NMR observables reflect the nature of the underlying state-to-state dynamics. This approach can be generalized to reflect specific features of the many and varied forms of biomolecular dynamics (11). Although most simulations were performed in the fast limit on the chemical shift time scale, the simulation methods themselves and the theoretical results derived have wider application; in particular the theoretical results are generally accurate if the site populations are highly skewed (4,13).

The numerical simulations using the strong-collision model enabled comparison of results for the discrete-state jump model (the telegraph model discussed above) commonly used to analyze NMR chemical exchange measurements and a model in which resonance frequencies vary within a potential energy well (the random-coil model discussed above). The results, for example, comparing Figs. 6 and 7 or Figs. 9 and 10, show that chemical shift averaging within potential wells contributes to motional narrowing of resonance linewidths, compared to an assumed two-state discrete jump. Thus, resonance linewidths or transverse relaxation rate constants are not interpretable in a simple manner, because all NMR and kinetic parameters contribute to the observable quantity. In contrast, the CPMG relaxation dispersion curves shown in Fig. 12 display nearly ideal discrete two-state behavior even for the random-coil model. In many experimental situations, the fast-pulsing plateau value of R_CPMD(τ_cp) is an adjustable parameter and the offset for the random-coil model in Fig. 12 would not be identified. Thus, the present results provide support for the use of discrete-state models for analysis of relaxation dispersion measurements. However, chemical exchange processes faster than the maximum pulsing rate in CPMG experiments can be detected as anomalously large plateau values for R_CPMG(τ_cp) or by R_1ρ relaxation dispersion experiments. Such processes typically are attributed to presence of additional discrete chemical or conformational states. The present work raises the possibility that such effects could arise from conformational fluctuations within a potential energy basin.

Appendix: Calculation of reaction rate constants for the biphasic potential with strong-collision dynamics

An extensive literature explores statistical mechanical theories for the absolute rate of barrier crossing (24). Transition state theory provides an upper bound for the rate of activated barrier crossing. For a bistable potential with harmonic wells separated by a dimensionless barrier of height Q >> 1, the sum of the forward and reverse barrier-crossing rate constants can be approximated as:

$$k_{TST}=\frac{{\omega }_A}{2\pi }{e}^{-Q}+\frac{{\omega }_B}{2\pi }{e}^{-(Q-V_B)}$$ (A1)

This estimate of the rate constant assumes (1) an equilibrium population of states and (2) that an activated state will undergo a transition and be deactivated without recrossing of the barrier. The transition state theory rate constant is determined entirely by equilibrium properties of the system and has no dependence on collision rate, friction, or any aspect of system dynamics.

Transition state theory can be corrected by accounting for the role of barrier recrossing in reducing the rate constant below k_TST:

$$k=\kappa k_{TST}$$ (A2)

in which k is the sum of the forward and reverse reaction rate constants and κ ≤ 1 is the transmission coefficient. As was first recognized by Kramers (25), over a wide range of collision rate or friction, the transmission coefficient shows a turnover between the energy diffusion regime (of low collision rate or low friction), where barrier recrossings occur due to inertial effects and slow energy dissipation, and the spatial diffusion regime (of high collision rate or high friction), where barrier recrossings occur due to slow spatial diffusion over the barrier (26).

For the strong-collision model, a transmission coefficient κ valid for all collision rate regimes has been derived by Weiss and coworkers (16):

$$\kappa ={\kappa }_{{}_1}{\kappa }_{{}_h}$$ (A3)

in which κ_l and κ_h are the transmission coefficients in the low-to-intermediate and intermediate-to-high collision rate regimes, respectively. In the low-to-intermediate collision rate regime, the transmission coefficient is given by:

$${\kappa }_1=\underset{0}{\overset{\infty }{\int }}\frac{\left(1-e^{-\alpha T_A(\epsilon )}\right)\left(1-e^{-\alpha T_B(\epsilon )}\right)}{1-e^{-\alpha (T_A(\epsilon )+T_B(\epsilon ))}}{e}^{-\epsilon }d\epsilon$$ (A4)

In the absence of collisions, the time period T_A(ε) for a particle located initially at q = 0 with kinetic energy ε and velocity < 0 to return to q = 0 with kinetic energy ε and velocity > 0 and the time period T_B(ε) for a particle located initially at q = 0 with kinetic energy ε and velocity > 0 to return to q = 0 with kinetic energy ε and velocity < 0 are given by:

$$\begin{gathered} T_A(\epsilon )=2\underset{q_{\text{min}}(\epsilon )}{\overset{0}{\int }}\frac{dq}{\sqrt{\epsilon +Q-V(q)}} \\ {T}_B(\epsilon )=2\underset{0}{\overset{q_{\max }(\epsilon )}{\int }}\frac{dq}{\sqrt{\epsilon +Q-V_B-V(q)}} \end{gathered}$$ (A5)

in which q_min(ε) and q_max(ε) are the classical turning points in the potential. Each of these integrals leads to a similar functional form:

$$T_X(\epsilon )=\frac{2\pi }{{\omega }_X}\left\{1-\frac{1}{\pi }{\tan }^{-1}\sqrt{U_X^2{\omega }_X^2+\epsilon U_X(1+{\omega }_X^2)}\right\}+\frac{1}{2}{\cosh }^{-1}\left(\frac{U_X{\omega }_X^2}{\epsilon (1+{\omega }_X^2)}+1\right)$$ (A6)

in which X= {A, B} and U_x = (Q, Q − V_B}. The energy diffusion regime is reached as α → 0. For well frequencies similar to the barrier frequency, i.e. 0.25 ≤ ω_x ≤ 1.5, κ_l = κ_ED = 2πα/(ω_A + ω_B). In the intermediate-to-high collision rate regime, the transmission coefficient is given by:

$${\kappa }_h=\underset{0}{\overset{\infty }{\int }}{e}^{-z}\tanh \left(z∕\alpha \right)dz$$ (A7)

As α → ∞, the spatial diffusion regime is reached and κ_h = κ_SD = 1/α (in dimensionless units).