ABSTRACT

A. Strong collision dynamics in the discrete state jump model

As an introduction to the approach used 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 2-state jump process (also called the telegraph process) is given by (5, 12) (1) $$\begin{array}{ll}\hfill s^{+}(t)& =\exp \left\{i{\displaystyle \underset{0}{\overset{t}{\int }}\left[q(t^{\prime }){\omega }_1+(1-q(t^{\prime })){\omega }_2\right]d{t}^{\prime }}\right\}\hfill \\ \hfill & =\exp \left\{-i\Delta \omega {\displaystyle \underset{0}{\overset{t}{\int }}\left[q(t^{\prime })-p_1\right]d{t}^{\prime }}\right\}\hfill \\ \hfill & =q(t)\exp \left\{-i{p}_2\Delta \omega t\right\}+\left[1-q(t)\right]\exp \left\{i{p}_1\Delta \omega t\right\}\hfill \end{array}$$ 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 p₁ = <q(t)> is the equilibrium population of the state 1 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 theoretic 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 (2) $$\begin{array}{ll}\hfill C(\tau )& =\langle s^{+}(t)s^{-}(t+\tau )\rangle \hfill \\ \hfill & =\underset{T\to \infty }{\lim }\frac{1}{T}{\displaystyle \underset{0}{\overset{T}{\int }}{s}^{+}(t)s^{-}(t+\tau )dt}\hfill \end{array}$$ 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 is given by (3) $$\begin{array}{ll}\hfill C(\tau )& =\langle \left(q(t)e^{-i{p}_2\Delta \omega t}+[1-q(t)]e^{i{p}_1\Delta \omega t}\right)\times \left(q(t+\tau )e^{i{p}_2\Delta \omega (t+\tau )}+[1-q(t+\tau )]e^{-i{p}_1\Delta \omega (t+\tau )}\right)\rangle \hfill \\ \hfill & =p_1{e}^{i{p}_2\Delta \omega \tau }+p_2{e}^{-i{p}_1\Delta \omega \tau }\hfill \end{array}$$ in which the third equality 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 (4) $$C(s)={\displaystyle \underset{0}{\overset{\infty }{\int }}C(\tau )e^{-s\tau }d\tau }=\frac{p_1}{s-i{p}_2\Delta \omega }+\frac{p_2}{s+i{p}_1\Delta \omega }$$

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 (s) is a function of the autocorrelation function in the absence of collisions C(s) (8, 9): (5) $$\tilde{C}(s)=\frac{C(s+\text{α})}{1-\text{α}C(s+\text{α})}$$ in which α is the collision rate. Substituting Eq. 4 into Eq. 5 gives (6) $$\tilde{C}(s)=\frac{s+\text{α}-i\Delta \omega \Delta p}{s^2+\left\{\text{α}-i\Delta \omega \Delta p\right\}s+p_1{p}_2\Delta {\omega }^2}$$ in which Δp = p₂ − p₁. Eq. 6 is the Laplace transform of the 2-state jump process in the strong collision model. The inverse Laplace transform of Eq. 6 yields the corresponding autocorrelation function in the time domain: (7) $$\tilde{C}(\tau )=e^{-(\alpha -i\Delta \omega \Delta p)\tau /2}\left\{\cosh \left(\frac{\tau }{2}\sqrt{{\alpha }^2-\Delta {\omega }^2-2i\alpha \Delta \omega \Delta p}\right)+\frac{\alpha -i\Delta \omega \Delta p}{\sqrt{{\alpha }^2-\Delta {\omega }^2-2i\alpha \Delta \omega \Delta p}}\times \sinh \left(\frac{\tau }{2}\sqrt{{\alpha }^2-\Delta {\omega }^2-2i\alpha \Delta \omega \Delta p}\right)\right\}$$ This equation is exact but is also complicated. Fortunately, useful limiting results can be obtained as described in the following.

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 (8) $$\tilde{C}(s)=\frac{1}{s+\frac{p_1{p}_2\Delta {\omega }^2}{s+\text{α}-i\Delta \omega \Delta p}}=\frac{1}{s+D(s)}$$ D(s) can be expanded in a Taylor series as (9) $$D(s)\approx \frac{p_1{p}_2\Delta {\omega }^2}{\text{α}-i\Delta \omega \Delta p}-s\frac{p_1{p}_2\Delta {\omega }^2}{{\left(\text{α}-i\Delta \omega \Delta p\right)}^2}+\dots$$ Keeping only the first 2 terms of the series, Eq. 8 becomes (10) $$\begin{array}{ll}\hfill \tilde{C}(s)& =\frac{1}{s\left(1-\frac{p_1{p}_2\Delta {\omega }^2}{{\left(\alpha -i\Delta \omega \Delta p\right)}^2}\right)+\frac{p_1{p}_2\Delta {\omega }^2}{\alpha -i\Delta \omega \Delta p}}\hfill \\ \hfill & =\frac{{\left(1-\frac{p_1{p}_2\Delta {\omega }^2}{{\left(\alpha -1\Delta \omega \Delta p\right)}^2}\right)}^{-1}}{s+{\left(1-\frac{p_1{p}_2\Delta {\omega }^2}{{\left(\alpha -1\Delta \omega \Delta p\right)}^2}\right)}^{-1}\frac{p_1{p}_2\Delta {\omega }^2}{\alpha -1\Delta \omega \Delta p}}\hfill \end{array}$$

This equation is isomorphic to a result derived from the Bloch–McConnell equations by Abergel and Palmer (Eq. 23 in (13)). The second equality of Eq. 10 is the Laplace transform of (τ) = Ae^iϕ exp[(iΩ_sc − R_sc)τ] in which (12) $$\begin{array}{l}a=\left|{\left(1-\frac{p_1{p}_2\Delta {\omega }^2}{{\left(\alpha -i\Delta \omega \Delta p\right)}^2}\right)}^{-1}\right|\\ \varphi =\text{Arg}\left[{\left(1-\frac{p_1{p}_2\Delta {\omega }^2}{{\left(\alpha -i\Delta \omega \Delta p\right)}^2}\right)}^{-1}\right]\\ {\Omega }_{\text{sc}}=\text{Im}\left[{\left(1-\frac{p_1{p}_2\Delta {\omega }^2}{{\left(\alpha -i\Delta \omega \Delta p\right)}^2}\right)}^{-1}\frac{p_1{p}_2\Delta {\omega }^2}{\alpha -i\Delta \omega \Delta p}\right]\\ R_{\text{sc}}=\text{Re}\left[{\left(1-\frac{p_1{p}_2\Delta {\omega }^2}{{\left(\alpha -i\Delta \omega \Delta p\right)}^2}\right)}^{-1}\frac{p_1{p}_2\Delta {\omega }^2}{\alpha -i\Delta \omega \Delta p}\right]\end{array}$$

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 2-state strong collision model. Following the procedure of Abergel and Palmer yields an explicit form for R_sc (Eq. 31 in (13)): (13) $$R_{\text{sc}}=p_1{p}_2\Delta {\omega }^2\alpha \times \left[\frac{{\alpha }^2+\Delta {\omega }^2}{{\left({\alpha }^2+\Delta {\omega }^2\right)}^2-p_1{p}_2\left(5{\alpha }^2+\Delta {\omega }^2\right)\Delta {\omega }^2}\right]$$ (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 timescale, α >> Δω, then the bracketed term in Eq. 13 approaches α⁻² and (14) $$R_{\text{sc}}=p_1{p}_2\Delta {\omega }^2/\alpha$$

Eq. 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 2-state telegraph model is a single exponential decay with rate constant R_sc. For comparison, either the random phase model or Bloch-Wangsness-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 2-state telegraph process, C(τ) = p₁p₂Δω²exp(−k_exτ), in which δω(t) = −Δω[q(t) − p₁] and C(0) = 〈δω²(t)〉 = p₁p₂Δω², yielding: (15) $$R_{\text{BWR}}={\displaystyle \underset{0}{\overset{\infty }{\int }}\langle \delta \omega (t)\delta \omega (t+\tau )\rangle d\tau }=p_1{p}_2\Delta {\omega }^2/k_{\text{ex}}$$

Eqs. 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 2-state telegraph model, k_ex is the only nonzero eigenvalue of the kinetic transition matrix and consequently the only rate constant appearing in Eq. 15, and by extension, Eqs. 13 and 14. Eqs. 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,b). 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,d). 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 τ longer than approximately (4–5)/α, after the initial fast decay of the autocorrelation function (Fig 2a), as well as fast exchange (Fig 2c,d). For the given illustrative parameters, Eq. 13 underestimates the relaxation rate constant obtained from line shape fitting by a maximum of 7.5% for intermediate exchange and becomes highly accurate in the fast exchange regime (see Figs 1 and 2 captions). 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 Figure 2, the approaches developed are more generally applicable outside this limit.

View Figure

• Download as Powerpoint
• Download Figure

Additional details regarding the Laplace transform approach used to determine theoretic 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 and Goldman et al. (12, 14).

B. Strong collision dynamics in the continuous reaction coordinate model

In the 2-state jump model discussed previously, 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 with residence times within the wells. A more realistic model of the chemical dynamics uses a continuous positional variable q(t) moving in a double-well (biphasic A–B) 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) (16) $$V(q)=\left\{\begin{array}{cc}\frac{1}{2}m{\omega }_{\text{A}}^2{(q+q_{\text{A}})}^2& q<-a\\ Q-\frac{1}{2}m{\omega }_1^2{q}^2& -a\le q\text{<}b\\ V_{\text{B}}+\frac{1}{2}m{\omega }_{\text{B}}^2{(q-q_{\text{B}})}^2& q\ge b\end{array}\right\}$$ in which Q is the energy barrier height between wells, V_B is the difference in energy between the well A and B 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 (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 (17) $$\begin{array}{l}a=2Q/(m{\omega }_1^2{q}_{\text{A}})\hfill \\ b=2(Q-V_{\text{B}})/(m{\omega }_1^2{q}_{\text{B}})\hfill \\ {\omega }_{\text{A}}={\omega }_1\sqrt{a/(q_{\text{A}}-a)}\hfill \\ {\omega }_{\text{B}}={\omega }_1\sqrt{b/(q_{\text{B}}-b)}\hfill \end{array}$$ ensure that V(q) and dV(q)/dq are continuous. The potential can be recast in dimensionless variables: (18) $$\widehat{V}(\widehat{q})=\left\{\begin{array}{cc}\frac{1}{2}{\widehat{\omega }}_{\text{A}}^2{(\widehat{q}+{\widehat{q}}_{\text{A}})}^2& \widehat{q}\text{ <}-\widehat{a}\\ \widehat{Q}-\frac{1}{2}{\widehat{q}}^2& -\widehat{a}\le \widehat{q}\text{ <}\widehat{b}\\ {\widehat{V}}_{\text{B}}+\frac{1}{2}{\widehat{\omega }}_{\text{B}}^2{(\widehat{q}-{\widehat{q}}_{\text{B}})}^2& \widehat{q}\ge \widehat{b}\end{array}\right\}$$ in which energy variables V̂_B = V_B/(k_BT) and Q̂ = Q/(k_BT) are in units of k_BT, length variables are redefined as x̂ = [$m{\omega }_1^2$/(k_BT)]^1/2x for x = {q, q_A, q_B, a, b}, and frequency variables are redefined as ω_γ = ω_γ/ω₁ for γ = {1, A, B}. Dimensionless time variables are defined as t̂ = ω₁t, and the dimensionless collision rate is â = α/ω₁ (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 sites 1 and 2 were obtained by integrating the canonical ensemble probability, assuming q < 0 corresponds to site 1 and q > 0 corresponds to site 2.

View Figure

• Download as Powerpoint
• Download Figure

A. Simulations for strong collisions

The previous theoretic discussion suggests a simple strategy for simulating chemical exchange dynamics by 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 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) by using Eq. 2 and the definition: $$s^{\pm }(t)=s^{\pm }(n\Delta t)=\exp \left(\pm i\Delta t{\displaystyle \sum _{k=0}^n{\omega }_k}\right)$$

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): (19) $$\begin{array}{l}q(t+\Delta t)=q(t)+v(t)\Delta t+\frac{F(t)}{2}\Delta t^2\hfill \\ v(t+\Delta t)=v(t)+\frac{F(t+\Delta t)+F(t)}{2}\Delta t\hfill \end{array}$$ in which the dimensionless instantaneous acceleration of a particle at a particular position q is given by (20) $$\begin{array}{ll}\hfill F(q)& =-\frac{dV(q)}{dq}\hfill \\ \hfill & =\left\{\begin{array}{cc}-{\omega }_{\text{A}}^2\left(q-q_{\text{A}}\right)& q\text{<}-a\\ q& -a\le q\text{<}b\\ -{\omega }_{\text{B}}^2\left(q-q_{\text{B}}\right)& q\ge b\end{array}\right\}\hfill \end{array}$$ 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.

View Figure

• Download as Powerpoint
• Download Figure

B. Mapping generalized coordinates to resonance frequencies

The mapping of the coordinate variable q(t) to the resonance frequency ω(t) was performed in 2 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>^1/2τ_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 2-state telegraph jump process analyzed theoretically, as mentioned previously.

In the second approach, the chemical shifts were modeled initially by 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 Figure 3. This function was then shifted and scaled so that <ω(t)> = 0 and <ω(t)²>^1/2τ_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 in the following.

To illustrate the previous models, the ¹³Cα spin has a secondary chemical shift, Δδ, of 2 to 4 ppm in the 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.

C. CPMG relaxation dispersion

CPMG and R_1ρ relaxation dispersion experiments have developed as powerful approaches for investigation of microsecond to millisecond timescale dynamic processes in proteins and other macromolecules (2). In the CPMG experiment, relaxation is measured during a train of spin–echo sequences: (τ_cp − 180° – τ_cp)_n, in which 180° is a radiofrequency refocusing pulse, τ_cp is the spin–echo delay time, the total relaxation delay period is T = 2nτ_cp, 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 previous 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_cp, prior to further analysis of ω(t). The autocorrelation function of s⁺(t) is calculated as described previously, 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_cp is called a CPMG relaxation dispersion curve. In the fast exchange limit, analysis of the relaxation dispersion curve for a 2-state model yields (21) $$R_{\text{CPMG}}({\tau }_{\text{cp}})=p_1{p}_2\Delta {\omega }^2{\tau }_{\text{ex}}\left[1-\frac{{\tau }_{\text{ex}}}{{\tau }_{\text{cp}}}\tanh \left(\frac{{\tau }_{\text{cp}}}{{\tau }_{\text{ex}}}\right)\right]$$ in which the exchange time τ_ex = 1/k_ex. More complex expressions for R_CPMG(τ_cp), valid for all chemical exchange timescales, have been reported and widely used in the analysis of experimental data (2).

A. Strong collision dynamics in the biphasic potential

Initial simulations were performed by 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 by using values of collision rate α, ranging between 10⁻³ and 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.

View Figure

• Download as Powerpoint
• Download Figure

In contrast to the 2-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 biexponential, with a short time constant, τ₁ = 9.4, arising from fluctuations within the potential energy wells and a longer time constant, τ₂ = 5,110, 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 theoretic 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 theoretic calculations are outlined in the appendix.

For the parameters given in Figure 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 (SD) over the barrier. The resulting time constant τ = 1/k = 4,980 agrees well with the fitted long time decay of the autocorrelation function, τ₂ = 5,110. Figure 5 shows a more extensive comparison between the transmission coefficients obtained from theoretic estimates of the rate constant, the reactive flux method, and the long time decay of the simulated autocorrelation function.

View Figure

• Download as Powerpoint
• Download Figure

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 Supplemental Movies S1 and S2. These parameters correspond to the energy dissipation (ED) and SD limits of the Kramers reaction rate theory discussed in the Appendix.

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 monoexponential. The long time constant from the autocorrelation function of q(t) and from the telegraph signal, τ_c = 5,110, agree well, as expected. The spin relaxation rate constant is obtained, using Eq. 15, as the integral of the autocorrelation function; from the monoexponential 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 2 results is consistent with the theoretic results previously mentioned. 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 Figure 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.

View Figure

• Download as Powerpoint
• Download Figure

A similar analysis was performed by 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 biexponential, 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 biexponential 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 biexponential fitting parameters, R_BWR = 1.10 × 10⁻⁶; the reduced relaxation rate constant, compared with 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 Figure 7d is monoexponential, even when the autocorrelation function of ω(t) in Figure 7b is biexponential in the fast exchange limit, as described previously and illustrated in Figure 2.

View Figure

• Download as Powerpoint
• Download Figure

A second simulation was performed for the parameters given in Figure 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 = 8,890. Similar plots as for the previously mentioned model for the case of equal well frequencies are shown in Figures 8–10. 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 Figure 8, is now τ₂ = 10,440, in agreement with the theoretic result. Similar levels of agreement are found between R_sc and R_BWR for telegraph (Fig 9) and random-coil models (Fig 10) for the resonance frequencies. Notably, the autocorrelation function for the frequency fluctuations ω(t) in the random-coil model (Fig 10b) is distinctly biexponential, reflecting the effects of chemical shift averaging in the broader potential well for state 2. Again, the autocorrelation of the NMR signal in Figure 10d is monoexponential, even when the autocorrelation function of ω(t) is biexponential in the fast exchange limit.

View Figure

• Download as Powerpoint
• Download Figure

View Figure

• Download as Powerpoint
• Download Figure

View Figure

• Download as Powerpoint
• Download Figure

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 with the leftmost well) reduces the transverse relaxation rate constant by approximately a factor of 2. The result that fast timescale 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).

C. CPMG relaxation dispersion for strong collision dynamics

CPMG relaxation dispersion was estimated by using the 2-state telegraph mapping of resonance frequencies applied to simulations by 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.

View Figure

• Download as Powerpoint
• Download Figure

An important aspect of CPMG experiments is illustrated by comparing relaxation dispersion curves acquired for the 2-state telegraph and random-coil mapping of resonance frequencies applied to simulations by using the strong collision model in the asymmetric biphasic double-well potential. The relaxation rate constants determined, as in Figure 11, are shown in Figure 12 for τ_cp values ranging from 327 to 4.19 × 10⁴. The simulated data were fit by using Eq. 21, with an added constant offset. As can be seen in the inset to Figure 12, the offset is zero for the data using the 2-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 Figure 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).

View Figure

• Download as Powerpoint
• Download Figure

A. Strong collision models for chemical exchange

A theoretic 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., BWR theory or the random phase model). Numeric simulations of the dynamics were performed for a particle moving stochastically on a classical one-dimensional biphasic double-well potential by using the strong collision model. Simulations of dynamics of simple models provide ready access to the kinetic rate constants on the chemical exchange timescale (microsecond to millisecond), 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 on the basis of the formal statistical mechanical theory of kinetic processes.

The theoretic derivation of the NMR spin relaxation rate constant for chemical exchange is commonly obtained by using discrete state jump models for the underlying chemical dynamics. Such approaches conform well to the theoretic strong collision model, remembering that collisions have a somewhat different meaning in the 2 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 with k_BT. Similar results to those presented for the strong collision model have been obtained by using Langevin dynamics (data not shown) (20, 21).

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 theoretic 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 theoretic derivations and numeric simulations have been presented in dimensionless units and can be rescaled to natural units, as desired by the 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 biologic macromolecules detected in actual NMR experiments (2).

The numeric simulations in the biphasic potential were performed for the most part by using parameters consistent with fast limit or BWR timescale 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 Figure 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 timescale, 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 Figure 12a.

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 Figures 11 and 12, the simulated relaxation dispersion data provides an excellent fit to Eq. 21. As an alternative approach, Xue et al. 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: (22) $$\begin{array}{l}{R}_{\text{CPMG}}({\tau }_{\text{cp}})\\ ={\displaystyle \underset{0}{\overset{\infty }{\int }}\langle \delta \omega (t)sq[\pi t/(2{\tau }_{\text{cp}})]\delta \omega (t+\tau )sq[\pi (t+\tau )/(2{\tau }_{\text{cp}})]\rangle d\tau }\\ ={\displaystyle \underset{0}{\overset{\infty }{\int }}\langle \delta \omega (t)\delta \omega (t+\tau )\rangle \langle sq[\pi t/(2{\tau }_{\text{cp}})]sq\left[\pi (t+\tau )/(2{\tau }_{\text{cp}})\right]\rangle d\tau }\\ ={\displaystyle \underset{0}{\overset{\infty }{\int }}\langle \delta \omega (t)\delta \omega (t+\tau )\rangle tri[\pi \tau /(2{\tau }_{\text{cp}})]d\tau }\end{array}$$ in which tri(x) is a triangle wave function that consists of linear segments connecting the extrema of the cosine function (10). The second equality of Eq. 22 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 $\langle \delta \omega \left(t\right)\delta \omega \left(t+\tau \right)\rangle =p_1{p}_2\Delta {\omega }^2{e}^{-\tau /{\tau }_{\text{ex}}}$ 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).

C. Generalization to dynamics in multiphasic many-well potentials

The 2-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 as described previously for a 2-well biphasic potential are extendable to more complex kinetic schemes. As one example, a linear 3-site potential function (C–A–B) can be designed in a similar fashion as for the biphasic potential, as follows (23) $$V(q)=\left\{\begin{array}{cc}{V}_{\text{B}}+\frac{1}{2}m{\omega }_{\text{B}}^2{(q-q_{\text{B}})}^2& q\text{ >}b\\ Q_{\text{AB}}-\frac{1}{2}m{\omega }_1^2{(q-q_{\text{AB}})}^2& b\ge q\text{ >}a\\ \frac{1}{2}m{\omega }_{\text{A}}^2{q}^2& a\ge q\text{ >}-c\\ Q_{\text{AC}}-\frac{1}{2}m{\omega }_1^2{(q+q_{\text{AC}})}^2& -c\ge q\text{ >}-d\\ V_{\text{C}}+\frac{1}{2}m{\omega }_{\text{C}}^2{(q+q_{\text{C}})}^2& q\le -d\end{array}\right\}$$ which includes parameters describing the 2 energy barrier heights (Q_AC, Q_AB), the energy difference between the main state and the 2 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). Eq. 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: (24) $$\begin{array}{ll}a\hfill & =(m{\omega }_1^2{q}_{\text{AB}}^2-2Q_{\text{AB}})/(m{\omega }_1^2{q}_{\text{AB}})\hfill \\ b\hfill & =\left\{2(Q_{\text{AB}}-V_{\text{B}})-m{\omega }_1^2{q}_{\text{AB}}(q_{\text{AB}}-q_{\text{B}})\right\}/(m{\omega }_1^2(q_{\text{B}}-q_{\text{AB}})\hfill \\ c\hfill & =(m{\omega }_1^2{q}_{\text{AC}}^2-2Q_{\text{AC}})/(m{\omega }_1^2{q}_{\text{AC}})\hfill \\ d\hfill & =\left\{2(Q_{\text{AC}}-V_{\text{C}})-m{\omega }_1^2{q}_{\text{AC}}(q_{\text{AC}}-q_{\text{C}})\right\}/(m{\omega }_1^2(q_{\text{C}}-q_{\text{AC}})\hfill \\ {\omega }_{\text{A}}\hfill & ={\omega }_1\sqrt{(q_{\text{AB}}-a)/a}\hfill \\ {\omega }_{\text{B}}\hfill & ={\omega }_1\sqrt{(q_{\text{AB}}-b)/(b-q_{\text{B}})}\hfill \\ {\omega }_{\text{C}}\hfill & ={\omega }_1\sqrt{(q_{\text{AC}}-d)/(d-q_{\text{C}})}\hfill \end{array}$$

Additionally, the A–C barrier location (q_AC) is constrained and is calculated by using the following equation: (25) $$q_{\text{AC}}=q_{\text{AB}}\sqrt{Q_{\text{AC}}/Q_{\text{AB}}}$$

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. Theoretic 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 2- and 3-state models, given by Eqs. 16 and 23 (10, 11).

D. Other possible mappings of coordinates to resonance frequencies

The models presented previously 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 2-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 2-state system, with averaged site properties for the 2 sets of states. As noted by Trott and Palmer (23), convergence to this simplified 2-state result depends on all parameters of the spin system in a 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 3-state potential, described previously, would allow for exploration of the convergence to a 2-state system by averaging of the states A and C, as the AC barrier is reduced relative to the AB barrier.