A theoretical framework for random accelerated molecular dynamics simulations

Abstract

The dissociation of a ligand bound to a receptor is a rare events occurring on a timescale that is far longer than can be afforded by standard simulation methodologies. A number of specialized approaches have been designed to overcome this challenge. One of these, Random Accelerated Molecular Dynamics (RAMD), is a biased simulation method that is particularly effective at accelerating the dissociation of ligands bound to protein receptors. However, no theoretical framework is currently available to extract unbiased estimates from the RAMD simulation results. In this work, we develop a simple theory for its interpretation based on the assumption that the RAMD simulations can be approximated by an overdamped Langevin dynamics with an additional force of fixed magnitude changing direction randomly at exponentially distributed times. We shown that the dynamics of the model is consistent with a Smoluchowski equation with an effective temperature and diffusion coefficient that depend quadratically on the magnitude of the random force. These observations explain why the RAMD simulations yields accelerated dynamics. Importantly, we derive an analytical expression for the unbiased escape time of a particle over a free-energy barrier based on the modified Smoluchowski equation. The theoretical framework was used to analyse RAMD simulations monitoring the dissociation of the Trypsin-Benzamidine protein-ligand complex. The results indicate that the theoretical framework is sound and could potentially be used to unbias the results from RAMD simulations.

I. INTRODUCTION

Many processes of interest in biophysics and biochemistry involve transitions over high free-energy barriers and occur over long time scales. In particular, the dissociation kinetics in protein-ligand interactions has recently gained attention as an important factor in drug discovery¹. Specifically, the residence time of a ligand, i.e., the average time it spends in contact with a protein receptor before unbinding appears to be closely related to the in vivo efficacy of potential drug candidates^2,3. However, these times can span a broad range, as the residence times of potential drug candidates may range from seconds to even minutes⁴, which makes their study particularly challenging.

In principle, molecular dynamics (MD) simulations represent a powerful tool to investigate the underlying mechanism of such processes. However, as the transitions of interest occur very infrequently on the time scale of molecular motions, a direct observation of ligand unbinding is computationally prohibitive using simple unbiased MD simulations. The extremely long time scales limit the usefulness of standard unbiased MD trajectories in predicting protein-ligand kinetics. For this reason, such slow processes are commonly described as “rare events” and their investigations require specialized computational methodologies^5,6. A wide class of approaches seek to accelerate the transitions across the free-energy barriers by modifying the potential energy, e.g., accelerated MD^7–10, solute tempering¹¹, or by introducing various biasing potentials, e.g., adaptive biasing force¹², local elevation¹³, conformational flooding¹⁴, and metadynamics^15–17. Other methods seek to combine short unbiased simulations to determine the long-time kinetics, e.g., Markov State Models^18–20 milestoning^21–23, and the adaptive multilevel splitting method²⁴. Another class of strategies seek to alter the inherent dynamical evolution of the system, e.g., as with the self-guided molecular dynamics²⁵ self-guided Langevin dynamics²⁶.

Perhaps in a class of its own, the Random Accelerated Molecular Dynamics (RAMD) was specifically designed to facilitate and accelerate the dissociation of a ligand from its receptor binding site^27–29. In RAMD, an external force of a fixed magnitude is applied to the center of mass (COM) of the ligand in a random direction that is periodically updated according to specific criteria in order to increases the propensity of the bound ligand to explore various regions of the binding pocket, and find an exit pathway²⁷. For example, in the so-called $\tau \text{RAMD}$ protocol³⁰, the COM of the ligand must cover a minimum threshold distance, otherwise a new direction of the force is generated and applied for a period of time $t_{\text{RAMD}}$. This process is repeated until the ligand dissociates from the binding pocket.

An important appeal of RAMD is its great effectiveness at accelerating the ligand unbinding process and its broad applicability^{1,3,29,31–34}. RAMD simulations is commonly used to estimate relative protein-ligand residence times, helping predict how long a ligand stays bound to its target. Additionally, it can be used to identify and define ligand unbinding pathways or determine collective variables for more complex systems. These identified collective variables delineating the dissociation pathway can then with other sampling techniques^12–15 to gain deeper quantitative insights into ligand binding.

An empirical analysis of RAMD previously showed that the logarithm of the ligand dissociation time correlates with the magnitude of the applied force in RAMD simulations³⁵. However, no theoretical framework currently explains the fundamental significance of RAMD or describes an efficient selection of its optimal parameters. Importantly, no established protocol currently exists to eliminate or correct the bias introduced by RAMD simulations. Motivated by these open questions, we develop a theoretical framework to elucidate the significance of RAMD simulations. In particular, we establish a theory that accounts for the influence of the random force in RAMD under Langevin dynamics. This formal analysis is then tested and validated with numerical simulations. Afterwards, the theoretical framework is illustrated in the context the protein-ligand complex Trypsin-Benzamidine, an experimentally well characterized system³⁶, that has been used as a benchmarking system for computational studies of unbinding kinetics. Although some previous computational estimates are close to experiment^24,37–39, the reported values span a wide range^{16–23,40–43}, reflecting the uncertainty in methods and force fields to accurately predict unbinding rates⁴⁴.

II. THEORY

A. The original RAMD protocol

In RAMD an external random force is applied to the COM of a ligand to accelerate its dissociation from its binding site²⁷. Let us write the RAMD force as ${\mathbf{F}}_{\text{RAMD}}(t)=F_{\text{RAMD}}\mathbf{u}(t)$, where $F_{\text{RAMD}}$ is the magnitude of the force and $\mathbf{u}(t)$ is a time-dependent unit vector. The magnitude of the external force is an input parameter that remains constant throughout the RAMD simulation, while its direction given by the unit vector $\mathbf{u}(t)$ is randomly changed when the ligand does not make sufficient progress toward dissociation according to a minimum distance threshold applied at a constant period of time $t_{\text{RAMD}}$^27–29. If the ligand travels a distance greater than a selected threshold over the time $t_{\text{RAMD}}$ then the direction $\mathbf{u}$ is kept constant for the subsequent period of time $t_{\text{RAMD}}$. However, if the distance traveled by the ligand is smaller than the threshold then a new random direction $\mathbf{u}$ is generated and applied for the subsequent period of time $t_{\text{RAMD}}$. The process is repeated until the ligand dissociates from the binding pocket. As designed, the applied random force ${\mathbf{F}}_{\text{RAMD}}(t)$ increases the propensity of the bound ligand to explore various regions of the binding pocket, and eventually find an exit pathway. Specifically, the “$\tau \text{RAMD}$ protocol”, outlining strategies for system preparation and statistical analysis, was developed to estimate the relative unbinding times of a set of drug-like molecules from a receptor³⁰. In the $\tau \text{RAMD}$ protocol, the COM of the ligand must cover a distance of at least 0.025 Å during 0.1 ps of RAMD simulation, otherwise a new direction of the force is selected. The RAMD simulations is carried out until the distance between the ligand and the protein COM is greater than 30 Å. The magnitude $F_{\text{RAMD}}$ is the only tunable input parameter of this protocol, and its value influences the computational time required for ligand unbinding.

B. Langevin dynamics model of RAMD

In RAMD simulations, the applied random force changes its direction randomly according to a minimum distance threshold. For this reason, the dynamical propagation comprises elements of spatial and temporal memory that are difficult to model mathematically in a simple form. To make progress, we consider a simpler dynamical model that mimics the true RAMD protocol while remaining mathematically tractable. To this end, we represent the RAMD dynamics of a ligand by the Langevin equation,

$$m\frac{d\mathbf{v}}{dt}=-𝛁W(\mathbf{r})-\gamma \mathbf{v}+{\mathbf{F}}_{\text{RAMD}}(t)+\mathbf{f}(t)$$ (1)

where $\mathbf{r}$ is the position in a $d$-dimensional space, $\mathbf{v}$ is the velocity, $W(\mathbf{r})$ is the potential of mean force (PMF), $\gamma$ is the dissipative friction, and $\mathbf{f}(t)$ is a gaussian-distributed random force obeying $⟨\mathbf{f}(t)⟩=0$, and $⟨\mathbf{f}(t)\cdot \mathbf{f}(0)⟩=2d\gamma k_{\text{B}}T\delta (t)$. $k_{\text{B}}$ is Boltzmann’s constant and $T$ is the temperature. In one-dimension, we recover the familiar fluctuation-dissipation relation, $⟨f(t)f(0)⟩=2\gamma k_{\text{B}}T\delta (t)$. Eq. (1) only provides an approximate description to the dynamics because the influence of the environment is modeled using the classic Langevin equation and memory effects are neglected. While one could, in principle, adopt a more detailed representation such as the Generalized Langevin Equation (GLE)⁴⁵, any potential benefit would be severely constrained by the strong approximations needed for the treatment of RAMD.

Two stochastic factors determine the dynamics of the system based on Eq. (1). The first is the random Langevin force, and the second is the RAMD force via the time-dependent random unit vector $\mathbf{u}$. In this model, we assume the RAMD force ${\mathbf{F}}_{\text{RAMD}}(t)$ to have a constant magnitude $F_{\text{RAMD}}$ in a random direction that persists for an exponentially distributed random time period. Therefore, the autocorrelation function of this RAMD force is $⟨{\mathbf{F}}_{\text{RAMD}}(t)\cdot {\mathbf{F}}_{\text{RAMD}}(0)⟩=F_{\text{RAMD}}^2⟨\mathbf{u}(t)\cdot \mathbf{u}(0)⟩=e^{-t/\tau }$ where $\mathbf{u}$ is a normalized unit vector $(\mathbf{u}\cdot \mathbf{u}=1)$ and $\tau$ is a correlation time. We note that this is an approximation because in the original RAMD protocol the direction of the force persists for a length of time that depends on a minimum distance threshold. This simplifying assumption will be illustrated and validated below using RAMD simulations of ligand dissociation simulations of the Trypsin-Benzamidine complex.

It can be anticipated that a system propagated via Eq. (1) will exhibit some peculiarities because it is a stochastic dynamics driven by a time-dependent random force ${\mathbf{F}}_{\text{RAMD}}(t)+(\mathbf{t})$ with a finite correlation time, which is counterbalanced by the Markovian friction $-\gamma \mathbf{v}$. This construction does not satisfy the generalized fluctuation-dissipation condition⁴⁵. Let us examine the dynamical consequences of Eq. (1) for a free particle in the absence of a potential $W$. Starting from Eq. (1), we use the Wiener-Khinchin theorem to express the Fourier transform of the velocity autocorrelation function as,

$${\tilde{C}}_{vv}(\omega )=\frac{1/m^2}{{\omega }^2+(\gamma /m{)}^2}\left({\tilde{C}}_{FF}(\omega )+{\tilde{C}}_{ff}(\omega )\right)$$ (2)

where ${\tilde{C}}_{FF}(\omega )$, and ${\tilde{C}}_{ff}(\omega )$ are the Fourier transform of the RAMD and Langevin force autocorrelation functions, respectively. Knowing that ${\tilde{C}}_{ff}=2\gamma k_{\text{B}}T$, and

$${\tilde{C}}_{FF}(\omega )=2F_{\text{RAMD}}^2\tau \frac{1}{(\omega \tau {)}^2+1}$$ (3)

It follows that

$${\tilde{C}}_{vv}(\omega )=\frac{2F_{\text{RAMD}}^2}{m^2}\frac{(1/\tau )}{\left({\omega }^2+(\gamma /m{)}^2\right)\left({\omega }^2+(1/\tau {)}^2\right)}+\frac{2d\gamma k_{\text{B}}T}{m^2}\frac{1}{{\omega }^2+(\gamma /m{)}^2}$$ (4)

The effective diffusion coefficient is given by

$$\begin{gathered} D_{\text{eff}}=\frac{1}{2d}{\tilde{C}}_{vv}\left(0\right) \\ =\frac{2F_{\text{RAMD}}^2}{d{m}^2}\frac{1/\tau }{\left({\left(\gamma /m\right)}^2\right)\left(1/{\tau }^2\right)}+\frac{2\gamma k_{\text{B}}T}{d{m}^2}\frac{1}{{\left(\gamma /m\right)}^2} \\ =\frac{1}{\gamma }\left(\frac{F_{\text{RAMD}}^2\tau }{d\gamma }+k_{\text{B}}T\right) \end{gathered}$$ (5)

Using the simple Einstein relation $D=k_{\text{B}}T/\gamma$, the effective diffusion coefficient can be expressed as,

$$D_{\text{eff}}=D\frac{k_{\text{B}}{T}_{\text{eff}}}{k_{\text{B}}T}$$ (6)

Assuming that an Einstein-like fluctuation-dissipation relation holds, $\gamma D_{\text{eff}}=k_{\text{B}}{T}_{\text{eff}}$, this analysis suggests the existence of an effective RAMD temperature,

$$k_{\text{B}}{T}_{\text{eff}}=\frac{F_{\text{RAMD}}^2\tau }{d\gamma }+k_{\text{B}}T$$ (7)

As expected, if $\gamma$ is very large then the Langevin terms (dissipation and random force) act as a strong thermostat on the system and the effect of the RAMD force becomes negligible and $T_{\text{eff}}\to T$. It is also possible to consider the average kinetic energy of the free particle. The inverse Fourier of ${\tilde{C}}_{vv}(\omega )$ yields the time-correlation function,

$$C_{vv}(t)=\frac{F_{\text{RAMD}}^2}{m^2}\frac{1/\tau }{\left(1/{\tau }^2-(\gamma /m{)}^2\right)}\left(\frac{e^{-\left|t\right|\gamma /m}}{\gamma /m}-\frac{e^{-\left|t\right|/\tau }}{1/\tau }\right)+\frac{k_{\text{B}}T}{m}{e}^{-\left|t\right|\gamma /m}$$ (8)

assuming that $\gamma /m\ne 1/\tau$. At $t=0$, we have,

$$m⟨{\mathbf{v}}^2⟩=m{C}_{vv}(0)$$ (9)

$$=\frac{F_{\text{RAMD}}^2\tau }{d\gamma }\frac{1}{1+(\gamma /m)\tau }+k_{\text{B}}T$$ (10)

We note that if the factor $(\gamma /m)\tau$ is small then the average kinetic energy of the particle is consistent with the effective temperature $T_{\text{eff}}$ identified in Eq. (7). While this analysis will not be pursued further, one could also derive formal results for a particle in a harmonic well based on Eq. (1).

C. Modified Smoluchowski equation

To make progress, we neglect the inertial term in the Langevin equation, $md\mathbf{v}/dt$, and consider an overdamped random diffusion model of RAMD,

$$\gamma \frac{d\mathbf{r}}{dt}=-𝛁W+{\mathbf{F}}_{\text{RAMD}}(t)+\mathbf{f}(t)$$ (11)

where $\gamma =k_{\text{B}}T/D$. The assumption is that the long-time evolution of the system can be pictured as some effective Brownian Dynamics (BD)⁴⁶. To determine the long-time evolution of the associated probability $P(\mathbf{r},t)$, we use the Kramers-Moyal expansion⁴⁷,

$$\frac{\partial P}{\partial t}=\frac{1}{\Delta t}𝛁\cdot \left(-{\mathbf{M}}_{1}P+\frac{1}{2}𝛁\left(M_{2}P\right)\right)$$ (12)

where ${\mathbf{M}}_1$ and $M_2$ are the first and second moments, respectively, with respect to the displacement $\Delta \mathbf{r}(\Delta t)=\mathbf{r}(\Delta t)-\mathbf{r}(0)$ over a time step $\Delta t$. To determine the first and second moments, we integrate Eq. (11) over a time-step $\Delta t$,

$$\mathbf{\text{r}}(\Delta t)=\mathbf{r}(0)+\frac{1}{\gamma }\left(-𝛁W\Delta t+{\int }_0^{\Delta t}\left({\mathbf{\text{F}}}_{\text{RAMD}}(t)+\mathbf{\text{f}}(t)\right)dt\right)$$ (13)

The first moment is the mean displacement over a time-step of $\Delta t$,

$$\begin{gathered} {\mathbf{M}}_1=⟨\Delta \mathbf{r}(\Delta t)⟩ \\ =-\frac{D}{k_{\text{B}}T}𝛁W\Delta t \end{gathered}$$

The second moment is the mean-square deviation of the displacement over a time-step of $\Delta t$,

$$\begin{gathered} M_2=\frac{1}{d}〈{\left(\Delta \mathbf{\text{r}}\left(\Delta t\right)\right)}^2〉 \\ =\frac{1}{d{\gamma }^2}\left({\int }_0^{\Delta t}d{t}^{\prime }{\int }_0^{\Delta t}d{t}^{″}(〈{\mathbf{\text{F}}}_{\text{RAMD}}\left(t^{\prime }\right)\cdot {\mathbf{\text{F}}}_{\text{RAMD}}\left(t^{″}\right)〉+〈\mathbf{\text{f}}\left(t^{\prime }\right)\cdot \mathbf{\text{f}}\left(t^{″}\right)〉\right) \end{gathered}$$ (14)

where $⟨{\mathbf{F}}_{\text{RAMD}}\left(t^{\prime }\right){\mathbf{F}}_{\text{RAMD}}\left(t^{″}\right)⟩=C_{FF}\left(t^{\prime }-t^{″}\right)$ is the time correlation function of the RAMD force. Assuming that $C_{\text{FF}}(t)=F_{\text{RAMD}}^2{e}^{-\left|t\right|/\tau }$, we find that

$$M_2=\frac{1}{d{\gamma }^2}{\int }_0^{\Delta t}d{t}^{\prime }{\int }_0^{\Delta t}d{t}^{″}{C}_{\text{FF}}\left(t^{\prime }-t^{″}\right)+2D\Delta t$$ (15)

where $D=k_{\text{B}}T/\gamma$. We perform a change of variables for the double integral $t_1=t^{\prime }-t^{\prime \prime }$ and $t_2=\left(t^{\prime }+t^{\prime \prime }\right)/2$,

$$\begin{gathered} M_2=\frac{1}{d{\gamma }^2}{\int }_{-\Delta t}^{\Delta t}d{t}_1{\int }_{\frac{1}{2}\left|t_1\right|}^{\Delta t-\frac{1}{2}\left|t_1\right|}d{t}_2{C}_{\text{FF}}\left(t_1\right)+2D\Delta t \\ \approx 2D\left(\frac{F_{\text{RAMD}}^2\tau }{d\gamma }+k_{\text{B}}T\right)\Delta t \end{gathered}$$ (16)

In the last step, it was assumed that $\Delta t\gg \tau$. In this approximation, the evolution of the probability $P(\mathbf{r},t)$ associated with the overdamped RAMD diffusion process follows the modified Smoluchowski equation,

$$\frac{\partial P}{\partial t}=𝛁\cdot \left(\frac{D_{\text{eff}}}{k_{\text{B}}{T}_{\text{eff}}}𝛁WP+D_{\text{eff}}𝛁P\right)$$ (17)

where $D_{\text{eff}}$ and $T_{\text{eff}}$ are consistent with Eqs. (5) and (7) from the analysis of the RAMD Langevin equation. The simple diffusion model based on the modified Smoluchowski Eq. (17) makes it possible to examine the relaxation of the system at very long time in the presence of a PMF. In the following, this theory is exploited to derive important results used for the analysis of RAMD simulations.

D. Analytical results from the modified Smoluchowski equation

1. A particle in a harmonic well

The modified Smoluchowski theory allows us to derive explicit expressions regarding the statistical behavior of a system propagated by the overdamped diffusion RAMD dynamics. The objective is to then exploit those expressions to extract the maximum of information from all-atom RAMD simulations. However, the modified Smoluchowski theory associated with the overdamped diffusion RAMD model involves an effective temperature and diffusion coefficient, which are defined by several parameters, $F_{\text{RAMD}}$, and $\tau$, as well as the underlying free energy landscape $W(\mathbf{r})$ and friction $\gamma$. While the present theoretical analysis provides coherent framework, it is important to keep in mind that it is an approximation to a real RAMD simulation. For this reason, there are hidden dependencies among the various parameters when trying to match the results of the (approximate) overdamped diffusion RAMD model to the real RAMD simulations. For example, the lag-time $\tau$ governing the period of time during which the RAMD force is kept constant may be affected by the magnitude of the random force in the true RAMD simulations. Furthermore, the PMF $W(\mathbf{r})$ and the friction $\gamma$ are unknown. All these may shift their value when the magnitude of the external force is altered in real RAMD simulations. For this reason, it may not be possible to extract all the parameters with high confidence. However, it is possible to bypass the uncertainties about these parameters by structuring the analysis around simple expressions corresponding to the situation of a ligand harmonically restrained in the binding site.

Let us first consider the simple situation where the COM of a ligand is restrained by a harmonic potential,

$$U_{\text{harm}}=\frac{1}{2}{K}_{\text{tot}}\Delta {\mathbf{r}}^2$$ (18)

where $K_{\text{tot}}$ is the total force constant affecting the ligand, and $\Delta \mathbf{r}$ is the displacement of the COM relative to its mean position. By the equipartition theorem, the fluctuations of the ligand from a RAMD simulation are,

$${⟨\Delta {\mathbf{r}}^2⟩}_{\text{RAMD}}=d\frac{k_{\text{B}}{T}_{\text{eff}}}{K_{\text{tot}}}$$ (19)

where $d$ is the dimension. In practice, when the analysis is carried out for a ligand in a binding site, $K_{\text{tot}}=W_{\text{min}}^{\prime \prime }+K$, where $W_{\text{min}}^{\prime \prime }$ is the intrinsic curvature of the PMF in the binding site and $K$ is the force constant from the harmonic restraining potential. It follows that the effective temperature can be expressed as,

$$k_{\text{B}}{T}_{\text{eff}}=K_{\text{tot}}\frac{1}{d}{⟨\Delta {\mathbf{r}}^2⟩}_{\text{RAMD}}$$ (20)

Similarly, the effective diffusion coefficient $D_{\text{eff}}$ can be estimated from the relaxation time of the harmonically restrained ligand in RAMD simulations,

$$D_{\text{eff}}=\frac{1}{{\tau }_{rr}d}{⟨\Delta {\mathbf{r}}^2⟩}_{\text{RAMD}}$$ (21)

where ${\tau }_{rr}$ is calculated from the position-position time-correlation function,

$$\begin{gathered} C_{rr}\left(t\right)={\langle \Delta \mathbf{\text{r}}\left(t\right)\cdot \Delta \mathbf{\text{r}}\left(0\right)\rangle }_{\text{RAMD}} \\ ={\langle \Delta {\mathbf{\text{r}}}^2\rangle }_{\text{RAM}{\text{D}}^{e^{-t/{\tau }_{rr}}}} \end{gathered}$$ (22)

using the identity,

$${\tau }_{\text{rr}}={\int }_0^{\infty }\frac{C_{rr}\left(t^{\prime }\right)}{C_{rr}(0)}d{t}^{\prime }$$ (23)

In the absence of RAMD, ${\tau }_{rr}$ reduces to the unperturbed diffusive relaxation time of the particle in the harmonic well, ${\tau }_{\text{harm}}=k_{\text{B}}T/D{K}_{\text{tot}}$. This simple analysis provides direct estimates of the effective temperature $T_{\text{eff}}$ and effective diffusion coefficient $D_{\text{eff}}$. This also makes possible to examine the linear relation between $T_{\text{eff}}$ and $D_{\text{eff}}$ to verify the validity and accuracy of the modified Smoluchowski overdamped dynamic RAMD model based on Eq. (17).

2. Mean escape time over a free-energy barrier

We now consider the mean escape time of a bound ligand over a free-energy barrier. This represents a critical step toward a rational interpretation of ligand residence times determined from real RAMD simulations. We consider a system with a bound (A) and unbound (B) states separated by a large energy barrier, and assume that the unbinding process occurs along the reaction coordinate $x$. Fluctuations orthogonal to the reaction coordinate $x$ are assumed to have no impact on the escape process. The mean escape time from state A can be derived following the classic Kramers-Smoluchowski theory^48,49, with a effective temperature $T_{\text{eff}}$ and diffusion coefficient $D_{\text{eff}}$,

$${\tau }_{\text{esc}}=\frac{1}{D_{\text{eff}}}{\int }_{x^{\prime }\in A}{e}^{-W\left(x^{\prime }\right)/k_{\text{B}}{T}_{\text{eff}}}d{x}^{\prime }{\int }_{x_{\text{A}}}^{x_{\text{B}}}{e}^{W\left(x\right)/k_{\text{B}}{T}_{\text{eff}}}dx$$ (24)

where the first integral is over the basin A and the second is over the barrier. The integrals are approximated using a quadratic expansion at the bottom of the reactant well at $x_A$ and the top of the free energy barrier located at $x^{†}$ is given by^50–52,

$$\begin{gathered} {\tau }_{\text{esc}}=\frac{k_{\text{B}}{T}_{\text{eff}}}{D_{\text{eff}}}\frac{2\pi }{\sqrt{W^{″}\left(x_A\right)W^{″}\left(x^{†}\right)}}{e}^{\Delta W^{†}/k_{\text{B}}{T}_{\text{eff}}} \\ ={\tau }_0{e}^{\Delta W^{†}/k_{\text{B}}{T}_{\text{eff}}} \end{gathered}$$ (25)

where $\Delta W^{†}=W\left(x^{†}\right)-W\left(x_A\right)$ is the activation free energy. According to the overdamped RAMD model, the dynamical prefactor ${\tau }_0$ is unaffected by the the random force $F_{\text{RAMD}}$ since $D_{\text{eff}}/k_{\text{B}}{T}_{\text{eff}}=D/k_{\text{B}}T$. The implication is that the mean escape time ${\tau }_{\text{esc}}$ in RAMD can be written as ${\tau }_0{e}^{\Delta W^{†}/k_{\text{B}}{T}_{\text{eff}}}$, where only the dimensionless quantity $\Delta W/k_{\text{B}}{T}_{\text{eff}}$ depends on the presence of the random force. If this assumption about the linear dependence of $D_{\text{eff}}$ and $k_{\text{B}}{T}_{\text{eff}}$ is not verified in practice, one can treat ${\tau }_0$ as some constant times $\text{ln}\left(D_{\text{eff}}/k_{\text{B}}{T}_{\text{eff}}\right)$.

We can rewrite the expression for the mean escape time from a well as,

$$\text{ln}\left({\tau }_{\text{esc}}\right)=\frac{\Delta W^{†}}{k_{\text{B}}T}\alpha +\text{ln}\left({\tau }_0\right)$$ (26)

where the dimensionless quantity $\alpha =T/T_{\text{eff}}$ corresponds to the inverse of the relative effective temperature. This result explains the relation between $F_{\text{RAMD}}$ and the logarithm of the ligand dissociation time that was noted previously³⁵. Because $\alpha <1$, Eq. (26) makes it clear that the RAMD algorithm reduces the mean escape time of a bound ligand via an increased effective temperature. Furthermore, this expression, displaying a linear relationship between $\text{ln}\left({\tau }_{\text{esc}}\right)$ and $\alpha$, is very convenient because it provides a simple route to determine an unbiased estimate of the mean escape time from the RAMD simulations.

E. Limitations of the overdamped RAMD approximation

The above results, obtained in the overdamped diffusive limit, assume implicitly that the RAMD correlation time $\tau$ is much shorter than ${\tau }_{\text{harm}}$, the unperturbed relaxation time of the ligand in the harmonic well. Further analysis (see Appendix) reveal that the effective temperature estimated from the quadratic fluctuations in a harmonic well has the form,

$$\begin{gathered} k_{\text{B}}{T}_{\text{eff}}=\frac{K_{\text{tot}}}{d}{\langle \Delta {\mathbf{\text{r}}}^2\rangle }_{\text{RAMD}}\text{} \\ =\frac{F_{\text{RAMD}}^2\tau }{d\gamma }\left(\frac{1}{\eta +1}\right)+k_{\text{B}}T \end{gathered}$$ (27)

where $\eta =\tau /{\tau }_{\text{harm}}$. Thus, Eq. (7) is valid only if the dimensionless parameter $\eta =\left(\tau /{\tau }_{\text{harm}}\right)\ll 1$. Eq. (27) shows that the effective temperature as defined is actually dependent on $\eta$. Similarly, the effective diffusion coefficient estimated from the relaxation time ${\tau }_{rr}$ in a harmonic retain the same form as in Eq. (21), but with ${\tau }_{rr}$ given by,

$${\tau }_{rr}={\tau }_{\text{harm}}\frac{k_{\text{B}}{T}_{\text{eff}}(\eta =0)}{k_{\text{B}}{T}_{\text{eff}}(\eta )}$$ (28)

If $\eta$ is negligible then ${\tau }_{rr}$ is equal to ${\tau }_{\text{harm}}$. A similar analysis would affect the imaginary relaxation time at the top of an inverted harmonic barrier in the case of an activated process⁵³.

These results, which will be illustrated with explicit BD simulations in the next section, shows the limitations of an analysis of RAMD simulations based on the overdamped diffusive model. For this reason, the analysis is expected to be more accurate in the limit of high friction and low curvature of the potential. Hopefully, those conditions are satisfied in the context of RAMD simulations of ligand unbinding, but should be verified for each particular case.

III. RESULTS

A. Simulations of the overdamped RAMD model

To verify and validate the theoretical analysis, simulations were carried out according to the simplified overdamped diffusive model of the RAMD algorithm. In these RAMD simulations, the particle is evolving in $d$ dimensions and an external force of fixed magnitude $F_{\text{RAMD}}$ is applied in a random direction over an exponentially distributed time period, after which a new direction is picked randomly. The exponential distribution is characterized by a correlation time $\tau$. In practice, the behavior of the system was examined for three simple scenarios. The simplest situation concerns the effective diffusion coefficient $D_{\text{eff}}$ of a free particle in $d$ dimension as expressed from its mean squared displacement (MSD),

$$D_{\text{eff}}=\underset{t\to \infty }{\text{lim}}\frac{1}{2dt}⟨(R(t)-R(0){)}^2⟩$$ (29)

According to Eq. (5), $D_{\text{eff}}$ depends quadratically on the magnitude of the applied random force $F_{\text{RAMD}}$. This was explicitly tested by carrying out simulations with different values of $F_{\text{RAMD}}$. The results are shown in Figure 1. Direct comparison of the effective diffusion coefficient, $D_{\text{eff}}$, as a function of the applied force, $F_{\text{RAMD}}$, with the theoretical prediction (Eq. 5) validates the theoretical analysis.

Figure 1.

Comparison between theoretical predictions and simulation results for the effective diffusion coefficient $D_{\text{eff}}$ as a function of the magnitude of the external force $F_{\text{RAMD}}$. The theory is from Eq. (5). In these simulations, $T=300\text{K},D={0.0005}^2/\text{ps}$. The integration time step is 0.01 ps, and the total simulation time is 100000 ps. The correlation time of the RAMD random force was set to 50 ps. Statistics are obtained by repeating the simulation 10 times for each value of $F_{\text{RAMD}}$.

The second situation concerns the fluctuations of a particle restrained by a harmonic potential. Both the effective temperature $T_{\text{eff}}$ and effective diffusion coefficient $D_{\text{eff}}$ depend quadratically on $F_{\text{RAMD}}$. A comparison between the theoretical analysis and the simulated results is shown in Figure 2. As with the freely diffusing particle, the good agreement confirms the validity of the theoretical analysis. One may note that the variations in the case of $D_{\text{eff}}$ are noticeably larger than for $T_{\text{eff}}$, suggesting that the effective temperature is perhaps a more robust feature of the system than the diffusion coefficient. Nonetheless, the comparison shows that the fluctuations of a harmonically restrained ligand can be analyzed using Eqs. (20) and (21), we can estimate to determine the effective temperatures and diffusion coefficient in RAMD simulations.

Figure 2.

Comparison of simulation and theoretical results for the effective temperature $T_{\text{eff}}$ (top) and effective diffusion constant $D_{\text{eff}}$ (bottom) of a harmonically restrained particle as a function of the magnitude of the random forces $F_{\text{RAMD}}$ and total restraining force constants $K_{\text{tot}}$. Dashed lines represent theoretical predictions, while solid markers with error bars represent simulation results. The theory is from Eq. (27) for $T_{\text{eff}}$, and from Eq. (21) for $D_{\text{eff}}$ with ${\tau }_{rr}$ from Eq. (28).

The last situation concerns the escape of a bound particle over an energy barrier. To model the system using the three-dimensional potential,

$$W(x,y,z)=\frac{1}{2}k\left(x^2+y^2\right)+50e^{-10z}+\Delta W^{†}\text{exp}\left(-\frac{(z-p{)}^2}{2w}\right)$$ (30)

where $k$ is the spring constant confining the particle in the $x-y$ plane, $\Delta W^{†}$ is the height of the free energy barrier along the $z$-axis, and $w$ determines the barrier width. We used $k=10,\Delta W^{†}=2.9575,p=5$, and $w=1$ in the simulations. The escape time as a function of $\alpha =T/T_{\text{eff}}$ are shown in Figure 3. Taking the dependence of the renormalized temperature $T_{\text{eff}}$ on $F_{\text{RAMD}}$ into account, it is observed that the results obtained from the RAMD simulations closely match the predictions of KS theory. The excellent agreement suggests that the theoretical analysis based on Eq. (26) provides a viable approach for estimating the effect of $F_{\text{RAMD}}$ on the escape times obtained from RAMD simulations.

Figure 3.

Comparison between simulations and theoretical predictions for the escape time over a potential energy barrier. Dash lines represent theoretical results, while markers with error bars represent simulation data. The result of linear regression is $y=5.00x+9.3405$, which is near the slope of 5 predicted from KS theory.

Based on these results, we propose the ensuing protocol to obtain unbiased estimates of the escape time ${\tau }_0$ from RAMD simulations:

1. Carry out several RAMD simulations to determine the biased escape time of the ligand, ${\tau }_{\text{esc}}$ as a function of $F_{\text{RAMD}}$. The values of the RAMD force should be sufficiently large to accelerate the unbinding within a reasonable simulation time, while still generating semi-realistic trajectories.

2. Carry out additional RAMD simulations for the same set of $F_{\text{RAMD}}$ with the ligand harmonically restrained in the binding pocket to determine the fluctuation of the ligand COM, hence the effective temperature $T_{\text{eff}}$ (Eq. (20)) as well as the effective diffusion coefficient $D_{\text{eff}}$ (Eq. (21)).

3. Perform a linear least-square fit using the pair of data points $\alpha =T_{\text{eff}}/T$ (as abscissa) and $\text{ln}\left({\tau }_{\text{esc}}\right)$ (as ordinate) determined for each value of $F_{\text{RAMD}}$ on the basis of Eq. (26). The slope of the linear relation corresponds to the activation free energy $\Delta W^{†}$, and the extrapolation $\alpha \approx 1$ yields the unbiased mean escape time along the ordinate.

While this itemized protocol can be rationally justified, it is important to recognize that the theory and analysis presented here is an approximate construct based on specific assumptions about the RAMD simulations. One of those assumptions is the exponential distribution of the time during which the random force remains constant before it changes its direction. Another potential weakness is the inherent friction affecting the the ligand dynamics, which translates into a sensitivity of the effective temperature on the dimensionless factor $\eta$ related to the correlation time $\tau$ and the relaxation time ${\tau }_{rr}$ of the ligand in the binding pocket. It shall generally make sense to complete the analysis by validating these assumptions and verify if they are valid in the RAMD simulations.

B. RAMD simulations of Trypsin-Benzamidine with explicit solvent

The RAMD simulation method is applied to the Trypsin-Benzamidine protein-ligand complex, depicted in Figure 4. The complex model was modeled starting from the crystallographic structure with PDB ID 3PTB⁵⁴. More information about the modeling can be found in Supplementary Information. The RAMD simulations were carried out using NAMD3⁵⁵, following the $\tau \text{RAMD}$ protocol describe previously³⁰. The temperature was kept constant at 300 K using a Langevin thermostat, and the pressure was kept constant at 1 atm using a Langevin barostat. The distance between the COMs of Trypsin and Benzamidine was evaluated every 0.1 ps (50 steps). If this distance exceeded a threshold of 0.025 Å, the current force direction was maintained; otherwise, a new random force direction was selected. This process was repeated until the distance between the protein and ligand COMs reached 40 Å.

Figure 4.

Structural representation of the Trypsin-Benzamidine complex. The ligand Benzamidine is depicted in light green, while the protein Trypsin is shown using a grey ribbon representation overlaid with a grey surface.

The most relevant parameter to define in $\tau \text{RAMD}$ is the magnitude of the random force, $F_{\text{RAMD}}$. Experimental measurements indicate that Benzamidine exhibits a short residence time in complex with Trypsin (≈ 0.0016 s); thus we selected a range of relatively small force magnitudes: $F_{\text{RAMD}}$ from 1.4 to 14 kcal/molÅ (from 58.5 to 585 kJ/mol nm). For each force value, ten independent RAMD simulations were performed to simulate the dissociation of Benzamidine from Trypsin. Different random seeds were used in the different replicas, generating different initial force directions. The escape time at each $F_{\text{RAMD}}$ (i.e., computational unbinding time) was defined as the bootstrap average of the escape times registered in the ten independent RAMD simulations (Table S1). The logarithm of the escape time as a function of $F_{\text{RAMD}}$ is shown in Figure S1. However, at the lowest force magnitude $(F_{\text{RAMD}}=1.4\text{kcal}/\text{mol}Å)$, the ligand did not dissociate in all simulated trajectories. The escape time for this force magnitude was estimated by accounting for the probability of the unbinding event to occur and not to occur within the simulated time, as better detailed in Supplementary Information.

To determine the fluctuation of the ligand COM, and hence $\alpha =T/T_{\text{eff}}$, additional RAMD simulations of the Trypsin-Benzamidine protein-ligand complex were carried out in the presence of a harmonic restraining potential to prevent the dissociation of the ligand. For each RAMD force in the considered range $(F_{\text{RAMD}}=1.4-14\text{kcal}/\text{mol}/Å)$, independent simulations were run with different force constants of the harmonic restraining potential $K$ applied to the ligand COM. The force constant varied from 1 to 20 kcal/mol Ų depending on the value of $F_{\text{RAMD}}$ to ensure that the ligand remains inside the binding pocket (more information about the combination of $F_{\text{RAMD}}$ and $K$ used can be found in Supplementary Information: Figures S2 and S3, Tables S2 and S3).

As shown in Figure 5, the magnitude of the RAMD force has a considerable impact on the distribution of the lag time $\tau$, i.e., the time interval between consecutive force changes. For large RAMD forces, only a limited number of fore directions are sample during the RAMD simulations. This occurs because the distance traveled by the ligand COM is in most cases higher than the defined threshold of 0.025 Å. In contrast, for smaller $F_{\text{RAMD}}$ a larger number of random directions is sampled during RAMD simulations, and the lag time $\tau$ exhibits an exponential distribution.

Figure 5.

Analysis of the lag time $\tau$ between consecutive changes in the applied RAMD force. Top) Probability distribution of the lag time extracted from a RAMD simulation with a force $F_{\text{RAMD}}$ equal to 1.4 kcal/molÅ. The escape time of the ligand is 113 ns, with 2245 changes in the direction of the RAMD force. Bottom) Probability distribution of the lag time extracted from a RAMD simulation with a a force $F_{\text{RAMD}}$ equal to 5.6 kcal/molÅ. The escape time of the ligand is 953 ps, with 23 changes in the direction of the force.

To better sample the distribution of lag times between consecutive force changes, we calculated the lag time $\tau$ from the RAMD simulations in the presence of a harmonic restraining potential. The logarithm of the correlation function $⟨{\mathbf{F}}_{\text{RAMD}}(t)\cdot {\mathbf{F}}_{\text{RAMD}}(0)⟩$ for different magnitudes of the RAMD force is shown in Figure 6. The result shows that the exponential decay with a single correlation time is remarkably robust, even when the magnitude of $F_{\text{RAMD}}$ is excessive (i.e., larger than 5 kcal/mol Å). This is encouraging because an important assumption of our approximate model of RAMD based on Eq. (1) is that the random force is a random variable that follows an exponential distribution with a fixed correlation time $\tau$. This assumption is strongly supported by Figure 6. This empirical finding validates the assumption underlying the theoretical development.

Figure 6.

Fitting the correlation time $\tau$ for different RAMD simulations. The dashed lines are simulation results, and the solid lines are the fits. $F_{\text{RAMD}}=1.4,\tau =48.03;F_{\text{RAMD}}=2.8,\tau =40.71;F_{\text{RAMD}}=4.2,\tau =42.82;F_{\text{RAMD}}=5.6,\tau =42.94;F_{\text{RAMD}}F=7,\tau =49.85;F_{\text{RAMD}}=10,\tau =39.13;F_{\text{RAMD}}=12,\tau =42.03;F_{\text{RAMD}}=14,\tau =45.34.F_{\text{RAMD}}$ is in kcal/mol Å, and $\tau$ is in ps.

It is possible to determine the stationary fluctuations of the COM of the ligand, ${⟨\Delta {\mathbf{r}}^2⟩}_{\text{RAMD}}$, from the RAMD simulations performed with the harmonic restraining potential. From the stationary fluctuations, one can estimate the effective temperature, $T_{\text{eff}}$, based on Eq. (20) and the effective diffusion coefficient, $D_{\text{eff}}$, based on Eq. (21), both as a function of the random force magnitude $F_{\text{RAMD}}$. The results are shown in Figure 7. The diffusion coefficient is not absolutely necessary for the unbiasing procedure, but it provides a valuable piece of information to better interpret the results. As shown in Figure 7, both the effective temperature and the effective diffusion coefficient exhibit a dependence on $F_{\text{RAMD}}^2$. However, while the estimated effective temperature appears to be consistent with Eq. (7), there are considerable variations with regards to the effective diffusion coefficient. In Figure 7, the estimated $D_{\text{eff}}$ for different values of the harmonic restraint as a function of $F_{\text{RAMD}}$ increase faster than predicted by Eq. (5). Since this analysis is based on an overdamped diffusion in a harmonic potential, we conjecture that these deviations in $D_{\text{eff}}$ from the prediction are due to the presence of some spatial ruggedness $\delta w$ in the underlying PMF felt by the ligand in the binding pocket. In one-dimension, the effect of $\delta w$ in a rugged potential gives rise to an effective diffusion coefficient,^56,57

$$D_{\text{rugged}}=D{⟨e^{-\delta w/k_{\text{B}}{T}_{\text{eff}}}⟩}^{-1}{⟨e^{+\delta w/k_{\text{B}}{T}_{\text{eff}}}⟩}^{-1}$$ (31)

where the bracket $⟨\dots ⟩$ represents a spatial average over the diffusion space and $D=k_{\text{B}}T/\gamma$ is the unperturbed diffusion coefficient from the overdamped BD dynamics Eq. (11). As the effective temperature rises, the diffusion coefficient increases because the effect of the ruggedness decreases. While the inaccuracy of the estimated diffusion coefficient from Eq. (5) shows the limitations of the approximate treatment, the consequences on the present analysis are limited. In practice, the value of $D_{\text{eff}}/D$ varies at most between 1 and around 3.5 in Figure 7. While the value $D_{\text{eff}}$ is helpful to assess the validity of the analysis, ultimately this quantity does not have a large effect on the mean escape time, which depends exponentially on $T_{\text{eff}}$. Therefore, the quantitative impact of $D_{\text{eff}}$ is limited.

Figure 7.

Estimates of $T_{\text{eff}}$ and $D_{\text{eff}}$ from RAMD simulations of the Trypsin-Benzamidine complex with a harmonic restraint using Eqs. (20) and (21). Simulations were performed with different values of the random force $\left(F_{\text{RAMD}}=\mathrm{1.4,2.8,4.2,5.6,7.0}\right.,\mathrm{10.0,12.0,14.0}\text{kcal}/\text{mol}Å)$ and of the force constant of harmonic restraining potential relative the equilibrium bound pose $(K=1,\mathrm{2,3},\mathrm{5,5},\mathrm{7,10,12,15,17,20}\text{kcal}/\text{mol}{Å}^2)$. The black line follows the effective temperature as $T_{\text{eff}}/T=\left(0.022F_{\text{RAMD}}^2+1.0\right)$.

Using the estimated effective temperature determined from Figure 7, we used the linear form Eq. (26) to analyze the results of the escape time obtained from the RAMD simulations. The natural logarithm of the escape time at each force magnitude is plotted against the corresponding $T/T_{\text{eff}}$ in Figure 8 and fitted with a linear regression, according to Eq. (26). The fitting was also performed considering only force magnitudes from 1.4 to 7 kcal/mol Å, as shown in Figure S4.

Figure 8.

Fitting the linear form based on Eq. (26). The solid line follows $\text{ln}\left({\tau }_{\text{esc}}\right)=12.29\alpha +0.38$, where $\alpha =\left(T/T_{\text{eff}}\right).\Delta W^{†}=7.32\text{kcal}/\text{mol}$ and ${\tau }_0=1.46\text{ps}$.

It is clear from Figure 8 that the linear form extracted from the theoretical analysis is satisfied, which is indicative that the natural logarithm of the average escape time is related to the magnitude of the random force $F_{\text{RAMD}}$. The slope is related to the activation free energy $\Delta W^{†}$, suggesting a value on the order of $12.3k_{\text{B}}T$. The unbiased residence time of the Trypsin-Benzamidine complex can be estimated by linearly extrapolating the data to $\alpha =1$, yielding a value about 320 ns (considering only $F_{\text{RAMD}}=1.4-7\text{kcal}/\text{mol}Å$, the unbiased ${\tau }_{\text{esc}}$ is 900 ns). While this estimate is much shorter than the experimental value of about 1.6 ms³⁶, reported values from various computational methods span a wide range, from $10\mu \text{s}$ to 300 ms^{16–24,37–43}.

Although the application of the proposed approach to the Tryspin-Benzamidine system resulted in an absolute unbinding time not consistent with the experimental data, it showcases the consistency of the theoretical framework, as demonstrated by the linear relationship between $\text{ln}\left({\tau }_{\text{esc}}\right)$ and $\alpha$ reported in Figure 8. Importantly, this application showed that a wide range of force magnitudes is not required to obtain this linear relation; it can also be captured using only relatively high random force values. The advantage is that instead of estimating the absolute unbinding time relying on few long simulations, several short and parallel RAMD trajectories can be used. Therefore, this theoretical framework represents a promising basis to improve the feasibility of absolute kinetics estimation in drug discovery.

For the Trypsin-Benzamidine system, considering the short experimental time that the ligand spends in contact with the protein (≈ 0.0016 s), it is expected that high force magnitudes results in very fast escape times. Indeed, only for force magnitudes lower than 10 kcal/mol Å the average escape time was in the order of nanoseconds, reaching 10¹ and 10² ns for smaller forces. Thus, simulations in the order of nanoseconds or shorter can be used to obtain the linear relationship shown in Figure 8.

IV. CONCLUSION

In this article, we have developed a novel theoretical framework to better understand the underlying effect of the RAMD simulation algorithm. In particular, a special focus was put on how RAMD simulations accelerate the mean escape time of a ligand bound to a receptor protein. In our analysis, we assumed that the RAMD simulations based on the $\tau \text{RAMD}$ protocol³⁰ can be approximated by an overdamped Langevin dynamics with an additional random force. By construction, we assumed that the force had a fixed magnitude $F_{\text{RAMD}}$ but changed its direction randomly at exponentially distributed random times. These simplifying assumptions allowed us to represent the RAMD dynamics of a bound ligand in terms of a modified overdamped Langevin equation. A number of formal results could be obtained on the basis of this simple theory. In particular, it was shown that dynamics of the model is roughly consistent with that of an effective Smoluchowski equation with effective temperature $T_{\text{eff}}$ and diffusion coefficient $D_{\text{eff}}$ which depend on the magnitude of the random force $F_{\text{RAMD}}$. This analysis shows that the overall RAMD dynamics is “accelerated” because the effective temperature and the diffusion coefficient increase quadratically with $F_{\text{RAMD}}$. In retrospect, the most useful result of the present analysis is the expression for the effective RAMD temperature given by Eq. (7). Following a classic Kramers treatment of the escape of a particle over a free energy barrier based on the effective Smoluchowski equation it becomes immediately clear how the RAMD simulations accelerate the dissociation rate of a bound ligand. To some extent, the effect of the effective temperature bears some similarities to what is achieved with the solute-tempering REST2 method⁵⁸, although RAMD is not explicitly scaling any component of the potential as is done in REST2. Furthermore, there are also additional kinetic effects as the diffusion coefficient is also increased. All these factors contribute to accelerate activated processes in RAMD simulations. The theoretical framework was used to analyze RAMD simulations monitoring the dissociation of the Trypsin-Benzamidine protein-ligand complex. The results show that the theoretical framework is sound and could potentially be used to unbias the results from RAMD simulations. The present treatment drawing from an effective Smoluchowski equation is based on the strong assumption of overdamped diffusive dynamics, and it is unavoidably imperfect. One possible source of inaccuracy may arise from the neglect non-Markovian effects related to the inherent relaxation times in the system relative to the RAMD lag time $\tau$. The test with the Trypsin-Benzamidine protein-ligand complex suggest that the analysis of RAMD simulations can overcome those limitations. Lastly, while the method has been only applied in the context of ligand unbinding, it could in principle be used to accelerate the dynamics of any selected degree of freedom.

Supplementary Material

The Supporting Information is available free of charge and includes:

• Modeling of the Trypsin-Benzamidine complex;

• Escape time for RAMD simulations with a low force magnitude;

• Escape times values at different force magnitudes;

• RAMD simulations with the ligand harmonically restrained in the binding pocket;

• Unbiased escape time from RAMD simulation with $F_{\text{RAMD}}=1.4-7\text{kcal}/\text{mol}\cdot Å$

APPENDIX: PARTICLE IN A HARMONIC WELL

Assuming that the overdamped RAMD dynamics of a particle in a harmonic well follows Eq. (11), the Fourier transform of the position-position time-correlation function is,

$${\tilde{C}}_{rr}(\omega )=\frac{1}{{\omega }^2+{\left(1/{\tau }_{\text{harm}}\right)}^2}\left(\frac{D}{k_{\text{B}}T}\frac{1}{\gamma }{\tilde{C}}_{FF}(\omega )+{\tilde{C}}_{ff}(\omega )\right)$$ (32)

where ${\tilde{C}}_{ff}(\omega )=2dD$, and ${\tilde{C}}_{\text{RAMD}}(\omega )=2F_{\text{RAMD}}^2\tau /\left(1+(\omega \tau {)}^2\right)$. The Fourier transform ${\tilde{C}}_{rr}(\omega )$ can be simplified as

$${\tilde{C}}_{rr}\left(\omega \right)=\frac{2F_{\text{RAMD}}^2}{\gamma \tau }\left(\frac{D}{k_{\text{B}}T}\right)\frac{1}{{\left(1/{\tau }_{\text{harm}}\right)}^2-{(1/\tau )}^2}\left(\frac{1}{{\omega }^2+{(1/\tau )}^2}\right)+\left(2dD-\frac{2F_{\text{RAMD}}^2}{\gamma \tau }\left(\frac{D}{k_{\text{B}}T}\right)\frac{1}{\left(1/{\tau }_{\text{harm}}\right)-{(1/\tau )}^2}\right)\left(\frac{1}{{\omega }^2+{\left(1/{\tau }_{\text{harm}}\right)}^2}\right)$$ (33)

The inverse Fourier of ${\tilde{C}}_{rr}(\omega )$ yields the time-correlation function,

$$C_{rr}(t)=\frac{F_{\text{RAMD}}^2{\tau }^2}{\gamma }\left(\frac{D}{k_{\text{B}}T}\right)\frac{1}{{\eta }^2-1}{e}^{-\left|t\right|/\tau }+\frac{1}{K}\left(d{k}_{\text{B}}T-\frac{F_{\text{RAMD}}^2\tau }{\gamma }\left(\frac{1}{{\eta }^2-1}\right)\right)e^{-\left|t\right|/{\tau }_{\text{harm}}}$$ (34)

where $\eta =\tau /{\tau }_{\text{harm}}$. At $t=0$, we have

$$\begin{gathered} C_{rr}\left(0\right)={〈\Delta {\mathbf{\text{r}}}^2〉}_{\text{RAMD}} \\ =\frac{d}{K}\left(\frac{F_{\text{RAMD}}^2\tau }{d\gamma }\frac{1}{\eta +1}+k_{\text{B}}T\right) \\ =\frac{d}{K}{k}_{\text{B}}{T}_{\text{eff}}\left(\eta \right) \end{gathered}$$ (35)

The effective diffusion coefficient estimated from the relaxation time ${\tau }_{rr}$ in a harmonic potential is given by Eq. (21), $D_{\text{eff}}={⟨\Delta {\mathbf{r}}^2⟩}_{\text{RAMD}}/\left({\tau }_{rr}d\right)$, with ${\tau }_{rr}$ given by,

$$\begin{gathered} {\tau }_{rr}=\frac{{\tilde{C}}_{rr}\left(0\right)/2}{{〈\Delta {\mathbf{\text{r}}}^2〉}_{\text{RAMD}}} \\ =\frac{\left(d{\tau }_{\text{harm}}^2\right)\left(\frac{D}{k_{\text{B}}T}\frac{F_{\text{RAMD}}^2\tau }{d\gamma }+D\right)}{{〈\Delta {\mathbf{\text{r}}}^2〉}_{\text{RAMD}}} \\ ={\tau }_{\text{harm}}\frac{T_{\text{eff}}\left(\eta =0\right)}{T_{\text{eff}}\left(\eta \right)} \end{gathered}$$ (36)

When $\eta$ is negligible, then ${\tau }_{rr}$ is identically equal to ${\tau }_{\text{harm}}$.