Free energy and kinetic rate calculation via non-equilibrium molecular simulation: application to biomolecules

Abstract

Non-equilibrium molecular dynamics (NEMD) simulation has been recognized as a powerful tool for examining biomolecules and provides fruitful insights into not only non-equilibrium but also equilibrium processes. We review recent advances in NEMD simulation and relevant, fundamental results of non-equilibrium statistical mechanics. We first introduce Crooks fluctuation theorem and Jarzynski equality that relate free energy difference to work done on a physical system during a non-equilibrium process. The theorems are beneficial for the analysis of NEMD trajectories. We then describe rate theory, a framework to calculate molecular kinetics from a non-equilibrium process; this theoretical framework enables us to calculate a reaction time—mean-first passage time—from NEMD trajectories. We, in turn, present recent NEMD techniques that apply an external force to a system to enhance molecular dissociation and introduce their application to biomolecules. Lastly, we show the current status of an appropriate selection of reaction coordinates for NEMD simulation.

Introduction

Classical molecular dynamics (MD) simulation has now become a core technology in biology since the first attempt of MD simulation in 1977 for a folded globular protein, bovine pancreatic trypsin inhibitor (McCammon et al. 1977). It allows us to investigate biomolecules in atomistic resolution and provides insights into structural fluctuations (Iida et al. 2020), a free energy profile (Iida et al. 2016), the kinetics of association and dissociation (Decherchi and Cavalli 2020), and drug design (De Vivo et al. 2016). It goes without saying that the effectiveness and the broad applications of MD simulation in biology are due not only to theoretical developments but also to the information contained within the database of protein structures, known as the Protein Data Bank (Burley et al. 2017).

Although equilibrium MD simulation can track conformational changes, it has difficulty sampling rare events. Rare events involve a high-energy barrier and are challenging to sample during MD simulation. The rare event problem is effectively addressed by the use of enhanced sampling methods, e.g., replica exchange MD (Sugita and Okamoto 1999), accelerated MD (Hamelberg et al. 2004; Miao et al. 2015; Miao and McCammon 2017), metadynamics (Laio and Parrinello 2002; Barducci et al. 2011), multicanonical MD (Berg and Neuhaus 1992; Hansmann and Okamoto 1993; Nakajima et al. 1997), and adapting biasing force MD (Darve et al. 2008). Recent advances in generalized ensemble approaches are reviewed in the following references (Yang et al. 2019; Liao 2020; Bussi and Laio 2020).

Non-equilibrium molecular dynamics (NEMD) simulation also plays a vital role in examining liquid or biomolecules (Ewen et al. 2018; Nunes-Alves et al. 2020). It adds external field, e.g., sheer, voltage, or time-dependent force, to observe the response of physical processes. In the 1990s, NEMD simulation and relevant results of statistical mechanics had been actively developed. Some of the pioneering works developed targeted MD (TMD) simulation (Schlitter et al. 1993, 1994) and steered MD (SMD) simulation (Izrailev et al. 1999). In the simulation methods, reaction coordinates (or collective variables) are defined along the direction in which a desired process is boosted; an appropriate selection of reaction coordinates is also an active topic in the field of NEMD. So far, variants of the methods have been proposed (Boubeta et al. 2019; Nunes-Alves et al. 2020; Ahmad et al.). Recent NEMD techniques are, for example, adaptive SMD (Ozer et al. 2010), dissipated-corrected TMD (Wolf and Stock 2018), random acceleration MD (RAMD) (Kokh et al. 2018), and constant-force SMD (Iida and Kameda 2022). These simulation techniques make rare events tractable, e.g., ligand unbinding (Truong and Li 2018; Kokh et al. 2018; Potterton et al. 2019), ion transport (Jalily Hasani et al. 2018; Jäger et al. 2022), and protein unfolding (Motta et al. 2021). It is also recognized that SMD is a powerful tool for drug design (Suan Li and Khanh Mai 2012; Do et al. 2018).

In addition to the development of NEMD, fluctuation theorems of non-equilibrium statistical mechanics were formulated in the 1990s. D. J. Evans et al. discovered a fluctuation theorem for the first time (Evans et al. 1993). Afterward, C. Jarzynski independently found equality that relates the free energy difference to work done on a physical system (Jarzynski 1997). The Jarzynski equality makes it possible to calculate a free energy difference from work obtained from a set of NEMD simulations. Quickly after the equality was formulated, G. E. Crooks found another fluctuation theorem, Crooks fluctuation theorem (Crooks 1998, 1999), that relates work distributions of forward and reverse non-equilibrium processes to free energy change. During this time, crucial results of non-equilibrium statistical mechanics were formulated, and they have been utilized for the analysis of NEMD simulation.

We can utilize rate theory to analyze non-equilibrium processes generated by NEMD. Rate theory is a framework for solving the problem of how long it takes for a reaction to escape from a stable state. Arrhenius law is, for example, the most famous phenomenological relation between rate constant (or escape rate) and an activation barrier. Kramers derived the law from the investigation of Brownian motion under a potential from first principles (Kramers 1940). Kramers rate theory has been extended to physical models that extract kinetic information from pulling experiments (Bell 1978; Evans and Ritchie 1997; Dudko et al. 2006, 2008; Friddle et al. 2012) and are helpful to interpret non-equilibrium dynamics. Rate theory and historical background were well reviewed in Hänggi et al. (1990).

The present review deals with the theoretical background of non-equilibrium statistical mechanics and NEMD techniques. Firstly, we describe fluctuation theorems—Crooks fluctuation theorem and Jarzynski equality—that underpin the free energy calculation of NEMD. Secondly, we explain Kramers rate theory and its extension (Bell’s and Dudko-Hummer-Szabo (DHS) models) and show how to extract rate constant from non-equilibrium processes. Thirdly, we describe a variety of NEMD simulations, e.g., SMD and TMD. Lastly, we introduce recent studies investigating an optimal selection of reaction coordinates.

Non-equilibrium statistical mechanics

In this section, we introduce Crooks fluctuation theorem and Jarzynski equality that underlie free energy calculation from NEMD simulation. Let us set notation. Suppose a classical system that is specified by a microscopic state coordinates $\mathit{r}(t)$ and momenta $\mathit{p}(t)$ and is in contact with a heat bath at a temperature $T$. A control parameter $\lambda (t)$, which depends on time, is introduced such that the Hamiltonian of the system $H(\mathit{r},\mathit{p};\lambda )$ is coupled with the parameter. The microscopic states are sampled from the distribution $Z_{\lambda }^{-1}\left(\beta \right)\exp \left(-,\beta ,H,(,\mathit{r},,,\mathit{p},;,\lambda ,)\right)$ where $Z_{\lambda }^{-1}\left(\beta \right)$ is the partition function and $\beta$ is the inverse temperature ($\beta =1/k_{\mathrm{B}}T$, where $k_{\mathrm{B}}$ is the Boltzmann constant). The free energy difference between $\lambda (t)$ and $\lambda (0)$ is then given by $\mathrm{\Delta }G=-{\beta }^{-1}\ln \left[Z_{\lambda (t)},/,Z_{\lambda (0)}\right]$. In a non-equilibrium process realized by changing the control parameter from $0$ to $t$, the work $W$ done on the system is given by (Jarzynski 2011)

$$W_{0\to t}={\int }_0^t\frac{\partial H\left(\mathit{r},,,\mathit{p},;,\lambda \right)}{\partial \lambda }\dot{\lambda }\left(t^{{}^{\prime }}\right)d{t}^{{}^{\prime }},$$ 1.1

where $\dot{\lambda }$ is the time derivative of $\lambda$.

Crooks fluctuation theorem

Crooks fluctuation theorem (CFT) (Crooks 1998) is expressed by

$$\frac{P_{\mathrm{F}}\left(W\right)}{P_{\mathrm{R}}\left(-,W\right)}=\exp \left[\beta ,\left(W,-,\mathrm{\Delta },G\right)\right],$$ 1.1.1

where $P_{\mathrm{F}}\left(W\right)$ and $P_{\mathrm{R}}\left(W\right)$ are the work distributions for forward and reverse non-equilibrium processes, respectively. This theorem indicates that once we obtain the work distributions for forward and reverse processes, we can identify the work $W^{{}^{\prime }}$ at which the two probability density functions are equal (Fig. 1), and then Eq. (1.1.1) becomes $W^{{}^{\prime }}=\mathrm{\Delta }G$. We now obtain the free energy difference. As an experimental verification, a single-molecule experiment was conducted where an RNA string was pulled forward and backward, calculating the free energy difference of unfolding of RNA(Collin et al. 2005), and other studies also verified the theorem (Douarche et al. 2005; Blickle et al. 2006).

Fig. 1.

Work distributions of forward (orange) and reverse (blue) processes. The intersection of the two distributions indicates free energy difference

From a practical viewpoint for MD simulation, it is important to note that CFT can be generalized for constant-volume and temperature (NVT) and constant-pressure and temperature (NPT) ensembles that are frequently used in MD simulation (Procacci et al. 2006; Chelli et al. 2007). Wolf et al. performed TMD simulations for an NaCl system, applying CFT to the dissociation and association processes of NaCl ions (Wolf and Stock 2018). Although CFT can be used for such processes of simple molecules, it has not been applied to more exciting systems, e.g., ligand binding and unbinding processes, to our knowledge. The reason may be that the two probability densities well sampled around the intersection are challenging to obtain. For such problems in the analysis of non-equilibrium processes, optimal protocols for non-equilibrium simulation have been studied to accurately calculate free energy (Minh and Adib 2008; Chelli and Procacci 2009; Dellago and Hummer 2013).

Jarzynski equality

Jarzynski equality is expressed by (Jarzynski 1997, 2011)

$$\mathrm{\Delta }G=-{\beta }^{-1}\ln ⟨\exp \left[-,\beta ,W\right]⟩,$$ 1.2.1

where $⟨·⟩$ indicates the average over a set of non-equilibrium trajectories. This equality connects the non-equilibrium quantity $W$ with the thermodynamic quantity $\mathrm{\Delta }G$. The equality has been verified by non-equilibrium experiments of RNA stretching (Liphardt et al. 2002).

Importantly, the Jarzynski equality is known as a generalization of the equations used in the free energy perturbation and thermodynamic integration methods (Hendrix and Jarzynski 2001; Dellago and Hummer 2013): if the non-equilibrium process between two states, e.g., folded and stretched RNA, is performed at an infinitely slow speed, the Jarzynski equality reduces to the core equation of the thermodynamic integration method (Kirkwood 1935); if the non-equilibrium process is performed instantly, it reduces to the Zwanzig equation of the free energy perturbation method (Zwanzig 1954).

It is recognized that the convergence of Eq. (1.2.1) is considered poor because of the exponential average. An adequate convergence needs an enormous number of NEMD simulations because the average is mainly influenced by the small values of work, which are infrequently observed. Research suggested that hundreds of NEMD trajectories were not enough to converge (Wolf et al. 2020; Kuang et al. 2020).

The sampling error may be avoided by considering the cumulant expansion of the equality (Jarzynski 1997; Park et al. 2003):

$$\mathrm{\Delta }G=⟨W⟩-\frac{\beta }{2}\left(⟨W^2⟩,-,{⟨W⟩}^2\right)+\cdots .$$ 1.2.2

Practically, we usually truncate higher-order terms in Eq. (1.2.2) and get the second-order cumulant expansion: $\mathrm{\Delta }G=⟨W⟩-\frac{\beta }{2}\left(⟨W^2⟩,-,{⟨W⟩}^2\right)$, which corresponds to the result of linear response theory. This approximation holds perfectly if the distribution of work is Gaussian. Even though the truncation causes a systematic error, it allows us to avoid taking the exponential average.

We visualize how the exponential average is more difficult than the second-order cumulant to converge (Fig. 2). We generated random variables that obey Gaussian distribution whose mean is 0.0 and variance is 16. The total sample size is 1000. As we gave the exact mean and variance a priori, we could check the asymptotical behavior to the true value depending on the increase in sample size. The second-order cumulant expansion converged to the exact value within the sample size, whereas the exponential average did not; the exponential average fluctuated in a zigzag. This is because the exponential average is significantly affected by a large value of the random variable that is not frequently observed.

Fig. 2.

a Random variables were generated to obey Gaussian distribution whose mean and variance were 0.0 and 16.0. b The convergence of Jarzynski equality (magenta), the second-order cumulant expansion (green), variance (orange), and mean (blue). The dotted horizontal line indicates the true value (8.0) of the exponential average and the second-order cumulant expansion. These figures were made by a library of Julia language, Plots.jl (Christ et al. 2022)

The work distribution is generally not Gaussian; hence, the second-order cumulant expansion is not generally valid. In fact, recent experimental studies demonstrated that the work did not obey Gaussian distribution if work fluctuation is much more significant than $k_{\mathrm{B}}T$; in the greater fluctuations, the work appeared to follow Gamma distribution (Arrar et al. 2019; Kuang et al. 2020). Their studies suggested that free energy difference was estimated more accurately by the Gamma estimator, which would be an alternative to Gaussian.

It is worth mentioning that we have a way to restrict the work distribution to Gaussian by a stiff-spring condition (Park and Schulten 2004). As we shall say, the stiff-spring condition was originally introduced for the calculation of the potential of mean force (PMF) of a biased system. It also benefits the calculation of the second-order cumulant expansion because the condition allows the work distribution to be Gaussian if a non-equilibrium process follows the overdamped Langevin equation. Namely, the stiff-spring condition guarantees the second-order cumulant expansion to be completely valid. Indeed, under the stiff-spring condition, resultant work distribution from SMD simulation obeyed Gaussian (Park and Schulten 2004).

Rate theory

The previous section deals with how we can calculate the free energy difference from work done on a classical system. Here, we shall explain how we can extract the kinetics of molecular dissociation from NEMD simulation.

Suppose a one-dimensional energy profile $G(x)$ of a physical system along a reaction coordinate $x$. The physical system is initially positioned at the bottom of the left well. The activation barrier ${\mathrm{\Delta }G}^{\ast }$ is located at the transition distance $x^{\ast }$ (Fig. 3a). The profile is influenced by a heat bath at the temperature $T$. Given these settings, we discuss the problem: how much time does it take for the system to escape from the left well in the presence of thermal fluctuations? For instance, this problem is addressed in the ligand unbinding from a protein (Fig. 3b).

Fig. 3.

a Free energy profile with respect to a reaction coordinate—the blue line is the profile in equilibrium, and the orange line is under a constant force. b The process of ligand unbinding can be characterized by the profile; under a constant force $f$, the activation barrier decreases, facilitating ligand unbinding

The escape rate, as we know, is related to the activation barrier height and temperature (Arrhenius law). Although the law was originally proposed phenomenologically, Kramers theory can derive it from underlying dynamics governed by the Smoluchowski equation or equivalently overdamped Langevin equation. The escape problem then reduces to solving a differential equation of mean first-passage time $\tau$ (MFPT) with a boundary condition (Zwanzig 2001; Pavliotis 2014). MFPT is then given by

$$\tau \left(x\right)=D^{-1}{\int }_x^{x^{\ast }}dy{e}^{\beta G\left(y\right)}{\int }_{-\infty }^{y}dz{e}^{-\beta G\left(z\right)},$$ 2.1

where $D$ is the diffusion coefficient. High-barrier approximation $(\mathrm{\Delta }{G}^{\ast }\gg {\beta }^{-1})$ allows the profile $G\left(x\right)$ to be expanded around the minimum and the maximum by Taylor series, and then the expansion is truncated by the second order. We obtain the following MFPT:

$$\tau =\frac{\pi \gamma }{{\omega }_{\mathrm{a}}{\omega }_{\mathrm{b}}}{e}^{\beta \mathrm{\Delta }{G}^{\ast }},$$ 2.2

where ${\omega }_{\mathrm{a}}$ and ${\omega }_{\mathrm{b}}$ are the surface curvatures of $G\left(x\right)$ at the minimum and the maximum (or frequencies) in Fig. 3a and $\gamma$ is a friction coefficient ($\gamma ={\left(D,\beta \right)}^{-1})$. This is also known as Kramers time, or the inverse is Kramers rate. For more detail, we recommend reading the following references (Hänggi et al. 1990; Zwanzig 2001; Pavliotis 2014):

Bell’s model

The Kramers theory has been extended to models widely used in single-molecule pulling experiments (Müller et al. 2021). One simple extension is Bell’s model (Bell 1978) where a constant force $f$ is added to an energy profile, reducing the barrier height $\mathrm{\Delta }{G}^{\ast }$:

$$\tau \left(f\right)={\tau }_0{e}^{-\beta f{x}^{\ast }},$$ 2.1.1

where ${\tau }_0$ is the MFPT in the absence of the force. The term $-f{x}^{\ast }$ acts to reduce the activation barrier at $x^{\ast }$ (Fig. 3a). This relation states that it is possible to estimate ${\tau }_0$ by fitting a dissociation time–force plot. However, the model assumes that the position of the activation barrier does not depend on the strength of a constant force, although it generally depends on it. Thus, Bell’s model reasonably predicts MFPT only if the applied force is weak. A general form of Bell’s model was formulated recently, which gives a more optimal estimation of MFPT (Kuznets-Speck and Limmer 2022).

Dudko-Hummer-Szabo model

Dudko et al. devised a model to treat the force dependence of the activation barrier position, called DHS model (Dudko et al. 2006, 2008). DHS model is formulated by

$$\tau \left(f\right)={\tau }_0{\left(1,-,\frac{\nu f{x}^{\ast }}{\mathrm{\Delta }{G}^{\ast }}\right)}^{\frac{1}{\nu }-1}\exp \left(-,\beta ,\mathrm{\Delta },G^{\ast },\left[1,-,{\left(1,-,\frac{\nu f{x}^{\ast }}{\mathrm{\Delta }{G}^{\ast }}\right)}^{\frac{1}{\nu }}\right]\right),$$ 2.2.1

where $\mathrm{\Delta }{G}^{\ast }$ is the activation barrier height, $x^{\ast }$ is the position of the activation barrier from the minimum, and $\nu$ is a parameter that specifies the shape of an energy surface. While Bell’s model has two kinetic parameters ${\tau }_0$ and $x^{\ast }$, DHS model has not only the two but also ${\mathrm{\Delta }G}^{\ast }$. Unlike Bell’s model, DHS model takes into account the dependency of the activation barrier position on an external force. As this model is still based on the high-barrier approximation, it is not applicable to the domain of a strong force.

This model has been used for atomic force spectroscopy measurements (Müller et al. 2021) and was originally tested for unzipping RNA hairpins in a nanopore and protein unfolding (Dudko et al. 2008). DHS model was also employed by a coarse-grained MD simulation of protein folding (Best and Hummer 2008).

Non-equilibrium molecular dynamics simulation

Having explained the results of non-equilibrium statistical mechanics, we shall introduce several NEMD simulation methods that enhance physical processes by adding external forces. The NEMD simulations provide insights into the kinetics of molecular association and dissociation. Although many simulation techniques that extract the kinetics have been devised so far, e.g., scaled MD (Tsujishita et al. 1993; Bianciotto et al. 2021), the present review focuses on pulling MD simulations.

Steered MD simulation

One of the most famous NEMD methods is SMD simulation which has been proven to be a powerful tool for the investigation of molecular interactions (Isralewitz et al. 2001). In the “Rate theory” section, the parameter $\lambda \left(t\right)$ is introduced to control a system externally, inducing a non-equilibrium process. In SMD simulation, the control parameter is usually defined by $\lambda \left(t\right)=x_0+vt$, where $x_0$ is the initial position on a reaction coordinate (e.g., the initial distance between center-of-mass points) and $v$ is the constant velocity; the Hamiltonian of a system is expressed as

$$H\left(\mathit{r},,,\mathit{p},;,\lambda \right)=H^{{}^{\prime }}\left(\mathit{r},,,\mathit{p}\right)+u\left(\mathit{r},;,\lambda \right),$$ 3.1.1

where the first term is the Hamiltonian of an unbiased system and the second term is called a guiding (bias) potential defined by the harmonic potential:

$$u\left(\mathit{r},;,\lambda \right)=\frac{k}{2}{\left(x,\left(\mathit{r}\right),-,\lambda ,\left(t\right)\right)}^2,$$ 3.1.2

where $k$ is a force constant, $x(\mathit{r})$ is a reaction coordinate, and $\lambda \left(t\right)$ is the control parameter, which is considered the time-dependent equilibrium position of the harmonic potential. The guiding potential confines $x$ near the control parameter $\lambda \left(t\right)$ in time development. From Eq. (1.1) and the concrete expression of $\lambda \left(t\right)$, the work is given by

$$W_{0\to t}=-kv{\int }_0^t\left(x,\left(t^{{}^{\prime }}\right),-,x_0,-,v,t^{{}^{\prime }}\right)d{t}^{{}^{\prime }}.$$ 3.1.3

Figure 4a shows the movement of the guiding potential along a reaction coordinate during SMD. Suppose that a reaction coordinate is the distance between the two circles (blue and red). The red circle initially binds to the blue one, and as time evolves, the inter-circle distance increases because the guiding potential Eq. (3.1.2) moves depending on the control parameter $\lambda (t)$. In this process, the guiding potential excites the system, leading to work generation (Fig. 4b).

Fig. 4.

a Schematic pulling movement in SMD simulation. b Work generation in SMD simulation. Since $x\left(\mathit{r}\right)=x\left(\mathit{r},(,t,)\right)$, we temporarily use $x\left(t\right)$ instead of $x(\mathit{r})$ in this figure to emphasize its time dependence. In general, $x\left(t,+,\mathrm{\Delta },t\right)$ does not instantly follow the control parameter $\lambda (t+\mathrm{\Delta }t)$ in time development. If their values are different at time $t+\mathrm{\Delta }t$, the guiding potential $u$ at $t+$ Δ $t$ excites the system (A); the system in the excited state feels the force towards the bottom of the guiding potential, and then the process generates the work $d^{{}^{\prime }}W$ done on the system accordingly (B)

The use of the biased (guiding) potential gives rise to a question: how can we obtain the potential of mean force (PMF) $\varphi (x)$ of the unbiased system (i.e., the system without the guiding potential)? Our interest is in the unbiased system; thus, we need a way to compute the unbiased PMF $\varphi (x)$. For this, it is proven that large-force constant gives a reasonable approximation to $\varphi (x)$ from the biased PMF ${\varphi }_b\left(t\right)$ of the biased system including the guiding potential (stiff-spring approximation) (Park et al. 2003; Park and Schulten 2004). Formally, the unbiased PMF $\varphi (x)$ is approximated by

$$\varphi \left(x,=,x,(,t,)\right)\approx {\varphi }_b\left(t\right)-\frac{1}{2k{v}^2}\left(\frac{{\varphi }_b^{{}^{″}}\left(t\right)}{\beta },-,{\varphi }_b^{{}^{\prime }},{\left(t\right)}^2\right).$$ 3.1.4

Equation (3.1.4) can be utilized for the calculation of the unbiased PMF in SMD simulation: multiple SMD simulations produce a work distribution. Jarzynski equality Eq. (1.2.1) or its second-order cumulant expansion Eq. (1.2.2) yields the biased PMF ${\varphi }_b\left(x\right)$ and its first and second derivatives (${\varphi }_b^{{}^{\prime }}$ and ${\varphi }_b^{{}^{″}}$) with respect to time $t$. An alternative way was also put forward to calculate the unbiased PMF $\varphi (x)$ with the same information but with more accuracy (Hummer and Szabo 2005, 2010).

Adaptive SMD simulation

As mentioned, the exponential average of Jarzynski equality is difficult to converge (Fig. 2). To overcome this difficulty, adaptive SMD simulation (ASMD) was devised (Ozer et al. 2010, 2012). In ASMD scheme, a reaction coordinate is divided into multiple stages: at first stage, multiple conventional SMD simulations are performed until each simulation reaches the end position of the first stage. Using the last conformations obtained, multiple SMD simulations start again. This is iterated until each simulation reaches a desired position. This scheme can reduce the width of work distribution, thereby avoiding generating trajectories that less contribute to the exponential average (i.e., large value of work). Since there are various manners to inherit initial conformations from the previous stage, such as Full-relaxation SMD, Partial-relaxation ASMD, Naïve ASMD, and Multiple-Branched ASMD (Zhuang et al. 2022). Further detail is provided in a recent review (Zhuang et al. 2021). ASMD has been applied to the unfolding pathways of neuropeptide Y (Ozer et al. 2010) and mutants of neuropeptide (Quirk et al. 2018), and beta hairpin (Bureau et al. 2016).

Targeted MD simulation

TMD simulation enhances conformational changes by applying constraint forces. It was, for example, used for the transition between a closed state and an open state of a protein (Schlitter et al. 1993, 1994). TMD simulation produces conformations $\mathit{r}$ at every step such that every conformation satisfies the holonomic constraint $\psi \left(\mathit{r}\right)={\rho }^2-{\left|\mathit{r},-,{\mathit{r}}_{\mathrm{f}}\right|}^2=0$, where $\rho (t)$ is a control parameter depending on time. To satisfy the constraint, a constraint force $f_{\mathrm{c}}$ is introduced such that ${\mathit{f}}_{\mathrm{c}}=\alpha \frac{d\psi \left(\mathit{r}\right)}{d\mathit{r}}=2\alpha (\mathit{r}-{\mathit{r}}_{\mathrm{f}})$, where $\alpha$ is the Lagrange multiplier. As TMD simulation evolves, $\rho (t)$ is made to approach zero, and thus, owing to the constraint, the trajectory approaches asymptotically to ${\mathit{r}}_{\mathrm{f}}$. TMD simulation solves a modified Newtonian equation that includes a constraint potential $\psi \left(\mathit{r}\right)$, requiring the finding of the parameter $\alpha$ by Lagrange multiplier method (same as SHAKE algorithm (Ryckaert et al. 1977)). The difference between SMD and TMD is that while SMD introduces harmonic potential, TMD does constraints, although on the stiff spring condition, both can be equivalent (Wolf and Stock 2018). It should also be noted that such a constraint force, in general, distorts the PMF of an unbiased system, but it does not do so in distance-based constraints (Mülders et al. 1996; Schlitter et al. 2001).

Another implementation of TMD is based on root-mean-square displacement (RMSD) (Apostolakis et al. 1999; Ovchinnikov and Karplus 2012). This is a simpler implementation than the original TMD because unlike the original TMD, the integrator does not have to be modified. In the RMSD-based TMD, the potential $V_{\mathrm{RMSD}}\left(\mathit{r}\right)=k{\left(\mathrm{RMSD},\left(t\right),-,{\rho }_{\mathrm{RMSD}},\left(t\right)\right)}^2$ was simply added where $k$ is the force constant, $\mathrm{RMSD}(t)$ is the RMSD of an MD snapshot at time $t$ for the final structure, and ${\rho }_{\mathrm{RMSD}}$ is a control parameter that is initialized to the RMSD between the initial structure and the final one.

One of the pioneering works of TMD investigated the protein calmodulin (Ovchinnikov and Karplus 2012). The protein is known to undergo open-closed transition depending on calcium ions. The TMD simulation was initiated from the open conformation. As time evolved, the control parameter $\rho (t)$ changed gradually to the closed states, and optimal transition pathways were obtained.

Dissipated-corrected targeted MD simulation

Wolf et al. proposed dissipated-corrected TMD (dcTMD) (Wolf and Stock 2018; Wolf et al. 2020). They derived an expression of dissipated work and calculated a free energy as the sum of the dissipated work and the average work. Their TMD is thus called dissipated corrected. The dissipated work is calculated from position-dependent friction coefficient along a reaction coordinate, the coefficient which is calculated from a constraint force.

Moreover, the authors demonstrated a novel method to estimate association and dissociation rates (Wolf et al. 2020). In their study, dcTMD produced a free energy profile and a friction profile at a temperature $T_1$. Subsequently, with the two profiles, one-dimensional Langevin dynamics simulation was performed at higher temperature $T_2\left(>,T_1\right)$, which aimed to boost binding and unbinding transitions (they call T-boosting) and to obtain association and dissociation rates at $T_2$. Each of the rates at $T_2$ was converted to those at $T_1$ via the expression $k_2=k_1\exp \left[-,\left({\beta }_2,-,{\beta }_1\right),\mathrm{\Delta },G^{\ast }\right]$, where $k_1$ and $k_2$ are the rates at $T_1$ and $T_2$, respectively. That is, rates at a higher temperature enabled the extrapolation to kinetic rates at a desired temperature.

dcTMD simulation was applied to several systems and produced $k_{\mathrm{on}}$ and $k_{\mathrm{off}}$ for NaCl, trypsin-benzamidine, and heat shock protein 90 (HSP90)-inhibitor systems (Wolf et al. 2020). Notably, the rates agreed well with those obtained by unbiased MD simulations or by experiments within a factor of 2–20. Also, the dissociation constant $K_{\mathrm{D}}=k_{\mathrm{off}}/k_{\mathrm{on}}$ was in agreement within a factor of 2–4. dcTMD was also recently applied to potassium diffusion through an ion channel (Jäger et al. 2022). A tutorial can be accessed from the following: https://github.com/floWneffetS/tutorial_dcTMD, accessed 28 November 2022 (Wolf and Stock 2022).

τ-random acceleration MD

Random acceleration MD (RAMD) is a powerful method to induce the unbinding of a molecule, which was originally referred to as random expulsion MD (Lüdemann et al. 2000). RAMD applies a random force to the center of mass of a molecule in a regular time interval to induce its unbinding. After every interval, it is judged whether the direction of the random force is updated: if the distance is more than a user-defined threshold, then the direction of the random force is kept and is applied again; otherwise, the direction is selected randomly. RAMD was utilized for investigating the dissociation of substrates of cytochrome P450 enzymes (Lüdemann et al. 2000; Schleinkofer et al. 2005).

RAMD was recently extended to τ-random acceleration MD (τRAMD) where a number of RAMD runs are initiated from different coordinates and velocities of equilibrated systems (Kokh et al. 2018, 2019, 2020). In the original paper of τRAMD, 70 inhibitors against the HSP90 were examined, producing relative dissociation times (Kokh et al. 2018). It was found that the dissociation times were correlated well ($R^2=0.86$) with experimental dissociation times obtained from surface plasmon resonance. This result showed that τRAMD was able to rank the set of inhibitors. Plus, τRAMD requires computational time 1–10 ns per inhibitor, indicating that it is, relatively speaking, computationally inexpensive. The authors recently published an article describing a workflow of τRAMD in combination with ligand interaction fingerprint (Kokh et al. 2020); τRAMD was applied to investigate dissociation times of ligands that bind to a G protein-coupled receptor (Kokh and Wade 2021). τRAMD is implemented in GROMACS and is publicly available on https://github.com/HITS-MCM/GROMACS-ramd, accessed 28 November 2022 (Kokh et al. 2022).

Constant-force SMD simulation

While SMD employs a time-dependent force, constant-force SMD (CF-SMD) employs a time-independent constant force. As explained in Fig. 3, a constant force reduces an activation barrier, enhancing barrier crossing. We have recently developed CF-SMD in conjunction with the rate theory, and in this section, we briefly introduce our recent investigation of CF-SMD (Iida and Kameda 2022).

CF-SMD is performed under various strengths of force, resulting in dissociation times at different strengths (Fig. 5). It is then straightforward to obtain MFPT by fitting dissociation time at each strength via Bell’s or DHS models (Eqs. (2.1.1) and (2.2.1)). We have applied CF-SMD to three systems—NaCl, FKBP-ligand, and streptavidin–biotin—and demonstrated the ability of CF-SMD combined with Bell’s and DHS models to predict a dissociation rate in equilibrium.

Fig. 5.

Schematic diagram of the dependency of a dissociation time on a constant force. Orange circles are observations at a constant force. The blue star indicates the dissociation time in the absence of a constant force, i.e., the time in equilibrium. The observations are fit by Bell’s model (blue line) and DHS model (orange line)

We first simulated the NaCl system as a simple case and made a comparison with an equilibrium MD of the same system. We found that the dissociation rate predicted by CF-SMD combined with DHS model was consistent with that of the equilibrium MD. For the FKBP-ligand systems, we showed that the predicted dissociation rate of each ligand was in agreement with that of an earlier computational study (Pan et al. 2017). For streptavidin–biotin system, we predicted a dissociation rate with reasonable accuracy to an experimental value. These results indicate the capability of CF-SMD with Bell’s and DHS models to predict dissociation rates. We emphasize that our method was able to estimate dissociation rates accurately in a wide range of dissociation rates: ${\approx 10}^9{\mathrm{s}}^{-1}$ (NaCl), ${\approx 10}^3{\mathrm{s}}^{-1}$ (FKBP-ligand), and $\approx {10}^{-3}{\mathrm{s}}^{-1}$ (streptavidin–biotin).

Selection of reaction coordinate

Reaction coordinates—also known as collective variables or order parameters—are the coordinates along which a physical process of interest progresses. Examples of reaction coordinates are the distance between two center-of-mass points, dihedral angles, potential energy, the number of hydrogen bonds, radius of gyration, principal component axis, etc. Dimensional reduction is particularly useful to generate reaction coordinates and investigate a conformational ensemble of biomolecules, reviewed in the reference (Sittel and Stock 2018).

The selection of a reasonable reaction coordinate is of importance in NEMD or enhanced sampling methods and depends on a process to be studied: for association and dissociation of two molecules, the center-of-mass distance would be an appropriate reaction coordinate to describe a PMF; for rotamer change, dihedral angle would be suitable to scrutinize. The selection is, however, not always straightforward and needs to be done carefully as it can influence the results of free energy calculation or rate estimate.

We present some attempts to identify reaction coordinates. As a reaction coordinate, a low-free energy path between two states on a two-dimensional free energy profile was examined. The path is assumed to describe the two-state reaction well (Branduardi et al. 2007; Barducci et al. 2011). Structural fluctuations obtained from unbiased MD simulation also provide insights into the selection of reaction coordinates (Mendels et al. 2018; Sultan and Pande 2018). Machine learning techniques have also been applied to find optimal reaction coordinates; it was shown that the problem of finding optimal reaction coordinates can reduce to Bayesian inference problem (Schöberl et al. 2019).

MD software—GROMACS (Abraham et al. 2015), PLUMED (Tribello et al. 2014; PLUMED consortium 2019), NAMD (Phillips et al. 2005, 2020), Amber (Case et al. 2005), Colvar (Fiorin et al. 2013), and CHARMM (Brooks et al. 2009)—provides the functionality for the use of a variety of reaction coordinates. Once one can guess the direction of a reaction process, one can perform NEMD simulations along the coordinate in the software.

Conclusions and outlook

In the last few decades, the fluctuation theorems of non-equilibrium statistical mechanics had been formulated. The theorems underlie the free energy calculation of NEMD simulation. NEMD methods combined with the theorems have been applied to a variety of biomolecular systems to estimate binding free energy. Also, NEMD simulations have been also utilized for drug design (Do et al. 2018). By virtue of the efficient computation, it is possible to screen out compounds that are unlikely to bind to a target protein.

Regardless of the active development, the application is somewhat limited to protein–ligand systems. The next step will be the applications to protein–protein interactions. A few studies attempted to analyze protein–protein interactions via SMD simulation and provided insights into the interactions in an atomic detail (Cuendet and Michielin 2008; Cao et al. 2021). Nonetheless, calculating the binding free energy for the interaction is still challenging because it is likely to deviate from experimental values (Cuendet and Michielin 2008). To resolve this, we should scrutinize an optimal protocol of a non-equilibrium process to improve the accuracy of free energy calculation of NEMD further.

Moreover, the calculation of association rates remains challenging. So far, many studies have succeeded in the observation of dissociation pathways and the calculation of dissociation rate; however, few studies achieved direct calculation of association rate (Wolf et al. 2020). One reason is that unlike a dissociation process, we do not have clues to the pulling direction of association pathways. It is, of course, possible to observe an association rate in conventional MD simulation, but it is achievable only if we use a specialized computer (Shan et al. 2011; Pan et al. 2017).

We have been seeing the stage when non-equilibrium statistical mechanics meets computer simulations and its diverse applications. Like the discovery of the fluctuation theorem from a computer simulation (Evans et al. 1993), we are still having the opportunity to import fruitful results of non-equilibrium statistical mechanics into molecular simulation, and we should be aware of the theoretical progress.

Funding

The authors were supported by JSPS KAKENHI Grant No. 22H00553.