15 Years of Molecular Simulation of Drug-Binding Kinetics

Abstract

Introduction:

Drug-binding kinetics has been increasingly recognized as an important factor to be considered in drug discovery. Long residence time could prolong the action of some drugs while produce toxicity on others. Early evaluation of the binding kinetics of drug candidates could reduce attrition rate late in the drug discovery process. Computational prediction of drug-binding kinetics is useful as compounds can be evaluated even before they are made. However, simulation of drug-binding kinetics is a challenging problem because of the long-time scale involved. Nevertheless, significant progress has been made.

Areas covered:

This review illustrates the rapid evolution of qualitative to quantitative methods that have developed over the last 15 years.

Expert opinion:

The development of new methods based on molecular dynamics simulations now enables computation of absolute association/dissociation rate constants. Cheaper methods capable of identifying candidates with fast or slow binding kinetics, or rank-ordering rate constants are also available. Together, these methods have generated useful insights into the molecular mechanisms of drug-binding kinetics, and the design of drug candidates with therapeutically favorable kinetics. Although predicting absolute rate constants is still expensive and challenging, rapid improvement is expected in the coming years with the continuing refinement of current technologies, development of new methodologies, and the utilization of machine learning.

1. Introduction

Accumulating evidence supports the idea that drug-binding kinetics can play an important role in determining whether a molecule can be a useful drug. Long residence time is a concept that has become popular. A number of studies advocate that good drugs should have long residence time in their biological receptors to produce long-lasting therapeutic effects.[1–8] Scientists usually define residence time as the inverse of the dissociation rate constant, $k_{off}$, of a drug candidate from its receptor.

In contrast, literature search and simulations using pathway models of cell signaling[9,10,11 ] have found drug-binding kinetics to influence drug efficacy in more ways than requiring long residence time. For example, fast off rate, or short residence time, is sometimes important to make a molecule a useful drug. The drug memantine for treating patients suffering from Alzheimer’s disease provides one example. Fast off rate plays a role in reducing its toxicity.[12] Memantine functions by blocking the N-methyl D aspartate (NMDA) receptor. This synaptic receptor mediates signaling by the neurotransmitter glutamate. Alzheimer patients produce overly activated glutamatergic transmission that over-excites several pathways leading to apoptosis, causing neurodegeneration. Memantine suppresses such unwanted triggers of cell death. However, it reduces toxic effects by not suppressing neurotransmission by the NMDA receptor completely because the receptor is also required for normal functions such as memory and learning. Memantine achieves this by having a fast off rate so that glutamate can still access the receptor when needed.

$k_{off}$ alone does not tell the whole story. For example, Wu et al provided an example in which $k_{on}$, rather than $k_{off}$, of engineered antibodies correlated better with the ability of the antibodies to neutralize the respiratory syncytial virus.[13] Simulation studies applying drugs to different protein kinase targets in quantitative cell signaling models also suggest that biological targets transmitting signals quickly to downstream molecules soon after they are activated require fast-binding drugs to block them.[9–11]

Therefore, to take full advantage of drug-binding kinetics in drug discovery, drug developers need to consider not only long residence time (or small $k_{off}$), but also short residence time (or large $k_{off}$), and association rate.

Although the view on how drug-binding kinetics influences drug efficacy is still under debate and continuously evolving,[14–16] computational scientists have made good progress on simulating drug-binding kinetics at the atomic level, including the estimation of absolute, not only relative, association and dissociation rate constants.

To help readers appreciate the rapid development and applications of powerful quantitative methods that have been developed and used in the past 15 years in studying drug-binding kinetics, this review starts by giving examples in the mid-2000s in which only crude simulation models were used. Limitations of current methods and further development will also be discussed. This review is not intended to be comprehensive, as many methods have now been developed and used to study drug-binding kinetics. Rather, it focuses on a subset of popular methods that have been used to study drug-binding kinetics to illustrate the variety of strategies that have been used to make simulating such long-time processes feasible with current computing technologies.

2. A mining-minima approach for qualitatively identifying binding/unbinding pathways

In the mid-2000s, when not many methods were available for studying long-time dynamics of complex protein-ligand systems, Huang and Wong introduced a mining-minima approach to finding approximate association and dissociation pathways and estimating the associated activation barriers.[17] The method employed several approximations to allow quick calculations. It first surveyed the energy landscape of a protein-ligand system thoroughly by mining minima and then connected the resulting structures to form pathways via clustering and similarity analysis. To survey the energy landscape, it drew upon the idea that the energy landscape of a protein-ligand system was complex containing many local minima.[18,19] Local minima were so numerous that joining them appropriately could approximate any pathway. To mine energy minima efficiently, they used a simulated annealing cycling protocol that had been shown to work well in molecular docking of protein kinase and phosphatase systems.[20–24] Each molecular dynamics (MD) simulation was composed of a sequence of many short simulated annealing cycles. In each cycle, a system was heated to a high temperature such as 1000 K and then cooled to 0 K rapidly in ~10 ps. The heating encouraged a system to move away from a local energy well to sample another region of the configurational space. The subsequent rapid cooling trapped the system into a new local minimum. One usually ran many such simulated annealing trajectories, starting with different initial atomic velocities, different placements of the ligand inside the protein, or/and different starting conformations of the protein. Because the trajectories were independent, many trajectories with different starting conditions could be run in parallel to reduce turn-around time.

The mining-minima approach is summarized in Figure 1.

Figure 1:

Workflow of the mining-minima approach

When this approach was applied to study the (un)docking pathways between para-nitrocatechol sulfate (pNCS) and the protein tyrosine phosphatase YopH of Yersinia pestis,[17] four major pathways were found. By analyzing the interaction patterns between the ligand and the protein along the pathways, useful insights into the design of molecules with certain desired drug-binding kinetics could be obtained.[25] For example, some amino acid residues were found to interact with the ligand only when the ligand was in the binding pocket, some participated only when the ligand was at the protein surface near the entrance to the docking pathway, but some interacted with the ligand almost all through the docking pathway inside the protein. Such analysis suggests some ways that a compound could be modified to achieve certain desired binding kinetics. For example, if one wished to find compounds with longer residence time, one might want to modify the portion of the compound that interacted with the residues that stabilized the bound form but not the transition state. If one modified the portion of the compound that interacted with an amino acid participating all through the docking pathway, for example, it was harder to predict without quantitative calculations whether such a modification would increase, decrease, or produce little effect on the activation barrier, because both the bound and the transition states could be affected by the modification.

In another application on the study of the dissociation of a hexapeptide from the insulin receptor tyrosine kinase,[26] multiple entrance/exit sites were found (Figure 2). The possibility of multiple docking pathways with multiple entry/exit sites could complicate the understanding of drug-binding kinetics and the rational design of drugs. For example, it is possible for different compounds to take different pathways, or they can use multiple pathways simultaneously but the flux through each pathway depends on the compound. These more complicated scenarios require quantitative models to produce better insights.

Figure 2:

Two entrance/exit sites of the association/dissociation between the hexapeptide GDYMNM and the catalytic domain of the insulin receptor tyrosine kinase. (Reproduced from Figure 3 of reference [26])

The mining-minima approach provides a quick way to identify the association and dissociation pathways of drug candidates in biomolecular targets. However, several approximations limit its reliability. A Feynman path integral formalism of diffusional dynamics was later introduced to quantify the approach.[25,27] One approximation in the approach was the neglect of energetic effects in determining whether two metastable states represented by clusters of structures should be connected to form a part of a pathway. To introduce energetic effects into the mining-minima approach without performing additional expensive simulations, one could draw upon the Feynman path integral formalism for diffusive motion, assuming the coupled protein dynamics and ligand association/dissociation followed a diffusive, or Brownian, process. Researchers often use this approximation to study configurational transitions in condensed and biomolecular systems.[28] Recently, Kramers’ theory[29] has also been used to study drug-binding kinetics with the Gaussian accelerated MD method (GaMD).[30,31] The Feynman path integral approach had only been applied to a simple model so far. Nevertheless, the equation derived for calculating the rate constant $k$ gives some useful insights:

$$k\approx T(\overrightarrow{x_t},t;\overrightarrow{x_0},\overrightarrow{t_0}=0)/t$$ Eq. 1

where the transition probability, $T(\overrightarrow{x_t},t;\overrightarrow{x_0},\overrightarrow{t_0}=0)$, in going from structure $\overrightarrow{x_0}$ at initial time $\overrightarrow{t_0}=0$ to final structure $\overrightarrow{x_t}$ reached at time $t$ can be approximated by the most probable path as

$$T{(\overrightarrow{x_t},t;\overrightarrow{x_0},t_0)}_{most\_probable\_path}=C_{most\_probable\_path}\exp \left\{-\frac{1}{4D}{\sum }_{i=1}^{3N_{atom}}{\sum }_{j=1}^N\Delta t_j{\left(\frac{x_i\left(j+1\right)-x_i\left(j\right)}{\Delta t_j}+\frac{D}{k_{B}T}\frac{\partial V}{\partial x_i}\right)}^2\right\}\exp \left\{\frac{1}{2}{\sum }_{\left(i=1\right)}^{3N_{atom}}{\sum }_{j=1}^N\left(\frac{D}{k_{B}T}\frac{{\partial }^{2}V}{\partial x_i^2}\right)\Delta t_j\right\}$$ Eq. 2

where $N$ is the number of discretized states used to represent a path, $N_{atom}$ is the number of atoms, $C_{most\_probable\_path}$ is a normalization constant, $D$ is an effective diffusion constant, and $x_i$’s are coordinates of representative structures for the clusters obtained from mining-minima simulations.

The probability of connecting two structural clusters now also depends on the interaction potentials, not just on structural similarity. Even when interaction potentials are ignored, this equation gives a quantitative expression on how the transition probability between two representative cluster structures should depend on their structural similarities.

3. Steered molecular dynamics (SMD)

Steered molecular dynamics (SMD) simulation[32–34] provides another fast method for studying protein-ligand (un)binding. SMD mimics the use of atomic force microcopy to study the conformational change of biomolecules by applying a force to speed up the process. The same idea can be applied to study the dissociation of ligands from their protein receptors. For example, Colizzi et al.[35] performed single-molecule force-pulling simulation on the unbinding of flavonoid inhibitors to β-hydroxyacyl-ACP dehydratase of Plasmodium falciparum and found active compounds to be harder to be pulled out than inactive ones. They also found that plots of force versus time clearly separated into two groups: one for active compounds, one for inactives. Thus, it was possible to use such plots to discern active inhibitors from inactive ones for some systems. In addition, by analyzing how the molecules interacted with the proteins along the undocking pathways, they found the 7-hydroxyl group of these compounds to not interact with the protein but the two hydroxyl groups on the phenyl ring gave the active compounds better resistance to be pulled out than the inactive compounds containing only one or no hydroxyl groups. Therefore, they added two hydroxyl groups to the phenyl ring of the inactive compound kaempferol to form rhamnetin and the profile obtained from steered MD simulation became similar to the active compounds. Rhamnetin was later confirmed experimentally to be active. This analysis resembled the examination of interaction patterns along docking pathways obtained by the mining-minima approach described above.

In another study on the unbinding of nine compounds from the cyclin-dependent kinase 5, Patel et al.[36] found that SMD was able to distinguish active from inactive compounds, although it could not quantitatively rank-order their activities well. They also showed that SMD could provide useful insights into how interactions changed along the pathways of ligand dissociation.

SMD was later used to compare the dissociation rate of compounds from their protein receptors. In studying the dissociation of ligands from the focal adhesion kinase,[37] steered MD simulations were first performed on one compound by applying pulling forces in several directions and with several different pulling rates to identify the pathway that might contribute most to ligand dissociation, such as the one with least resistance to dissociation. Other ligands were then pulled along the same direction producing this pathway and the dissociation time was found to be able to distinguish ligands with fast dissociating kinetics from those with slow dissociating kinetics[37,38]

4. τ-random accelerated molecular dynamics (τ-RAMD)

In τ-RAMD,[39] replicas of the bound state are prepared and allowed to dissociate quickly by applying a randomly applied force with a fixed magnitude. The random force is applied after a chosen number of dynamics steps if the ligand does not move by a pre-specified distance. The simulations are inexpensive and Kokh et al.[39] were able to use the method to study the dissociation of 70 compounds from the N-terminal domain of the heat shock protein HSP90α. The correlation between the simulated dissociation time and the experimental values measured by surface plasma resonance was good, especially when outliners were removed. The outliners might result from inaccurate force fields for about 22% of the compounds. Analyzing the dissociation trajectories yielded useful insights into factors affecting ligand unbinding rates. Factors that appeared to play a role included transient polar interactions and steric hindrance. Water was not found to play a dominant role in these systems. Furthermore, the mechanisms of dissociation were found to be diverse -- no single leading factor was found to determine the transition barriers of dissociation for all the compounds.

In a later study, Kokh et al.[40] used machine learning to improve the prediction of residence time of compounds in HSP90α by τ-RAMD. In this study, they included additional compounds to give a total of 94 for their analyses. The compounds encompassed 11 different scaffolds. Using protein-ligand contacts or interaction fingerprints observed in τ-RAMD trajectories as features in two machine-learning models, they were able to obtain residence times that correlated better with experimental values than using the dissociation time directly obtained from the simulations. Improvement was achieved even when including the outliners found in the previous study.[39]

In another study of the dissociation of inhibitors from the focal adhesion kinase (FAK) and the protein kinase PYK2 encoded by the PTK2B (protein tyrosine kinase 2 beta) gene,[41] τ-RAMD provided useful insights into why some inhibitors gave rise to longer residence times for FAK than for PYK2. The involvement of the DFG motif was highlighted.

5. Scaled molecular dynamics

Scaled MD provides another quick method for estimating residence times. Mollica et al.[42] sped up the simulation of ligand dissociation by scaling down the potential energy function to weaken interactions. To prevent a protein receptor from unfolding when a uniform scaling factor was applied to the potential, restraints were applied to the receptor except the residues near the ligand in the bound structure. Twenty simulations were run for each ligand and the averaged dissociation time was used to rank-order ligands. In applying the method to study three systems -- heat shock protein 90, 78 kDa glucose-regulated protein, and adenosine A_2A receptor – they obtained good Pearson’s coefficients between experimental residence time and simulated dissociation time: 0.95, 0.85, and 0.95 respectively. The method was further tested on glucokinase activators later.[43] By comparing the results obtained by using different scaling factors, they concluded that scaling down interactions more extensively was useful for fast rank-ordering of dissociation kinetics whereas scaling down interactions less extensively could give better results on the expense of longer simulation time. Even though scaling potential down lost some details of molecular interactions, they found that such simulations could still help to understand structure-kinetic relationships. Besides differences in protein-ligand interactions, they also found that the shape of the ligand could play a role in influencing residence times. To find residues on the glucose kinase that might be important in affecting ligand dissociation, they identified residues that were in contact with the ligand in more than 66% of the replicas. Besides residues in the binding pocket, a group of residues further away from the binding pocket were identified.

Four years later, another paper[44] was published on the use of scaled MD simulations to compare the off rates of a structurally diverse set of inhibitors of heat shock protein 90 (Hsp90). Although the experimental residence time spanned 3 orders of magnitude, the simulation results rank-ordered the ligands well. Including all the compounds gave a Spearman’s rank correlation coefficient (ρ) of 0.89 with a p-value < 0.05, and a Pearson’s correlation coefficient of 0.73. Excluding an outliner gave a ρ value of 0.94 with a p-value < 0.01.

Bianciotto et al.[45] used scaled MD to study 27 ligands of Hsp90 belonging to 7 chemical series. Although the simulations did not reproduce the trend of the residence time for the whole series, the method was able to identify the slowest-dissociating ligands. The authors suggested that selectively scaled MD simulations[46] might produce better results.

In selectively scaled MD simulations, only specific energy terms in the potential energy function are “scaled” to accelerate ligand dissociation. No restraint needs to be applied to the protein to prevent it from being unfolded. Deb and Frank[46] illustrated this approach by scaling up ligand-water interactions to speed up ligand unbinding in several cyclin-dependent kinase-inhibitor complexes.

6. Brute-force unbiased simulations with specialized computers

In 2008, D.E. Shaw Research introduced the supercomputer, Anton, designed for performing MD simulations. Anton could run MD simulations hundreds to thousands times faster than other machines at the time.[47]

Dror et al.[48] were able to use Anton to perform unbiased MD simulations that enabled several beta blockers to find their docking poses to the β1- and β2-adrenergic receptors. The simulations found that the beta blockers used the same dominant docking pathways, and desolvation of the beta blockers created the largest energetic barrier for association.

Shan et al. used Anton to study the binding of dasatinib and the kinase inhibitor PP1 to the c-Src kinase.[49] By placing the ligands randomly on the surface of the protein, they were able to find the correct docking poses from long unbiased MD simulation made feasible by Anton. Dasatinib found its docking site after 2.3 μs in one of four simulations. PP1 did so in three of seven simulations in 15.1, 1.9, and 0.6 μs. Water shells were found around the ligands before the experimental binding poses were reached.

Guo et al.[50] used Anton to simulate the dissociation of the antagonist ZM241385 from the adenosine A_2A receptor. From the simulations, they identified residues contacted by the antagonist during dissociation. Mutating some of these residues and performing kinetic radioligand binding experiments confirmed some of these residues to influence dissociation rate. Some of these mutants affected dissociation rate but not binding affinity substantially, indicating that interactions that controlled ligand dissociation kinetics did not necessarily occur in the bound state. The simulations also found the ligand pausing transiently in several locations, suggesting a multi-step dissociation pathway.

Despite these interesting findings from simulations with Anton, many scientists do not have access to such computers. Furthermore, many drugs or drug candidates’ dissociation times are longer than seconds and they are still difficult to be studied by unbiased MD simulations with Anton. Calculating dissociation rates quantitatively requires even more computing time as multiple dissociation events need to be observed from the simulations. We next describe simulations that use different methods to study long-time events that can be accomplished by more wildly available CPU/GPU computing systems.

7. Umbrella sampling simulations

Umbrella sampling simulations have been used for several decades to study long-time events occurring in protein systems. McCammon and Karpus[51] applied this method to study the rotation of a tyrosine residue in bovine pancreatic trypsin inhibitor early on. Umbrella sampling simulations have been used recently to study drug-binding kinetics.

In this method, one or more progress coordinates is(are) selected. Ideally, the progress coordinate(s) capture(s) the slowest motions whereas their orthogonal coordinates describe dynamics at much faster time scales. Restraining windows, usually in the form of harmonic potentials, are applied to different value(s) of the progress coordinate(s) in performing MD simulations. The restraining potentials make it easy to sample regions with high free energies along the progress coordinate(s). Results from the simulations from different windows can then be pieced together to obtain the potential of mean force along the progress coordinate(s) using methods such as WHAM[52] and MBAR.[53] Transition-state theory[54] can be used to estimate association/dissociation rates from the activation barrier estimated from the potential of mean force. Transmission coefficients can also be computed such as using the method introduced by Chandler et al.[55]

One or more progress coordinates need to be chosen for carrying out umbrella sampling simulations. Steered MD provides one way to estimate a progress coordinate. In an application studying ligand dissociation from the focal adhesion kinase, a ligand was pulled out in several directions and with several different pulling rates to identify the pathway that might contribute most to ligand dissociation.[37] Umbrella sampling windows could then be placed along the pathway.[38] Multiple pathways can also be used. For example, Spiriti and Wong[56] placed windows along several pathways obtained by τ-random accelerated MD.[41] The Weighted Histogram Analysis Method (WHAM)[52] could then be used to compute the potential of mean force on grid points on the protein through which a ligand dissociated. The values of the potential of mean force allowed them to draw free energy contours on top of a protein structure to show the location of the bound and intermediate states. From the contours, the activation energy of dissociation could be estimated. Using the activation energy in transition-state theory and assuming a unit transmission coefficient over-estimated the dissociation rate constants of several compounds by more than six orders of magnitude for the focal adhesion kinase.[38] However, when they applied the same method to the protein kinase PYK2, the computed rate constant differed from the experimental value by only three orders of magnitude.[56]

You et al.[57] discussed some issues in performing umbrella sampling simulations using two systems: 1). the study of the dissociation of aspirin and 1-batanol from β-cyclodextrin and 2). the study of the dissociation of the inhibitor SB2 from the p38α mitogen-activated protein kinase. In the β-cyclodextrin systems, they found that the calculated potential of mean force could depend on the starting conformation of the receptor. Once the ligand was bound, the restrained receptor could be difficult to convert from one conformation to another during the simulation. For the protein-ligand dissociation, the results could be sensitive to the pathways along which umbrella sampling simulations were conducted.

As a method depending on the choice of progress coordinate(s), the results from umbrella sampling simulations are sensitive to the progress coordinate(s) chosen. Good progress coordinate(s) capture(s) the mechanism of dissociation well and reveal(s) important intermediates along the dissociation pathway. The time scale along the progress coordinate(s) should be substantially longer than those orthogonal to the progress coordinate(s). Methods have been developed to help to identify good progress coordinate(s). For example, an autoencoder has been used.[58] In another publication, Smith et al.[59] reported a method called multi-dimensional spectral gap optimization of order parameters (SGOOP) for finding good progress coordinates expressed in terms of a linear or non-linear combination of ordered parameters.

8. Markov state model (MSM)

The MSM[60,61] does not require one to define a progress coordinate beforehand. The pathway of ligand dissociation can be deduced from many short unbiased MD simulations sampling the ligand movement inside a protein. A protocol for using a MSM includes the following steps:

1. Run many short unbiased MD simulations starting with the ligand placed at different distances from the binding pocket. The starting structures can come from other simulations such as steered MD. The Fluctuation Amplification of Specific Traits (FAST) approach[62] can also be used to obtain initial dissociation pathways along which many unbiased MD simulations can be spawned. The FAST approach can start with the ligand in its bound form and quickly discover dissociation pathways.

2. Select features to simplify the description of the protein-ligand (un)binding process. For example, the minimum distances between the heavy atoms of the ligand and the heavy atoms of each amino acid residue in the protein and some internal coordinates of the ligand can be used.

3. Reduce the dimension of the problem by identifying the slowest modes most important in describing protein-ligand association or dissociation. Time-lagged independent component analysis (TICA)[63] is useful here. It is more useful than the principal component analysis (PCA) method[64] for this purpose. PCA identifies the modes corresponding to the largest amplitude motion by solving the eigenvalue problem:

   $$C\left(0\right)u_i={\sigma }_i^2{\text{u}}_{\text{i}}$$ Eq. 3

   in which $C(0)$ is the covariance matrix of structural variations casted in the feature space. On the other hand, TICA identifies the modes corresponding to the slowest motion by solving the eigenvalue problem:

   $$C\left(\tau \right)u_i={C(0)\gamma }_i\left(\tau \right){\text{u}}_{\text{i}}$$ Eq. 4

   in which $\tau$ is a user-chosen lag time.

4. Structures from the MD simulation are clustered in a subspace of the TICA components.

5. The transition probabilities $T_{ij}$ between the clusters are calculated.

6. The matrix containing the elements $T_{ij}$ allows one to obtain the binding affinity of the protein-ligand complex, and to calculate association and dissociation rates using, for example, the transition-path theory.[65–68]

Buch et al.[69] used 495 unbiased MD simulations lasting 100 ns each to construct Markov State Models to study the binding between benzamidine and trypsin. The total simulation time amounted to 50 μs. The long-aggregated simulation time was achieved by using a GPU grid. They obtained a $k_{on}$ of 15.0 ± 2.0 × 10⁷ M⁻¹ s⁻¹, about a factor of 3 larger than the experimental value of 2.9 × 10⁷ M⁻¹s⁻¹. The calculated $k_{off}$ was 9.5 ± 3.3 s⁻¹, about 60 times smaller than the experimental value of 600 s⁻¹.

Plattner and Noé[70] revisited this problem in 2015, using an aggregate of 150 μs of MD simulations. In addition to obtaining values of $k_{on}$ and $k_{off}$ comparable to experimental ones, they identified two main binding channels and multiple long-lived conformations of trypsin that coupled with the ligand dissociation process.

The MSM was also applied to study the binding between the T99A mutant of T4 lysozyme and benzene, using 59 μs of simulation data. The calculated $k_{on}$ and $k_{off}$ were in line with the experimental values: 0.21 ± 0.09 × 10⁷ M⁻¹ s⁻¹ versus 0.08−0.1 × 10⁷ M⁻¹ s⁻¹ for $k_{on}$ and 310 ± 130 s⁻¹ versus 950 ± 200 s⁻¹ for $k_{off}$.

Different methods have also been introduced to reduce the computational costs in using the MSM. For example, the transition-based reweighting analysis method (TRAM)[71] combines biased and unbiased MD simulations. Unbiased simulations are not efficient in sampling uphill processes. By introducing biased simulations connecting metastable states, unbiased simulations only need to sample downhill processes well. Biased simulations can be performed in different ways such as by carrying out umbrella sampling simulations. Wu et al.[71] worked out equations for calculating the transition probabilities between metastable states using a combination of biased and unbiased MD trajectories. The transition probabilities $p_{ij}^k$ between states $i$ and $j$ in an unbiased or a biased simulation labelled $k$ can be calculated by

$$p_{ij}^k=\frac{c_{ij}^k+c_{ji}^k}{{\nu }_j^k\exp \left(f_j^k-f_i^k\right)+{\nu }_i^k}$$ Eq. 5

In the equation, $c_{ij}^k$ is an element of a count matrix. It counts the number of trajectories starting from state $i$ that reaches state $j.f_j^k$ is the local free energy at state $j.{\nu }_j^k$’s are Lagrangian undetermined multipliers. $f_j^k$’s and ${\nu }_j^k$’s can be obtained by solving a set of non-linear algebraic equations described by Wu et al.[71]

In applying TRAM to study the dissociation of a ligand from the focal adhesion kinase,[56] $k_{off}$ ranging from 0.17 to 1.39 $s^{-1}$ was obtained when different number of clusters and different lag times were used to compute the transition probabilities. These numbers agree well with the experimental value of $0.07\pm 0.02s^{-1}$. The paper also reported a calculated $k_{off}$ of $33.44s^{-1}$ that deviated more from the experimental value when the largest number of clusters and the longest lag time were used. The authors attributed this larger discrepancy to the fewer snapshots available for doing the calculations properly for this system.

Other methods introduced to reduce costs in using the MSM include an adaptive restarting strategies[72–76] and the MBAR variant of TRAM[77]

Applying the MSM effectively requires substantial expertise, as a user needs to choose appropriate features to describe the ligand dissociation process, to choose a lag time for performing time-lagged independent component analysis to help to reduce dimension, to choose the number of clusters to define metastable states, and to choose a lag time for calculating transition probabilities for the MSM. Machine learning has been introduced to alleviate this problem. For example, the variational approach for Markov processes (VAMP)[78] has been used to develop the neural networks VAMPnets for this purpose.[79]

9. The milestoning method

In the milestoning method,[80,81] space is divided into subspaces separated by milestones after choosing a progress coordinate. Figure 3 illustrates one way to create milestones along a progress coordinate obtained from a SMD simulation. Anchors are first placed along the progress coordinate. Initial milestones are formed between successive anchors. MD simulations are run from these milestones and new milestones are discovered. This process can be repeated until calculated quantities such as mean first passage time (MFPT) converge. The calculation of quantities such as MPT requires the probabilities of trajectories moving from milestones to milestones and the averaged times in doing so to be calculated. This approach benefits by starting unbiased MD simulations directly from milestones that might be difficult to reach by unbiased MD simulations.

Figure 3:

Schematic of one way to perform milestoning simulations. (Reproduced from Figure 1 of reference [84])

A progress coordinate is first estimated, by performing a SMD simulation for example (long curvy line). Anchors (shown as black dots) are then placed along the progress coordinate. The anchors become centers of Voronoi cells. A milestone separated two Voronoi cells. An ensemble of trajectories is then started from each initial milestone (a dashed line) and the number of trajectories reaching other milestones for the first time counted. Some of these trajectories discover new milestones (marked with 4-point stars) as they move along directions orthogonal to the progress coordinate.

However, assumptions are usually made in starting trajectories from each milestone because the distributions of the starting phase points are not known for most milestones. Earlier milestoning simulations started trajectories from snapshots obtained from an equilibrium MD simulation constrained to each milestone.[80–83] This assumption works well for some systems. Recent work using this assumption includes the simulation of the dissociation of the drug Gleevec from the Abl kinase[84] and the simulation of the dissociation of a drug candidate from the Glycogen Synthase Kinase 3β.[84] In both cases, the calculated dissociation constants were within an order of magnitude of the experimental values.

Nevertheless, Vanden-Eijnden et al.[85] introduced ways to start the trajectories from more realistic distributions. In the overdamped limit, they derived an expression for the first hitting point probability density at milestone $j$ by

$${\rho }_j\left(x\right)=Z_j^{-1}{e}^{-\beta V\left(x\right)}|\nabla q(x)|$$ Eq. 6

where the committor function $q(x)$ was the probability that trajectories started at the milestone would reach the product milestone before the reactant milestone, $V\left(x\right)$ was the potential on point $x$ located in the milestone, and $Z_j^{-1}$ was a normalization factor. In systems for which inertia cannot be ignored, no closed form solution has been derived. To provide a practical solution, Vanden-Eijnden et al.[86] introduced Markovian Milestoning with Voronoi Tessellations (MMVT). In this method, a user runs a trajectory in each cell confined within milestones and monitors how many times the trajectory reaches milestone $i$ and then milestone $j$. The trajectory does not go outside the Voronoi cell. For example, when the trajectory hits a milestone, it can be made to change course by reversing its atomic velocities. One can also introduce repulsive walls to the milestones.[87]

Vanden-Eijnden et al.[85,86] also discussed the conditions under which the Markovian milestoning method was valid. For the transition probability from an initial milestone to a final milestone to be described by a Markovian process involving a sequence of transitions between milestones, the transitions need to be statistically independent. Vanden-Eijnden et al. showed that this condition could be satisfied by choosing optimal milestones formed from isocommittor surfaces. At each surface, the probability of the system starting from any point on the surface to reach the final milestone before the first milestone was the same. They also suggested that the string method[88–92] provided a practical way to obtain approximate isocommittor surfaces.

Vanden-Eijnden et al. [86] also derived useful equations for obtaining mean first message time from simulations carried out in Voronoi cells enclosed by milestones:

$$\widehat{\mathit{Q}}{\mathit{T}}^N=-1$$ Eq. 7

in which the vector ${\mathit{T}}^N$ contained the mean first passage time from milestone 1 through $N-1$ to milestone $N$, the target milestone. Each element of $Q_{ij}$ was given by

$$\frac{N_{ij}}{R_i}if{R}_i\ne 0,i\ne j;or0if{R}_i=0,i\ne j;or-{\sum }_{i\ne j}{Q}_{ij}ifi=j$$ Eq. 8

where

$$N_{ij}=T{\sum }_{\alpha =1}^{\Lambda }{\pi }_{\alpha }(\frac{N_{ij}^{\alpha }}{T_{\alpha }})$$ Eq. 9

$$R_i=T{\sum }_{\alpha =1}^{\Lambda }{\pi }_{\alpha }(\frac{R_i^{\alpha }}{T_{\alpha }})$$ Eq. 10

in which $N_{ij}^{\alpha }$ was the number of times the trajectory in Voronoi cell $\alpha$ reached milestone $i$ and then milestone $j,T_{\alpha }$ was the total length the trajectory spent in Voronoi cell $\alpha ,R_i^{\alpha }$ was the accumulated time for all sub-trajectories spanning between milestones $i$ and $j$.

${\pi }_{\alpha }$ was the probability of finding Voronoi cell $\alpha$. Under steady state conditions ${\pi }_{\alpha }$ could be obtained by solving the equations

$${\sum }_{\beta =1,\beta \ne \alpha }^{\Lambda }{\pi }_{\beta }{k}_{\beta ,\alpha }={\sum }_{\beta =1,\beta \ne \alpha }^{\Lambda }{\pi }_{\alpha }{k}_{\alpha ,\beta }$$ Eq. 11

$${\sum }_{\alpha =1}^{\Lambda }{\pi }_{\alpha }=1$$ Eq. 12

$$k_{\alpha ,\beta }=\frac{N_{\alpha ,\beta }}{T_{\alpha }}$$ Eq. 13

in which $k_{\alpha ,\beta }=\frac{N_{\alpha ,\beta }}{T_{\alpha }}$ where $N_{\alpha ,\beta }$ counted the number of collisions between Voronoi cells $\alpha$ and $\beta$.

One advantage of this approach is that one only needs to run one trajectory per Voronoi cell rather than start many trajectories from each milestone, and there are many more milestones than Voronoi cells.

Milestoning simulations require the choice of a good progress coordinate describing the slowest motions. As mentioned above, some researchers used steered MD to obtain the progress coordinate.[84,93] Alternatively, Tang et al.[94] used the two principal components accounting for the slowest motions obtained from metadynamics simulations.

10. SEEKR and SEEKR2: software packages for performing milestoning simulations

SEEKR is a software package developed for performing milestoning simulations. It uses a multi-scale approach in which the system is treated differently depending on whether the protein and the ligand are close to each other or far apart. When the protein and the ligand are close to each other, they are represented as flexible molecules immersed in explicit water molecules and ions. When the protein and ligand are far apart, they are treated as rigid molecules diffusing relative to each other in an implicit solvent environment. The diffusional motion is carried out by using the software package BrownDyn.[95] Milestones are constructed as concentric spheres around the binding site. To start trajectories from a milestone for calculating the transition kernels to other milestones and the associated mean transition times, a simulation is first performed with restrain applied. After equilibration, selected snapshots are propagated backward in time. Snapshots that hit another milestone before re-crossing the milestone are kept for starting forward trajectories for calculating the transition kernels and the mean transition times. In applying this approach to study the binding between trypsin and benzamidine, good agreement with experimental measurements were obtained for $k_{on}(2.1\pm {10}^7{M}^{-1}{s}^{-1}$ versus $2.9\pm {10}^7{M}^{-1}{s}^{-1}$) and $k_{off}(83\pm 14s^{-1}$ versus $600\pm 300s^{-1}$).

Later MMVT SEEKR[96] was introduced by incorporating Markovian Milestoning with Voronoi Tesselations (MMVT).[86] Unlike SEEKR, which used snapshots obtained from an equilibrated simulation at a milestone to start trajectories, MMVT SEEKR runs a single simulation within each Voronoi cell to obtain the transition kernels between milestones surrounding a Voronoi cell.

The SEEKR2 package[97] was later developed to further improve SEEKR and MMVT SEEKR. SEEKR2 improves performance further by using the OpenMM package[98] for running MD simulations and using hydrogen mass repartitioning to allow a larger time step of 4 fs to be used.

When applying these packages to calculate $k_{off}$ for the trypsin-benzamidine system, agreement within an order of magnitude of the experimental value was obtained - experiment: $600\pm 300/s$, SEEKR: $83\pm 14/s$, MMVT SEEKR: $174\pm 9/s$, SEEKR2: $990\pm 130/s$, SEEKR2 with hydrogen mass repartitioning: $310\pm 30/s$.

11. The weighted ensemble method

The weighted ensemble method was first introduced to improve sampling in Brownian dynamics simulations of the diffusional encounter between protein molecules.[99] In Brownian dynamics simulation of diffusional encounter, one usually runs many trajectories starting with the two molecules far apart and calculating the probability of the trajectories forming the desired molecular complex rather than diffusing further apart to an outer surface. Theories such as the Northrup-Allison-McCammon approach[100] can then be used to estimate the diffusion-controlled complexation rate constant. The weighted ensemble method improves sampling by allowing trajectories to split or merge at regular intervals. Trajectories in over-represented regions are merged and under-represented regions are split so that a diverse configurational space can be sampled thoroughly with less computing time. The trajectories are initially given equal weights that sum to unity. When two trajectories merge, the weight of the resulting trajectory is the sum of the weights of the parent trajectories. When a trajectory splits, each daughter trajectory receives one half of the weight of the parent trajectory. Figure 4 uses a simple example to illustrate the idea.

Figure 4:

Illustration of the weighted ensemble method

This schematic picture illustrates the simulation of the diffusion of particles from the leftmost to the rightmost bin. At $t_1$, a single trajectory is started in the leftmost bin, or bin 1. It is then split into two trajectories, each carries one half of the weight of the original trajectory to improve the sampling of space. The dotted arrows show the movement of the split trajectories, and they occupy different bins at $t_2$. The two trajectories are then further split into four trajectories to further improve sampling, and this simulation uses a maximum of four trajectories. At $t_3$, three trajectories end up in bin 1 and one reaches bin 3. To reduce over-representation of trajectories in bin 1, two trajectories occupying similar space are merged. As few trajectories sample the other bins, the trajectory in bin 3 is split. At $t_4$, one trajectory reaches the last bin, bin 4, and it is warped back to bin 1 at $t_5$ to maintain a steady state of particles diffusing from the leftmost to the rightmost bin. Further merging or splitting of trajectories to produce a more balanced sampling continues from $t_5$ to $t_6$. (The sizes of the darked circles reflect the weights carried by the trajectories.)

The weighted ensemble method was subsequently extended to MD simulation of long-time events, including the exploration of the RNA interhelical conformations[101], simulation of the folding of a small WW domain[102], study of protein-peptide binding[103], and investigation of the permeation of ion through an ion channel[104]. Applications to studying drug-binding kinetics then followed[105–108]

Earlier simulation used fixed bins along a pre-selected progress coordinate. A user needed to determine how to select the progress coordinate and the bins at the outset. The user also needed to choose parameters such as the number of trajectories to be run simultaneously, and the interval between resampling. Zwier et al.[109] provided useful recommendations on choosing these parameters.

Later, other methods have been introduced to further improve the weighted ensemble method. WEplore and REVO are described here.

11.1. WExplore

As in the original weighted ensemble method, WExplore[110] runs multiple trajectories sampled by the so called “walkers”. The trajectories are run in parallel and each of them carries a statistical weight. The statistical weights of all the walkers sum to 1. Every pre-chosen number of time steps, some walkers are cloned or merged to improve the sampling of configuration space. Walkers in overrepresented regions are merged. Walkers in underrepresented regions are cloned.

However, WExplore introduces additional features to improve performance. 1). In simulating ligand unbinding, it discovers the dissociation path automatically without pre-defining a progress coordinate. WExplore divides space into Voronoi polyhedra. Given a set of anchors placed in space, a Voronoi polyhedron associated with an anchor encloses all points which distances to the anchor are closer to the anchor than to any other anchor. Each anchor represents a conformation of a protein-ligand system. WExplore discovers these anchors automatically. It starts with one anchor from which trajectories are started. As the trajectories move away from the anchor and exceed a certain chosen distance from the anchor, new anchors and new Voronoi polyhedra are formed, and this process is repeated as the trajectories progress. Different distance measures can be used. For example, it can be the root mean squared distance (RMSD) between two conformations of the ligand after the corresponding conformations of the protein have been structurally aligned.

WExplore also scales better with the number of progress coordinates than previous implementation of the weighted ensemble method. The previous bin implementation requires many regions and many walkers for a high-dimensional system. WExplore alleviates this problem by defining regions in a hierarchical fashion[110], which allows a small number of walkers to be distributed across a high-dimensional space. The number of walkers can be smaller than the number of regions. (The original weighted ensemble method using binning needs to have walkers in every region.) In addition, unlike previous implementations of the weighted ensemble method which keeps the number of sampling regions and the number of walkers in each region constant, WExplore allows these numbers to change during a simulation.

WExplore was used successfully to simulate the unbinding of three ligands from FKBP [105] The residence times obtained, in the nanosecond timescale, were close to the values obtained by Huang et al.[111] from unbiased MD simulations. In using WExplore to study the dissociation of benzamidine from trypsin, Dickson and Lotz[108] obtained a residence time of 180 μs, within an order of magnitude of the experimental value of 1700 μs. Analyzing the dissociation trajectories identified three major exit channels.

Lotz and Dickson[107] used WExplore to study the dissociation of 1-trifluoromethoxyphenyl-3-(1-propionylpiperidin-propionylpiperidin-4-yl)-urea, or TPPU, from soluble epoxide hydrolase, which occurred in the 11-minute timescale. From only 6 μs of simulation time, they observed 75 unbinding events from which they estimated the residence time to be 42 s with an uncertainty spanning from 23 s to 280 s. They also constructed a free-energy landscape of ligand dissociation by building a conformation space network (CSN), and they characterized the stabilizing interactions in the ensemble of the transition states identified.

11.2. Resampling of Ensembles by Variation Optimization (REVO)

Later, Resampling of Ensembles by Variation Optimization (REVO)[112] was developed to further improve WExplore. WExplore has three limitations: 1). Many cloning events of a single trajectory could occur in quick succession, 2). Definition of regions is not optimal and once the regions are created, they are fixed, and 3). The distribution of sampling regions is uneven. REVO clones and merges trajectories without defining regions. Instead, REVO optimizes a measure of variation that depends on the pairwise distances between walkers to increase the diversity of trajectories to sample large space and increase the probability of observing dissociation events.

REVO maximizes the following measure of trajectory variation

$$V={\sum }_i{V}_i={\sum }_i{\sum }_j{\left(\frac{d_{ij}}{d_o}\right)}^{\alpha }{\varphi }_i{\varphi }_j$$ Eq. 14

in which $d_{ij}$ measures the distance between two walkers and $d_0$ is a reference distance. ${\varphi }_i$ measures the relative importance of walkers such as favoring those that carry higher statistical weights, with constraints that the weights will not be too high to have a single walker dominate, or too low that the walker’s contribution to different calculated quantities will be small. $\alpha$ controls how sensitive $V$ depends on changes in the distances.

Dixon et al.[113] used REVO to study the dissociation of the neuroimaging agent PK-11195 from the translocator protein encoded by the TSPO gene. They found that the ligand could dissociate from the membrane protein via the membrane. Mean first passage time calculated from five different docking poses gave values of 260 minutes, 28 minutes, 4.1 minutes, 2.6 minutes, and 0.015 s. Four of these were within about an order of magnitude of the experimental value of 34 minutes.

Roussey et al.[106] later introduced Cutoff-REVO to further improve the efficiency of the REVO method. Cutoff-REVO avoided cloning trajectories when the ligand had already moved away from the binding pocket and committed to dissociating. They first ran trial REVO simulations to determine a suitable cutoff protein-ligand distance beyond which trajectories would not be cloned before running productive Cutoff-REVO simulations. Walkers reaching the unbound form carried larger statistical weights and dissociation rates calculated from such walkers gave smaller statistical errors than normal REVO.

12. Weighted ensemble milestoning

Ray and Andricioaei[114] combined key concepts of the weighted ensemble and the milestoning method to reduce the costs in computing protein-ligand binding kinetics and thermodynamics. The main idea was to use the weighted ensemble method to run trajectories between milestones. This allowed milestones to be placed sufficiently far apart to maintain Markovianity without significantly increasing computational time to run trajectories between milestones. In testing the method on a 1-dimensional double-well model, a 11-dimension potential with 10 degrees of freedom coupled to the reaction coordinate, and an atomistic model of alanine dipeptide, weighted ensemble milestoning computed mean first passage time an order of magnitude faster than the weighted ensemble method.

Later, Ray et al.[115] developed the Markovian Weighted Ensemble Milestoning Method (M-WEM) which used the Markovian Milestoning with Voronoi Tessellations[86] to further speed up the weighted ensemble milestoning method. In simulating the trypsin-benzamidine system, M-WEM was able to obtain $k_{off}$ and $k_{on}$ with an order of magnitude of simulation time less than MMVT SEEKR.[96] It took only a few hundred nanoseconds of simulation time to simulate systems with residence times four orders of magnitude longer.

13. Metadynamics

Metadynamics facilitates the simulation of long-time dynamics by continuously depositing Gaussian bias functions to help a system escape from local energy minima and discourage it from revisiting previously visited regions. The Gaussian functions are defined in terms of a small set of n collective variables $S=(S_1,S_2,\dots .,S_n$)[116,117] by

$$V\left(S,t\mathrm{\text{'}}\right)=\frac{\omega \left(t^{\mathrm{\text{'}}}\right)\exp \left(-{\sum }_{\left(s_i\right)}^n{(\left(s-s_i\left(t^{\mathrm{\text{'}}}\right)\right)}^2\right)}{2{\sigma }_i^2}$$ Eq. 15

where the Gaussian bias has a width of ${\sigma }_i$ and its height, $\omega (t^{\text{'}})$, changes with time in well-tempered metadynamics[118] as

$$\omega \left(t^{\mathrm{\text{'}}}\right)={\omega }_0\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{V\left(S,t^{\mathrm{\text{'}}}\right)}{k_{B}T\left(1-\gamma \right)})$$ Eq. 16

where $k_{B}T$ is the Boltzmann constant multiplied by the absolute temperature, ${\omega }_0$ is the initial height, and $\gamma$ is a bias factor.

Infrequent metadynamics[119,120] was introduced to calculate dissociation time by

$$t={\int }_0^{t_{total}}d{t}^{\mathrm{\text{'}}}{e}^{\frac{V\left(S,t^{\mathrm{\text{'}}}\right)}{k_{B}T}}$$ Eq. 17

where $t_{total}$ was the time taken for the simulation to observe a ligand dissociation.

To start a metadynamics simulation, a small number of collective variables (CVs) is first chosen. The chosen CVs should be able to discriminate relevant metastable states and include the slowest modes of a process. The number of CVs should be as small as possible because sampling time increases rapidly with this number.

Tiwary et al.[121] performed 12 independent metadynamics runs to study the unbinding of dasatinib from the c-Src kinase. Two collective variables were used in the metadynamics simulations: the distance of the drug to the protein and the solvation state of the binding pocket. Infrequent metadynamics estimated the residence time to be 21 ± 10 s, close to the experimental value of 18 s. From the intermediate states identified from the metadynamics simulations, the authors found that ligand dissociation was preceded by coupled protein-water movements and the breaking of the Lys36-Glu46 salt bridge when the drug was still inside the pocket. The $\alpha$C helix also moved from the in to the out conformation. The conformational change of the protein created space for more water molecules to move into the pocket before the drug started to dissociate. By solving a master equation, they found two slow steps in the dissociation process. The rate-limiting one involved the coupling between the movement of the drug and the breaking of its interactions with Glu75 and Met77 when the $\alpha$C helix was already rotated to the out conformation.

In another study on the unbinding of 1-(3-(tert-butyl)-1-(p-tolyl)-1H-pyrazol-r-yl)urea from the mitogen-activated protein kinase p38α[122] infrequent metadynamics predicted a $k_{off}$ of 0.02 ± 0.01 s⁻¹ in comparison to the experimental value of 0.14 s⁻¹. Two sets of two collective variables were used and similar results were obtained.

14. Influence of force field on simulation results

In MD simulations, insufficiently accurate force field for a system is always a concern. Cappelli et al.[123] used two ways of calculating atomic charges for the drug iperoxo and computed $k_{off}$ for its dissociation from the muscarinic receptor M2. When they computed the atomic partial charges of the drug by using the RESP model[124] with Hartree-Fock calculations employing the 6–31G* basis set, the activation barrier was extremely high, giving a residence time in years. Computing the atomic charges using density functional theory with the density functional B3LYP[125,126] employing the same basis set gave a $k_{off}$ of $3.7\pm 0.7\times {10}^{-4}{s}^{-1}$, about two orders of magnitude smaller than the experimental value of $1.0\pm 0.2\times {10}^{-2}{s}^{-1}$.

Haldar et al.[127] compared the free energy differences obtained between Molecular Mechanics (MM) and Quantum Mechanics/Molecular Mechanics (QM/MM) models for four structures along the pathways of dissociation of imatinib from the protein kinase c-Src. The structures were chosen from the dissociation paths obtained from the transition-state-partial path transition interface sampling method[128] using a classical force field. The four structures corresponded to the bound, unbound, and transition states, as well as an encounter complex. Free energy perturbation calculations[129] were performed to calculate the free energy differences between the MM and QM/MM models for each structure. The quantum mechanical calculations were performed at the BLYP[130,131]/VDZ level. The differences of free energy for the four structures ranged from −1.9 to −4.7 kcal/mol. These relatively large discrepancies in free energies suggested that effects such as explicit polarization and charge transfer not captured well by classical force fields could hinder the latter from giving highly accurate dissociation rate constants.

15. Using machine learning in molecular simulations for studying drug-binding kinetics

Machine learning can help to improve the prediction of $k_{on}$ or $k_{off}$ from molecular simulations.

Chiu and Xie[132] tested the use of protein-ligand pairwise interaction energies obtained from MD simulations of the bound states of complexes formed between 39 inhibitors and the HIV-1 protease, and information from coarse-grained normal model analyses as features in a machine-learning model (random forest predictive clustering) to classify the inhibitors into four groups with $k_{on}$ and $k_{off}$ each separated into two groups based on a cutoff value. From the normal model analyses, they calculated the relative movement between ligand and the residues in the binding pocket, and the relative movement between residues upon ligand binding to be used as features in the machine-learning model. They found that using pairwise electrostatic energies and the relative movement between ligand and residues in the binding site as features allowed the machine-learning model to classify the ligands into the four groups with 74% accuracy. They also found that the use of pairwise van der Waals energies as features was less predictive.

In the work by Nunes-Alves et al.,[133] several intermediate states were found along dissociation paths obtained from τ-random accelerated MD simulations, suggesting that features beyond the bound state might also be important for predicting residence time in the general case. As mentioned above, Kokh et al.[40] used features obtained from analyzing many whole dissociation trajectories from τ-RAMD simulations for 95 inhibitors of HSP90 in machine-learning models to obtain better prediction of residence time.

Huang et al. [134] employed protein-ligand interaction profile along dissociation pathways obtained by steered MD to predict $k_{off}$ for complexes formed between HIV-1 protease and 37 inhibitors. Three types of interactions were calculated between the ligand and the residues lying along the dissociation pathways: total interaction energy, electrostatic interaction energy, and van der Waals interaction energy. These interactions were calculated at 300-ps intervals. Using these interaction fingerprints in a partial-least-square regression model gave good prediction of $k_{off}$. In addition, the authors identified key interactions affecting ligand dissociation. These interactions were found to occur during the first half of the dissociation process.

Machine learning has also been used to find good progress coordinate(s) for predicting long-time events. For example, Tiwary and co-workers introduced the Reweighted Autoencoded Variational Bayes for Enhanced Sampling (RAVE) method[58] that used an autoencoder to discover good progress coordinate(s). Starting from an unbiased MD simulation, the autoencoder found a latent variable $z$ that best captured long-time dynamics and the associated probability distribution $P(z)$. The best progress coordinate $\chi$, expressed in terms of a pre-chosen set of ordered parameters, was then found by requiring its probability distribution $P(\chi )$ to be as close to $P(z)$ as possible. A biased potential $V_b\left(\chi \right)=k_{B}Tln(P(\chi )$ could then be used in a biased MD simulation to better sample along the progress coordinate. Repeating this process found increasingly better progress coordinate for describing long-time events such as the dissociation of a ligand from its protein receptor. Using a refined version of RAVE, Lamin et al.[135] were able to observe dissociation of benzene from the T99A mutant of T4 lysozyme with simulations lasting only 3–50 ns, when the experimental time scale for this process was in the millisecond range.

16. Conclusions

As accumulated evidence supports drug-binding kinetics as an important factor to be considered in drug discovery, computational scientists have been actively developing and testing MD-based methods for studying drug-binding kinetics. This review illustrates the rapid progress in the last 15 years. Although only crude qualitative models were used at the beginning, quantitative estimates of absolute, not only relative, rates are now feasible. Many methods have been developed or/and tested. This review covers only a subset of popular methods as examples. These methods range from those requiring progress coordinates to be pre-defined to those that could discover progress coordinates. Currently, identifying compounds with fast or slow drug-binding kinetics, or rank-ordering association/dissociation constants is an easier task. However, predicting absolute association/dissociation rate is still challenging. Although some simulations obtained results within an order of magnitude of experimental values, some could deviate by more than two orders of magnitude. Insufficient sampling for simulating long-time events even with enhanced sampling methods and deficiency in current force fields are two contributing factors. Nevertheless, considering the rapid progress in recent years, significant improvement of current methodologies and the emergence of new methodologies are expected in the coming years.

17. Expert Opinion

The important role of drug-binding kinetics is now widely recognized. Molecular simulation should play an increasingly useful role in studying drug-binding kinetics and designing compounds with the desired kinetic parameters. An advantage of molecular simulation is that it can provide atomistic insights into drug association/dissociation processes. By analyzing in atomic detail the changes in the ligand, the protein, their surrounding water molecules, and their interactions along association/dissociation pathways, useful insights into drug design can be obtained. Analyzing pathways is more general than analyzing bound structures alone, as drug-binding kinetics could be influenced by factors beyond the bound states. Features of the transition states, for example, can be crucial in guiding compound design.

Predicting absolute association/dissociation rates is still challenging. Although rapid progress has been made in the past 15 years, large deviations from experimental values can still be observed for some systems. Besides the accuracy of force fields in the simulations, efficient sampling of relevant phase space is still challenging. Methods that enhance samplings along pre-selected progress coordinates depend on how well the coordinates describe long-time events related to drug association/dissociation. Encouragingly, methods for finding good progress coordinates have been introduced. Methods that do not require pre-selection of progress coordinates can suffer from inadequate sampling of or missing regions essential for describing association/dissociation dynamics. Methods that can iteratively guide the sampling of important regions, for example, can alleviate these problems.

As calculating absolute rates are still expensive, many publications reported the simulation of only one or a few systems. Several systems have been used frequently for evaluation of methods. The total number of distinct systems for which drug-binding kinetics has been simulated is still small. Evaluation of methods on more systems will provide better assessment of the performance of available methodologies. The rapid increase in the number of publications in this area is a good sign. The collective experiences by the increasing number of research groups working in this field should continue to advance the field at a rapid pace.

Machine learning is expected to play a more extensive role in studying drug-binding kinetics. This review has already given examples. It has been used to improve the prediction of rate constants by using features obtained from MD simulations, to help choose parameters in simulation models such as those in the Markov State Model and to discover better progress coordinates for methods that require them to be specified. With rapid advances in machine learning, the study of drug-binding kinetics should benefit from the wide array of tools available.

In practical drug discovery, it is also important to know what kinetics parameters useful drug candidates need to have. Although more focus has been placed on long residence time, the Introduction section of this review mentions that long residence time is not always desirable. Some drug developers in the pharmaceutical industry avoid compounds with extremely long residence times because of toxicity risks. Models for simulating drug action closer to the systems level are not yet accurate enough for predicting how drug-binding kinetics affect drug efficacy. Improvement in this area could increase the power of computational techniques in suggesting promising compounds with desirable drug-binding kinetics for experimental testing.

Article highlights.

• Drug-binding kinetics has become an important factor for consideration in drug discovery.

• Molecular dynamics-based simulations have helped to decipher the molecular mechanisms of drug-binding kinetics and design compounds with therapeutically useful kinetic parameters.

• The last 15 years have seen significant progress in moving from qualitative to quantitative models. This review examines how a subset of methods have been used to study drug-binding kinetics. These methods include a mining-minima approach, steered molecular dynamics, τ-random accelerated molecular dynamics, scaled molecular dynamics, umbrella sampling simulation, Markov State Model, milestoning simulation, weighted ensemble simulation, and metadynamics simulation. Some of these methods allow absolute, not only relative, association/dissociation rates to be computed.

• Machine learning has been used with molecular dynamics to improve the study of drug-binding kinetics.

• Because it is still expensive to compute absolute association/dissociation rates, the number of systems studied is still small. However, the increase in the number of research groups tackling this problem should help to validate methodologies at a more rapid pace.