SEEKR: Simulation Enabled Estimation of Kinetic Rates, A Computational Tool to Estimate Molecular Kinetics and Its Application to Trypsin-Benzamidine Binding

Toward Grid-Based Models for Molecular Association

This paper presents a grid-based approach to model molecular association processes as an alternative to sampling-based Markov models. Our method discretizes the six-dimensional space of relative translation and orientation into grid cells. By discretizing the Fokker-Planck operator governing the system dynamics via the square-root approximation, we derive analytical expressions for the transition rate constants between grid cells. These expressions depend on geometric properties of the grid, such as the cell surface area and volume, which we provide. In addition, one needs only the molecular energy at the grid cell center, circumventing the need for extensive MD simulations and reducing the number of energy evaluations to the number of grid cells. The resulting rate matrix is closely related to the Markov state model transition matrix, offering insights into metastable states and association kinetics. We validate the accuracy of the model in identifying metastable states and binding mechanisms, though improvements are necessary to address limitations like ignoring bulk transitions and anisotropic rotational diffusion. The flexibility of this grid-based method makes it applicable to a variety of molecular systems and energy functions, including those derived from quantum mechanical calculations. The software package MolGri, which implements this approach, offers a systematic and computationally efficient tool for studying molecular association processes.

Advances and Challenges in Milestoning Simulations for Drug-Target Kinetics

Molecular dynamics simulations have become indispensable for exploring complex biological processes, yet their limitations in capturing rare events hinder our understanding of drug-target kinetics. In this Perspective, we investigate the domain of milestoning simulations to understand this challenge. The milestoning approach divides the phase space of the drug-target complex into discrete cells, offering extended time scale insights. This Perspective traces the history, applications, and future potential of milestoning simulations in the context of drug-target kinetics. It explores the fundamental principles of milestoning, highlighting the importance of probabilistic transitions and transition time independence. Markovian milestoning with Voronoi tessellations is revisited to address the traditional milestoning challenges. While observing the advancements in this field, this Perspective also addresses impending challenges in estimating drug-target unbinding rate constants through milestoning simulations, paving the way for more effective drug design strategies.

Prediction of Threonine-Tyrosine Kinase Receptor-Ligand Unbinding Kinetics with Multiscale Milestoning and Metadynamics

Accurately describing protein-ligand binding and unbinding kinetics remains challenging. Computational calculations are difficult and costly, while experimental measurements often lack molecular detail and can be unobtainable. Here, we extend our multiscale milestoning method, Simulation-Enabled Estimation of Kinetics Rates (SEEKR), with metadynamics molecular dynamics simulations to yield accurate small molecule drug residence times. Using the pharmaceutically relevant threonine-tyrosine kinase (TTK) and eight long-residence-time (tens of seconds to hours) inhibitors, we demonstrate accurate prediction of absolute and rank-ordered ligand residence times and free energies of binding.

A methodology of quantifying membrane permeability based on returning probability theory and molecular dynamics simulation

We propose a theoretical approach to estimate the permeability coefficients of substrates (permeants) for crossing membranes from donor (D) phase to acceptor (A) phase by means of molecular dynamics (MD) simulation. A fundamental aspect of our approach involves reformulating the returning probability (RP) theory, a rigorous bimolecular reaction theory, to describe permeation phenomena. This reformulation relies on the parallelism between permeation and bimolecular reaction processes. In the present method, the permeability coefficient is represented in terms of the thermodynamic and kinetic quantities for the reactive (R) phase that exists within the inner region of a membrane. One can evaluate these quantities using multiple MD trajectories starting from phase R. We apply the RP theory to the permeation of ethanol and methylamine at different concentrations (infinitely dilute and 1 mol % conditions of permeants). Under the 1 mol% condition, the present method yields a larger permeability coefficient for ethanol (0.12 ± 0.01 cm s-1) than for methylamine (0.069 ± 0.006 cm s-1), while the values of the permeability coefficient are satisfactorily close to those obtained from the brute-force MD simulations (0.18 ± 0.03 and 0.052 ± 0.005 cm s-1 for ethanol and methylamine, respectively). Moreover, upon analyzing the thermodynamic and kinetic contributions to the permeability, we clarify that a higher concentration dependency of permeability for ethanol, as compared to methylamine, arises from the sensitive nature of ethanol's free-energy barrier within the inner region of the membrane against ethanol concentration.

Learning Protein-Ligand Unbinding Pathways via Single-Parameter Community Detection

Understanding the dynamics of biomolecular complexes, e.g., of protein-ligand (un)binding, requires the comprehension of paths such systems take between metastable states. In MD simulations, paths are usually not observable per se, but they need to be inferred from simulation trajectories. Here, we present a novel approach to cluster trajectories based on a community detection algorithm that necessitates only the definition of a single parameter. The unbinding of the streptavidin-biotin complex is used as a benchmark system and the A2a adenosine receptor in complex with the inhibitor ZM241385 as an elaborate application. We demonstrate how such clusters of trajectories correspond to pathways and how the approach helps in the identification of reaction coordinates for a considered (un)binding process.

Understanding the impact of binding free energy and kinetics calculations in modern drug discovery

INTRODUCTION

For rational drug design, it is crucial to understand the receptor-drug binding processes and mechanisms. A new era for the use of computer simulations in predicting drug-receptor interactions at an atomic level has begun with remarkable advances in supercomputing and methodological breakthroughs.

AREAS COVERED

End-point free energy calculation methods such as Molecular Mechanics/Poisson Boltzmann Surface Area (MM/PBSA) or Molecular-Mechanics/Generalized Born Surface Area (MM/GBSA), free energy perturbation (FEP), and thermodynamic integration (TI) are commonly used for binding free energy calculations in drug discovery. In addition, kinetic dissociation and association rate constants (k off and k on ) play critical roles in the function of drugs. Nowadays, Molecular Dynamics (MD) and enhanced sampling simulations are increasingly being used in drug discovery. Here, the authors provide a review of the computational techniques used in drug binding free energy and kinetics calculations.

EXPERT OPINION

The applications of computational methods in drug discovery and design are expanding, thanks to improved predictions of the binding free energy and kinetic rates of drug molecules. Recent microsecond-timescale enhanced sampling simulations have made it possible to accurately capture repetitive ligand binding and dissociation, facilitating more efficient and accurate calculations of ligand binding free energy and kinetics.

Quantitative Prediction of Dissociation Rates of PYK2 Ligands Using Umbrella Sampling and Milestoning

We used umbrella sampling and the milestoning simulation method to study the dissociation of multiple ligands from protein kinase PYK2. The activation barriers obtained from the potential of mean force of the umbrella sampling simulations correlated well with the experimental dissociation rates. Using the zero-temperature string method, we obtained optimized paths along the free-energy surfaces for milestoning simulations of three ligands with a similar structure. The milestoning simulations gave an absolute dissociation rate within 2 orders of magnitude of the experimental value for two ligands but at least 3 orders of magnitude too high for the third. Despite the similarity in their structures, the ligands took different pathways to exit from the binding site of PYK2, making contact with different sets of residues. In addition, the protein experienced different conformational changes for dissociation of the three ligands.

Investigation of Rare Protein Conformational Transitions via Dissipation-Corrected Targeted Molecular Dynamics

To sample rare events, dissipation-corrected targeted molecular dynamics (dcTMD) applies a constant velocity constraint along a one-dimensional reaction coordinate s, which drives an atomistic system from an initial state into a target state. Employing a cumulant approximation of Jarzynski's identity, the free energy ΔG(s) is calculated from the mean external work and dissipated work of the process. By calculating the friction coefficient Γ(s) from the dissipated work, in a second step, the equilibrium dynamics of the process can be studied by propagating a Langevin equation. While so far dcTMD has been mostly applied to study the unbinding of protein-ligand complexes, here its applicability to rare conformational transitions within a protein and the prediction of their kinetics are investigated. As this typically requires the introduction of multiple collective variables {xj} = x, a theoretical framework is outlined to calculate the associated free energy ΔG(x) and friction Γ(x) from dcTMD simulations along coordinate s. Adopting the α-β transition of alanine dipeptide as well as the open-closed transition of T4 lysozyme as representative examples, the virtues and shortcomings of dcTMD to predict protein conformational transitions and the related kinetics are studied.

QMrebind: incorporating quantum mechanical force field reparameterization at the ligand binding site for improved drug-target kinetics through milestoning simulations

Understanding the interaction of ligands with biomolecules is an integral component of drug discovery and development. Challenges for computing thermodynamic and kinetic quantities for pharmaceutically relevant receptor-ligand complexes include the size and flexibility of the ligands, large-scale conformational rearrangements of the receptor, accurate force field parameters, simulation efficiency, and sufficient sampling associated with rare events. Our recently developed multiscale milestoning simulation approach, SEEKR2 (Simulation Enabled Estimation of Kinetic Rates v.2), has demonstrated success in predicting unbinding (koff) kinetics by employing molecular dynamics (MD) simulations in regions closer to the binding site. The MD region is further subdivided into smaller Voronoi tessellations to improve the simulation efficiency and parallelization. To date, all MD simulations are run using general molecular mechanics (MM) force fields. The accuracy of calculations can be further improved by incorporating quantum mechanical (QM) methods into generating system-specific force fields through reparameterizing ligand partial charges in the bound state. The force field reparameterization process modifies the potential energy landscape of the bimolecular complex, enabling a more accurate representation of the intermolecular interactions and polarization effects at the bound state. We present QMrebind (Quantum Mechanical force field reparameterization at the receptor-ligand binding site), an ORCA-based software that facilitates reparameterizing the potential energy function within the phase space representing the bound state in a receptor-ligand complex. With SEEKR2 koff estimates and experimentally determined kinetic rates, we compare and interpret the receptor-ligand unbinding kinetics obtained using the newly reparameterized force fields for model host-guest systems and HSP90-inhibitor complexes. This method provides an opportunity to achieve higher accuracy in predicting receptor-ligand koff rate constants.

Elucidating protein-ligand binding kinetics based on returning probability theory

The returning probability (RP) theory, a rigorous diffusion-influenced reaction theory, enables us to analyze the binding process systematically in terms of thermodynamics and kinetics using molecular dynamics (MD) simulations. Recently, the theory was extended to atomistically describe binding processes by adopting the host-guest interaction energy as the reaction coordinate. The binding rate constants can be estimated by computing the thermodynamic and kinetic properties of the reactive state existing in the binding processes. Here, we propose a methodology based on the RP theory in conjunction with the energy representation theory of solution, applicable to complex binding phenomena, such as protein-ligand binding. The derived scheme of calculating the equilibrium constant between the reactive and dissociate states, required in the RP theory, can be used for arbitrary types of reactive states. We apply the present method to the bindings of small fragment molecules [4-hydroxy-2-butanone (BUT) and methyl methylthiomethyl sulphoxide (DSS)] to FK506 binding protein (FKBP) in an aqueous solution. Estimated binding rate constants are consistent with those obtained from long-timescale MD simulations. Furthermore, by decomposing the rate constants to the thermodynamic and kinetic contributions, we clarify that the higher thermodynamic stability of the reactive state for DSS causes the faster binding kinetics compared with BUT.

Free energy along drug-protein binding pathways interactively sampled in virtual reality

We describe a two-step approach for combining interactive molecular dynamics in virtual reality (iMD-VR) with free energy (FE) calculation to explore the dynamics of biological processes at the molecular level. We refer to this combined approach as iMD-VR-FE. Stage one involves using a state-of-the-art 'human-in-the-loop' iMD-VR framework to generate a diverse range of protein-ligand unbinding pathways, benefitting from the sophistication of human spatial and chemical intuition. Stage two involves using the iMD-VR-sampled pathways as initial guesses for defining a path-based reaction coordinate from which we can obtain a corresponding free energy profile using FE methods. To investigate the performance of the method, we apply iMD-VR-FE to investigate the unbinding of a benzamidine ligand from a trypsin protein. The binding free energy calculated using iMD-VR-FE is similar for each pathway, indicating internal consistency. Moreover, the resulting free energy profiles can distinguish energetic differences between pathways corresponding to various protein-ligand conformations (e.g., helping to identify pathways that are more favourable) and enable identification of metastable states along the pathways. The two-step iMD-VR-FE approach offers an intuitive way for researchers to test hypotheses for candidate pathways in biomolecular systems, quickly obtaining both qualitative and quantitative insight.

Perspectives on Ligand/Protein Binding Kinetics Simulations: Force Fields, Machine Learning, Sampling, and User-Friendliness

Computational techniques applied to drug discovery have gained considerable popularity for their ability to filter potentially active drugs from inactive ones, reducing the time scale and costs of preclinical investigations. The main focus of these studies has historically been the search for compounds endowed with high affinity for a specific molecular target to ensure the formation of stable and long-lasting complexes. Recent evidence has also correlated the in vivo drug efficacy with its binding kinetics, thus opening new fascinating scenarios for ligand/protein binding kinetic simulations in drug discovery. The present article examines the state of the art in the field, providing a brief summary of the most popular and advanced ligand/protein binding kinetics techniques and evaluating their current limitations and the potential solutions to reach more accurate kinetic models. Particular emphasis is put on the need for a paradigm change in the present methodologies toward ligand and protein parametrization, the force field problem, characterization of the transition states, the sampling issue, and algorithms' performance, user-friendliness, and data openness.

Predicting Protein-Ligand Binding and Unbinding Kinetics with Biased MD Simulations and Coarse-Graining of Dynamics: Current State and Challenges

The prediction of drug-target binding and unbinding kinetics that occur on time scales between milliseconds and several hours is a prime challenge for biased molecular dynamics simulation approaches. This Perspective gives a concise summary of the theory and the current state-of-the-art of such predictions via biased simulations, of insights into the molecular mechanisms defining binding and unbinding kinetics as well as of the extraordinary challenges predictions of ligand kinetics pose in comparison to binding free energy predictions.

Selectivity and Ranking of Tight-Binding JAK-STAT Inhibitors Using Markovian Milestoning with Voronoi Tessellations

Janus kinases (JAK), a group of proteins in the nonreceptor tyrosine kinase (NRTKs) family, play a crucial role in growth, survival, and angiogenesis. They are activated by cytokines through the Janus kinase-signal transducer and activator of a transcription (JAK-STAT) signaling pathway. JAK-STAT signaling pathways have significant roles in the regulation of cell division, apoptosis, and immunity. Identification of the V617F mutation in the Janus homology 2 (JH2) domain of JAK2 leading to myeloproliferative disorders has stimulated great interest in the drug discovery community to develop JAK2-specific inhibitors. However, such inhibitors should be selective toward JAK2 over other JAKs and display an extended residence time. Recently, novel JAK2/STAT5 axis inhibitors (N-(1H-pyrazol-3-yl)pyrimidin-2-amino derivatives) have displayed extended residence times (hours or longer) on target and adequate selectivity excluding JAK3. To facilitate a deeper understanding of the kinase-inhibitor interactions and advance the development of such inhibitors, we utilize a multiscale Markovian milestoning with Voronoi tessellations (MMVT) approach within the Simulation-Enabled Estimation of Kinetic Rates v.2 (SEEKR2) program to rank order these inhibitors based on their kinetic properties and further explain the selectivity of JAK2 inhibitors over JAK3. Our approach investigates the kinetic and thermodynamic properties of JAK-inhibitor complexes in a user-friendly, fast, efficient, and accurate manner compared to other brute force and hybrid-enhanced sampling approaches.

Predicting Biomolecular Binding Kinetics: A Review

Biomolecular binding kinetics including the association (k_on) and dissociation (k_off) rates are critical parameters for therapeutic design of small-molecule drugs, peptides, and antibodies. Notably, the drug molecule residence time or dissociation rate has been shown to correlate with their efficacies better than binding affinities. A wide range of modeling approaches including quantitative structure-kinetic relationship models, Molecular Dynamics simulations, enhanced sampling, and Machine Learning has been developed to explore biomolecular binding and dissociation mechanisms and predict binding kinetic rates. Here, we review recent advances in computational modeling of biomolecular binding kinetics, with an outlook for future improvements.

Path separation of dissipation-corrected targeted molecular dynamics simulations of protein-ligand unbinding

Protein-ligand (un)binding simulations are a recent focus of biased molecular dynamics simulations. Such binding and unbinding can occur via different pathways in and out of a binding site. Here, we present a theoretical framework on how to compute kinetics along separate paths and on how to combine the path-specific rates into global binding and unbinding rates for comparison with experimental results. Using dissipation-corrected targeted molecular dynamics in combination with temperature-boosted Langevin equation simulations [S. Wolf et al., Nat. Commun. 11, 2918 (2020)] applied to a two-dimensional model and the trypsin-benzamidine complex as test systems, we assess the robustness of the procedure and discuss the aspects of its practical applicability to predict multisecond kinetics of complex biomolecular systems.

Milestoning simulation of ligand dissociation from the glycogen synthase kinase 3β

As drug-binding kinetics has become an important factor to be considered in modern drug discovery, this work evaluated the ability of the Milestoning method in computing the absolute dissociation rate of a ligand from the serine-threonine kinase, glycogen synthase kinase 3β, which is a target for designing drugs to treat diseases such as neurodegenerative disorders and diabetes. We found that the Milestoning method gave good agreement with experiment with modest computational costs. Although the time scale for dissociation lasted tens of seconds, the collective molecular dynamics simulations total less than 1μs. Computing the committor function helped to identify the transition states (TSs), in which the ligand moved substantially away from the binding pocket. The glycine-rich loop with a serine residue attaching to its tips was found to undergo large movement from the bound to the TSs and might play a role in controlling drug-dissociation kinetics.

Myofilament-associated proteins with intrinsic disorder (MAPIDs) and their resolution by computational modeling

The cardiac sarcomere is a cellular structure in the heart that enables muscle cells to contract. Dozens of proteins belong to the cardiac sarcomere, which work in tandem to generate force and adapt to demands on cardiac output. Intriguingly, the majority of these proteins have significant intrinsic disorder that contributes to their functions, yet the biophysics of these intrinsically disordered regions (IDRs) have been characterized in limited detail. In this review, we first enumerate these myofilament-associated proteins with intrinsic disorder (MAPIDs) and recent biophysical studies to characterize their IDRs. We secondly summarize the biophysics governing IDR properties and the state-of-the-art in computational tools toward MAPID identification and characterization of their conformation ensembles. We conclude with an overview of future computational approaches toward broadening the understanding of intrinsic disorder in the cardiac sarcomere.

Advances in computational methods for ligand binding kinetics

Binding kinetic parameters can be correlated with drug efficacy, which in recent years led to the development of various computational methods for predicting binding kinetic rates and gaining insight into protein-drug binding paths and mechanisms. In this review, we introduce and compare computational methods recently developed and applied to two systems, trypsin-benzamidine and kinase-inhibitor complexes. Methods involving enhanced sampling in molecular dynamics simulations or machine learning can be used not only to predict kinetic rates, but also to reveal factors modulating the duration of residence times, selectivity, and drug resistance to mutations. Methods which require less computational time to make predictions are highlighted, and suggestions to reduce the error of computed kinetic rates are presented.

Water Network in the Binding Pocket of Fluorinated BPTI-Trypsin Complexes─Insights from Simulation and Experiment

Structural waters in the S1 binding pocket of β-trypsin are critical for the stabilization of the complex of β-trypsin with its inhibitor bovine pancreatic trypsin inhibitor (BPTI). The inhibitor strength of BPTI can be modulated by replacing the critical lysine residue at the P1 position by non-natural amino acids. We study BPTI variants in which the critical Lys15 in BPTI has been replaced by α-aminobutyric acid (Abu) and its fluorinated derivatives monofluoroethylglycine (MfeGly), difluoroethylglycine (DfeGly), and trifluoroethylglycine (TfeGly). We investigate the hypothesis that additional water molecules in the binding pocket can form specific noncovalent interactions with the fluorinated side chains and thereby act as an extension of the inhibitors. We report potentials of mean force (PMF) of the unbinding process for all four complexes and enzyme activity inhibition assays. Additionally, we report the protein crystal structure of the Lys15MfeGly-BPTI-β-trypsin complex (pdb: 7PH1). Both experimental and computational data show a stepwise increase in inhibitor strength with increasing fluorination of the Abu side chain. The PMF additionally shows a minimum for the encounter complex and an intermediate state just before the bound state. In the bound state, the computational analysis of the structure and dynamics of the water molecules in the S1 pocket shows a highly dynamic network of water molecules that does not indicate a rigidification or stabilizing trend in regard to energetic properties that could explain the increase in inhibitor strength. The analysis of the energy and the entropy of the water molecules in the S1 binding pocket using grid inhomogeneous solvation theory confirms this result. Overall, fluorination systematically changes the binding affinity, but the effect cannot be explained by a persistent water network in the binding pocket. Other effects, such as the hydrophobicity of fluorinated amino acids and the stability of the encounter complex as well as the additional minimum in the potential of mean force in the bound state, likely influence the affinity more directly.

Estimation of Drug-Target Residence Time by Targeted Molecular Dynamics Simulations

Drug-target residence time has emerged as a key selection factor in drug discovery since the binding duration of a drug molecule to its protein target can significantly impact its in vivo efficacy. The challenge in studying the residence time, in early drug discovery stages, lies in how to cost-effectively determine the residence time for the systematic assessment of compounds. Currently, there is still a lack of computational protocols to quickly estimate such a measure, particularly for large and flexible protein targets and drugs. Here, we report an efficient computational protocol, based on targeted molecular dynamics, to rank drug candidates by their residence time and to obtain insights into ligand-target dissociation mechanisms. The method was assessed on a dataset of 10 arylpyrazole inhibitors of CDK8, a large, flexible, and clinically important target, for which the experimental residence time of the inhibitors ranges from minutes to hours. The compounds were correctly ranked according to their estimated residence time scores compared to their experimental values. The analysis of protein-ligand interactions along the dissociation trajectories highlighted the favorable contribution of hydrophobic contacts to residence time and revealed key residues that strongly affect compound residence time.

Ligand Unbinding Pathway and Mechanism Analysis Assisted by Machine Learning and Graph Methods

We present two methods to reveal protein-ligand unbinding mechanisms in biased unbinding simulations by clustering trajectories into ensembles representing unbinding paths. The first approach is based on a contact principal component analysis for reducing the dimensionality of the input data, followed by identification of unbinding paths and training a machine learning model for trajectory clustering. The second approach clusters trajectories according to their pairwise mean Euclidean distance employing the neighbor-net algorithm, which takes into account input data bias in the distances set and is superior to dendrogram construction. Finally, we describe a more complex case where the reaction coordinate relevant for path identification is a single intraligand hydrogen bond, highlighting the challenges involved in unbinding path reaction coordinate detection.

SEEKR2: Versatile Multiscale Milestoning Utilizing the OpenMM Molecular Dynamics Engine

We present SEEKR2 (simulation-enabled estimation of kinetic rates version 2)—the latest iteration in the family of SEEKR programs for using multiscale simulation methods to computationally estimate the kinetics and thermodynamics of molecular processes, in particular, ligand-receptor binding. SEEKR2 generates equivalent, or improved, results compared to the earlier versions of SEEKR but with significant increases in speed and capabilities. SEEKR2 has also been built with greater ease of usability and with extensible features to enable future expansions of the method. Now, in addition to supporting simulations using NAMD, calculations may be run with the fast and extensible OpenMM simulation engine. The Brownian dynamics portion of the calculation has also been upgraded to Browndye 2. Furthermore, this version of SEEKR supports hydrogen mass repartitioning, which significantly reduces computational cost, while showing little, if any, loss of accuracy in the predicted kinetics.

Enhanced-Sampling Simulations for the Estimation of Ligand Binding Kinetics: Current Status and Perspective

The dissociation rate (k off) associated with ligand unbinding events from proteins is a parameter of fundamental importance in drug design. Here we review recent major advancements in molecular simulation methodologies for the prediction of k off. Next, we discuss the impact of the potential energy function models on the accuracy of calculated k off values. Finally, we provide a perspective from high-performance computing and machine learning which might help improve such predictions.

GeomBD3: Brownian Dynamics Simulation Software for Biological and Engineered Systems

GeomBD3 is a robust Brownian dynamics simulation package designed to easily handle natural or engineered systems in diverse environments and arrangements. The software package described herein allows users to design, execute, and analyze BD simulations. The simulations use all-atom, rigid molecular models that diffuse according to overdamped Langevin dynamics and interact through electrostatic, Lennard-Jones, and ligand desolvation potentials. The program automatically calculates molecular association rates, surface residence times, and association statistics for any number of user-defined association criteria. Users can also extract molecular association pathways, diffusion coefficients, intermolecular interaction energies, intermolecular contact probability maps, and more using the provided supplementary analysis scripts. We detail the use of the package from start to finish and apply it to a protein-ligand system and a large nucleic acid biosensor. GeomBD3 provides a versatile tool for researchers from various disciplines that can aid in rational design of engineered systems or play an explanatory role as a complement to experiments. GeomBD version 3 is available on our website at http://chemcha-gpu0.ucr.edu/geombd3/ and KBbox at https://kbbox.h-its.org/toolbox/methods/molecular-simulation/geombd/.

Accelerating All-Atom Simulations and Gaining Mechanistic Understanding of Biophysical Systems through State Predictive Information Bottleneck

An effective implementation of enhanced sampling algorithms for molecular dynamics simulations requires a priori knowledge of the approximate reaction coordinate describing the relevant mechanisms in the system. In this work, we focus on the recently developed artificial intelligence-based State Predictive Information Bottleneck (SPIB) approach and demonstrate how SPIB can learn such a reaction coordinate as a deep neural network even from undersampled trajectories. We exemplify its usefulness by achieving more than 40 times acceleration in simulating two model biophysical systems through well-tempered metadynamics performed by biasing along the SPIB-learned reaction coordinate. These include left- to right-handed chirality transitions in a synthetic helical peptide (Aib)9 and permeation of a small benzoic acid molecule through a synthetic, symmetric phospholipid bilayer. In addition to significantly accelerating the dynamics and achieving back and forth movement between different metastable states, the SPIB-based reaction coordinate gives mechanistic insights into the processes driving these two important problems.

Determination of Multidirectional Pathways for Ligand Release from the Receptor: A New Approach Based on Differential Evolution

Steered molecular dynamics (SMD) simulation is a powerful method in computer-aided drug design as it can be used to access the relative binding affinity with high precision but with low computational cost. The success of SMD depends on the choice of the direction along which the ligand is pulled from the receptor-binding site. In most simulations, the unidirectional pathway was used, but in some cases, this choice resulted in the ligand colliding with the complex surface of the exit tunnel. To overcome this difficulty, several variants of SMD with multidirectional pulling have been proposed, but they are not completely devoid of disadvantages. Here, we have proposed to determine the direction of pulling with a simple scoring function that minimizes the receptor–ligand interaction, and an optimization algorithm called differential evolution is used for energy minimization. The effectiveness of our protocol was demonstrated by finding expulsion pathways of Huperzine A and camphor from the binding site of Torpedo California acetylcholinesterase and P450cam proteins, respectively, and comparing them with the previous results obtained using memetic sampling and random acceleration molecular dynamics. In addition, by applying this protocol to a set of ligands bound with LSD1 (lysine specific demethylase 1), we obtained a much higher correlation between the work of pulling force and experimental data on the inhibition constant IC50 compared to that obtained using the unidirectional approach based on minimal steric hindrance.

A Companion Guide to the String Method with Swarms of Trajectories: Characterization, Performance, and Pitfalls

The string method with swarms of trajectories (SMwST) is an algorithm that identifies a physically meaningful transition pathway─a one-dimensional curve, embedded within a high-dimensional space of selected collective variables. The SMwST algorithm leans on a series of short, unbiased molecular dynamics simulations spawned at different locations of the discretized path, from whence an average dynamic drift is determined to evolve the string toward an optimal pathway. However conceptually simple in both its theoretical formulation and practical implementation, the SMwST algorithm is computationally intensive and requires a careful choice of parameters for optimal cost-effectiveness in applications to challenging problems in chemistry and biology. In this contribution, the SMwST algorithm is presented in a self-contained manner, discussing with a critical eye its theoretical underpinnings, applicability, inherent limitations, and use in the context of path-following free-energy calculations and their possible extension to kinetics modeling. Through multiple simulations of a prototypical polypeptide, combining the search of the transition pathway and the computation of the potential of mean force along it, several practical aspects of the methodology are examined with the objective of optimizing the computational effort, yet without sacrificing accuracy. In light of the results reported here, we propose some general guidelines aimed at improving the efficiency and reliability of the computed pathways and free-energy profiles underlying the conformational transitions at hand.

Markovian Weighted Ensemble Milestoning (M-WEM): Long-Time Kinetics from Short Trajectories

We introduce a rare-event sampling scheme, named Markovian Weighted Ensemble Milestoning (M-WEM), which inlays a weighted ensemble framework within a Markovian milestoning theory to efficiently calculate thermodynamic and kinetic properties of long-time-scale biomolecular processes from short atomistic molecular dynamics simulations. M-WEM is tested on the Müller-Brown potential model, the conformational switching in alanine dipeptide, and the millisecond time-scale protein-ligand unbinding in a trypsin-benzamidine complex. Not only can M-WEM predict the kinetics of these processes with quantitative accuracy but it also allows for a scheme to reconstruct a multidimensional free-energy landscape along additional degrees of freedom, which are not part of the milestoning progress coordinate. For the ligand-receptor system, the experimental residence time, association and dissociation kinetics, and binding free energy could be reproduced using M-WEM within a simulation time of a few hundreds of nanoseconds, which is a fraction of the computational cost of other currently available methods, and close to 4 orders of magnitude less than the experimental residence time. Due to the high accuracy and low computational cost, the M-WEM approach can find potential applications in kinetics and free-energy-based computational drug design.

SGOOP-d: Estimating Kinetic Distances and Reaction Coordinate Dimensionality for Rare Event Systems from Biased/Unbiased Simulations

Understanding kinetics including reaction pathways and associated transition rates is an important yet difficult problem in numerous chemical and biological systems, especially in situations with multiple competing pathways. When these high-dimensional systems are projected on low-dimensional coordinates, which are often needed for enhanced sampling or for interpretation of simulations and experiments, one can end up losing the kinetic connectivity of the underlying high-dimensional landscape. Thus, in the low-dimensional projection, metastable states might appear closer or further than they actually are. To deal with this issue, in this work, we develop a formalism that learns a multidimensional yet minimally complex reaction coordinate (RC) for generic high-dimensional systems. When projected along this RC, all possible kinetically relevant pathways can be demarcated and the true high-dimensional connectivity is maintained. One of the defining attributes of our method lies in that it can work on long unbiased simulations as well as biased simulations often needed for rare event systems. We demonstrate the utility of the method by studying a range of model systems including conformational transitions in a small peptide Ace-Ala₃-Nme, where we show how two-dimensional and three-dimensional RCs found by our previously published spectral gap optimization method "SGOOP" [Tiwary, P. and Berne, B. J. Proc. Natl. Acad. Sci. 2016, 113, 2839] can capture the kinetics for 23 and all 28 out of the 28 dominant state-to-state transitions, respectively.

Multiscale Approach for Computing Gated Ligand Binding from Molecular Dynamics and Brownian Dynamics Simulations

We develop an approach to characterize the effects of gating by a multiconformation protein consisting of macrostate conformations that are either accessible or inaccessible to ligand binding. We first construct a Markov state model of the apo-protein from atomistic molecular dynamics simulations from which we identify macrostates and their conformations, compute their relative macrostate populations and interchange kinetics, and structurally characterize them in terms of ligand accessibility. We insert the calculated first-order rate constants for conformational transitions into a multistate gating theory from which we derive a gating factor γ that quantifies the degree of conformational gating. Applied to HIV-1 protease, our approach yields a kinetic network of three accessible (semi-open, open, and wide-open) and two inaccessible (closed and a newly identified, "parted") macrostate conformations. The parted conformation sterically partitions the active site, suggesting a possible role in product release. We find that the binding kinetics of drugs and drug-like inhibitors to HIV-1 protease falls in the slow gating regime. However, because γ = 0.75, conformational gating only modestly slows ligand binding. Brownian dynamics simulations of the diffusional association of eight inhibitors to the protease─having a wide range of experimental association constants (∼10⁴-10¹⁰ M^-1 s^-1)─yields gated rate constants in the range of ∼0.5-5.7 × 10⁸ M^-1 s^-1. This indicates that, whereas the association rate of some inhibitors could be described by the model, for many inhibitors either subsequent conformational transitions or alternate binding mechanisms may be rate-limiting. For systems known to be modulated by conformational gating, the approach could be scaled computationally efficiently to screen association kinetics for a large number of ligands.

Multiscale molecular kinetics by coupling Markov state models and reaction-diffusion dynamics

A novel approach to simulate simple protein-ligand systems at large time and length scales is to couple Markov state models (MSMs) of molecular kinetics with particle-based reaction-diffusion (RD) simulations, MSM/RD. Currently, MSM/RD lacks a mathematical framework to derive coupling schemes, is limited to isotropic ligands in a single conformational state, and lacks multiparticle extensions. In this work, we address these needs by developing a general MSM/RD framework by coarse-graining molecular dynamics into hybrid switching diffusion processes. Given enough data to parameterize the model, it is capable of modeling protein-protein interactions over large time and length scales, and it can be extended to handle multiple molecules. We derive the MSM/RD framework, and we implement and verify it for two protein-protein benchmark systems and one multiparticle implementation to model the formation of pentameric ring molecules. To enable reproducibility, we have published our code in the MSM/RD software package.

New perspectives in cancer drug development: computational advances with an eye to design

Computational chemistry has come of age in drug discovery. Indeed, most pharmaceutical development programs rely on computer-based data and results at some point. Herein, we discuss recent applications of advanced simulation techniques to difficult challenges in drug discovery. These entail the characterization of allosteric mechanisms and the identification of allosteric sites or cryptic pockets determined by protein motions, which are not immediately evident in the experimental structure of the target; the study of ligand binding mechanisms and their kinetic profiles; and the evaluation of drug-target affinities. We analyze different approaches to tackle challenging and emerging biological targets. Finally, we discuss the possible perspectives of future application of computation in drug discovery.

Reduced efficacy of a Src kinase inhibitor in crowded protein solution

The inside of a cell is highly crowded with proteins and other biomolecules. How proteins express their specific functions together with many off-target proteins in crowded cellular environments is largely unknown. Here, we investigate an inhibitor binding with c-Src kinase using atomistic molecular dynamics (MD) simulations in dilute as well as crowded protein solution. The populations of the inhibitor, 4-amino-5-(4-methylphenyl)-7-(t-butyl)pyrazolo[3,4-d]pyrimidine (PP1), in bulk solution and on the surface of c-Src kinase are reduced as the concentration of crowder bovine serum albumins (BSAs) increases. This observation is consistent with the reduced PP1 inhibitor efficacy in experimental c-Src kinase assays in addition with BSAs. The crowded environment changes the major binding pathway of PP1 toward c-Src kinase compared to that in dilute solution. This change is explained based on the population shift mechanism of local conformations near the inhibitor binding site in c-Src kinase.

Molecular Mechanics Study of Flow and Surface Influence in Ligand-Protein Association

Ligand–protein association is the first and critical step for many biological and chemical processes. This study investigated the molecular association processes under different environments. In biology, cells have different compartments where ligand–protein binding may occur on a membrane. In experiments involving ligand–protein binding, such as the surface plasmon resonance and continuous flow biosynthesis, a substrate flow and surface are required in experimental settings. As compared with a simple binding condition, which includes only the ligand, protein, and solvent, the association rate and processes may be affected by additional ligand transporting forces and other intermolecular interactions between the ligand and environmental objects. We evaluated these environmental factors by using a ligand xk263 binding to HIV protease (HIVp) with atomistic details. Using Brownian dynamics simulations, we modeled xk263 and HIVp association time and probability when a system has xk263 diffusion flux and a non-polar self-assembled monolayer surface. We also examined different protein orientations and accessible surfaces for xk263. To allow xk263 to access to the dimer interface of immobilized HIVp, we simulated the system by placing the protein 20Å above the surface because immobilizing HIVp on a surface prevented xk263 from contacting with the interface. The non-specific interactions increased the binding probability while the association time remained unchanged. When the xk263 diffusion flux increased, the effective xk263 concentration around HIVp, xk263–HIVp association time and binding probability decreased non-linearly regardless of interacting with the self-assembled monolayer surface or not. The work sheds light on the effects of the solvent flow and surface environment on ligand–protein associations and provides a perspective on experimental design.

An Appraisal of Computer Simulation Approaches in Elucidating Biomolecular Recognition Pathways

Computer simulation approaches in biomolecular recognition processes have come a long way. In this Perspective, we highlight a series of recent success stories in which computer simulations have played a remarkable role in elucidating the atomic resolution mechanism of kinetic processes of protein-ligand binding in a quantitative fashion. In particular, we show that a robust combination of unbiased simulation, harnessed by a high-fidelity computing environment, and Markov state modeling approaches has been instrumental in revealing novel protein-ligand recognition pathways in multiple systems. We also elucidate the role of recent developments in enhanced sampling approaches in providing the much-needed impetus in accelerating simulation of the ligand recognition process. We identify multiple key issues, including force fields and the sampling bottleneck, which are currently preventing the field from achieving quantitative reconstruction of experimental measurements. Finally, we suggest a possible way forward via adoption of multiscale approaches and coarse-grained simulations as next steps toward efficient elucidation of ligand binding kinetics.

Molecular Dynamics Simulations in Drug Discovery and Pharmaceutical Development

Molecular dynamics (MD) simulations have become increasingly useful in the modern drug development process. In this review, we give a broad overview of the current application possibilities of MD in drug discovery and pharmaceutical development. Starting from the target validation step of the drug development process, we give several examples of how MD studies can give important insights into the dynamics and function of identified drug targets such as sirtuins, RAS proteins, or intrinsically disordered proteins. The role of MD in antibody design is also reviewed. In the lead discovery and lead optimization phases, MD facilitates the evaluation of the binding energetics and kinetics of the ligand-receptor interactions, therefore guiding the choice of the best candidate molecules for further development. The importance of considering the biological lipid bilayer environment in the MD simulations of membrane proteins is also discussed, using G-protein coupled receptors and ion channels as well as the drug-metabolizing cytochrome P450 enzymes as relevant examples. Lastly, we discuss the emerging role of MD simulations in facilitating the pharmaceutical formulation development of drugs and candidate drugs. Specifically, we look at how MD can be used in studying the crystalline and amorphous solids, the stability of amorphous drug or drug-polymer formulations, and drug solubility. Moreover, since nanoparticle drug formulations are of great interest in the field of drug delivery research, different applications of nano-particle simulations are also briefly summarized using multiple recent studies as examples. In the future, the role of MD simulations in facilitating the drug development process is likely to grow substantially with the increasing computer power and advancements in the development of force fields and enhanced MD methodologies.

Thermodynamics and Kinetics of Drug-Target Binding by Molecular Simulation

Computational studies play an increasingly important role in chemistry and biophysics, mainly thanks to improvements in hardware and algorithms. In drug discovery and development, computational studies can reduce the costs and risks of bringing a new medicine to market. Computational simulations are mainly used to optimize promising new compounds by estimating their binding affinity to proteins. This is challenging due to the complexity of the simulated system. To assess the present and future value of simulation for drug discovery, we review key applications of advanced methods for sampling complex free-energy landscapes at near nonergodicity conditions and for estimating the rate coefficients of very slow processes of pharmacological interest. We outline the statistical mechanics and computational background behind this research, including methods such as steered molecular dynamics and metadynamics. We review recent applications to pharmacology and drug discovery and discuss possible guidelines for the practitioner. Recent trends in machine learning are also briefly discussed. Thanks to the rapid development of methods for characterizing and quantifying rare events, simulation's role in drug discovery is likely to expand, making it a valuable complement to experimental and clinical approaches.

Ranking of Ligand Binding Kinetics Using a Weighted Ensemble Approach and Comparison with a Multiscale Milestoning Approach

To improve lead optimization efforts in finding the right ligand, pharmaceutical industries need to know the ligand's binding kinetics, such as binding and unbinding rate constants, which often correlate with the ligand's efficacy in vivo. To predict binding kinetics efficiently, enhanced sampling methods, such as milestoning and the weighted ensemble (WE) method, have been used in molecular dynamics (MD) simulations of these systems. However, a comparison of these enhanced sampling methods in ranking ligands has not been done. Hence, a WE approach called the concurrent adaptive sampling (CAS) algorithm that uses MD simulations was used to rank seven ligands for β-cyclodextrin, a system in which a multiscale milestoning approach called simulation enabled estimation of kinetic rates (SEEKR) was also used, which uses both MD and Brownian dynamics simulations. Overall, the CAS algorithm can successfully rank ligands using the unbinding rate constant k_off values and binding free energy ΔG values, as SEEKR did, with reduced computational cost that is about the same as SEEKR. We compare the CAS algorithm simulations with different parameters and discuss the impact of parameters in ranking ligands and obtaining rate constant and binding free energy estimates. We also discuss similarities and differences and advantages and disadvantages of SEEKR and the CAS algorithm for future use.

Kinetics and free energy of ligand dissociation using weighted ensemble milestoning

We consider the recently developed weighted ensemble milestoning (WEM) scheme [D. Ray and I. Andricioaei, J. Chem. Phys. 152, 234114 (2020)] and test its capability of simulating ligand-receptor dissociation dynamics. We performed WEM simulations on the following host-guest systems: Na⁺/Cl^- ion pair and 4-hydroxy-2-butanone ligand with FK506 binding protein. As a proof of principle, we show that the WEM formalism reproduces the Na⁺/Cl^- ion pair dissociation timescale and the free energy profile obtained from long conventional MD simulation. To increase the accuracy of WEM calculations applied to kinetics and thermodynamics in protein-ligand binding, we introduced a modified WEM scheme called weighted ensemble milestoning with restraint release (WEM-RR), which can increase the number of starting points per milestone without adding additional computational cost. WEM-RR calculations obtained a ligand residence time and binding free energy in agreement with experimental and previous computational results. Moreover, using the milestoning framework, the binding time and rate constants, dissociation constants, and committor probabilities could also be calculated at a low computational cost. We also present an analytical approach for estimating the association rate constant (k_on) when binding is primarily diffusion driven. We show that the WEM method can efficiently calculate multiple experimental observables describing ligand-receptor binding/unbinding and is a promising candidate for computer-aided inhibitor design.

Enhanced Jarzynski free energy calculations using weighted ensemble

The free energy of transitions between stable states is the key thermodynamic quantity that governs the relative probabilities of the forward and reverse reactions and the ratio of state probabilities at equilibrium. The binding free energy of a drug and its receptor is of particular interest, as it serves as an optimization function for drug design. Over the years, many computational methods have been developed to calculate binding free energies, and while many of these methods have a long history, issues such as convergence of free energy estimates and the projection of a binding process onto order parameters remain. Over 20 years ago, the Jarzynski equality was derived with the promise to calculate equilibrium free energies by measuring the work applied to short nonequilibrium trajectories. However, these calculations were found to be dominated by trajectories with low applied work that occur with extremely low probability. Here, we examine the combination of weighted ensemble algorithms with the Jarzynski equality. In this combined method, an ensemble of nonequilibrium trajectories are run in parallel, and cloning and merging operations are used to preferentially sample low-work trajectories that dominate the free energy calculations. Two additional methods are also examined: (i) a novel weighted ensemble resampler that samples trajectories directly according to their importance to the work of work and (ii) the diffusion Monte Carlo method using the applied work as the selection potential. We thoroughly examine both the accuracy and efficiency of unbinding free energy calculations for a series of model Lennard-Jones atom pairs with interaction strengths ranging from 2 kcal/mol to 20 kcal/mol. We find that weighted ensemble calculations can more efficiently determine accurate binding free energies, especially for deeper Lennard-Jones well depths.

Multisecond ligand dissociation dynamics from atomistic simulations

Coarse-graining of fully atomistic molecular dynamics simulations is a long-standing goal in order to allow the description of processes occurring on biologically relevant timescales. For example, the prediction of pathways, rates and rate-limiting steps in protein-ligand unbinding is crucial for modern drug discovery. To achieve the enhanced sampling, we perform dissipation-corrected targeted molecular dynamics simulations, which yield free energy and friction profiles of molecular processes under consideration. Subsequently, we use these fields to perform temperature-boosted Langevin simulations which account for the desired kinetics occurring on multisecond timescales and beyond. Adopting the dissociation of solvated sodium chloride, trypsin-benzamidine and Hsp90-inhibitor protein-ligand complexes as test problems, we reproduce rates from molecular dynamics simulation and experiments within a factor of 2–20, and dissociation constants within a factor of 1–4. Analysis of friction profiles reveals that binding and unbinding dynamics are mediated by changes of the surrounding hydration shells in all investigated systems.

Protein-ligand unbinding processes are out of reach for atomistic simulations due to time-scale involved. Here the authors demonstrate an approach relying on dissipation-corrected targeted molecular dynamics that enables to provide binding and unbinding rates with a speed-up of several orders of magnitude.

On Calculating Free Energy Differences Using Ensembles of Transition Paths

The free energy of a process is the fundamental quantity that determines its spontaneity or propensity at a given temperature. In particular, the binding free energy of a drug candidate to its biomolecular target is used as an objective quantity in drug design. Recently, binding kinetics—rates of association (k_on) and dissociation (k_off)—have also demonstrated utility for their ability to predict efficacy and in some cases have been shown to be more predictive than the binding free energy alone. Some methods exist to calculate binding kinetics from molecular simulations, although these are typically more difficult to calculate than the binding affinity as they depend on details of the transition path ensemble. Assessing these rate constants can be difficult, due to uncertainty in the definition of the bound and unbound states, large error bars and the lack of experimental data. As an additional consistency check, rate constants from simulation can be used to calculate free energies (using the log of their ratio) which can then be compared to free energies obtained experimentally or using alchemical free energy perturbation. However, in this calculation it is not straightforward to account for common, practical details such as the finite simulation volume or the particular definition of the “bound” and “unbound” states. Here we derive a set of correction terms that can be applied to calculations of binding free energies using full reactive trajectories. We apply these correction terms to revisit the calculation of binding free energies from rate constants for a host-guest system that was part of a blind prediction challenge, where significant deviations were observed between free energies calculated with rate ratios and those calculated from alchemical perturbation. The correction terms combine to significantly decrease the error with respect to computational benchmarks, from 3.4 to 0.76 kcal/mol. Although these terms were derived with weighted ensemble simulations in mind, some of the correction terms are generally applicable to free energies calculated using physical pathways via methods such as Markov state modeling, metadynamics, milestoning, or umbrella sampling.

Multiscale simulation approaches to modeling drug-protein binding

Simulations can provide detailed insight into the molecular processes involved in drug action, such as protein-ligand binding, and can therefore be a valuable tool for drug design and development. Processes with a large range of length and timescales may be involved, and understanding these different scales typically requires different types of simulation methodology. Ideally, simulations should be able to connect across scales, to analyze and predict how changes at one scale can influence another. Multiscale simulation methods, which combine different levels of treatment, are an emerging frontier with great potential in this area. Here we review multiscale frameworks of various types, and selected applications to biomolecular systems with a focus on drug-ligand binding.

Transient States and Barriers from Molecular Simulations and the Milestoning Theory: Kinetics in Ligand-Protein Recognition and Compound Design

This study presents a novel computational approach to study molecular recognition and binding kinetics for drug-like compounds dissociating from a flexible protein system. The intermediates and their free energy profile during ligand association and dissociation processes control ligand-protein binding kinetics and bring a more complete picture of ligand-protein binding. The method applied the milestoning theory to extract kinetics and thermodynamics information from running short classical molecular dynamics (MD) simulations for frames from a given dissociation path. High-dimensional ligand-protein motions (3N-6 degrees of freedom) during ligand dissociation were reduced by use of principal component modes for assigning more than 100 milestones, and classical MD runs were allowed to travel multiple milestones to efficiently obtain ensemble distribution of initial structures for MD simulations and estimate the transition time and rate during ligand traveling between milestones. We used five pyrazolourea ligands and cyclin-dependent kinase 8 with cyclin C (CDK8/CycC) as our model system as well as metadynamics and a pathway search method to sample dissociation pathways. With our strategy, we constructed the free energy profile for highly mobile biomolecular systems. The computed binding free energy and residence time correctly ranked the pyrazolourea ligand series, in agreement with experimental data. Guided by a barrier of a ligand passing an αC helix and activation loop, we introduced one hydroxyl group to parent compounds to design our ligands with increased residence time and validated our prediction by experiments. This work provides a novel and robust approach to investigate dissociation kinetics of large and flexible systems for understanding unbinding mechanisms and designing new small-molecule drugs with desired binding kinetics.