Exact milestoning

Precisely controlled colloids: a playground for path-wise non-equilibrium physics

We investigate path-wise observables in experiments on driven colloids in a periodic light field to dissect selected intricate transport features, kinetics, and transition-path time statistics out of thermodynamic equilibrium. These observables directly reflect the properties of individual paths in contrast to the properties of an ensemble of particles, such as radial distribution functions or mean-squared displacements. In particular, we present two distinct albeit equivalent formulations of the underlying stochastic equation of motion, highlight their respective practical relevance, and show how to interchange between them. We discuss conceptually different notions of local velocities and interrogate one- and two-sided first-passage and transition-path time statistics in and out of equilibrium. Our results reiterate how path-wise observables may be employed to systematically assess the quality of experimental data and demonstrate that, given sufficient control and sampling, one may quantitatively verify subtle theoretical predictions.

Experiment-Guided Refinement of Milestoning Network

Milestoning is an efficient method for calculating rare event kinetics by constructing a continuous-time kinetic network that connects the reactant and product states. Its accuracy depends on both the quality of the underlying force fields and the trajectory sampling. The sampling error can be effectively controlled through various methods. However, the force fields are often not accurate enough, leading to quantitative discrepancies between simulations and experimental data. To address this challenge, we present a refinement approach for Milestoning network based on the maximum caliber (MaxCal), a general variational principle for dynamical systems, to combine simulations and experimental data. The Kullback-Leibler divergence rate between two Milestoning networks is analytically evaluated and minimized as the loss function. Meanwhile, experimental thermodynamic (equilibrium constants) and kinetic (rate constants) data are incorporated as constraints. The use of MaxCal implies that the refined kinetic network is minimally perturbed from the original one while satisfying the experimental constraints. The refined network is expected to align better with available experimental data. The refinement approach is demonstrated using the binding and unbinding dynamics of a series of six small molecule ligands for the model host system, β-cyclodextrin.

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.

Identifying the Most Probable Transition Path with Constant Advance Replicas

Locating plausible transition paths and enhanced sampling of rare events are fundamental to understanding the functional dynamics of biomolecules. Here, a constraint-based constant advance replicas (CAR) formalism of reaction paths is reported for identifying the most probable transition path (MPTP) between two given states. We derive the temporal-integrated effective dynamics governing the projected subsystem under the holonomic CAR path constraints and show that a dynamical action functional can be defined and used for optimizing the MPTP. We further demonstrate how the CAR MPTP can be located by asymptotically minimizing an upper bound of the CAR action functional using a variational expectation-maximization framework. Essential thermodynamics and kinetic observables are retrieved by integrating the boxed molecular dynamics on the CAR MPTP using a newly proposed adaptive reflecting boundary condition. The efficiency of the proposed method is demonstrated for the Müller potential, the alanine dipeptide isomerization, and the DNA base pairing transition (Watson-Crick to Hoogsteen) in explicit solvent. The CAR representation of transition paths constitutes a robust and extensible platform that can be combined with diverse enhanced sampling methods to aid future flexible and reliable biomolecular simulations.

Transition-path sampling for run-and-tumble particles

We elaborate and validate a generalization of the renowned transition-path-sampling algorithm for a paradigmatic model of active particles, namely, the run-and-tumble particles. Notwithstanding the nonequilibrium character of these particles, we show how the consequent lack of the microscopical reversibility property, which is usually required by transition-path sampling, can be circumvented by identifying reasonable backward dynamics with a well-defined path-probability density. Our method is then applied to characterize the structure and kinetics of rare transition pathways undergone by run-and-tumble particles having to cross a potential barrier in order to find a target.

Generating Protein Structures for Pathway Discovery Using Deep Learning

Resolving the intricate details of biological phenomena at the molecular level is fundamentally limited by both length- and time scales that can be probed experimentally. Molecular dynamics (MD) simulations at various scales are powerful tools frequently employed to offer valuable biological insights beyond experimental resolution. However, while it is relatively simple to observe long-lived, stable configurations of, for example, proteins, at the required spatial resolution, simulating the more interesting rare transitions between such states often takes orders of magnitude longer than what is feasible even on the largest supercomputers available today. One common aspect of this challenge is pathway discovery, where the start and end states of a scientific phenomenon are known or can be approximated, but the mechanistic details in between are unknown. Here, we propose a representation-learning-based solution that uses interpolation and extrapolation in an abstract representation space to synthesize potential transition states, which are automatically validated using MD simulations. The new simulations of the synthesized transition states are subsequently incorporated into the representation learning, leading to an iterative framework for targeted path sampling. Our approach is demonstrated by recovering the transition of a RAS-RAF protein domain (CRD) from membrane-free to interacting with the membrane using coarse-grain MD simulations.

BAD-NEUS: Rapidly converging trajectory stratification

An issue for molecular dynamics simulations is that events of interest often involve timescales that are much longer than the simulation time step, which is set by the fastest timescales of the model. Because of this timescale separation, direct simulation of many events is prohibitively computationally costly. This issue can be overcome by aggregating information from many relatively short simulations that sample segments of trajectories involving events of interest. This is the strategy of Markov state models (MSMs) and related approaches, but such methods suffer from approximation error because the variables defining the states generally do not capture the dynamics fully. By contrast, once converged, the weighted ensemble (WE) method aggregates information from trajectory segments so as to yield unbiased estimates of both thermodynamic and kinetic statistics. Unfortunately, errors decay no faster than unbiased simulation in WE as originally formulated and commonly deployed. Here, we introduce a theoretical framework for describing WE that shows that the introduction of an approximate stationary distribution on top of the stratification, as in nonequilibrium umbrella sampling (NEUS), accelerates convergence. Building on ideas from MSMs and related methods, we generalize the NEUS approach in such a way that the approximation error can be reduced systematically. We show that the improved algorithm can decrease the simulation time required to achieve the desired precision by orders of magnitude.

Kinetic Network in Milestoning: Clustering, Reduction, and Transition Path Analysis

We present a reduction of the Milestoning (ReM) algorithm to analyze the high-dimensional Milestoning kinetic network. The algorithm reduces the Milestoning network to low dimensions but preserves essential kinetic information, such as local residence time, exit time, and mean first passage time between any two states. This is achieved in three steps. First, nodes (milestones) in the high-dimensional Milestoning network are grouped into clusters based on the metastability identified by an auxiliary continuous-time Markov chain. Our clustering method is applicable not only to time-reversible networks but also to nonreversible networks generated from practical simulations with statistical fluctuations. Second, a reduced network is established via network transformation, containing only the core sets of clusters as nodes. Finally, transition pathways are analyzed in the reduced network based on the transition path theory. The algorithm is illustrated using a toy model and a solvated alanine dipeptide in two and four dihedral angles.

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.

Exploring Biomolecular Conformational Dynamics with Polarizable Force Field AMOEBA and Enhanced Sampling Method Milestoning

Conformational dynamics play a crucial role in determining the behavior of the biomolecules. Polarizable force fields, such as AMOEBA, can accurately capture electrostatic interactions underlying the conformational space. However, applying a polarizable force field in molecular dynamics (MD) simulations can be computationally expensive, especially in studying long-time-scale dynamics. To overcome this challenge, we incorporated the AMOEBA potential with Milestoning, an enhanced sampling method in this work. This integration allows us to efficiently sample the rare and important conformational states of a biomolecule by using many short and independent molecular dynamics trajectories with the AMOEBA force field. We applied this method to investigate the conformational dynamics of alanine dipeptide, DNA, and RNA A-B form conversion. Well-converged thermodynamic and kinetic properties were obtained, including the free energy difference, mean first passage time, and critical transitions between states. Our results demonstrate the power of integrating polarizable force fields with enhanced sampling methods in quantifying the thermodynamic and kinetic properties of biomolecules at the atomic level.

Milestoning estimators of dissipation in systems observed at a coarse resolution

Many nonequilibrium, active processes are observed at a coarse-grained level, where different microscopic configurations are projected onto the same observable state. Such "lumped" observables display memory, and in many cases, the irreversible character of the underlying microscopic dynamics becomes blurred, e.g., when the projection hides dissipative cycles. As a result, the observations appear less irreversible, and it is very challenging to infer the degree of broken time-reversal symmetry. Here we show, contrary to intuition, that by ignoring parts of the already coarse-grained state space we may-via a process called milestoning-improve entropy-production estimates. We present diverse examples where milestoning systematically renders observations "closer to underlying microscopic dynamics" and thereby improves thermodynamic inference from lumped data assuming a given range of memory, and we hypothesize that this effect is quite general. Moreover, whereas the correct general physical definition of time reversal in the presence of memory remains unknown, we here show by means of physically relevant examples that at least for semi-Markov processes of first and second order, waiting-time contributions arising from adopting a naive Markovian definition of time reversal generally must be discarded.

Comparison of Accuracy and Efficiency of Milestoning Variants: Introducing Buffer Milestoning

The Milestoning algorithm is a method for long-time molecular dynamics simulations. It enables the sampling of rare events. The precise calculations of observables depend on accurately determining the first hitting point distribution (FHPD) for each milestone. There is no analytical expression for FHPD, which is estimated numerically. Several variants of Milestoning offer approximations to the FHPD. Here, we examine in detail the FHPD of an exact calculation and Milestoning variants. We also introduce a new version of the Milestoning algorithm, buffer Milestoning, with a comparable cost to conventional Milestoning but higher accuracy. We use the mean first passage time and the free energy to assess the simulation quality, and we compare the accuracy and efficiency of buffer Milestoning to exact calculations, conventional Milestoning, local-passage-time-weighted Milestoning, Markovian Milestoning with Voronoi tessellation, and exact Milestoning. Conventional Milestoning requires milestone decorrelation. If this condition is not satisfied, it is the least accurate approach of all the techniques we examined. We conclude that for a small increase in cost compared to conventional Milestoning, buffer Milestoning provides accurate results for a range of problems, including more correlated milestones and is, therefore, versatile compared to other variants. Local-passage-time-weighted Milestoning provides accuracy similar to that of buffer Milestoning but at an increased simulation cost. Markovian Milestoning with Voronoi tessellation is the most accurate compared with other approximations, but it is less stable for high barriers and more expensive.

On committor functions in milestoning

As an optimal one-dimensional reaction coordinate, the committor function not only describes the probability of a trajectory initiated at a phase space point first reaching the product state before reaching the reactant state but also preserves the kinetics when utilized to run a reduced dynamics model. However, calculating the committor function in high-dimensional systems poses significant challenges. In this paper, within the framework of milestoning, exact expressions for committor functions at two levels of coarse graining are given, including committor functions of phase space point to point (CFPP) and milestone to milestone (CFMM). When combined with transition kernels obtained from trajectory analysis, these expressions can be utilized to accurately and efficiently compute the committor functions. Furthermore, based on the calculated committor functions, an adaptive algorithm is developed to gradually refine the transition state region. Finally, two model examples are employed to assess the accuracy of these different formulations of committor functions.

Error-Controlled Coarse-Graining Dynamics with Mean-Field Randomization

In order to comprehend the stochastic behavior of biological systems, it is essential to accurately infer the dynamics of chemical reaction networks. However, computation of the likelihood remains a bottleneck. In this study, we propose the mean-field randomization procedure as a means of efficiently generating error-controlled coarse-graining dynamics. The error is measured by mutual information between the generated trajectories and the coarse-graining procedure. We demonstrate that the exact dynamics can be recovered by resampling, which eliminates the correlation between the dynamics and the procedure. We developed three algorithms to efficiently generate exact or coarse-graining trajectories within a specified error range. By subjecting our algorithms to testing on chemical reaction systems of varying complexities and scales, we observe that they outperform existing state-of-the-art algorithms, and the efficiency of coarse-graining trajectory generation is only weakly dependent on system scales.

Approximating First Hitting Point Distribution in Milestoning for Rare Event Kinetics

Milestoning is an efficient method for rare event kinetics calculation using short trajectory parallelization. Mean first passage time (MFPT) is the key kinetic output of Milestoning, whose accuracy crucially depends on the initial distribution of the short trajectory ensemble. The true initial distribution, i.e., the first hitting point distribution (FHPD), has no analytic expression in the general case. Here, we introduce two algorithms, local passage time weighted Milestoning (LPT-M) and Bayesian inference Milestoning (BI-M), to accurately and efficiently approximate FHPD for systems at equilibrium condition. Starting from sampling the Boltzmann distribution on milestones, we calculate the proper weighting factor for the short trajectory ensemble. The methods are tested on two model examples for illustration purpose. Both methods improve significantly over the widely used classical Milestoning (CM) method in terms of the accuracy of MFPT. In particular, BI-M covers the directional Milestoning method as a special case in deterministic Hamiltonian dynamics. LPT-M is especially advantageous in terms of computational costs and robustness with respect to the increasing number of intermediate milestones. Furthermore, a locally iterative correction algorithm for nonequilibrium stationary FHPD is developed for exact MFPT calculation, which can be combined with LPT-M/BI-M and is much cheaper than the exact Milestoning (ExM) method.

Kinetics from Metadynamics: Principles, Applications, and Outlook

Metadynamics is a popular enhanced sampling algorithm for computing the free energy landscape of rare events by using molecular dynamics simulation. Ten years ago, Tiwary and Parrinello introduced the infrequent metadynamics approach for calculating the kinetics of transitions across free energy barriers. Since then, metadynamics-based methods for obtaining rate constants have attracted significant attention in computational molecular science. Such methods have been applied to study a wide range of problems, including protein-ligand binding, protein folding, conformational transitions, chemical reactions, catalysis, and nucleation. Here, we review the principles of elucidating kinetics from metadynamics-like approaches, subsequent methodological developments in this area, and successful applications on chemical, biological, and material systems. We also highlight the challenges of reconstructing accurate kinetics from enhanced sampling simulations and the scope of future developments.

Targeted Protein Degradation: Advances, Challenges, and Prospects for Computational Methods

The therapeutic approach of targeted protein degradation (TPD) is gaining momentum due to its potentially superior effects compared with protein inhibition. Recent advancements in the biotech and pharmaceutical sectors have led to the development of compounds that are currently in human trials, with some showing promising clinical results. However, the use of computational tools in TPD is still limited, as it has distinct characteristics compared with traditional computational drug design methods. TPD involves creating a ternary structure (protein-degrader-ligase) responsible for the biological function, such as ubiquitination and subsequent proteasomal degradation, which depends on the spatial orientation of the protein of interest (POI) relative to E2-loaded ubiquitin. Modeling this structure necessitates a unique blend of tools initially developed for small molecules (e.g., docking) and biologics (e.g., protein-protein interaction modeling). Additionally, degrader molecules, particularly heterobifunctional degraders, are generally larger than conventional small molecule drugs, leading to challenges in determining drug-like properties like solubility and permeability. Furthermore, the catalytic nature of TPD makes occupancy-based modeling insufficient. TPD consists of multiple interconnected yet distinct steps, such as POI binding, E3 ligase binding, ternary structure interactions, ubiquitination, and degradation, along with traditional small molecule properties. A comprehensive set of tools is needed to address the dynamic nature of the induced proximity ternary complex and its implications for ubiquitination. In this Perspective, we discuss the current state of computational tools for TPD. We start by describing the series of steps involved in the degradation process and the experimental methods used to characterize them. Then, we delve into a detailed analysis of the computational tools employed in TPD. We also present an integrative approach that has proven successful for degrader design and its impact on project decisions. Finally, we examine the future prospects of computational methods in TPD and the areas with the greatest potential for impact.

Weighted ensemble: Recent mathematical developments

Weighted ensemble (WE) is an enhanced sampling method based on periodically replicating and pruning trajectories generated in parallel. WE has grown increasingly popular for computational biochemistry problems due, in part, to improved hardware and accessible software implementations. Algorithmic and analytical improvements have played an important role, and progress has accelerated in recent years. Here, we discuss and elaborate on the WE method from a mathematical perspective, highlighting recent results that enhance the computational efficiency. The mathematical theory reveals a new strategy for optimizing trajectory management that approaches the best possible variance while generalizing to systems of arbitrary dimension.

ScMiles2: A Script to Conduct and Analyze Milestoning Trajectories for Long Time Dynamics

Milestoning is a theory and an algorithm that computes kinetics and thermodynamics at long time scales. It is based on partitioning the (phase) space into cells and running a large number of short trajectories between the boundaries of the cells. The termination points of the trajectories are analyzed with the Milestoning theory to obtain kinetic and thermodynamic information. Managing the tens to hundreds of thousands of Milestoning trajectories is a challenge, which we handle with a python script, ScMiles. Here, we introduce a new version of the python script ScMiles2 to conduct Milestoning simulations. Major enhancements are: (i) post analysis of Milestoning trajectories to obtain the free energy, mean first passage time, the committor function, and exit times; (ii) similar to (i) but the post analysis is for a single long trajectory; (iii) we support the use of the GROMACS software in addition to NAMD; (iv) a restart option; (v) the automated finding, sampling, and launching trajectories from new milestones that are found on the fly; and (vi) support Milestoning calculations with several coarse variables and for complex reaction coordinates. We also evaluate the simulation parameters and suggest new algorithmic features to enhance the rate of convergence of observables. We propose the use of an iteration-averaged kinetic matrix for a rapid approach to asymptotic values. Illustrations are provided for small systems and one large example.

Perspective: Path Sampling Methods Applied to Enzymatic Catalysis

This Perspective reviews the use of Transition Path Sampling methods to study enzymatically catalyzed chemical reactions. First applied by our group to an enzymatic reaction over 15 years ago, the method has uncovered basic principles in enzymatic catalysis such as the protein promoting vibration, and it has also helped harmonize such ideas as electrostatic preorganization with dynamic views of enzyme function. It is now being used to help uncover principles of protein design necessary to artificial enzyme creation.

Computing transition path theory quantities with trajectory stratification

Transition path theory computes statistics from ensembles of reactive trajectories. A common strategy for sampling reactive trajectories is to control the branching and pruning of trajectories so as to enhance the sampling of low probability segments. However, it can be challenging to apply transition path theory to data from such methods because determining whether configurations and trajectory segments are part of reactive trajectories requires looking backward and forward in time. Here, we show how this issue can be overcome efficiently by introducing simple data structures. We illustrate the approach in the context of nonequilibrium umbrella sampling, but the strategy is general and can be used to obtain transition path theory statistics from other methods that sample segments of unbiased trajectories.

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.

Practical guide to replica exchange transition interface sampling and forward flux sampling

Path sampling approaches have become invaluable tools to explore the mechanisms and dynamics of the so-called rare events that are characterized by transitions between metastable states separated by sizable free energy barriers. Their practical application, in particular to ever more complex molecular systems, is, however, not entirely trivial. Focusing on replica exchange transition interface sampling (RETIS) and forward flux sampling (FFS), we discuss a range of analysis tools that can be used to assess the quality and convergence of such simulations, which is crucial to obtain reliable results. The basic ideas of a step-wise evaluation are exemplified for the study of nucleation in several systems with different complexities, providing a general guide for the critical assessment of RETIS and FFS simulations.

An ergodic theorem for the weighted ensemble method

We study weighted ensemble, an interacting particle method for sampling distributions of Markov chains that has been used in computational chemistry since the 1990s. Many important applications of weighted ensemble require the computation of long time averages. We establish the consistency of weighted ensemble in this setting by proving an ergodic theorem for time averages. As part of the proof, we derive explicit variance formulas that could be useful for optimizing the method.

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.

Gaussian-Accelerated Molecular Dynamics with the Weighted Ensemble Method: A Hybrid Method Improves Thermodynamic and Kinetic Sampling

Gaussian-accelerated molecular dynamics (GaMD) is a well-established enhanced sampling method for molecular dynamics simulations that effectively samples the potential energy landscape of the system by adding a boost potential, which smoothens the surface and lowers the energy barriers between states. GaMD is unable to give time-dependent properties such as kinetics directly. On the other hand, the weighted ensemble (WE) method can efficiently sample transitions between states with its many weighted trajectories, which directly yield rates and pathways. However, convergence to equilibrium conditions remains a challenge for the WE method. Hence, we have developed a hybrid method that combines the two methods, wherein GaMD is first used to sample the potential energy landscape of the system and WE is subsequently used to further sample the potential energy landscape and kinetic properties of interest. We show that the hybrid method can sample both thermodynamic and kinetic properties more accurately and quickly compared to using either method alone.

Dissecting Multiple Pathways in the Relaxation Dynamics of Helix <==> Coil Transitions with Optimum Dimensionality Reduction

Fast kinetic experiments with dramatically improved time resolution have contributed significantly to understanding the fundamental processes in protein folding pathways involving the formation of a-helices and b-hairpin, contact formation, and overall collapse of the peptide chain. Interpretation of experimental results through application of a simple statistical mechanical model was key to this understanding. Atomistic description of all events observed in the experimental findings was challenging. Recent advancements in theory, more sophisticated algorithms, and a true long-term trajectory made way for an atomically detailed description of kinetics, examining folding pathways, validating experimental results, and reporting new findings for a wide range of molecular processes in biophysical chemistry. This review describes how optimum dimensionality reduction theory can construct a simplified coarse-grained model with low dimensionality involving a kinetic matrix that captures novel insights into folding pathways. A set of metastable states derived from molecular dynamics analysis generate an optimally reduced dimensionality rate matrix following transition pathway analysis. Analysis of the actual long-term simulation trajectory extracts a relaxation time directly comparable to the experimental results and confirms the validity of the combined approach. The application of the theory is discussed and illustrated using several examples of helix <==> coil transition pathways. This paper focuses primarily on a combined approach of time-resolved experiments and long-term molecular dynamics simulation from our ongoing work.

Impact of the Protonation State of Phosphatidylinositol 4,5-Bisphosphate (PIP2) on the Binding Kinetics and Thermodynamics to Transient Receptor Potential Vanilloid (TRPV5): A Milestoning Study

The binding of phosphatidylinositol 4,5-bisphosphate (PIP2) to the ion channel transient receptor potential vanilloid 5 (TRPV5) is critical for its function. We use atomically detailed simulations and the milestoning theory to compute the free energy profile and the kinetics of PIP2 binding to TRPV5. We estimate the rate of binding and the impact of the protonation state on the process. Several channel residues are identified as influential in the association event and will be interesting targets for mutation analysis. Our simulations reveal that PIP2 binds to TRPV5 in an unprotonated state and is protonated in the membrane. The switch between the protonation state of PIP2 is modeled as a diabatic transition and occurs about halfway through the reaction.

Computational methods for exploring protein conformations

Proteins are dynamic molecules that can transition between a potentially wide range of structures comprising their conformational ensemble. The nature of these conformations and their relative probabilities are described by a high-dimensional free energy landscape. While computer simulation techniques such as molecular dynamics simulations allow characterisation of the metastable conformational states and the transitions between them, and thus free energy landscapes, to be characterised, the barriers between states can be high, precluding efficient sampling without substantial computational resources. Over the past decades, a dizzying array of methods have emerged for enhancing conformational sampling, and for projecting the free energy landscape onto a reduced set of dimensions that allow conformational states to be distinguished, known as collective variables (CVs), along which sampling may be directed. Here, a brief description of what biomolecular simulation entails is followed by a more detailed exposition of the nature of CVs and methods for determining these, and, lastly, an overview of the myriad different approaches for enhancing conformational sampling, most of which rely upon CVs, including new advances in both CV determination and conformational sampling due to machine learning.

What Markov State Models Can and Cannot Do: Correlation versus Path-Based Observables in Protein-Folding Models

Markov state models (MSMs) have been widely applied to study the kinetics and pathways of protein conformational dynamics based on statistical analysis of molecular dynamics (MD) simulations. These MSMs coarse-grain both configuration space and time in ways that limit what kinds of observables they can reproduce with high fidelity over different spatial and temporal resolutions. Despite their popularity, there is still limited understanding of which biophysical observables can be computed from these MSMs in a robust and unbiased manner, and which suffer from the space-time coarse-graining intrinsic in the MSM model. Most theoretical arguments and practical validity tests for MSMs rely on long-time equilibrium kinetics, such as the slowest relaxation time scales and experimentally observable time-correlation functions. Here, we perform an extensive assessment of the ability of well-validated protein folding MSMs to accurately reproduce path-based observable such as mean first-passage times (MFPTs) and transition path mechanisms compared to a direct trajectory analysis. We also assess a recently proposed class of history-augmented MSMs (haMSMs) that exploit additional information not accounted for in standard MSMs. We conclude with some practical guidance on the use of MSMs to study various problems in conformational dynamics of biomolecules. In brief, MSMs can accurately reproduce correlation functions slower than the lag time, but path-based observables can only be reliably reproduced if the lifetimes of states exceed the lag time, which is a much stricter requirement. Even in the presence of short-lived states, we find that haMSMs reproduce path-based observables more reliably.

Computer Simulations of the Dissociation Mechanism of Gleevec from Abl Kinase with Milestoning

Gleevec (a.k.a., imatinib) is an important anticancer (e.g., chronic myeloid leukemia) chemotherapeutic drug due to its inhibitory interaction with the Abl kinase. Here, we use atomically detailed simulations within the Milestoning framework to study the molecular dissociation mechanism of Gleevec from Abl kinase. We compute the dissociation free energy profile, the mean first passage time for unbinding, and explore the transition state ensemble of conformations. The milestones form a multidimensional network with average connectivity of about 2.93, which is significantly higher than the connectivity for a one-dimensional reaction coordinate. The free energy barrier for Gleevec dissociation is estimated to be ∼10 kcal/mol, and the exit time is ∼55 ms. We examined the transition state conformations using both, the committor and transition function. We show that near the transition state the highly conserved salt bridge K217 and E286 is transiently broken. Together with the calculated free energy profile, these calculations can advance the understanding of the molecular interaction mechanisms between Gleevec and Abl kinase and play a role in future drug design and optimization studies.

Target Search of Active Agents Crossing High Energy Barriers

Target search by active agents in rugged energy landscapes has remained a challenge because standard enhanced sampling methods do not apply to irreversible dynamics. We overcome this nonequilibrium rare-event problem by developing an algorithm generalizing transition-path sampling to active Brownian dynamics. This method is exemplified and benchmarked for a paradigmatic two-dimensional potential with a high barrier. We find that even in such a simple landscape the structure and kinetics of the ensemble of transition paths changes drastically in the presence of activity. Indeed, active Brownian particles reach the target more frequently than passive Brownian particles, following longer and counterintuitive search patterns.

Milestoning: An Efficient Approach for Atomically Detailed Simulations of Kinetics in Biophysics

Recent advances in theory and algorithms for atomically detailed simulations open the way to the study of the kinetics of a wide range of molecular processes in biophysics. The theories propose a shift from the traditionally very long molecular dynamic trajectories, which are exact but may not be efficient in the study of kinetics, to the use of a large number of short trajectories. The short trajectories exploit a mapping to a mesh in coarse space and allow for efficient calculations of kinetics and thermodynamics. In this review, I focus on one theory: Milestoning is a theory and an algorithm that offers a hierarchical calculation of properties of interest, such as the free energy profile and the mean first passage time. Approximations to the true long-time dynamics can be computed efficiently and assessed at different steps of the investigation. The theory is discussed and illustrated using two biophysical examples: ion permeation through a phospholipid membrane and protein translocation through a channel.

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.

Value of Temporal Information When Analyzing Reaction Coordinates

Reaction coordinates chart pathways from reactants to products of chemical reactions. Determination of reaction coordinates from ensembles of molecular trajectories has thus been the focus of many studies. A widely used and insightful choice of a reaction coordinate is the committor function, defined as the probability that a trajectory will reach the product before the reactant. Here, we consider alternatives to the committor function that add useful mechanistic information, the mean first passage time, and the exit time to the product. We further derive a simple relationship between the functions of the committor, the mean first passage time, and the exit time. We illustrate the diversity of mechanisms predicted by alternative reaction coordinates with several toy problems and with a simple model of protein searching for a specific DNA motif.

Capacities and efficient computation of first-passage probabilities

A reversible diffusion process is initialized at position x_{0} and run until it first hits any of several targets. What is the probability that it terminates at a particular target? We propose a computationally efficient approach for estimating this probability, focused on those situations in which it takes a long time to hit any target. In these cases, direct simulation of the hitting probabilities becomes prohibitively expensive. On the other hand, if the timescales are sufficiently long, then the system will essentially "forget" its initial condition before it encounters a target. In these cases the hitting probabilities can be accurately approximated using only local simulations around each target, obviating the need for direct simulations. In empirical tests, we find that these local estimates can be computed in the same time it would take to compute a single direct simulation, but that they achieve an accuracy that would require thousands of direct simulation runs.

Improving double-ended transition state searches for soft-matter systems

Transitions between different stable configurations of biomolecules are important in understanding disease mechanisms, structure-function relations, and novel molecular-scale engineering. The corresponding pathways can be characterized efficiently using geometry optimization schemes based on double-ended transition state searches. An interpolation is first constructed between the known states and then refined, yielding a band that contains transition state candidates. Here, we analyze an example where various interpolation schemes lead to bands with a single step transition, but the correct pathway actually proceeds via an intervening, low-energy minimum. We compare a number of different interpolation schemes for this problem. We systematically alter the number of discrete images in the interpolations and the spring constants used in the optimization and test two schemes for adjusting the spring constants and image distribution, resulting in a total of 2760 different connection attempts. Our results confirm that optimized bands are not necessarily a good description of the transition pathways in themselves, and further refinement to actually converge transition states and establish their connectivity is required. We see an improvement in the optimized bands if we employ the adjustment of spring constants with doubly-nudged elastic band and a smaller improvement from the image redistribution. The example we consider is representative of numerous cases we have encountered in a wide variety of molecular and condensed matter systems.

Milestoning with wind: Exploring the impact of a biasing potential in exact calculation of kinetics

We consider the algorithm wind-assisted reweighted Milestoning of Grazioli and Andricioaei [J. Chem. Phys 149(8), 084103 (2018)], expand it, and assess its performance. We derive exact expressions for underdamped and overdamped Langevin dynamics and examine its efficiency for a simple model system (Mueller's potential) and for bond breaking in solution. The use of a biasing force (wind) significantly enhances the sampling of otherwise rare trajectories but also introduces an exponential weight to the trajectories that significantly impact the value of the statistics. In our examples, computing averages and standard deviations are not better using wind compared to straightforward Milestoning. However, the biasing force is useful for highly steep energy landscapes. On these landscapes, the probability of sampling straightforward Milestoning trajectories, which overcome the barrier, is low and the biasing force enables the observations of these rare events.

Exploring the Reaction Mechanism of HIV Reverse Transcriptase with a Nucleotide Substrate

Enzymatic reactions consist of several steps: (i) a weak binding event of the substrate to the enzyme, (ii) an induced fit or a protein conformational transition upon ligand binding, (iii) the chemical reaction, and (iv) the release of the product. Here we focus on step iii of the reaction of a DNA polymerase, HIV RT, with a nucleotide. We determine the rate and the free energy profile for the addition of a nucleotide to a DNA strand using a combination of a QM/MM model, the string method, and exact Milestoning. The barrier height and the time scale of the reaction are consistent with experiment. We show that the observables (free energies and mean first passage time) converge rapidly, as a function of the Milestoning iteration number. We also consider the substitution of an oxygen of the incoming nucleotide by a nonbridging sulfur atom and its impact on the enzymatic reaction. This substitution has been suggested in the past as a tool to examine the influence of the chemical step on the overall rate. Our joint computational and experimental study suggests that the impact of the substitution is small. Computationally, the differences between the two are within the estimated error bars. Experiments suggest a small difference. Finally, we examine step i, the weak binding of the nucleotide to the protein surface. We suggest that this step has only a small contribution to the selectivity of the enzyme. Comments are made on the impact of these steps on the overall mechanism.

OPTIMIZING WEIGHTED ENSEMBLE SAMPLING OF STEADY STATES

We propose parameter optimization techniques for weighted ensemble sampling of Markov chains in the steady-state regime. Weighted ensemble consists of replicas of a Markov chain, each carrying a weight, that are periodically resampled according to their weights inside of each of a number of bins that partition state space. We derive, from first principles, strategies for optimizing the choices of weighted ensemble parameters, in particular the choice of bins and the number of replicas to maintain in each bin. In a simple numerical example, we compare our new strategies with more traditional ones and with direct Monte Carlo.

The transition between active and inactive conformations of Abl kinase studied by rock climbing and Milestoning

BACKGROUND

Kinases are a family of enzymes that catalyze the transfer of the ɤ-phosphate group from ATP to a protein's residue. Malfunctioning kinases are involved in many health problems such as cardiovascular diseases, diabetes, and cancer. Kinases transitions between multiple conformations of inactive to active forms attracted considerable interest.

METHOD

A reaction coordinate is computed for the transition between the active to inactive conformation in Abl kinase with a focus on the DFG-in to DFG-out flip. The method of Rock Climbing is used to construct a path locally, which is subsequently optimized using a functional of the entire path. The discrete coordinate sets along the reaction path are used in a Milestoning calculation of the free energy landscape and the rate of the transition.

RESULTS

The estimated transition times are between a few milliseconds and seconds, consistent with simulations of the kinetics and with indirect experimental data. The activation requires the transient dissociation of the salt bridge between Lys271 and Glu286. The salt bridge reforms once the DFG motif is stabilized by a locked conformation of Phe382. About ten residues are identified that contribute significantly to the process and are included as part of the reaction space.

CONCLUSIONS

The transition from DFG-in to DFG-out in Abl kinase was simulated using atomic resolution of a fully solvated protein yielding detailed description of the kinetics and the mechanism of the DFG flip. The results are consistent with other computational methods that simulate the kinetics and with some indirect experimental measurements.

GENERAL SIGNIFICANCE

The activation of kinases includes a conformational transition of the DFG motif that is important for enzyme activity but is not accessible to conventional Molecular Dynamics. We propose a detailed mechanism for the transition, at a timescale longer than conventional MD, using a combination of reaction path and Milestoning algorithms. The mechanism includes local structural adjustments near the binding site as well as collective interactions with more remote residues.

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.