Defining an Optimal Metric for the Path Collective Variables

Spectral Map for Slow Collective Variables, Markovian Dynamics, and Transition State Ensembles

Understanding the behavior of complex molecular systems is a fundamental problem in physical chemistry. To describe the long-time dynamics of such systems, which is responsible for their most informative characteristics, we can identify a few slow collective variables (CVs) while treating the remaining fast variables as thermal noise. This enables us to simplify the dynamics and treat it as diffusion in a free-energy landscape spanned by slow CVs, effectively rendering the dynamics Markovian. Our recent statistical learning technique, spectral map [Rydzewski, J. J. Phys. Chem. Lett. 2023, 14(22), 5216-5220], explores this strategy to learn slow CVs by maximizing a spectral gap of a transition matrix. In this work, we introduce several advancements into our framework, using a high-dimensional reversible folding process of a protein as an example. We implement an algorithm for coarse-graining Markov transition matrices to partition the reduced space of slow CVs kinetically and use it to define a transition state ensemble. We show that slow CVs learned by spectral map closely approach the Markovian limit for an overdamped diffusion. We demonstrate that coordinate-dependent diffusion coefficients only slightly affect the constructed free-energy landscapes. Finally, we present how spectral maps can be used to quantify the importance of features and compare slow CVs with structural descriptors commonly used in protein folding. Overall, we demonstrate that a single slow CV learned by spectral map can be used as a physical reaction coordinate to capture essential characteristics of protein folding.

Data-Driven Path Collective Variables

Identifying optimal collective variables to model transformations using atomic-scale simulations is a long-standing challenge. We propose a new method for the generation, optimization, and comparison of collective variables that can be thought of as a data-driven generalization of the path collective variable concept. It consists of a kernel ridge regression of the committor probability, which encodes a transformation's progress. The resulting collective variable is one-dimensional, interpretable, and differentiable, making it appropriate for enhanced sampling simulations requiring biasing. We demonstrate the validity of the method on two different applications: a precipitation model and the association of Li+ and F- in water. For the former, we show that global descriptors such as the permutation invariant vector allow reaching an accuracy far from the one achieved via simpler, more intuitive variables. For the latter, we show that information correlated with the transformation mechanism is contained in the first solvation shell only and that inertial effects prevent the derivation of optimal collective variables from the atomic positions only.

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.

OneOPES, a Combined Enhanced Sampling Method to Rule Them All

Enhanced sampling techniques have revolutionized molecular dynamics (MD) simulations, enabling the study of rare events and the calculation of free energy differences in complex systems. One of the main families of enhanced sampling techniques uses physical degrees of freedom called collective variables (CVs) to accelerate a system's dynamics and recover the original system's statistics. However, encoding all the relevant degrees of freedom in a limited number of CVs is challenging, particularly in large biophysical systems. Another category of techniques, such as parallel tempering, simulates multiple replicas of the system in parallel, without requiring CVs. However, these methods may explore less relevant high-energy portions of the phase space and become computationally expensive for large systems. To overcome the limitations of both approaches, we propose a replica exchange method called OneOPES that combines the power of multireplica simulations and CV-based enhanced sampling. This method efficiently accelerates the phase space sampling without the need for ideal CVs, extensive parameters fine tuning nor the use of a large number of replicas, as demonstrated by its successful applications to protein-ligand binding and protein folding benchmark systems. Our approach shows promise as a new direction in the development of enhanced sampling techniques for molecular dynamics simulations, providing an efficient and robust framework for the study of complex and unexplored problems.

Discovering Reaction Pathways, Slow Variables, and Committor Probabilities with Machine Learning

A significant challenge faced by atomistic simulations is the difficulty, and often impossibility, to sample the transitions between metastable states of the free-energy landscape associated with slow molecular processes. Importance-sampling schemes represent an appealing option to accelerate the underlying dynamics by smoothing out the relevant free-energy barriers, but require the definition of suitable reaction-coordinate (RC) models expressed in terms of compact low-dimensional sets of collective variables (CVs). While most computational studies of slow molecular processes have traditionally relied on educated guesses based on human intuition to reduce the dimensionality of the problem at hand, a variety of machine-learning (ML) algorithms have recently emerged as powerful alternatives to discover meaningful CVs capable of capturing the dynamics of the slowest degrees of freedom. Considering a simple paradigmatic situation in which the long-time dynamics is dominated by the transition between two known metastable states, we compare two variational data-driven ML methods based on Siamese neural networks aimed at discovering a meaningful RC model─the slowest decorrelating CV of the molecular process, and the committor probability to first reach one of the two metastable states. One method is the state-free reversible variational approach for Markov processes networks (VAMPnets), or SRVs─the other, inspired by the transition path theory framework, is the variational committor-based neural networks, or VCNs. The relationship and the ability of these methodologies to discover the relevant descriptors of the slow molecular process of interest are illustrated with a series of simple model systems. We also show that both strategies are amenable to importance-sampling schemes through an appropriate reweighting algorithm that approximates the kinetic properties of the transition.

Open Binding Pose Metadynamics: An Effective Approach for the Ranking of Protein-Ligand Binding Poses

Predicting the correct pose of a ligand binding to a protein and its associated binding affinity is of great importance in computer-aided drug discovery. A number of approaches have been developed to these ends, ranging from the widely used fast molecular docking to the computationally expensive enhanced sampling molecular simulations. In this context, methods such as coarse-grained metadynamics and binding pose metadynamics (BPMD) use simulations with metadynamics biasing to probe the binding affinity without trying to fully converge the binding free energy landscape in order to decrease the computational cost. In BPMD, the metadynamics bias perturbs the ligand away from the initial pose. The resistance of the ligand to this bias is used to calculate a stability score. The method has been shown to be useful in reranking predicted binding poses from docking. Here, we present OpenBPMD, an open-source Python reimplementation and reinterpretation of BPMD. OpenBPMD is powered by the OpenMM simulation engine and uses a revised scoring function. The algorithm was validated by testing it on a wide range of targets and showing that it matches or exceeds the performance of the original BPMD. We also investigated the role of accurate water positioning on the performance of the algorithm and showed how the combination with a grand-canonical Monte Carlo algorithm improves the accuracy of the predictions.

Unraveling the Abnormal Molecular Mechanism of Suicide Inhibition of Cytochrome P450 3A4

Suicide inhibition of the CYP3A4 enzyme by a drug inactivates the enzyme in the drug biotransformation process and often shows safety concerns about the drug. Despite extensive experimental studies, the abnormal molecular mechanism of a suicide inhibitor that forms a covalent bond with the residue far away from the catalytically active center of CYP3A4 inactivating the enzyme remains elusive. Here, the authors used molecular simulation approaches to study in detail how diquinone methide (DQR), the metabolite product of raloxifene, unbinds from CYP3A4 and inactivates the enzyme at the atomistic level. The results clearly indicate that in one of the intermediate states formed in its unbinding process, DQR covalently binds to Cys239, a residue far away from the catalytically active center of CYP3A4, and hinders the substrate from entering or leaving the enzyme. This work therefore provides an unprecedented way of clarifying the abnormal mechanism of suicide inhibition of the CYP3A4 enzyme.

Iterative Landmark-Based Umbrella Sampling (ILBUS) Protocol for Sampling of Conformational Space of Biomolecules

Computer simulations of biomolecules such as molecular dynamics often suffer from insufficient sampling. Due to limited computational resources, insufficient sampling prevents obtaining proper equilibrium distributions of observed properties. To deal with this problem, we proposed a simulation protocol for efficient resampling of collected off-equilibrium trajectories. These trajectories are utilized for the initial mapping of the conformational space, which is later properly resampled by the introduced Iterative Landmark-Based Umbrella Sampling (ILBUS) method. Reconstruction of static equilibrium properties is achieved by the multistate Bennett acceptance ratio (MBAR) method, which enables efficient use of simulated data. The ILBUS protocol is geometry-based and does not demand any additional collective variable or a dimensional-reduction technique. The only requirement is a set of suitably spaced reference conformations, which serve as landmarks in the mapped conformational space. Additionally, the ILBUS protocol encompasses an iterative process that optimizes the force constant used in the umbrella sampling simulation. Such tuning is an inherent feature of the protocol and does not need to be performed by the user in advance. Furthermore, even the simulations with suboptimal force constants can be used in estimates by MBAR. We demonstrate the feasibility and the performance of this approach in the study of the conformational landscape of the alanine dipeptide, met-enkephalin, and adenylate kinase.

Multitask Machine Learning of Collective Variables for Enhanced Sampling of Rare Events

Computing accurate reaction rates is a central challenge in computational chemistry and biology because of the high cost of free energy estimation with unbiased molecular dynamics. In this work, a data-driven machine learning algorithm is devised to learn collective variables with a multitask neural network, where a common upstream part reduces the high dimensionality of atomic configurations to a low dimensional latent space and separate downstream parts map the latent space to predictions of basin class labels and potential energies. The resulting latent space is shown to be an effective low-dimensional representation, capturing the reaction progress and guiding effective umbrella sampling to obtain accurate free energy landscapes. This approach is successfully applied to model systems including a 5D Müller Brown model, a 5D three-well model, the alanine dipeptide in vacuum, and an Au(110) surface reconstruction unit reaction. It enables automated dimensionality reduction for energy controlled reactions in complex systems, offers a unified and data-efficient framework that can be trained with limited data, and outperforms single-task learning approaches, including autoencoders.

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.

Combined Free-Energy Calculation and Machine Learning Methods for Understanding Ligand Unbinding Kinetics

The determination of drug residence times, which define the time an inhibitor is in complex with its target, is a fundamental part of the drug discovery process. Synthesis and experimental measurements of kinetic rate constants are, however, expensive and time consuming. In this work, we aimed to obtain drug residence times computationally. Furthermore, we propose a novel algorithm to identify molecular design objectives based on ligand unbinding kinetics. We designed an enhanced sampling technique to accurately predict the free-energy profiles of the ligand unbinding process, focusing on the free-energy barrier for unbinding. Our method first identifies unbinding paths determining a corresponding set of internal coordinates (ICs) that form contacts between the protein and the ligand; it then iteratively updates these interactions during a series of biased molecular dynamics (MD) simulations to reveal the ICs that are important for the whole of the unbinding process. Subsequently, we performed finite-temperature string simulations to obtain the free-energy barrier for unbinding using the set of ICs as a complex reaction coordinate. Importantly, we also aimed to enable the further design of drugs focusing on improved residence times. To this end, we developed a supervised machine learning (ML) approach with inputs from unbiased “downhill” trajectories initiated near the transition state (TS) ensemble of the string unbinding path. We demonstrate that our ML method can identify key ligand–protein interactions driving the system through the TS. Some of the most important drugs for cancer treatment are kinase inhibitors. One of these kinase targets is cyclin-dependent kinase 2 (CDK2), an appealing target for anticancer drug development. Here, we tested our method using two different CDK2 inhibitors for the potential further development of these compounds. We compared the free-energy barriers obtained from our calculations with those observed in available experimental data. We highlighted important interactions at the distal ends of the ligands that can be targeted for improved residence times. Our method provides a new tool to determine unbinding rates and to identify key structural features of the inhibitors that can be used as starting points for novel design strategies in drug discovery.

An Efficient Gaussian-Accelerated Molecular Dynamics (GaMD) Multilevel Enhanced Sampling Strategy: Application to Polarizable Force Fields Simulations of Large Biological Systems

We introduce a novel multilevel enhanced sampling strategy grounded on Gaussian-accelerated Molecular Dynamics (GaMD). First, we propose a GaMD multi-GPUs-accelerated implementation within the Tinker-HP molecular dynamics package. We introduce the new "dual-water" mode and its use with the flexible AMOEBA polarizable force field. By adding harmonic boosts to the water stretching and bonding terms, it accelerates the solvent-solute interactions while enabling speedups, thanks to the use of fast multiple-time step integrators. To further reduce the time-to-solution, we couple GaMD to Umbrella Sampling (US). The GaMD─US/dual-water approach is tested on the 1D Potential of Mean Force (PMF) of the solvated CD2-CD58 system (168 000 atoms), allowing the AMOEBA PMF to converge within 1 kcal/mol of the experimental value. Finally, Adaptive Sampling (AS) is added, enabling AS-GaMD capabilities but also the introduction of the new Adaptive Sampling-US-GaMD (ASUS-GaMD) scheme. The highly parallel ASUS-GaMD setup decreases time to convergence by, respectively, 10 and 20 times, compared to GaMD-US and US. Overall, beside the acceleration of PMF computations, Tinker-HP now allows for the simultaneous use of Adaptive Sampling and GaMD-"dual water" enhanced sampling approaches increasing the applicability of polarizable force fields to large-scale simulations of biological systems.

Computer-aided design of molecular machines: techniques, paradigms and difficulties

With their development in the past decade, molecular machines, which achieve specific tasks by responding to external stimuli, have gradually come to be regarded as powerful tools for a wide range of applications, rather than interesting molecular toys. This conceptual change in turn motivates scientists to design molecular machines with complex architectures. Due to the lack of general principles bridging the functions and the chemical structures of molecular machines, experience-based design becomes difficult with the increase of size and complexity of the architectures. Computer-aided molecular-machine design, therefore, has attracted widespread attention on account of its ability to model and investigate complex molecular architectures without too much time and expense required for synthetic experiments. Using leading-edge numerical-simulation techniques, the mechanisms underlying achieving tasks through response to external stimuli of a large number of existing molecular machines have been successfully explored. Based on the experience of studying existing molecular machines, generalized methodologies of predicting the properties and working principles of molecular candidates have been established, paving the way for de novo computer-aided design of molecular machines. In this perspective, we introduce cutting-edge techniques that have been applied for investigating and designing molecular machines. We show paradigms of computer-aided design of molecular machines, which can serve as guidelines for the investigation of new supramolecular architectures. Moreover, we discuss the limitations and possible future developments of current techniques and methodologies in the field of computer-aided design of molecular machines.

The prediction of protein-ligand unbinding for modern drug discovery

INTRODUCTION

Drug-target thermodynamic and kinetic information have perennially important roles in drug design. The prediction of protein-ligand unbinding, which can provide important kinetic information, in experiments continues to face great challenges. Uncovering protein-ligand unbinding through molecular dynamics simulations has become efficient and inexpensive with the progress and enhancement of computing power and sampling methods.

AREAS COVERED

In this review, various sampling methods for protein-ligand unbinding and their basic principles are firstly briefly introduced. Then, their applications in predicting aspects of protein-ligand unbinding, including unbinding pathways, dissociation rate constants, residence time and binding affinity, are discussed.

EXPERT OPINION

Although various sampling methods have been successfully applied in numerous systems, they still have shortcomings and deficiencies. Most enhanced sampling methods require researchers to possess a wealth of prior knowledge of collective variables or reaction coordinates. In addition, most systems studied at present are relatively simple, and the study of complex systems in real drug research remains greatly challenging. Through the combination of machine learning and enhanced sampling methods, prediction accuracy can be further improved, and some problems encountered in complex systems also may be solved.

Metadynamics-Based Approaches for Modeling the Hypoxia-Inducible Factor 2α Ligand Binding Process

Several methods based on enhanced-sampling molecular dynamics have been proposed for studying ligand binding processes. Here, we developed a protocol that combines the advantages of steered molecular dynamics (SMD) and metadynamics. While SMD is proposed for investigating possible unbinding pathways of the ligand and identifying the preferred one, metadynamics, with the path collective variable (PCV) formalism, is suggested to explore the binding processes along the pathway defined on the basis of SMD, by using only two CVs. We applied our approach to the study of binding of two known ligands to the hypoxia-inducible factor 2α, where the buried binding cavity makes simulation of the process a challenging task. Our approach allowed identification of the preferred entrance pathway for each ligand, highlighted the features of the bound and intermediate states in the free-energy surface, and provided a binding affinity scale in agreement with experimental data. Therefore, it seems to be a suitable tool for elucidating ligand binding processes of similar complex systems.

Ligand binding free-energy calculations with funnel metadynamics

The accurate resolution of the binding mechanism of a ligand to its molecular target is fundamental to develop a successful drug design campaign. Free-energy calculations, which provide the energy value of the ligand-protein binding complex, are essential for resolving the binding mode of the ligand. The accuracy of free-energy calculation methods is counteracted by their poor user-friendliness, which hampers their broad application. Here we present the Funnel-Metadynamics Advanced Protocol (FMAP), which is a flexible and user-friendly graphical user interface (GUI)-based protocol to perform funnel metadynamics, a binding free-energy method that employs a funnel-shape restraint potential to reveal the ligand binding mode and accurately calculate the absolute ligand-protein binding free energy. FMAP guides the user through all phases of the free-energy calculation process, from preparation of the input files, to production simulation, to analysis of the results. FMAP delivers the ligand binding mode and the absolute protein-ligand binding free energy as outputs. Alternative binding modes and the role of waters are also elucidated, providing a detailed description of the ligand binding mechanism. The entire protocol on the paradigmatic system benzamidine-trypsin, composed of ~105 k atoms, took ~2.8 d using the Cray XC50 piz Daint cluster at the Swiss National Supercomputing Centre.

Combining Machine Learning and Enhanced Sampling Techniques for Efficient and Accurate Calculation of Absolute Binding Free Energies

Calculating absolute binding free energies is challenging and important. In this paper, we test some recently developed metadynamics-based methods and develop a new combination with a Hamiltonian replica-exchange approach. The methods were tested on 18 chemically diverse ligands with a wide range of different binding affinities to a complex target; namely, human soluble epoxide hydrolase. The results suggest that metadynamics with a funnel-shaped restraint can be used to calculate, in a computationally affordable and relatively accurate way, the absolute binding free energy for small fragments. When used in combination with an optimal pathlike variable obtained using machine learning or with the Hamiltonian replica-exchange algorithm SWISH, this method can achieve reasonably accurate results for increasingly complex ligands, with a good balance of computational cost and speed. An additional benefit of using the combination of metadynamics and SWISH is that it also provides useful information about the role of water in the binding mechanism.

The Maximum Caliber Variational Principle for Nonequilibria

Ever since Clausius in 1865 and Boltzmann in 1877, the concepts of entropy and of its maximization have been the foundations for predicting how material equilibria derive from microscopic properties. But, despite much work, there has been no equally satisfactory general variational principle for nonequilibrium situations. However, in 1980, a new avenue was opened by E.T. Jaynes and by Shore and Johnson. We review here maximum caliber, which is a maximum-entropy-like principle that can infer distributions of flows over pathways, given dynamical constraints. This approach is providing new insights, particularly into few-particle complex systems, such as gene circuits, protein conformational reaction coordinates, network traffic, bird flocking, cell motility, and neuronal firing.

Synergistic Approach of Ultrafast Spectroscopy and Molecular Simulations in the Characterization of Intramolecular Charge Transfer in Push-Pull Molecules

The comprehensive characterization of Intramolecular Charge Transfer (ICT) stemming in push-pull molecules with a delocalized π-system of electrons is noteworthy for a bespoke design of organic materials, spanning widespread applications from photovoltaics to nanomedicine imaging devices. Photo-induced ICT is characterized by structural reorganizations, which allows the molecule to adapt to the new electronic density distribution. Herein, we discuss recent photophysical advances combined with recent progresses in the computational chemistry of photoactive molecular ensembles. We focus the discussion on femtosecond Transient Absorption Spectroscopy (TAS) enabling us to follow the transition from a Locally Excited (LE) state to the ICT and to understand how the environment polarity influences radiative and non-radiative decay mechanisms. In many cases, the charge transfer transition is accompanied by structural rearrangements, such as the twisting or molecule planarization. The possibility of an accurate prediction of the charge-transfer occurring in complex molecules and molecular materials represents an enormous advantage in guiding new molecular and materials design. We briefly report on recent advances in ultrafast multidimensional spectroscopy, in particular, Two-Dimensional Electronic Spectroscopy (2DES), in unraveling the ICT nature of push-pull molecular systems. A theoretical description at the atomistic level of photo-induced molecular transitions can predict with reasonable accuracy the properties of photoactive molecules. In this framework, the review includes a discussion on the advances from simulation and modeling, which have provided, over the years, significant information on photoexcitation, emission, charge-transport, and decay pathways. Density Functional Theory (DFT) coupled with the Time-Dependent (TD) framework can describe electronic properties and dynamics for a limited system size. More recently, Machine Learning (ML) or deep learning approaches, as well as free-energy simulations containing excited state potentials, can speed up the calculations with transferable accuracy to more complex molecules with extended system size. A perspective on combining ultrafast spectroscopy with molecular simulations is foreseen for optimizing the design of photoactive compounds with tunable properties.

An approach to calculate the free energy changes of surface reactions using free energy decomposition on ab initio brute-force molecular dynamics trajectories

To understand the mechanisms and kinetics of catalytic reactions in heterogeneous catalysis, ab initio molecular dynamics is one of the powerful methods used to explore the free energy surface (FES) of surface elementary steps.

Exploration of Chemical Compound, Conformer, and Reaction Space with Meta-Dynamics Simulations Based on Tight-Binding Quantum Chemical Calculations

The semiempirical tight-binding based quantum chemistry method GFN2-xTB is used in the framework of meta-dynamics (MTD) to globally explore chemical compound, conformer, and reaction space. The biasing potential given as a sum of Gaussian functions is expressed with the root-mean-square-deviation (RMSD) in Cartesian space as a metric for the collective variables. This choice makes the approach robust and generally applicable to three common problems (i.e., conformer search, chemical reaction space exploration in a virtual nanoreactor, and for guessing reaction paths). Because of the inherent locality of the atomic RMSD, functional group or fragment selective treatments are possible facilitating the investigation of catalytic processes where, for example, only the substrate is thermally activated. Due to the approximate character of the GFN2-xTB method, the resulting structure ensembles require further refinement with more sophisticated, for example, density functional or wave function theory methods. However, the approach is extremely efficient running routinely on common laptop computers in minutes to hours of computation time even for realistically sized molecules with a few hundred atoms. Furthermore, the underlying potential energy surface for molecules containing almost all elements ( Z = 1-86) is globally consistent including the covalent dissociation process and electronically complicated situations in, for example, transition metal systems. As examples, thermal decomposition, ethyne oligomerization, the oxidation of hydrocarbons (by oxygen and a P450 enzyme model), a Miller-Urey model system, a thermally forbidden dimerization, and a multistep intramolecular cyclization reaction are shown. For typical conformational search problems of organic drug molecules, the new MTD(RMSD) algorithm yields lower energy structures and more complete conformer ensembles at reduced computational effort compared with its already well performing predecessor.