Spectral gap optimization of order parameters for sampling complex molecular systems

Simulating Crystallization in a Colloidal System Using State Predictive Information Bottleneck Based Enhanced Sampling

We investigate crystal nucleation in supersaturated colloid suspensions using enhanced molecular dynamics simulations augmented with machine learning techniques. The simulations reveal that crystallization in the model colloidal system studied here, with particles interacting through a repulsive screened Coulomb Yukawa potential, proceeds from vapor to dense liquid droplet to crystalline phases across multiple high barriers. Employing a one-dimensional reaction coordinate derived from the State Predictive Information Bottleneck framework, our simulations capture back-and-forth phase transitions across multiple barriers effectively in biased metadynamics simulations. We obtain relative free energy differences between different phases and also quantify the roles of different molecular level features in driving the phase changes.

Melting entropy of crystals determined by electron-beam-induced configurational disordering

Upon melting, the molecules in a crystal explore numerous configurations, reflecting an increase in disorder. The molar entropy of disorder can be defined by Boltzmann's formula ΔSd = Rln(Wd), where Wd is the increase in the number of microscopic states, so far inaccessible experimentally. We found that the Arrhenius frequency factor A of the electron diffraction signal decay provides Wd through an experimental equation A = AINTWd, where AINT is an inelastic scattering cross section. The method connects Clausius and Boltzmann experimentally and supplements the Clausius approach, being applicable to a femtogram quantity of thermally unstable and biomolecular crystals. The data also showed that crystal disordering and crystallization of melt are reciprocal, both governed by the entropy change but manifesting in opposite directions.

Dynamical Reweighting for Biased Rare Event Simulations

Dynamical reweighting techniques aim to recover the correct molecular dynamics from a simulation at a modified potential energy surface. They are important for unbiasing enhanced sampling simulations of molecular rare events. Here, we review the theoretical frameworks of dynamical reweighting for modified potentials. Based on an overview of kinetic models with increasing level of detail, we discuss techniques to reweight two-state dynamics, multistate dynamics, and path integrals. We explore the natural link to transition path sampling and how the effect of nonequilibrium forces can be reweighted. We end by providing an outlook on how dynamical reweighting integrates with techniques for optimizing collective variables and with modern potential energy surfaces.

Enhanced Sampling with Machine Learning

Molecular dynamics (MD) enables the study of physical systems with excellent spatiotemporal resolution but suffers from severe timescale limitations. To address this, enhanced sampling methods have been developed to improve the exploration of configurational space. However, implementing these methods is challenging and requires domain expertise. In recent years, integration of machine learning (ML) techniques into different domains has shown promise, prompting their adoption in enhanced sampling as well. Although ML is often employed in various fields primarily due to its data-driven nature, its integration with enhanced sampling is more natural with many common underlying synergies. This review explores the merging of ML and enhanced MD by presenting different shared viewpoints. It offers a comprehensive overview of this rapidly evolving field, which can be difficult to stay updated on. We highlight successful strategies such as dimensionality reduction, reinforcement learning, and flow-based methods. Finally, we discuss open problems at the exciting ML-enhanced MD interface.

Generating High-Precision Force Fields for Molecular Dynamics Simulations to Study Chemical Reaction Mechanisms Using Molecular Configuration Transformer

Theoretical studies on chemical reaction mechanisms have been crucial in organic chemistry. Traditionally, calculating the manually constructed molecular conformations of transition states for chemical reactions using quantum chemical calculations is the most commonly used method. However, this way is heavily dependent on individual experience and chemical intuition. In our previous study, we proposed a research paradigm that used enhanced sampling in molecular dynamics simulations to study chemical reactions. This approach can directly simulate the entire process of a chemical reaction. However, the computational speed limited the use of high-precision potential energy functions for simulations. To address this issue, we presented a scheme for training high-precision force fields for molecular modeling using a previously developed graph-neural-network-based molecular model, molecular configuration transformer. This potential energy function allowed for highly accurate simulations at a low computational cost, leading to more precise calculations of the mechanism of chemical reactions. We applied this approach to study a Claisen rearrangement reaction and a carbonyl insertion reaction catalyzed by manganese.

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.

Unbiasing Enhanced Sampling on a High-Dimensional Free Energy Surface with a Deep Generative Model

Biased enhanced sampling methods that utilize collective variables (CVs) are powerful tools for sampling conformational ensembles. Due to their large intrinsic dimensions, efficiently generating conformational ensembles for complex systems requires enhanced sampling on high-dimensional free energy surfaces. While temperature-accelerated molecular dynamics (TAMD) can trivially adopt many CVs in a simulation, unbiasing the simulation to generate unbiased conformational ensembles requires accurate modeling of a high-dimensional CV probability distribution, which is challenging for traditional density estimation techniques. Here we propose an unbiasing method based on the score-based diffusion model, a deep generative learning method that excels in density estimation across complex data landscapes. We demonstrate that this unbiasing approach, tested on multiple TAMD simulations, significantly outperforms traditional unbiasing methods and can generate accurate unbiased conformational ensembles. With the proposed approach, TAMD can adopt CVs that focus on improving sampling efficiency and the proposed unbiasing method enables accurate evaluation of ensemble averages of important chemical features.

Surface-Accelerated String Method for Locating Minimum Free Energy Paths

We present a surface-accelerated string method (SASM) to efficiently optimize low-dimensional reaction pathways from the sampling performed with expensive quantum mechanical/molecular mechanical (QM/MM) Hamiltonians. The SASM accelerates the convergence of the path using the aggregate sampling obtained from the current and previous string iterations, whereas approaches like the string method in collective variables (SMCV) or the modified string method in collective variables (MSMCV) update the path only from the sampling obtained from the current iteration. Furthermore, the SASM decouples the number of images used to perform sampling from the number of synthetic images used to represent the path. The path is optimized on the current best estimate of the free energy surface obtained from all available sampling, and the proposed set of new simulations is not restricted to being located along the optimized path. Instead, the umbrella potential placement is chosen to extend the range of the free energy surface and improve the quality of the free energy estimates near the path. In this manner, the SASM is shown to improve the exploration for a minimum free energy pathway in regions where the free energy surface is relatively flat. Furthermore, it improves the quality of the free energy profile when the string is discretized with too few images. We compare the SASM, SMCV, and MSMCV using 3 QM/MM applications: a ribozyme methyltransferase reaction using 2 reaction coordinates, the 2'-O-transphosphorylation reaction of Hammerhead ribozyme using 3 reaction coordinates, and a tautomeric reaction in B-DNA using 5 reaction coordinates. We show that SASM converges the paths using roughly 3 times less sampling than the SMCV and MSMCV methods. All three algorithms have been implemented in the FE-ToolKit package made freely available.

Learning Markovian dynamics with spectral maps

The long-time behavior of many complex molecular systems can often be described by Markovian dynamics in a slow subspace spanned by a few reaction coordinates referred to as collective variables (CVs). However, determining CVs poses a fundamental challenge in chemical physics. Depending on intuition or trial and error to construct CVs can lead to non-Markovian dynamics with long memory effects, hindering analysis. To address this problem, we continue to develop a recently introduced deep-learning technique called spectral map [J. Rydzewski, J. Phys. Chem. Lett. 14, 5216-5220 (2023)]. Spectral map learns slow CVs by maximizing a spectral gap of a Markov transition matrix describing anisotropic diffusion. Here, to represent heterogeneous and multiscale free-energy landscapes with spectral map, we implement an adaptive algorithm to estimate transition probabilities. Through a Markov state model analysis, we validate that spectral map learns slow CVs related to the dominant relaxation timescales and discerns between long-lived metastable states.

A slow feature analysis approach for the optimization of collective variables

Molecular dynamics simulations have become increasingly important in understanding the microscopic mechanisms of various molecular systems. However, the high energy barriers in complicated molecules often make it difficult to observe events of interest within a reasonable timescale. To address this issue, researchers have developed a variety of enhanced sampling methods to explore configuration space by adding bias potentials along the slowly changing collective variables (CVs). In this study, we have developed a new tool that combines slow feature analysis and biasing-enhanced sampling methods to identify effective CVs and enhance the sampling efficiency of configuration space. We have demonstrated the effectiveness of this tool through three general examples.

Addressing Suboptimal Poses in Nonequilibrium Alchemical Calculations

Alchemical transformations can be used to quantitatively estimate absolute binding free energies at a reasonable computational cost. However, most of the approaches currently in use require knowledge of the correct (crystallographic) pose. In this paper, we present a combined Hamiltonian replica exchange nonequilibrium alchemical method that allows us to reliably calculate absolute binding free energies, even when starting from suboptimal initial binding poses. Performing a preliminary Hamiltonian replica exchange enhances the sampling of slow degrees of freedom of the ligand and the target, allowing the system to populate the correct binding pose when starting from an approximate docking pose. We apply the method on 6 ligands of the first bromodomain of the BRD4 bromodomain-containing protein. For each ligand, we start nonequilibrium alchemical transformations from both the crystallographic pose and the top-scoring docked pose that are often significantly different. We show that the method produces statistically equivalent binding free energies, making it a useful tool for computational drug discovery pipelines.

Collective Variable-Based Enhanced Sampling: From Human Learning to Machine Learning

Enhanced-sampling algorithms relying on collective variables (CVs) are extensively employed to study complex (bio)chemical processes that are not amenable to brute-force molecular simulations. The selection of appropriate CVs characterizing the slow movement modes is of paramount importance for reliable and efficient enhanced-sampling simulations. In this Perspective, we first review the application and limitations of CVs obtained from chemical and geometrical intuition. We also introduce path-sampling algorithms, which can identify path-like CVs in a high-dimensional free-energy space. Machine-learning algorithms offer a viable approach to finding suitable CVs by analyzing trajectories from preliminary simulations. We discuss both the performance of machine-learning-derived CVs in enhanced-sampling simulations of experimental models and the challenges involved in applying these CVs to realistic, complex molecular assemblies. Moreover, we provide a prospective view of the potential advancements of machine-learning algorithms for the development of CVs in the field of enhanced-sampling simulations.

Is the Local Ion Density Sufficient to Drive NaCl Nucleation from the Melt and Aqueous Solution?

Even though nucleation is ubiquitous in different science and engineering problems, investigating nucleation is extremely difficult due to the complicated ranges of time and length scales involved. In this work, we simulate NaCl nucleation in both molten and aqueous environments using enhanced sampling of all-atom molecular dynamics with deep-learning-based estimation of reaction coordinates. By incorporating various structural order parameters and learning the reaction coordinate as a function thereof, we achieve significantly improved sampling relative to traditional ad hoc descriptions of what drives nucleation, particularly in an aqueous medium. Our results reveal a one-step nucleation mechanism in both environments, with reaction coordinate analysis highlighting the importance of local ion density in distinguishing solid and liquid states. However, although fluctuations in the local ion density are necessary to drive nucleation, they are not sufficient. Our analysis shows that near the transition states, descriptors such as enthalpy and local structure become crucial. Our protocol proposed here enables robust nucleation analysis and phase sampling and could offer insights into nucleation mechanisms for generic small molecules in different environments.

The calculus of codes - From entropy, complexity, and information to life

Exploring the core components that define living systems and their operational mechanisms within emerging biological entities is a complex endeavor. In the realm of biological systems literature, the terms matter, energy, information, complexity, and entropy are frequently referenced. However, possessing these concepts alone does not guarantee a comprehensive understanding or the ability to reconstruct the intricate nature of life. This study aims to illuminate the trajectory of these organic attributes, presenting a theoretical framework that delves into the integrated role of these concepts in biology. We assert that Code Biology serves as a pivotal steppingstone for unraveling the mechanisms underlying life. Biological codes (BCs) emerge not only from the interplay of matter and energy but also from Information. Contrary to deriving information from the former elements, we propose that information holds its place as a fundamental physical aspect. Consequently, we propose a continuum perspective called Calculus of Fundamentals involving three fundamentals: Matter, Energy, and Information, to depict the dynamics of BCs. To achieve this, we emphasize the necessity of studying Entropy and Complexity as integral organic descriptors. This perspective also facilitates the introduction of a mathematical theoretical framework that aids in comprehending continuous changes, the driving dynamics of biological fundamentals. We posit that Energy, Matter, and Information constitute the essential building blocks of living systems, and their interactions are governed by Entropy and Complexity analyses, redefined as biological descriptors. This interdisciplinary perspective of Code Biology sheds light on the intricate interplay between the controversial phenomenon of life and advances the idea of constructing a theory rooted in information as an organic fundamental.

Deep-Learning-Assisted Enhanced Sampling for Exploring Molecular Conformational Changes

We present a novel strategy to explore conformational changes and identify stable states of molecular objects, eliminating the need for a priori knowledge. The approach applies a deep learning method to extract information about the movement modes of the molecular object from a short, high-dimensional, and parameter-free preliminary enhanced-sampling simulation. The gathered information is described by a small set of deep-learning-based collective variables (dCVs), which steer the production-enhanced-sampling simulation. Considering the challenge of adequately exploring the configurational space using the low-dimensional, suboptimal dCVs, we incorporate a method designed for ergodic sampling, namely, Gaussian-accelerated molecular dynamics (MD), into the framework of CV-based enhanced sampling. MD simulations on both toy models and nontrivial examples demonstrate the remarkable computational efficiency of the strategy in capturing the conformational changes of molecular objects without a priori knowledge. Specifically, we achieved the blind folding of two fast folders, chignolin and villin, within a time scale of hundreds of nanoseconds and successfully reconstructed the free-energy landscapes that characterize their reversible folding. All in all, the presented strategy holds significant promise for investigating conformational changes in macromolecules, and it is anticipated to find extensive applications in the fields of chemistry and biology.

Understanding the free-energy landscape of phase separation in lipid bilayers using molecular dynamics

Liquid-liquid phase separation inside the cell often results in biological condensates that can critically affect cell homeostasis. Such phase separation events occur in multiple parts of cells, including the cell membranes, where the "lipid raft" hypothesis posits the formation of ordered domains floating in a sea of disordered lipids. The resulting lipid domains often have functional roles. However, the thermodynamics of lipid phase separation and their resulting mechanistic effects on cell function and dysfunction are poorly understood. Understanding such complex phenomena in cell membranes, with their diverse lipid compositions, is exceptionally difficult. For these reasons, simple model systems that can recapitulate similar behavior are widely used to study this phenomenon. Despite these simplifications, the timescale and length scales of domain formation pose a challenge for molecular dynamics (MD) simulations. Thus, most MD studies focus on spontaneous lipid phase separation-essentially measuring the sign (but not the amplitude) of the free-energy change upon separation-rather than directly interrogating the thermodynamics. Here, we propose a proof-of-concept pipeline that can directly measure this free energy by combining coarse-grained MD with enhanced sampling protocols using a novel collective variable. This approach will be a useful tool to help connect the thermodynamics of phase separation with the mechanistic insights already available from MD simulations.

Vendi sampling for molecular simulations: Diversity as a force for faster convergence and better exploration

Molecular dynamics (MD) is the method of choice for understanding the structure, function, and interactions of molecules. However, MD simulations are limited by the strong metastability of many molecules, which traps them in a single conformation basin for an extended amount of time. Enhanced sampling techniques, such as metadynamics and replica exchange, have been developed to overcome this limitation and accelerate the exploration of complex free energy landscapes. In this paper, we propose Vendi Sampling, a replica-based algorithm for increasing the efficiency and efficacy of the exploration of molecular conformation spaces. In Vendi sampling, replicas are simulated in parallel and coupled via a global statistical measure, the Vendi Score, to enhance diversity. Vendi sampling allows for the recovery of unbiased sampling statistics and dramatically improves sampling efficiency. We demonstrate the effectiveness of Vendi sampling in improving molecular dynamics simulations by showing significant improvements in coverage and mixing between metastable states and convergence of free energy estimates for four common benchmarks, including Alanine Dipeptide and Chignolin.

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.

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.

Optimal Reaction Coordinates and Kinetic Rates from the Projected Dynamics of Transition Paths

Finding optimal reaction coordinates and predicting accurate kinetic rates for activated processes are two of the foremost challenges of molecular simulations. We introduce an algorithm that tackles the two problems at once: starting from a limited number of reactive molecular dynamics trajectories (transition paths), we automatically generate with a Monte Carlo approach a sequence of different reaction coordinates that progressively reduce the kinetic rate of their projected effective dynamics. Based on a variational principle, the minimal rate accurately approximates the exact one, and it corresponds to the optimal reaction coordinate. After benchmarking the method on an analytic double-well system, we apply it to complex atomistic systems: the interaction of carbon nanoparticles of different sizes in water.

Determination of the Structure and Dynamics of the Fuzzy Coat of an Amyloid Fibril of IAPP Using Cryo-Electron Microscopy

In recent years, major advances in cryo-electron microscopy (cryo-EM) have enabled the routine determination of complex biomolecular structures at atomistic resolution. An open challenge for this approach, however, concerns large systems that exhibit continuous dynamics. To address this problem, we developed the metadynamic electron microscopy metainference (MEMMI) method, which incorporates metadynamics, an enhanced conformational sampling approach, into the metainference method of integrative structural biology. MEMMI enables the simultaneous determination of the structure and dynamics of large heterogeneous systems by combining cryo-EM density maps with prior information through molecular dynamics, while at the same time modeling the different sources of error. To illustrate the method, we apply it to elucidate the dynamics of an amyloid fibril of the islet amyloid polypeptide (IAPP). The resulting conformational ensemble provides an accurate description of the structural variability of the disordered region of the amyloid fibril, known as fuzzy coat. The conformational ensemble also reveals that in nearly half of the structural core of this amyloid fibril, the side chains exhibit liquid-like dynamics despite the presence of the highly ordered network backbone of hydrogen bonds characteristic of the cross-β structure of amyloid fibrils.

Identifying and Overcoming the Sampling Challenges in Relative Binding Free Energy Calculations of a Model Protein:Protein Complex

Relative alchemical binding free energy calculations are routinely used in drug discovery projects to optimize the affinity of small molecules for their drug targets. Alchemical methods can also be used to estimate the impact of amino acid mutations on protein:protein binding affinities, but these calculations can involve sampling challenges due to the complex networks of protein and water interactions frequently present in protein:protein interfaces. We investigate these challenges by extending a graphics processing unit (GPU)-accelerated open-source relative free energy calculation package (Perses) to predict the impact of amino acid mutations on protein:protein binding. Using the well-characterized model system barnase:barstar, we describe analyses for identifying and characterizing sampling problems in protein:protein relative free energy calculations. We find that mutations with sampling problems often involve charge-changes, and inadequate sampling can be attributed to slow degrees of freedom that are mutation-specific. We also explore the accuracy and efficiency of current state-of-the-art approaches─alchemical replica exchange and alchemical replica exchange with solute tempering─for overcoming relevant sampling problems. By employing sufficiently long simulations, we achieve accurate predictions (RMSE 1.61, 95% CI: [1.12, 2.11] kcal/mol), with 86% of estimates within 1 kcal/mol of the experimentally determined relative binding free energies and 100% of predictions correctly classifying the sign of the changes in binding free energies. Ultimately, we provide a model workflow for applying protein mutation free energy calculations to protein:protein complexes, and importantly, catalog the sampling challenges associated with these types of alchemical transformations. Our free open-source package (Perses) is based on OpenMM and is available at https://github.com/choderalab/perses.

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.

Reaction Coordinates for Conformational Transitions Using Linear Discriminant Analysis on Positions

In this work, we demonstrate that Linear Discriminant Analysis (LDA) applied to atomic positions in two different states of a biomolecule produces a good reaction coordinate between those two states. Atomic coordinates of a macromolecule are a direct representation of a macromolecular configuration, and yet, they are not used in enhanced sampling studies due to a lack of rotational and translational invariance. We resolve this issue using the technique of our prior work, whereby a molecular configuration is considered a member of an equivalence class in size-and-shape space, which is the set of all configurations that can be translated and rotated to a single point within a reference multivariate Gaussian distribution characterizing a single molecular state. The reaction coordinates produced by LDA applied to positions are shown to be good reaction coordinates both in terms of characterizing the transition between two states of a system within a long molecular dynamics (MD) simulation and also ones that allow us to readily produce free energy estimates along that reaction coordinate using enhanced sampling MD techniques.

Selecting High-Dimensional Representations of Physical Systems by Reweighted Diffusion Maps

Constructing reduced representations of high-dimensional systems is a fundamental problem in physical chemistry. Many unsupervised machine learning methods can automatically find such low-dimensional representations. However, an often overlooked problem is what high-dimensional representation should be used to describe systems before dimensionality reduction. Here, we address this issue using a recently developed method called the reweighted diffusion map [J. Chem. Theory Comput. 2022, 18, 7179-7192]. We show how high-dimensional representations can be quantitatively selected by exploring the spectral decomposition of Markov transition matrices built from data obtained from standard or enhanced sampling atomistic simulations. We demonstrate the performance of the method in several high-dimensional examples.

Efficient Interrogation of the Kinetic Barriers Demarcating Catalytic States of a Tyrosine Kinase with Optimal Physical Descriptors and Mixture Models

Computer simulations are increasingly used to access thermo-kinetic information underlying structural transformation of protein kinases. Such information are necessary to probe their roles in disease progression and interactions with drug targets. However, the investigations are frequently challenged by forbiddingly high computational expense, and by the lack of standard protocols for the design of low dimensional physical descriptors that encode system features important for transitions. Here, we consider the demarcating characteristics of the different states of Abelson tyrosine kinase associated with distinct catalytic activity to construct a set of physically meaningful, orthogonal collective variables that preserve the slow modes of the system. Independent sampling of each metastable state is followed by the estimation of global partition function along the appropriate physical descriptors using the modified Expectation Maximized Molecular Dynamics method. The resultant free energy barriers are in excellent agreement with experimentally known rate-limiting dynamics and activation energy computed with conventional enhanced sampling methods. We discuss possible directions for further development and applications.

Identifying and overcoming the sampling challenges in relative binding free energy calculations of a model protein:protein complex

Relative alchemical binding free energy calculations are routinely used in drug discovery projects to optimize the affinity of small molecules for their drug targets. Alchemical methods can also be used to estimate the impact of amino acid mutations on protein:protein binding affinities, but these calculations can involve sampling challenges due to the complex networks of protein and water interactions frequently present in protein:protein interfaces. We investigate these challenges by extending a GPU-accelerated opensource relative free energy calculation package (Perses) to predict the impact of amino acid mutations on protein:protein binding. Using the well-characterized model system barnase:barstar, we describe analyses for identifying and characterizing sampling problems in protein:protein relative free energy calculations. We find that mutations with sampling problems often involve charge-changes, and inadequate sampling can be attributed to slow degrees of freedom that are mutation-specific. We also explore the accuracy and efficiency of current state-of-the-art approaches-alchemical replica exchange and alchemical replica exchange with solute tempering-for overcoming relevant sampling problems. By employing sufficiently long simulations, we achieve accurate predictions (RMSE 1.61, 95% CI: [1.12, 2.11] kcal/mol), with 86% of estimates within 1 kcal/mol of the experimentally-determined relative binding free energies and 100% of predictions correctly classifying the sign of the changes in binding free energies. Ultimately, we provide a model workflow for applying protein mutation free energy calculations to protein:protein complexes, and importantly, catalog the sampling challenges associated with these types of alchemical transformations. Our free open-source package (Perses) is based on OpenMM and available at https://github.com/choderalab/perses .

Long-time-step molecular dynamics can retard simulation of protein-ligand recognition process

Molecular dynamics (MD) simulation of biologically relevant processes at realistic time scale and atomistic precision is generally limited by prohibitively large computational cost, due to its restriction of using an ultrashort integration time step (1-2 fs). A popular numerical recipe to reduce the associated computational burden is adopting schemes that would allow relatively longer-time-step for MD propagation. Here, we explore the perceived potential of one of the most frequently used long-time-step protocols, namely the hydrogen mass repartitioning (HMR) approach, in alleviating the computational overhead associated with simulation of the kinetic process of protein-ligand recognition events. By repartitioning the mass of heavier atoms to their linked hydrogen atoms, HMR leverages around twofold longer time step than regular simulation, holding promise of significant performance boost. However, our probe into direct simulation of the protein-ligand recognition event, one of the computationally most challenging processes, shows that long-time-step HMR MD simulations do not necessarily translate to a computationally affordable solution. Our investigations spanning cumulative 176 μs in three independent proteins (T4 lysozyme, sensor domain of MopR, and galectin-3) show that long-time-step HMR-based MD simulations can catch the ligand in its act of recognizing the native cavity. But, as a major caveat, the ligand is found to require significantly longer time to identify buried native protein cavity in an HMR MD simulation than regular simulation, thereby defeating the purpose of its usage for performance upgrade. A molecular analysis shows that the longer time required by a ligand to recognize the protein in HMR is rooted in faster diffusion of the ligand, which reduces the survival probability of decisive on-pathway metastable intermediates, thereby slowing down the eventual recognition process at the native cavity. Together, the investigation stresses careful assessment of pitfalls of long-time-step algorithms before attempting to utilize them for higher performance for biomolecular recognition simulations.

Alchemical Metadynamics: Adding Alchemical Variables to Metadynamics to Enhance Sampling in Free Energy Calculations

Performing alchemical transformations, in which one molecular system is nonphysically changed to another system, is a popular approach adopted in performing free energy calculations associated with various biophysical processes, such as protein-ligand binding or the transfer of a molecule between environments. While the sampling of alchemical intermediate states in either parallel (e.g., Hamiltonian replica exchange) or serial manner (e.g., expanded ensemble) can bridge the high-probability regions in the configurational space between two end states of interest, alchemical methods can fail in scenarios where the most important slow degrees of freedom in the configurational space are, in large part, orthogonal to the alchemical variable, or if the system gets trapped in a deep basin extending in both the configurational and alchemical space. To alleviate these issues, we propose to use alchemical variables as an additional dimension in metadynamics, making it possible to both sample collective variables and to enhance sampling in free energy calculations simultaneously. In this study, we validate our implementation of "alchemical metadynamics" in PLUMED with test systems and alchemical processes with varying complexities and dimensionalities of collective variable space, including the interconversion between the torsional metastable states of a toy system and the methylation of a nucleoside both in the isolated form and in a duplex. We show that multidimensional alchemical metadynamics can address the challenges mentioned above and further accelerate sampling by introducing configurational collective variables. The method can trivially be combined with other metadynamics-based algorithms implemented in PLUMED. The necessary PLUMED code changes have already been released for general use in PLUMED 2.8.

Collective Variables for Conformational Polymorphism in Molecular Crystals

Controlling polymorphism in molecular crystals is crucial in the pharmaceutical, dye, and pesticide industries. However, its theoretical description is extremely challenging, due to the associated long time scales (>1 μs). We present an efficient procedure for identifying collective variables that promote transitions between conformational polymorphs in molecular dynamics simulations. It involves applying a simple dimensionality reduction algorithm to data from short (∼ps) simulations of the isolated conformers that correspond to each polymorph. We demonstrate the utility of our method in the challenging case of the important energetic material, CL-20, which has three anhydrous conformational polymorphs at ambient pressure. Using these collective variables in Metadynamics simulations, we observe transitions between all solid polymorphs in the biased trajectories. We reconstruct the free energy surface and identify previously unknown defect and intermediate forms in the transition from one known polymorph to another. Our method provides insights into complex conformational polymorphic transitions of flexible molecular crystals.

Predicting efficacy of drug-carrier nanoparticle designs for cancer treatment: a machine learning-based solution

Molecular Dynamic (MD) simulations are very effective in the discovery of nanomedicines for treating cancer, but these are computationally expensive and time-consuming. Existing studies integrating machine learning (ML) into MD simulation to enhance the process and enable efficient analysis cannot provide direct insights without the complete simulation. In this study, we present an ML-based approach for predicting the solvent accessible surface area (SASA) of a nanoparticle (NP), denoting its efficacy, from a fraction of the MD simulations data. The proposed framework uses a time series model for simulating the MD, resulting in an intermediate state, and a second model to calculate the SASA in that state. Empirically, the solution can predict the SASA value 260 timesteps ahead 7.5 times faster with a very low average error of 1956.93. We also introduce the use of an explainability technique to validate the predictions. This work can reduce the computational expense of both processing and data size greatly while providing reliable solutions for the nanomedicine design process.

Collective Variables for Crystallization Simulations-from Early Developments to Recent Advances

Crystallization is an important physicochemical process which has relevance in material science, biology, and the environment. Decades of experimental and theoretical efforts have been made to understand this fundamental symmetry-breaking transition. While experiments provide equilibrium structures and shapes of crystals, they are limited to unraveling how molecules aggregate to form crystal nuclei that subsequently transform into bulk crystals. Computer simulations, mainly molecular dynamics (MD), can provide such microscopic details during the early stage of a crystallization event. Crystallization is a rare event that takes place in time scales much longer than a typical equilibrium MD simulation can sample. This inadequate sampling of the MD method can be easily circumvented by the use of enhanced sampling (ES) simulations. In most of the ES methods, the fluctuations of a system's slow degrees of freedom, called collective variables (CVs), are enhanced by applying a bias potential. This transforms the system from one state to the other within a short time scale. The most crucial part of such CV-based ES methods is to find suitable CVs, which often needs intuition and several trial-and-error optimization steps. Over the years, a plethora of CVs has been developed and applied in the study of crystallization. In this review, we provide a brief overview of CVs that have been developed and used in ES simulations to study crystallization from melt or solution. These CVs can be categorized mainly into four types: (i) spherical particle-based, (ii) molecular template-based, (iii) physical property-based, and (iv) CVs obtained from dimensionality reduction techniques. We present the context-based evolution of CVs, discuss the current challenges, and propose future directions to further develop effective CVs for the study of crystallization of complex systems.

Chasing collective variables using temporal data-driven strategies

The convergence of free-energy calculations based on importance sampling depends heavily on the choice of collective variables (CVs), which in principle, should include the slow degrees of freedom of the biological processes to be investigated. Autoencoders (AEs), as emerging data-driven dimension reduction tools, have been utilised for discovering CVs. AEs, however, are often treated as black boxes, and what AEs actually encode during training, and whether the latent variables from encoders are suitable as CVs for further free-energy calculations remains unknown. In this contribution, we review AEs and their time-series-based variants, including time-lagged AEs (TAEs) and modified TAEs, as well as the closely related model variational approach for Markov processes networks (VAMPnets). We then show through numerical examples that AEs learn the high-variance modes instead of the slow modes. In stark contrast, time series-based models are able to capture the slow modes. Moreover, both modified TAEs with extensions from slow feature analysis and the state-free reversible VAMPnets (SRVs) can yield orthogonal multidimensional CVs. As an illustration, we employ SRVs to discover the CVs of the isomerizations of N-acetyl-N'-methylalanylamide and trialanine by iterative learning with trajectories from biased simulations. Last, through numerical experiments with anisotropic diffusion, we investigate the potential relationship of time-series-based models and committor probabilities.

Predictive Molecular Models for Charged Materials Systems: From Energy Materials to Biomacromolecules

Electrostatic interactions play a dominant role in charged materials systems. Understanding the complex correlation between macroscopic properties with microscopic structures is of critical importance to develop rational design strategies for advanced materials. But the complexity of this challenging task is augmented by interfaces present in the charged materials systems, such as electrode-electrolyte interfaces or biological membranes. Over the last decades, predictive molecular simulations that are founded in fundamental physics and optimized for charged interfacial systems have proven their value in providing molecular understanding of physicochemical properties and functional mechanisms for diverse materials. Novel design strategies utilizing predictive models have been suggested as promising route for the rational design of materials with tailored properties. Here, an overview of recent advances in the understanding of charged interfacial systems aided by predictive molecular simulations is presented. Focusing on three types of charged interfaces found in energy materials and biomacromolecules, how the molecular models characterize ion structure, charge transport, morphology relation to the environment, and the thermodynamics/kinetics of molecular binding at the interfaces is discussed. The critical analysis brings two prominent field of energy materials and biological science under common perspective, to stimulate crossover in both research field that have been largely separated.

Force-Field-Dependent DNA Breathing Dynamics: A Case Study of Hoogsteen Base Pairing in A6-DNA

The Hoogsteen (HG) base pairing conformation, commonly observed in damaged and mutated DNA helices, facilitates DNA repair and DNA recognition. The free energy difference between HG and Watson-Crick (WC) base pairs has been computed in previous studies. However, the mechanism of the conformational transition is not well understood. A detailed understanding of the process of WC to HG base pair transition can provide a deeper understanding of DNA repair and recognition. In an earlier study, we explored the free energy landscape for this process using extensive computer simulation with the CHARMM36 force field. In this work, we study the impact of force field models in describing the WC to HG base pairing transition using meta-eABF enhanced sampling, quasi-harmonic entropy calculation, and nonbonded energy analysis. The secondary structures of both base pairing forms and the topology of the free energy landscapes were consistent over different force field models, although the relative free energy, entropy, and the interaction energies tend to vary. The relative stability of the WC and HG conformations is dictated by a delicate balance between the enthalpic stabilization and the reduced entropy of the structurally rigid HG structure. These findings highlight the impact that subtleties in force field models can have on accurately modeling DNA base pair dynamics and should stimulate further computational investigations into other dynamically important motions in DNA.

Reweighted Manifold Learning of Collective Variables from Enhanced Sampling Simulations

Enhanced sampling methods are indispensable in computational chemistry and physics, where atomistic simulations cannot exhaustively sample the high-dimensional configuration space of dynamical systems due to the sampling problem. A class of such enhanced sampling methods works by identifying a few slow degrees of freedom, termed collective variables (CVs), and enhancing the sampling along these CVs. Selecting CVs to analyze and drive the sampling is not trivial and often relies on chemical intuition. Despite routinely circumventing this issue using manifold learning to estimate CVs directly from standard simulations, such methods cannot provide mappings to a low-dimensional manifold from enhanced sampling simulations, as the geometry and density of the learned manifold are biased. Here, we address this crucial issue and provide a general reweighting framework based on anisotropic diffusion maps for manifold learning that takes into account that the learning data set is sampled from a biased probability distribution. We consider manifold learning methods based on constructing a Markov chain describing transition probabilities between high-dimensional samples. We show that our framework reverts the biasing effect, yielding CVs that correctly describe the equilibrium density. This advancement enables the construction of low-dimensional CVs using manifold learning directly from the data generated by enhanced sampling simulations. We call our framework reweighted manifold learning. We show that it can be used in many manifold learning techniques on data from both standard and enhanced sampling simulations.

Exact reaction coordinates for flap opening in HIV-1 protease

The primary goal of protein science is to understand how proteins function, which requires understanding the functional dynamics responsible for transitions between different functional structures of a protein. A central concept is the exact reaction coordinates that can determine the value of committor for any protein configuration, which provide the optimal description of functional dynamics. Despite intensive efforts, identifying the exact reaction coordinates (RCs) in complex molecules remains a formidable challenge. Using the recently developed generalized work functional, we report the discovery of the exact RCs for an important functional process-the flap opening of HIV-1 protease. Our results show that this process has six RCs, each one is a linear combination of ~240 backbone dihedrals, providing the precise definition of collectivity and cooperativity in the functional dynamics of a protein. Applying bias potentials along each RC can accelerate flap opening by [Formula: see text] to [Formula: see text] folds. The success in identifying the RCs of a protein with 198 residues represents a significant progress beyond that of the alanine dipeptide, currently the only other complex molecule for which the exact RCs for its conformational changes are known. Our results suggest that the generalized work functional (GWF) might be the fundamental operator of mechanics that controls protein dynamics.

Markov field models: Scaling molecular kinetics approaches to large molecular machines

With recent advances in structural biology, including experimental techniques and deep learning-enabled high-precision structure predictions, molecular dynamics methods that scale up to large biomolecular systems are required. Current state-of-the-art approaches in molecular dynamics modeling focus on encoding global configurations of molecular systems as distinct states. This paradigm commands us to map out all possible structures and sample transitions between them, a task that becomes impossible for large-scale systems such as biomolecular complexes. To arrive at scalable molecular models, we suggest moving away from global state descriptions to a set of coupled models that each describe the dynamics of local domains or sites of the molecular system. We describe limitations in the current state-of-the-art global-state Markovian modeling approaches and then introduce Markov field models as an umbrella term that includes models from various scientific communities, including Independent Markov decomposition, Ising and Potts models, and (dynamic) graphical models, and evaluate their use for computational molecular biology. Finally, we give a few examples of early adoptions of these ideas for modeling molecular kinetics and thermodynamics.

Extending and validating bubble nucleation rate predictions in a Lennard-Jones fluid with enhanced sampling methods and transition state theory

We calculate bubble nucleation rates in a Lennard-Jones fluid through explicit molecular dynamics simulations. Our approach—based on a recent free energy method (dubbed reweighted Jarzynski sampling), transition state theory, and a simple recrossing correction—allows us to probe a fairly wide range of rates in several superheated and cavitation regimes in a consistent manner. Rate predictions from this approach bridge disparate independent literature studies on the same model system. As such, we find that rate predictions based on classical nucleation theory, direct brute force molecular dynamics simulations, and seeding are consistent with our approach and one another. Published rates derived from forward flux sampling simulations are, however, found to be outliers. This study serves two purposes: First, we validate the reliability of common modeling techniques and extrapolation approaches on a paradigmatic problem in materials science and chemical physics. Second, we further test our highly generic recipe for rate calculations, and establish its applicability to nucleation processes.

Committor-Consistent Variational String Method

The treatment of slow and rare transitions in the simulation of complex systems poses a great computational challenge. A powerful approach to tackle this challenge is the string method, which represents the transition path as a one-dimensional curve in a multidimensional space of collective variables. Commonly used strategies for pathway optimization include aligning the tangent of the string to the local mean force or to the mean drift determined from swarms of short trajectories. Here, a novel strategy is proposed, allowing the string to be optimized based on a variational principle involving the unidirectional reactive flux expressed in terms of the time-correlation function of the committor. The method is illustrated with model systems and then probed with the alanine dipeptide and a coarse-grained model of the barstar-barnase protein complex. Successive iterations variationally refine the string toward an optimal transition pathway following the gradient of the committor between two metastable states.

LINES: Log-Probability Estimation via Invertible Neural Networks for Enhanced Sampling

It is very challenging to sample a molecular process with large activation energies using molecular dynamics simulations. Current enhanced sampling methodologies, such as umbrella sampling and metadynamics, rely on the identification of appropriate reaction coordinates for a system. In this paper, we developed a method for log-probability estimation via invertible neural networks for enhanced sampling (LINES). This iterative scheme utilizes a normalizing flow machine learning model to learn the underlying free energy surface (FES) of a system as a function of molecular coordinates and then applies a gradient-based optimization method to the learned normalizing flow to identify reaction coordinates. A biasing potential is then evaluated over a tabulated grid of the reaction coordinate values, which can be applied to the next round of simulations for enhanced sampling, resulting in more efficient sampling. We tested the accuracy and efficiency of the LINES method in sampling the FES using the alanine dipeptide system. We also demonstrated the effectiveness of identification of reaction coordinates through simulation of cyclobutanol unbinding from β-cyclodextrin and the folding/unfolding of CLN025─a variant of the peptide Chignolin. The LINES method can be extended to the study of large-scale protein systems with complex nonlinear reaction pathways.

Collective variable discovery in the age of machine learning: reality, hype and everything in between

Understanding the kinetics and thermodynamics profile of biomolecules is necessary to understand their functional roles which has a major impact in mechanism driven drug discovery. Molecular dynamics simulation has been routinely used to understand conformational dynamics and molecular recognition in biomolecules. Statistical analysis of high-dimensional spatiotemporal data generated from molecular dynamics simulation requires identification of a few low-dimensional variables which can describe the essential dynamics of a system without significant loss of information. In physical chemistry, these low-dimensional variables are often called collective variables. Collective variables are used to generate reduced representations of free energy surfaces and calculate transition probabilities between different metastable basins. However the choice of collective variables is not trivial for complex systems. Collective variables range from geometric criteria such as distances and dihedral angles to abstract ones such as weighted linear combinations of multiple geometric variables. The advent of machine learning algorithms led to increasing use of abstract collective variables to represent biomolecular dynamics. In this review, I will highlight several nuances of commonly used collective variables ranging from geometric to abstract ones. Further, I will put forward some cases where machine learning based collective variables were used to describe simple systems which in principle could have been described by geometric ones. Finally, I will put forward my thoughts on artificial general intelligence and how it can be used to discover and predict collective variables from spatiotemporal data generated by molecular dynamics simulations.

Free Energy Landscapes, Diffusion Coefficients, and Kinetic Rates from Transition Paths

We address the problem of constructing accurate mathematical models of the dynamics of complex systems projected on a collective variable. To this aim we introduce a conceptually simple yet effective algorithm for estimating the parameters of Langevin and Fokker-Planck equations from a set of short, possibly out-of-equilibrium molecular dynamics trajectories, obtained for instance from transition path sampling or as relaxation from high free-energy configurations. The approach maximizes the model likelihood based on any explicit expression of the short-time propagator, hence it can be applied to different evolution equations. We demonstrate the numerical efficiency and robustness of the algorithm on model systems, and we apply it to reconstruct the projected dynamics of pairs of C₆₀ and C₂₄₀ fullerene molecules in explicit water. Our methodology allows reconstructing the accurate thermodynamics and kinetics of activated processes, namely free energy landscapes, diffusion coefficients, and kinetic rates. Compared to existing enhanced sampling methods, we directly exploit short unbiased trajectories at a competitive computational cost.

Protein Flexibility and Dissociation Pathway Differentiation Can Explain Onset of Resistance Mutations in Kinases

Understanding how mutations render a drug ineffective is a problem of immense relevance. Often the mechanism through which mutations cause drug resistance can be explained purely through thermodynamics. However, the more perplexing situation is when two proteins have the same drug binding affinities but different residence times. In this work, we demonstrate how all-atom molecular dynamics simulations using recent developments grounded in statistical mechanics can provide a detailed mechanistic rationale for such variances. We discover dissociation mechanisms for the anti-cancer drug Imatinib (Gleevec) against wild-type and the N368S mutant of Abl kinase. We show how this point mutation triggers far-reaching changes in the protein's flexibility and leads to a different, much faster, drug dissociation pathway. We believe that this work marks an efficient and scalable approach to obtain mechanistic insight into resistance mutations in biomolecular receptors that are hard to explain using a structural perspective.

Thermal Fluctuations of Singular Bar-Joint Mechanisms

A bar-joint mechanism is a deformable assembly of freely rotating joints connected by stiff bars. Here we develop a formalism to study the equilibration of common bar-joint mechanisms with a thermal bath. When the constraints in a mechanism cease to be linearly independent, singularities can appear in its shape space, which is the part of its configuration space after discarding rigid motions. We show that the free-energy landscape of a mechanism at low temperatures is dominated by the neighborhoods of points that correspond to these singularities. We consider two example mechanisms with shape-space singularities and find that they are more likely to be found in configurations near the singularities than others. These findings are expected to help improve the design of nanomechanisms for various applications.

Multidimensional minimum-work control of a 2D Ising model

A system's configurational state can be manipulated using dynamic variation of control parameters, such as temperature, pressure, or magnetic field; for finite-duration driving, excess work is required above the equilibrium free-energy change. Minimum-work protocols in multidimensional control-parameter space have potential to significantly reduce work relative to one-dimensional control. By numerically minimizing a linear-response approximation to the excess work, we design protocols in control-parameter spaces of a 2D Ising model that efficiently drive the system from the all-down to all-up configuration. We find that such designed multidimensional protocols take advantage of more flexible control to avoid control-parameter regions of high system resistance, heterogeneously input and extract work to make use of system relaxation, and flatten the energy landscape, making accessible many configurations that would otherwise have prohibitively high energy and thus decreasing spin correlations. Relative to one-dimensional protocols, this speeds up the rate-limiting spin-inversion reaction, thereby keeping the system significantly closer to equilibrium for a wide range of protocol durations, and significantly reducing resistance and hence work.

Local Ion Densities can Influence Transition Paths of Molecular Binding

Improper reaction coordinates can pose significant problems for path-based binding free energy calculations. Particularly, omission of long timescale motions can lead to over-estimation of the energetic barriers between the bound and unbound states. Many methods exist to construct the optimal reaction coordinate using a pre-defined basis set of features. Although simulations are typically conducted in explicit solvent, the solvent atoms are often excluded by these feature sets—resulting in little being known about their role in reaction coordinates, and ultimately, their role in determining (un)binding rates and free energies. In this work, analysis is done on an extensive set of host-guest unbinding trajectories, working to characterize differences between high and low probability unbinding trajectories with a focus on solvent-based features, including host-ion interactions, guest-ion interactions and location-dependent ion densities. We find that differences in ion densities as well as guest-ion interactions strongly correlate with differences in the probabilities of reactive paths that are used to determine free energies of (un)binding and play a significant role in the unbinding process.

Solvent's Role in Cavity-Ligand Recognition Would Depend on the Mode of Ligand Diffusion

Solvent is known to play crucial roles in dictating the thermodynamics and kinetics of the biomolecular recognition process. Here, we show that the extent of significance of water in modulating the ligand recognition process is critically contingent on the ligand diffusion and on the constraints introduced on it. Toward the end, we use a well-known prototypical system of spherical ligand diffusing freely toward a hydrophobic concave cavity in explicit water. We analyze a large series of adaptively sampled unbiased molecular dynamics simulation trajectories within the framework of time-structured independent component analysis (TICA). Our quantitative investigations reveal that water would play a significant role in the ligand recognition process, provided that the ligand is constricted to diffuse along a centro-symmetric fashion. On the contrary, water's contribution in the ligand recognition process would diminish to a negligible value if the ligand freely diffuses toward the pocket. A Markov state model (MSM) constructed using the simulated trajectories identifies a set of transiently populated metastable states comprising partially ligand-unbound macro states, alongside ligand-bound and ligand-unbound pose and gives rise to multiple transition paths of ligand in its way to the hydrophobic cavity. Lifting the restriction on ligand movement changes its binding pathway, time scales, and the extent of the role of solvent in modulating the recognition process.

Transition rate theory, spectral analysis, and reactive paths

The kinetics of a dynamical system dominated by two metastable states is examined from the perspective of the activated-dynamics reactive flux formalism, Markov state eigenvalue spectral decomposition, and committor-based transition path theory. Analysis shows that the different theoretical formulations are consistent, clarifying the significance of the inherent microscopic lag-times that are implicated, and that the most meaningful one-dimensional reaction coordinate in the region of the transition state is along the gradient of the committor in the multidimensional subspace of collective variables. It is shown that the familiar reactive flux activated dynamics formalism provides an effective route to calculate the transition rate in the case of a narrow sharp barrier but much less so in the case of a broad flat barrier. In this case, the standard reactive flux correlation function decays very slowly to the plateau value that corresponds to the transmission coefficient. Treating the committor function as a reaction coordinate does not alleviate all issues caused by the slow relaxation of the reactive flux correlation function. A more efficient activated dynamics simulation algorithm may be achieved from a modified reactive flux weighted by the committor. Simulation results on simple systems are used to illustrate the various conceptual points.