Improving collective variables: The case of crystallization

tICA-Metadynamics for Identifying Slow Dynamics in Membrane Permeation

Molecular simulations are commonly used to understand the mechanism of membrane permeation of small molecules, particularly for biomedical and pharmaceutical applications. However, despite significant advances in computing power and algorithms, calculating an accurate permeation free energy profile remains elusive for many drug molecules because it can require identifying the rate-limiting degrees of freedom (i.e., appropriate reaction coordinates). To resolve this issue, researchers have developed machine learning approaches to identify slow system dynamics. In this work, we apply time-lagged independent component analysis (tICA), an unsupervised dimensionality reduction algorithm, to molecular dynamics simulations with well-tempered metadynamics to find the slowest collective degrees of freedom of the permeation process of trimethoprim through a multicomponent membrane. We show that tICA-metadynamics yields translational and orientational collective variables (CVs) that increase convergence efficiency ∼1.5 times. However, crossing the periodic boundary is shown to introduce artifacts in the translational CV that can be corrected by taking absolute values of molecular features. Additionally, we find that the convergence of the tICA CVs is reached with approximately five membrane crossings and that data reweighting is required to avoid deviations in the translational CV.

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

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

tICA-Metadynamics for Identifying Slow Dynamics in Membrane Permeation

Molecular simulations are commonly used to understand the mechanism of membrane permeation of small molecules, particularly for biomedical and pharmaceutical applications. However, despite significant advances in computing power and algorithms, calculating an accurate permeation free energy profile remains elusive for many drug molecules because it can require identifying the rate-limiting degrees of freedom (i.e., appropriate reaction coordinates). To resolve this issue, researchers have developed machine learning approaches to identify slow system dynamics. In this work, we apply time-lagged independent component analysis (tICA), an unsupervised dimensionality reduction algorithm, to molecular dynamics simulations with well-tempered metadynamics to find the slowest collective degrees of freedom of the permeation process of trimethoprim through a multicomponent membrane. We show that tICA-metadynamics yields translational and orientational collective variables (CVs) that increase convergence efficiency ∼1.5 times. However, crossing the periodic boundary is shown to introduce artefacts in the translational CV that can be corrected by taking absolute values of molecular features. Additionally, we find that the convergence of the tICA CVs is reached with approximately five membrane crossings, and that data reweighting is required to avoid deviations in the translational CV.

Anisotropic Collective Variables with Machine Learning Potential for Ab Initio Crystallization of Complex Ceramics

Enhanced sampling molecular dynamics (MD) simulations have been extensively used in the phase transition study of simple crystalline materials, such as aluminum, silica, and ice. However, MD simulation of the crystallization process for complex crystalline materials still faces a formidable challenge due to their multicomponent induced multiphase problem. Here, we realize the ab initio accuracy MD crystallization simulations of complex ceramics by using anisotropic collective variables (CVs) and machine learning (ML) potential. The anisotropic X-ray diffraction intensity CVs provide precise identification of complex crystal structures with detailed crystallography information, while the ML potential makes it feasible to further perform enhanced sampling simulations with ab initio accuracy. We verify the universality and accuracy of this method through complex ceramics with three kinds of representative structures, i.e., Ti3SiC2 for the MAX structure, zircon for the mineral structure, and lead zirconate titanate for the perovskite structure. It demonstrates exceptional efficiency and ab initio quality in achieving crystallization and generating free energy surfaces of all these ceramics, facilitating the analysis and design of complex crystalline materials.

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.

An exploration of machine learning models for the determination of reaction coordinates associated with conformational transitions

Determining collective variables (CVs) for conformational transitions is crucial to understanding their dynamics and targeting them in enhanced sampling simulations. Often, CVs are proposed based on intuition or prior knowledge of a system. However, the problem of systematically determining a proper reaction coordinate (RC) for a specific process in terms of a set of putative CVs can be achieved using committor analysis (CA). Identifying essential degrees of freedom that govern such transitions using CA remains elusive because of the high dimensionality of the conformational space. Various schemes exist to leverage the power of machine learning (ML) to extract an RC from CA. Here, we extend these studies and compare the ability of 17 different ML schemes to identify accurate RCs associated with conformational transitions. We tested these methods on an alanine dipeptide in vacuum and on a sarcosine dipeptoid in an implicit solvent. Our comparison revealed that the light gradient boosting machine method outperforms other methods. In order to extract key features from the models, we employed Shapley Additive exPlanations analysis and compared its interpretation with the "feature importance" approach. For the alanine dipeptide, our methodology identifies ϕ and θ dihedrals as essential degrees of freedom in the C7ax to C7eq transition. For the sarcosine dipeptoid system, the dihedrals ψ and ω are the most important for the cisαD to transαD transition. We further argue that analysis of the full dynamical pathway, and not just endpoint states, is essential for identifying key degrees of freedom governing transitions.

Systematic modification of functionality in disordered elastic networks through free energy surface tailoring

A combined machine learning-physics-based approach is explored for molecular and materials engineering. Specifically, collective variables, akin to those used in enhanced sampled simulations, are constructed using a machine learning model trained on data gathered from a single system. Through the constructed collective variables, it becomes possible to identify critical molecular interactions in the considered system, the modulation of which enables a systematic tailoring of the system's free energy landscape. To explore the efficacy of the proposed approach, we use it to engineer allosteric regulation and uniaxial strain fluctuations in a complex disordered elastic network. Its successful application in these two cases provides insights regarding how functionality is governed in systems characterized by extensive connectivity and points to its potential for design of complex molecular systems.

Intrinsic Disorder in α-Synuclein Regulates the Exocytotic Fusion Pore Transition

Today, it is widely accepted that intrinsic disorder is strongly related to the cell cycle, during mitosis, differentiation, and apoptosis. Of particular interest are hybrid proteins possessing both structured and unstructured domains that are critical in human health and disease, such as α-synuclein. In this work, we describe how α-synuclein interacts with the nascent fusion pore as it evolves toward expansion. We unveil the key role played by its intrinsically disordered region as a thermodynamic regulator of the nucleation-expansion energy barrier. By analyzing a truncated variant of α-synuclein that lacks the disordered region, we find that the landscape of protein interactions with PIP₂ and POPS lipids is highly altered, ultimately increasing the energy cost for the fusion pore to transit from nucleation to expansion. We conclude that the intrinsically disordered region in full-length α-synuclein recognizes and allocates pivotal protein:lipid interactions during membrane remodeling in the first stages of the fusion pore.

A variational approach to assess reaction coordinates for two-step crystallization

Molecule- and particle-based simulations provide the tools to test, in microscopic detail, the validity of classical nucleation theory. In this endeavor, determining nucleation mechanisms and rates for phase separation requires an appropriately defined reaction coordinate to describe the transformation of an out-of-equilibrium parent phase for which myriad options are available to the simulator. In this article, we describe the application of the variational approach to Markov processes to quantify the suitability of reaction coordinates to study crystallization from supersaturated colloid suspensions. Our analysis indicates that collective variables (CVs) that correlate with the number of particles in the condensed phase, the system potential energy, and approximate configurational entropy often feature as the most appropriate order parameters to quantitatively describe the crystallization process. We apply time-lagged independent component analysis to reduce high-dimensional reaction coordinates constructed from these CVs to build Markov State Models (MSMs), which indicate that two barriers separate a supersaturated fluid phase from crystals in the simulated environment. The MSMs provide consistent estimates for crystal nucleation rates, regardless of the dimensionality of the order parameter space adopted; however, the two-step mechanism is only consistently evident from spectral clustering of the MSMs in higher dimensions. As the method is general and easily transferable, the variational approach we adopt could provide a useful framework to study controls for crystal nucleation.

Evidence of Multiple Crystallization Pathways in Lithium Disilicate: A Metadynamics Investigation

Metadynamics simulations driven by using two X-ray diffraction peaks identified three alternative crystallization pathways of the lithium disilicate crystal from the melt. The most favorable one passes through the formation of disordered layered structures undergoing internal ordering in a second step. The second pathway involves the formation of phase-separated structures composed of nuclei of β-cristobalite crystals surrounded by lithium-rich phases in which metasilicate chains are formed. The conversion of these structures to the stable lithium disilicate crystal involves an intermediate structure whose silicate layers are connected by silicate rings with the energy barrier of 2.5 kJ/mol per formula unit (f.u.). The third pathway is highly unlikely because of the huge energy barrier involved (20 kJ/mol per f.u.). This path also involves the passage through a phase-separated structure of an indefinite silica region surrounded mainly by amorphous lithium oxide.

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.

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.

DeepCV: A Deep Learning Framework for Blind Search of Collective Variables in Expanded Configurational Space

We present Deep learning for Collective Variables (DeepCV), a computer code that provides an efficient and customizable implementation of the deep autoencoder neural network (DAENN) algorithm that has been developed in our group for computing collective variables (CVs) and can be used with enhanced sampling methods to reconstruct free energy surfaces of chemical reactions. DeepCV can be used to conveniently calculate molecular features, train models, generate CVs, validate rare events from sampling, and analyze a trajectory for chemical reactions of interest. We use DeepCV in an example study of the conformational transition of cyclohexene, where metadynamics simulations are performed using DAENN-generated CVs. The results show that the adopted CVs give free energies in line with those obtained by previously developed CVs and experimental results. DeepCV is open-source software written in Python/C++ object-oriented languages, based on the TensorFlow framework and distributed free of charge for noncommercial purposes, which can be incorporated into general molecular dynamics software. DeepCV also comes with several additional tools, i.e., an application program interface (API), documentation, and tutorials.

Computer simulation of the early stages of self-assembly and thermal decomposition of ZIF-8

We employ all-atom well-tempered metadynamics simulations to study the mechanistic details of both the early stages of nucleation and crystal decomposition for the benchmark metal–organic framework (MOF) ZIF-8. To do so, we developed and validated a force field that reliably models the modes of coordination bonds via a Morse potential functional form and employs cationic and anionic dummy atoms to capture coordination symmetry. We also explored a set of physically relevant collective variables and carefully selected an appropriate subset for our problem at hand. After a rapid increase of the Zn–N connectivity, we observe the evaporation of small clusters in favor of a few large clusters, which leads to the formation of an amorphous highly connected aggregate. [Formula: see text] and [Formula: see text] complexes are observed with lifetimes in the order of a few picoseconds, while larger structures, such as four-, five-, and six-membered rings, have substantially longer lifetimes of a few nanoseconds. The free ligands act as “templating agents” for the formation of sodalite cages. ZIF-8 crystal decomposition results in the formation of a vitreous phase. Our findings contribute to a fundamental understanding of MOF’s synthesis that paves the way to controlling synthesis products. Furthermore, our developed force field and methodology can be applied to model solution processes that require coordination bond reactivity for other ZIFs besides ZIF-8.

Enhanced Sampling Simulation Reveals How Solvent Influences Chirogenesis of the Intra-Molecular Diels-Alder Reaction

The timescale involved in chemical reactions is quite often beyond that of normal molecular dynamics simulations. Here, we combine metadynamics with selective integrated tempering sampling to simulate an intra-molecular Diels-Alder reaction in explicit solvents. Based on a one-dimensional collective variable obtained from harmonic linear discriminant analysis, four chiral isomers of products were observed in the simulation. Analyses of reactive trajectories showed that this reaction follows a concerted mechanism in all four solvents. In addition, the hydrogen bond between the reactant and water solvent plays an important role in the water-accelerated reaction mechanism. The dynamics of chirality formation varies significantly with solvents. The chirality of products forms significantly before the transition state, especially in ionic liquid.

Stable Binding Conformations of Polymaleic and Polyacrylic Acids at a Calcite Surface in the Presence of Countercations: A Metadynamics Study

Elucidating the stable binding conformations of additives at the surface of CaCO3 crystals is essential to biomineralization, scale inhibition, and materials technology. However, accomplishing this by experimental means is rather difficult. In this study, molecular dynamics simulations based on a metadynamics approach were conducted to elucidate the stable binding conformations of a deprotonated polymaleic acid (PMA) additive and two deprotonated poly(acrylic acid) (PAA) additives with different polymerization degrees in the presence of various countercations at a hydrated calcite (104) surface. The simulated free-energy surfaces suggested the existence of several slightly different stable binding conformations for each additive. The appearance of these distinct binding conformations is speculated to originate from different balances of interactions between the additive, the calcite surface, and the countercations. The binding conformations and binding stabilities at the calcite surface were affected by the countercations, with Ca2+ ions producing a more pronounced effect than Na+ ions. Furthermore, the simulation results suggested that the binding stability at the calcite surface was higher for the PMA additive than for the PAA additives, and the PAA additive with a polymerization degree of 10 displayed a binding stability that was similar to or lower than that of the PAA additive with a polymerization degree of 5. The present simulation method provides a new strategy for analyzing the binding conformations of complex additives at material surfaces, developing additives that stably bind to these surfaces, and designing additives to control crystal growth.

Collective Variables for Free Energy Surface Tailoring: Understanding and Modifying Functionality in Systems Dominated by Rare Events

We introduce a method for elucidating and modifying the functionality of systems dominated by rare events that relies on the semiautomated tuning of their underlying free energy surface. The proposed approach seeks to construct collective variables (CVs) that encode the essential information regarding the rare events of the system of interest. The appropriate CVs are identified using harmonic linear discriminant analysis (HLDA), a machine-learning-based method that is trained solely on data collected from short ordinary simulations in the relevant metastable states of the system. Utilizing the interpretable form of the resulting CVs, the critical interaction potentials that determine the system's rare transitions are identified and purposely modified to tailor the free energy surface in a manner that alters functionality as desired. The applicability of the method is illustrated in the context of three different systems, thereby demonstrating that thermodynamic and kinetic properties can be tractably modified with little to no prior knowledge or intuition.

Deep learning the slow modes for rare events sampling

The development of enhanced sampling methods has greatly extended the scope of atomistic simulations, allowing long-time phenomena to be studied with accessible computational resources. Many such methods rely on the identification of an appropriate set of collective variables. These are meant to describe the system's modes that most slowly approach equilibrium under the action of the sampling algorithm. Once identified, the equilibration of these modes is accelerated by the enhanced sampling method of choice. An attractive way of determining the collective variables is to relate them to the eigenfunctions and eigenvalues of the transfer operator. Unfortunately, this requires knowing the long-term dynamics of the system beforehand, which is generally not available. However, we have recently shown that it is indeed possible to determine efficient collective variables starting from biased simulations. In this paper, we bring the power of machine learning and the efficiency of the recently developed on the fly probability-enhanced sampling method to bear on this approach. The result is a powerful and robust algorithm that, given an initial enhanced sampling simulation performed with trial collective variables or generalized ensembles, extracts transfer operator eigenfunctions using a neural network ansatz and then accelerates them to promote sampling of rare events. To illustrate the generality of this approach, we apply it to several systems, ranging from the conformational transition of a small molecule to the folding of a miniprotein and the study of materials crystallization.

Nucleation rates from small scale atomistic simulations and transition state theory

The evaluation of nucleation rates from molecular dynamics trajectories is hampered by the slow nucleation time scale and impact of finite size effects. Here, we show that accurate nucleation rates can be obtained in a very general fashion relying only on the free energy barrier, transition state theory, and a simple dynamical correction for diffusive recrossing. In this setup, the time scale problem is overcome by using enhanced sampling methods, in casu metadynamics, whereas the impact of finite size effects can be naturally circumvented by reconstructing the free energy surface from an appropriate ensemble. Approximations from classical nucleation theory are avoided. We demonstrate the accuracy of the approach by calculating macroscopic rates of droplet nucleation from argon vapor, spanning 16 orders of magnitude and in excellent agreement with literature results, all from simulations of very small (512 atom) systems.

Improving the accuracy and convergence of drug permeation simulations via machine-learned collective variables

Understanding the permeation of biomolecules through cellular membranes is critical for many biotechnological applications, including targeted drug delivery, pathogen detection, and the development of new antibiotics. To this end, computer simulations are routinely used to probe the underlying mechanisms of membrane permeation. Despite great progress and continued development, permeation simulations of realistic systems (e.g., more complex drug molecules or biologics through heterogeneous membranes) remain extremely challenging if not intractable. In this work, we combine molecular dynamics simulations with transition-tempered metadynamics and techniques from the variational approach to conformational dynamics to study the permeation mechanism of a drug molecule, trimethoprim, through a multicomponent membrane. We show that collective variables (CVs) obtained from an unsupervised machine learning algorithm called time-structure based Independent Component Analysis (tICA) improve performance and substantially accelerate convergence of permeation potential of mean force (PMF) calculations. The addition of cholesterol to the lipid bilayer is shown to increase both the width and height of the free energy barrier due to a condensing effect (lower area per lipid) and increase bilayer thickness. Additionally, the tICA CVs reveal a subtle effect of cholesterol increasing the resistance to permeation in the lipid head group region, which is not observed when canonical CVs are used. We conclude that the use of tICA CVs can enable more efficient PMF calculations with additional insight into the permeation mechanism.

Efficient approaches to solutions of partition function for condensed matters

The key problem of statistical physics standing over one hundred years is how to exactly calculate the partition function (or free energy), which severely hinders the theory to be applied to predict the thermodynamic properties of condensed matters. Very recently, we developed a direct integral approach (DIA) to the solutions and achieved ultrahigh computational efficiency and precision. In the present work, the background and the limitations of DIA were examined in details, and another method with the same efficiency was established to overcome the shortage of DIA for condensed system with lower density. The two methods were demonstrated with empirical potentials for solid and liquid cooper, solid argon and C₆₀ molecules by comparing the derived internal energy or pressure with the results of vast molecular dynamics simulations, showing that the precision is about ten times higher than previous methods in a temperature range up to melting point. The ultrahigh efficiency enables the two methods to be performed with ab initio calculations and the experimental equation of state of solid copper up to ∼600 GPa was well reproduced, for the first time, from the partition function via density functional theory implemented.

Out-of-Equilibrium Polymorph Selection in Nanoparticle Freezing

The ability to design synthesis processes that are out of equilibrium has opened the possibility of creating nanomaterials with remarkable physicochemical properties, choosing from a much richer palette of possible atomic architectures compared to equilibrium processes in extended systems. In this work, we employ atomistic simulations to demonstrate how to control polymorph selection via the cooling rate during nanoparticle freezing in the case of Ni₃Al, a material with a rich structural landscape. State-of-the-art free-energy calculations allow us to rationalize the complex nucleation process, discovering a switch between two kinetic pathways, yielding the equilibrium structure at room temperature and an alternative metastable one at higher temperature. Our findings address the key challenge in the synthesis of nanoalloys for technological applications, i.e., rationally exploiting the competition between kinetics and thermodynamics by designing a treatment history that forces the system into desirable metastable states.

Machine Learning Force Fields and Coarse-Grained Variables in Molecular Dynamics: Application to Materials and Biological Systems

Machine learning encompasses tools and algorithms that are now becoming popular in almost all scientific and technological fields. This is true for molecular dynamics as well, where machine learning offers promises of extracting valuable information from the enormous amounts of data generated by simulation of complex systems. We provide here a review of our current understanding of goals, benefits, and limitations of machine learning techniques for computational studies on atomistic systems, focusing on the construction of empirical force fields from ab initio databases and the determination of reaction coordinates for free energy computation and enhanced sampling.

Ab initio phase diagram and nucleation of gallium

Elemental gallium possesses several intriguing properties, such as a low melting point, a density anomaly and an electronic structure in which covalent and metallic features coexist. In order to simulate this complex system, we construct an ab initio quality interaction potential by training a neural network on a set of density functional theory calculations performed on configurations generated in multithermal–multibaric simulations. Here we show that the relative equilibrium between liquid gallium, α-Ga, β-Ga, and Ga-II is well described. The resulting phase diagram is in agreement with the experimental findings. The local structure of liquid gallium and its nucleation into α-Ga and β-Ga are studied. We find that the formation of metastable β-Ga is kinetically favored over the thermodinamically stable α-Ga. Finally, we provide insight into the experimental observations of extreme undercooling of liquid Ga.

Exploring nucleation processes of gallium by molecular simulation is extremely challenging due to its structural complexity. Here the authors demonstrate a neural network potential trained on a multithermal–multibaric DFT data for the study of the phase diagram of gallium in a wide temperature and pressure range.

Data-Driven Collective Variables for Enhanced Sampling

Designing an appropriate set of collective variables is crucial to the success of several enhanced sampling methods. Here we focus on how to obtain such variables from information limited to the metastable states. We characterize these states by a large set of descriptors and employ neural networks to compress this information in a lower-dimensional space, using Fisher's linear discriminant as an objective function to maximize the discriminative power of the network. We test this method on alanine dipeptide, using the nonlinearly separable data set composed by atomic distances. We then study an intermolecular aldol reaction characterized by a concerted mechanism. The resulting variables are able to promote sampling by drawing nonlinear paths in the physical space connecting the fluctuations between metastable basins. Lastly, we interpret the behavior of the neural network by studying its relation to the physical variables. Through the identification of its most relevant features, we are able to gain chemical insight into the process.

Enhanced sampling in molecular dynamics

Although molecular dynamics simulations have become a useful tool in essentially all fields of chemistry, condensed matter physics, materials science, and biology, there is still a large gap between the time scale which can be reached in molecular dynamics simulations and that observed in experiments. To address the problem, many enhanced sampling methods were introduced, which effectively extend the time scale being approached in simulations. In this perspective, we review a variety of enhanced sampling methods. We first discuss collective-variables-based methods including metadynamics and variationally enhanced sampling. Then, collective variable free methods such as parallel tempering and integrated tempering methods are presented. At last, we conclude with a brief introduction of some newly developed combinatory methods. We summarize in this perspective not only the theoretical background and numerical implementation of these methods but also the new challenges and prospects in the field of the enhanced sampling.

Accurate Quantum Chemical Free Energies at Affordable Cost

Free energy sampling methods allow studying the full dynamics of activated processes. Unfortunately, the affordable accuracy of the potential describing the energy and forces of the system is usually rather low. Here we introduce a new method that by combining metadynamics and free energy perturbation allows calculating accurate quantum chemical free energies for chemical reactions. To prove the effectiveness of this new approach we study the S_N2 reaction of CH₃F + Cl^- → CH₃Cl + F^- in vacuo and solvated by water. Comparisons are made with harmonic transition-state theory to show how this method could provide accurate equilibrium and rate constants for complex systems.