Path finding on high-dimensional free energy landscapes

Finite Temperature String with Order Parameter as Collective Variables for Molecular Crystal: A Case of Polymorphic Transformation of TNT under External Electric Field

An external electric field is an effective tool to induce the polymorphic transformation of molecular crystals, which is important practically in the chemical, material, and energy storage industries. However, the understanding of this mechanism is poor at the molecular level. In this work, two types of order parameters (OPs) were constructed for the molecular crystal based on the intermolecular distance, bond orientation, and molecular orientation. Using the K-means clustering algorithm for the sampling of OPs based on the Euclidean distance and density weight, the polymorphic transformation of TNT was investigated using a finite temperature string (FTS) under external electric fields. The potential of mean force (PMF) was obtained, and the essence of the polymorphic transformation between o-TNT and m-TNT was revealed, which verified the effectiveness of the FTS method based on K-means clustering to OPs. The differences in PMFs between the o-TNT and transition state were decreased under external electric fields in comparison with those in no field. The fields parallel to the c-axis obviously affected the difference in PMF, and the relationship between the changes in PMFs and field strengths was found. Although the external electric field did not promote the convergence, the time of the polymorphic transformation was reduced under the external electric field in comparison to its absence. Moreover, under the external electric field, the polymorphic transformation from o-TNT to m-TNT occurred while that from m-TNT to o-TNT was prevented, which was explained by the dipole moment of molecule, relative permittivity, chemical potential difference, nucleation work and nucleation rate. This confirmed that the polymorphic transformation orientation of the molecular crystal could be controlled by the external electric field. This work provides an effective way to explore the polymorphic transformation of the molecular crystals at a molecular level, and it is useful to control the production process and improve the performance of energetic materials by using the external electric fields.

Exploring Product Release from Yeast Cytosine Deaminase with Metadynamics

The yeast cytosine deaminase (yCD) enzyme/5-fluorocytosine prodrug system is a promising candidate for targeted chemotherapeutics. After conversion of the prodrug into the toxic chemotherapeutic drug, 5-fluorouracil (5-FU), the slow product release from the enzyme limits the overall catalytic efficiency of the enzyme/prodrug system. Here, we present a computational study of the product release of the anticancer drug, 5-FU, from yCD using metadynamics. We present a comparison of the 5-FU drug to the natural enzyme product, uracil. We use volume-based metadynamics to compute the free energy landscape for product release and show a modest binding affinity for the product to the enzyme, consistent with experiments. Next, we use infrequent metadynamics to estimate the unbiased release rate from Kramers time-dependent rate theory and find a favorable comparison to experiment with a slower rate of product release for the 5-FU system. Our work demonstrates how adaptive sampling methods can be used to study the protein-ligand unbinding process for engineering enzyme/prodrug systems and gives insights into the molecular mechanism of product release for the yCD/5-FU system.

Experimental and Theoretical Investigation of External Electric-Field-Induced Crystallization of TKX-50 from Solution by Finite-Temperature String with Order Parameters as Collective Variables for Ionic Crystals

External electric fields are an effective tool to induce phase transformations. The crystallization of ionic crystals from solution is a common phase transformation. However, understanding of mechanisms is poor at the molecular level. In this work, we carried out an experimental and theoretical investigation of the external electric-field-induced crystallization of TKX-50 from saturated formic acid solution by finite-temperature string (FTS) with order parameters (OPs) as collective variables for ionic crystals. The minimum-free-energy path was sketched by the string method in collective variables. The results show that the K-means clustering algorithm based on Euclidean distance and density weights can be used for enhanced sampling of the OPs in external electric-field-induced crystallization of ionic crystal from solution, which improves the conventional FTS. The crystallization from solution is a process of surface-mediated nucleation. The external electric field can accelerate the evolution of the string and decrease the difference in the potential of mean forces between the crystal and the transition state. Due to the significant change in OPs induced by the external electric field in nucleation, the crystalline quality was enhanced, which explains the experimental results that the external electric field enhanced the density, detonation velocity, and detonation pressure of TKX-50. This work provides an effective way to explore the crystallization of ionic crystals from solution at the molecular level, and it is useful for improving the properties of ionic crystal explosives by using external electric fields.

Fine-tuning activation specificity of G-protein-coupled receptors via automated path searching

Physics-based simulation methods can grant atomistic insights into the molecular origin of the function of biomolecules. However, the potential of such approaches has been hindered by their low efficiency, including in the design of selective agonists where simulations of myriad protein-ligand combinations are necessary. Here, we describe an automated input-free path searching protocol that offers (within 14 d using Graphics Processing Unit servers) a minimum free energy path (MFEP) defined in high-dimension configurational space for activating sphingosine-1-phosphate receptors (S1PRs) by arbitrary ligands. The free energy distributions along the MFEP for four distinct ligands and three S1PRs reached a remarkable agreement with Bioluminescence Resonance Energy Transfer (BRET) measurements of G-protein dissociation. In particular, the revealed transition state structures pointed out toward two S1PR3 residues F263/I284, that dictate the preference of existing agonists CBP307 and BAF312 on S1PR1/5. Swapping these residues between S1PR1 and S1PR3 reversed their response to the two agonists in BRET assays. These results inspired us to design improved agonists with both strong polar head and bulky hydrophobic tail for higher selectivity on S1PR1. Through merely three in silico iterations, our tool predicted a unique compound scaffold. BRET assays confirmed that both chiral forms activate S1PR1 at nanomolar concentration, 1 to 2 orders of magnitude less than those for S1PR3/5. Collectively, these results signify the promise of our approach in fine agonist design for G-protein-coupled receptors.

Perspectives on Computational Enzyme Modeling: From Mechanisms to Design and Drug Development

Understanding enzyme mechanisms is essential for unraveling the complex molecular machinery of life. In this review, we survey the field of computational enzymology, highlighting key principles governing enzyme mechanisms and discussing ongoing challenges and promising advances. Over the years, computer simulations have become indispensable in the study of enzyme mechanisms, with the integration of experimental and computational exploration now established as a holistic approach to gain deep insights into enzymatic catalysis. Numerous studies have demonstrated the power of computer simulations in characterizing reaction pathways, transition states, substrate selectivity, product distribution, and dynamic conformational changes for various enzymes. Nevertheless, significant challenges remain in investigating the mechanisms of complex multistep reactions, large-scale conformational changes, and allosteric regulation. Beyond mechanistic studies, computational enzyme modeling has emerged as an essential tool for computer-aided enzyme design and the rational discovery of covalent drugs for targeted therapies. Overall, enzyme design/engineering and covalent drug development can greatly benefit from our understanding of the detailed mechanisms of enzymes, such as protein dynamics, entropy contributions, and allostery, as revealed by computational studies. Such a convergence of different research approaches is expected to continue, creating synergies in enzyme research. This review, by outlining the ever-expanding field of enzyme research, aims to provide guidance for future research directions and facilitate new developments in this important and evolving field.

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.

Finite temperature string by K-means clustering sampling with order parameters as collective variables for molecular crystals: application to polymorphic transformation between β-CL-20 and ε-CL-20

Polymorphic transformation of molecular crystals is a fundamental phase transition process, and it is important practically in the chemical, material, biopharmaceutical, and energy storage industries. However, understanding of the transformation mechanism at the molecular level is poor due to the extreme simulating challenges in enhanced sampling and formulating order parameters (OPs) as the collective variables that can distinguish polymorphs with quite similar and complicated structures so as to describe the reaction coordinate. In this work, two kinds of OPs for CL-20 were constructed by the bond distances, bond orientations and relative orientations. A K-means clustering algorithm based on the Euclidean distance and sample weight was used to smooth the initial finite temperature string (FTS), and the minimum free energy path connecting β-CL-20 and ε-CL-20 was sketched by the string method in collective variables, and the free energy profile along the path and the nucleation kinetics were obtained by Markovian milestoning with Voronoi tessellations. In comparison with the average-based sampling, the K-means clustering algorithm provided an improved convergence rate of FTS. The simulation of transformation was independent of OP types but was affected greatly by finite-size effects. A surface-mediated local nucleation mechanism was confirmed and the configuration located at the shoulder of potential of mean force, rather than overall maximum, was confirmed to be the critical nucleus formed by the cooperative effect of the intermolecular interactions. This work provides an effective way to explore the polymorphic transformation of caged molecular crystals at the molecular level.

Improved Sampling of Adaptive Path Collective Variables by Stabilized Extended-System Dynamics

Because of the complicated multistep nature of many biocatalytic reactions, an a priori definition of reaction coordinates is difficult. Therefore, we apply enhanced sampling algorithms along with adaptive path collective variables (PCVs), which converge to the minimum free energy path (MFEP) during the simulation. We show how PCVs can be combined with the highly efficient well-tempered metadynamics extended-system adaptive biasing force (WTM-eABF) hybrid sampling algorithm, offering dramatically increased sampling efficiency due to its fast adaptation to path updates. For this purpose, we address discontinuities of PCVs that can arise due to path shortcutting or path updates with a novel stabilization algorithm for extended-system methods. In addition, we show how the convergence of simulations can be further accelerated by utilizing the multistate Bennett's acceptance ratio (MBAR) estimator. These methods are applied to the first step of the enzymatic reaction mechanism of pseudouridine synthases, where the ability of path WTM-eABF to efficiently explore intricate molecular transitions is demonstrated.

Free Energy Surface and Molecular Mechanism of Slow Structural Transitions in Lipid Bilayers

Lipid membrane remodeling, crucial for many cellular processes, is governed by the coupling of membrane structure and shape fluctuations. Given the importance of the ∼ nm length scale, details of the transition intermediates for conformational change are not fully captured by a continuum-mechanical description. Slow dynamics and the lack of knowledge of reaction coordinates (RCs) for biasing methods pose a challenge for all-atom (AA) simulations. Here, we map system dynamics on Langevin dynamics in a normal mode space determined from an elastic network model representation for the lipid-water Hamiltonian. AA molecular dynamics (MD) simulations are used to determine model parameters, and Langevin dynamics predictions for bilayer structural, mechanical, and dynamic properties are validated against MD simulations and experiments. Transferability to describe the dynamics of a larger lipid bilayer and a heterogeneous membrane-protein system is assessed. A set of generic RCs for pore formation in two tensionless bilayers is obtained by coupling Langevin dynamics to the underlying energy landscape for membrane deformations. Structure evolution is carried out by AA MD, wherein the generic RCs are used in a path metadynamics or an umbrella sampling simulation to determine the thermodynamics of pore formation and its molecular determinants, such as the role of distinct bilayer motions, lipid solvation, and lipid packing.

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.

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.

Binding of human serum albumin with uranyl ion at various pH: an all atom molecular dynamics study

Uranium is routinely handled in various stages of nuclear fuel cycle and its association with human serum albumin (HSA) has been reported in literature, however, their binding characteristics still remains obscure. The present study aims to understand interaction of uranium with HSA by employing all atom molecular dynamics simulation of the HSA-metal ion complex. His67, His247 and Asp249 residues constitute the major binding site of HSA, which capture the uranyl ion (UO22+). A total of six sets of initial coordinates are used for Zn2+-HSA and UO22+-HSA system at pH = 4, 7.4 and 9, respectively. Enhance sampling method, namely, well-tempered meta-dynamics (WT-MtD) is employed to study the binding and un-binding processes of UO22+ and Zn2+ ions. Potential of mean force (PMF) profiles are generated for all the six sets of complexes from the converged WT-MtD run. Various basins and barriers are observed along the (un)binding pathways. Hydrogen bond dynamics and short-range Coulomb interactions are evaluated from the equilibrium run at each basins and barriers for both the ions at all pH values. The binding of UO22+ ion with HSA is the result of the dynamical balance between UO22+-HSA and UO22+-water short range Coulomb interactions. Zn2+ ion interact more strongly than UO22+ at all pH through short range Coulomb interactions. PMF values further concludes that UO22+ cannot associate to the Zn2+ bound HSA protein but can be captured by free HSA at all pH values i.e. endosomal, alkaline and physiological pH.Communicated by Ramaswamy H. Sarma.

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

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

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

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

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.

Sequence dependence of transient Hoogsteen base pairing in DNA

Hoogsteen (HG) base pairing is characterized by a 180° rotation of the purine base with respect to the Watson-Crick-Franklin (WCF) motif. Recently, it has been found that both conformations coexist in a dynamical equilibrium and that several biological functions require HG pairs. This relevance has motivated experimental and computational investigations of the base-pairing transition. However, a systematic simulation of sequence variations has remained out of reach. Here, we employ advanced path-based methods to perform unprecedented free-energy calculations. Our methodology enables us to study the different mechanisms of purine rotation, either remaining inside or after flipping outside of the double helix. We study seven different sequences, which are neighbor variations of a well-studied A⋅T pair in A6-DNA. We observe the known effect of A⋅T steps favoring HG stability, and find evidence of triple-hydrogen-bonded neighbors hindering the inside transition. More importantly, we identify a dominant factor: the direction of the A rotation, with the 6-ring pointing either towards the longer or shorter segment of the chain, respectively relating to a lower or higher barrier. This highlights the role of DNA's relative flexibility as a modulator of the WCF/HG dynamic equilibrium. Additionally, we provide a robust methodology for future HG proclivity studies.

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

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

Machine Learning-Assisted Discovery of Hidden States in Expanded Free Energy Space

Collective variables (CVs) are crucial parameters in enhanced sampling calculations and strongly impact the quality of the obtained free energy surface. However, many existing CVs are unique to and dependent on the system they are constructed with, making the developed CV non-transferable to other systems. Herein, we develop a non-instructor-led deep autoencoder neural network (DAENN) for discovering general-purpose CVs. The DAENN is used to train a model by learning molecular representations upon unbiased trajectories that contain only the reactant conformers. The prior knowledge of nonconstraint reactants coupled with the here-introduced topology variable and loss-like penalty function are only required to make the biasing method able to expand its configurational (phase) space to unexplored energy basins. Our developed autoencoder is efficient and relatively inexpensive to use in terms of a priori knowledge, enabling one to automatically search for hidden CVs of the reaction of interest.

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

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

Force-Correction Analysis Method for Derivation of Multidimensional Free-Energy Landscapes from Adaptively Biased Replica Simulations

A methodology is proposed for the calculation of multidimensional free-energy landscapes of molecular systems, based on analysis of multiple molecular dynamics trajectories wherein adaptive biases have been applied to enhance the sampling of different collective variables. In this approach, which we refer to as the Force-Correction Analysis Method (FCAM), local averages of the total and biasing forces are evaluated post hoc, and the latter are subtracted from the former to obtain unbiased estimates of the mean force across collective-variable space. Multidimensional free-energy surfaces and minimum free-energy pathways are then derived by integrating the mean-force landscape with a kinetic Monte Carlo algorithm. To evaluate the proposed method, a series of numerical tests and comparisons with existing approaches were carried out for small molecules, peptides, and proteins, based on all-atom trajectories generated with standard, concurrent, and replica-exchange metadynamics in collective-variable spaces ranging from one to six dimensional. The tests confirm the correctness of the FCAM formulation and demonstrate that calculated mean forces and free energies converge rapidly and accurately, outperforming other methods used to unbias this kind of simulation data.

Closer Look at Inverse Electron Demand Diels–Alder and Nucleophilic Addition Reactions on s-Tetrazines Using Enhanced Sampling Methods

Inverse electron demand [4+2] Diels–Alder (iEDDA) reactions as well as unprecedented nucleophilic (azaphilic) additions of R-substituted silyl-enol ethers (where R is Phenyl, Methyl, and Hydrogen) to 1,2,4,5-tetrazine (s-tetrazine) catalyzed by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {BF}_{3}$$\end{document}BF3 have recently been discovered (Simon et al. in Org Lett 23(7):2426–2430, 2021), where static calculations were employed for calculation of activation energies. In order to have a more realistic dynamic description of these reactions in explicit solution at ambient conditions, in this work we use a semiempirical tight-binding method combined with enhanced sampling techniques to calculate free energy surfaces (FESs) of the iEDDA and azaphilic addition reactions. Relevant products of not only s-tetrazine but also its derivatives such as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {BF}_{3}$$\end{document}BF3-mediated s-tetrazine adducts are investigated. We reconstruct the FESs of the iEDDA and azaphilic addition reactions using metadynamics and blue moon ensemble, and compare the ability of different collective variables (CVs) including bond distances, Social PeRmutation INvarianT (SPRINT) coordinates, and path-CV to describe the reaction pathway. We find that when a bulky Phenyl is used as a substituent at the dienophile the azaphilic addition is preferred over the iEDDA reaction. In addition, we also investigate the effect of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {BF}_{3}$$\end{document}BF3 in the diene and steric hindrance in the dienophile on the competition between the iEDDA and azaphilic addition reactions, providing chemical insight for reaction design.

String Method with Swarms-of-Trajectories, Mean Drifts, Lag Time, and Committor

The kinetics of a dynamical system comprising two metastable states is formulated in terms of a finite-time propagator in phase space (position and velocity) adapted to the underdamped Langevin equation. Dimensionality reduction to a subspace of collective variables yields familiar expressions for the propagator, committor, and steady-state flux. A quadratic expression for the steady-state flux between the two metastable states can serve as a robust variational principle to determine an optimal approximate committor expressed in terms of a set of collective variables. The theoretical formulation is exploited to clarify the foundation of the string method with swarms-of-trajectories, which relies on the mean drift of short trajectories to determine the optimal transition pathway. It is argued that the conditions for Markovity within a subspace of collective variables may not be satisfied with an arbitrary short time-step and that proper kinetic behaviors appear only when considering the effective propagator for longer lag times. The effective propagator with finite lag time is amenable to an eigenvalue-eigenvector spectral analysis, as elaborated previously in the context of position-based Markov models. The time-correlation functions calculated by swarms-of-trajectories along the string pathway constitutes a natural extension of these developments. The present formulation provides a powerful theoretical framework to characterize the optimal pathway between two metastable states of a system.

Assessing the Performance of Traveling-salesman based Automated Path Searching (TAPS) on Complex Biomolecular Systems

Though crucial for understanding the function of large biomolecular systems, locating the minimum free energy paths (MFEPs) between their key conformational states is far from trivial due to their high-dimensional nature. Most existing path-searching methods require a static collective variable space as input, encoding intuition or prior knowledge of the transition mechanism. Such information is, however, hardly available a priori and expensive to validate. To alleviate this issue, we have previously introduced a Traveling-salesman based Automated Path Searching method (TAPS) and demonstrated its efficiency on simple peptide systems. Having implemented a parallel version of this method, here we assess the performance of TAPS on three realistic systems (tens to hundreds of residues) in explicit solvents. We show that TAPS successfully located the MFEP for the ground/excited state transition of the T4 lysozyme L99A variant, consistent with previous findings. TAPS also helped identifying the important role of the two polar contacts in directing the loop-in/loop-out transition of the mitogen-activated protein kinase kinase (MEK1), which explained previous mutant experiments. Remarkably, at a minimal cost of 126 ns sampling, TAPS revealed that the Ltn40/Ltn10 transition of lymphotactin needs no complete unfolding/refolding of its β-sheets and that five polar contacts are sufficient to stabilize the various partially unfolded intermediates along the MFEP. These results present TAPS as a general and promising tool for studying the functional dynamics of complex biomolecular systems.

Global Free-Energy Landscapes as a Smoothly Joined Collection of Local Maps

Enhanced sampling techniques have become an essential tool in computational chemistry and physics, where they are applied to sample activated processes that occur on a time scale that is inaccessible to conventional simulations. Despite their popularity, it is well known that they have constraints that hinder their application to complex problems. The core issue lies in the need to describe the system using a small number of collective variables (CVs). Any slow degree of freedom that is not properly described by the chosen CVs will hinder sampling efficiency. However, the exploration of configuration space is also hampered by including variables that are not relevant for the activated process under study. This paper presents the Adaptive Topography of Landscape for Accelerated Sampling (ATLAS), a new biasing method capable of working with many CVs. The root idea of ATLAS is to apply a divide-and-conquer strategy, where the high-dimensional CVs space is divided into basins, each of which is described by an automatically determined, low-dimensional set of variables. A well-tempered metadynamics-like bias is constructed as a function of these local variables. Indicator functions associated with the basins switch on and off the local biases so that the sampling is performed on a collection of low-dimensional CV spaces that are smoothly combined to generate an effectively high-dimensional bias. The unbiased Boltzmann distribution is recovered through reweighing, making the evaluation of conformational and thermodynamic properties straightforward. The decomposition of the free-energy landscape in local basins can be updated iteratively as the simulation discovers new (meta)stable states.

Discovering Collective Variables of Molecular Transitions via Genetic Algorithms and Neural Networks

With the continual improvement of computing hardware and algorithms, simulations have become a powerful tool for understanding all sorts of (bio)molecular processes. To handle the large simulation data sets and to accelerate slow, activated transitions, a condensed set of descriptors, or collective variables (CVs), is needed to discern the relevant dynamics that describes the molecular process of interest. However, proposing an adequate set of CVs that can capture the intrinsic reaction coordinate of the molecular transition is often extremely difficult. Here, we present a framework to find an optimal set of CVs from a pool of candidates using a combination of artificial neural networks and genetic algorithms. The approach effectively replaces the encoder of an autoencoder network with genes to represent the latent space, i.e., the CVs. Given a selection of CVs as input, the network is trained to recover the atom coordinates underlying the CV values at points along the transition. The network performance is used as an estimator of the fitness of the input CVs. Two genetic algorithms optimize the CV selection and the neural network architecture. The successful retrieval of optimal CVs by this framework is illustrated at the hand of two case studies: the well-known conformational change in the alanine dipeptide molecule and the more intricate transition of a base pair in B-DNA from the classic Watson-Crick pairing to the alternative Hoogsteen pairing. Key advantages of our framework include the following: optimal interpretable CVs, avoiding costly calculation of committor or time-correlation functions, and automatic hyperparameter optimization. In addition, we show that applying a time-delay between the network input and output allows for enhanced selection of slow variables. Moreover, the network can also be used to generate molecular configurations of unexplored microstates, for example, for augmentation of the simulation data.

Frontiers in Multiscale Modeling of Photoreceptor Proteins

This perspective article highlights the challenges in the theoretical description of photoreceptor proteins using multiscale modeling, as discussed at the CECAM workshop in Tel Aviv, Israel. The participants have identified grand challenges and discussed the development of new tools to address them. Recent progress in understanding representative proteins such as green fluorescent protein, photoactive yellow protein, phytochrome, and rhodopsin is presented, along with methodological developments.

Free Energy Landscape and Conformational Kinetics of Hoogsteen Base Pairing in DNA vs. RNA

Genetic information is encoded in the DNA double helix, which, in its physiological milieu, is characterized by the iconical Watson-Crick nucleo-base pairing. Recent NMR relaxation experiments revealed the transient presence of an alternative, Hoogsteen (HG) base pairing pattern in naked DNA duplexes, and estimated its relative stability and lifetime. In contrast with DNA, such structures were not observed in RNA duplexes. Understanding HG base pairing is important because the underlying "breathing" motion between the two conformations can significantly modulate protein binding. However, a detailed mechanistic insight into the transition pathways and kinetics is still missing. We performed enhanced sampling simulation (with combined metadynamics and adaptive force-bias method) and Markov state modeling to obtain accurate free energy, kinetics, and the intermediates in the transition pathway between Watson-Crick and HG base pairs for both naked B-DNA and A-RNA duplexes. The Markov state model constructed from our unbiased MD simulation data revealed previously unknown complex extrahelical intermediates in the seemingly simple process of base flipping in B-DNA. Extending our calculation to A-RNA, for which HG base pairing is not observed experimentally, resulted in relatively unstable, single-hydrogen-bonded, distorted Hoogsteen-like bases. Unlike B-DNA, the transition pathway primarily involved base paired and intrahelical intermediates with transition timescales much longer than that of B-DNA. The seemingly obvious flip-over reaction coordinate (i.e., the glycosidic torsion angle) is unable to resolve the intermediates. Instead, a multidimensional picture involving backbone dihedral angles and distance between hydrogen bond donor and acceptor atoms is required to gain insight into the molecular mechanism.

Scalable molecular dynamics on CPU and GPU architectures with NAMD

NAMDis a molecular dynamics program designed for high-performance simulations of very large biological objects on CPU- and GPU-based architectures. NAMD offers scalable performance on petascale parallel supercomputers consisting of hundreds of thousands of cores, as well as on inexpensive commodity clusters commonly found in academic environments. It is written in C++ and leans on Charm++ parallel objects for optimal performance on low-latency architectures. NAMD is a versatile, multipurpose code that gathers state-of-the-art algorithms to carry out simulations in apt thermodynamic ensembles, using the widely popular CHARMM, AMBER, OPLS, and GROMOS biomolecular force fields. Here, we review the main features of NAMD that allow both equilibrium and enhanced-sampling molecular dynamics simulations with numerical efficiency. We describe the underlying concepts utilized by NAMD and their implementation, most notably for handling long-range electrostatics; controlling the temperature, pressure, and pH; applying external potentials on tailored grids; leveraging massively parallel resources in multiple-copy simulations; and hybrid quantum-mechanical/molecular-mechanical descriptions. We detail the variety of options offered by NAMD for enhanced-sampling simulations aimed at determining free-energy differences of either alchemical or geometrical transformations and outline their applicability to specific problems. Last, we discuss the roadmap for the development of NAMD and our current efforts toward achieving optimal performance on GPU-based architectures, for pushing back the limitations that have prevented biologically realistic billion-atom objects to be fruitfully simulated, and for making large-scale simulations less expensive and easier to set up, run, and analyze. NAMD is distributed free of charge with its source code at www.ks.uiuc.edu.

FSATOOL: A useful tool to do the conformational sampling and trajectory analysis work for biomolecules

Reliable conformational sampling and trajectory analysis are always important to the study of the folding or binding mechanisms of biomolecules. Generally, one has to prepare many complicated parameters and follow a lot of steps to obtain the final data. The whole process is too complicated to new users. In this article, we provide a convenient and user-friendly tool that is compatible to AMBER, called fast sampling and analysis tool (FSATOOL). FSATOOL has some useful features. First and the most important, the whole work is extremely simplified into two steps, one is the fast sampling procedure and the other is the trajectory analysis procedure. Second, it contains several powerful sampling methods for the simulation on graphics process unit, including our previous mixing replica exchange molecular dynamics method. The method combines the advantages of the biased and unbiased simulations. Finally, it extracts the dominant transition pathways automatically from the folding network by Markov state model. Users do not need to do the tedious intermediate steps by hand. To illustrate the usage of FSATOOL in practice, we perform one simulation for a RNA hairpin in explicit solvent. All the results are presented. © 2019 Wiley Periodicals, Inc.

Combining the biased and unbiased sampling strategy into one convenient free energy calculation method

Constructing a free energy landscape for a large molecule is difficult. One has to use either a high temperature or a strong driving force to enhance the sampling on the free energy barriers. In this work, we propose a mixed method that combines these two kinds of acceleration strategies into one simulation. First, it applies an adaptive biasing potential to some replicas of the molecule. These replicas are particularly accelerated in a collective variable space. Second, it places some unbiased and exchangeable replicas at various temperature levels. These replicas generate unbiased sampling data in the canonical ensemble. To improve the sampling efficiency, biased replicas transfer their state variables to the unbiased replicas after equilibrium by Monte Carlo trial moves. In comparison to previous integrated methods, it is more convenient for users. It does not need an initial reference biasing potential to guide the sampling of the molecule. And it is also unnecessary to insert many replicas for the requirement of passing the free energy barriers. The free energy calculation is accomplished in a single stage. It samples the data as fast as a biased simulation and it processes the data as simple as an unbiased simulation. The method provides a minimalist approach to the construction of the free energy landscape. © 2019 Wiley Periodicals, Inc.

The Adaptive Path Collective Variable: A Versatile Biasing Approach to Compute the Average Transition Path and Free Energy of Molecular Transitions

In the past decade, great progress has been made in the development of enhanced sampling methods, aimed at overcoming the time-scale limitations of molecular dynamics (MD) simulations. Many sampling schemes rely on adding an external bias to favor the sampling of transitions and to estimate the underlying free energy landscape. Nevertheless, sampling molecular processes described by many order parameters, or collective variables (CVs), such as complex biomolecular transitions, remains often very challenging. The computational cost has a prohibitive scaling with the dimensionality of the CV-space. Inspiration can be taken from methods that focus on localizing transition pathways: the CV-space can be projected onto a path-CV that connects two stable states, and a bias can be exerted onto a one-dimensional parameter that captures the progress of the transition along the path-CV. In principle, such a sampling scheme can handle an arbitrarily large number of CVs. A standard enhanced sampling technique combined with an adaptive path-CV can then locate the mean transition pathway and obtain the free energy profile along the path. In this chapter, we discuss the adaptive path-CV formalism and its numerical implementation. We apply the path-CV with several enhanced sampling methods-steered MD, metadynamics, and umbrella sampling-to a biologically relevant process: the Watson-Crick to Hoogsteen base-pairing transition in double-stranded DNA. A practical guide is provided on how to recognize and circumvent possible pitfalls during the calculation of a free energy landscape that contains multiple pathways. Examples are presented on how to perform enhanced sampling simulations using PLUMED, a versatile plugin that can work with many popular MD engines.

Analyzing and Biasing Simulations with PLUMED

This chapter discusses how the PLUMED plugin for molecular dynamics can be used to analyze and bias molecular dynamics trajectories. The chapter begins by introducing the notion of a collective variable and by then explaining how the free energy can be computed as a function of one or more collective variables. A number of practical issues mostly around periodic boundary conditions that arise when these types of calculations are performed using PLUMED are then discussed. Later parts of the chapter discuss how PLUMED can be used to perform enhanced sampling simulations that introduce simulation biases or multiple replicas of the system and Monte Carlo exchanges between these replicas. This section is then followed by a discussion on how free-energy surfaces and associated error bars can be extracted from such simulations by using weighted histogram and block averaging techniques.

Defining an Optimal Metric for the Path Collective Variables

Path Collective Variables (PCVs) are a set of path-like variables that have been successfully used to investigate complex chemical and biological processes and compute their associated free energy surfaces and kinetics. Their current implementation relies on general, but at times inefficient, metrics (such as RMSD or DRMSD) to evaluate the distance between the instantaneous conformational state during the simulation and the reference coordinates defining the path. In this work, we present a new algorithm to construct optimal PCVs metrics as linear combinations of different CVs weighted through a spectral gap optimization procedure. The method was tested first on a simple model, trialanine peptide, in vacuo and then on a more complex path of an anticancer inhibitor binding to its pharmacological target. We also compared the results to those obtained with other path-based algorithms. We find that not only our proposed approach is able to automatically select relevant CVs for the PCVs metric but also that the resulting PCVs allow for reconstructing the associated free energy very efficiently. What is more, at difference with other path-based methods, our algorithm is able to explore nonlocally the reaction path space.

Structural insight into precursor tRNA processing by yeast ribonuclease P

Ribonuclease P (RNase P) is a universal ribozyme responsible for processing the 5'-leader of pre-transfer RNA (pre-tRNA). Here, we report the 3.5-angstrom cryo-electron microscopy structures of Saccharomyces cerevisiae RNase P alone and in complex with pre-tRNA^Phe The protein components form a hook-shaped architecture that wraps around the RNA and stabilizes RNase P into a "measuring device" with two fixed anchors that recognize the L-shaped pre-tRNA. A universally conserved uridine nucleobase and phosphate backbone in the catalytic center together with the scissile phosphate and the O3' leaving group of pre-tRNA jointly coordinate two catalytic magnesium ions. Binding of pre-tRNA induces a conformational change in the catalytic center that is required for catalysis. Moreover, simulation analysis suggests a two-metal-ion S_N2 reaction pathway of pre-tRNA cleavage. These results not only reveal the architecture of yeast RNase P but also provide a molecular basis of how the 5'-leader of pre-tRNA is processed by eukaryotic RNase P.

Transition path theory from biased simulations

Transition Path Theory (TPT) provides a rigorous framework to investigate the dynamics of rare thermally activated transitions. In this theory, a central role is played by the forward committor function q⁺(x), which provides the ideal reaction coordinate. Furthermore, the reactive dynamics and kinetics are fully characterized in terms of two time-independent scalar and vector distributions. In this work, we develop a scheme which enables all these ingredients of TPT to be efficiently computed using the short non-equilibrium trajectories generated by means of a specific combination of enhanced path sampling techniques. In particular, first we further extend the recently introduced self-consistent path sampling algorithm in order to compute the committor q⁺(x). Next, we show how this result can be exploited in order to define efficient algorithms which enable us to directly sample the transition path ensemble.

Liberation of H2 from (o-C6H4Me)3P-H(+) + (-)H-B(p-C6F4H)3 ion-pair: A transition-state in the minimum energy path versus the transient species in Born-Oppenheimer molecular dynamics

Using Born-Oppenheimer molecular dynamics (BOMD) with density functional theory, transition-state (TS) calculations, and the quantitative energy decomposition analysis (EDA), we examined the mechanism of H₂-liberation from LB-H⁽⁺⁾ + ^(-)H-LA ion-pair, 1, in which the Lewis base (LB) is (o-C₆H₄Me)₃P and the Lewis acid (LA) is B(p-C₆F₄H)₃. BOMD simulations indicate that the path of H₂ liberation from the ion-pair 1 goes via the short-lived transient species, LB⋯H₂⋯LA, which are structurally reminiscent of the TS-structure in the minimum-energy-path describing the reversible reaction between H₂ and (o-C₆H₄Me)₃P/B(p-C₆F₄H)₃ frustrated Lewis pair (FLP). With electronic structure calculations performed on graphics processing units, our BOMD data-set covers more than 1 ns of evolution of the ion-pair 1 at temperature T ≈ 400 K. BOMD simulations produced H₂-recombination events with various durations of H₂ remaining fully recombined as a molecule within a LB/LA attractive "pocket"-from very short vibrational-time scale to time scales in the range of a few hundred femtoseconds. With the help of perturbational approach to trajectory-propagation over a saddle-area, we directly examined dynamics of H₂-liberation. Using EDA, we elucidated interactions between the cationic and anionic fragments in the ion-pair 1 and between the molecular fragments in the TS-structure. We have also considered a model that qualitatively takes into account the potential energy characteristics of H-H recombination and H₂-release plus inertia of molecular motion of the (o-C₆H₄Me)₃P/B(p-C₆F₄H)₃ FLP.

Surprisingly Flexible Oxonium/Borohydride Ion Pair Configurations

We investigate the geometry of oxonium/borohydride ion pairs [ether-H⁽⁺⁾-ether][LA-H^(-)] with dioxane, THF, and Et₂O as ethers and B(C₆F₅)₃ as the Lewis acid (LA). The question is about possible location of the disolvated proton, [ether-H⁽⁺⁾-ether], with respect to the hydride of the structurally complex [LA-H^(-)] anion. Using Born-Oppenheimer molecular dynamics and a comparison of the potential and free energies of the optimized configurations, we show that herein considered ion pairs are much more flexible geometrically than previously thought. Conformers with different locations of cations with respect to anions are governed by a flat energy-landscape. We found a novel configuration in which oxonium is below [LA-H^(-)], with respect to the direction of borane → hydride vector, and the proton-hydride distance is ca. 6 Å. With calculations of the vibrational spectra of [ether-H⁽⁺⁾-ether][(C₆F₅)₃B-H^(-)] for dioxane, THF, and Et₂O as ethers, we investigate the manifestation of SSLB-type (short, strong, low-barrier) hydrogen bonding in the OHO motif of an oxonium cation.

Molecular mechanisms of RNA polymerase II transcription elongation elucidated by kinetic network models

Transcription elongation cycle (TEC) of RNA polymerase II (Pol II) is a process of adding a nucleoside triphosphate to the growing messenger RNA chain. Due to the long timescale events in Pol II TEC, an advanced computational technique, such as Markov State Model (MSM), is needed to provide atomistic mechanism and reaction rates. The combination of MSM and experimental results can be used to build a kinetic network model (KNM) of the whole TEC. This review provides a brief protocol to build MSM and KNM of the whole TEC, along with the latest findings of MSM and other computational studies of Pol II TEC. Lastly, we offer a perspective on potentially using a sequence dependent KNM to predict genome-wide transcription error.

Constructing a multidimensional free energy surface like a spider weaving a web

Complete free energy surface in the collective variable space provides important information of the reaction mechanisms of the molecules. But, sufficient sampling in the collective variable space is not easy. The space expands quickly with the number of the collective variables. To solve the problem, many methods utilize artificial biasing potentials to flatten out the original free energy surface of the molecule in the simulation. Their performances are sensitive to the definitions of the biasing potentials. Fast-growing biasing potential accelerates the sampling speed but decreases the accuracy of the free energy result. Slow-growing biasing potential gives an optimized result but needs more simulation time. In this article, we propose an alternative method. It adds the biasing potential to a representative point of the molecule in the collective variable space to improve the conformational sampling. And the free energy surface is calculated from the free energy gradient in the constrained simulation, not given by the negative of the biasing potential as previous methods. So the presented method does not require the biasing potential to remove all the barriers and basins on the free energy surface exactly. Practical applications show that the method in this work is able to produce the accurate free energy surfaces for different molecules in a short time period. The free energy errors are small in the cases of various biasing potentials. © 2017 Wiley Periodicals, Inc.

Fast exploration of an optimal path on the multidimensional free energy surface

In a reaction, determination of an optimal path with a high reaction rate (or a low free energy barrier) is important for the study of the reaction mechanism. This is a complicated problem that involves lots of degrees of freedom. For simple models, one can build an initial path in the collective variable space by the interpolation method first and then update the whole path constantly in the optimization. However, such interpolation method could be risky in the high dimensional space for large molecules. On the path, steric clashes between neighboring atoms could cause extremely high energy barriers and thus fail the optimization. Moreover, performing simulations for all the snapshots on the path is also time-consuming. In this paper, we build and optimize the path by a growing method on the free energy surface. The method grows a path from the reactant and extends its length in the collective variable space step by step. The growing direction is determined by both the free energy gradient at the end of the path and the direction vector pointing at the product. With fewer snapshots on the path, this strategy can let the path avoid the high energy states in the growing process and save the precious simulation time at each iteration step. Applications show that the presented method is efficient enough to produce optimal paths on either the two-dimensional or the twelve-dimensional free energy surfaces of different small molecules.

Ligand diffusion in proteins via enhanced sampling in molecular dynamics

Computational simulations in biophysics describe the dynamics and functions of biological macromolecules at the atomic level. Among motions particularly important for life are the transport processes in heterogeneous media. The process of ligand diffusion inside proteins is an example of a complex rare event that can be modeled using molecular dynamics simulations. The study of physical interactions between a ligand and its biological target is of paramount importance for the design of novel drugs and enzymes. Unfortunately, the process of ligand diffusion is difficult to study experimentally. The need for identifying the ligand egress pathways and understanding how ligands migrate through protein tunnels has spurred the development of several methodological approaches to this problem. The complex topology of protein channels and the transient nature of the ligand passage pose difficulties in the modeling of the ligand entry/escape pathways by canonical molecular dynamics simulations. In this review, we report a methodology involving a reconstruction of the ligand diffusion reaction coordinates and the free-energy profiles along these reaction coordinates using enhanced sampling of conformational space. We illustrate the above methods on several ligand-protein systems, including cytochromes and G-protein-coupled receptors. The methods are general and may be adopted to other transport processes in living matter.

A Networks Approach to Modeling Enzymatic Reactions

Modeling enzymatic reactions is a demanding task due to the complexity of the system, the many degrees of freedom involved and the complex, chemical, and conformational transitions associated with the reaction. Consequently, enzymatic reactions are not determined by precisely one reaction pathway. Hence, it is beneficial to obtain a comprehensive picture of possible reaction paths and competing mechanisms. By combining individually generated intermediate states and chemical transition steps a network of such pathways can be constructed. Transition networks are a discretized representation of a potential energy landscape consisting of a multitude of reaction pathways connecting the end states of the reaction. The graph structure of the network allows an easy identification of the energetically most favorable pathways as well as a number of alternative routes.