Hierarchical Folding Free Energy Landscape of HP35 Revealed by Most Probable Path Clustering

Toward a Benchmark for Markov State Models: The Folding of HP35

Adopting a 300 μs long MD trajectory of the folding of villin headpiece (HP35) by D. E. Shaw Research, we recently constructed a Markov state model (MSM) based on inter-residue contacts. The model reproduces the folding time and predicts that the native basin and unfolded region consist of metastable substates that are structurally well-characterized. Recognizing the need to establish well-defined benchmark problems, we study to what extent and in what sense this MSM can be employed as a reference model. Hence, we test the robustness of the MSM by comparing it to models that use alternative combinations of features, dimensionality reduction methods, and clustering schemes. The study suggests some main characteristics of the folding of HP35 that should be reproduced by other competitive models. Moreover, the discussion reveals which parts of the MSM workflow matter most for the considered problem and illustrates the promises and pitfalls of state-based models for the interpretation of biomolecular simulations.

15 Years of molecular simulation of drug-binding kinetics

INTRODUCTION

Drug-binding kinetics has been increasingly recognized as an important factor to be considered in drug discovery. Long residence time could prolong the action of some drugs while produce toxicity on others. Early evaluation of the binding kinetics of drug candidates could reduce attrition rate late in the drug discovery process. Computational prediction of drug-binding kinetics is useful as compounds can be evaluated even before they are made. However, simulation of drug-binding kinetics is a challenging problem because of the long-time scale involved. Nevertheless, significant progress has been made.

AREAS COVERED

This review illustrates the rapid evolution of qualitative to quantitative molecular dynamics-based methods that have been developed over the last 15 years.

EXPERT OPINION

The development of new methods based on molecular dynamics simulations now enables computation of absolute association/dissociation rate constants. Cheaper methods capable of identifying candidates with fast or slow binding kinetics, or rank-ordering rate constants are also available. Together, these methods have generated useful insights into the molecular mechanisms of drug-binding kinetics, and the design of drug candidates with therapeutically favorable kinetics. Although predicting absolute rate constants is still expensive and challenging, rapid improvement is expected in the coming years with the continuing refinement of current technologies, development of new methodologies, and the utilization of machine learning.

Selecting Features for Markov Modeling: A Case Study on HP35

Markov state models represent a popular means to interpret molecular dynamics trajectories in terms of memoryless transitions between metastable conformational states. To provide a mechanistic understanding of the considered biomolecular process, these states should reflect structurally distinct conformations and ensure a time scale separation between fast intrastate and slow interstate dynamics. Adopting the folding of villin headpiece (HP35) as a well-established model problem, here we discuss the selection of suitable input coordinates or "features", such as backbone dihedral angles and interresidue distances. We show that dihedral angles account accurately for the structure of the native energy basin of HP35, while the unfolded region of the free energy landscape and the folding process are best described by tertiary contacts of the protein. To construct a contact-based model, we consider various ways to define and select contact distances and introduce a low-pass filtering of the feature trajectory as well as a correlation-based characterization of states. Relying on input data that faithfully account for the mechanistic origin of the studied process, the states of the resulting Markov model are clearly discriminated by the features, describe consistently the hierarchical structure of the free energy landscape, and─as a consequence─correctly reproduce the slow time scales of the process.

Revealing Free Energy Landscape From MD Data via Conditional Angle Partition Tree

Deciphering the free energy landscape of biomolecular structure space is crucial for understanding many complex molecular processes, such as protein-protein interaction, RNA folding, and protein folding. A major source of current dynamic structure data is Molecular Dynamics (MD) simulations. Several methods have been proposed to investigate the free energy landscape from MD data, but all of them rely on the assumption that kinetic similarity is associated with global geometric similarity, which may lead to unsatisfactory results. In this paper, we proposed a new method called Conditional Angle Partition Tree to reveal the hierarchical free energy landscape by correlating local geometric similarity with kinetic similarity. Its application on the benchmark alanine dipeptide MD data showed a much better performance than existing methods in exploring and understanding the free energy landscape. We also applied it to the MD data of Villin HP35. Our results are more reasonable on various aspects than those from other methods and very informative on the hierarchical structure of its energy landscape.

On the Uses of PCA to Characterise Molecular Dynamics Simulations of Biological Macromolecules: Basics and Tips for an Effective Use

Principal Component Analysis (PCA) is a procedure widely used to examine data collected from molecular dynamics simulations of biological macromolecules. It allows for greatly reducing the dimensionality of their configurational space, facilitating further qualitative and quantitative analysis. Its simplicity and relatively low computational cost explain its extended use. However, a judicious implementation of PCA requires the knowledge of its theoretical grounds as well as its weaknesses and capabilities. In this article, we review these issues and discuss several strategies developed over the last years to mitigate the main PCA flaws and enhance the reproducibility of its results.

Conformational fluctuations in the molten globule state of α-lactalbumin

A molten globule (MG) state is an intermediate state of a protein observed during the unfolding of the native structure. The MG state of the protein is induced by various denaturing agents (like urea), extreme pH, pressure, and heat. Experiments suggest that the MG state of some proteins is functionally relevant even if there is no well-defined tertiary structure. Earlier experimental and theoretical studies show that the MG state of a protein is dynamic in nature, where conformational states are interconverted on nanosecond time scales. These observations lead us to study and compare the conformational fluctuations of the MG state to those of intrinsic disordered proteins (IDPs). We consider a milk protein, α-lactalbumin (aLA), which shows an MG state at low pH upon removal of the calcium (Ca²⁺) ion. We use the constant pH molecular dynamics (CpHMD) simulation to maintain the protonation state of titratable residues at a low pH during the simulation. We use the dihedral principal component analysis, the density based clustering method, and the machine learning technique to identify the conformational fluctuations. We observe metastable states in the MG state. The residues containing the essential coordinates responsible for metastability belong to a stable helix in the crystal structure, but most of them prefer unstructured or bent conformation in the MG state. These residues control the exposure of the putative binding residues for fatty acids. Thus, the MG state of a protein behaves as an intrinsic disorder protein, although the disorder here is induced by external conditions.

Singular Interface Dynamics of the SARS-CoV-2 Delta Variant Explained with Contact Perturbation Analysis

Emerging SARS-CoV-2 variants raise concerns about our ability to withstand the Covid-19 pandemic, and therefore, understanding mechanistic differences of those variants is crucial. In this study, we investigate disparities between the SARS-CoV-2 wild type and five variants that emerged in late 2020, focusing on the structure and dynamics of the spike protein interface with the human angiotensin-converting enzyme 2 (ACE2) receptor, by using crystallographic structures and extended analysis of microsecond molecular dynamics simulations. Dihedral angle principal component analysis (PCA) showed the strong similarities in the spike receptor binding domain (RBD) dynamics of the Alpha, Beta, Gamma, and Delta variants, in contrast with those of WT and Epsilon. Dynamical perturbation networks and contact PCA identified the peculiar interface dynamics of the Delta variant, which cannot be directly imputable to its specific L452R and T478K mutations since those residues are not in direct contact with the human ACE2 receptor. Our outcome shows that in the Delta variant the L452R and T478K mutations act synergistically on neighboring residues to provoke drastic changes in the spike/ACE2 interface; thus a singular mechanism of action eventually explains why it dominated over preceding variants.

Size-and-Shape Space Gaussian Mixture Models for Structural Clustering of Molecular Dynamics Trajectories

Determining the optimal number and identity of structural clusters from an ensemble of molecular configurations continues to be a challenge. Recent structural clustering methods have focused on the use of internal coordinates due to the innate rotational and translational invariance of these features. The vast number of possible internal coordinates necessitates a feature space supervision step to make clustering tractable but yields a protocol that can be system type-specific. Particle positions offer an appealing alternative to internal coordinates but suffer from a lack of rotational and translational invariance, as well as a perceived insensitivity to regions of structural dissimilarity. Here, we present a method, denoted shape-GMM, that overcomes the shortcomings of particle positions using a weighted maximum likelihood alignment procedure. This alignment strategy is then built into an expectation maximization Gaussian mixture model (GMM) procedure to capture metastable states in the free-energy landscape. The resulting algorithm distinguishes between a variety of different structures, including those indistinguishable by root-mean-square displacement and pairwise distances, as demonstrated on several model systems. Shape-GMM results on an extensive simulation of the fast-folding HP35 Nle/Nle mutant protein support a four-state folding/unfolding mechanism, which is consistent with previous experimental results and provides kinetic details comparable to previous state-of-the art clustering approaches, as measured by the VAMP-2 score. Currently, training of shape-GMMs is recommended for systems (or subsystems) that can be represented by ≲200 particles and ≲100k configurations to estimate high-dimensional covariance matrices and balance computational expense. Once a shape-GMM is trained, it can be used to predict the cluster identities of millions of configurations.

Topological data analysis gives two folding paths in HP35(nle-nle), double mutant of villin headpiece subdomain

The folding dynamics of proteins is a primary area of interest in protein science. We carried out topological data analysis (TDA) of the folding process of HP35(nle-nle), a double-mutant of the villin headpiece subdomain. Using persistent homology and non-negative matrix factorization, we reduced the dimension of protein structure and investigated the flow in the reduced space. We found this protein has two folding paths, distinguished by the pairings of inter-helix residues. Our analysis showed the excellent performance of TDA in capturing the formation of tertiary structure.

CATBOSS: Cluster Analysis of Trajectories Based on Segment Splitting

Molecular dynamics (MD) simulations are an exceedingly and increasingly potent tool for molecular behavior prediction and analysis. However, the enormous wealth of data generated by these simulations can be difficult to process and render in a human-readable fashion. Cluster analysis is a commonly used way to partition data into structurally distinct states. We present a method that improves on the state of the art by taking advantage of the temporal information of MD trajectories to enable more accurate clustering at a lower memory cost. To date, cluster analysis of MD simulations has generally treated simulation snapshots as a mere collection of independent data points and attempted to separate them into different clusters based on structural similarity. This new method, cluster analysis of trajectories based on segment splitting (CATBOSS), applies density-peak-based clustering to classify trajectory segments learned by change detection. Applying the method to a synthetic toy model as well as four real-life data sets–trajectories of MD simulations of alanine dipeptide and valine dipeptide as well as two fast-folding proteins–we find CATBOSS to be robust and highly performant, yielding natural-looking cluster boundaries and greatly improving clustering resolution. As the classification of points into segments emphasizes density gaps in the data by grouping them close to the state means, CATBOSS applied to the valine dipeptide system is even able to account for a degree of freedom deliberately omitted from the input data set. We also demonstrate the potential utility of CATBOSS in distinguishing metastable states from transition segments as well as promising application to cases where there is little or no advance knowledge of intrinsic coordinates, making for a highly versatile analysis tool.

Volume-scaled common nearest neighbor clustering algorithm with free-energy hierarchy

The combination of Markov state modeling (MSM) and molecular dynamics (MD) simulations has been shown in recent years to be a valuable approach to unravel the slow processes of molecular systems with increasing complexity. While the algorithms for intermediate steps in the MSM workflow such as featurization and dimensionality reduction have been specifically adapted to MD datasets, conventional clustering methods are generally applied to the discretization step. This work adds to recent efforts to develop specialized density-based clustering algorithms for the Boltzmann-weighted data from MD simulations. We introduce the volume-scaled common nearest neighbor (vs-CNN) clustering that is an adapted version of the common nearest neighbor (CNN) algorithm. A major advantage of the proposed algorithm is that the introduced density-based criterion directly links to a free-energy notion via Boltzmann inversion. Such a free-energy perspective allows a straightforward hierarchical scheme to identify conformational clusters at different levels of a generally rugged free-energy landscape of complex molecular systems.

MSMPathfinder: Identification of Pathways in Markov State Models

Markov state models represent a popular means to interpret biomolecular processes in terms of memoryless transitions between metastable conformational states. To gain insight into the underlying mechanism, it is instructive to determine all relevant pathways between initial and final states of the process. Currently available methods, such as Markov chain Monte Carlo and transition path theory, are convenient for identifying the most frequented pathways. They are less suited to account for the typically huge amount of pathways with low probability which, though, may dominate the cumulative flux of the reaction. On the basis of a systematic construction of all possible pathways, the here proposed method MSMPathfinder is able to characterize the multitude of unique pathways (say, up to 10¹⁰) in a complex system and to quantitatively calculate their correct weights and associated waiting times with predefined accuracy. Adopting the chiral transitions of a peptide helix and the folding of the villin headpiece as model problems, mechanisms and associated waiting times of these processes are discussed using a kinetic network representation. The analysis reveals that the waiting time distribution may yield only little insight into the diversity of pathways, because the measured folding times do typically not reflect the most probable path lengths but rather the cumulative effect of many different pathways.

Real-time observation of ligand-induced allosteric transitions in a PDZ domain

While allostery is of paramount importance for protein regulation, the underlying dynamical process of ligand (un)binding at one site, resulting time evolution of the protein structure, and change of the binding affinity at a remote site are not well understood. Here the ligand-induced conformational transition in a widely studied model system of allostery, the PDZ2 domain, is investigated by transient infrared spectroscopy accompanied by molecular dynamics simulations. To this end, an azobenzene-derived photoswitch is linked to a peptide ligand in a way that its binding affinity to the PDZ2 domain changes upon switching, thus initiating an allosteric transition in the PDZ2 domain protein. The subsequent response of the protein, covering four decades of time, ranging from ∼1 ns to ∼μs, can be rationalized by a remodeling of its rugged free-energy landscape, with very subtle shifts in the populations of a small number of structurally well-defined states. It is proposed that structurally and dynamically driven allostery, often discussed as limiting scenarios of allosteric communication, actually go hand-in-hand, allowing the protein to adapt its free-energy landscape to incoming signals.

Identifying metastable states of biomolecules by trajectory mapping and density peak clustering

Efficiently and accurately analyzing high-dimensional time series, such as the molecular dynamics (MD) trajectory of biomolecules, is a long-standing and intriguing task. Two different but related techniques, i.e., dimension reduction methods and clustering algorithms, have been developed and applied widely in this field. Here we show that the combination of these techniques enables further improvement of the analyses, especially with very complicated data. Specifically, we present an approach that combines the trajectory mapping (TM) method, which constructs slow collective variables of a time series, with density peak clustering (DPC) [A. Rodriguez and A. Laio, Science 344, 1492 (2014)SCIEAS0036-807510.1126/science.1242072], which identifies similar data points to form clusters in a static data set. We illustrate the application of the TMDPC approach with hundreds of microseconds of all-atomic MD trajectories of two proteins, the villin headpiece and protein G. The results show that TMDPC is a powerful tool for achieving the metastable states and slow dynamics of these high-dimensional time series due to the efficient consideration of the time successiveness and the geometric distances between data points.

Unrolr: Structural analysis of protein conformations using stochastic proximity embedding

Molecular dynamics (MD) simulations are widely used to explore the conformational space of biological macromolecules. Advances in hardware, as well as in methods, make the generation of large and complex MD datasets much more common. Although different clustering and dimensionality reduction methods have been applied to MD simulations, there remains a need for improved strategies that handle nonlinear data and/or can be applied to very large datasets. We present an original implementation of the pivot-based version of the stochastic proximity embedding method aimed at large MD datasets using the dihedral distance as a metric. The advantages of the algorithm in terms of data storage and computational efficiency are presented, as well as the implementation realized. Application and testing through the analysis of a 200 ns accelerated MD simulation of a 35-residue villin headpiece is discussed. Analysis of the simulation shows the promise of this method to organize large conformational ensembles. © 2018 Wiley Periodicals, Inc.

Perspective: Identification of collective variables and metastable states of protein dynamics

The statistical analysis of molecular dynamics simulations requires dimensionality reduction techniques, which yield a low-dimensional set of collective variables (CVs) {x _i } = x that in some sense describe the essential dynamics of the system. Considering the distribution P( x ) of the CVs, the primal goal of a statistical analysis is to detect the characteristic features of P( x ), in particular, its maxima and their connection paths. This is because these features characterize the low-energy regions and the energy barriers of the corresponding free energy landscape ΔG( x ) = -k _B T ln P( x ), and therefore amount to the metastable states and transition regions of the system. In this perspective, we outline a systematic strategy to identify CVs and metastable states, which subsequently can be employed to construct a Langevin or a Markov state model of the dynamics. In particular, we account for the still limited sampling typically achieved by molecular dynamics simulations, which in practice seriously limits the applicability of theories (e.g., assuming ergodicity) and black-box software tools (e.g., using redundant input coordinates). We show that it is essential to use internal (rather than Cartesian) input coordinates, employ dimensionality reduction methods that avoid rescaling errors (such as principal component analysis), and perform density based (rather than k-means-type) clustering. Finally, we briefly discuss a machine learning approach to dimensionality reduction, which highlights the essential internal coordinates of a system and may reveal hidden reaction mechanisms.

An efficient Bayesian kinetic lumping algorithm to identify metastable conformational states via Gibbs sampling

Markov State Model (MSM) has become a popular approach to study the conformational dynamics of complex biological systems in recent years. Built upon a large number of short molecular dynamics simulation trajectories, MSM is able to predict the long time scale dynamics of complex systems. However, to achieve Markovianity, an MSM often contains hundreds or thousands of states (microstates), hindering human interpretation of the underlying system mechanism. One way to reduce the number of states is to lump kinetically similar states together and thus coarse-grain the microstates into macrostates. In this work, we introduce a probabilistic lumping algorithm, the Gibbs lumping algorithm, to assign a probability to any given kinetic lumping using the Bayesian inference. In our algorithm, the transitions among kinetically distinct macrostates are modeled by Poisson processes, which will well reflect the separation of time scales in the underlying free energy landscape of biomolecules. Furthermore, to facilitate the search for the optimal kinetic lumping (i.e., the lumped model with the highest probability), a Gibbs sampling algorithm is introduced. To demonstrate the power of our new method, we apply it to three systems: a 2D potential, alanine dipeptide, and a WW protein domain. In comparison with six other popular lumping algorithms, we show that our method can persistently produce the lumped macrostate model with the highest probability as well as the largest metastability. We anticipate that our Gibbs lumping algorithm holds great promise to be widely applied to investigate conformational changes in biological macromolecules.

Principal component analysis on a torus: Theory and application to protein dynamics

A dimensionality reduction method for high-dimensional circular data is developed, which is based on a principal component analysis (PCA) of data points on a torus. Adopting a geometrical view of PCA, various distance measures on a torus are introduced and the associated problem of projecting data onto the principal subspaces is discussed. The main idea is that the (periodicity-induced) projection error can be minimized by transforming the data such that the maximal gap of the sampling is shifted to the periodic boundary. In a second step, the covariance matrix and its eigendecomposition can be computed in a standard manner. Adopting molecular dynamics simulations of two well-established biomolecular systems (Aib₉ and villin headpiece), the potential of the method to analyze the dynamics of backbone dihedral angles is demonstrated. The new approach allows for a robust and well-defined construction of metastable states and provides low-dimensional reaction coordinates that accurately describe the free energy landscape. Moreover, it offers a direct interpretation of covariances and principal components in terms of the angular variables. Apart from its application to PCA, the method of maximal gap shifting is general and can be applied to any other dimensionality reduction method for circular data.

Machine Learning of Biomolecular Reaction Coordinates

We present a systematic approach to reduce the dimensionality of a complex molecular system. Starting with a data set of molecular coordinates (obtained from experiment or simulation) and an associated set of metastable conformational states (obtained from clustering the data), a supervised machine learning model is trained to assign unknown molecular structures to the set of metastable states. In this way, the model learns to determine the features of the molecular coordinates that are most important to discriminate the states. Using a new algorithm that exploits this feature importance via an iterative exclusion principle, we identify the essential internal coordinates (such as specific interatomic distances or dihedral angles) of the system, which are shown to represent versatile reaction coordinates that account for the dynamics of the slow degrees of freedom and explain the mechanism of the underlying processes. Moreover, these coordinates give rise to a free energy landscape that may reveal previously hidden intermediate states of the system.

Imaging Metastable States and Transitions in Proteins by Trajectory Map

It has been a long-standing and intriguing issue to develop robust methods to identify metastable states and interstate transitions from simulations or experimental data to understand the functional conformational changes of proteins. It is usually hard to define the complicated boundaries of the states in the conformational space using most of the existing methods, and they often lead to parameter-sensitive results. Here, we present a new approach, visualized Trajectory Map (vTM), to identify the metastable states and the rare interstate transitions, by considering both the conformational similarity and the temporal successiveness of conformations. The vTM is able to give a nonambiguous description of slow dynamics. The case study of a β-hairpin peptide shows that the vTM can reveal the states and transitions from all-atom MD trajectory data even when a single observable (i.e, one-dimensional reaction coordinate) is used. We also use the vTM to refine the folding/unfolding mechanism of HP35 in explicit water by analyzing a 125 μs all-atom MD trajectory and obtain folding/unfolding rates of about 1/μs, which are in good agreement with the experimental values.

Robust Density-Based Clustering To Identify Metastable Conformational States of Proteins

A density-based clustering method is proposed that is deterministic, computationally efficient, and self-consistent in its parameter choice. By calculating a geometric coordinate space density for every point of a given data set, a local free energy is defined. On the basis of these free energy estimates, the frames are lumped into local free energy minima, ultimately forming microstates separated by local free energy barriers. The algorithm is embedded into a complete workflow to robustly generate Markov state models from molecular dynamics trajectories. It consists of (i) preprocessing of the data via principal component analysis in order to reduce the dimensionality of the problem, (ii) proposed density-based clustering to generate microstates, and (iii) dynamical clustering via the most probable path algorithm to construct metastable states. To characterize the resulting state-resolved conformational distribution, dihedral angle content color plots are introduced which identify structural differences of protein states in a concise way. To illustrate the performance of the method, three well-established model problems are adopted: conformational transitions of hepta-alanine, folding of villin headpiece, and functional dynamics of bovine pancreatic trypsin inhibitor.

PyEMMA 2: A Software Package for Estimation, Validation, and Analysis of Markov Models

Markov (state) models (MSMs) and related models of molecular kinetics have recently received a surge of interest as they can systematically reconcile simulation data from either a few long or many short simulations and allow us to analyze the essential metastable structures, thermodynamics, and kinetics of the molecular system under investigation. However, the estimation, validation, and analysis of such models is far from trivial and involves sophisticated and often numerically sensitive methods. In this work we present the open-source Python package PyEMMA ( http://pyemma.org ) that provides accurate and efficient algorithms for kinetic model construction. PyEMMA can read all common molecular dynamics data formats, helps in the selection of input features, provides easy access to dimension reduction algorithms such as principal component analysis (PCA) and time-lagged independent component analysis (TICA) and clustering algorithms such as k-means, and contains estimators for MSMs, hidden Markov models, and several other models. Systematic model validation and error calculation methods are provided. PyEMMA offers a wealth of analysis functions such that the user can conveniently compute molecular observables of interest. We have derived a systematic and accurate way to coarse-grain MSMs to few states and to illustrate the structures of the metastable states of the system. Plotting functions to produce a manuscript-ready presentation of the results are available. In this work, we demonstrate the features of the software and show new methodological concepts and results produced by PyEMMA.

Dynamical footprint of falcipain-2 catalytic triad in hemoglobin-β-bound state

Falcipain-2 (FP-2) is a member of papain family of cysteine proteases and the major hemoglobinase of the hemoglobin detoxification and hemozoin polymerization complex localized in the food vacuole of the plasmodium species. FP-2 is currently gaining clinical significance as the drug target of choice in combating malaria epidemic. Here, a theoretical FP-2/hemoglobin complex has been proposed and the dynamical footprint and energetics of binding have been investigated using molecular and quantum mechanics approaches. The mapped interaction interface comprises residues 34-51 of hemoglobin and cysteine-42/histidine-174/glutamine-36/asparagine-173/204 and subsites S1, S1', and S3 of FP-2. In hemoglobin-bound FP-2, asparagine-173 preferentially partners histidine-174, while glutamine-36 is preferred in ligand-free state. Cysteine-42 exhibits dihedral switch from 110° to 30° in free and bound states, respectively, with exclusion of water from the binding core upon hemoglobin binding. Hemoglobin similarly exhibits high occupancy within .2 nm distance with charged amido acid-rich subsites S1 and S3 of FP-2 functioning in tandem to reduce conformational flexibility of hemoglobin and facilitate the formation of a stabilizing anti-parallel β-sheet between Leucine-172-valine-176 of FP-2 and phenylalanine-45-asparate-47 of hemoglobin and to overcome the + 1.13e + 5 eV activation energy required to optimize the FP-2/hemoglobin-β conformation that precedes hydrolysis.

Markov state models of biomolecular conformational dynamics

It has recently become practical to construct Markov state models (MSMs) that reproduce the long-time statistical conformational dynamics of biomolecules using data from molecular dynamics simulations. MSMs can predict both stationary and kinetic quantities on long timescales (e.g. milliseconds) using a set of atomistic molecular dynamics simulations that are individually much shorter, thus addressing the well-known sampling problem in molecular dynamics simulation. In addition to providing predictive quantitative models, MSMs greatly facilitate both the extraction of insight into biomolecular mechanism (such as folding and functional dynamics) and quantitative comparison with single-molecule and ensemble kinetics experiments. A variety of methodological advances and software packages now bring the construction of these models closer to routine practice. Here, we review recent progress in this field, considering theoretical and methodological advances, new software tools, and recent applications of these approaches in several domains of biochemistry and biophysics, commenting on remaining challenges.