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

Accurate estimates of dynamical statistics using memory

Many chemical reactions and molecular processes occur on time scales that are significantly longer than those accessible by direct simulations. One successful approach to estimating dynamical statistics for such processes is to use many short time series of observations of the system to construct a Markov state model, which approximates the dynamics of the system as memoryless transitions between a set of discrete states. The dynamical Galerkin approximation (DGA) is a closely related framework for estimating dynamical statistics, such as committors and mean first passage times, by approximating solutions to their equations with a projection onto a basis. Because the projected dynamics are generally not memoryless, the Markov approximation can result in significant systematic errors. Inspired by quasi-Markov state models, which employ the generalized master equation to encode memory resulting from the projection, we reformulate DGA to account for memory and analyze its performance on two systems: a two-dimensional triple well and the AIB9 peptide. We demonstrate that our method is robust to the choice of basis and can decrease the time series length required to obtain accurate kinetics by an order of magnitude.

Genetic modification of inflammation- and clonal hematopoiesis-associated cardiovascular risk

Clonal hematopoiesis of indeterminate potential (CHIP) is associated with an increased risk of cardiovascular diseases (CVDs), putatively via inflammasome activation. We pursued an inflammatory gene modifier scan for CHIP-associated CVD risk among 424,651 UK Biobank participants. We identified CHIP using whole-exome sequencing data of blood DNA and modeled as a composite, considering all driver genes together, as well as separately for common drivers (DNMT3A, TET2, ASXL1, and JAK2). We developed predicted gene expression scores for 26 inflammasome-related genes and assessed how they modify CHIP-associated CVD risk. We identified IL1RAP as a potential key molecule for CHIP-associated CVD risk across genes and increased AIM2 gene expression leading to heightened JAK2- and ASXL1-associated CVD risk. We show that CRISPR-induced Asxl1-mutated murine macrophages had a particularly heightened inflammatory response to AIM2 agonism, associated with an increased DNA damage response, as well as increased IL-10 secretion, mirroring a CVD-protective effect of IL10 expression in ASXL1 CHIP. Our study supports the role of inflammasomes in CHIP-associated CVD and provides evidence to support gene-specific strategies to address CHIP-associated CVD risk.

Fluctuation-Dominated Ligand Binding in Molten Globule Protein

A molten globule (MG) state is an intermediate state of protein observed during the unfolding of the native structure. In MG states, milk protein α-lactalbumin (aLA) binds to oleic acid (OLA). This MG-aLA-OLA complex, popularly known as XAMLET, performs cytotoxic activities against cancer cell lines. However, the microscopic understanding of ligand recognition ability in the MG state of the protein has not yet been explored. Motivated by this, we explore the binding of bovine aLA with OLA using all-atom molecular dynamics (MD) simulations. We find the binding mode between MG-aLA and OLA using the conformational thermodynamics method. We also estimate the binding free energy using the umbrella sampling (US) method for both the MG state and the neutral state. We find that the binding free energy obtained from US is comparable with earlier experimental results. We characterize the dihedral fluctuations as the ligand is liberated from the active site of the protein using steered MD. The low energy fluctuations occur near the ligand binding site, which eventually transfer toward the Ca2+-binding site as the ligand is taken away from the protein.

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.

Inexact iterative numerical linear algebra for neural network-based spectral estimation and rare-event prediction

Understanding dynamics in complex systems is challenging because there are many degrees of freedom, and those that are most important for describing events of interest are often not obvious. The leading eigenfunctions of the transition operator are useful for visualization, and they can provide an efficient basis for computing statistics, such as the likelihood and average time of events (predictions). Here, we develop inexact iterative linear algebra methods for computing these eigenfunctions (spectral estimation) and making predictions from a dataset of short trajectories sampled at finite intervals. We demonstrate the methods on a low-dimensional model that facilitates visualization and a high-dimensional model of a biomolecular system. Implications for the prediction problem in reinforcement learning are discussed.

Beta-KTx14.3, a scorpion toxin, blocks the human potassium channel KCNQ1

Potassium channels play a key role in regulating many physiological processes, thus, alterations in their proper functioning can lead to the development of several diseases. Hence, the search for compounds capable of regulating the activity of these channels constitutes an intense field of investigation. Potassium scorpion toxins are grouped into six subfamilies (α, β, γ, κ, δ, and λ). However, experimental structures and functional analyses of the long chain β-KTx subfamily are lacking. In this study, we recombinantly produced the toxins TcoKIK and beta-KTx14.3 present in the venom of Tityus costatus and Lychas mucronatus scorpions, respectively. The 3D structures of these β-KTx toxins were determined by nuclear magnetic resonance. In both toxins, the N-terminal region is unstructured, while the C-terminal possesses the classic CSα/β motif. TcoKIK did not show any clear activity against frog Shaker and human KCNQ1 potassium channels; however, beta-KTx14.3 was able to block the KCNQ1 channel. The toxin-channel interaction mode was investigated using molecular dynamics simulations. The results showed that this toxin could form a stable network of polar-to-polar and hydrophobic interactions with KCNQ1, involving key conserved residues in both molecular partners. The discovery and characterization of a toxin capable of inhibiting KCNQ1 pave the way for the future development of novel drugs for the treatment of human diseases caused by the malfunction of this potassium channel. STATEMENT OF SIGNIFICANCE: Scorpion toxins have been shown to rarely block human KCNQ1 channels, which participate in the regulation of cardiac processes. In this study, we obtained recombinant beta-KTx14.3 and TcoKIK toxins and determined their 3D structures by nuclear magnetic resonance. Electrophysiological studies and molecular dynamics models were employed to examine the interactions between these two toxins and the human KCNQ1, which is the major driver channel of cardiac repolarization; beta-KTx14.3 was found to block effectively this channel. Our findings provide insights for the development of novel toxin-based drugs for the treatment of cardiac channelopathies involving KCNQ1-like channels.

Anisotropic Friction in a Ligand-Protein Complex

The effect of an externally applied directional force on molecular friction is so far poorly understood. Here, we study the force-driven dissociation of the ligand-protein complex biotin-streptavidin and identify anisotropic friction as a not yet described type of molecular friction. Using AFM-based stereographic single molecule force spectroscopy and targeted molecular dynamics simulations, we find that the rupture force and friction for biotin-streptavidin vary with the pulling angle. This observation holds true for friction extracted from Kramers' rate expression and by dissipation-corrected targeted molecular dynamics simulations based on Jarzynski's identity. We rule out ligand solvation and protein-internal friction as sources of the angle-dependent friction. Instead, we observe a heterogeneity in free energy barriers along an experimentally uncontrolled orientation parameter, which increases the rupture force variance and therefore the overall friction. We anticipate that anisotropic friction needs to be accounted for in a complete understanding of friction in biomolecular dynamics and anisotropic mechanical environments.

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.

Conserved Conformational Dynamics Reveal a Key Dynamic Residue in the Gatekeeper Loop of Human Cyclophilins

Cyclophilins are ubiquitous human enzymes that catalyze peptidyl-prolyl cis-trans isomerization in protein substrates. Of the 17 unique isoforms, five closely related isoforms (CypA-E) are found in various environments and participate in diverse cellular processes, yet all have similar structures and the same core catalytic function. The question is what key residues are behind the conserved function of these enzymes. Here, conformational dynamics are compared across these isoforms to detect conserved dynamics essential for the catalytic activity of cyclophilins. A set of key dynamic residues, defined by the most dynamically conserved positions, are identified in the gatekeeper 2 region. The highly conserved glycine (Gly80) in this region is predicted to underlie the local flexibility, which is further tested by molecular dynamics simulations performed on mutants (G80A) of CypE and CypA. The mutation leads to decreased flexibility of CypE and CypA during substrate binding but increased flexibility during catalysis. Dynamical changes occur in the mutated region and a distal loop downstream of the mutation site in sequence. Examinations of the mutational effect on catalysis show that both mutated CypE and CypA exhibit shifted binding free energies of the substrate under distinct isomer conformations. The results suggest a loss of function in the mutated CypE and CypA. These catalytic changes by the mutation are likely independent of the substrate sequence, at least in CypA. Our work presents a method to identify function-related key residues in proteins.

Path separation of dissipation-corrected targeted molecular dynamics simulations of protein-ligand unbinding

Protein-ligand (un)binding simulations are a recent focus of biased molecular dynamics simulations. Such binding and unbinding can occur via different pathways in and out of a binding site. Here, we present a theoretical framework on how to compute kinetics along separate paths and on how to combine the path-specific rates into global binding and unbinding rates for comparison with experimental results. Using dissipation-corrected targeted molecular dynamics in combination with temperature-boosted Langevin equation simulations [S. Wolf et al., Nat. Commun. 11, 2918 (2020)] applied to a two-dimensional model and the trypsin-benzamidine complex as test systems, we assess the robustness of the procedure and discuss the aspects of its practical applicability to predict multisecond kinetics of complex biomolecular systems.

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.

Measured Blood Pressure, Genetically Predicted Blood Pressure, and Cardiovascular Disease Risk in the UK Biobank

Importance

Hypertension remains the major cardiovascular disease risk factor globally, but variability in measured blood pressure may result in suboptimal management. Whether genetic contributors to elevated blood pressure may complementarily inform cardiovascular disease risk assessment is unknown.

Objective

To examine incident cardiovascular disease by blood pressure polygenic risk score independent of measured blood pressures and antihypertensive medication prescriptions.

Design, Setting, and Participants

The cohort study (UK Biobank) recruited UK residents aged 40 to 69 years between March 2006 and August 2010. Participants without a prior physician diagnosis of cardiovascular disease, including myocardial infarction, stroke, or heart failure, were included. Excluded were individuals with mismatch between self-reported and genotypically inferred sex, sex aneuploidy, missing genotype rates of 1% or greater, and excess genotypic heterozygosity. Data analyses were performed from September 25, 2021, to July 21, 2022.

Exposures

Measured blood pressure and externally derived blood pressure polygenic risk score stratified by hypertension diagnosis and management, which included normal blood pressure (<130/80 mm Hg without antihypertensives), untreated hypertension (systolic blood pressure ≥130 mm Hg or diastolic blood pressure ≥80 mm Hg without antihypertensives), and treated hypertension (current antihypertensives prescriptions).

Main Outcomes and Measures

Composite of first incident myocardial infarction, stroke, heart failure, or cardiovascular-related death.

Results

Of the 331 078 study participants included (mean [SD] age at enrollment, 56.9 [8.1] years; 178 824 female [54.0%]), 83 094 (25.1%) had normal blood pressure, 197 597 (59.7%) had untreated hypertension, and 50 387 (15.2%) had treated hypertension. Over a median (IQR) follow-up of 11.1 (10.4-11.8) years, the primary outcome occurred in 15 293 participants. Among those with normal blood pressure, untreated hypertension, and treated hypertension, each SD increase in measured systolic blood pressure was associated with hazard ratios of 1.08 (95% CI, 0.93-1.25), 1.20 (95% CI, 1.16-1.23), and 1.16 (95% CI, 1.11-1.20), respectively, for the primary outcome. Among these same categories, each SD increase in genetically predicted systolic blood pressure was associated with increased hazard ratios of 1.13 (95% CI, 1.05-1.20), 1.04 (95% CI, 1.01-1.07), and 1.06 (95% CI, 1.02-1.10), respectively, for the primary outcome independent of measured blood pressures and other covariates. Findings were similar for measured and genetically predicted diastolic blood pressure.

Conclusions and Relevance

Blood pressure polygenic risk score may augment identification of individuals at heightened cardiovascular risk, including those with both normal blood pressure and hypertension. Whether it may also guide antihypertensive initiation or intensification requires further study.

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.

Correlation-Based Feature Selection to Identify Functional Dynamics in Proteins

To interpret molecular dynamics simulations of biomolecular systems, systematic dimensionality reduction methods are commonly employed. Among others, this includes principal component analysis (PCA) and time-lagged independent component analysis (TICA), which aim to maximize the variance and the time scale of the first components, respectively. A crucial first step of such an analysis is the identification of suitable and relevant input coordinates (the so-called features), such as backbone dihedral angles and interresidue distances. As typically only a small subset of those coordinates is involved in a specific biomolecular process, it is important to discard the remaining uncorrelated motions or weakly correlated noise coordinates. This is because they may exhibit large amplitudes or long time scales and therefore will be erroneously considered important by PCA and TICA, respectively. To discriminate collective motions underlying functional dynamics from uncorrelated motions, the correlation matrix of the input coordinates is block-diagonalized by a clustering method. This strategy avoids possible bias due to presumed functional observables and conformational states or variation principles that maximize variance or time scales. Considering several linear and nonlinear correlation measures and various clustering algorithms, it is shown that the combination of linear correlation and the Leiden community detection algorithm yields excellent results for all considered model systems. These include the functional motion of T4 lysozyme to demonstrate the successful identification of collective motion, as well as the folding of the villin headpiece to highlight the physical interpretation of the correlated motions in terms of a functional mechanism.

Principal Component Analysis and Related Methods for Investigating the Dynamics of Biological Macromolecules

Principal component analysis (PCA) is used to reduce the dimensionalities of high-dimensional datasets in a variety of research areas. For example, biological macromolecules, such as proteins, exhibit many degrees of freedom, allowing them to adopt intricate structures and exhibit complex functions by undergoing large conformational changes. Therefore, molecular simulations of and experiments on proteins generate a large number of structure variations in high-dimensional space. PCA and many PCA-related methods have been developed to extract key features from such structural data, and these approaches have been widely applied for over 30 years to elucidate macromolecular dynamics. This review mainly focuses on the methodological aspects of PCA and related methods and their applications for investigating protein dynamics.

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.

Accelerating All-Atom Simulations and Gaining Mechanistic Understanding of Biophysical Systems through State Predictive Information Bottleneck

An effective implementation of enhanced sampling algorithms for molecular dynamics simulations requires a priori knowledge of the approximate reaction coordinate describing the relevant mechanisms in the system. In this work, we focus on the recently developed artificial intelligence-based State Predictive Information Bottleneck (SPIB) approach and demonstrate how SPIB can learn such a reaction coordinate as a deep neural network even from undersampled trajectories. We exemplify its usefulness by achieving more than 40 times acceleration in simulating two model biophysical systems through well-tempered metadynamics performed by biasing along the SPIB-learned reaction coordinate. These include left- to right-handed chirality transitions in a synthetic helical peptide (Aib)₉ and permeation of a small benzoic acid molecule through a synthetic, symmetric phospholipid bilayer. In addition to significantly accelerating the dynamics and achieving back and forth movement between different metastable states, the SPIB-based reaction coordinate gives mechanistic insights into the processes driving these two important problems.

Structural Consequence of Non-Synonymous Single-Nucleotide Variants in the N-Terminal Domain of LIS1

Disruptive neuronal migration during early brain development causes severe brain malformation. Characterized by mislocalization of cortical neurons, this condition is a result of the loss of function of migration regulating genes. One known neuronal migration disorder is lissencephaly (LIS), which is caused by deletions or mutations of the LIS1 (PAFAH1B1) gene that has been implicated in regulating the microtubule motor protein cytoplasmic dynein. Although this class of diseases has recently received considerable attention, the roles of non-synonymous polymorphisms (nsSNPs) in LIS1 on lissencephaly progression remain elusive. Therefore, the present study employed combined bioinformatics and molecular modeling approach to identify potential damaging nsSNPs in the LIS1 gene and provide atomic insight into their roles in LIS1 loss of function. Using this approach, we identified three high-risk nsSNPs, including rs121434486 (F31S), rs587784254 (W55R), and rs757993270 (W55L) in the LIS1 gene, which are located on the N-terminal domain of LIS1. Molecular dynamics simulation highlighted that all variants decreased helical conformation, increased the intermonomeric distance, and thus disrupted intermonomeric contacts in the LIS1 dimer. Furthermore, the presence of variants also caused a loss of positive electrostatic potential and reduced dimer binding potential. Since self-dimerization is an essential aspect of LIS1 to recruit interacting partners, thus these variants are associated with the loss of LIS1 functions. As a corollary, these findings may further provide critical insights on the roles of LIS1 variants in brain malformation.

Identification of Pulsatilla chinensis (Bge.) Regel and look-alike species by ultra-performance liquid chromatography/time-of-flight mass spectrometry using multivariate statistical analysis

Pulsatillae Radix, the root of Pulsatilla chinensis (Bge.) Regel, is recorded in the Pharmacopoeia of the People's Republic of China and has been widely used for its pharmacological activities, such as anti-inflammatory, antioxidant, antibacterial, antitumor, and cardiovascular benefits. However, there are several look-alike species that can be marketed as Pulsatillae Radix. To distinguish Pulsatilla chinensis (Bge.) Regel from its look-alikes, viz. Pulsatilla cernua (Thunb.) Bercht et Opiz., Pulsatilla dahurica (Fisch.) Spreng., Anemone tomeutosa (Maxim.) Pei., and Rhaponticum uniflorum (L.) DC, we used ultra-performance liquid chromatography/time-of-flight mass spectrometry combined with principal component analysis to compare their chemical compositions. Four ions, a (RT 8.98 min, m/z 1381.6671), b (RT 10.64 min, m/z 1219.6143), c (RT 11.52 min, m/z 1217.5978), and d (RT 13.6 min, m/z 749.4463) from Pulsatillae chinensis (Bge.) Regel were identified as potential chemical markers to distinguish it from look-alike species using an unsupervised statistical model combined with orthogonal partial least-squares discriminant analysis. The results of this study provide an effective method for identifying and distinguishing Pulsatilla chinensis (Bge.) Regel from similar plants. This article is protected by copyright. All rights reserved.

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

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

Data driven collective variable discovery methods to capture conformational dynamics in biological macromolecules.

Predicting Ion Channel Conductance via Dissipation-Corrected Targeted Molecular Dynamics and Langevin Equation Simulations

Ion channels are important proteins for physiological information transfer and functional control. To predict the microscopic origins of their voltage-conductance characteristics, here we applied dissipation-corrected targeted molecular dynamics in combination with Langevin equation simulations to potassium diffusion through the gramicidin A channel as a test system. Performing a nonequilibrium principal component analysis on backbone dihedral angles, we find coupled protein-ion dynamics to occur during ion transfer. The dissipation-corrected free energy profiles correspond well to predictions from other biased simulation methods. The incorporation of an external electric field in Langevin simulations enables the prediction of macroscopic observables in the form of I-V characteristics.

Data-Driven Langevin Modeling of Nonequilibrium Processes

Given nonstationary data from molecular dynamics simulations, a Markovian Langevin model is constructed that aims to reproduce the time evolution of the underlying process. While at equilibrium the free energy landscape is sampled, nonequilibrium processes can be associated with a biased energy landscape, which accounts for finite sampling effects and external driving. When the data-driven Langevin equation (dLE) approach [Phys. Rev. Lett. 2015, 115, 050602] is extended to the modeling of nonequilibrium processes, an efficient way to calculate multidimensional Langevin fields is outlined. The dLE is shown to correctly account for various nonequilibrium processes, including the enforced dissociation of sodium chloride in water, the pressure-jump induced nucleation of a liquid of hard spheres, and the conformational dynamics of a helical peptide sampled from nonstationary short trajectories.

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.

Binding Ensembles of p53-MDM2 Peptide Inhibitors by Combining Bayesian Inference and Atomistic Simulations

Designing peptide inhibitors of the p53-MDM2 interaction against cancer is of wide interest. Computational modeling and virtual screening are a well established step in the rational design of small molecules. But they face challenges for binding flexible peptide molecules that fold upon binding. We look at the ability of five different peptides, three of which are intrinsically disordered, to bind to MDM2 with a new Bayesian inference approach (MELD × MD). The method is able to capture the folding upon binding mechanism and differentiate binding preferences between the five peptides. Processing the ensembles with statistical mechanics tools depicts the most likely bound conformations and hints at differences in the binding mechanism. Finally, the study shows the importance of capturing two driving forces to binding in this system: the ability of peptides to adopt bound conformations (ΔGconformation) and the interaction between interface residues (ΔGinteraction).

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.

Revealing the selective mechanisms of inhibitors to PARP-1 and PARP-2 via multiple computational methods

Background

Research has shown that Poly-ADP-ribose polymerases 1 (PARP-1) is a potential therapeutic target in the clinical treatment of breast cancer. An increasing number of studies have focused on the development of highly selective inhibitors that target PARP-1 over PARP-2, its closest isoform, to mitigate potential side effects. However, due to the highly conserved and similar binding sites of PARP-1 and PARP-2, there is a huge challenge for the discovery and design of PARP-1 inhibitors. Recently, it was reported that a potent PARP-1 inhibitor named NMS-P118 exhibited greater selectivity to PARP-1 over PARP-2 compared with a previously reported drug (Niraparib). However, the mechanisms underlying the effect of this inhibitor remains unclear.

Methods

In the present study, classical molecular dynamics (MD) simulations and accelerated molecular dynamics (aMD) simulations combined with structural and energetic analysis were used to investigate the structural dynamics and selective mechanisms of PARP-1 and PARP-2 that are bound to NMS-P118 and Niraparib with distinct selectivity.

Results

The results from classical MD simulations indicated that the selectivity of inhibitors may be controlled by electrostatic interactions, which were mainly due to the residues of Gln-322, Ser-328, Glu-335, and Tyr-455 in helix αF. The energetic differences were corroborated by the results from aMD simulations.

Conclusion

This study provides new insights about how inhibitors specifically bind to PARP-1 over PARP-2, which may help facilitate the design of highly selective PARP-1 inhibitors in the future.

Detecting Functional Dynamics in Proteins with Comparative Perturbed-Ensembles Analysis

Recent advances have made all-atom molecular dynamics (MD) a powerful tool to sample the conformational energy landscape. There are still however three major challenges in the application of MD to biological systems: accuracy of force field, time scale, and the analysis of simulation trajectories. Significant progress in addressing the first two challenges has been made and extensively reviewed previously. This Account focuses on strategies of analyzing simulation data of biomolecules that also covers ways to properly design simulations and validate simulation results. In particular, we examine an approach named comparative perturbed-ensembles analysis, which we developed to efficiently detect dynamics in protein MD simulations that can be linked to biological functions. In our recent studies, we implemented this approach to understand allosteric regulations in several disease-associated human proteins. The central task of a comparative perturbed-ensembles analysis is to compare two or more conformational ensembles of a system generated by MD simulations under distinct perturbation conditions. Perturbations can be different sequence variations, ligand-binding conditions, and other physical/chemical modifications of the system. Each simulation is long enough (e.g., microsecond-long) to ensure sufficient sampling of the local substate. Then, sophisticated bioinformatic and statistical tools are applied to extract function-related information from the simulation data, including principal component analysis, residue-residue contact analysis, difference contact network analysis (dCNA) based on the graph theory, and statistical analysis of side-chain conformations. Computational findings are further validated with experimental data. By comparing distinct conformational ensembles, functional micro- to millisecond dynamics can be inferred. In contrast, such a time scale is difficult to reach in a single simulation; even when reached for a single condition of a system, it is elusive as to what dynamical motions are related to functions without, for example, comparing free and substrate-bound proteins at the minimum. We illustrate our approach with three examples. First, we discuss using the approach to identify allosteric pathways in cyclophilin A (CypA), a member of a ubiquitous class of peptidyl-prolyl cis-trans isomerase enzymes. By comparing side-chain torsion-angle distributions of CypA in wild-type and mutant forms, we identified three pathways: two are consistent with recent nuclear magnetic resonance experiments, whereas the third is a novel pathway. Second, we show how the approach enables a dynamical-evolution analysis of the human cyclophilin family. In the analysis, both conserved and divergent conformational dynamics across three cyclophilin isoforms (CypA, CypD, and CypE) were summarized. The conserved dynamics led to the discovery of allosteric networks resembling those found in CypA. A residue wise determinant underlying the unique dynamics in CypD was also detected and validated with additional mutational MD simulations. In the third example, we applied the approach to elucidate a peptide sequence-dependent allosteric mechanism in human Pin 1, a phosphorylation-dependent peptidyl-prolyl isomerase. We finally present our outlook of future directions. Especially, we envisage how the approach could help open a new avenue in drug discovery.

Principal component analysis of nonequilibrium molecular dynamics simulations

Principal component analysis (PCA) represents a standard approach to identify collective variables {x_i} = x, which can be used to construct the free energy landscape ΔG(x) of a molecular system. While PCA is routinely applied to equilibrium molecular dynamics (MD) simulations, it is less obvious as to how to extend the approach to nonequilibrium simulation techniques. This includes, e.g., the definition of the statistical averages employed in PCA as well as the relation between the equilibrium free energy landscape ΔG(x) and the energy landscapes ΔG(x) obtained from nonequilibrium MD. As an example for a nonequilibrium method, "targeted MD" is considered which employs a moving distance constraint to enforce rare transitions along some biasing coordinate s. The introduced bias can be described by a weighting function P(s), which provides a direct relation between equilibrium and nonequilibrium data, and thus establishes a well-defined way to perform PCA on nonequilibrium data. While the resulting distribution P(x) and energy ΔG∝lnP will not reflect the equilibrium state of the system, the nonequilibrium energy landscape ΔG(x) may directly reveal the molecular reaction mechanism. Applied to targeted MD simulations of the unfolding of decaalanine, for example, a PCA performed on backbone dihedral angles is shown to discriminate several unfolding pathways. Although the formulation is in principle exact, its practical use depends critically on the choice of the biasing coordinate s, which should account for a naturally occurring motion between two well-defined end-states of the system.

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.

A spin-1 representation for dual-funnel energy landscapes

The interconversion between the left- and right-handed helical folds of a polypeptide defines a dual-funneled free energy landscape. In this context, the funnel minima are connected through a continuum of unfolded conformations, evocative of the classical helix-coil transition. Physical intuition and recent conjectures suggest that this landscape can be mapped by assigning a left- or right-handed helical state to each residue. We explore this possibility using all-atom replica exchange molecular dynamics and an Ising-like model, demonstrating that the energy landscape architecture is at odds with a two-state picture. A three-state model-left, right, and unstructured-can account for most key intermediates during chiral interconversion. Competing folds and excited conformational states still impose limitations on the scope of this approach. However, the improvement is stark: Moving from a two-state to a three-state model decreases the fit error from 1.6 kBT to 0.3 kBT along the left-to-right interconversion pathway.

A non-equilibrium approach to allosteric communication

While the theory of protein folding is well developed, including concepts such as rugged energy landscape, folding funnel, etc., the same degree of understanding has not been reached for the description of the dynamics of allosteric transitions in proteins. This is not only due to the small size of the structural change upon ligand binding to an allosteric site, but also due to challenges in designing experiments that directly observe such an allosteric transition. On the basis of recent pump-probe-type experiments (Buchli et al. 2013 Proc. Natl Acad. Sci. USA110, 11 725-11 730. (doi:10.1073/pnas.1306323110)) and non-equilibrium molecular dynamics simulations (Buchenberg et al. 2017 Proc. Natl Acad. Sci. USA114, E6804-E6811. (doi:10.1073/pnas.1707694114)) studying an photoswitchable PDZ2 domain as model for an allosteric transition, we outline in this perspective how such a description of allosteric communication might look. That is, calculating the dynamical content of both experiment and simulation (which agree remarkably well with each other), we find that allosteric communication shares some properties with downhill folding, except that it is an 'order-order' transition. Discussing the multiscale and hierarchical features of the dynamics, the validity of linear response theory as well as the meaning of 'allosteric pathways', we conclude that non-equilibrium experiments and simulations are a promising way to study dynamical aspects of allostery.This article is part of a discussion meeting issue 'Allostery and molecular machines'.

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.

MELD-Path Efficiently Computes Conformational Transitions, Including Multiple and Diverse Paths

The molecular actions of proteins occur along reaction coordinates. Current computer methods have limited ability to explore them. We describe a fast protocol called MELD-path that (1) efficiently samples relevant conformational states via MELD, an accelerator of Molecular Dynamics (MD), (2) seeds multiple short MD trajectories from MELD states, and then (3) constructs Markov State Models (MSM) that give the routes and kinetics. We tested the method against extensive (multi μs) MD simulations of the right-handed- to left-handed-helix transition of a 9-mer peptide of AIB, the symmetry of which allows us to establish convergence. MELD-path finds all the metastable states, their correct relative populations, and the full ensemble of routes, not just a single assumed route. For this transition, we find a very broad route structure. MELD-path is highly parallelizable and efficient, yielding the full route map in a few days of computation. We believe MELD-path could be a general and rapid way to explore mechanistic processes in biomolecules on the computer.

Metadynamics Enhanced Markov Modeling of Protein Dynamics

Enhanced sampling techniques represent a versatile approach to account for rare conformational transitions in biomolecules. A particularly promising strategy is to combine massive parallel computing of short molecular dynamics (MD) trajectories (to sample the free energy landscape of the system) with Markov state modeling (to rebuild the kinetics from the sampled data). To obtain well-distributed initial structures for the short trajectories, it is proposed to employ metadynamics MD, which quickly sweeps through the entire free energy landscape of interest. Being only used to generate initial conformations, the implementation of metadynamics can be simple and fast. The conformational dynamics of helical peptide Aib₉ is adopted to discuss various technical issues of the approach, including metadynamics settings, minimal number and length of short MD trajectories, and the validation of the resulting Markov models. Using metadynamics to launch some thousands of nanosecond trajectories, several Markov state models are constructed that reveal that previous unbiased MD simulations of in total 16 μs length cannot provide correct equilibrium populations or qualitative features of the pathway distribution of the short peptide.