Identification of kinetic order parameters for non-equilibrium dynamics

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

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

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.

A deep encoder-decoder framework for identifying distinct ligand binding pathways

The pathway(s) that a ligand would adopt en route to its trajectory to the native pocket of the receptor protein act as a key determinant of its biological activity. While Molecular Dynamics (MD) simulations have emerged as the method of choice for modeling protein-ligand binding events, the high dimensional nature of the MD-derived trajectories often remains a barrier in the statistical elucidation of distinct ligand binding pathways due to the stochasticity inherent in the ligand's fluctuation in the solution and around the receptor. Here, we demonstrate that an autoencoder based deep neural network, trained using an objective input feature of a large matrix of residue-ligand distances, can efficiently produce an optimal low-dimensional latent space that stores necessary information on the ligand-binding event. In particular, for a system of L99A mutant of T4 lysozyme interacting with its native ligand, benzene, this deep encoder-decoder framework automatically identifies multiple distinct recognition pathways, without requiring user intervention. The intermediates involve the spatially discrete location of the ligand in different helices of the protein before its eventual recognition of native pose. The compressed subspace derived from the autoencoder provides a quantitatively accurate measure of the free energy and kinetics of ligand binding to the native pocket. The investigation also recommends that while a linear dimensional reduction technique, such as time-structured independent component analysis, can do a decent job of state-space decomposition in cases where the intermediates are long-lived, autoencoder is the method of choice in systems where transient, low-populated intermediates can lead to multiple ligand-binding pathways.

Conformational Plasticity in α-Synuclein and How Crowded Environment Modulates It

A 140-residue intrinsically disordered protein (IDP), α-synuclein (αS), is known to adopt conformations that are vastly plastic and susceptible to environmental cues and crowders. However, the inherently heterogeneous nature of αS has precluded a clear demarcation of its monomeric precursor between aggregation-prone and functionally relevant aggregation-resistant states and how a crowded environment could modulate their mutual dynamic equilibrium. Here, we identify an optimal set of distinct metastable states of αS in aqueous media by dissecting a 73 μs-long molecular dynamics ensemble via building a comprehensive Markov state model (MSM). Notably, the most populated metastable state corroborates with the dimension obtained from PRE-NMR studies of αS monomer, and it undergoes kinetic transition at diverse time scales with a weakly populated random-coil-like ensemble and a globular protein-like state. However, subjecting αS to a crowded environment results in a nonmonotonic compaction of these metastable conformations, thereby skewing the ensemble by either introducing new tertiary contacts or by reinforcing the innate contacts. The early stage of dimerization process is found to be considerably expedited in the presence of crowders, albeit promoting nonspecific interactions. Together with this, using an extensively sampled ensemble of αS, this exposition demonstrates that crowded environments can potentially modulate the conformational preferences of IDP that can either promote or inhibit aggregation events.

Resolving Protein Conformational Plasticity and Substrate Binding via Machine Learning

A long-standing target in elucidating the biomolecular recognition process is the identification of binding-competent conformations of the receptor protein. However, protein conformational plasticity and the stochastic nature of the recognition processes often preclude the assignment of a specific protein conformation to an individual ligand-bound pose. Here, we demonstrate that a computational framework coined as RF-TICA-MD, which integrates an ensemble decision-tree-based Random Forest (RF) machine learning (ML) technique with an unsupervised dimension reduction approach time-structured independent component analysis (TICA), provides an efficient and unambiguous solution toward resolving protein conformational plasticity and the substrate binding process. In particular, we consider multimicrosecond-long molecular dynamics (MD) simulation trajectories of a ligand recognition process in solvent-inaccessible cavities of archetypal proteins T4 lysozyme and cytochrome P450cam. We show that in a scenario in which clear correspondence between protein conformation and binding-competent macrostates could not be obtained via an unsupervised dimension reduction approach, an a priori decision-tree-based supervised classification of the simulated recognition trajectories via RF would help characterize key amino acid residue pairs of the protein that are deemed sensitive for ligand binding. A subsequent unsupervised dimensional reduction of the selected residue pairs via TICA would then delineate a conformational landscape of protein which is able to demarcate ligand-bound poses from unbound ones. The proposed RF-TICA-MD approach is shown to be data agnostic and found to be robust when using other ML-based classification methods such as XGBoost. As a promising spinoff of the protocol, the framework is found to be capable of identifying distal protein locations which would be allosterically important for ligand binding and would characterize their roles in recognition pathways. A Python implementation of a proposed ML workflow is available in GitHub https://github.com/navjeet0211/rf-tica-md.

Mechanism of Calcium Permeation in a Glutamate Receptor Ion Channel

The α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptors (AMPARs) are neurotransmitter-activated cation channels ubiquitously expressed in vertebrate brains. The regulation of calcium flux through the channel pore by RNA-editing is linked to synaptic plasticity while excessive calcium influx poses a risk for neurodegeneration. Unfortunately, the molecular mechanisms underlying this key process are mostly unknown. Here, we investigated calcium conduction in calcium-permeable AMPAR using Molecular Dynamics (MD) simulations with recently introduced multisite force-field parameters for Ca²⁺. Our calculations are consistent with experiment and explain the distinct calcium permeability in different RNA-edited forms of GluA2. For one of the identified metal binding sites, multiscale Quantum Mechanics/Molecular Mechanics (QM/MM) simulations further validated the results from MD and revealed small but reproducible charge transfer between the metal ion and its first solvation shell. In addition, the ion occupancy derived from MD simulations independently reproduced the Ca²⁺ binding profile in an X-ray structure of an NaK channel mimicking the AMPAR selectivity filter. This integrated study comprising X-ray crystallography, multisite MD, and multiscale QM/MM simulations provides unprecedented insights into Ca²⁺ permeation mechanisms in AMPARs, and paves the way for studying other biological processes in which Ca²⁺ plays a pivotal role.

Stochastic Lag Time Parameterization for Markov State Models of Protein Dynamics

Markov state models (MSMs) play a key role in studying protein conformational dynamics. A sliding count window with a fixed lag time is widely used to sample sub-trajectories for transition counting and MSM construction. However, sub-trajectories sampled with a fixed lag time may not perform well under different selections of lag time, which requires strong prior practice and leads to less robust estimation. To alleviate it, we propose a novel stochastic method from a Poisson process to generate perturbative lag time for sub-trajectory sampling and utilize it to construct a Markov chain. Comprehensive evaluations on the double-well system, WW domain, BPTI, and RBD-ACE2 complex of SARS-CoV-2 reveal that our algorithm significantly increases the robustness and power of a constructed MSM without disturbing the Markovian properties. Furthermore, the superiority of our algorithm is amplified for slow dynamic modes in complex biological processes.

A deep autoencoder framework for discovery of metastable ensembles in biomacromolecules

Biomacromolecules manifest dynamic conformational fluctuation and involve mutual interconversion among metastable states. A robust mapping of their conformational landscape often requires the low-dimensional projection of the conformational ensemble along optimized collective variables (CVs). However, the traditional choice for the CV is often limited by user-intuition and prior knowledge about the system, and this lacks a rigorous assessment of their optimality over other candidate CVs. To address this issue, we propose an approach in which we first choose the possible combinations of inter-residue Cα-distances within a given macromolecule as a set of input CVs. Subsequently, we derive a non-linear combination of latent space embedded CVs via auto-encoding the unbiased molecular dynamics simulation trajectories within the framework of the feed-forward neural network. We demonstrate the ability of the derived latent space variables in elucidating the conformational landscape in four hierarchically complex systems. The latent space CVs identify key metastable states of a bead-in-a-spring polymer. The combination of the adopted dimensional reduction technique with a Markov state model, built on the derived latent space, reveals multiple spatially and kinetically well-resolved metastable conformations for GB1 β-hairpin. A quantitative comparison based on the variational approach-based scoring of the auto-encoder-derived latent space CVs with the ones obtained via independent component analysis (principal component analysis or time-structured independent component analysis) confirms the optimality of the former. As a practical application, the auto-encoder-derived CVs were found to predict the reinforced folding of a Trp-cage mini-protein in aqueous osmolyte solution. Finally, the protocol was able to decipher the conformational heterogeneities involved in a complex metalloenzyme, namely, cytochrome P450.

Unsupervised Learning Methods for Molecular Simulation Data

Unsupervised learning is becoming an essential tool to analyze the increasingly large amounts of data produced by atomistic and molecular simulations, in material science, solid state physics, biophysics, and biochemistry. In this Review, we provide a comprehensive overview of the methods of unsupervised learning that have been most commonly used to investigate simulation data and indicate likely directions for further developments in the field. In particular, we discuss feature representation of molecular systems and present state-of-the-art algorithms of dimensionality reduction, density estimation, and clustering, and kinetic models. We divide our discussion into self-contained sections, each discussing a specific method. In each section, we briefly touch upon the mathematical and algorithmic foundations of the method, highlight its strengths and limitations, and describe the specific ways in which it has been used-or can be used-to analyze molecular simulation data.

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.

Reconciling conformational heterogeneity and substrate recognition in cytochrome P450

Cytochrome P450, the ubiquitous metalloenzyme involved in detoxification of foreign components, has remained one of the most popular systems for substrate-recognition process. However, despite being known for its high substrate specificity, the mechanistic basis of substrate-binding by archetypal system cytochrome P450cam has remained at odds with the contrasting reports of multiple diverse crystallographic structures of its substrate-free form. Here, we address this issue by elucidating the probability of mutual dynamical transition to the other crystallographic pose of cytochrome P450cam and vice versa via unbiased all-atom computer simulation. A robust Markov state model, constructed using adaptively sampled 84-μs-long molecular dynamics simulation trajectories, maps the broad and heterogenous P450cam conformational landscape into five key substates. In particular, the Markov state model identifies an intermediate-assisted dynamic equilibrium between a pair of conformations of P450cam, in which the substrate-recognition sites remain "closed" and "open," respectively. However, the estimate of a significantly higher stationary population of closed conformation, coupled with faster rate of open → closed transition than its reverse process, dictates that the net conformational equilibrium would be swayed in favor of "closed" conformation. Together, the investigation quantitatively infers that although a potential substrate of cytochrome P450cam would, in principle, explore a diverse array of conformations of substrate-free protein, it would mostly encounter a "closed" or solvent-occluded conformation and hence would follow an induced-fit-based recognition process. Overall, the work reconciles multiple precedent crystallographic, spectroscopic investigations and establishes how a statistical elucidation of conformational heterogeneity in protein would provide crucial insights in the mechanism of potential substrate-recognition process.

Comparison between slow anisotropic LE4PD fluctuations and the principal component analysis modes of ubiquitin

The biological function and folding mechanisms of proteins are often guided by large-scale slow motions, which involve crossing high energy barriers. In a simulation trajectory, these slow fluctuations are commonly identified using a principal component analysis (PCA). Despite the popularity of this method, a complete analysis of its predictions based on the physics of protein motion has been so far limited. This study formally connects the PCA to a Langevin model of protein dynamics and analyzes the contributions of energy barriers and hydrodynamic interactions to the slow PCA modes of motion. To do so, we introduce an anisotropic extension of the Langevin equation for protein dynamics, called the LE4PD-XYZ, which formally connects to the PCA "essential dynamics." The LE4PD-XYZ is an accurate coarse-grained diffusive method to model protein motion, which describes anisotropic fluctuations in the alpha carbons of the protein. The LE4PD accounts for hydrodynamic effects and mode-dependent free-energy barriers. This study compares large-scale anisotropic fluctuations identified by the LE4PD-XYZ to the mode-dependent PCA predictions, starting from a microsecond-long alpha carbon molecular dynamics atomistic trajectory of the protein ubiquitin. We observe that the inclusion of free-energy barriers and hydrodynamic interactions has important effects on the identification and timescales of ubiquitin's slow modes.

Molecular Simulations of Hydrophobic Gating of Pentameric Ligand Gated Ion Channels: Insights into Water and Ions

Ion channels are proteins which form gated nanopores in biological membranes. Many channels exhibit hydrophobic gating, whereby functional closure of a pore occurs by local dewetting. The pentameric ligand gated ion channels (pLGICs) provide a biologically important example of hydrophobic gating. Molecular simulation studies comparing additive vs polarizable models indicate predictions of hydrophobic gating are robust to the model employed. However, polarizable models suggest favorable interactions of hydrophobic pore-lining regions with chloride ions, of relevance to both synthetic carriers and channel proteins. Electrowetting of a closed pLGIC hydrophobic gate requires too high a voltage to occur physiologically but may inform designs for switchable nanopores. Global analysis of ∼200 channels yields a simple heuristic for structure-based prediction of (closed) hydrophobic gates. Simulation-based analysis is shown to provide an aid to interpretation of functional states of new channel structures. These studies indicate the importance of understanding the behavior of water and ions within the nanoconfined environment presented by ion channels.

Accelerated Estimation of Long-Timescale Kinetics from Weighted Ensemble Simulation via Non-Markovian "Microbin" Analysis

The weighted ensemble (WE) simulation strategy provides unbiased sampling of nonequilibrium processes, such as molecular folding or binding, but the extraction of rate constants relies on characterizing steady-state behavior. Unfortunately, WE simulations of sufficiently complex systems will not relax to steady state on observed simulation times. Here, we show that a postsimulation clustering of molecular configurations into "microbins" using methods developed in the Markov State Model (MSM) community can yield unbiased kinetics from WE data before steady-state convergence of the WE simulation itself. Because WE trajectories are directional and not equilibrium distributed, the history-augmented MSM (haMSM) formulation can be used, which yields the mean first-passage time (MFPT) without bias for arbitrarily small lag times. Accurate kinetics can be obtained while bypassing the often prohibitive convergence requirements of the nonequilibrium weighted ensemble. We validate the method in a simple diffusive process on a two-dimensional (2D) random energy landscape and then analyze atomistic protein folding simulations using WE molecular dynamics. We report significant progress toward the unbiased estimation of protein folding times and pathways, though key challenges remain.

Identification of Druggable Kinase Target Conformations Using Markov Model Metastable States Analysis of apo-Abl

Kinases are important targets for drug development. However, accounting for the impact of possible structural rearrangements on the binding of kinase inhibitors is complicated by the extensive flexibility of their catalytic domain. The dynamic N-lobe contains four particular mobile structural elements: the Asp-Phe-Gly (DFG) motif, the phosphate (P) positioning loop, the activation (A) loop, and the αC helix. In our previous study [Meng et al. J. Chem. Theory Comput. 2018 14, 2721-2732], we combined various simulation techniques with Markov state modeling (MSM) to explore the free energy landscape of Abl kinase beyond conformations that are known from X-ray crystallography. Here we examine the resulting Markov model in greater detail by analyzing its metastable states. A characterization of the states in terms of their DFG state, P-loop, and αC conformations is presented and compared to existing classification schemes. Several metastable states are found to be structurally close to known crystal structures of different kinases in complex with a variety of inhibitors. These results suggest that the set of conformations accessible to tyrosine kinases may be shared within the entire family and that the conformational dynamics of one kinase in the absence of any ligand can provide meaningful information about possible target conformations for inhibitors of any member of the kinase family.

IR Spectroscopy Can Reveal the Mechanism of K+ Transport in Ion Channels

Ion channels like KcsA enable ions to move across cell membranes at near diffusion-limited rates and with very high selectivity. Various mechanisms have been proposed to explain this phenomenon. Broadly, there is disagreement among the proposed mechanisms about whether ions occupy adjacent sites in the channel during the transport process. Here, using a mixed quantum-classical approach to calculate theoretical infrared spectra, we propose a set of infrared spectroscopy experiments that can discriminate between mechanisms with and without adjacent ions. These experiments differ from previous ones in that they independently probe specific ion binding sites within the selectivity filter. When ions occupy adjacent sites in the selectivity filter, the predicted spectra are significantly redshifted relative to when ions do not occupy adjacent sites. Comparisons between theoretical and experimental peak frequencies will therefore discriminate the mechanisms.