Variational cross-validation of slow dynamical modes in molecular kinetics

Information Bottleneck Approach for Markov Model Construction

Markov state models (MSMs) have proven valuable in studying the dynamics of protein conformational changes via statistical analysis of molecular dynamics simulations. In MSMs, the complex configuration space is coarse-grained into conformational states, with dynamics modeled by a series of Markovian transitions among these states at discrete lag times. Constructing the Markovian model at a specific lag time necessitates defining states that circumvent significant internal energy barriers, enabling internal dynamics relaxation within the lag time. This process effectively coarse-grains time and space, integrating out rapid motions within metastable states. Thus, MSMs possess a multiresolution nature, where the granularity of states can be adjusted according to the time-resolution, offering flexibility in capturing system dynamics. This work introduces a continuous embedding approach for molecular conformations using the state predictive information bottleneck (SPIB), a framework that unifies dimensionality reduction and state space partitioning via a continuous, machine learned basis set. Without explicit optimization of the VAMP-based scores, SPIB demonstrates state-of-the-art performance in identifying slow dynamical processes and constructing predictive multiresolution Markovian models. Through applications to well-validated mini-proteins, SPIB showcases unique advantages compared to competing methods. It autonomously and self-consistently adjusts the number of metastable states based on a specified minimal time resolution, eliminating the need for manual tuning. While maintaining efficacy in dynamical properties, SPIB excels in accurately distinguishing metastable states and capturing numerous well-populated macrostates. This contrasts with existing VAMP-based methods, which often emphasize slow dynamics at the expense of incorporating numerous sparsely populated states. Furthermore, SPIB's ability to learn a low-dimensional continuous embedding of the underlying MSMs enhances the interpretation of dynamic pathways. With these benefits, we propose SPIB as an easy-to-implement methodology for end-to-end MSM construction.

CoVAMPnet: Comparative Markov State Analysis for Studying Effects of Drug Candidates on Disordered Biomolecules

Computational study of the effect of drug candidates on intrinsically disordered biomolecules is challenging due to their vast and complex conformational space. Here, we developed a comparative Markov state analysis (CoVAMPnet) framework to quantify changes in the conformational distribution and dynamics of a disordered biomolecule in the presence and absence of small organic drug candidate molecules. First, molecular dynamics trajectories are generated using enhanced sampling, in the presence and absence of small molecule drug candidates, and ensembles of soft Markov state models (MSMs) are learned for each system using unsupervised machine learning. Second, these ensembles of learned MSMs are aligned across different systems based on a solution to an optimal transport problem. Third, the directional importance of inter-residue distances for the assignment to different conformational states is assessed by a discriminative analysis of aggregated neural network gradients. This final step provides interpretability and biophysical context to the learned MSMs. We applied this novel computational framework to assess the effects of ongoing phase 3 therapeutics tramiprosate (TMP) and its metabolite 3-sulfopropanoic acid (SPA) on the disordered Aβ42 peptide involved in Alzheimer's disease. Based on adaptive sampling molecular dynamics and CoVAMPnet analysis, we observed that both TMP and SPA preserved more structured conformations of Aβ42 by interacting nonspecifically with charged residues. SPA impacted Aβ42 more than TMP, protecting α-helices and suppressing the formation of aggregation-prone β-strands. Experimental biophysical analyses showed only mild effects of TMP/SPA on Aβ42 and activity enhancement by the endogenous metabolization of TMP into SPA. Our data suggest that TMP/SPA may also target biomolecules other than Aβ peptides. The CoVAMPnet method is broadly applicable to study the effects of drug candidates on the conformational behavior of intrinsically disordered biomolecules.

An Information Bottleneck Approach for Markov Model Construction

Markov state models (MSMs) have proven valuable in studying dynamics of protein conformational changes via statistical analysis of molecular dynamics (MD) simulations. In MSMs, the complex configuration space is coarse-grained into conformational states, with dynamics modeled by a series of Markovian transitions among these states at discrete lag times. Constructing the Markovian model at a specific lag time necessitates defining states that circumvent significant internal energy barriers, enabling internal dynamics relaxation within the lag time. This process effectively coarse-grains time and space, integrating out rapid motions within metastable states. Thus, MSMs possess a multi-resolution nature, where the granularity of states can be adjusted according to the time-resolution, offering flexibility in capturing system dynamics. This work introduces a continuous embedding approach for molecular conformations using the state predictive information bottleneck (SPIB), a framework that unifies dimensionality reduction and state space partitioning via a continuous, machine learned basis set. Without explicit optimization of the VAMP-based scores, SPIB demonstrates state-of-the-art performance in identifying slow dynamical processes and constructing predictive multi-resolution Markovian models. Through applications to well-validated mini-proteins, SPIB showcases unique advantages compared to competing methods. It autonomously and self-consistently adjusts the number of metastable states based on specified minimal time resolution, eliminating the need for manual tuning. While maintaining efficacy in dynamical properties, SPIB excels in accurately distinguishing metastable states and capturing numerous well-populated macrostates. This contrasts with existing VAMP-based methods, which often emphasize slow dynamics at the expense of incorporating numerous sparsely populated states. Furthermore, SPIB's ability to learn a low-dimensional continuous embedding of the underlying MSMs enhances the interpretation of dynamic pathways. With these benefits, we propose SPIB as an easy-to-implement methodology for end-to-end MSMs construction.

Functional protein dynamics in a crystal

Proteins are molecular machines and to understand how they work, we need to understand how they move. New pump-probe time-resolved X-ray diffraction methods open up ways to initiate and observe protein motions with atomistic detail in crystals on biologically relevant timescales. However, practical limitations of these experiments demands parallel development of effective molecular dynamics approaches to accelerate progress and extract meaning. Here, we establish robust and accurate methods for simulating dynamics in protein crystals, a nontrivial process requiring careful attention to equilibration, environmental composition, and choice of force fields. With more than seven milliseconds of sampling of a single chain, we identify critical factors controlling agreement between simulation and experiments and show that simulated motions recapitulate ligand-induced conformational changes. This work enables a virtuous cycle between simulation and experiments for visualizing and understanding the basic functional motions of proteins.

Understanding the Fast-Triggering Unfolding Dynamics of FK-11 upon Photoexcitation of Azobenzene

Photoswitchable molecules can control the activity and functions of biomolecules by triggering conformational changes. However, it is still challenging to fully understand such fast-triggering conformational evolution from nonequilibrium to equilibrium distribution at the molecular level. Herein, we successfully simulated the unfolding of the FK-11 peptide upon the photoinduced trans-to-cis isomerization of azobenzene based on the Markov state model. We found that the ensemble of FK-11 contains five conformational states, constituting two unfolding pathways. More intriguingly, we observed the microsecond-scale conformational propagation of the FK-11 peptide from the fully folded state to the equilibrium populations of the five states. The computed CD spectra match well with the experimental data, validating our simulation method. Overall, our study not only offers a protocol to study the photoisomerization-induced conformational changes of enzymes but also could orientate the rational design of a photoswitchable molecule to manipulate biological functions.

Tutorial on how to build non-Markovian dynamic models from molecular dynamics simulations for studying protein conformational changes

Protein conformational changes play crucial roles in their biological functions. In recent years, the Markov State Model (MSM) constructed from extensive Molecular Dynamics (MD) simulations has emerged as a powerful tool for modeling complex protein conformational changes. In MSMs, dynamics are modeled as a sequence of Markovian transitions among metastable conformational states at discrete time intervals (called lag time). A major challenge for MSMs is that the lag time must be long enough to allow transitions among states to become memoryless (or Markovian). However, this lag time is constrained by the length of individual MD simulations available to track these transitions. To address this challenge, we have recently developed Generalized Master Equation (GME)-based approaches, encoding non-Markovian dynamics using a time-dependent memory kernel. In this Tutorial, we introduce the theory behind two recently developed GME-based non-Markovian dynamic models: the quasi-Markov State Model (qMSM) and the Integrative Generalized Master Equation (IGME). We subsequently outline the procedures for constructing these models and provide a step-by-step tutorial on applying qMSM and IGME to study two peptide systems: alanine dipeptide and villin headpiece. This Tutorial is available at https://github.com/xuhuihuang/GME_tutorials. The protocols detailed in this Tutorial aim to be accessible for non-experts interested in studying the biomolecular dynamics using these non-Markovian dynamic models.

PopShift: A Thermodynamically Sound Approach to Estimate Binding Free Energies by Accounting for Ligand-Induced Population Shifts from a Ligand-Free Markov State Model

Obtaining accurate binding free energies from in silico screens has been a long-standing goal for the computational chemistry community. However, accuracy and computational cost are at odds with one another, limiting the utility of methods that perform this type of calculation. Many methods achieve massive scale by explicitly or implicitly assuming that the target protein adopts a single structure, or undergoes limited fluctuations around that structure, to minimize computational cost. Others simulate each protein-ligand complex of interest, accepting lower throughput in exchange for better predictions of binding affinities. Here, we present the PopShift framework for accounting for the ensemble of structures a protein adopts and their relative probabilities. Protein degrees of freedom are enumerated once, and then arbitrarily many molecules can be screened against this ensemble. Specifically, we use Markov state models (MSMs) as a compressed representation of a protein's thermodynamic ensemble. We start with a ligand-free MSM and then calculate how addition of a ligand shifts the populations of each protein conformational state based on the strength of the interaction between that protein conformation and the ligand. In this work we use docking to estimate the affinity between a given protein structure and ligand, but any estimator of binding affinities could be used in the PopShift framework. We test PopShift on the classic benchmark pocket T4 Lysozyme L99A. We find that PopShift is more accurate than common strategies, such as docking to a single structure and traditional ensemble docking─producing results that compare favorably with alchemical binding free energy calculations in terms of RMSE but not correlation─and may have a more favorable computational cost profile in some applications. In addition to predicting binding free energies and ligand poses, PopShift also provides insight into how the probability of different protein structures is shifted upon addition of various concentrations of ligand, providing a platform for predicting affinities and allosteric effects of ligand binding. Therefore, we expect PopShift will be valuable for hit finding and for providing insight into phenomena like allostery.

Folding-upon-binding pathways of an intrinsically disordered protein from a deep Markov state model

A central challenge in the study of intrinsically disordered proteins is the characterization of the mechanisms by which they bind their physiological interaction partners. Here, we utilize a deep learning-based Markov state modeling approach to characterize the folding-upon-binding pathways observed in a long timescale molecular dynamics simulation of a disordered region of the measles virus nucleoprotein NTAIL reversibly binding the X domain of the measles virus phosphoprotein complex. We find that folding-upon-binding predominantly occurs via two distinct encounter complexes that are differentiated by the binding orientation, helical content, and conformational heterogeneity of NTAIL. We observe that folding-upon-binding predominantly proceeds through a multi-step induced fit mechanism with several intermediates and do not find evidence for the existence of canonical conformational selection pathways. We observe four kinetically separated native-like bound states that interconvert on timescales of eighty to five hundred nanoseconds. These bound states share a core set of native intermolecular contacts and stable NTAIL helices and are differentiated by a sequential formation of native and non-native contacts and additional helical turns. Our analyses provide an atomic resolution structural description of intermediate states in a folding-upon-binding pathway and elucidate the nature of the kinetic barriers between metastable states in a dynamic and heterogenous, or "fuzzy", protein complex.

Analyzing Molecular Dynamics Trajectories Thermodynamically through Artificial Intelligence

Molecular dynamics simulations produce trajectories that correspond to vast amounts of structure when exploring biochemical processes. Extracting valuable information, e.g., important intermediate states and collective variables (CVs) that describe the major movement modes, from molecular trajectories to understand the underlying mechanisms of biological processes presents a significant challenge. To achieve this goal, we introduce a deep learning approach, coined DIKI (deep identification of key intermediates), to determine low-dimensional CVs distinguishing key intermediate conformations without a-priori assumptions. DIKI dynamically plans the distribution of latent space and groups together similar conformations within the same cluster. Moreover, by incorporating two user-defined parameters, namely, coarse focus knob and fine focus knob, to help identify conformations with low free energy and differentiate the subtle distinctions among these conformations, resolution-tunable clustering was achieved. Furthermore, the integration of DIKI with a path-finding algorithm contributes to the identification of crucial intermediates along the lowest free-energy pathway. We postulate that DIKI is a robust and flexible tool that can find widespread applications in the analysis of complex biochemical processes.

Markov State Models: To Optimize or Not to Optimize

Markov state models (MSM) are a popular statistical method for analyzing the conformational dynamics of proteins including protein folding. With all statistical and machine learning (ML) models, choices must be made about the modeling pipeline that cannot be directly learned from the data. These choices, or hyperparameters, are often evaluated by expert judgment or, in the case of MSMs, by maximizing variational scores such as the VAMP-2 score. Modern ML and statistical pipelines often use automatic hyperparameter selection techniques ranging from the simple, choosing the best score from a random selection of hyperparameters, to the complex, optimization via, e.g., Bayesian optimization. In this work, we ask whether it is possible to automatically select MSM models this way by estimating and analyzing over 16,000,000 observations from over 280,000 estimated MSMs. We find that differences in hyperparameters can change the physical interpretation of the optimization objective, making automatic selection difficult. In addition, we find that enforcing conditions of equilibrium in the VAMP scores can result in inconsistent model selection. However, other parameters that specify the VAMP-2 score (lag time and number of relaxation processes scored) have only a negligible influence on model selection. We suggest that model observables and variational scores should be only a guide to model selection and that a full investigation of the MSM properties should be undertaken when selecting hyperparameters.

Revealing the conformational dynamics of UDP-GlcNAc recognition by O-GlcNAc transferase via Markov state model

The O-linked N-acetylglucosamine (O-GlcNAc) glycosylation is a critical post-translational modification and closely linked to various physiological and pathological conditions. The O-GlcNAc transferase (OGT) functions as the only glycosyltransferase of O-GlcNAc glycosylation by transferring GlcNAc from UDP-GlcNAc to serine or threonine residues on protein substrates. The interaction mode of UDP-GlcNAc against OGT has been preliminarily revealed by the crystal structures, yet an atomic-level comprehension for the conformational dynamics of the recognition process remains elusive. Here, we construct the Markov state model based on extensive all-atom molecular dynamics (MD) simulations with an aggregated simulation time of ∼9 μs, and reveal that the UDP-GlcNAc recognition process by OGT encompasses four key metastable states, occurring within an estimated timescale of ∼10 μs. During UDP-GlcNAc recognition process, we find the pyrophosphate moiety (P2O52-) initially anchors to the active pocket via salt bridge and hydrogen bonds, facilitating subsequent binding of the uridine and GlcNAc moieties. Furthermore, the functional roles of K842 involved in the salt bridge with P2O52- were evaluated through extra mutant MD simulations. Overall, our study provides valuable insights into the UDP-GlcNAc recognition mechanism by OGT, which could further aid in mechanistic studies of O-GlcNAc glycosylation and drug development targeting on OGT.

Resolving binding pathways and solvation thermodynamics of plant hormone receptors

Plant hormones are small molecules that regulate plant growth, development, and responses to biotic and abiotic stresses. They are specifically recognized by the binding site of their receptors. In this work, we resolved the binding pathways for eight classes of phytohormones (auxin, jasmonate, gibberellin, strigolactone, brassinosteroid, cytokinin, salicylic acid, and abscisic acid) to their canonical receptors using extensive molecular dynamics simulations. Furthermore, we investigated the role of water displacement and reorganization at the binding site of the plant receptors through inhomogeneous solvation theory. Our findings predict that displacement of water molecules by phytohormones contributes to free energy of binding via entropy gain and is associated with significant free energy barriers for most systems analyzed. Also, our results indicate that displacement of unfavorable water molecules in the binding site can be exploited in rational agrochemical design. Overall, this study uncovers the mechanism of ligand binding and the role of water molecules in plant hormone perception, which creates new avenues for agrochemical design to target plant growth and development.

Uncertainties in Markov State Models of Small Proteins

Markov state models are widely used to describe and analyze protein dynamics based on molecular dynamics simulations, specifically to extract functionally relevant characteristic time scales and motions. Particularly for larger biomolecules such as proteins, however, insufficient sampling is a notorious concern and often the source of large uncertainties that are difficult to quantify. Furthermore, there are several other sources of uncertainty, such as choice of the number of Markov states and lag time, choice and parameters of dimension reduction preprocessing step, and uncertainty due to the limited number of observed transitions; the latter is often estimated via a Bayesian approach. Here, we quantified and ranked all of these uncertainties for four small globular test proteins. We found that the largest uncertainty is due to insufficient sampling and initially increases with the total trajectory length T up to a critical tipping point, after which it decreases as 1 / T , thus providing guidelines for how much sampling is required for given accuracy. We also found that single long trajectories yielded better sampling accuracy than many shorter trajectories starting from the same structure. In comparison, the remaining sources of the above uncertainties are generally smaller by a factor of about 5, rendering them less of a concern but certainly not negligible. Importantly, the Bayes uncertainty, commonly used as the only uncertainty estimate, captures only a relatively small part of the true uncertainty, which is thus often drastically underestimated.

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.

Folding-upon-binding pathways of an intrinsically disordered protein from a deep Markov state model

A central challenge in the study of intrinsically disordered proteins is the characterization of the mechanisms by which they bind their physiological interaction partners. Here, we utilize a deep learning based Markov state modeling approach to characterize the folding-upon-binding pathways observed in a long-time scale molecular dynamics simulation of a disordered region of the measles virus nucleoprotein NTAIL reversibly binding the X domain of the measles virus phosphoprotein complex. We find that folding-upon-binding predominantly occurs via two distinct encounter complexes that are differentiated by the binding orientation, helical content, and conformational heterogeneity of NTAIL. We do not, however, find evidence for the existence of canonical conformational selection or induced fit binding pathways. We observe four kinetically separated native-like bound states that interconvert on time scales of eighty to five hundred nanoseconds. These bound states share a core set of native intermolecular contacts and stable NTAIL helices and are differentiated by a sequential formation of native and non-native contacts and additional helical turns. Our analyses provide an atomic resolution structural description of intermediate states in a folding-upon-binding pathway and elucidate the nature of the kinetic barriers between metastable states in a dynamic and heterogenous, or "fuzzy", protein complex.

An Efficient Path Classification Algorithm Based on Variational Autoencoder to Identify Metastable Path Channels for Complex Conformational Changes

Conformational changes (i.e., dynamic transitions between pairs of conformational states) play important roles in many chemical and biological processes. Constructing the Markov state model (MSM) from extensive molecular dynamics (MD) simulations is an effective approach to dissect the mechanism of conformational changes. When combined with transition path theory (TPT), MSM can be applied to elucidate the ensemble of kinetic pathways connecting pairs of conformational states. However, the application of TPT to analyze complex conformational changes often results in a vast number of kinetic pathways with comparable fluxes. This obstacle is particularly pronounced in heterogeneous self-assembly and aggregation processes. The large number of kinetic pathways makes it challenging to comprehend the molecular mechanisms underlying conformational changes of interest. To address this challenge, we have developed a path classification algorithm named latent-space path clustering (LPC) that efficiently lumps parallel kinetic pathways into distinct metastable path channels, making them easier to comprehend. In our algorithm, MD conformations are first projected onto a low-dimensional space containing a small set of collective variables (CVs) by time-structure-based independent component analysis (tICA) with kinetic mapping. Then, MSM and TPT are constructed to obtain the ensemble of pathways, and a deep learning architecture named the variational autoencoder (VAE) is used to learn the spatial distributions of kinetic pathways in the continuous CV space. Based on the trained VAE model, the TPT-generated ensemble of kinetic pathways can be embedded into a latent space, where the classification becomes clear. We show that LPC can efficiently and accurately identify the metastable path channels in three systems: a 2D potential, the aggregation of two hydrophobic particles in water, and the folding of the Fip35 WW domain. Using the 2D potential, we further demonstrate that our LPC algorithm outperforms the previous path-lumping algorithms by making substantially fewer incorrect assignments of individual pathways to four path channels. We expect that LPC can be widely applied to identify the dominant kinetic pathways underlying complex conformational changes.

Allosterically coupled conformational dynamics in solution prepare the sterol transfer protein StarD4 to release its cargo upon interaction with target membranes

Complex mechanisms regulate the cellular distribution of cholesterol, a critical component of eukaryote membranes involved in regulation of membrane protein functions directly and through the physiochemical properties of membranes. StarD4, a member of the steroidogenic acute regulator-related lipid-transfer (StART) domain (StARD)-containing protein family, is a highly efficient sterol-specific transfer protein involved in cholesterol homeostasis. Its mechanism of cargo loading and release remains unknown despite recent insights into the key role of phosphatidylinositol phosphates in modulating its interactions with target membranes. We have used large-scale atomistic Molecular dynamics (MD) simulations to study how the dynamics of cholesterol bound to the StarD4 protein can affect interaction with target membranes, and cargo delivery. We identify the two major cholesterol (CHL) binding modes in the hydrophobic pocket of StarD4, one near S136&S147 (the Ser-mode), and another closer to the putative release gate located near W171, R92&Y117 (the Trp-mode). We show that conformational changes of StarD4 associated directly with the transition between these binding modes facilitate the opening of the gate. To understand the dynamics of this connection we apply a machine-learning algorithm for the detection of rare events in MD trajectories (RED), which reveals the structural motifs involved in the opening of a front gate and a back corridor in the StarD4 structure occurring together with the spontaneous transition of CHL from the Ser-mode of binding to the Trp-mode. Further analysis of MD trajectory data with the information-theory based NbIT method reveals the allosteric network connecting the CHL binding site to the functionally important structural components of the gate and corridor. Mutations of residues in the allosteric network are shown to affect the performance of the allosteric connection. These findings outline an allosteric mechanism which prepares the CHL-bound StarD4 to release and deliver the cargo when it is bound to the target membrane.

Distinct activation mechanisms regulate subtype selectivity of Cannabinoid receptors

Design of cannabinergic subtype selective ligands is challenging because of high sequence and structural similarities of cannabinoid receptors (CB₁ and CB₂). We hypothesize that the subtype selectivity of designed selective ligands can be explained by the ligand binding to the conformationally distinct states between cannabinoid receptors. Analysis of ~ 700 μs of unbiased simulations using Markov state models and VAMPnets identifies the similarities and distinctions between the activation mechanism of both receptors. Structural and dynamic comparisons of metastable intermediate states allow us to observe the distinction in the binding pocket volume change during CB₁ and CB₂ activation. Docking analysis reveals that only a few of the intermediate metastable states of CB₁ show high affinity towards CB₂ selective agonists. In contrast, all the CB₂ metastable states show a similar affinity for these agonists. These results mechanistically explain the subtype selectivity of these agonists by deciphering the activation mechanism of cannabinoid receptors.

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

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

Drug specificity and affinity are encoded in the probability of cryptic pocket opening in myosin motor domains

The design of compounds that can discriminate between closely related target proteins remains a central challenge in drug discovery. Specific therapeutics targeting the highly conserved myosin motor family are urgently needed as mutations in at least six of its members cause numerous diseases. Allosteric modulators, like the myosin-II inhibitor blebbistatin, are a promising means to achieve specificity. However, it remains unclear why blebbistatin inhibits myosin-II motors with different potencies given that it binds at a highly conserved pocket that is always closed in blebbistatin-free experimental structures. We hypothesized that the probability of pocket opening is an important determinant of the potency of compounds like blebbistatin. To test this hypothesis, we used Markov state models (MSMs) built from over 2 ms of aggregate molecular dynamics simulations with explicit solvent. We find that blebbistatin's binding pocket readily opens in simulations of blebbistatin-sensitive myosin isoforms. Comparing these conformational ensembles reveals that the probability of pocket opening correctly identifies which isoforms are most sensitive to blebbistatin inhibition and that docking against MSMs quantitatively predicts blebbistatin binding affinities (R²=0.82). In a blind prediction for an isoform (Myh7b) whose blebbistatin sensitivity was unknown, we find good agreement between predicted and measured IC50s (0.67 μM vs. 0.36 μM). Therefore, we expect this framework to be useful for the development of novel specific drugs across numerous protein targets.

Mechanistic Basis for Enhanced Strigolactone Sensitivity in KAI2 Triple Mutant

Striga hermonthica is a parasitic weed that destroys billions of dollars' worth of staple crops every year. Its rapid proliferation stems from an enhanced ability to metabolize strigolactones (SLs), plant hormones that direct root branching and shoot growth. Striga's SL receptor, ShHTL7, bears more similarity to the staple crop karrikin receptor KAI2 than to SL receptor D14, though KAI2 variants in plants like Arabidopsis thaliana show minimal SL sensitivity. Recently, studies have indicated that a small number of point mutations to HTL7 residues can confer SL sensitivity to AtKAI2. Here, we analyze both wild-type AtKAI2 and SL-sensitive mutant Var64 through all-atom, long-timescale molecular dynamics simulations to determine the effects of these mutations on receptor function at a molecular level. We demonstrate that the mutations stabilize SL binding by about 2 kcal/mol. They also result in a doubling of the average pocket volume, and eliminate the dependence of binding on certain pocket conformational arrangements. While the probability of certain non-binding SL-receptor interactions increases in the mutant compared with the wild-type, the rate of binding also increases by a factor of ten. All these changes account for the increased SL sensitivity in mutant KAI2, and suggest mechanisms for increasing functionality of host crop SL receptors.

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.

Deep learning to decompose macromolecules into independent Markovian domains

The increasing interest in modeling the dynamics of ever larger proteins has revealed a fundamental problem with models that describe the molecular system as being in a global configuration state. This notion limits our ability to gather sufficient statistics of state probabilities or state-to-state transitions because for large molecular systems the number of metastable states grows exponentially with size. In this manuscript, we approach this challenge by introducing a method that combines our recent progress on independent Markov decomposition (IMD) with VAMPnets, a deep learning approach to Markov modeling. We establish a training objective that quantifies how well a given decomposition of the molecular system into independent subdomains with Markovian dynamics approximates the overall dynamics. By constructing an end-to-end learning framework, the decomposition into such subdomains and their individual Markov state models are simultaneously learned, providing a data-efficient and easily interpretable summary of the complex system dynamics. While learning the dynamical coupling between Markovian subdomains is still an open issue, the present results are a significant step towards learning Ising models of large molecular complexes from simulation data.

Investigating the Bioactive Conformation of Angiotensin II Using Markov State Modeling Revisited with Web-Scale Clustering

Molecular dynamics simulation is a powerful technique for studying the structure and dynamics of biomolecules in atomic-level detail by sampling their various conformations in real time. Because of the long timescales that need to be sampled to study biomolecular processes and the big and complex nature of the corresponding data, relevant analyses of important biophysical phenomena are challenging. Clustering and Markov state models (MSMs) are efficient computational techniques that can be used to extract dominant conformational states and to connect those with kinetic information. In this work, we perform Molecular Dynamics simulations to investigate the free energy landscape of Angiotensin II (AngII) in order to unravel its bioactive conformations using different clustering techniques and Markov state modeling. AngII is an octapeptide hormone, which binds to the AT1 transmembrane receptor, and plays a vital role in the regulation of blood pressure, conservation of total blood volume, and salt homeostasis. To mimic the water-membrane interface as AngII approaches the AT1 receptor and to compare our findings with available experimental results, the simulations were performed in water as well as in water-ethanol mixtures. Our results show that in the water-ethanol environment, AngII adopts more compact U-shaped (folded) conformations than in water, which resembles its structure when bound to the AT1 receptor. For clustering of the conformations, we validate the efficiency of an inverted-quantized k-means algorithm, as a fast approximate clustering technique for web-scale data (millions of points into thousands or millions of clusters) compared to k-means, on data from trajectories of molecular dynamics simulations with reasonable trade-offs between time and accuracy. Finally, we extract MSMs using various clustering techniques for the generation of microstates and macrostates and for the selection of the macrostate representatives.

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.

GraphVAMPNet, using graph neural networks and variational approach to Markov processes for dynamical modeling of biomolecules

Finding low dimensional representation of data from long-timescale trajectories of biomolecular processes such as protein-folding or ligand-receptor binding is of fundamental importance and kinetic models such as Markov modeling have proven useful in describing the kinetics of these systems. Recently, an unsupervised machine learning technique called VAMPNet was introduced to learn the low dimensional representation and linear dynamical model in an end-to-end manner. VAMPNet is based on variational approach to Markov processes (VAMP) and relies on neural networks to learn the coarse-grained dynamics. In this contribution, we combine VAMPNet and graph neural networks to generate an end-to-end framework to efficiently learn high-level dynamics and metastable states from the long-timescale molecular dynamics trajectories. This method bears the advantages of graph representation learning and uses graph message passing operations to generate an embedding for each datapoint which is used in the VAMPNet to generate a coarse-grained representation. This type of molecular representation results in a higher resolution and more interpretable Markov model than the standard VAMPNet enabling a more detailed kinetic study of the biomolecular processes. Our GraphVAMPNet approach is also enhanced with an attention mechanism to find the important residues for classification into different metastable states.

Activation Pathways of Neurotensin Receptor 1 Elucidated Using Statistical Machine Learning

Neurotensin receptor 1 (NTSR1) is a G-protein coupled receptor (GPCR) that mediates many biological processes through its interaction with the neurotensin (NTS) peptide. The NTSR1 protein is a clinically significant target as it is involved in the proliferation of cancer cells. Understanding the activation mechanism of NTSR1 is an important prerequisite for exploring the therapeutic potential of targeting NTSR1 and the development of drug molecules specific to NTSR1. Previous studies have been aimed at elucidating the structure of NTSR1 in the active and inactive conformations; however, the intermediate molecular pathway for NTSR1 activation dynamics is largely unknown. In this study, we performed extensive molecular dynamics (MD) simulations of the NTSR1 protein and analyzed its kinetic conformational changes to determine the microswitches that drive NTSR1 activation. To biophysically interpret the high-dimensional simulation trajectories, we used Markov state models and machine learning to elucidate the important and detailed conformational changes in NTSR1. Through the analysis of identified microswitches, we propose a mechanistic pathway for NTSR1 activation.

Mechanistic Kinetic Model Reveals How Amyloidogenic Hydrophobic Patches Facilitate the Amyloid-β Fibril Elongation

Abnormal aggregation of amyloid β (Aβ) peptides into fibrils plays a critical role in the development of Alzheimer's disease. A two-stage "dock-lock" model has been proposed for the Aβ fibril elongation process. However, the mechanisms of the Aβ monomer-fibril binding process have not been elucidated with the necessary molecular-level precision, so it remains unclear how the lock phase dynamics leads to the overall in-register binding of the Aβ monomer onto the fibril. To gain mechanistic insights into this critical step during the fibril elongation process, we used molecular dynamics (MD) simulations with a physics-based coarse-grained UNited-RESidue (UNRES) force field and sampled extensively the dynamics of the lock phase process, in which a fibril-bound Aβ_(9-40) peptide rearranged to establish the native docking conformation. Analysis of the MD trajectories with Markov state models was used to quantify the kinetics of the rearrangement process and the most probable pathways leading to the overall native docking conformation of the incoming peptide. These revealed a key intermediate state in which an intra-monomer hairpin is formed between the central core amyloidogenic patch ₁₈VFFA₂₁ and the C-terminal hydrophobic patch ₃₄LMVG₃₇. This hairpin structure is highly favored as a transition state during the lock phase of the fibril elongation. We propose a molecular mechanism for facilitation of the Aβ fibril elongation by amyloidogenic hydrophobic patches.

Dual Role of Strigolactone Receptor Signaling Partner in Inhibiting Substrate Hydrolysis

Plant branch and root growth relies on metabolism of the strigolactone (SL) hormone. The interaction between the SL molecule, Oryza sativa DWARF14 (D14) SL receptor, and D3 F-box protein has been shown to play a critical role in SL perception. Previously, it was believed that D3 only interacts with the closed form of D14 to induce downstream signaling, but recent experiments indicate that D3, as well as its C-terminal helix (CTH), can interact with the open form as well to inhibit strigolactone signaling. Two hypotheses for the CTH induced inhibition are that either the CTH affects the conformational ensemble of D14 by stabilizing catalytically inactive states or the CTH interacts with SLs in a way that prevents them from entering the binding pocket. In this study, we have performed molecular dynamics (MD) simulations to assess the validity of these hypotheses. We used an apo system with only D14 and the CTH to test the active site conformational stability and a holo system with D14, the CTH, and an SL molecule to test the interaction between the SL and CTH. Our simulations show that the CTH affects both active site conformation and the ability of SLs to move into the binding pocket. In the apo system, the CTH allosterically stabilized catalytic residues into their inactive conformation. In the holo system, significant interactions between SLs and the CTH hindered the ability of SLs to enter the D14 binding pocket. These two mechanisms account for the observed decrease in SL binding to D14 and subsequent ligand hydrolysis in the presence of the CTH.

Deeptime: a Python library for machine learning dynamical models from time series data

Generation and analysis of time-series data is relevant to many quantitative fields ranging from economics to fluid mechanics. In the physical sciences, structures such as metastable and coherent sets, slow relaxation processes, collective variables, dominant transition pathways or manifolds and channels of probability flow can be of great importance for understanding and characterizing the kinetic, thermodynamic and mechanistic properties of the system. Deeptime is a general purpose Python library offering various tools to estimate dynamical models based on time-series data including conventional linear learning methods, such as Markov state models (MSMs), Hidden Markov Models and Koopman models, as well as kernel and deep learning approaches such as VAMPnets and deep MSMs. The library is largely compatible with scikit-learn, having a range of Estimator classes for these different models, but in contrast to scikit-learn also provides deep Model classes, e.g. in the case of an MSM, which provide a multitude of analysis methods to compute interesting thermodynamic, kinetic and dynamical quantities, such as free energies, relaxation times and transition paths. The library is designed for ease of use but also easily maintainable and extensible code. In this paper we introduce the main features and structure of the deeptime software. Deeptime can be found under https://deeptime-ml.github.io/.

Activation Mechanism of Strigolactone Receptors and Its Impact on Ligand Selectivity between Host and Parasitic Plants

Parasitic weeds such as Striga have led to significant losses in agricultural productivity worldwide. These weeds use the plant hormone strigolactone as a germination stimulant. Strigolactone signaling involves substrate hydrolysis followed by a conformational change of the receptor to a "closed" or "active" state that associates with a signaling partner, MAX2/D3. Crystal structures of active and inactive AtD14 receptors have helped elucidate the structural changes involved in activation. However, the mechanism by which the receptor activates remains unknown. The ligand dependence of AtD14 activation has been disputed by mutagenesis studies showing that enzymatically inactive receptors are able to associate with MAX2 proteins. Furthermore, activation differences between strigolactone receptor in Striga, ShHTL7, and AtD14 could contribute to the high sensitivity to strigolactones exhibited by parasitic plants. Using molecular dynamics simulations, we demonstrate that both AtD14 and ShHTL7 could adopt an active conformation in the absence of ligand. However, ShHTL7 exhibits a higher population in the inactive apo state as compared to the AtD14 receptor. We demonstrate that this difference in inactive state population is caused by sequence differences between their D-loops and interactions with the catalytic histidine that prevent full binding pocket closure in ShHTL7. These results indicate that ligand hydrolysis would enhance the active state population by destabilizing the inactive state in ShHTL7 as compared to AtD14. We also show that the mechanism of activation is more concerted in AtD14 than in ShHTL7 and that the main barrier to activation in ShHTL7 is closing of the binding pocket.

RPnet: a reverse-projection-based neural network for coarse-graining metastable conformational states for protein dynamics

The Markov State Model (MSM) is a powerful tool for modeling long timescale dynamics based on numerous short molecular dynamics (MD) simulation trajectories, which makes it a useful tool for elucidating the conformational changes of biological macromolecules. By partitioning the phase space into discretized states and estimating the probabilities of inter-state transitions based on short MD trajectories, one can construct a kinetic network model that could be used to extrapolate long-timescale kinetics if the Markovian condition is met. However, meeting the Markovian condition often requires hundreds or even thousands of states (microstates), which greatly hinders the comprehension of the conformational dynamics of complex biomolecules. Kinetic lumping algorithms can coarse grain numerous microstates into a handful of metastable states (macrostates), which would greatly facilitate the elucidation of biological mechanisms. In this work, we have developed a reverse-projection-based neural network (RPnet) to lump microstates into macrostates, by making use of a physics-based loss function that is based on the projection operator framework of conformational dynamics. By recognizing that microstate and macrostate transition modes can be related through a projection process, we have developed a reverse-projection scheme to directly compare the microstate and macrostate dynamics. Based on this reverse-projection scheme, we designed a loss function that allows the effective assessment of the quality of a given kinetic lumping. We then make use of a neural network to efficiently minimize this loss function to obtain an optimized set of macrostates. We have demonstrated the power of our RPnet in analyzing the dynamics of a numerical 2D potential, alanine dipeptide, and the clamp opening of an RNA polymerase. In all these systems, we have illustrated that our method could yield comparable or better results than competing methods in terms of state partitioning and reproduction of slow dynamics. We expect that our RPnet holds promise in analyzing the conformational dynamics of biological macromolecules.

Progress in deep Markov state modeling: Coarse graining and experimental data restraints

Recent advances in deep learning frameworks have established valuable tools for analyzing the long-timescale behavior of complex systems, such as proteins. In particular, the inclusion of physical constraints, e.g., time-reversibility, was a crucial step to make the methods applicable to biophysical systems. Furthermore, we advance the method by incorporating experimental observables into the model estimation showing that biases in simulation data can be compensated for. We further develop a new neural network layer in order to build a hierarchical model allowing for different levels of details to be studied. Finally, we propose an attention mechanism, which highlights important residues for the classification into different states. We demonstrate the new methodology on an ultralong molecular dynamics simulation of the Villin headpiece miniprotein.

Critical role of backbone coordination in the mRNA recognition by RNA induced silencing complex

Despite its functional importance, the molecular mechanism underlying target mRNA recognition by Argonaute (Ago) remains largely elusive. Based on extensive all-atom molecular dynamics simulations, we constructed quasi-Markov State Model (qMSM) to reveal the dynamics during recognition at position 6-7 in the seed region of human Argonaute 2 (hAgo2). Interestingly, we found that the slowest mode of motion therein is not the gRNA-target base-pairing, but the coordination of the target phosphate groups with a set of positively charged residues of hAgo2. Moreover, the ability of Helix-7 to approach the PIWI and MID domains was found to reduce the effective volume accessible to the target mRNA and therefore facilitate both the backbone coordination and base-pair formation. Further mutant simulations revealed that alanine mutation of the D358 residue on Helix-7 enhanced a trap state to slow down the loading of target mRNA. Similar trap state was also observed when wobble pairs were introduced in g6 and g7, indicating the role of Helix-7 in suppressing non-canonical base-paring. Our study pointed to a general mechanism for mRNA recognition by eukaryotic Agos and demonstrated the promise of qMSM in investigating complex conformational changes of biomolecular systems.

Molecular Simulations and Markov State Modeling Reveal Inactive Form of Quorum Sensing Regulator SdiA of Escherichia Coli

Enteropathogenic Escherichia coli remains one of the most important pathogens infecting children and it is one of the main causes of persistent diarrhea worldwide. Enteropathogenic Escherichia coli is capable of forming biofilms. Several E. coli mechanisms are regulated by quorum sensing, including virulence factors and biofilm formation. Quorum sensing is the communication system of bacteria with the ability to respond to chemical molecules known as autoinducers. Suppressor of division inhibitor (SdiA) is a quorum sensing receptor present in enteropathogenic E. coli in humans that detect acyl-homoserine lactone type autoinducers. SdiA receptor can also respond to autoinducers produced by other bacterial species that control cell division and virulence. SdiA is regulated by 1-octanoyl-rac-glycerol, which serves as an energy source, signaling molecule, and substrate for membrane biogenesis. SdiA is a potential target, which can be used as an anti-infectious technique. Current crystallographic structures for virtual screening may not be sufficient for molecular docking. So they are not very predictive, because the structures are in the active form. It has been shown that SdiA protein is not activated without a ligand. Generally, ligands bind to the ligand binding domain of SdiA. We employ Markov modeling and molecular dynamics simulations to understand the behaviour of SdiA protein and find the possible inactive form. We find an unknown conformation after 24 molecular dynamics simulation runs with random initial velocities and Markov state modeling. In summary, using molecular simulations and Markov state modeling, we have obtained an unknown conformation, which is not available in the crystallographic structures of SdiA. This unknown conformation could be the structure of the inactive form without a ligand. The obtained ensemble structures could be used for virtual screening.

Crystallization in Microgravity and the Atomic-Resolution Structure of Uridine Phosphorylase from Vibrio cholerae

Abstract

Uridine phosphorylases are known as key targets for the development of new anticancer and antiparasitic agents. Crystals of uridine phosphorylase from the pathogenic bacterium Vibrio cholerae were grown in microgravity by the capillary counter-diffusion method on board of the International Space Station. The three-dimensional structure of this enzyme was determined at atomic (1.04 Å) resolution (RCSB PDB ID: 6Z9Z). Alternative conformations of long fragments (β-strands and adjacent loops) of the protein molecule were found for the first time in the three-dimensional structure of uridine phosphorylase in the absence of specific bound ligands. Apparently, these alternative conformations are related to the enzyme function. Conformational analysis with Markov state models demonstrated that conformational rearrangements can occur in the ligand-free state of the enzyme.

Exploring the Regulatory Function of the N-terminal Domain of SARS-CoV-2 Spike Protein through Molecular Dynamics Simulation

SARS-CoV-2 is what has caused the COVID-19 pandemic. Early viral infection is mediated by the SARS-CoV-2 homo-trimeric Spike (S) protein with its receptor binding domains (RBDs) in the receptor-accessible state. Molecular dynamics simulation on the S protein with a focus on the function of its N-terminal domains (NTDs) is performed. The study reveals that the NTD acts as a "wedge" and plays a crucial regulatory role in the conformational changes of the S protein. The complete RBD structural transition is allowed only when the neighboring NTD that typically prohibits the RBD's movements as a wedge detaches and swings away. Based on this NTD "wedge" model, it is proposed that the NTD-RBD interface should be a potential drug target.

Markov State Models to Study the Functional Dynamics of Proteins in the Wake of Machine Learning

Markov state models (MSMs) based on molecular dynamics (MD) simulations are routinely employed to study protein folding, however, their application to functional conformational changes of biomolecules is still limited. In the past few years, the field of computational chemistry has experienced a surge of advancements stemming from machine learning algorithms, and MSMs have not been left out. Unlike global processes, such as protein folding, the application of MSMs to functional conformational changes is challenging because they mostly consist of localized structural transitions. Therefore, it is critical to properly select a subset of structural features that can describe the slowest dynamics of these functional conformational changes. To address this challenge, we recommend several automatic feature selection methods such as Spectral-OASIS. To identify states in MSMs, the chosen features can be subject to dimensionality reduction methods such as TICA or deep learning based VAMPNets to project MD conformations onto a few collective variables for subsequent clustering. Another challenge for the application of MSMs to the study of functional conformational changes is the ability to comprehend their biophysical mechanisms, as MSMs built for these processes often require a large number of states. We recommend the recently developed quasi-MSMs (qMSMs) to address this issue. Compared to MSMs, qMSMs encode the non-Markovian dynamics via the generalized master equation and can significantly reduce the number of states. As a result, qMSMs can be built with a handful of states to facilitate the interpretation of functional conformational changes. In the wake of machine learning, we believe that the rapid advancement in the MSM methodology will lead to their wider application in studying functional conformational changes of biomolecules.

Markov state models from hierarchical density-based assignment

Markov state models (MSMs) have become one of the preferred methods for the analysis and interpretation of molecular dynamics (MD) simulations of conformational transitions in biopolymers. While there is great variation in terms of implementation, a well-defined workflow involving multiple steps is often adopted. Typically, molecular coordinates are first subjected to dimensionality reduction and then clustered into small "microstates," which are subsequently lumped into "macrostates" using the information from the slowest eigenmodes. However, the microstate dynamics is often non-Markovian, and long lag times are required to converge the relevant slow dynamics in the MSM. Here, we propose a variation on this typical workflow, taking advantage of hierarchical density-based clustering. When applied to simulation data, this type of clustering separates high population regions of conformational space from others that are rarely visited. In this way, density-based clustering naturally implements assignment of the data based on transitions between metastable states, resulting in a core-set MSM. As a result, the state definition becomes more consistent with the assumption of Markovianity, and the timescales of the slow dynamics of the system are recovered more effectively. We present results of this simplified workflow for a model potential and MD simulations of the alanine dipeptide and the FiP35 WW domain.

Role of substrate recognition in modulating strigolactone receptor selectivity in witchweed

Witchweed, or Striga hermonthica, is a parasitic weed that destroys billions of dollars' worth of crops globally every year. Its germination is stimulated by strigolactones exuded by its host plants. Despite high sequence, structure, and ligand-binding site conservation across different plant species, one strigolactone receptor in witchweed, ShHTL7, uniquely exhibits a picomolar EC50 for downstream signaling. Previous biochemical and structural analyses have hypothesized that this unique ligand sensitivity can be attributed to a large binding pocket volume in ShHTL7 resulting in enhanced ability to bind substrates, but additional structural details of the substrate-binding process would help explain its role in modulating the ligand selectivity. Using long-timescale molecular dynamics simulations, we demonstrate that mutations at the entrance of the binding pocket facilitate a more direct ligand-binding pathway to ShHTL7, whereas hydrophobicity at the binding pocket entrance results in a stable “anchored” state. We also demonstrate that several residues on the D-loop of AtD14 stabilize catalytically inactive conformations. Finally, we show that strigolactone selectivity is not modulated by binding pocket volume. Our results indicate that while ligand binding is not the sole modulator of strigolactone receptor selectivity, it is a significant contributing factor. These results can be used to inform the design of selective antagonists for strigolactone receptors in witchweed.

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.

Temperature-dependent kinetic pathways of heterogeneous ice nucleation competing between classical and non-classical nucleation

Ice nucleation on the surface plays a vital role in diverse areas, ranging from physics and cryobiology to atmospheric science. Compared to ice nucleation in the bulk, the water-surface interactions present in heterogeneous ice nucleation complicate the nucleation process, making heterogeneous ice nucleation less comprehended, especially the relationship between the kinetics and the structures of the critical ice nucleus. Here we combine Markov State Models and transition path theory to elucidate the ensemble pathways of heterogeneous ice nucleation. Our Markov State Models reveal that the classical one-step and non-classical two-step nucleation pathways can surprisingly co-exist with comparable fluxes at T = 230 K. Interestingly, we find that the disordered mixing of rhombic and hexagonal ice leads to a favorable configurational entropy that stabilizes the critical nucleus, facilitating the non-classical pathway. In contrast, the favorable energetics promotes the formation of hexagonal ice, resulting in the classical pathway. Furthermore, we discover that, at elevated temperatures, the nucleation process prefers to proceed via the classical pathway, as opposed to the non-classical pathway, since the potential energy contributions override the configurational entropy compensation. This study provides insights into the mechanisms of heterogeneous ice nucleation and sheds light on the rational designs to control crystallization processes.

Independent Markov decomposition: Toward modeling kinetics of biomolecular complexes

To advance the mission of in silico cell biology, modeling the interactions of large and complex biological systems becomes increasingly relevant. The combination of molecular dynamics (MD) simulations and Markov state models (MSMs) has enabled the construction of simplified models of molecular kinetics on long timescales. Despite its success, this approach is inherently limited by the size of the molecular system. With increasing size of macromolecular complexes, the number of independent or weakly coupled subsystems increases, and the number of global system states increases exponentially, making the sampling of all distinct global states unfeasible. In this work, we present a technique called independent Markov decomposition (IMD) that leverages weak coupling between subsystems to compute a global kinetic model without requiring the sampling of all combinatorial states of subsystems. We give a theoretical basis for IMD and propose an approach for finding and validating such a decomposition. Using empirical few-state MSMs of ion channel models that are well established in electrophysiology, we demonstrate that IMD models can reproduce experimental conductance measurements with a major reduction in sampling compared with a standard MSM approach. We further show how to find the optimal partition of all-atom protein simulations into weakly coupled subunits.

Solution-State Preorganization of Cyclic β-Hairpin Ligands Determines Binding Mechanism and Affinities for MDM2

Understanding mechanisms of protein folding and binding is crucial to designing their molecular function. Molecular dynamics (MD) simulations and Markov state model (MSM) approaches provide a powerful way to understand complex conformational change that occurs over long time scales. Such dynamics are important for the design of therapeutic peptidomimetic ligands, whose affinity and binding mechanism are dictated by a combination of folding and binding. To examine the role of preorganization in peptide binding to protein targets, we performed massively parallel explicit-solvent MD simulations of cyclic β-hairpin ligands designed to mimic the p53 transactivation domain and competitively bind mouse double minute 2 homologue (MDM2). Disrupting the MDM2-p53 interaction is a therapeutic strategy to prevent degradation of the p53 tumor suppressor in cancer cells. MSM analysis of over 3 ms of aggregate trajectory data enabled us to build a detailed mechanistic model of coupled folding and binding of four cyclic peptides which we compare to experimental binding affinities and rates. The results show a striking relationship between the relative preorganization of each ligand in solution and its affinity for MDM2. Specifically, changes in peptide conformational populations predicted by the MSMs suggest that entropy loss upon binding is the main factor influencing affinity. The MSMs also enable detailed examination of non-native interactions which lead to misfolded states and comparison of structural ensembles with experimental NMR measurements. In contrast to an MSM study of p53 transactivation domain (TAD) binding to MDM2, MSMs of cyclic β-hairpin binding show a conformational selection mechanism. Finally, we make progress toward predicting accurate off rates of cyclic peptides using multiensemble Markov models (MEMMs) constructed from unbiased and biased simulated trajectories.

A comprehensive mechanism for 5-carboxylcytosine-induced transcriptional pausing revealed by Markov state models

RNA polymerase II (Pol II) surveils the genome, pausing as it encounters DNA lesions and base modifications and initiating signals for DNA repair among other important regulatory events. Recent work suggests that Pol II pauses at 5-carboxycytosine (5caC), an epigenetic modification of cytosine, because of a specific hydrogen bond between the carboxyl group of 5caC and a specific residue in fork loop 3 of Pol II. This hydrogen bond compromises productive NTP binding and slows down elongation. Apart from this specific interaction, the carboxyl group of 5caC can potentially interact with numerous charged residues in the cleft of Pol II. However, it is not clear how other interactions between Pol II and 5caC contribute to pausing. In this study, we use Markov state models (a type of kinetic network models) built from extensive molecular dynamics simulations to comprehensively study the impact of 5caC on Pol II translocation. We describe two translocation intermediates with specific interactions that prevent the template base from loading into the Pol II active site. In addition to the previously observed state with 5caC constrained by fork loop 3, we discovered a new intermediate state with a hydrogen bond between 5caC and fork loop 2. Surprisingly, we find that 5caC may curb translocation by suppressing kinking of the helix bordering the active site (the bridge helix) because its high flexibility is critical to translocation. Our work provides new insights into how epigenetic modifications of genomic DNA can modulate Pol II translocation, inducing pauses in transcription.

What Markov State Models Can and Cannot Do: Correlation versus Path-Based Observables in Protein-Folding Models

Markov state models (MSMs) have been widely applied to study the kinetics and pathways of protein conformational dynamics based on statistical analysis of molecular dynamics (MD) simulations. These MSMs coarse-grain both configuration space and time in ways that limit what kinds of observables they can reproduce with high fidelity over different spatial and temporal resolutions. Despite their popularity, there is still limited understanding of which biophysical observables can be computed from these MSMs in a robust and unbiased manner, and which suffer from the space-time coarse-graining intrinsic in the MSM model. Most theoretical arguments and practical validity tests for MSMs rely on long-time equilibrium kinetics, such as the slowest relaxation time scales and experimentally observable time-correlation functions. Here, we perform an extensive assessment of the ability of well-validated protein folding MSMs to accurately reproduce path-based observable such as mean first-passage times (MFPTs) and transition path mechanisms compared to a direct trajectory analysis. We also assess a recently proposed class of history-augmented MSMs (haMSMs) that exploit additional information not accounted for in standard MSMs. We conclude with some practical guidance on the use of MSMs to study various problems in conformational dynamics of biomolecules. In brief, MSMs can accurately reproduce correlation functions slower than the lag time, but path-based observables can only be reliably reproduced if the lifetimes of states exceed the lag time, which is a much stricter requirement. Even in the presence of short-lived states, we find that haMSMs reproduce path-based observables more reliably.

Role of bacterial RNA polymerase gate opening dynamics in DNA loading and antibiotics inhibition elucidated by quasi-Markov State Model

To initiate transcription, the holoenzyme (RNA polymerase [RNAP] in complex with σ factor) loads the promoter DNA via the flexible loading gate created by the clamp and β-lobe, yet their roles in DNA loading have not been characterized. We used a quasi-Markov State Model (qMSM) built from extensive molecular dynamics simulations to elucidate the dynamics of Thermus aquaticus holoenzyme's gate opening. We showed that during gate opening, β-lobe oscillates four orders of magnitude faster than the clamp, whose opening depends on the Switch 2's structure. Myxopyronin, an antibiotic that binds to Switch 2, was shown to undergo a conformational selection mechanism to inhibit clamp opening. Importantly, we reveal a critical but undiscovered role of β-lobe, whose opening is sufficient for DNA loading even when the clamp is partially closed. These findings open the opportunity for the development of antibiotics targeting β-lobe of RNAP. Finally, we have shown that our qMSMs, which encode non-Markovian dynamics based on the generalized master equation formalism, hold great potential to be widely applied to study biomolecular dynamics.

Analyzing the Long Time-Scale Dynamics of Uremic Toxins Bound to Sudlow Site II in Human Serum Albumin

Protein bound uremic toxins (PBUTs), a series of chemicals that remain a challenge for removal strategies used on patients suffering with chronic kidney disease, could be strong candidates for MD study in order to better understand the interactions and time scales associated with binding mode transitions. Currently, traditional dialysis methods cannot satisfactorily remove PBUTs from the bloodstream. This is at least partly due to these toxin's high level of affinity for protein binding sites, particularly the prominent human serum albumin (HSA) and two of its drug binding sites (Sudlow site I and II). We investigate the dynamics of binding site transitions and interactions by MD simulations targeting four well-known toxins: indoxyl sulfate (IS), p-cresyl sulfate (PCS), indole-3-acetic acid (IAA), and hippurate acid (HIP). Long-time scale dynamics are obtained by the use of time-structure independent component analysis (tICA) for dimensionality reduction followed by spectral analysis of a Markov state model (MSM) scored using the generalized matrix Rayleigh quotient (GMRQ). Our results add new insights to prior findings related to the key role of charge-pairing in governing toxin-protein interactions. We find that IAA, the bulkiest hydrophobic toxin studied, observes the slowest process of at least 3 times slower than the smaller, less hydrophobic toxins. In general, we find that the processes slower than 15 ns are correlated with a transition from dominantly hydrophobic interactions deep in the binding pocket to a gain in hydrogen bonding partners near the mouth of the pocket. Our results indicate that aromatic residues such as PHE play a part in a type of toxin stabilization akin to π-stacking. In conclusion, this work presents mechanistic descriptions of interactions/transitions for a set of important PBUTs that bind Sudlow site II on time scales relevant to the underlying binding kinetics of most interest.