Active Learning of the Conformational Ensemble of Proteins Using Maximum Entropy VAMPNets.

Rapid computational exploration of the free energy landscape of biological molecules remains an active area of research due to the difficulty of sampling rare state transitions in molecular dynamics (MD) simulations. In recent years, an increasing number of studies have exploited machine learning (ML) models to enhance and analyze MD simulations. Notably, unsupervised models that extract kinetic information from a set of parallel trajectories have been proposed including the variational approach for Markov processes (VAMP), VAMPNets, and time-lagged variational autoencoders (TVAE). In this work, we propose a combination of adaptive sampling with active learning of kinetic models to accelerate the discovery of the conformational landscape of biomolecules. In particular, we introduce and compare several techniques that combine kinetic models with two adaptive sampling regimes (least counts and multiagent reinforcement learning-based adaptive sampling) to enhance the exploration of conformational ensembles without introducing biasing forces. Moreover, inspired by the active learning approach of uncertainty-based sampling, we also present MaxEnt VAMPNet. This technique consists of restarting simulations from the microstates that maximize the Shannon entropy of a VAMPNet trained to perform the soft discretization of metastable states. By running simulations on two test systems, the WLALL pentapeptide and the villin headpiece subdomain, we empirically demonstrate that MaxEnt VAMPNet results in faster exploration of conformational landscapes compared with the baseline and other proposed methods.

Markov state modeling of membrane transport proteins.

The flux of ions and molecules in and out of the cell is vital for maintaining the basis of various biological processes. The permeation of substrates across the cellular membrane is mediated through the function of specialized integral membrane proteins commonly known as membrane transporters. These proteins undergo a series of structural rearrangements that allow a primary substrate binding site to be accessed from either side of the membrane at a given time. Structural insights provided by experimentally resolved structures of membrane transporters have aided in the biophysical characterization of these important molecular drug targets. However, characterizing the transitions between conformational states remains challenging to achieve both experimentally and computationally. Though molecular dynamics simulations are a powerful approach to provide atomistic resolution of protein dynamics, a recurring challenge is its ability to efficiently obtain relevant timescales of large conformational transitions as exhibited in transporters. One approach to overcome this difficulty is to adaptively guide the simulation to favor exploration of the conformational landscape, otherwise known as adaptive sampling. Furthermore, such sampling is greatly benefited by the statistical analysis of Markov state models. Historically, the use of Markov state models has been effective in quantifying slow dynamics or long timescale behaviors such as protein folding. Here, we review recent implementations of adaptive sampling and Markov state models to not only address current limitations of molecular dynamics simulations, but to also highlight how Markov state modeling can be applied to investigate the structure-function mechanisms of large, complex membrane transporters.

Maximizing Kinetic Information Gain of Markov State Models for Optimal Design of Spectroscopy Experiments.

Spectroscopic techniques such as Trp-Tyr quenching, luminescence resonance energy transfer, and triplet-triplet energy transfer are widely used for understanding the dynamic behavior of proteins. These experiments measure the relaxation of a particular labeled set of residue pairs, and the choice of residue pairs requires careful thought. As a result, experimentalists must pick residue pairs from a large pool of possibilities. In the current work, we show that molecular simulation datasets of protein dynamics can be used to systematically select an optimal set of residue positions to place probes for conducting spectroscopic experiments. The method described in this work, called Optimal Probes, can be used to rank trial sets of residue pairs in terms of their ability to capture the conformational dynamics of the protein. Optimal probes ensures two conditions: residue pairs capture the slow dynamics of the protein and their dynamics is not correlated for maximum information gain to score each trial set. Eventually, the highest scored set can be used for biophysical experiments to study the kinetics of the protein. The scoring methodology is based on kinetic network models of protein dynamics and a variational principle for molecular kinetics to optimize the hyperparameters used for the model. We also discuss that the scoring strategy used by Optimal Probes is the best possible way to ensure the ideal choice of residue pairs for experiments. We predict the best experimental probe positions for proteins λ-repressor, ß2-adrenergic receptor, and villin headpiece domain. These proteins have been well-studied and allow for a rigorous comparison of Optimal Probes predictions with already available experiments. Additionally, we also illustrate that our method can be used to predict the best choice for experiments by including any previous experiment choices available from other studies on the same protein. We consistently find that the best choice cannot be based on intuition or structural information such as distance difference between few known stable structures of the protein. Therefore, we show that incorporating protein dynamics could be used to maximize the information gain from experiments.

Enhanced unbiased sampling of protein dynamics using evolutionary coupling information.

One of the major challenges in atomistic simulations of proteins is efficient sampling of pathways associated with rare conformational transitions. Recent developments in statistical methods for computation of direct evolutionary couplings between amino acids within and across polypeptide chains have allowed for inference of native residue contacts, informing accurate prediction of protein folds and multimeric structures. In this study, we assess the use of distances between evolutionarily coupled residues as natural choices for reaction coordinates which can be incorporated into Markov state model-based adaptive sampling schemes and potentially used to predict not only functional conformations but also pathways of conformational change, protein folding, and protein-protein association. We demonstrate the utility of evolutionary couplings in sampling and predicting activation pathways of the ß 2-adrenergic receptor (ß 2-AR), folding of the FiP35 WW domain, and dimerization of the E. coli molybdopterin synthase subunits. We find that the time required for ß 2-AR activation and folding of the WW domain are greatly diminished using evolutionary couplings-guided adaptive sampling. Additionally, we were able to identify putative molybdopterin synthase association pathways and near-crystal structure complexes from protein-protein association simulations.

Predicting Optimal DEER Label Positions to Study Protein Conformational Heterogeneity.

Double electron-electron resonance (DEER) spectroscopy is a powerful experimental technique for understanding the conformational heterogeneity of proteins. It involves attaching nitroxide spin labels to two residues in the protein to obtain a distance distribution between them. However, the choice of residue pairs to label in the protein requires careful thought, as experimentalists must pick label positions from a large set of all possible residue-pair combinations in the protein. In this article, we address the problem of the choice of DEER spin-label positions in a protein. For this purpose, we utilize all-atom molecular dynamics simulations of protein dynamics, to rank the sets of labeled residue pairs in terms of their ability to capture the conformational dynamics of the protein. Our design methodology is based on the following two criteria: (1) An ideal set of DEER spin-label positions should capture the slowest conformational-change processes observed in the protein dynamics, and (2) any two sets of residue pairs should describe orthogonal conformational-change processes to maximize the overall information gain and reduce the number of labeled residue pairs. We utilize Markov state models of protein dynamics to identify slow dynamical processes and a genetic-algorithm-based approach to predict the optimal choices of residue pairs with limited computational time requirements. We predict the optimal residue pairs for DEER spectroscopy in ß2 adrenergic receptor, the C-terminal domain of calmodulin, and peptide transporter PepTSo. We find that our choices were ranked higher than those used to perform DEER experiments on the proteins investigated in this study. Hence, the predicted choices of DEER residue pairs determined by our method provide maximum insight into the conformational heterogeneity of the protein while using the minimum number of labeled residues.

Application of Hidden Markov Models in Biomolecular Simulations.

Hidden Markov models (HMMs) provide a framework to analyze large trajectories of biomolecular simulation datasets. HMMs decompose the conformational space of a biological molecule into finite number of states that interconvert among each other with certain rates. HMMs simplify long timescale trajectories for human comprehension, and allow comparison of simulations with experimental data. In this chapter, we provide an overview of building HMMs for analyzing bimolecular simulation datasets. We demonstrate the procedure for building a Hidden Markov model for Met-enkephalin peptide simulation dataset and compare the timescales of the process.

Pricing policy for declining demand using item preservation technology.

We have designed an inventory model for seasonal products in which deterioration can be controlled by item preservation technology investment. Demand for the product is considered price sensitive and decreases linearly. This study has shown that the profit is a concave function of optimal selling price, replenishment time and preservation cost parameter. We simultaneously determined the optimal selling price of the product, the replenishment cycle and the cost of item preservation technology. Additionally, this study has shown that there exists an optimal selling price and optimal preservation investment to maximize the profit for every business set-up. Finally, the model is illustrated by numerical examples and sensitive analysis of the optimal solution with respect to major parameters.

Transition path theory analysis of c-Src kinase activation.

Nonreceptor tyrosine kinases of the Src family are large multidomain allosteric proteins that are crucial to cellular signaling pathways. In a previous study, we generated a Markov state model (MSM) to simulate the activation of c-Src catalytic domain, used as a prototypical tyrosine kinase. The long-time kinetics of transition predicted by the MSM was in agreement with experimental observations. In the present study, we apply the framework of transition path theory (TPT) to the previously constructed MSM to characterize the main features of the activation pathway. The analysis indicates that the activating transition, in which the activation loop first opens up followed by an inward rotation of the αC-helix, takes place via a dense set of intermediate microstates distributed within a fairly broad "transition tube" in a multidimensional conformational subspace connecting the two end-point conformations. Multiple microstates with negligible equilibrium probabilities carry a large transition flux associated with the activating transition, which explains why extensive conformational sampling is necessary to accurately determine the kinetics of activation. Our results suggest that the combination of MSM with TPT provides an effective framework to represent conformational transitions in complex biomolecular systems.

Conformational heterogeneity of the calmodulin binding interface.

Calmodulin (CaM) is a ubiquitous Ca(2+) sensor and a crucial signalling hub in many pathways aberrantly activated in disease. However, the mechanistic basis of its ability to bind diverse signalling molecules including G-protein-coupled receptors, ion channels and kinases remains poorly understood. Here we harness the high resolution of molecular dynamics simulations and the analytical power of Markov state models to dissect the molecular underpinnings of CaM binding diversity. Our computational model indicates that in the absence of Ca(2+), sub-states in the folded ensemble of CaM's C-terminal domain present chemically and sterically distinct topologies that may facilitate conformational selection. Furthermore, we find that local unfolding is off-pathway for the exchange process relevant for peptide binding, in contrast to prior hypotheses that unfolding might account for binding diversity. Finally, our model predicts a novel binding interface that is well-populated in the Ca(2+)-bound regime and, thus, a candidate for pharmacological intervention.

Markov State Models and tICA Reveal a Nonnative Folding Nucleus in Simulations of NuG2.

After reanalyzing simulations of NuG2-a designed mutant of protein G-generated by Lindorff-Larsen et al. with time structure-based independent components analysis and Markov state models as well as performing 1.5 ms of additional sampling on Folding@home, we found an intermediate with a register-shift in one of the ß-sheets that was visited along a minor folding pathway. The minor folding pathway was initiated by the register-shifted sheet, which is composed of solely nonnative contacts, suggesting that for some peptides, nonnative contacts can lead to productive folding events. To confirm this experimentally, we suggest a mutational strategy for stabilizing the register shift, as well as an infrared experiment that could observe the nonnative folding nucleus.

Heat dissipation guides activation in signaling proteins.

Life is fundamentally a nonequilibrium phenomenon. At the expense of dissipated energy, living things perform irreversible processes that allow them to propagate and reproduce. Within cells, evolution has designed nanoscale machines to do meaningful work with energy harnessed from a continuous flux of heat and particles. As dictated by the Second Law of Thermodynamics and its fluctuation theorem corollaries, irreversibility in nonequilibrium processes can be quantified in terms of how much entropy such dynamics produce. In this work, we seek to address a fundamental question linking biology and nonequilibrium physics: can the evolved dissipative pathways that facilitate biomolecular function be identified by their extent of entropy production in general relaxation processes? We here synthesize massive molecular dynamics simulations, Markov state models (MSMs), and nonequilibrium statistical mechanical theory to probe dissipation in two key classes of signaling proteins: kinases and G-protein-coupled receptors (GPCRs). Applying machinery from large deviation theory, we use MSMs constructed from protein simulations to generate dynamics conforming to positive levels of entropy production. We note the emergence of an array of peaks in the dynamical response (transient analogs of phase transitions) that draw the proteins between distinct levels of dissipation, and we see that the binding of ATP and agonist molecules modifies the observed dissipative landscapes. Overall, we find that dissipation is tightly coupled to activation in these signaling systems: dominant entropy-producing trajectories become localized near important barriers along known biological activation pathways. We go on to classify an array of equilibrium and nonequilibrium molecular switches that harmonize to promote functional dynamics.

A network of molecular switches controls the activation of the two-component response regulator NtrC.

Recent successes in simulating protein structure and folding dynamics have demonstrated the power of molecular dynamics to predict the long timescale behaviour of proteins. Here, we extend and improve these methods to predict molecular switches that characterize conformational change pathways between the active and inactive state of nitrogen regulatory protein C (NtrC). By employing unbiased Markov state model-based molecular dynamics simulations, we construct a dynamic picture of the activation pathways of this key bacterial signalling protein that is consistent with experimental observations and predicts new mutants that could be used for validation of the mechanism. Moreover, these results suggest a novel mechanistic paradigm for conformational switching.

Elucidating Ligand-Modulated Conformational Landscape of GPCRs Using Cloud-Computing Approaches.

G-protein-coupled receptors (GPCRs) are a versatile family of membrane-bound signaling proteins. Despite the recent successes in obtaining crystal structures of GPCRs, much needs to be learned about the conformational changes associated with their activation. Furthermore, the mechanism by which ligands modulate the activation of GPCRs has remained elusive. Molecular simulations provide a way of obtaining detailed an atomistic description of GPCR activation dynamics. However, simulating GPCR activation is challenging due to the long timescales involved and the associated challenge of gaining insights from the "Big" simulation datasets. Here, we demonstrate how cloud-computing approaches have been used to tackle these challenges and obtain insights into the activation mechanism of GPCRs. In particular, we review the use of Markov state model (MSM)-based sampling algorithms for sampling milliseconds of dynamics of a major drug target, the G-protein-coupled receptor ß2-AR. MSMs of agonist and inverse agonist-bound ß2-AR reveal multiple activation pathways and how ligands function via modulation of the ensemble of activation pathways. We target this ensemble of conformations with computer-aided drug design approaches, with the goal of designing drugs that interact more closely with diverse receptor states, for overall increased efficacy and specificity. We conclude by discussing how cloud-based approaches present a powerful and broadly available tool for studying the complex biological systems routinely.

Markov state models provide insights into dynamic modulation of protein function.

CONSPECTUS: Protein function is inextricably linked to protein dynamics. As we move from a static structural picture to a dynamic ensemble view of protein structure and function, novel computational paradigms are required for observing and understanding conformational dynamics of proteins and its functional implications. In principle, molecular dynamics simulations can provide the time evolution of atomistic models of proteins, but the long time scales associated with functional dynamics make it difficult to observe rare dynamical transitions. The issue of extracting essential functional components of protein dynamics from noisy simulation data presents another set of challenges in obtaining an unbiased understanding of protein motions. Therefore, a methodology that provides a statistical framework for efficient sampling and a human-readable view of the key aspects of functional dynamics from data analysis is required. The Markov state model (MSM), which has recently become popular worldwide for studying protein dynamics, is an example of such a framework. In this Account, we review the use of Markov state models for efficient sampling of the hierarchy of time scales associated with protein dynamics, automatic identification of key conformational states, and the degrees of freedom associated with slow dynamical processes. Applications of MSMs for studying long time scale phenomena such as activation mechanisms of cellular signaling proteins has yielded novel insights into protein function. In particular, from MSMs built using large-scale simulations of GPCRs and kinases, we have shown that complex conformational changes in proteins can be described in terms of structural changes in key structural motifs or "molecular switches" within the protein, the transitions between functionally active and inactive states of proteins proceed via multiple pathways, and ligand or substrate binding modulates the flux through these pathways. Finally, MSMs also provide a theoretical toolbox for studying the effect of nonequilibrium perturbations on conformational dynamics. Considering that protein dynamics in vivo occur under nonequilibrium conditions, MSMs coupled with nonequilibrium statistical mechanics provide a way to connect cellular components to their functional environments. Nonequilibrium perturbations of protein folding MSMs reveal the presence of dynamically frozen glass-like states in their conformational landscape. These frozen states are also observed to be rich in ß-sheets, which indicates their possible role in the nucleation of ß-sheet rich aggregates such as those observed in amyloid-fibril formation. Finally, we describe how MSMs have been used to understand the dynamical behavior of intrinsically disordered proteins such as amyloid-ß, human islet amyloid polypeptide, and p53. While certainly not a panacea for studying functional dynamics, MSMs provide a rigorous theoretical foundation for understanding complex entropically dominated processes and a convenient lens for viewing protein motions.

Complex pathways in folding of protein G explored by simulation and experiment.

The B1 domain of protein G has been a classic model system of folding for decades, the subject of numerous experimental and computational studies. Most of the experimental work has focused on whether the protein folds via an intermediate, but the evidence is mostly limited to relatively slow kinetic observations with a few structural probes. In this work we observe folding on the submillisecond timescale with microfluidic mixers using a variety of probes including tryptophan fluorescence, circular dichroism, and photochemical oxidation. We find that each probe yields different kinetics and compare these observations with a Markov State Model constructed from large-scale molecular dynamics simulations and find a complex network of states that yield different kinetics for different observables. We conclude that there are many folding pathways before the final folding step and that these paths do not have large free energy barriers.

Cloud-based simulations on Google Exacycle reveal ligand modulation of GPCR activation pathways.

Simulations can provide tremendous insight into the atomistic details of biological mechanisms, but micro- to millisecond timescales are historically only accessible on dedicated supercomputers. We demonstrate that cloud computing is a viable alternative that brings long-timescale processes within reach of a broader community. We used Google's Exacycle cloud-computing platform to simulate two milliseconds of dynamics of a major drug target, the G-protein-coupled receptor ß2AR. Markov state models aggregate independent simulations into a single statistical model that is validated by previous computational and experimental results. Moreover, our models provide an atomistic description of the activation of a G-protein-coupled receptor and reveal multiple activation pathways. Agonists and inverse agonists interact differentially with these pathways, with profound implications for drug design.

Coagulation of nanoparticles in reverse micellar systems: a Monte Carlo model.

The process of formation of nanoparticles obtained by mixing two micellized, aqueous solutions has been simulated using the Monte Carlo technique. The model includes the phenomena of finite nucleation, growth via intermicellar exchange, and coagulation of nanoparticles after their formation. Using the model, an exploratory study has been conducted to analyze whether the coagulation of nanoparticles is the reason for the formation of nanoparticles whose sizes are comparable to the size of the reverse micelles. The model explains the possible mechanism of coagulation of semiconductor nanoparticles formed within reverse micelles and its effect on the evolution of their size with time. The model is predictive in nature, and the simulation results compare well with those observed experimentally.