A critical perspective on Markov state model treatments of protein-protein association using coarse-grained simulations.

Atomic-level information is essential to explain the specific interactions governing protein-protein recognition in terms of structure and dynamics. Of particular interest is a characterization of the time-dependent kinetic aspects of protein-protein association and dissociation. A powerful framework to characterize the dynamics of complex molecular systems is provided by Markov State Models (MSMs). The central idea is to construct a reduced stochastic model of the full system by defining a set of conformational featured microstates and determining the matrix of transition probabilities between them. While a MSM framework can sometimes be very effective, different combinations of input featurization and simulation methods can significantly affect the robustness and the quality of the information generated from MSMs in the context of protein association. Here, a systematic examination of a variety of MSMs methodologies is undertaken to clarify these issues. To circumvent the uncertainties caused by sampling issues, we use a simplified coarse-grained model of the barnase-barstar protein complex. A sensitivity analysis is proposed to identify the microstates of an MSM that contribute most to the error in conjunction with the transition-based reweighting analysis method for a more efficient and accurate MSM construction.

Combining experimental and simulation data of molecular processes via augmented Markov models.

Accurate mechanistic description of structural changes in biomolecules is an increasingly important topic in structural and chemical biology. Markov models have emerged as a powerful way to approximate the molecular kinetics of large biomolecules while keeping full structural resolution in a divide-and-conquer fashion. However, the accuracy of these models is limited by that of the force fields used to generate the underlying molecular dynamics (MD) simulation data. Whereas the quality of classical MD force fields has improved significantly in recent years, remaining errors in the Boltzmann weights are still on the order of a few [Formula: see text], which may lead to significant discrepancies when comparing to experimentally measured rates or state populations. Here we take the view that simulations using a sufficiently good force-field sample conformations that are valid but have inaccurate weights, yet these weights may be made accurate by incorporating experimental data a posteriori. To do so, we propose augmented Markov models (AMMs), an approach that combines concepts from probability theory and information theory to consistently treat systematic force-field error and statistical errors in simulation and experiment. Our results demonstrate that AMMs can reconcile conflicting results for protein mechanisms obtained by different force fields and correct for a wide range of stationary and dynamical observables even when only equilibrium measurements are incorporated into the estimation process. This approach constitutes a unique avenue to combine experiment and computation into integrative models of biomolecular structure and dynamics.

The mechanism of RNA base fraying: Molecular dynamics simulations analyzed with core-set Markov state models.

The process of RNA base fraying (i.e., the transient opening of the termini of a helix) is involved in many aspects of RNA dynamics. We here use molecular dynamics simulations and Markov state models to characterize the kinetics of RNA fraying and its sequence and direction dependence. In particular, we first introduce a method for determining biomolecular dynamics employing core-set Markov state models constructed using an advanced clustering technique. The method is validated on previously reported simulations. We then use the method to analyze extensive trajectories for four different RNA model duplexes. Results obtained using D. E. Shaw research and AMBER force fields are compared and discussed in detail and show a non-trivial interplay between the stability of intermediate states and the overall fraying kinetics.

Variational selection of features for molecular kinetics.

The modeling of atomistic biomolecular simulations using kinetic models such as Markov state models (MSMs) has had many notable algorithmic advances in recent years. The variational principle has opened the door for a nearly fully automated toolkit for selecting models that predict the long time-scale kinetics from molecular dynamics simulations. However, one yet-unoptimized step of the pipeline involves choosing the features, or collective variables, from which the model should be constructed. In order to build intuitive models, these collective variables are often sought to be interpretable and familiar features, such as torsional angles or contact distances in a protein structure. However, previous approaches for evaluating the chosen features rely on constructing a full MSM, which in turn requires additional hyperparameters to be chosen, and hence leads to a computationally expensive framework. Here, we present a method to optimize the feature choice directly, without requiring the construction of the final kinetic model. We demonstrate our rigorous preprocessing algorithm on a canonical set of 12 fast-folding protein simulations and show that our procedure leads to more efficient model selection.

Identification of kinetic order parameters for non-equilibrium dynamics.

A popular approach to analyze the dynamics of high-dimensional many-body systems, such as macromolecules, is to project the trajectories onto a space of slowly varying collective variables, where subsequent analyses are made, such as clustering or estimation of free energy profiles or Markov state models. However, existing "dynamical" dimension reduction methods, such as the time-lagged independent component analysis (TICA), are only valid if the dynamics obeys detailed balance (microscopic reversibility) and typically require long, equilibrated simulation trajectories. Here, we develop a dimension reduction method for non-equilibrium dynamics based on the recently developed Variational Approach for Markov Processes (VAMP) by Wu and Noé. VAMP is illustrated by obtaining a low-dimensional description of a single file ion diffusion model and by identifying long-lived states from molecular dynamics simulations of the KcsA channel protein in an external electrochemical potential. This analysis provides detailed insights into the coupling of conformational dynamics, the configuration of the selectivity filter, and the conductance of the channel. We recommend VAMP as a replacement for the less general TICA method.

Multiensemble Markov models of molecular thermodynamics and kinetics.

We introduce the general transition-based reweighting analysis method (TRAM), a statistically optimal approach to integrate both unbiased and biased molecular dynamics simulations, such as umbrella sampling or replica exchange. TRAM estimates a multiensemble Markov model (MEMM) with full thermodynamic and kinetic information at all ensembles. The approach combines the benefits of Markov state models-clustering of high-dimensional spaces and modeling of complex many-state systems-with those of the multistate Bennett acceptance ratio of exploiting biased or high-temperature ensembles to accelerate rare-event sampling. TRAM does not depend on any rate model in addition to the widely used Markov state model approximation, but uses only fundamental relations such as detailed balance and binless reweighting of configurations between ensembles. Previous methods, including the multistate Bennett acceptance ratio, discrete TRAM, and Markov state models are special cases and can be derived from the TRAM equations. TRAM is demonstrated by efficiently computing MEMMs in cases where other estimators break down, including the full thermodynamics and rare-event kinetics from high-dimensional simulation data of an all-atom protein-ligand binding model.

Identification of novel regulators of developmental hematopoiesis using Endoglin regulatory elements as molecular probes.

Enhancers are the primary determinants of cell identity, and specific promoter/enhancer combinations of Endoglin (ENG) have been shown to target blood and endothelium in the embryo. Here, we generated a series of embryonic stem cell lines, each targeted with reporter constructs driven by specific promoter/enhancer combinations of ENG, to evaluate their discriminative potential and value as molecular probes of the corresponding transcriptome. The Eng promoter (P) in combination with the -8/+7/+9-kb enhancers, targeted cells in FLK1 mesoderm that were enriched for blast colony forming potential, whereas the P/-8-kb enhancer targeted TIE2+/c-KIT+/CD41- endothelial cells that were enriched for hematopoietic potential. These fractions were isolated using reporter expression and their transcriptomes profiled by RNA-seq. There was high concordance between our signatures and those from embryos with defects at corresponding stages of hematopoiesis. Of the 6 genes that were upregulated in both hemogenic mesoderm and hemogenic endothelial fractions targeted by the reporters, LRP2, a multiligand receptor, was the only gene that had not previously been associated with hematopoiesis. We show that LRP2 is indeed involved in definitive hematopoiesis and by doing so validate the use of reporter gene-coupled enhancers as probes to gain insights into transcriptional changes that facilitate cell fate transitions.

How to Distinguish Conformational Selection and Induced Fit Based on Chemical Relaxation Rates.

Protein binding often involves conformational changes. Important questions are whether a conformational change occurs prior to a binding event ('conformational selection') or after a binding event ('induced fit'), and how conformational transition rates can be obtained from experiments. In this article, we present general results for the chemical relaxation rates of conformational-selection and induced-fit binding processes that hold for all concentrations of proteins and ligands and, thus, go beyond the standard pseudo-first-order approximation of large ligand concentration. These results allow to distinguish conformational-selection from induced-fit processes-also in cases in which such a distinction is not possible under pseudo-first-order conditions-and to extract conformational transition rates of proteins from chemical relaxation data.

Variational Koopman models: Slow collective variables and molecular kinetics from short off-equilibrium simulations.

Markov state models (MSMs) and master equation models are popular approaches to approximate molecular kinetics, equilibria, metastable states, and reaction coordinates in terms of a state space discretization usually obtained by clustering. Recently, a powerful generalization of MSMs has been introduced, the variational approach conformation dynamics/molecular kinetics (VAC) and its special case the time-lagged independent component analysis (TICA), which allow us to approximate slow collective variables and molecular kinetics by linear combinations of smooth basis functions or order parameters. While it is known how to estimate MSMs from trajectories whose starting points are not sampled from an equilibrium ensemble, this has not yet been the case for TICA and the VAC. Previous estimates from short trajectories have been strongly biased and thus not variationally optimal. Here, we employ the Koopman operator theory and the ideas from dynamic mode decomposition to extend the VAC and TICA to non-equilibrium data. The main insight is that the VAC and TICA provide a coefficient matrix that we call Koopman model, as it approximates the underlying dynamical (Koopman) operator in conjunction with the basis set used. This Koopman model can be used to compute a stationary vector to reweight the data to equilibrium. From such a Koopman-reweighted sample, equilibrium expectation values and variationally optimal reversible Koopman models can be constructed even with short simulations. The Koopman model can be used to propagate densities, and its eigenvalue decomposition provides estimates of relaxation time scales and slow collective variables for dimension reduction. Koopman models are generalizations of Markov state models, TICA, and the linear VAC and allow molecular kinetics to be described without a cluster discretization.

Estimation and uncertainty of reversible Markov models.

Reversibility is a key concept in Markov models and master-equation models of molecular kinetics. The analysis and interpretation of the transition matrix encoding the kinetic properties of the model rely heavily on the reversibility property. The estimation of a reversible transition matrix from simulation data is, therefore, crucial to the successful application of the previously developed theory. In this work, we discuss methods for the maximum likelihood estimation of transition matrices from finite simulation data and present a new algorithm for the estimation if reversibility with respect to a given stationary vector is desired. We also develop new methods for the Bayesian posterior inference of reversible transition matrices with and without given stationary vector taking into account the need for a suitable prior distribution preserving the meta-stable features of the observed process during posterior inference. All algorithms here are implemented in the PyEMMA software--http://pyemma.org--as of version 2.0.

Identification of slow molecular order parameters for Markov model construction.

A goal in the kinetic characterization of a macromolecular system is the description of its slow relaxation processes via (i) identification of the structural changes involved in these processes and (ii) estimation of the rates or timescales at which these slow processes occur. Most of the approaches to this task, including Markov models, master-equation models, and kinetic network models, start by discretizing the high-dimensional state space and then characterize relaxation processes in terms of the eigenvectors and eigenvalues of a discrete transition matrix. The practical success of such an approach depends very much on the ability to finely discretize the slow order parameters. How can this task be achieved in a high-dimensional configuration space without relying on subjective guesses of the slow order parameters? In this paper, we use the variational principle of conformation dynamics to derive an optimal way of identifying the "slow subspace" of a large set of prior order parameters - either generic internal coordinates or a user-defined set of parameters. Using a variational formulation of conformational dynamics, it is shown that an existing method-the time-lagged independent component analysis-provides the optional solution to this problem. In addition, optimal indicators-order parameters indicating the progress of the slow transitions and thus may serve as reaction coordinates-are readily identified. We demonstrate that the slow subspace is well suited to construct accurate kinetic models of two sets of molecular dynamics simulations, the 6-residue fluorescent peptide MR121-GSGSW and the 30-residue intrinsically disordered peptide kinase inducible domain (KID). The identified optimal indicators reveal the structural changes associated with the slow processes of the molecular system under analysis.

Balancing WNT signalling in early forebrain development: The role of LRP4 as a modulator of LRP6 function.

The specification of the forebrain relies on the precise regulation of WNT/ß-catenin signalling to support neuronal progenitor cell expansion, patterning, and morphogenesis. Imbalances in WNT signalling activity in the early neuroepithelium lead to congenital disorders, such as neural tube defects (NTDs). LDL receptor-related protein (LRP) family members, including the well-studied receptors LRP5 and LRP6, play critical roles in modulating WNT signalling capacity through tightly regulated interactions with their co-receptor Frizzled, WNT ligands, inhibitors and intracellular WNT pathway components. However, little is known about the function of LRP4 as a potential modulator of WNT signalling in the central nervous system. In this study, we investigated the role of LRP4 in the regulation of WNT signalling during early mouse forebrain development. Our results demonstrate that LRP4 can modulate LRP5- and LRP6-mediated WNT signalling in the developing forebrain prior to the onset of neurogenesis at embryonic stage 9.5 and is therefore essential for accurate neural tube morphogenesis. Specifically, LRP4 functions as a genetic modifier for impaired mitotic activity and forebrain hypoplasia, but not for NTDs in LRP6-deficient mutants. In vivo and in vitro data provide evidence that LRP4 is a key player in fine-tuning WNT signalling capacity and mitotic activity of mouse neuronal progenitors and of human retinal pigment epithelial (hTERT RPE-1) cells. Our data demonstrate the crucial roles of LRP4 and LRP6 in regulating WNT signalling and forebrain development and highlight the need to consider the interaction between different signalling pathways to understand the underlying mechanisms of disease. The findings have significant implications for our mechanistic understanding of how LRPs participate in controlling WNT signalling.

Bayesian inference of kinetic schemes for ion channels by Kalman filtering.

Inferring adequate kinetic schemes for ion channel gating from ensemble currents is a daunting task due to limited information in the data. We address this problem by using a parallelized Bayesian filter to specify hidden Markov models for current and fluorescence data. We demonstrate the flexibility of this algorithm by including different noise distributions. Our generalized Kalman filter outperforms both a classical Kalman filter and a rate equation approach when applied to patch-clamp data exhibiting realistic open-channel noise. The derived generalization also enables inclusion of orthogonal fluorescence data, making unidentifiable parameters identifiable and increasing the accuracy of the parameter estimates by an order of magnitude. By using Bayesian highest credibility volumes, we found that our approach, in contrast to the rate equation approach, yields a realistic uncertainty quantification. Furthermore, the Bayesian filter delivers negligibly biased estimates for a wider range of data quality. For some data sets, it identifies more parameters than the rate equation approach. These results also demonstrate the power of assessing the validity of algorithms by Bayesian credibility volumes in general. Finally, we show that our Bayesian filter is more robust against errors induced by either analog filtering before analog-to-digital conversion or by limited time resolution of fluorescence data than a rate equation approach.

Diversity of Long-Lived Intermediates along the Binding Pathway of Imatinib to Abl Kinase Revealed by MD Simulations.

Imatinib, a drug used for the treatment of chronic myeloid leukemia and other cancers, works by blocking the catalytic site of pathological constitutively active Abl kinase. While the binding pose is known from X-ray crystallography, the different steps leading to the formation of the complex are not well understood. The results from extensive molecular dynamics simulations show that imatinib can primarily exit the known crystallographic binding pose through the cleft of the binding site or by sliding under the αC helix. Once displaced from the crystallographic binding pose, imatinib becomes trapped in intermediate states. These intermediates are characterized by a high diversity of ligand orientations and conformations, and relaxation timescales within this region may exceed 3-4 ms. Analysis indicates that the metastable intermediate states should be spectroscopically indistinguishable from the crystallographic binding pose, in agreement with tryptophan stopped-flow fluorescence experiments.

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

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

Polymerization and editing modes of a high-fidelity DNA polymerase are linked by a well-defined path.

Proofreading by replicative DNA polymerases is a fundamental mechanism ensuring DNA replication fidelity. In proofreading, mis-incorporated nucleotides are excised through the 3'-5' exonuclease activity of the DNA polymerase holoenzyme. The exonuclease site is distal from the polymerization site, imposing stringent structural and kinetic requirements for efficient primer strand transfer. Yet, the molecular mechanism of this transfer is not known. Here we employ molecular simulations using recent cryo-EM structures and biochemical analyses to delineate an optimal free energy path connecting the polymerization and exonuclease states of E. coli replicative DNA polymerase Pol III. We identify structures for all intermediates, in which the transitioning primer strand is stabilized by conserved Pol III residues along the fingers, thumb and exonuclease domains. We demonstrate switching kinetics on a tens of milliseconds timescale and unveil a complete pol-to-exo switching mechanism, validated by targeted mutational experiments.

Identifying Conformational-Selection and Induced-Fit Aspects in the Binding-Induced Folding of PMI from Markov State Modeling of Atomistic Simulations.

Unstructured proteins and peptides typically fold during binding to ligand proteins. A challenging problem is to identify the mechanism and kinetics of these binding-induced folding processes in experiments and atomistic simulations. In this Article, we present a detailed picture for the folding of the inhibitor peptide PMI into a helix during binding to the oncoprotein fragment 25-109Mdm2 obtained from atomistic, explicit-water simulations and Markov state modeling. We find that binding-induced folding of PMI is highly parallel and can occur along a multitude of pathways. Some pathways are induced-fit-like with binding occurring prior to PMI helix formation, while other pathways are conformational-selection-like with binding after helix formation. On the majority of pathways, however, binding is intricately coupled to folding, without clear temporal ordering. A central feature of these pathways is PMI motion on the Mdm2 surface, along the binding groove of Mdm2 or over the rim of this groove. The native binding groove of Mdm2 thus appears as an asymmetric funnel for PMI binding. Overall, binding-induced folding of PMI does not fit into the classical picture of induced fit or conformational selection that implies a clear temporal ordering of binding and folding events. We argue that this holds in general for binding-induced folding processes because binding and folding events in these processes likely occur on similar time scales and do exhibit the time-scale separation required for temporal ordering.

Author Correction: Protein-peptide association kinetics beyond the seconds timescale from atomistic simulations.

In the original version of this Article, the Acknowledgement section omitted financial support from the Deutsche Forschungsgemeinschaft grant SFB 958/A4. This error has now been corrected in both the PDF and HTML versions of the Article.

Predicting the Kinetics of RNA Oligonucleotides Using Markov State Models.

Nowadays different experimental techniques, such as single molecule or relaxation experiments, can provide dynamic properties of biomolecular systems, but the amount of detail obtainable with these methods is often limited in terms of time or spatial resolution. Here we use state-of-the-art computational techniques, namely, atomistic molecular dynamics and Markov state models, to provide insight into the rapid dynamics of short RNA oligonucleotides, to elucidate the kinetics of stacking interactions. Analysis of multiple microsecond-long simulations indicates that the main relaxation modes of such molecules can consist of transitions between alternative folded states, rather than between random coils and native structures. After properly removing structures that are artificially stabilized by known inaccuracies of the current RNA AMBER force field, the kinetic properties predicted are consistent with the time scales of previously reported relaxation experiments.