Milestoning simulation of ligand dissociation from the glycogen synthase kinase 3ß.

As drug-binding kinetics has become an important factor to be considered in modern drug discovery, this work evaluated the ability of the Milestoning method in computing the absolute dissociation rate of a ligand from the serine-threonine kinase, glycogen synthase kinase 3ß, which is a target for designing drugs to treat diseases such as neurodegenerative disorders and diabetes. We found that the Milestoning method gave good agreement with experiment with modest computational costs. Although the time scale for dissociation lasted tens of seconds, the collective molecular dynamics simulations total less than 1µs. Computing the committor function helped to identify the transition states (TSs), in which the ligand moved substantially away from the binding pocket. The glycine-rich loop with a serine residue attaching to its tips was found to undergo large movement from the bound to the TSs and might play a role in controlling drug-dissociation kinetics.

Calcium-Lipid Interactions Observed with Isotope-Edited Infrared Spectroscopy.

Calcium ions bind to lipid membranes containing anionic lipids; however, characterizing the specific ion-lipid interactions in multicomponent membranes has remained challenging because it requires nonperturbative lipid-specific probes. Here, using a combination of isotope-edited infrared spectroscopy and molecular dynamics simulations, we characterize the effects of a physiologically relevant (2 mM) Ca2+ concentration on zwitterionic phosphatidylcholine and anionic phosphatidylserine lipids in mixed lipid membranes. We show that Ca2+ alters hydrogen bonding between water and lipid headgroups by forming a coordination complex involving the lipid headgroups and water. These interactions distort interfacial water orientations and prevent hydrogen bonding with lipid ester carbonyls. We demonstrate, experimentally, that these effects are more pronounced for the anionic phosphatidylserine lipids than for zwitterionic phosphatidylcholine lipids in the same membrane.

Milestoning with wind: Exploring the impact of a biasing potential in exact calculation of kinetics.

We consider the algorithm wind-assisted reweighted Milestoning of Grazioli and Andricioaei [J. Chem. Phys 149(8), 084103 (2018)], expand it, and assess its performance. We derive exact expressions for underdamped and overdamped Langevin dynamics and examine its efficiency for a simple model system (Mueller's potential) and for bond breaking in solution. The use of a biasing force (wind) significantly enhances the sampling of otherwise rare trajectories but also introduces an exponential weight to the trajectories that significantly impact the value of the statistics. In our examples, computing averages and standard deviations are not better using wind compared to straightforward Milestoning. However, the biasing force is useful for highly steep energy landscapes. On these landscapes, the probability of sampling straightforward Milestoning trajectories, which overcome the barrier, is low and the biasing force enables the observations of these rare events.

Computer simulations of a heterogeneous membrane with enhanced sampling techniques.

Computational determination of the equilibrium state of heterogeneous phospholipid membranes is a significant challenge. We wish to explore the rich phase diagram of these multi-component systems. However, the diffusion and mixing times in membranes are long compared to typical time scales of computer simulations. Here, we evaluate the combination of the enhanced sampling techniques molecular dynamics with alchemical steps and Monte Carlo with molecular dynamics with a coarse-grained model of membranes (Martini) to reduce the number of steps and force evaluations that are needed to reach equilibrium. We illustrate a significant gain compared to straightforward molecular dynamics of the Martini model by factors between 3 and 10. The combination is a useful tool to enhance the study of phase separation and the formation of domains in biological membranes.

Revealing the distinct folding phases of an RNA three-helix junction.

Remarkable new insight has emerged into the biological role of RNA in cells. RNA folding and dynamics enable many of these newly discovered functions, calling for an understanding of RNA self-assembly and conformational dynamics. Because RNAs pass through multiple structures as they fold, an ensemble perspective is required to visualize the flow through fleetingly populated sets of states. Here, we combine microfluidic mixing technology and small angle X-ray scattering (SAXS) to measure the Mg-induced folding of a small RNA domain, the tP5abc three helix junction. Our measurements are interpreted using ensemble optimization to select atomically detailed structures that recapitulate each experimental curve. Structural ensembles, derived at key stages in both time-resolved studies and equilibrium titrations, reproduce the features of known intermediates, and more importantly, offer a powerful new structural perspective on the time-progression of folding. Distinct collapse phases along the pathway appear to be orchestrated by specific interactions with Mg ions. These key interactions subsequently direct motions of the backbone that position the partners of tertiary contacts for later bonding, and demonstrate a remarkable synergy between Mg and RNA across numerous time-scales.

Conformations of an RNA Helix-Junction-Helix Construct Revealed by SAXS Refinement of MD Simulations.

RNA is involved in a broad range of biological processes that extend far beyond translation. Many of RNA's recently discovered functions rely on folding to a specific conformation or transitioning between conformations. The RNA structure contains rigid, short basepaired regions connected by more flexible linkers. Studies of model constructs such as small helix-junction-helix (HJH) motifs are useful in understanding how these elements work together to determine RNA conformation. Here, we reveal the full ensemble of solution structures assumed by a model RNA HJH. We apply small-angle x-ray scattering and an ensemble optimization method to selectively refine models generated by all-atom molecular dynamics simulations. The expectation of a broad distribution of helix orientations, at and above physiological ionic strength, is not met. Instead, this analysis shows that the HJH structures are dominated by two distinct conformations at moderate to high ionic strength. Atomic structures, selected from the molecular dynamics simulations, reveal strong base-base interactions in the junction that critically constrain the conformational space available to the HJH molecule and lead to a surprising re-extension at high salt. These results are corroborated by comparison with previous single-molecule fluorescence resonance energy transfer experiments on the same constructs.

A new paradigm for atomically detailed simulations of kinetics in biophysical systems.

The kinetics of biochemical and biophysical events determined the course of life processes and attracted considerable interest and research. For example, modeling of biological networks and cellular responses relies on the availability of information on rate coefficients. Atomically detailed simulations hold the promise of supplementing experimental data to obtain a more complete kinetic picture. However, simulations at biological time scales are challenging. Typical computer resources are insufficient to provide the ensemble of trajectories at the correct length that is required for straightforward calculations of time scales. In the last years, new technologies emerged that make atomically detailed simulations of rate coefficients possible. Instead of computing complete trajectories from reactants to products, these approaches launch a large number of short trajectories at different positions. Since the trajectories are short, they are computed trivially in parallel on modern computer architecture. The starting and termination positions of the short trajectories are chosen, following statistical mechanics theory, to enhance efficiency. These trajectories are analyzed. The analysis produces accurate estimates of time scales as long as hours. The theory of Milestoning that exploits the use of short trajectories is discussed, and several applications are described.

Physiological Calcium Concentrations Slow Dynamics at the Lipid-Water Interface.

Phospholipids can interact strongly with ions at physiological concentrations, and these interactions can alter membrane properties. Here, we describe the effects of calcium ions on the dynamics in phospholipid membranes. We used a combination of time-resolved ultrafast two-dimensional infrared spectroscopy and molecular dynamics simulations. We found that millimolar Ca2+ concentrations lead to slower fluctuations in the local environment at the lipid-water interface of membranes with phosphatidylserine. The effect was only observed in bilayers containing anionic phosphatidylserine; membranes composed of only zwitterionic phosphatidylcholine did not experience a slowdown. Local water dynamics were measured using the ester groups as label-free probes and were found to be up to 50% slower with 2.5 mM Ca2+. Molecular dynamics simulations show that Ca2+ primarily binds to the carboxylate group of phosphatidylserines. These findings have implications for apoptotic and diseased cells in which phosphatidylserine is exposed to extracellular calcium and for the biophysical effects of divalent cations on lipid bilayers.

Why Does RNA Collapse? The Importance of Water in a Simulation Study of Helix-Junction-Helix Systems.

Using computer simulations, we consider the balance of thermodynamic forces that collapse RNA. A model helix-junction-helix (HJH) construct is used to investigate the transition from an extended to a collapsed conformation. Conventional Molecular Dynamics and Milestoning Simulations are used to study the free energy profile of the process for two ion concentrations. We illustrate that HJH folds to a collapsed state with two types of counterions (Mg2+ and K+). By dissecting the free energy landscape into energetic and entropic contributions, we illustrate that the electrostatic forces between the RNA and the mobile ions do not drive the RNA to a collapsed state. Instead, entropy gains from water expulsion near the neighborhood of the RNA provide the stabilization free energy that tilt HJH into more compact structures. Further simulations of a three-helix hammerhead ribozyme show a similar behavior and support the idea of collapse due to increased gain in water entropy.

A mixed alchemical and equilibrium dynamics to simulate heterogeneous dense fluids: Illustrations for Lennard-Jones mixtures and phospholipid membranes.

An algorithm to efficiently simulate multi-component fluids is proposed and illustrated. The focus is on biological membranes that are heterogeneous and challenging to investigate quantitatively. To achieve rapid equilibration of spatially inhomogeneous fluids, we mix conventional molecular dynamics simulations with alchemical trajectories. The alchemical trajectory switches the positions of randomly selected pairs of molecules and plays the role of an efficient Monte Carlo move. It assists in accomplishing rapid spatial de-correlations. Examples of phase separation and mixing are given in two-dimensional binary Lennard-Jones fluid and a DOPC-POPC membrane. The performance of the algorithm is analyzed, and tools to maximize its efficiency are provided. It is concluded that the algorithm is vastly superior to conventional molecular dynamics for the equilibrium study of biological membranes.

The Impact of Protonation on Early Translocation of Anthrax Lethal Factor: Kinetics from Molecular Dynamics Simulations and Milestoning Theory.

We report atomically detailed molecular dynamics simulations of the permeation of the lethal factor (LF) N-terminal segment through the anthrax channel. The N-terminal chain is unstructured and leads the permeation process for the LF protein. The simulations were conducted in explicit solvent with milestoning theory, making it possible to extract kinetic information from nanosecond to millisecond time scales. We illustrate that the initial event is strongly influenced by the protonation states of the permeating amino acids. While the N-terminal segment passes easily at high protonation state through the anthrax channel (and the ϕ clamp), the initial permeation represents a critical step, which can be irreversible and establishes a hook in the channel mouth.

Rock climbing: A local-global algorithm to compute minimum energy and minimum free energy pathways.

The calculation of minimum energy or minimum free energy paths is an important step in the quantitative and qualitative studies of chemical and physical processes. The computations of these coordinates present a significant challenge and have attracted considerable theoretical and computational interest. Here we present a new local-global approach to study reaction coordinates, based on a gradual optimization of an action. Like other global algorithms, it provides a path between known reactants and products, but it uses a local algorithm to extend the current path in small steps. The local-global approach does not require an initial guess to the path, a major challenge for global pathway finders. Finally, it provides an exact answer (the steepest descent path) at the end of the calculations. Numerical examples are provided for the Mueller potential and for a conformational transition in a solvated ring system.

Calculating Iso-Committor Surfaces as Optimal Reaction Coordinates with Milestoning.

Reaction coordinates are vital tools for qualitative and quantitative analysis of molecular processes. They provide a simple picture of reaction progress and essential input for calculations of free energies and rates. Iso-committor surfaces are considered the optimal reaction coordinate. We present an algorithm to compute efficiently a sequence of isocommittor surfaces. These surfaces are considered an optimal reaction coordinate. The algorithm analyzes Milestoning results to determine the committor function. It requires only the transition probabilities between the milestones, and not transition times. We discuss the following numerical examples: (i) a transition in the Mueller potential; (ii) a conformational change of a solvated peptide; and (iii) cholesterol aggregation in membranes.

Prediction of homoprotein and heteroprotein complexes by protein docking and template-based modeling: A CASP-CAPRI experiment.

We present the results for CAPRI Round 30, the first joint CASP-CAPRI experiment, which brought together experts from the protein structure prediction and protein-protein docking communities. The Round comprised 25 targets from amongst those submitted for the CASP11 prediction experiment of 2014. The targets included mostly homodimers, a few homotetramers, and two heterodimers, and comprised protein chains that could readily be modeled using templates from the Protein Data Bank. On average 24 CAPRI groups and 7 CASP groups submitted docking predictions for each target, and 12 CAPRI groups per target participated in the CAPRI scoring experiment. In total more than 9500 models were assessed against the 3D structures of the corresponding target complexes. Results show that the prediction of homodimer assemblies by homology modeling techniques and docking calculations is quite successful for targets featuring large enough subunit interfaces to represent stable associations. Targets with ambiguous or inaccurate oligomeric state assignments, often featuring crystal contact-sized interfaces, represented a confounding factor. For those, a much poorer prediction performance was achieved, while nonetheless often providing helpful clues on the correct oligomeric state of the protein. The prediction performance was very poor for genuine tetrameric targets, where the inaccuracy of the homology-built subunit models and the smaller pair-wise interfaces severely limited the ability to derive the correct assembly mode. Our analysis also shows that docking procedures tend to perform better than standard homology modeling techniques and that highly accurate models of the protein components are not always required to identify their association modes with acceptable accuracy. Proteins 2016; 84(Suppl 1):323-348. © 2016 Wiley Periodicals, Inc.

Simulations of thermodynamics and kinetics on rough energy landscapes with milestoning.

We investigated by computational means the kinetics and stationary behavior of stochastic dynamics on an ensemble of rough two-dimensional energy landscapes. There are no obvious separations of temporal scales in these systems, which constitute a simple model for the behavior of glasses and some biomaterials. Even though there are significant computational challenges present in these systems due to the large number of metastable states, the Milestoning method is able to compute their kinetic and thermodynamic properties exactly. We observe two clearly distinguished regimes in the overall kinetics: one in which diffusive behavior dominates and another that follows an Arrhenius law (despite the absence of a dominant barrier). We compare our results with those obtained with an exactly-solvable one-dimensional model, and with the results from the rough one-dimensional energy model introduced by Zwanzig. © 2015 Wiley Periodicals, Inc.

Perspective: Computer simulations of long time dynamics.

Atomically detailed computer simulations of complex molecular events attracted the imagination of many researchers in the field as providing comprehensive information on chemical, biological, and physical processes. However, one of the greatest limitations of these simulations is of time scales. The physical time scales accessible to straightforward simulations are too short to address many interesting and important molecular events. In the last decade significant advances were made in different directions (theory, software, and hardware) that significantly expand the capabilities and accuracies of these techniques. This perspective describes and critically examines some of these advances.

A MATHEMATICAL FRAMEWORK FOR EXACT MILESTONING.

We give a mathematical framework for Exact Milestoning, a recently introduced algorithm for mapping a continuous time stochastic process into a Markov chain or semi-Markov process that can be efficiently simulated and analyzed. We generalize the setting of Exact Milestoning and give explicit error bounds for the error in the Milestoning equation for mean first passage times.

Molecular dynamics studies of modular polyketide synthase ketoreductase stereospecificity.

Ketoreductases (KRs) from modular polyketide synthases (PKSs) can perform stereospecific catalysis, selecting a polyketide with a D- or L-α-methyl substituent for NADPH-mediated reduction. In this report, molecular dynamics (MD) simulations were performed to investigate the interactions that control stereospecificity. We studied the A1-type KR from the second module of the amphotericin PKS (A1), which is known to be stereospecific for a D-α-methyl-substituted diketide substrate (dkD). MD simulations of two ternary complexes comprised of the enzyme, NADPH, and either the correct substrate, dkD, or its enantiomer (dkL) were performed. The coordinates for the A1/NADPH binary complex were obtained from a crystal structure (PDB entry 3MJS), and substrates were modeled in the binding pocket in conformations appropriate for reduction. Simulations were intended to reproduce the initial weak binding of the polyketide substrate to the enzyme. Long (tens of nanoseconds) MD simulations show that the correct substrate is retained in a conformation closer to the reactive configuration. Many short (up to a nanosecond) MD runs starting from the initial structures display evidence that Q364, three residues N-terminal to the catalytic tyrosine, forms a hydrogen bond to the incorrect dkL substrate to yield an unreactive conformation that is more favorable than the reactive configuration. This interaction is not as strong for dkD, as the D-α-methyl substituent is positioned between the glutamine and the reactive site. This result correlates with experimental findings [Zheng, J., et al. (2010) Structure 18, 913-922] in which a Q364H mutant was observed to lose stereospecificity.

Extension of a protein docking algorithm to membranes and applications to amyloid precursor protein dimerization.

Novel adjustments are introduced to the docking algorithm, DOCK/PIERR, for the purpose of predicting structures of transmembrane protein complexes. Incorporating knowledge about the membrane environment is shown to significantly improve docking accuracy. The extended version of DOCK/PIERR is shown to perform comparably to other leading docking packages. This membrane version of DOCK/PIERR is applied to the prediction of coiled-coil homodimer structures of the transmembrane region of the C-terminal peptide of amyloid precursor protein (C99). Results from MD simulation of the C99 homodimer in POPC bilayer and docking are compared. Docking results are found to capture key aspects of the homodimer ensemble, including the existence of three topologically distinct conformers. Furthermore, the extended version of DOCK/PIERR is successful in capturing the effects of solvation in membrane and micelle. Specifically, DOCK/PIERR reproduces essential differences in the homodimer ensembles simulated in POPC bilayer and DPC micelle, where configurational entropy and surface curvature effects bias the handedness and topology of the homodimer ensemble.

Extracting the diffusion tensor from molecular dynamics simulation with Milestoning.

We propose an algorithm to extract the diffusion tensor from Molecular Dynamics simulations with Milestoning. A Kramers-Moyal expansion of a discrete master equation, which is the Markovian limit of the Milestoning theory, determines the diffusion tensor. To test the algorithm, we analyze overdamped Langevin trajectories and recover a multidimensional Fokker-Planck equation. The recovery process determines the flux through a mesh and estimates local kinetic parameters. Rate coefficients are converted to the derivatives of the potential of mean force and to coordinate dependent diffusion tensor. We illustrate the computation on simple models and on an atomically detailed system-the diffusion along the backbone torsions of a solvated alanine dipeptide.