Adaptive Umbrella Sampling Research Articles

Uncovering the folding mechanism of pertactin: A comparative study of isolated and vectorial folding

Autotransporters are a large family of virulence factors found in Gram-negative bacteria that play important roles in their pathogenesis. The passenger domain of autotransporters is almost always composed of a large β-helix, with only a small portion of it being relevant to its virulence function. This has led to the hypothesis that the folding of the β-helical structure aids the secretion of the passenger domain across the Gram-negative outer membrane. In this study, we used molecular dynamics simulations and enhanced sampling methods to investigate the stability and folding of the passenger domain of pertactin, an autotransporter from Bordetella pertussis. Specifically, we employed steered molecular dynamics to simulate the unfolding of the entire passenger domain as well as self-learning adaptive umbrella sampling to compare the energetics of folding rungs of the β-helix independently (“isolated folding”) versus folding rungs on top of a previously folded rung (“vectorial folding”). Our results showed that vectorial folding is highly favorable compared with isolated folding; moreover, our simulations showed that the C-terminal rung of the β-helix is the most resistant to unfolding, in agreement with previous studies that found the C-terminal half of the passenger domain to be more stable than the N-terminal one. Overall, this study provides new insights into the folding process of an autotransporter passenger domain and its potential role in secretion across the outer membrane.

Calculation of CYP450 protein-ligand binding and dissociation free energy paths.

The function of an enzyme depends on its dynamic structure, and the catalytic mechanism has long been an active focus of research. The principle for interpreting protein selectivity and fidelity stems from optimization of the active site upon protein-substrate complexation, i.e., a lock-and-key configuration, on which most protein-substrate molecule binding recognition, and hence drug discovery, relies. Yet another thought has been to incorporate the protein folding interior tunnels for stereo- and regio-selectivity along the protein-substrate or protein-ligand/inhibitor binding process. Free energy calculations provide valuable information for molecular recognition and protein-ligand binding dynamics and kinetics. In this study, we focused on the kinetics of cytochrome P450 proteins (CYP450s) and the protein interior tunnel structure-dynamics relationship in terms of the substrate binding and leaving mechanism. A case in point is given by the prostaglandin H2 (PGH2) homologous isomerase of prostacyclin synthase. To calculate the reactant and product traversing the tunnels to and from the heme site, the free energy paths and tunnel potentials of mean force are constructed from steered molecular dynamics simulations and adaptive basing force umbrella sampling simulations. We explore the binding tunnels and critical residue lining characteristics for the ligand traverse and the underlying mechanism of CYP450 activity. Our theoretical analysis provides insights into the decisive role of the substrate tunnel binding process of the CYP450 mechanism and may be useful in drug design and protein engineering contexts.

MyPresto/omegagene 2020: a molecular dynamics simulation engine for virtual-system coupled sampling.

The molecular dynamics (MD) method is a promising approach for investigating the molecular mechanisms of microscopic phenomena. In particular, generalized ensemble MD methods can efficiently explore the conformational space with a rugged free-energy surface. However, the implementation and acquisition of technical knowledge for each generalized ensemble MD method are not straightforward for end-users. Here, we present a new version of the myPresto/omegagene software, which is an MD simulation engine tailored for a series of generalized ensemble methods, which are virtual-system coupled multicanonical MD (V-McMD), virtual-system coupled adaptive umbrella sampling (V-AUS), and virtual-system coupled canonical MD (VcMD). This program has been applied in several studies analyzing free-energy landscapes of a variety of molecular systems with all-atom simulations. The updated version provides new functionality for coarse-grained simulations powered by the hydrophobicity scale method. The software package includes a step-by-step tutorial document for enhanced conformational sampling of the poly-glutamine (poly-Q) oligomer expressed as a one-bead per residue model. The myPresto/omegagene software is freely available at the following URL: https://github.com/kotakasahara/omegagene under the Apache2 license.

Multidimensional virtual-system coupled canonical molecular dynamics to compute free-energy landscapes of peptide multimer assembly.

An enhanced-sampling method termed multidimensional virtual-system coupled canonical molecular dynamics (mD-VcMD) method is developed. In many cases, generalized-ensemble methods realizing enhanced sampling, for example, adaptive umbrella sampling, apply an effective potential, which is derived from temporarily assumed canonical distribution as a function of one or more arbitrarily defined reaction coordinates. However, it is not straightforward to estimate the appropriate canonical distribution, especially for cases applying multiple reaction coordinates. The current method, mD-VcMD, does not rely on the form of the canonical distribution. Therefore, it is practically useful to explore a high-dimensional reaction-coordinate space. In this article, formulation of mD-VcMD and its evaluation with the simple molecular models consisting of three or four alanine peptides are presented. We confirmed that mD-VcMD efficiently searched 2D and 3D reaction-coordinate spaces defined as interpeptide distances. Direct comparisons with results of long-term canonical MD simulations revealed that mD-VcMD produces correct canonical ensembles. © 2019 Wiley Periodicals, Inc.

Molecular dynamics coupled with a virtual system for effective conformational sampling.

An enhanced conformational sampling method is proposed: virtual-system coupled canonical molecular dynamics (VcMD). Although VcMD enhances sampling along a reaction coordinate, this method is free from estimation of a canonical distribution function along the reaction coordinate. This method introduces a virtual system that does not necessarily obey a physical law. To enhance sampling the virtual system couples with a molecular system to be studied. Resultant snapshots produce a canonical ensemble. This method was applied to a system consisting of two short peptides in an explicit solvent. Conventional molecular dynamics simulation, which is ten times longer than VcMD, was performed along with adaptive umbrella sampling. Free-energy landscapes computed from the three simulations mutually converged well. The VcMD provided quicker association/dissociation motions of peptides than the conventional molecular dynamics did. The VcMD method is applicable to various complicated systems because of its methodological simplicity. © 2018 Wiley Periodicals, Inc.

Flexible binding simulation by a novel and improved version of virtual-system coupled adaptive umbrella sampling

Virtual-system coupled adaptive umbrella sampling (VAUS) enhances sampling along a reaction coordinate by using a virtual degree of freedom. However, VAUS and regular adaptive umbrella sampling (AUS) methods are yet computationally expensive. To decrease the computational burden further, improvements of VAUS for all-atom explicit solvent simulation are presented here. The improvements include probability distribution calculation by a Markov approximation; parameterization of biasing forces by iterative polynomial fitting; and force scaling. These when applied to study Ala-pentapeptide dimerization in explicit solvent showed advantage over regular AUS. By using improved VAUS larger biological systems are amenable.

Faster Binding Free-Energy Landscape Calculation by Virtual-State Coupled Adaptive Umbrella Sampling

Accurate calculations of binding free-energy landscapes provide useful thermodynamic quantities to understand a given biological process. Adaptive umbrella sampling (AUS) aims to calculate such binding free-energy landscape along a given reaction coordinate. Typically, AUS requires a biasing force, which is defined from a canonical probability distribution of the reaction coordinate; and the distribution is updated iteratively until the system randomly walks along the reaction coordinate. However, convergence of the above iterative process may require enormous sampling, and hence computationally expensive. We established a novel method “virtual-state coupled adaptive umbrella sampling” (V-AUS) by enhancing the sampling along a virtual degree of freedom. The V-AUS quickly estimates free-energy landscape compared to AUS. We applied this method successfully to obtain the free-energy landscape for flexible binding between two Aβ-peptides in explicit water at room temperature (J. Comput. Chem. vol. 36, p1489, 2015). To further improve the V-AUS ensuring even faster convergence to uniform sampling we designed a set of methods on top of V-AUS. We define grids around a receptor in which its ligand is placed to obtain a better initial set of conformations. Secondly, sampling errors from the estimated canonical probability distribution due to inadequate sampling are removed by Markov state modeling. Thirdly, we introduce a set of novel fitting schemes to parameterize the biasing force in terms of reaction coordinate. These set of methods to improve V-AUS are tested for Ala-pentapeptide dimerization. We conclude that by using such advanced set of methods faster free-energy calculation of flexible molecular docking is possible. Currently a study for larger biological system is ongoing.

Virtual-system-coupled adaptive umbrella sampling to compute free-energy landscape for flexible molecular docking.

A novel enhanced conformational sampling method, virtual-system-coupled adaptive umbrella sampling (V-AUS), was proposed to compute 300-K free-energy landscape for flexible molecular docking, where a virtual degrees of freedom was introduced to control the sampling. This degree of freedom interacts with the biomolecular system. V-AUS was applied to complex formation of two disordered amyloid-β (Aβ30-35 ) peptides in a periodic box filled by an explicit solvent. An interpeptide distance was defined as the reaction coordinate, along which sampling was enhanced. A uniform conformational distribution was obtained covering a wide interpeptide distance ranging from the bound to unbound states. The 300-K free-energy landscape was characterized by thermodynamically stable basins of antiparallel and parallel β-sheet complexes and some other complex forms. Helices were frequently observed, when the two peptides contacted loosely or fluctuated freely without interpeptide contacts. We observed that V-AUS converged to uniform distribution more effectively than conventional AUS sampling did.

Theory of Adaptive Optimization for Umbrella Sampling.

We present a theory of adaptive optimization for umbrella sampling. With the analytical bias force constant obtained from the constrained thermodynamic length along the reaction coordinate, the windows are distributed to optimize the overlap between neighbors. Combining with the replica exchange method, we propose a method of adaptive window exchange umbrella sampling. The efficiency gain in sampling by the present method originates from the optimal window distribution, in which windows are concentrated to the region where the free energy is steep, as well as consequently improved random walk.

Well-Tempered Metadynamics Converges Asymptotically

Metadynamics is a versatile and capable enhanced sampling method for the computational study of soft matter materials and biomolecular systems. However, over a decade of application and several attempts to give this adaptive umbrella sampling method a firm theoretical grounding prove that a rigorous convergence analysis is elusive. This Letter describes such an analysis, demonstrating that well-tempered metadynamics converges to the final state it was designed to reach and, therefore, that the simple formulas currently used to interpret the final converged state of tempered metadynamics are correct and exact. The results do not rely on any assumption that the collective variable dynamics are effectively Brownian or any idealizations of the hill deposition function; instead, they suggest new, more permissive criteria for the method to be well behaved. The results apply to tempered metadynamics with or without adaptive Gaussians or boundary corrections and whether the bias is stored approximately on a grid or exactly.

Dissociation rates of H2 on a Ni(100) surface: the role of the physisorbed state.

The dissociation and recombination rates of physisorbed H2, and the total dissociation rate of gas phase H2 on the rigid Ni(100) surface, as well as the corresponding kinetic isotope effects, are calculated by using the quantum instanton method, together with path integral Monte Carlo and adaptive umbrella sampling techniques. Both the dissociation and recombination rates of physisorbed H2 are dramatically enhanced by the quantum motions of H2 at low temperatures, for instance, the quantum rates are 43 and 7.5 times larger than the classical ones at 200 K, respectively. For the dissociation of gas phase H2, at high temperatures, the H2 can fly over the physisorbed state and dissociate directly, however, at low temperatures, the H2 is first physisorbed and then dissociates under steady state approximation. The total dissociation rate of gas phase H2 can be expressed as a combination of the direct and steady state dissociation rates. It has the form of an inverted bell with a minimum value at about 400 K, and detailed analysis shows that the dissociation of gas phase H2 is dominated by a steady state process below 400 K, however, both the steady state and direct processes are important above 400 K. The calculated kinetic isotope effects reveal that H2 always has larger rates than D2 no matter which dissociative process they undergo.

Effect of Lattice Motion on Dissociation and Recombination Rates of H2 on Ni(100) Surface

The rate constants and kinetic isotope effects of H2 dissociation and recombination on Ni(100) surface are calculated by using the quantum instanton method, together with path integral Monte Carlo and adaptive umbrella sampling techniques. The Ni(100) surface model containing 104 nickel atoms and the potential energy surface based on the embedded diatomics in molecules are used. For the H2 dissociation, the results on the rigid lattice are consistent with experimental data. Compared to the rigid lattice, the classical and quantum motions of the lattice further enhance the dissociation rates by 18 and 49% at 300 K. The calculated kinetic isotope effects show that the H2 always has the largest rate, while the D2 has the smallest one. For the H2 recombination, however, the effects of lattice motions on the rates are different from those for the dissociation, that is, compared to the rigid lattice, both the classical and quantum motions of the lattice lower the recombination rates. The possible mechanism is analyzed by the corresponding free energy profiles.

Self-Learning Adaptive Umbrella Sampling Method for the Determination of Free Energy Landscapes in Multiple Dimensions.

The potential of mean force describing conformational changes of biomolecules is a central quantity that determines the function of biomolecular systems. Calculating an energy landscape of a process that depends on three or more reaction coordinates might require a lot of computational power, making some of multidimensional calculations practically impossible. Here, we present an efficient automatized umbrella sampling strategy for calculating multidimensional potential of mean force. The method progressively learns by itself, through a feedback mechanism, which regions of a multidimensional space are worth exploring and automatically generates a set of umbrella sampling windows that is adapted to the system. The self-learning adaptive umbrella sampling method is first explained with illustrative examples based on simplified reduced model systems, and then applied to two non-trivial situations: the conformational equilibrium of the pentapeptide Met-enkephalin in solution and ion permeation in the KcsA potassium channel. With this method, it is demonstrated that a significant smaller number of umbrella windows needs to be employed to characterize the free energy landscape over the most relevant regions without any loss in accuracy.

DNA Bending through Roll Angles Is Independent of Adjacent Base Pairs.

We have studied DNA bending for a wide range of DNA sequences by two-dimensional adaptive umbrella sampling simulations on adjacent roll angles. Calculated free energy surfaces are largely additive and can be well approximated by the sum of the one-dimensional free energy surfaces. Cooperativity between adjacent roll angles was found to be negligible: less than 1.0 kcal/mol and a small fraction of the overall bending energy. Our calculations validate the assumptions underlying many popular coarse-grained models for DNA bending, and demonstrate their theoretical validity for investigating DNA bending.

DNA Bending through Large Angles Is Aided by Ionic Screening

We used adaptive umbrella sampling on a modified version of the roll angle to simulate the bending of DNA dodecamers. Simulations were carried out with the AMBER and CHARMM force fields for 10 sequences in which the central base pair step was varied. On long length scales, the DNA behavior was found to be consistent with the worm-like chain model. Persistence lengths calculated directly from the simulated structures and indirectly through the use of sequence-independent coarse-grained models based on simulation data were similar to literature values. On short length scales, the free energy cost of bending DNA was found to be consistent with the worm-like chain model for small and intermediate bending angles. At large angles, the bending free energy as a function of the roll angle became linear, suggesting a relative increase in flexibility at larger roll angles. Counterions congregated on the concave side of the highly bent DNA and screened the repulsion of the phosphate groups, facilitating the bending.

Enhanced and effective conformational sampling of protein molecular systems for their free energy landscapes.

Protein folding and protein–ligand docking have long persisted as important subjects in biophysics. Using multicanonical molecular dynamics (McMD) simulations with realistic expressions, i.e., all-atom protein models and an explicit solvent, free-energy landscapes have been computed for several systems, such as the folding of peptides/proteins composed of a few amino acids up to nearly 60 amino-acid residues, protein–ligand interactions, and coupled folding and binding of intrinsically disordered proteins. Recent progress in conformational sampling and its applications to biophysical systems are reviewed in this report, including descriptions of several outstanding studies. In addition, an algorithm and detailed procedures used for multicanonical sampling are presented along with the methodology of adaptive umbrella sampling. Both methods control the simulation so that low-probability regions along a reaction coordinate are sampled frequently. The reaction coordinate is the potential energy for multicanonical sampling and is a structural identifier for adaptive umbrella sampling. One might imagine that this probability control invariably enhances conformational transitions among distinct stable states, but this study examines the enhanced conformational sampling of a simple system and shows that reasonably well-controlled sampling slows the transitions. This slowing is induced by a rapid change of entropy along the reaction coordinate. We then provide a recipe to speed up the sampling by loosening the rapid change of entropy. Finally, we report all-atom McMD simulation results of various biophysical systems in an explicit solvent.

Simulations of DNA Bending using Adaptive Umbrella Sampling on Roll Angles

Many proteins bend DNA significantly upon binding. To simulate this bending within the time scales accessible to molecular dynamics, we used adaptive umbrella sampling on the roll angles. We studied the inherent flexibility of bare DNA by simulating dodecamers with varying sequences and made comparisons to the worm-like chain model. Simulations of bare DNA were also performed to assess the co-operativity of the bending of adjacent roll angles. A study of DNA bending by the archaeal DNA packaging protein Sac7d will be presented as well.

Virtual states introduced for overcoming entropic barriers in conformational space

Free-energy landscape is an important quantity to study large-scale motions of a biomolecular system because it maps possible pathways for the motions. When the landscape consists of thermodynamically stable states (low-energy basins), which are connected by narrow conformational pathways (i.e., bottlenecks), the narrowness slows the inter-basin round trips in conformational sampling. This results in inaccuracy of free energies for the basins. This difficulty is not cleared out even when an enhanced conformational sampling is fairly performed along a reaction coordinate. In this study, to enhance the inter-basin round trips we introduced a virtual state that covers the narrow pathways. The probability distribution function for the virtual state was controlled based on detailed balance condition for the inter-state transitions (transitions between the real-state basins and the virtual state). To mimic the free-energy landscape of a real biological system, we introduced a simple model where a wall separates two basins and a narrow hole is pierced in the wall to connect the basins. The sampling was done based on Monte Carlo (MC). We examined several hole-sizes and inter-state transition probabilities. For a small hole-size, a small inter-state transition probability produced a sampling efficiency 100 times higher than a conventional MC does. This result goes against ones intuition, because one considers generally that the sampling efficiency increases with increasing the transition probability. The present method is readily applicable to enhanced conformational sampling such as multi-canonical or adaptive umbrella sampling, and extendable to molecular dynamics.

MMC: A Monte Carlo and Analysis Program

The poster describes the program MMC that performs Monte Carlo simulations on a molecular assembly in the canonical, isothermal-isobaric and grand-canonical ensembles. Both all-atom and continuum treatment of the solvent environment is available; solvents can also be limited to a primary hydration shell that adjusts to the solute's shape. A variety of free-energy methodologies are implemented, including polynomial thermodynamic integration and adaptive umbrella sampling. The forcefields implemented include Charmm, Amber and Gromos/Gromacs. Extensive consistency checks and a large test suite are employed to keep the code as bug-free as possible. The analysis options include determination of generic solvent sites, cavity and invagination detection, energy partitioning, and a wide range of analyses under the Proximity Criterion. The Proximity Criterion partitions the space around the solute bases on bisectors or radical planes and for each region calculates, among others, radial distribution functions, coordination numbers, angular distributions and energy distributions. The analysis options can use as input simulation trajectories, besides those in MMC's own format, trajectories in Charmm or Amber formats as well.

Cy3-DNA Stacking Interactions Strongly Depend on the Identity of the Terminal Basepair

We characterized the effect of the first basepair on the conformational dynamics of the fluorescent dye Cy3 attached to the 5′ end of double-stranded DNA using Gaussian-mixture adaptive umbrella sampling simulations. In the simulations, the sampling of all five dihedral angles along the linker was enhanced, so that both stacked and unstacked states were sampled. The affinity of Cy3 for a T·A basepair (with the dye attached to T) was found to be significantly less than for the other basepairs. This was verified experimentally by measuring the activation energies for cis-trans isomerization of the dye. The simulation and experimental results indicate the existence of partially unstacked conformations amenable to photoisomerization. The simulations also showed that stacking of Cy3 straightens the DNA while stabilizing the first basepair. Our findings indicate that fluorescence is modulated by Cy3-DNA interactions in a sequence-dependent manner.