Reversible folding simulation by hybrid Hamiltonian replica exchange

Reversible foldings of a beta-hairpin peptide, chignolin, by recently invented hybrid Hamiltonian replica exchange molecular dynamics simulations based on Poisson-Boltzmann model in explicit water are demonstrated. Initiated from extended structures the peptide folded and unfolded a couple of times in seven out of eight replica trajectories during 100 nanoseconds simulation. The folded states have the lowest all-atom root mean squared deviation of 1.3 A with respect to the NMR structures. At T=300 K the occurrence of folded states was converged to 62% during 80 ns simulation which agrees well with experimental data. Especially, a detailed structural evolution map was constructed based on 800,000 structural snapshots and from where a unique folding doorway emerges. Compared with 130 ns standard replica exchange simulation using 24 replicas on the same system, the hybrid Hamiltonian replica exchange molecular dynamics simulation presents consistent results.     

Multiple scaling replica exchange for the conformational sampling of biomolecules in explicit water

A multiple scaling replica exchange method for the efficient conformational sampling of biomolecular systems in explicit solvent is presented. The method is a combination of the replica exchange with solute tempering (REST) technique and a Tsallis biasing potential. The Tsallis biasing increases the sampling efficiency, while the REST minimizes the number of replicas needed. Unbiased statistics can be obtained by reweighting of the data using a weighted histogram analysis technique. The method is illustrated by its application to a ten residue peptide in explicit water.     

Simulation studies of the fidelity of biomolecular structure ensemble recreation

We examine the ability of Bayesian methods to recreate structural ensembles for partially folded molecules from averaged data. Specifically we test the ability of various algorithms to recreate different transition state ensembles for folding proteins using a multiple replica simulation algorithm using input from "gold standard" reference ensembles that were first generated with a Go-like Hamiltonian having nonpairwise additive terms. A set of low resolution data, which function as the "experimental" phi values, were first constructed from this reference ensemble. The resulting phi values were then treated as one would treat laboratory experimental data and were used as input in the replica reconstruction algorithm. The resulting ensembles of structures obtained by the replica algorithm were compared to the gold standard reference ensemble, from which those "data" were, in fact, obtained. It is found that for a unimodal transition state ensemble with a low barrier, the multiple replica algorithm does recreate the reference ensemble fairly successfully when no experimental error is assumed. The Kolmogorov-Smirnov test as well as principal component analysis show that the overlap of the recovered and reference ensembles is significantly enhanced when multiple replicas are used. Reduction of the multiple replica ensembles by clustering successfully yields subensembles with close similarity to the reference ensembles. On the other hand, for a high barrier transition state with two distinct transition state ensembles, the single replica algorithm only samples a few structures of one of the reference ensemble basins. This is due to the fact that the phi values are intrinsically ensemble averaged quantities. The replica algorithm with multiple copies does sample both reference ensemble basins. In contrast to the single replica case, the multiple replicas are constrained to reproduce the average phi values, but allow fluctuations in phi for each individual copy. These fluctuations facilitate a more faithful sampling of the reference ensemble basins. Finally, we test how robustly the reconstruction algorithm can function by introducing errors in phi comparable in magnitude to those suggested by some authors. In this circumstance we observe that the chances of ensemble recovery with the replica algorithm are poor using a single replica, but are improved when multiple copies are used. A multimodal transition state ensemble, however, turns out to be more sensitive to large errors in phi (if appropriately gauged) and attempts at successful recreation of the reference ensemble with simple replica algorithms can fall short.     

Hydrophobic aided replica exchange: an efficient algorithm for protein folding in explicit solvent

A hydrophobic aided replica exchange method (HAREM) is introduced to accelerate the simulation of all-atom protein folding in explicit solvent. This method is based on exaggerating the hydrophobic effect of various protein amino acids in water by attenuating the protein-water attractive interactions (mimicking the Chaperon effect) while leaving other interactions among protein atoms and water molecules unchanged. The method is applied to a small representative protein, the alpha-helix 3K(I), and it is found that the HAREM method successfully folds the protein within 4 ns, while the regular replica exchange method does not fold the same protein within 5 ns, even with many more replicas.     

Replica exchange with dynamical scaling

A replica exchange method is presented which requires fewer replicas and is designed to be used for large systems. In this method, dynamically scaled replicas are placed between conventional replicas at broadly spaced temperatures. The potential of the scaled replicas is linearly scaled by a dynamical variable which varies between 0 and 1. When the variable is near either end point the replica can undergo exchanges with one of its neighboring replicas. Two different versions of the method are presented for a model system of a small peptide in water. The scaled replica can replace many replicas and the method can be up to ten times more efficient than conventional replica exchange.     

Replica exchange with guided annealing for accelerated sampling of disordered protein conformations

We critically examine a recently proposed convective replica exchange (cRE) method for enhanced sampling of protein conformation based on theoretical and numerical analysis. The results demonstrate that cRE and related replica exchange with guided annealing (RE‐GA) schemes lead to unbalanced exchange attempt probabilities and break detailed balance whenever the system undergoes slow conformational transitions (relative to the temperature diffusion timescale). Nonetheless, numerical simulations suggest that approximate canonical ensembles can be generated for systems with small conformational transition barriers. This suggests that RE‐GA maybe suitable for simulating intrinsically disordered proteins, an important class of newly recognized functional proteins. The efficacy of RE‐GA is demonstrated by calculating the conformational ensembles of intrinsically disordered kinase inducible domain protein. The results show that RE‐GA helps the protein to escape nonspecific compact states more efficiently and provides several fold speedups in generating converged and largely correct ensembles compared to the standard temperature RE. © 2014 Wiley Periodicals, Inc.     

Coulomb replica‐exchange method: Handling electrostatic attractive and repulsive forces for biomolecules

We propose a new type of the Hamiltonian replica‐exchange method (REM) for molecular dynamics (MD) and Monte Carlo simulations, which we refer to as the Coulomb REM (CREM). In this method, electrostatic charge parameters in the Coulomb interactions are exchanged among replicas while temperatures are exchanged in the usual REM. By varying the atom charges, the CREM overcomes free‐energy barriers and realizes more efficient sampling in the conformational space than the REM. Furthermore, this method requires only a smaller number of replicas because only the atom charges of solute molecules are used as exchanged parameters. We performed Coulomb replica‐exchange MD simulations of an alanine dipeptide in explicit water solvent and compared the results with those of the conventional canonical, replica exchange, and van der Waals REMs. Two force fields of AMBER parm99 and AMBER parm99SB were used. As a result, the CREM sampled all local‐minimum free‐energy states more frequently than the other methods for both force fields. Moreover, the Coulomb, van der Waals, and usual REMs were applied to a fragment of an amyloid‐β peptide (Aβ) in explicit water solvent to compare the sampling efficiency of these methods for a larger system. The CREM sampled structures of the Aβ fragment more efficiently than the other methods. We obtained β‐helix, α‐helix, 3₁₀‐helix, β‐hairpin, and β‐sheet structures as stable structures and deduced pathways of conformational transitions among these structures from a free‐energy landscape. © 2012 Wiley Periodicals, Inc.     

A combined treatment of hydration and dynamical effects for the modeling of host–guest binding thermodynamics: the SAMPL5 blinded challenge

As part of the SAMPL5 blinded experiment, we computed the absolute binding free energies of 22 host–guest complexes employing a novel approach based on the BEDAM single-decoupling alchemical free energy protocol with parallel replica exchange conformational sampling and the AGBNP2 implicit solvation model specifically customized to treat the effect of water displacement as modeled by the Hydration Site Analysis method with explicit solvation. Initial predictions were affected by the lack of treatment of ionic charge screening, which is very significant for these highly charged hosts, and resulted in poor relative ranking of negatively versus positively charged guests. Binding free energies obtained with Debye–Hückel treatment of salt effects were in good agreement with experimental measurements. Water displacement effects contributed favorably and very significantly to the observed binding affinities; without it, the modeling predictions would have grossly underestimated binding. The work validates the implicit/explicit solvation approach employed here and it shows that comprehensive physical models can be effective at predicting binding affinities of molecular complexes requiring accurate treatment of conformational dynamics and hydration.     

Optimized parameters for continuum solvation calculations with carbohydrates

Continuum solvent models have been shown to be an efficient method for the calculation of the energetics of biomolecules in solution. However, for these methods to produce accurate results, an appropriate set of atomic radii or volumes is needed. While these have been developed for proteins and nucleic acids, the same is not true of carbohydrates. Here, a set of optimized parameters for continuum solvation calculations of carbohydrates in conjunction with the Carbohydrate Solution Force Field are presented. Explicit solvent free-energy perturbation simulations were performed on a set of hexapyranose sugars and used to fit atomic radii for Poisson-Boltzmann and generalized-Born calculations, and to fit atomic volumes for use with the analytical continuum electrostatics model. The solvation energetics computed with the optimized radii and a Poisson-Boltzmann model show remarkable agreement with explicit solvent simulation, with a root-mean-square error of 1.19 kcal/mol over a large test set of sugars in many conformations. The generalized-Born model gives slightly poorer agreement, but still correlates very strongly, with an error of 1.69 kcal/mol. The analytical continuum electrostatics model correlates well with the explicit solvent results, but gives a larger error of 4.71 kcal/mol. The remarkable agreement between the solvation free energies computed in explicit and implicit solvent provides strong motivation for the use of implicit solvent models in the simulation of carbohydrate-containing systems.     

Influence of water-protein hydrogen bonding on the stability of Trp-cage miniprotein. A comparison between the TIP3P and TIP4P-Ew water models

We report extensive replica exchange molecular dynamics (REMD) simulations on the folding/unfolding equilibrium of Trp-cage miniprotein using the Amber ff99SB all atom forcefield and TIP3P and TIP4P-Ew explicit water solvent models. REMD simulation-lengths in the 500 ns to the microsecond regime per replica are required to adequately sample the folding/unfolding equilibrium. We observe that this equilibrium is significantly affected by the choice of the water model. Compared with experimental data, simulations using the TIP3P solvent describe the stability of the Trp-cage quite realistically, providing a melting point which is just a few Kelvins above the experimental transition temperature of 317 K. The TIP4P-Ew model shifts the equilibrium towards the unfolded state and lowers the free energy of unfolding by about 3 kJ mol(-1) at 280 K, demonstrating the need to fine-tune the protein-forcefield depending on the chosen water model. We report evidence that the main difference between the two water models is mostly due to the different solvation of polar groups of the peptide. The unfolded state of the Trp-cage is stabilized by an increasing number of hydrogen bonds, destabilizing the α-helical part of the molecule and opening the R-D salt bridge. By reweighting the strength of solvent-peptide hydrogen bonds by adding a hydrogen bond square well potential, we can fully recover the effect of the different water models and estimate the shift in population as due to a difference in hydrogen bond-strength of about 0.4 kJ mol(-1) per hydrogen bond.     

Serial replica exchange

Parallel tempering (or the replica exchange method (REM)) is a powerful method for speeding up the sampling of conformational states of systems with rough energy landscapes, like proteins, where stable conformational states can be separated by large energy barriers. The usual implementation of the REM is performed on local computer clusters (or parallel processors) where the different replicas must be run synchronously. Here, we present serial replica exchange (SREM), a method that is equivalent to the standard REM in terms of efficiency yet runs asynchronously on a distributed network of computers. A second advantage is the method's greatly enhanced fault tolerance, which enables the study of biological systems on worldwide distributed computing environments, such as Folding@Home. For proof of concept, we apply the SREM to a single alanine dipeptide molecule in explicit water. We show that the SREM reproduces the thermodynamic and structural properties determined by the REM.     

Smart resolution replica exchange: an efficient algorithm for exploring complex energy landscapes

A coarse-grained representation of a condensed phase system can significantly reduce the number of system degrees of freedom, making coarse-grained simulations very computationally efficient. Moreover, coarse graining can smoothen the free energy landscape of the system. Thus coarse-grained dynamics is usually faster than its fully atomistic counterpart. In this work, the smart resolution replica exchange method is introduced that incorporates the information from coarse-grained simulations into atomistic simulations in order to accelerate the sampling of rough, complex atomistic energy landscapes. Within this methodology, interactions between particles are defined by a potential energy that interpolates between a fully atomistic potential and a fully coarse-grained effective potential according to a parameter lambda. Instead of exchanging the configurations from neighboring resolutions directly, as has been done in the resolution replica exchange methods [E. Lyman et al., Phys. Rev. Lett. 96, 028105 (2006); M. Christen and W. F. v. Gunsteren, J. Chem. Phys. 124, 154106 (2006)], the configuration described at the coarser resolution is first relaxed before an exchange is attempted, similar to the smart walking method [R. Zhou and B. J. Berne, J. Chem. Phys. 107, 9185 (1997)]. This approach greatly increases the acceptance ratio of exchange and only two replicas, one at the atomistic level and one at the coarse-grained level, are usually required (although more can be implemented if desired). This new method can approximately obtain the correct canonical sampling if the exchange interval is sufficiently large to allow the system to explore the local energy landscape. The method is demonstrated for a two-dimensional model system, where the ideal population distribution can be recovered, and also for an alanine polypeptide (Ala(15)) model with explicit water, where its native structure, an alpha helix, is obtained from the extended structure within 1 ns.     

Polarizable atomic multipole solutes in a Poisson-Boltzmann continuum

Modeling the change in the electrostatics of organic molecules upon moving from vacuum into solvent, due to polarization, has long been an interesting problem. In vacuum, experimental values for the dipole moments and polarizabilities of small, rigid molecules are known to high accuracy; however, it has generally been difficult to determine these quantities for a polar molecule in water. A theoretical approach introduced by Onsager [J. Am. Chem. Soc. 58, 1486 (1936)] used vacuum properties of small molecules, including polarizability, dipole moment, and size, to predict experimentally known permittivities of neat liquids via the Poisson equation. Since this important advance in understanding the condensed phase, a large number of computational methods have been developed to study solutes embedded in a continuum via numerical solutions to the Poisson-Boltzmann equation. Only recently have the classical force fields used for studying biomolecules begun to include explicit polarization in their functional forms. Here the authors describe the theory underlying a newly developed polarizable multipole Poisson-Boltzmann (PMPB) continuum electrostatics model, which builds on the atomic multipole optimized energetics for biomolecular applications (AMOEBA) force field. As an application of the PMPB methodology, results are presented for several small folded proteins studied by molecular dynamics in explicit water as well as embedded in the PMPB continuum. The dipole moment of each protein increased on average by a factor of 1.27 in explicit AMOEBA water and 1.26 in continuum solvent. The essentially identical electrostatic response in both models suggests that PMPB electrostatics offers an efficient alternative to sampling explicit solvent molecules for a variety of interesting applications, including binding energies, conformational analysis, and pK(a) prediction. Introduction of 150 mM salt lowered the electrostatic solvation energy between 2 and 13 kcalmole, depending on the formal charge of the protein, but had only a small influence on dipole moments.     

Velocity-scaling optimized replica exchange molecular dynamics of proteins in a hybrid explicit/implicit solvent

We propose a scheme for replica exchange molecular dynamics of proteins in explicit solvent that minimizes the number of required replicas using velocity rescaling. Our approach relies on a hybrid method where the protein evolves at each temperature in an explicit solvent, but replica exchange moves utilize an implicit solvent term. The two terms are coupled through the velocity rescaling. We test the efficiency of this approach for a common test case, the trp-cage protein.     

Replica exchange with solute tempering: efficiency in large scale systems

We apply the recently developed replica exchange with solute tempering (REST) to three large solvated peptide systems: an alpha-helix, a beta-hairpin, and a TrpCage, with these peptides defined as the "central group". We find that our original implementation of REST is not always more efficient than the replica exchange method (REM). Specifically, we find that exchanges between folded (F) and unfolded (U) conformations with vastly different structural energies are greatly reduced by the nonappearance of the water self-interaction energy in the replica exchange acceptance probabilities. REST, however, is expected to remain useful for a large class of systems for which the energy gap between the two states is not large, such as weakly bound protein-ligand complexes. Alternatively, a shell of water molecules can be incorporated into the central group, as discussed in the original paper.     

Continuum solvation models in the linear interaction energy method

The linear interaction energy (LIE) method in combination with two different continuum solvent models has been applied to calculate protein-ligand binding free energies for a set of inhibitors against the malarial aspartic protease plasmepsin II. Ligand-water interaction energies are calculated from both Poisson-Boltzmann (PB) and Generalized Born (GB) continuum models using snapshots from explicit solvent simulations of the ligand and protein-ligand complex. These are compared to explicit solvent calculations, and we find close agreement between the explicit water and PB solvation models. The GB model overestimates the change in solvation energy, and this is caused by consistent underestimation of the effective Born radii in the protein-ligand complex. The explicit solvent LIE calculations and LIE-PB, with our standard parametrization, reproduce absolute experimental binding free energies with an average unsigned error of 0.5 and 0.7 kcal/mol, respectively. The LIE-GB method, however, requires a constant offset to approach the same level of accuracy.     

How well does Poisson-Boltzmann implicit solvent agree with explicit solvent? A quantitative analysis

We have quantitatively studied the performance of a finite-difference Poisson-Boltzmann implicit solvent with respect to the TIP3P explicit solvent in a range of systems of biochemical interest. An overall agreement was found between the tested implicit and explicit solvents for hydrogen-bonding/salt-bridging dimers and peptide monomers and dimers of different conformations and different lengths. These comparative analyses also indicate a good transferability of empirically optimized parameters for the implicit solvent from small training molecules to large testing peptides. However, deviations between the two tested solvents are also apparent. Specifically, a consistent deviation was observed when hydrogen-bonding or salt-bridging dimers are within 4-6 A. The deviation reaches a maximum at about 5.5 A, the so-called water-bridging distance. The tested implicit solvent, even with optimized parameters, cannot capture the subtle fluctuation in the distance-dependent reaction field energy profiles, although smoothed profiles can still be obtained and are in overall agreement with those in the explicit solvent. Interestingly, the same mechanism underlining the above discrepancy is also responsible for the larger deviations of certain peptide conformations, such as parallel beta-strand dimers. It is likely that the observed discrepancy may cause improper conformational distributions in simulations with the implicit solvent when hydrogen-bonding or salt-bridging interactions are crucial, such as secondary structure populations in proteins. Validation of the implicit solvent with optimized parameters in dynamics simulations will be the next step to study the influences of the observed discrepancy at biological conditions.     

How hot? Systematic convergence of the replica exchange method using multiple reservoirs

We have devised a systematic approach to converge a replica exchange molecular dynamics simulation by dividing the full temperature range into a series of higher temperature reservoirs and a finite number of lower temperature subreplicas. A defined highest temperature reservoir of equilibrium conformations is used to help converge a lower but still hot temperature subreplica, which in turn serves as the high‐temperature reservoir for the next set of lower temperature subreplicas. The process is continued until an optimal temperature reservoir is reached to converge the simulation at the target temperature. This gradual convergence of subreplicas allows for better and faster convergence at the temperature of interest and all intermediate temperatures for thermodynamic analysis, as well as optimizing the use of multiple processors. We illustrate the overall effectiveness of our multiple reservoir replica exchange strategy by comparing sampling and computational efficiency with respect to replica exchange, as well as comparing methods when converging the structural ensemble of the disordered Aβ_21‐30 peptide simulated with explicit water by comparing calculated Rotating Overhauser Effect Spectroscopy intensities to experimentally measured values. © 2009 Wiley Periodicals, Inc. J Comput Chem 31: 620–627, 2010     

Recognition of RNA by amide modified backbone nucleic acids: molecular dynamics simulations of DNA-RNA hybrids in aqueous solution

Thermodynamic and structural properties of a chemically modified DNA-RNA hybrid in which a phosphodiester linkage is replaced by a neutral amide-3 linkage (3'-CH(2)-CONH-5') were investigated using UV melting experiments, molecular dynamics simulations in explicit water, and continuum solvent models. van't Hoff analysis of the experimental UV melting curves suggests that the significant increase of the thermodynamic stability of a 15-mer DNA-RNA with seven alternated amide-3 modifications (+11 degrees C) is mainly due to an increased binding enthalpy. To further evaluate the origin in the observed affinities differences, the electrostatic contribution to the binding free energy was calculated by solving the Poisson-Boltzmann equation numerically. The nonelectrostatic contribution was estimated as the product of a hydrophobic surface tension coefficient and the surface area that is buried upon double strand formation. Structures were taken from 10 ns molecular dynamics simulations computed in a consistent fashion using explicit solvent, counterions, and the particle-mesh Ewald procedure. The present preliminary thermodynamic study suggests that the favorable binding free energy of the amide-3 DNA single strand to the complementary RNA is equally driven by electrostatic and nonpolar contributions to the binding compared to their natural analogues. In addition, molecular dynamics simulations in explicit water were performed on an amide-3 DNA single strand and the corresponding natural DNA. Results from the conformations cluster analysis of the simulated amide-3 DNA single strand ensembles suggest that the 25% of the population sampled within 10 ns has a pre-organized conformation where the sugar C3' endo pucker is favored at the 3'-flanking nucleotides. These structural and thermodynamic features contribute to the understanding of the observed increased affinities of the amide-3 DNA-RNA hybrids at the microscopic level.