A new combination of replica exchange Monte Carlo and histogram analysis for protein folding and thermodynamics

Combining coarse-grained protein models with replica-exchange all-atom molecular dynamics

We describe a combination of all-atom simulations with CABS, a well-established coarse-grained protein modeling tool, into a single multiscale protocol. The simulation method has been tested on the C-terminal beta hairpin of protein G, a model system of protein folding. After reconstructing atomistic details, conformations derived from the CABS simulation were subjected to replica-exchange molecular dynamics simulations with OPLS-AA and AMBER99sb force fields in explicit solvent. Such a combination accelerates system convergence several times in comparison with all-atom simulations starting from the extended chain conformation, demonstrated by the analysis of melting curves, the number of native-like conformations as a function of time and secondary structure propagation. The results strongly suggest that the proposed multiscale method could be an efficient and accurate tool for high-resolution studies of protein folding dynamics in larger systems.

Finite-length effects on the coil-globule transition of a strongly charged polyelectrolyte chain in a salt-free solvent

The nature of coil-globule transition and scaling behavior of a strongly charged polyelectrolyte chain in a solution system with explicit neutralizing counterions and solvent molecules are studied using replica-exchange Monte Carlo simulations, focusing on the effects of finite chain length. The results reveal that at the thermodynamic limit of infinitely long chain length, the coil-globule transition may remain first order. Phase transition temperatures at various ion concentrations are obtained by extrapolating the values obtained at finite chain lengths. Furthermore, it is found that the exponent ν of the radius of gyration, <R(g)(2)> ~ N(2ν), can be slightly larger than 1 under some conditions.

Finding multiple minimum-energy conformations of the hydrophobic-polar protein model via multidomain sampling

We demonstrate the efficiency of the multidomain sampler (MDS) in finding multiple distinct global minima and low-energy local minima in the hydrophobic-polar (HP) lattice protein model. Extending the idea of partitioning energy space in the Wang-Landau algorithm, our approach introduces an additional partitioning scheme to divide the protein conformation space into local basins of attraction. This double-partitioning design is very powerful in guiding the sampler to visit the basins of unexplored local minima. An H-residue subchain distance is used to merge the basins of similar local minima into one domain, which increases the diversity among identified minimum-energy conformations. Moreover, a visit-enhancement factor is introduced for long protein chains to facilitate jumps between basins. Results on three benchmark protein sequences reveal that our approach is capable of finding multiple global minima and hundreds of low-energy local minima of great diversity.

Conformation transitions of a polyelectrolyte chain: a replica-exchange Monte-Carlo study

The thermodynamic behavior of a strongly charged polyelectrolyte chain immersed in a salt-free solution is studied using replica-exchange Monte-Carlo simulations. The results reveal that the chain can assume a variety of conformations, and it undergoes two phase transitions upon cooling. The first transition is identified as a continuous counterion condensation transition while the second one as a first-order coil-globule transition. In the globular state, the counterions and the charged chain segments are densely packed forming a three-dimensional Wigner crystal.

Protein folding: then and now

Over the past three decades the protein folding field has undergone monumental changes. Originally a purely academic question, how a protein folds has now become vital in understanding diseases and our abilities to rationally manipulate cellular life by engineering protein folding pathways. We review and contrast past and recent developments in the protein folding field. Specifically, we discuss the progress in our understanding of protein folding thermodynamics and kinetics, the properties of evasive intermediates, and unfolded states. We also discuss how some abnormalities in protein folding lead to protein aggregation and human diseases.

Calculation of adsorption free energy for solute-surface interactions using biased replica-exchange molecular dynamics

The adsorption behavior of a biomolecule, such as a peptide or protein, to a functionalized surface is of fundamental importance for a broad range of applications in biotechnology. The adsorption free energy for these types of interactions can be determined from a molecular dynamics simulation using the partitioning between adsorbed and nonadsorbed states, provided that sufficient sampling of both states is obtained. However, if interactions between the solute and the surface are strong, the solute will tend to be trapped near the surface during the simulation, thus preventing the adsorption free energy from being calculated by this method. This situation occurs even when using an advanced sampling algorithm such as replica-exchange molecular dynamics (REMD). In this paper, the authors demonstrate the fundamental basis of this problem using a model system consisting of one sodium ion (Na(+)) as the solute positioned over a surface functionalized with one negatively charged group (COO(-)) in explicit water. With this simple system, the authors show that sufficient sampling in the coordinate normal to the surface cannot be obtained by conventional REMD alone. The authors then present a method to overcome this problem through the use of an adaptive windowed-umbrella sampling technique to develop a biased-energy function that is combined with REMD. This approach provides an effective method for the calculation of adsorption free energy for solute-surface interactions.

A replica exchange Monte Carlo algorithm for protein folding in the HP model

Background

The ab initio protein folding problem consists of predicting protein tertiary structure from a given amino acid sequence by minimizing an energy function; it is one of the most important and challenging problems in biochemistry, molecular biology and biophysics. The ab initio protein folding problem is computationally challenging and has been shown to be NP MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFneVtcqqGqbauaaa@3961@-hard even when conformations are restricted to a lattice. In this work, we implement and evaluate the replica exchange Monte Carlo (REMC) method, which has already been applied very successfully to more complex protein models and other optimization problems with complex energy landscapes, in combination with the highly effective pull move neighbourhood in two widely studied Hydrophobic Polar (HP) lattice models.

Results

We demonstrate that REMC is highly effective for solving instances of the square (2D) and cubic (3D) HP protein folding problem. When using the pull move neighbourhood, REMC outperforms current state-of-the-art algorithms for most benchmark instances. Additionally, we show that this new algorithm provides a larger ensemble of ground-state structures than the existing state-of-the-art methods. Furthermore, it scales well with sequence length, and it finds significantly better conformations on long biological sequences and sequences with a provably unique ground-state structure, which is believed to be a characteristic of real proteins. We also present evidence that our REMC algorithm can fold sequences which exhibit significant interaction between termini in the hydrophobic core relatively easily.

Conclusion

We demonstrate that REMC utilizing the pull move neighbourhood significantly outperforms current state-of-the-art methods for protein structure prediction in the HP model on 2D and 3D lattices. This is particularly noteworthy, since so far, the state-of-the-art methods for 2D and 3D HP protein folding – in particular, the pruned-enriched Rosenbluth method (PERM) and, to some extent, Ant Colony Optimisation (ACO) – were based on chain growth mechanisms. To the best of our knowledge, this is the first application of REMC to HP protein folding on the cubic lattice, and the first extension of the pull move neighbourhood to a 3D lattice.

T-Pile--a package for thermodynamic calculations for biomolecules

Molecular dynamics and Monte Carlo, usually conducted in canonical ensemble, deliver a plethora of biomolecular conformations. Proper analysis of the simulation data is a crucial part of biophysical and bioinformatics studies. Sequence alignment problem can be also formulated in terms of Boltzmann distribution. Therefore tools for efficient analysis of canonical ensemble data become extremely valuable. T-Pile package, presented here provides a user-friendly implementation of most important algorithms such as multihistogram analysis and reweighting technique. The package can be used in studies of virtually any system governed by Boltzmann distribution.

AVAILABILITY

T-Pile can be downloaded from: http://biocomp.chem.uw.edu.pl/services/tpile. These pages provide a comprehensive tutorial and documentation with illustrative examples of applications.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Replica Exchange and Multicanonical Algorithms with the coarse-grained UNRES force field

Three algorithms, namely a Replica Exchange method (REM), a Replica Exchange Multicanonical method (REMUCA), and Replica Exchange Multicanonical with Replica Exchange (REMUCAREM), were implemented with the coarse-grained united-residue force field (UNRES) in both Monte Carlo and Molecular Dynamics versions. The MD algorithms use the constant-temperature Berendsen thermostat, with the velocity Verlet algorithm and variable time step. The algorithms were applied to one peptide (20 residues of Alanine with free ends; ala(20)) and two small proteins, namely an α-helical protein of 46 residues (the B-domain of the staphylococal protein A; 1BDD), and an α+β-protein of 48 residues (the E. Coli Mltd Lysm Domain; 1E0G). Calculated thermodynamic averages, such as canonical average energy and heat capacity, are in good agreement among all simulations for poly-L-alanine, showing that the algorithms were implemented correctly, and that all three algorithms are equally effective for small systems. For protein A, all algorithms performed reasonably well, although some variability in the calculated results was observed whereas, for a more complicated α+β-protein (1E0G), only Replica Exchange was capable of producing reliable statistics for calculating thermodynamic quantities. Finally, from the Replica Exchange molecular dynamics results, we calculated free energy maps as functions of RMSD and radius of gyration for different temperatures. The free energy calculations show correct folding behavior for poly-L-alanine and protein A while, for 1E0G, the native structure had the lowest free energy only at very low temperatures. Hence, the entropy contribution for 1E0G is larger than that for protein A at the same temperature. A larger contribution from entropy means that there are more accessible conformations at a given temperature, making it more difficult to obtain an efficient coverage of conformational space to obtain reliable thermodynamic properties. At the same temperature, ala(20) has the smallest entropy contribution, followed by protein A, and then by 1E0G.

Thermodynamics of Gō-type models for protein folding

Go-type potentials, based on the inter-residue contacts present in the native structure of a protein, are frequently used to predict dynamic and structural features of the folding pathways through computer simulations. However, the mathematical form used to define the model interactions includes several arbitrary choices, whose consequences are not usually analyzed. In this work, we use a simple off-lattice protein model and a parallel tempering Monte Carlo simulation technique to carry out such analysis, centered in the thermodynamic characteristics of the folding transition. We show how the definition of a native contact has a deep impact on the presence of simple or complex transitions, with or without thermodynamic intermediates. In addition, we have checked that the width of the attractive wells has a profound effect on the free-energy barrier between the folded and unfolded states, mainly through its influence on the entropy of the denatured state.

Protein structure prediction with the UNRES force-field using Replica-Exchange Monte Carlo-with-Minimization; Comparison with MCM, CSA, and CFMC

Two current methods of global optimization are coupled to produce the Replica-Exchange method together with Monte Carlo-with-Minimization (REMCM). Its performance is compared with each separate component and with other global optimization techniques. REMCM was applied to search the conformational space of coarse grain protein systems described by the UNRES force field. The method consists of several noninteracting copies of Monte Carlo simulation, and minimization was used after every perturbation to enhance the sampling of low-energy conformations. REMCM was applied to five proteins of different topology, and the results were compared to those from other optimization methods, namely Monte Carlo-with-Minimization (MCM), Conformational Space Annealing (CSA), and Conformational Family Monte Carlo (CFMC). REMCM located global minima for four proteins faster and more consistently than either MCM or CFMC, and it converged faster than CSA for three of the five proteins tested. A performance comparison was also carried out between REMCM and the traditional Replica Exchange method (REM) for one protein, with REMCM showing a significant improvement. Moreover, because of its simplicity, REMCM was easy to implement, thereby offering an alternative to other global optimization methods used in protein structure prediction.

MMTSB Tool Set: enhanced sampling and multiscale modeling methods for applications in structural biology

We describe the Multiscale Modeling Tools for Structural Biology (MMTSB) Tool Set (https://mmtsb.scripps.edu/software/mmtsbToolSet.html), which is a novel set of utilities and programming libraries that provide new enhanced sampling and multiscale modeling techniques for the simulation of proteins and nucleic acids. The tool set interfaces with the existing molecular modeling packages CHARMM and Amber for classical all-atom simulations, and with MONSSTER for lattice-based low-resolution conformational sampling. In addition, it adds new functionality for the integration and translation between both levels of detail. The replica exchange method is implemented to allow enhanced sampling of both the all-atom and low-resolution models. The tool set aims at applications in structural biology that involve protein or nucleic acid structure prediction, refinement, and/or extended conformational sampling. With structure prediction applications in mind, the tool set also implements a facility that allows the control and application of modeling tasks on a large set of conformations in what we have termed ensemble computing. Ensemble computing encompasses loosely coupled, parallel computation on high-end parallel computers, clustered computational grids and desktop grid environments. This paper describes the design and implementation of the MMTSB Tool Set and illustrates its utility with three typical examples--scoring of a set of predicted protein conformations in order to identify the most native-like structures, ab initio folding of peptides in implicit solvent with the replica exchange method, and the prediction of a missing fragment in a larger protein structure.

The incomplete beta function law for parallel tempering sampling of classical canonical systems

We show that the acceptance probability for swaps in the parallel tempering Monte Carlo method for classical canonical systems is given by a universal function that depends on the average statistical fluctuations of the potential and on the ratio of the temperatures. The law, called the incomplete beta function law, is valid in the limit that the two temperatures involved in swaps are close to one another. An empirical version of the law, which involves the heat capacity of the system, is developed and tested on a Lennard-Jones cluster. We argue that the best initial guess for the distribution of intermediate temperatures for parallel tempering is a geometric progression and we also propose a technique for the computation of optimal temperature schedules. Finally, we demonstrate that the swap efficiency of the parallel tempering method for condensed-phase systems decreases naturally to zero at least as fast as the inverse square root of the dimensionality of the physical system.

A simple lattice model that exhibits a protein-like cooperative all-or-none folding transition

In a recent paper (D. Gront et al., Journal of Chemical Physics, Vol. 115, pp. 1569, 2001) we applied a simple combination of the Replica Exchange Monte Carlo and the Histogram methods in the computational studies of a simplified protein lattice model containing hydrophobic and polar units and sequence-dependent local stiffness. A well-defined, relatively complex Greek-key topology, ground (native) conformations was found; however, the cooperativity of the folding transition was very low. Here we describe a modified minimal model of the same Greek-key motif for which the folding transition is very cooperative and has all the features of the "all-or-none" transition typical of real globular proteins. It is demonstrated that the all-or-none transition arises from the interplay between local stiffness and properly defined tertiary interactions. The tertiary interactions are directional, mimicking the packing preferences seen in proteins. The model properties are compared with other minimal protein-like models, and we argue that the model presented here captures essential physics of protein folding (structurally well-defined protein-like native conformation and cooperative all-or-none folding transition).

Thermodynamical properties of simple models of protein-like heteropolymers

The lattice approximation of a heteropolymer chain as a model of a single polypeptide was used in the computer simulation. The residues of a model polypeptide were represented by the chain of alpha-carbons located on a very flexible [310] lattice. The force field that mimic the intramolecular interactions contained the long-range contact potential between the residues and the local preferences in forming helical structures. The chain consisted of two types of residues that had different hydrophobicity. The simulations were performed by means of the Replica Exchange Monte Carlo method combined with the Histogram method. The series of simulations were carried out to investigate the influence of both components of the force field on the transition temperature and the characteristics of the coil-to-globule transition. The properties of low-temperature ordered structures were determined. The thermodynamical description of the model chain was also given. The phase transition was found to be sharp and cooperative for longer chains and strong helical potential. The collapsed globule contained the strongly hydrophobic residues inside the globule while the remaining residues were mainly located close to the globule surface.

A minimal physically realistic protein-like lattice model: designing an energy landscape that ensures all-or-none folding to a unique native state

A simple protein model restricted to the face-centered cubic lattice has been studied. The model interaction scheme includes attractive interactions between hydrophobic (H) residues, repulsive interactions between hydrophobic and polar (P) residues, and orientation-dependent P-P interactions. Additionally, there is a potential that favors extended beta-type conformations. A sequence has been designed that adopts a native structure, consisting of an antiparallel, six-member Greek-key beta-barrel with protein-like structural degeneracy. It has been shown that the proposed model is a minimal one, i.e., all the above listed types of interactions are necessary for cooperative (all-or-none) type folding to the native state. Simulations were performed via the Replica Exchange Monte Carlo method and the numerical data analyzed via a multihistogram method.