Comparison of three Monte Carlo conformational search strategies for a proteinlike homopolymer model: Folding thermodynamics and identification of low-energy structures

Protein structure predictions by enhanced conformational sampling methods

In this Special Festschrift Issue for the celebration of Professor Nobuhiro Gō's 80th birthday, we review enhanced conformational sampling methods for protein structure predictions. We present several generalized-ensemble algorithms such as multicanonical algorithm, replica-exchange method, etc. and parallel Monte Carlo or molecular dynamics method with genetic crossover. Examples of the results of these methods applied to the predictions of protein tertiary structures are also presented.

Parallel Implementation of a Sequential Markov Chain in Monte Carlo Simulations of Physical Systems with Pairwise Interactions

In molecular simulations performed by Markov Chain Monte Carlo (typically employing the Metropolis criterion), each state of a system is obtained by a small random modification of the previous state. Therefore, the process consists of an immense number of small, quick to calculate steps, which are inherently sequential and hence considered to be very hard to parallelise. Here, we present a novel protocol for efficient calculation of multiple sequential steps in parallel. To this end, we first precompute in parallel energy components of all states achievable in a sequence of steps. Then we select a single path through all achievable states, which is identical with the path obtained with the sequential algorithm. As an example, we carried out simulations of the TIP5P water model with the new protocol and compared results with those obtained using the standard Metropolis Monte Carlo scheme. The implementation on the Titan X (Pascal) graphic processor (GPU) architectures allows for a 30-fold speedup in comparison with a simulation on a single core of a multicore CPU. The protocol is general and not limited to the GPU; it can also be used on multicore CPU when the longest possible length of the single simulation is required.

Enhanced sampling algorithms

In biomolecular systems (especially all-atom models) with many degrees of freedom such as proteins and nucleic acids, there exist an astronomically large number of local-minimum-energy states. Conventional simulations in the canonical ensemble are of little use, because they tend to get trapped in states of these energy local minima. Enhanced conformational sampling techniques are thus in great demand. A simulation in generalized ensemble performs a random walk in potential energy space and can overcome this difficulty. From only one simulation run, one can obtain canonical-ensemble averages of physical quantities as functions of temperature by the single-histogram and/or multiple-histogram reweighting techniques. In this article we review uses of the generalized-ensemble algorithms in biomolecular systems. Three well-known methods, namely, multicanonical algorithm, simulated tempering, and replica-exchange method, are described first. Both Monte Carlo and molecular dynamics versions of the algorithms are given. We then present various extensions of these three generalized-ensemble algorithms. The effectiveness of the methods is tested with short peptide and protein systems.

Application of Monte Carlo-based receptor ensemble docking to virtual screening for GPCR ligands

Receptor ensemble docking (RED) is an effective strategy to account for receptor flexibility in the course of a docking-based virtual screening campaign. Such an approach can be applied when multiple crystal structures of a receptor have been solved, but it can also be applied when only a single crystal structure is available. In this case, alternative structures can be generated from the latter by computational means and subsequently applied to RED. Here, we illustrate how such conformers can be generated by subjecting a crystal structure to Monte Carlo conformational searches. Through a controlled virtual screening experiment, we then show the applicability of such a strategy to the identification of ligands of the β(2) adrenergic receptor, a G protein-coupled receptor activated by epinephrine. Requiring the availability of one crystal structure only, this strategy is applicable to all systems for which multiple experimentally elucidated structures are not available.

Density of states-based molecular simulations

One of the central problems in statistical mechanics is that of finding the density of states of a system. Knowledge of the density of states of a system is equivalent to knowledge of its fundamental equation, from which all thermodynamic quantities can be obtained. Over the past several years molecular simulations have made considerable strides in their ability to determine the density of states of complex fluids and materials. In this review we discuss some of the more promising approaches proposed in the recent literature along with their advantages and limitations.

A comparative study of the reported performance of ab initio protein structure prediction algorithms

Protein structure prediction is one of the major challenges in bioinformatics today. Throughout the past five decades, many different algorithmic approaches have been attempted, and although progress has been made the problem remains unsolvable even for many small proteins. While the general objective is to predict the three-dimensional structure from primary sequence, our current knowledge and computational power are simply insufficient to solve a problem of such high complexity. Some prediction algorithms do, however, appear to perform better than others, although it is not always obvious which ones they are and it is perhaps even less obvious why that is. In this review, the reported performance results from 18 different recently published prediction algorithms are compared. Furthermore, the general algorithmic settings most likely responsible for the difference in the reported performance are identified, and the specific settings of each of the 18 prediction algorithms are also compared. The average normalized r.m.s.d. scores reported range from 11.17 to 3.48. With a performance measure including both r.m.s.d. scores and CPU time, the currently best-performing prediction algorithm is identified to be the I-TASSER algorithm. Two of the algorithmic settings--protein representation and fragment assembly--were found to have definite positive influence on the running time and the predicted structures, respectively. There thus appears to be a clear benefit from incorporating this knowledge in the design of new prediction algorithms.

An adaptive bin framework search method for a beta-sheet protein homopolymer model

BACKGROUND

The problem of protein structure prediction consists of predicting the functional or native structure of a protein given its linear sequence of amino acids. This problem has played a prominent role in the fields of biomolecular physics and algorithm design for over 50 years. Additionally, its importance increases continually as a result of an exponential growth over time in the number of known protein sequences in contrast to a linear increase in the number of determined structures. Our work focuses on the problem of searching an exponentially large space of possible conformations as efficiently as possible, with the goal of finding a global optimum with respect to a given energy function. This problem plays an important role in the analysis of systems with complex search landscapes, and particularly in the context of ab initio protein structure prediction.

RESULTS

In this work, we introduce a novel approach for solving this conformation search problem based on the use of a bin framework for adaptively storing and retrieving promising locally optimal solutions. Our approach provides a rich and general framework within which a broad range of adaptive or reactive search strategies can be realized. Here, we introduce adaptive mechanisms for choosing which conformations should be stored, based on the set of conformations already stored in memory, and for biasing choices when retrieving conformations from memory in order to overcome search stagnation.

CONCLUSION

We show that our bin framework combined with a widely used optimization method, Monte Carlo search, achieves significantly better performance than state-of-the-art generalized ensemble methods for a well-known protein-like homopolymer model on the face-centered cubic lattice.

Monte Carlo refinement of rigid-body protein docking structures with backbone displacement and side-chain optimization

Structures of hitherto unknown protein complexes can be predicted by docking the solved protein monomers. Here, we present a method to refine initial docking estimates of protein complex structures by a Monte Carlo approach including rigid-body moves and side-chain optimization. The energy function used is comprised of van der Waals, Coulomb, and atomic contact energy terms. During the simulation, we gradually shift from a novel smoothed van der Waals potential, which prevents trapping in local energy minima, to the standard Lennard-Jones potential. Following the simulation, the conformations are clustered to obtain the final predictions. Using only the first 100 decoys generated by a fast Fourier transform (FFT)-based rigid-body docking method, our refinement procedure is able to generate near-native structures (interface RMSD <2.5 A) as first model in 14 of 59 cases in a benchmark set. In most cases, clear binding funnels around the native structure can be observed. The results show the potential of Monte Carlo refinement methods and emphasize their applicability for protein-protein docking.

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.

Comparative modeling without implicit sequence alignments

MOTIVATION

The number of known protein sequences is about thousand times larger than the number of experimentally solved 3D structures. For more than half of the protein sequences a close or distant structural analog could be identified. The key starting point in a classical comparative modeling is to generate the best possible sequence alignment with a template or templates. With decreasing sequence similarity, the number of errors in the alignments increases and these errors are the main causes of the decreasing accuracy of the molecular models generated. Here we propose a new approach to comparative modeling, which does not require the implicit alignment - the model building phase explores geometric, evolutionary and physical properties of a template (or templates).

RESULTS

The proposed method requires prior identification of a template, although the initial sequence alignment is ignored. The model is built using a very efficient reduced representation search engine CABS to find the best possible superposition of the query protein onto the template represented as a 3D multi-featured scaffold. The criteria used include: sequence similarity, predicted secondary structure consistency, local geometric features and hydrophobicity profile. For more difficult cases, the new method qualitatively outperforms existing schemes of comparative modeling. The algorithm unifies de novo modeling, 3D threading and sequence-based methods. The main idea is general and could be easily combined with other efficient modeling tools as Rosetta, UNRES and others.

All-atom ab initio folding of a diverse set of proteins

Natural proteins fold to a unique, thermodynamically dominant state. Modeling of the folding process and prediction of the native fold of proteins are two major unsolved problems in biophysics. Here, we show successful all-atom ab initio folding of a representative diverse set of proteins by using a minimalist transferable-energy model that consists of two-body atom-atom interactions, hydrogen bonding, and a local sequence-energy term that models sequence-specific chain stiffness. Starting from a random coil, the native-like structure was observed during replica exchange Monte Carlo (REMC) simulation for most proteins regardless of their structural classes; the lowest energy structure was close to native-in the range of 2-6 A root-mean-square deviation (rmsd). Our results demonstrate that the successful folding of a protein chain to its native state is governed by only a few crucial energetic terms.

Thermodynamics of folding and association of lattice-model proteins

Closely related to the "protein folding problem" is the issue of protein misfolding and aggregation. Protein aggregation has been associated with the pathologies of nearly 20 human diseases and presents serious difficulties during the manufacture of pharmaceutical proteins. Computational studies of multiprotein systems have recently emerged as a powerful complement to experimental efforts aimed at understanding the mechanisms of protein aggregation. We describe the thermodynamics of systems containing two lattice-model 64-mers. A parallel tempering algorithm abates problems associated with glassy systems and the weighted histogram analysis method improves statistical quality. The presence of a second chain has a substantial effect on single-chain conformational preferences. The melting temperature is substantially reduced, and the increase in the population of unfolded states is correlated with an increase in interactions between chains. The transition from two native chains to a non-native aggregate is entropically favorable. Non-native aggregates receive approximately 25% of their stabilizing energy from intraprotein contacts not found in the lowest-energy structure. Contact maps show that for non-native dimers, nearly 50% of the most probable interprotein contacts involve pairs of residues that form native contacts, suggesting that a domain-swapping mechanism is involved in self-association.

Effect of single-point sequence alterations on the aggregation propensity of a model protein

Sequences of contemporary proteins are believed to have evolved through a process that optimized their overall fitness, including their resistance to deleterious aggregation. Biotechnological processing may expose therapeutic proteins to conditions that are much more conducive to aggregation than those encountered in a cellular environment. An important task of protein engineering is to identify alternative sequences that would protect proteins when processed at high concentrations without altering their native structure associated with specific biological function. Our computational studies exploit parallel tempering simulations of coarse-grained model proteins to demonstrate that isolated amino acid residue substitutions can result in significant changes in the aggregation resistance of the protein in a crowded environment while retaining protein structure in isolation. A thermodynamic analysis of protein clusters subject to competing processes of folding and association shows that moderate mutations can produce effects similar to those caused by changes in system conditions, including temperature, concentration, and solvent composition, that affect the aggregation propensity. The range of conditions where a protein can resist aggregation can therefore be tuned by sequence alterations, although the protein generally may retain its generic ability for aggregation.

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.

Free energy surfaces of miniproteins with a ββα motif: Replica exchange molecular dynamics simulation with an implicit solvation model

Designed miniproteins with a betabetaalpha motif, such as BBA5, 1FSD, and 1PSV can serve as a benchmark set to test the validity of all-atom force fields with computer simulation, because they contain all the basic structural elements in protein folding. Unfortunately, it was found that the standard all-atom force fields with the generalized Born (GB) implicit solvation model tend to produce distorted free energy surfaces for the betabetaalpha proteins, not only because energetically those proteins need to be described by more balanced weights of the alpha- and beta-strands, but also because the GB implicit solvation model suffers from overestimated salt bridge effects. In an attempt to resolve these problems, we have modified one of the standard all-atom force fields in conjunction with the GB model, such that each native state of the betabetaalpha proteins is in its free energy minimum state with reasonable energy barriers separating local minima. With this modified energy model, the free energy contour map in each protein was constructed from the replica exchange molecular dynamics REMD simulation. The resulting free energy surfaces are significantly improved in comparison with previous simulation results and consistent with general views on small protein folding behaviors with realistic topology and energetics of all three proteins.

Protein-folding landscapes in multichain systems

Computational studies of proteins have significantly improved our understanding of protein folding. These studies are normally carried out by using chains in isolation. However, in many systems of practical interest, proteins fold in the presence of other molecules. To obtain insight into folding in such situations, we compare the thermodynamics of folding for a Miyazawa-Jernigan model 64-mer in isolation to results obtained in the presence of additional chains. The melting temperature falls as the chain concentration increases. In multichain systems, free-energy landscapes for folding show an increased preference for misfolded states. Misfolding is accompanied by an increase in interprotein interactions; however, near the folding temperature, the transition from folded chains to misfolded and associated chains is entropically driven. A majority of the most probable interprotein contacts are also native contacts, suggesting that native topology plays a role in early stages of aggregation.

Conformational transition free energy profiles of an adsorbed, lattice model protein by multicanonical Monte Carlo simulation

Proteins often undergo changes in internal conformation upon interacting with a surface. We investigate the thermodynamics of surface induced conformational change in a lattice model protein using a multicanonical Monte Carlo method. The protein is a linear heteropolymer of 27 segments (of types A and B) confined to a cubic lattice. The segmental order and nearest neighbor contact energies are chosen to yield, in the absence of an adsorbing surface, a unique 3x3x3 folded structure. The surface is a plane of sites interacting either equally with A and B segments (equal affinity surface) or more strongly with the A segments (A affinity surface). We use a multicanonical Monte Carlo algorithm, with configuration bias and jump walking moves, featuring an iteratively updated sampling function that converges to the reciprocal of the density of states 1/Omega(E), E being the potential energy. We find inflection points in the configurational entropy, S(E)=k ln Omega(E), for all but a strongly adsorbing equal affinity surface, indicating the presence of free energy barriers to transition. When protein-surface interactions are weak, the free energy profiles F(E)=E-TS(E) qualitatively resemble those of a protein in the absence of a surface: a free energy barrier separates a folded, lowest energy state from globular, higher energy states. The surface acts in this case to stabilize the globular states relative to the folded state. When the protein surface interactions are stronger, the situation differs markedly: the folded state no longer occurs at the lowest energy and free energy barriers may be absent altogether.

Exploring protein energy landscapes with hierarchical clustering

In this work we present a new method for investigating local energy minima on a protein energy landscape. The CABS (CAlpha, CBeta and the center of mass of the Side chain) method was employed for generating protein models, but any other method could be used instead. Cα traces from an ensemble of models are hierarchical clustered with the HCPM (Hierarchical Clustering of Protein Models) method. The efficiency of this method for sampling and analyzing energy landscapes is shown.

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.

Lattice models of peptide aggregation: Evaluation of conformational search algorithms

We present a series of conformational search calculations on the aggregation of short peptide fragments that form fibrils similar to those seen in many protein mis-folding diseases. The proteins were represented by a face-centered cubic lattice model with the conformational energies calculated using the Miyazawa-Jernigan potential. The searches were performed using algorithms based on the Metropolis Monte Carlo method, including simulated annealing and replica exchange. We also present the results of searches using the tabu search method, an algorithm that has been used for many optimization problems, but has rarely been used in protein conformational searches. The replica exchange algorithm consistently found more stable structures then the other algorithms, and was particularly effective for the octamers and larger systems.

Generalized-ensemble algorithms: enhanced sampling techniques for Monte Carlo and molecular dynamics simulations

In complex systems with many degrees of freedom such as spin glass and biomolecular systems, conventional simulations in canonical ensemble suffer from the quasi-ergodicity problem. A simulation in generalized ensemble performs a random walk in potential energy space and overcomes this difficulty. From only one simulation run, one can obtain canonical ensemble averages of physical quantities as functions of temperature by the single-histogram and/or multiple-histogram reweighting techniques. In this article we review the generalized ensemble algorithms. Three well-known methods, namely, multicanonical algorithm (MUCA), simulated tempering (ST), and replica-exchange method (REM), are described first. Both Monte Carlo (MC) and molecular dynamics (MD) versions of the algorithms are given. We then present five new generalized-ensemble algorithms which are extensions of the above methods.

Generalized simulated tempering realized on expanded ensembles of non-Boltzmann weights

A generalized version of the simulated tempering operated in the expanded ensembles of non-Boltzmann weights has been proposed to mitigate a quasiergodicity problem occurring in simulations of rough energy landscapes. In contrast to conventional simulated tempering employing the Boltzmann weight, our method utilizes a parametrized, generalized distribution as a workhorse for stochastic exchanges of configurations and subensembles transitions, which allows a considerable enhancement for the rate of convergence of Monte Carlo and molecular dynamics simulations using delocalized weights. A feature of our method is that the exploration of the parameter space encouraging subensembles transitions is greatly accelerated using the dynamic update scheme for the weight via the average guide specific to the energy distribution. The performance and characteristic feature of our method have been validated in the liquid-solid transition of Lennard-Jones clusters and the conformational sampling of alanine dipeptide by taking two types of Tsallis [C. Tsallis, J. Stat. Phys. 52, 479 (1988)] expanded ensembles associated with different parametrization schemes.

Dynamical origin of enhanced conformational searches of Tsallis statistics sampling

The characteristic sampling dynamics of importance samplings driven by the Tsallis weight [C. Tsallis, J. Stat. Phys. 52, 479 (1988)] has been analyzed in terms of recently developed Langevin stochastic model by considering the effects of the density of states and the potential smoothing of the Tsallis transformation. Our study reveals that the fixed points, which are determined by the crossing points of the statistical temperature and the Tsallis effective temperature, play a critical role in overall dynamics of the Tsallis statistics sampling. The dynamical origin of enhanced conformational searches of the Tsallis weight has been investigated by unveiling the intimate relationship between the sampling dynamics and the stability change of corresponding fixed points. Based on this stochastic analysis, we propose one effective method to realize a broad energy distribution in the Tsallis statistics sampling by determining optimal Tsallis parameters systematically based on preliminary canonical samplings. The effectiveness of our method has been validated in the folding simulation of Met-Enkephalin and liquid-solid transition simulation of Lennard-Jones cluster systems.

Configurational temperature density of states simulations of proteins

A novel method has been implemented to compute the density of states of proteins. A united atom representation and the CHARMM (Brooks et al., 1983) force-field parameters have been adopted for all the simulations. In this approach, an intrinsic temperature is computed based on configurational information about the protein. A random walk is performed in potential energy space and the configurational temperature is collected as a function of potential energy of the system. The density of states is then calculated by integrating the reciprocal of temperature. Unlike previously available methods, this approach does not involve calculations based on histograms of stochastic visits to distinct energy states. It is found that the proposed method is more efficient than earlier, related schemes for simulation of protein folding. Furthermore, it directly provides thermodynamic information, including free energies. The usefulness of the method is discussed by presenting results of simulations of the 16-residue beta-hairpin taken from the C-terminal fragment (41-56) of protein G.

Replica-exchange method using the generalized effective potential

We propose an effective scheme for fast conformational searches by combining the replica exchange method (REM) with the generalized effect potential concept. The present method introduces the "q" value from the effective potential as a coupling parameter. It is found that the new method not only requires a much smaller number of replicas than the conventional REM, but also makes it possible to perform effective conformational sampling of complex systems with correct distributions maintained. The advantage of the present method has been demonstrated with in vacuo alanine dipeptide using a molecular dynamics simulation.

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.

Ab initio protein structure prediction via a combination of threading, lattice folding, clustering, and structure refinement

A combination of sequence comparison, threading, lattice, and off-lattice Monte Carlo (MC) simulations and clustering of MC trajectories was used to predict the structure of all (but one) targets of the CASP4 experiment on protein structure prediction. Although this method is automated and is operationally the same regardless of the level of uniqueness of the query proteins, here we focus on the more difficult targets at the border of the fold recognition and new fold categories. For a few targets (T0110 is probably the best example), the ab initio method produced more accurate models than models obtained by the fold recognition techniques. For the most difficult targets from the new fold categories, substantial fragments of structures have been correctly predicted. Possible improvements of the method are briefly discussed.

Local energy landscape flattening: parallel hyperbolic Monte Carlo sampling of protein folding

Among the major difficulties in protein structure prediction is the roughness of the energy landscape that must be searched for the global energy minimum. To address this issue, we have developed a novel Monte Carlo algorithm called parallel hyperbolic sampling (PHS) that logarithmically flattens local high-energy barriers and, therefore, allows the simulation to tunnel more efficiently through energetically inaccessible regions to low-energy valleys. Here, we show the utility of this approach by applying it to the SICHO (SIde-CHain-Only) protein model. For the same CPU time, the parallel hyperbolic sampling method can identify much lower energy states and explore a larger region phase space than the commonly used replica sampling (RS) Monte Carlo method. By clustering the simulated structures obtained in the PHS implementation of the SICHO model, we can successfully predict, among a representative benchmark 65 proteins set, 50 cases in which one of the top 5 clusters have a root-mean-square deviation (RMSD) from the native structure below 6.5 A. Compared with our previous calculations that used RS as the conformational search procedure, the number of successful predictions increased by four and the CPU cost is reduced. By comparing the structure clusters produced by both PHS and RS, we find a strong correlation between the quality of predicted structures and the minimum relative RMSD (mrRMSD) of structures clusters identified by using different search engines. This mrRMSD correlation may be useful in blind prediction as an indicator of the likelihood of successful folds.

Three-dimensional modeling of the I-TevI homing endonuclease catalytic domain, a GIY-YIG superfamily member, using NMR restraints and Monte Carlo dynamics

Using a recent version of the SICHO algorithm for in silico protein folding, we made a blind prediction of the tertiary structure of the N-terminal, independently folded, catalytic domain (CD) of the I-TevI homing endonuclease, a representative of the GIY-YIG superfamily of homing endonucleases. The secondary structure of the I-TevI CD has been determined using NMR spectroscopy, but computational sequence analysis failed to detect any protein of known tertiary structure related to the GIY-YIG nucleases (Kowalski et al., Nucleic Acids Res., 1999, 27, 2115-2125). To provide further insight into the structure-function relationships of all GIY-YIG superfamily members, including the recently described subfamily of type II restriction enzymes (Bujnicki et al., Trends Biochem. Sci., 2000, 26, 9-11), we incorporated the experimentally determined and predicted secondary and tertiary restraints in a reduced (side chain only) protein model, which was minimized by Monte Carlo dynamics and simulated annealing. The subsequently elaborated full atomic model of the I-TevI CD allows the available experimental data to be put into a structural context and suggests that the GIY-YIG domain may dimerize in order to bring together the conserved residues of the active site.

Generalized-ensemble algorithms for molecular simulations of biopolymers

In complex systems with many degrees of freedom such as peptides and proteins, there exists a huge number of local-minimum-energy states. Conventional simulations in the canonical ensemble are of little use, because they tend to get trapped in states of these energy local minima. A simulation in generalized ensemble performs a random walk in potential energy space and can overcome this difficulty. From only one simulation run, one can obtain canonical-ensemble averages of physical quantities as functions of temperature by the single-histogram and/or multiple-histogram reweighting techniques. In this article we review uses of the generalized-ensemble algorithms in biomolecular systems. Three well-known methods, namely, multicanonical algorithm, simulated tempering, and replica-exchange method, are described first. Both Monte Carlo and molecular dynamics versions of the algorithms are given. We then present three new generalized-ensemble algorithms that combine the merits of the above methods. The effectiveness of the methods for molecular simulations in the protein folding problem is tested with short peptide systems.

TOUCHSTONE: an ab initio protein structure prediction method that uses threading-based tertiary restraints

The successful prediction of protein structure from amino acid sequence requires two features: an efficient conformational search algorithm and an energy function with a global minimum in the native state. As a step toward addressing both issues, a threading-based method of secondary and tertiary restraint prediction has been developed and applied to ab initio folding. Such restraints are derived by extracting consensus contacts and local secondary structure from at least weakly scoring structures that, in some cases, can lack any global similarity to the sequence of interest. Furthermore, to generate representative protein structures, a reduced lattice-based protein model is used with replica exchange Monte Carlo to explore conformational space. We report results on the application of this methodology, termed TOUCHSTONE, to 65 proteins whose lengths range from 39 to 146 residues. For 47 (40) proteins, a cluster centroid whose rms deviation from native is below 6.5 (5) A is found in one of the five lowest energy centroids. The number of correctly predicted proteins increases to 50 when atomic detail is added and a knowledge-based atomic potential is combined with clustered and nonclustered structures for candidate selection. The combination of the ratio of the relative number of contacts to the protein length and the number of clusters generated by the folding algorithm is a reliable indicator of the likelihood of successful fold prediction, thereby opening the way for genome-scale ab initio folding.