Reduced models of proteins and their applications

Determination of side-chain-rotamer and side-chain and backbone virtual-bond-stretching potentials of mean force from AM1 energy surfaces of terminally-blocked amino-acid residues, for coarse-grained simulations of protein structure and folding. I. The method

In this and the accompanying article, we report the development of new physics-based side-chain-rotamer and virtual-bond-deformation potentials which now replace the respective statistical potentials used so far in our physics-based united-reside UNRES force field for large-scale simulations of protein structure and dynamics. In this article, we describe the methodology for determining the corresponding potentials of mean force (PMF's) from the energy surfaces of terminally-blocked amino-acid residues calculated with the AM1 quantum-mechanical semiempirical method. The approach is based on minimization of the AM1 energy for fixed values of the angles lambda for rotation of the peptide groups about the C(alpha)...C(alpha) virtual bonds, and for fixed values of the side-chain dihedral angles chi, which formed a multidimensional grid. A harmonic-approximation approach was developed to extrapolate from the energy at a given grid point to other points of the conformational space to compute the respective contributions to the PMF. To test the applicability of the harmonic approximation, the rotamer PMF's of alanine and valine obtained with this approach have been compared with those obtained by using a Metropolis Monte Carlo method. The PMF surfaces computed with the harmonic approximation are more rugged and have more pronounced minima than the MC-calculated surfaces but the harmonic-approximation- and MC-calculated PMF values are linearly correlated. The potentials derived with the harmonic approximation are, therefore, appropriate for UNRES for which the weights (scaling factors) of the energy terms are determined by force-field optimization for foldability.

Implementation of molecular dynamics and its extensions with the coarse-grained UNRES force field on massively parallel systems; towards millisecond-scale simulations of protein structure, dynamics, and thermodynamics

We report the implementation of our united-residue UNRES force field for simulations of protein structure and dynamics with massively parallel architectures. In addition to coarse-grained parallelism already implemented in our previous work, in which each conformation was treated by a different task, we introduce a fine-grained level in which energy and gradient evaluation are split between several tasks. The Message Passing Interface (MPI) libraries have been utilized to construct the parallel code. The parallel performance of the code has been tested on a professional Beowulf cluster (Xeon Quad Core), a Cray XT3 supercomputer, and two IBM BlueGene/P supercomputers with canonical and replica-exchange molecular dynamics. With IBM BlueGene/P, about 50 % efficiency and 120-fold speed-up of the fine-grained part was achieved for a single trajectory of a 767-residue protein with use of 256 processors/trajectory. Because of averaging over the fast degrees of freedom, UNRES provides an effective 1000-fold speed-up compared to the experimental time scale and, therefore, enables us to effectively carry out millisecond-scale simulations of proteins with 500 and more amino-acid residues in days of wall-clock time.

Study of a structurally similar kappa opioid receptor agonist and antagonist pair by molecular dynamics simulations

Among the structurally similar guanidinonaltrindole (GNTI) compounds, 5'-GNTI is an antagonist while 6'-GNTI is an agonist of the kappaOR opioid receptor. To explore how a subtle alteration of the ligand structure influences the receptor activity, we investigated two concurrent processes: the final steps of ligand binding at the receptor binding site and the initial steps of receptor activation. To trace these early activation steps, the membranous part of the receptor was built on an inactive receptor template while the extracellular loops were built using the ab initio CABS method. We used the simulated annealing procedure for ligand docking and all-atom molecular dynamics simulations to determine the immediate changes in the structure of the ligand-receptor complex. The binding of an agonist, in contrast to an antagonist, induced the breakage of the "3-7 lock" between helices TM3 and TM7. We also observed an action of the extended rotamer toggle switch which suggests that those two switches are interdependent.

Construction of an implicit membrane environment for the lattice Monte Carlo simulation of transmembrane protein

Due to the complexity of biological membrane, computer simulation of transmembrane protein's folding is challenging. In this paper, an implicit biological membrane environment has been constructed in lattice space, in which the lipid chains and water molecules were represented by the unoccupied lattice sites. The biological membrane was characterized with three features: stronger hydrogen bonding interaction, membrane lateral pressure, and lipophobicity index for the amino acid residues. In addition to the hydrocarbon core spanning region and the water solution, the lipid interface has also been represented in this implicit membrane environment, which was proved to be effective for the transmembrane protein's folding. The associated Monte Carlo simulations have been performed for SARS-CoV E protein and M2 protein segment (residues 18–60) of influenza A virus. It was found that the coil–helix transition of the transmembrane segment occurred earlier than the coil–globule transition of the two terminal domains. The folding process and final orientation of the amphipathic helical block in water solution are obviously influenced by its corresponding hydrophobicity/lipophobicity. Therefore, this implicit membrane environment, though in lattice space, can make an elaborate balance between different driving forces for the membrane protein's folding, thus offering a potential means for the simulation of transmembrane protein oligomers in feasible time.

A Transferable Coarse Grain Non-bonded Interaction Model For Amino Acids

The large quantity of protein sequences being generated from genomic data has greatly outpaced the throughput of experimental protein structure determining methods and consequently brought urgency to the need for accurate protein structure prediction tools. Reduced resolution, or coarse grained (CG) models, have become a mainstay in computational protein structure prediction perfoming among the best tools available. The quest for high quality generalized CG models presents an extremely challenging yet popular endeavor. To this point, a CG based interaction potential is presented here for the naturally occurring amino acids. In the present approach, three to four heavy atoms and associated hydrogens are condensed into a single CG site. The parameterization of the site-site interaction potential relies on experimental data thus providing a novel approach that is neither based on all-atom (AA) simulations nor experimental protein structural data. Specifically, intermolecular potentials, which are based on Lennard-Jones (LJ) style functional forms, are parameterized using thermodynamic data including surface tension and density. Using this approach, an amino acid potential dataset has been developed for use in modeling peptides and proteins. The potential is evaluated here by comparing the solvent accessible surface area (SASA) to AA representations and ranking of protein decoy data sets provided by Decoys 'R' Us. The model is shown to perform very well compared to other existing prediction models for these properties.

An improved functional form for the temperature scaling factors of the components of the mesoscopic UNRES force field for simulations of protein structure and dynamics

Coarse-grained or mesoscopic models of proteins and the corresponding force fields are of great importance because they enable us to reduce the folding simulation time by several orders of magnitude compared to the all-atom approach and, consequently, reach the millisecond time scale of simulations. In the coarse-grained UNRES model for simulations of protein structure and dynamics, developed by our group, each amino acid residue is represented by a united side chain and a united peptide group located in the middle between the two neighboring alpha-carbon atoms, which assist only in the definition of the geometry. The prototype of the UNRES force field has been defined as a potential of mean force or restricted free-energy function corresponding to averaging out the degrees of freedom not present in the coarse-grained representation, which has further been approximated by a truncated Kubo cumulant series to enable us to derive analytical expressions for the corresponding terms. This force field should depend on temperature, and in its simplest form, a term corresponding to the cumulant of order n should be multiplied by f(n) = 1/T(n-1). The temperature dependence has been introduced in recent work ( J. Phys. Chem. B , 2007 , 111 , 260 - 285 ), and in order to prevent too steep a variation with temperature, the factors at the nth order cumulant terms were assumed to have a form f(n) = ln[exp(1) + exp(-1)]/ln{exp[(T/T(0))(n-1)] + exp[-(T/T(0))(n-1)]}, where T(0) = 300 K is the reference temperature. In this work, we have introduced a modified scaling factor f(n) = ln[exp(c) + exp(-c)]/ln{exp[c(T/T(0))(n-1)] + exp[-c(T/T(0))(n-1)]}, where c is an adjustable parameter, and determined c by fitting the analytical approximation of the temperature dependence of the virtual bond torsional term corresponding to rotation about the C(alpha)...C(alpha) virtual bond in terminally blocked dialanine to the respective potential of mean force calculated from the MP2/6-31G(d, p) ab initio energy surfaces of terminally blocked alanine (Ac-Ala-NHMe) and, independently, by optimizing it to obtain a sharp heat capacity curve and the lowest ensemble-averaged root-mean-square deviation over the C(alpha) atoms of 1GAB used as a training protein. Both approaches gave consistent results, and c = 1.4 has been selected as the optimal value of this parameter. The force field with the new temperature scaling factors has been optimized using 1GAB as the training protein. The new force field has been tested on a series of medium size alpha-helical proteins and found to perform better than that with the original temperature scaling factors.

Towards temperature-dependent coarse-grained potentials of side-chain interactions for protein folding simulations. I: molecular dynamics study of a pair of methane molecules in water at various temperatures

By means of molecular dynamics simulations of a pair of methane molecules in a TIP3P periodic water box with the NVT scheme at six temperatures and, additionally, the NPT scheme at three temperatures ranging from T = 283 to 373 K, we determined the potential of mean force (PMF) of pairs of interacting methane molecules in water as functions of distance between the methane molecules. The PMFs converge to a single baseline only for r> 11 A at all temperatures. The curves of the dimensionless PMF obtained at different temperatures with the NVT scheme overlap almost perfectly in the region of the contact minimum and still very well in the regions of the desolvation maximum and the solvent-separated minimum, which suggests that the temperature-dependent hydrophobic interaction potentials at constant volume in united-residue force fields can be obtained by scaling the respective dimensionless potentials by RT, R being the universal gas constant. For the dimensionless potentials of mean force obtained with the NPT scheme, the depth of the contact minimum increases, whereas the height of the desolvation maximum and the depth of the solvent-separated minimum decrease with temperature, in agreement with results reported in the literature.

Versatile approach to access the low temperature thermodynamics of lattice polymers and proteins

We show that Wang-Landau sampling, combined with suitable Monte Carlo trial moves, provides a powerful method for both the ground state search and the determination of the density of states for the hydrophobic-polar (HP) protein model and the interacting self-avoiding walk (ISAW) model for homopolymers. We obtain accurate estimates of thermodynamic quantities for HP sequences with >100 monomers and for ISAWs up to >500 monomers. Our procedure possesses an intrinsic simplicity and overcomes the limitations inherent in more tailored approaches making it interesting for a broad range of protein and polymer models.

All-or-none proteinlike folding transition of a flexible homopolymer chain

Here we report a first-order all-or-none transition from an expanded coil to a compact crystallite for a flexible polymer chain. Wang-Landau sampling is used to construct the complete density of states for square-well chains up to length 256. Analysis within both the microcanonical and canonical ensembles shows a direct freezing transition for finite length chains with sufficiently short-range interactions. This type of transition is a distinctive feature of "one-step" protein folding and our findings demonstrate that a simple homopolymer model can exhibit protein-folding thermodynamics.

Identification of amyloidogenic peptide sequences using a coarse-grained physicochemical model

Cross-beta amyloid is implicated in over 20 human diseases. Experiments suggest that specific sequence elements within amyloidogenic proteins play a major role in seeding amyloid formation. Identifying these seeding sequences is important for rationalizing the molecular mechanisms of amyloid formation and for elaborating therapeutic strategies that target amyloid. Theoretical techniques play an important role in facilitating the identification and structural characterization of putative seeding sequences; most amyloid species are not amenable to high resolution experimental structure techniques. In this study we have combined a coarse-grained physicochemical protein model with a highly efficient Monte Carlo sampling technique to identify amyloidogenic sequences in four proteins for which respective experimental peptide fragmentation data exist. Peptide sequences were defined as amyloidogenic if the ensemble structure predicted for three interacting peptides described a stable and regular three-stranded beta-sheet. For such peptides, free energies were calculated to provide a measure of amyloid propensity. The overall agreement between the experimental and predicted data is good, and we correctly identify several self-recognition motifs proposed to define the cross-beta amyloid fibril architectures of two of the proteins. Our results compare very favorably with those obtained using atomistic molecular dynamics methods, though our simulations are 30-40 times faster.

CRANKITE: A fast polypeptide backbone conformation sampler

BACKGROUND

CRANKITE is a suite of programs for simulating backbone conformations of polypeptides and proteins. The core of the suite is an efficient Metropolis Monte Carlo sampler of backbone conformations in continuous three-dimensional space in atomic details.

METHODS

In contrast to other programs relying on local Metropolis moves in the space of dihedral angles, our sampler utilizes local crankshaft rotations of rigid peptide bonds in Cartesian space.

RESULTS

The sampler allows fast simulation and analysis of secondary structure formation and conformational changes for proteins of average length.

Simple continuous and discrete models for simulating replica exchange simulations of protein folding

The efficiency of temperature replica exchange (RE) simulations hinge on their ability to enhance conformational sampling at physiological temperatures by taking advantage of more rapid conformational interconversions at higher temperatures. While temperature RE is a parallel simulation technique that is relatively straightforward to implement, kinetics in the RE ensemble is complicated, and there is much to learn about how best to employ RE simulations in computational biophysics. Protein folding rates often slow down above a certain temperature due to entropic bottlenecks. This "anti-Arrhenius" behavior represents a challenge for RE. However, it is far from straightforward to systematically explore the impact of this on RE by brute force molecular simulations, since RE simulations of protein folding are very difficult to converge. To understand some of the basic mechanisms that determine the efficiency of RE, it is useful to study simplified low dimensionality systems that share some of the key characteristics of molecular systems. Results are presented concerning the efficiency of temperature RE on a continuous two-dimensional potential that contains an entropic bottleneck. Optimal efficiency was obtained when the temperatures of the replicas did not exceed the temperature at which the harmonic mean of the folding and unfolding rates is maximized. This confirms a result we previously obtained using a discrete network model of RE. Comparison of the efficiencies obtained using the continuous and discrete models makes it possible to identify non-Markovian effects, which slow down equilibration of the RE ensemble on the more complex continuous potential. In particular, the rate of temperature diffusion and also the efficiency of RE is limited by the time scale of conformational rearrangements within free energy basins.

Probing protein fold space with a simplified model

We probe the stability and near-native energy landscape of protein fold space using powerful conformational sampling methods together with simple reduced models and statistical potentials. Fold space is represented by a set of 280 protein domains spanning all topological classes and having a wide range of lengths (33-300 residues) amino acid composition and number of secondary structural elements. The degrees of freedom are taken as the loop torsion angles. This choice preserves the native secondary structure but allows the tertiary structure to change. The proteins are represented by three-point per residue, three-dimensional models with statistical potentials derived from a knowledge-based study of known protein structures. When this space is sampled by a combination of parallel tempering and equi-energy Monte Carlo, we find that the three-point model captures the known stability of protein native structures with stable energy basins that are near-native (all alpha: 4.77 A, all beta: 2.93 A, alpha/beta: 3.09 A, alpha+beta: 4.89 A on average and within 6 A for 71.41%, 92.85%, 94.29% and 64.28% for all-alpha, all-beta, alpha/beta and alpha+beta, classes, respectively). Denatured structures also occur and these have interesting structural properties that shed light on the different landscape characteristics of alpha and beta folds. We find that alpha/beta proteins with alternating alpha and beta segments (such as the beta-barrel) are more stable than proteins in other fold classes.

Prediction of partition coefficients by multiscale hybrid atomic-level/coarse-grain simulations

Coarse-grain models are becoming an increasingly important tool in computer simulations of a wide variety of molecular processes. In many instances it is, however, desirable to describe key portions of a molecular system at the atomic level. There is therefore a strong interest in the development of simulation methodologies that allow representations of matter with mixed granularities in a multiscale fashion. We report here a strategy to conduct mixed atomic-level and coarse-grain simulations of molecular systems with a recently developed coarse-grain model. The methodology is validated by computing partition coefficients of small molecules described in atomic detail and solvated by water or octane, both of which are represented by coarse-grain models. Because the present coarse-grain force field retains electrostatic interactions, the simplified solvent particles can interact realistically with the all-atom solutes. The partition coefficients computed by this approach rival the accuracy of fully atomistic simulations and are obtained at a fraction of their computational cost. The present methodology is simple, robust and applicable to a wide variety of molecular systems.

Modeling of protein misfolding in disease

A short review of the results of molecular modeling of prion disease is presented in this chapter. According to the "one-protein theory" proposed by Prusiner, prion proteins are misfolded naturally occurring proteins, which, on interaction with correctly folded proteins may induce misfolding and propagate the disease, resulting in insoluble amyloid aggregates in cells of affected specimens. Because of experimental difficulties in measurements of origin and growth of insoluble amyloid aggregations in cells, theoretical modeling is often the only one source of information regarding the molecular mechanism of the disease. Replica exchange Monte Carlo simulations presented in this chapter indicate that proteins in the native state, N, on interaction with an energetically higher structure, R, can change their conformation into R and form a dimer, R(2). The addition of another protein in the N state to R(2) may lead to spontaneous formation of a trimer, R(3). These results reveal the molecular basis for a model of prion disease propagation or conformational diseases in general.

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.

Reduced C(beta) statistical potentials can outperform all-atom potentials in decoy identification

We developed a series of statistical potentials to recognize the native protein from decoys, particularly when using only a reduced representation in which each side chain is treated as a single C(beta) atom. Beginning with a highly successful all-atom statistical potential, the Discrete Optimized Protein Energy function (DOPE), we considered the implications of including additional information in the all-atom statistical potential and subsequently reducing to the C(beta) representation. One of the potentials includes interaction energies conditional on backbone geometries. A second potential separates sequence local from sequence nonlocal interactions and introduces a novel reference state for the sequence local interactions. The resultant potentials perform better than the original DOPE statistical potential in decoy identification. Moreover, even upon passing to a reduced C(beta) representation, these statistical potentials outscore the original (all-atom) DOPE potential in identifying native states for sets of decoys. Interestingly, the backbone-dependent statistical potential is shown to retain nearly all of the information content of the all-atom representation in the C(beta) representation. In addition, these new statistical potentials are combined with existing potentials to model hydrogen bonding, torsion energies, and solvation energies to produce even better performing potentials. The ability of the C(beta) statistical potentials to accurately represent protein interactions bodes well for computational efficiency in protein folding calculations using reduced backbone representations, while the extensions to DOPE illustrate general principles for improving knowledge-based potentials.

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.

Protein-Folding Dynamics: Overview of Molecular Simulation Techniques

Molecular dynamics (MD) is an invaluable tool with which to study protein folding in silico. Although just a few years ago the dynamic behavior of a protein molecule could be simulated only in the neighborhood of the experimental conformation (or protein unfolding could be simulated at high temperature), the advent of distributed computing, new techniques such as replica-exchange MD, new approaches (based on, e.g., the stochastic difference equation), and physics-based reduced models of proteins now make it possible to study protein-folding pathways from completely unfolded structures. In this review, we present algorithms for MD and their extensions and applications to protein-folding studies, using all-atom models with explicit and implicit solvent as well as reduced models of polypeptide chains.

Lattice Monte Carlo simulation of coil-helix transition

An improved self-avoiding lattice model was put forward to study coil-helix transition via dynamic Monte Carlo simulation. Each residue occupies eight simple lattices. By introducing a virtual-imino group and a virtual-carbonyl group, the helical period is not necessarily to be an integer. The Gö-like restriction of hydrogen bonding interaction has also been partially released. The coil-helix transition was well reproduced and consistent with the Zimm-Bragg theory. Non-native hydrogen bonding interaction seems significant around transition temperature. In characterization of a statistically helical structure, a novel correlation function was put forward and verified.

A minimal proteinlike lattice model: an alpha-helix motif

A simple protein model of a four-helix bundle motif on a face-centered cubic lattice has been studied. Total energy of a conformation includes attractive interactions between hydrophobic residues, repulsive interactions between hydrophobic and polar residues, and a potential that favors helical turns. Using replica exchange Monte Carlo simulations we have estimated a set of parameters for which the native structure is a global minimum of conformational energy. Then we have shown that all the above types of interactions are necessary to guarantee the cooperativity of folding transition and to satisfy the thermodynamic hypothesis.

Evaluation of coarse grained models for hydrogen bonds in proteins

Backbone hydrogen bonds contribute very importantly to the stability of proteins and therefore they must be appropriately represented in protein folding simulations. Simple models are frequently used in theoretical approaches to this process, but their simplifications are often confronted with the need to be true to the physics of the interactions. Here we study the effects of different levels of coarse graining in the modeling of backbone hydrogen bonds. We study three different models taken from the bibliography in a twofold fashion. First, we calculate the hydrogen bonds in 2gb1, an (alpha + beta)-protein, and see how different backbone representations and potentials can mimic the effects of real hydrogen bonds both in helices and sheets. Second, we use an evolutionary method for protein fragment assembly to locate the global energy minimum for a set of small beta-proteins with these models. This way, we assess the effects of coarse graining in hydrogen bonding models and show what can be expected from them when used in simulation experiments.

Molecular dynamics simulation suggests possible interaction patterns at early steps of beta2-microglobulin aggregation

Early events in aggregation of proteins are not easily accessible by experiments. In this work, we perform a 5-ns molecular dynamics simulation of an ensemble of 27 copies of beta(2)-microglobulin in explicit solvent. During the simulation, the formation of intermolecular contacts is observed. The simulation highlights the importance of apical residues and, in particular, of those at the N-terminus end of the molecule. The most frequently found pattern of interaction involves a head-to-head contact arrangement of molecules. Hydrophobic contacts appear to be important for the establishment of long-lived (on the simulation timescale) contacts. Although early events on the pathway to aggregation and fibril formation are not directly related to the end-state of the process, which is reached on a much longer timescale, simulation results are consistent with experimental data and in general with a parallel arrangement of intermolecular beta-strand pairs.

Monte Carlo sampling of near-native structures of proteins with applications

Since a protein's dynamic fluctuation inside cells affects the protein's biological properties, we present a novel method to study the ensemble of near-native structures (NNS) of proteins, namely, the conformations that are very similar to the experimentally determined native structure. We show that this method enables us to (i) quantify the difficulty of predicting a protein's structure, (ii) choose appropriate simplified representations of protein structures, and (iii) assess the effectiveness of knowledge-based potential functions. We found that well-designed simple representations of protein structures are likely as accurate as those more complex ones for certain potential functions. We also found that the widely used contact potential functions stabilize NNS poorly, whereas potential functions incorporating local structure information significantly increase the stability of NNS.

Dependence of folding dynamics and structural stability on the location of a hydrophobic pair in beta-hairpins

We study the dependence of folding time, nucleation site, and stability of a model beta-hairpin on the location of a cross-strand hydrophobic pair, using a coarse-grained off-lattice model with the aid of Monte Carlo simulations. Our simulations have produced 6500 independent folding trajectories dynamically, forming the basis for extensive statistical analysis. Four folding pathways, zipping-out, middle-out, zipping-in, and reptation, have been closely monitored and discussed in all seven sequences studied. A hydrophobic pair placed near the beta-turn or in the middle section effectively speed up folding; a hydrophobic pair placed close to the terminal ends or next to the beta-turn encourages stability of the entire chain.

Empirical potential function for simplified protein models: combining contact and local sequence-structure descriptors

An effective potential function is critical for protein structure prediction and folding simulation. Simplified protein models such as those requiring only Calpha or backbone atoms are attractive because they enable efficient search of the conformational space. We show residue-specific reduced discrete-state models can represent the backbone conformations of proteins with small RMSD values. However, no potential functions exist that are designed for such simplified protein models. In this study, we develop optimal potential functions by combining contact interaction descriptors and local sequence-structure descriptors. The form of the potential function is a weighted linear sum of all descriptors, and the optimal weight coefficients are obtained through optimization using both native and decoy structures. The performance of the potential function in a test of discriminating native protein structures from decoys is evaluated using several benchmark decoy sets. Our potential function requiring only backbone atoms or Calpha atoms have comparable or better performance than several residue-based potential functions that require additional coordinates of side-chain centers or coordinates of all side-chain atoms. By reducing the residue alphabets down to size 10 for contact descriptors, the performance of the potential function can be further improved. Our results also suggest that local sequence-structure correlation may play important role in reducing the entropic cost of protein folding.

Side chain placement using estimation of distribution algorithms

OBJECTIVE

This paper presents an algorithm for the solution of the side chain placement problem.

METHODS AND MATERIALS

The algorithm combines the application of the Goldstein elimination criterion with the univariate marginal distribution algorithm (UMDA), which stochastically searches the space of possible solutions. The suitability of the algorithm to address the problem is investigated using a set of 425 proteins.

RESULTS

For a number of difficult instances where inference algorithms do not converge, it has been shown that UMDA is able to find better structures.

CONCLUSIONS

The results obtained show that the algorithm can achieve better structures than those obtained with other state-of-the-art methods like inference-based techniques. Additionally, a theoretical and empirical analysis of the computational cost of the algorithm introduced has been presented.

Study of the Villin headpiece folding dynamics by combining coarse-grained Monte Carlo evolution and all-atom molecular dynamics

The folding mechanism of the Villin headpiece (HP36) is studied by means of a novel approach which entails an initial coarse-grained Monte Carlo (MC) scheme followed by all-atom molecular dynamics (MD) simulations in explicit solvent. The MC evolution occurs in a simplified free-energy landscape and allows an efficient selection of marginally-compact structures which are taken as viable initial conformations for the MD. The coarse-grained MC structural representation is connected to the one with atomic resolution through a "fine-graining" reconstruction algorithm. This two-stage strategy is used to select and follow the dynamics of seven different unrelated conformations of HP36. In a notable case the MD trajectory rapidly evolves towards the folded state, yielding a typical root-mean-square deviation (RMSD) of the core region of only 2.4 A from the closest NMR model (the typical RMSD over the whole structure being 4.0 A). The analysis of the various MC-MD trajectories provides valuable insight into the details of the folding and mis-folding mechanisms and particularly about the delicate influence of local and nonlocal interactions in steering the folding process.

A coarse-grained model for the formation of α helix with a noninteger period on simple cubic lattices

Periodicity is an important parameter in the characterization of a helix in proteins. In this work, a coarse-grained model for a homopolypeptide in simple cubic lattices has been extended to build an alpha helix with a noninteger period. The lattice model is based on the bond fluctuation algorithm in which bond lengths and orientations are altered in a quasicontinuous way, and the simulation is performed via dynamic Monte Carlo simulation. Hydrogen bonds are assumed to be formed between a virtual-carbonyl group in a residue and a downstream virtual-imino group in another residue. Consequently, the period of the formed alpha helix is a noninteger. An improved spatial correlation function has been suggested to quantitatively describe the periodicity of the helical conformation, by which helical period and helical persistent length can be calculated by statistics. The resultant periods are very close to 3.6 residues; the persistent length based upon the improved definition can be larger or smaller than the chain length and reflect the inherent regularity of a chain including one or multiple helical blocks. The simulation outputs agree with the calculation of the Zimm-Bragg theory based upon the associated nucleation parameter and propagation parameter as well.

Conserved quantitative stability/flexibility relationships (QSFR) in an orthologous RNase H pair

Many reports qualitatively describe conserved stability and flexibility profiles across protein families, but biophysical modeling schemes have not been available to robustly quantify both. Here we investigate an orthologous RNase H pair by using a minimal distance constraint model (DCM). The DCM is an all atom microscopic model [Jacobs and Dallakyan, Biophys J 2005;88(2):903-915] that accurately reproduces heat capacity measurements [Livesay et al., FEBS Lett 2004;576(3):468-476], and is unique in its ability to harmoniously calculate thermodynamic stability and flexibility in practical computing times. Consequently, quantified stability/flexibility relationships (QSFR) can be determined using the DCM. For the first time, a comparative QSFR analysis is performed, serving as a paradigm study to illustrate the utility of a QSFR analysis for elucidating evolutionarily conserved stability and flexibility profiles. Despite global conservation of QSFR profiles, distinct enthalpy-entropy compensation mechanisms are identified between the RNase H pair. In both cases, local flexibility metrics parallel H/D exchange experiments by correctly identifying the folding core and several flexible regions. Remarkably, at appropriately shifted temperatures (e.g., melting temperature), these differences lead to a global conservation in Landau free energy landscapes, which directly relate thermodynamic stability to global flexibility. Using ensemble-based sampling within free energy basins, rigidly, and flexibly correlated regions are quantified through cooperativity correlation plots. Five conserved flexible regions are identified within the structures of the orthologous pair. Evolutionary conservation of these flexibly correlated regions is strongly suggestive of their catalytic importance. Conclusions made herein are demonstrated to be robust with respect to the DCM parameterization.

Implicit solvent simulation models for biomembranes

Fully atomic simulation strategies are infeasible for the study of many processes of interest to membrane biology, biophysics and biochemistry. We review various coarse-grained simulation methodologies with special emphasis on methods and models that do not require the explicit simulation of water. Examples from our own research demonstrate that such models have potential for simulating a variety of biologically relevant phenomena at the membrane surface.

Theoretical model of prion propagation: a misfolded protein induces misfolding

There is a hypothesis that dangerous diseases such as bovine spongiform encephalopathy, Creutzfeldt-Jakob, Alzheimer's, fatal familial insomnia, and several others are induced by propagation of wrong or misfolded conformations of some vital proteins. If for some reason the misfolded conformations were acquired by many such protein molecules it might lead to a "conformational" disease of the organism. Here, a theoretical model of the molecular mechanism of such a conformational disease is proposed, in which a metastable (or misfolded) form of a protein induces a similar misfolding of another protein molecule (conformational autocatalysis). First, a number of amino acid sequences composed of 32 aa have been designed that fold rapidly into a well defined native-like alpha-helical conformation. From a large number of such sequences a subset of 14 had a specific feature of their energy landscape, a well defined local energy minimum (higher than the global minimum for the alpha-helical fold) corresponding to beta-type structure. Only one of these 14 sequences exhibited a strong autocatalytic tendency to form a beta-sheet dimer capable of further propagation of protofibril-like structure. Simulations were done by using a reduced, although of high resolution, protein model and the replica exchange Monte Carlo sampling procedure.

Development of novel statistical potentials for protein fold recognition

The need to perform large-scale studies of protein fold recognition, structure prediction and protein-protein interactions has led to novel developments of residue-level minimal models of proteins. A minimum requirement for useful protein force-fields is that they be successful in the recognition of native conformations. The balance between the level of detail in describing the specific interactions within proteins and the accuracy obtained using minimal protein models is the focus of many current protein studies. Recent results suggest that the introduction of explicit orientation dependence in a coarse-grained, residue-level model improves the ability of inter-residue potentials to recognize the native state. New statistical and optimization computational algorithms can be used to obtain accurate residue-dependent potentials for use in protein fold recognition and, more importantly, structure prediction.

Protein folding pathways from replica exchange simulations and a kinetic network model

We present an approach to the study of protein folding that uses the combined power of replica exchange simulations and a network model for the kinetics. We carry out replica exchange simulations to generate a large ( approximately 10(6)) set of states with an all-atom effective potential function and construct a kinetic model for folding, using an ansatz that allows kinetic transitions between states based on structural similarity. We use this network to perform random walks in the state space and examine the overall network structure. Results are presented for the C-terminal peptide from the B1 domain of protein G. The kinetics is two-state after small temperature perturbations. However, the coil-to-hairpin folding is dominated by pathways that visit metastable helical conformations. We propose possible mechanisms for the alpha-helix/beta-hairpin interconversion.

Density guided importance sampling: application to a reduced model of protein folding

MOTIVATION

Monte Carlo methods are the most effective means of exploring the energy landscapes of protein folding. The rugged topography of folding energy landscapes causes sampling inefficiencies however, particularly at low, physiological temperatures.

RESULTS

A hybrid Monte Carlo method, termed density guided importance sampling (DGIS), is presented that overcomes these sampling inefficiencies. The method is shown to be highly accurate and efficient in determining Boltzmann weighted structural metrics of a discrete off-lattice protein model. In comparison to the Metropolis Monte Carlo method, and the hybrid Monte Carlo methods, jump-walking, smart-walking and replica-exchange, the DGIS method is shown to be more efficient, requiring no parameter optimization. The method guides the simulation towards under-sampled regions of the energy spectrum and recognizes when equilibrium has been reached, avoiding arbitrary and excessively long simulation times.

AVAILABILITY

Fortran code available from authors upon request.

CONTACT

m.j.parker@leeds.ac.uk.

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.

Comparing folding codes in simple heteropolymer models of protein evolutionary landscape: robustness of the superfunnel paradigm

Understanding the evolution of biopolymers is a key element in rationalizing their structures and functions. Simple exact models (SEMs) are well-positioned to address general principles of evolution as they permit the exhaustive enumeration of both sequence and structure (conformational) spaces. The physics-based models of the complete mapping between genotypes and phenotypes afforded by SEMs have proven valuable for gaining insight into how adaptation and selection operate among large collections of sequences and structures. This study compares the properties of evolutionary landscapes of a variety of SEMs to delineate robust predictions and possible model-specific artifacts. Among the models studied, the ruggedness of evolutionary landscape is significantly model-dependent; those derived from more protein-like models appear to be smoother. We found that a common practice of restricting protein structure space to maximally compact lattice conformations results in (i.e., "designs in") many encodable (designable) structures that are not otherwise encodable in the corresponding unrestrained structure space. This discrepancy is especially severe for model potentials that seek to mimic the major role of hydrophobic interactions in protein folding. In general, restricting conformations to be maximally compact leads to larger changes in the model genotype-phenotype mapping than a moderate shifting of reference state energy of the model potential function to allow for more specific encoding via the "designing out" effects of repulsive interactions. Despite these variations, the superfunnel paradigm applies to all SEMs we have tested: For a majority of neutral nets across different models, there exists a funnel-like organization of native stabilities for the sequences in a neutral net encoding for the same structure, and the thermodynamically most stable sequence is also the most robust against mutation.