Improvement of the treatment of loop structures in the UNRES force field by inclusion of coupling between backbone- and side-chain-local conformational states

The UNited RESidue (UNRES) coarse-grained model of polypeptide chains, developed in our laboratory, enables us to carry out millisecond-scale molecular-dynamics simulations of large proteins effectively. It performs well in ab initio predictions of protein structure, as demonstrated in the last Community Wide Experiment on the Critical Assessment of Techniques for Protein Structure Prediction (CASP10). However, the resolution of the simulated structure is too coarse, especially in loop regions, which results from insufficient specificity of the model of local interactions. To improve the representation of local interactions, in this work we introduced new side-chain-backbone correlation potentials, derived from a statistical analysis of loop regions of 4585 proteins. To obtain sufficient statistics, we reduced the set of amino-acid-residue types to five groups, derived in our earlier work on structurally optimized reduced alphabets, based on a statistical analysis of the properties of amino-acid structures. The new correlation potentials are expressed as one-dimensional Fourier series in the virtual-bond-dihedral angles involving side-chain centroids. The weight of these new terms was determined by a trial-and-error method, in which Multiplexed Replica Exchange Molecular Dynamics (MREMD) simulations were run on selected test proteins. The best average root-mean-square deviations (RMSDs) of the calculated structures from the experimental structures below the folding-transition temperatures were obtained with the weight of the new side-chain-backbone correlation potentials equal to 0.57. The resulting conformational ensembles were analyzed in detail by using the Weighted Histogram Analysis Method (WHAM) and Ward's minimum-variance clustering. This analysis showed that the RMSDs from the experimental structures dropped by 0.5 Å on average, compared to simulations without the new terms, and the deviation of individual residues in the loop region of the computed structures from their counterparts in the experimental structures (after optimum superposition of the calculated and experimental structure) decreased by up to 8 Å. Consequently, the new terms improve the representation of local structure.

Local vs global motions in protein folding

It is of interest to know whether local fluctuations in a polypeptide chain play any role in the mechanism by which the chain folds to the native structure of a protein. This question is addressed by analyzing folding and non-folding trajectories of a protein; as an example, the analysis is applied to the 37-residue triple β-strand WW domain from the Formin binding protein 28 (FBP28) (PDB ID: 1E0L). Molecular dynamics (MD) trajectories were generated with the coarse-grained united-residue force field, and one- and two-dimensional free-energy landscapes (FELs) along the backbone virtual-bond angle θ and backbone virtual-bond-dihedral angle γ of each residue, and principal components, respectively, were analyzed. The key residues involved in the folding of the FBP28 WW domain are elucidated by this analysis. The correlations between local and global motions are found. It is shown that most of the residues in the folding trajectories of the system studied here move in a concerted fashion, following the dynamics of the whole system. This demonstrates how the choice of a pathway has to involve concerted movements in order for this protein to fold. This finding also sheds light on the effectiveness of principal component analysis (PCA) for the description of the folding dynamics of the system studied. It is demonstrated that the FEL along the PCs, computed by considering only several critically-placed residues, can correctly describe the folding dynamics.

Mean-field interactions between nucleic-acid-base dipoles can drive the formation of a double helix

A proposed coarse-grained model of nucleic acids demonstrates that average interactions between base dipoles, together with chain connectivity and excluded-volume interactions, are sufficient to form double-helical structures of DNA and RNA molecules. Additionally, local interactions determine helix handedness and direction of strand packing. This result, and earlier research on reduced protein models, suggests that mean-field multipole-multipole interactions are the principal factors responsible for the formation of regular structure of biomolecules.

Coexistence of phases in a protein heterodimer

A heterodimer consisting of two or more different kinds of proteins can display an enormous number of distinct molecular architectures. The conformational entropy is an essential ingredient in the Helmholtz free energy and, consequently, these heterodimers can have a very complex phase structure. Here, it is proposed that there is a state of proteins, in which the different components of a heterodimer exist in different phases. For this purpose, the structures in the protein data bank (PDB) have been analyzed, with radius of gyration as the order parameter. Two major classes of heterodimers with their protein components coexisting in different phases have been identified. An example is the PDB structure 3DXC. This is a transcriptionally active dimer. One of the components is an isoform of the intra-cellular domain of the Alzheimer-disease related amyloid precursor protein (AICD), and the other is a nuclear multidomain adaptor protein in the Fe65 family. It is concluded from the radius of gyration that neither of the two components in this dimer is in its own collapsed phase, corresponding to a biologically active protein. The UNRES energy function has been utilized to confirm that, if the two components are separated from each other, each of them collapses. The results presented in this work show that heterodimers whose protein components coexist in different phases, can have intriguing physical properties with potentially important biological consequences.

Extension of UNRES force field to treat polypeptide chains with D-amino-acid residues

Coarse-grained force fields for protein simulations are usually designed and parameterized to treat proteins composed of natural L-amino-acid residues. However, D-amino-acid residues occur in bacterial, fungal (e.g., gramicidins), as well as human-designed proteins. For this reason, we have extended the UNRES coarse-grained force field developed in our laboratory to treat systems with D-amino-acid residues. We developed the respective virtual-bond-torsional and double-torsional potentials for rotation about the C α · · · C α virtual-bond axis and two consecutive C α · · · C α virtual-bond axes, respectively, as functions of virtual-bond-dihedral angles γ. In turn, these were calculated as potentials of mean force (PMFs) from the diabatic energy surfaces of terminally-blocked model compounds for glycine, alanine, and proline. The potential-energy surfaces were calculated by using the ab initio method of molecular quantum mechanics at the Møller-Plesset (MP2) level of theory and the 6-31G(d,p) basis set, with the rotation angles of the peptide groups about [Formula: see text] and [Formula: see text] used as variables, and the energy was minimized with respect to the remaining degrees of freedom. The PMFs were calculated by numerical integration for all pairs and triplets with all possible combinations of types (glycine, alanine, and proline) and chirality (D or L); however, symmetry relations reduce the number of non-equivalent torsional potentials to 13 and the number of double-torsional potentials to 63 for a given C-terminal blocking group. Subsequently, one- (for torsional) and two-dimensional (for double-torsional potentials) Fourier series were fitted to the PMFs to obtain analytical expressions. It was found that the torsional potentials of the x-Y and X-y types, where X and Y are Ala or Pro, respectively, and a lowercase letter denotes D-chirality, have global minima for small absolute values of γ, accounting for the double-helical structure of gramicidin A, which is a dimer of two chains, each possessing an alternating D-Tyr-L-Tyr sequence, and similar peptides. The side-chain and correlation potentials for D-amino-acid residues were obtained by applying the reflection about the [Formula: see text] plane to the respective potentials for the L-amino-acid residues.

Determination of equilibrium parameters by minimization of an extended sum of squares

A method is proposed for the evaluation of equilibrium parameters from potentiometric data, in which all the sources of random error are taken into account. This means including in the minimized sum the residuals in all the quantities subject to error (titrant volume, emf, parameters of composition of the titrant and titrand, parameters of electrode characteristics, and equilibrium constants known from other experiments). The method of preparation of the solutions is represented by a directed, acyclic graph which allows consideration of the sources of error connected with the composition of the solutions, and avoidance of inconsistency in the composition characteristics. Numerical examples are presented.

Folding and self-assembly of a small protein complex

The synthetic homotetrameric ββα (BBAT1) protein possesses a stable quaternary structure with a ββα fold. Because of its small size (a total of 84 residues), the homotetramer is an excellent model system with which to study the self-assembly and protein-protein interactions. We find from replica exchange molecular dynamics simulations with the coarse-grain UNRES force field that the folding and association pathway consists of three well-separated steps, where that association to a tetramer precedes and facilitates folding of the four chains. At room temperature the tetramer exists in an ensemble of diverse structures. The crystal structure becomes energetically favored only when the molecule is put in a dense and crystal-like environment. The observed picture of folding promoted by association may mirror the mechanism according to which intrinsically unfolded proteins assume their functional structure.

Simulation of the opening and closing of Hsp70 chaperones by coarse-grained molecular dynamics

Heat-shock proteins 70 (Hsp70s) are key molecular chaperones which assist in the folding and refolding/disaggregation of proteins. Hsp70s, which consist of a nucleotide-binding domain (NBD, consisting of NBD-I and NBD-II subdomains) and a substrate-binding domain [SBD, further split into the β-sheet (SBD-β) and α-helical (SBD-α) subdomains], occur in two major conformations having (a) a closed SBD, in which the SBD and NBD domains do not interact, (b) an open SBD, in which SBD-α interacts with NBD-I and SBD-β interacts with the top parts of NBD-I and NBD-II. In the SBD-closed conformation, SBD is bound to a substrate protein, with release occurring after transition to the open conformation. While the transition from the closed to the open conformation is triggered efficiently by binding of adenosine triphosphate (ATP) to the NBD, it also occurs, although less frequently, in the absence of ATP. The reverse transition occurs after ATP hydrolysis. Here, we report canonical and multiplexed replica exchange simulations of the conformational dynamics of Hsp70s using a coarse-grained molecular dynamics approach with the UNRES force field. The simulations were run in the following three modes: (i) with the two halves of the NBD unrestrained relative to each other, (ii) with the two halves of the NBD restrained in an "open" geometry as in the SBD-closed form of DnaK (2KHO), and (iii) the two halves of NBD restrained in a "closed" geometry as in known experimental structures of ATP-bound NBD forms of Hsp70. Open conformations, in which the SBD interacted strongly with the NBD, formed spontaneously during all simulations; the number of transitions was largest in simulations carried out with the "closed" NBD domain, and smallest in those carried out with the "open" NBD domain; this observation is in agreement with the experimentally-observed influence of ATP-binding on the transition of Hsp70's from the SBD-closed to the SBD-open form. Two kinds of open conformations were observed: one in which SBD-α interacts with NBD-I and SBD-β interacts with the top parts of NBD-I and NBD-II (as observed in the structures of nucleotide exchange factors), and another one in which this interaction pattern is swapped. A third type of motion, in which SBD-α binds to NBD without dissociating from SBD-β was also observed. It was found that the first stage of interdomain communication (approach of SBD-β, to NBD) is coupled with the rotation of the long axes of NBD-I and NBD-II towards each other. To the best of our knowledge, this is the first successful simulation of the full transition of an Hsp70 from the SBD-closed to the SBD-open conformation.

Effects of mutation, truncation, and temperature on the folding kinetics of a WW domain

The purpose of this work is to show how mutation, truncation, and change of temperature can influence the folding kinetics of a protein. This is accomplished by principal component analysis of molecular-dynamics-generated folding trajectories of the triple β-strand WW domain from formin binding protein 28 (FBP28) (Protein Data Bank ID: 1E0L) and its full-size, and singly- and doubly-truncated mutants at temperatures below and very close to the melting point. The reasons for biphasic folding kinetics [i.e., coexistence of slow (three-state) and fast (two-state) phases], including the involvement of a solvent-exposed hydrophobic cluster and another delocalized hydrophobic core in the folding kinetics, are discussed. New folding pathways are identified in free-energy landscapes determined in terms of principal components for full-size mutants. Three-state folding is found to be a main mechanism for folding the FBP28 WW domain and most of the full-size and truncated mutants. The results from the theoretical analysis are compared to those from experiment. Agreements and discrepancies between the theoretical and experimental results are discussed. Because of its importance in understanding protein kinetics and function, the diffusive mechanism by which the FBP28 WW domain and its full-size and truncated mutants explore their conformational space is examined in terms of the mean-square displacement and principal component analysis eigenvalue spectrum analyses. Subdiffusive behavior is observed for all studied systems.

Toward temperature-dependent coarse-grained potentials of side-chain interactions for protein folding simulations. II. Molecular dynamics study of pairs of different types of interactions in water at various temperatures

By means of molecular dynamics simulations of 15 pairs of molecules selected to model the interactions of nonpolar, nonpolar and polar, nonpolar and charged, polar, and polar and charged side chains in water, we determined the potentials of mean force (PMFs) of pairs of interacting molecules in water as functions of distance between the interacting particles or their distance and orientations at three temperatures: 283, 323, and 373 K, respectively. The systems were found to fall into the following four categories as far as the temperature dependence of the PMF is concerned: (i) pairs for which association is entropy-driven, (ii) pairs for which association is energy-driven, (iii) pairs of positively charged solute molecules, for which association is energy-driven with unfavorable entropy change, and (iv) the remaining systems for which temperature dependence is weak. For each pair of PMFs, entropic and energetic contributions have been discussed.

Determination of effective potentials for the stretching of C(α) ⋯ C(α) virtual bonds in polypeptide chains for coarse-grained simulations of proteins from ab initio energy surfaces of N-methylacetamide and N-acetylpyrrolidine

The potentials of mean force (PMF's) for the deformation of the C(α) ⋯ C(α) virtual bonds in polypeptide chains were determined from the diabatic energy surfaces of N-methylacetamide (modeling regular peptide groups) and N-acetylpyrrolidine (modeling the peptide groups preceding proline), calculated at the Møller-Plesset (MP2) ab initio level of theory with the 6-31G(d,p) basis set. The energy surfaces were expressed in the C(α) ⋯ C(α) virtual-bond length (d) and the H-N-C(α) ⋯ C' improper dihedral angle (α) that describes the pyramidicity of the amide nitrogen, or in the C(α)-C'(O)-N-C(α) dihedral angle (ω) and the angle α. For each grid point, the potential energy was minimized with respect to all remaining degrees of freedom. The PMF's obtained from the (d, α) energy surfaces produced realistic free-energy barriers to the trans-cis transition (10 kcal/mol and 13 kcal/mol for the regular and proline peptide groups, respectively, compared to 12.6 - 13.9 kcal/mol and 17.3 - 19.6 kcal/mol determined experimentally for glycylglycine and N-acylprolines, respectively), while those obtained from the (ω, α) energy maps produced either low-quality PMF curves when direct Boltzmann summation was implemented to compute the PMF's or too-flat curves with too-low free-energy barriers to the trans-cis transition if harmonic extrapolation was used to estimate the contributions to the partition function. An analytical bimodal logarithmic-Gaussian expression was fitted to the PMF's, and the potentials were implemented in the UNRES force field. Test Langevin-dynamics simulations were carried out for the Gly-Gly and Gly-Pro dipeptides, which showed a 10(6)-fold increase of the simulated rate of the trans-cis isomerization with respect to that measured experimentally; effectively the same result was obtained with the analytical Kramers theory of reaction rate applied to the UNRES representation of the peptide groups. Application of Kramers' theory to compute the rate constants from the all-atom ab initio energy surfaces of the model compounds studied resulted in isomerization rates close to the experimental values, which demonstrates that the increase of the isomerization rate in UNRES simulations results solely from averaging out the secondary degrees of freedom.

Like-charged residues at the ends of oligoalanine sequences might induce a chain reversal

We have examined the effect of like-charged residues on the conformation of an oligoalanine sequence. This was facilitated by circular dichroism (CD) and NMR spectroscopic and differential scanning calorimetric (DSC) measurements, and molecular dynamics calculations of the following three alanine-based peptides: Ac-K-(A)(5) -K-NH(2) (KAK5), Ac-K-(A)(4) -K-NH(2) (KAK4), Ac-K-(A)(3) -K-NH(2) (KAK3), where A and K denote alanine and lysine residues, respectively. Our earlier studies suggested that the presence of like-charged residues at the end of a short polypeptide chain composed of nonpolar residues can induce a chain reversal. For all three peptides, canonical molecular dynamics simulations with NMR-derived restraints demonstrate the presence of ensembles of structures with a tendency to form a chain reversal. The KAK3 peptide exhibits a bent shape with its ends close to each other, while KAK4 and KAK5 are more extended. In the KAK5 peptide, the lysine residues do not have any influence on each other and are very mobile. Nevertheless, the tendency to form a more or less pronounced chain reversal is observed and it seems to be stable in all three peptides. This chain reversal seems to be caused by screening of the nonpolar core from the solvent by the hydrated charged residues.

Hidden protein folding pathways in free-energy landscapes uncovered by network analysis

A network analysis is used to uncover hidden folding pathways in free-energy landscapes usually defined in terms of such arbitrary order parameters as root-mean-square deviation from the native structure, radius of gyration, etc. The analysis has been applied to molecular dynamics (MD) trajectories of the B-domain of staphylococcal protein A, generated with the coarse-grained united-residue (UNRES) force field in a broad range of temperatures (270K ≤ T ≤ 325K). Thousands of folding pathways have been identified at each temperature. Out of these many folding pathways, several most probable ones were selected for investigation of the conformational transitions during protein folding. Unlike other conformational space network (CSN) methods, a node in the CSN variant implemented in this work is defined according to the nativelikeness class of the structure, which defines the similarity of segments of the compared structures in terms of secondary-structure, contact-pattern, and local geometry, as well as the overall geometric similarity of the conformation under consideration to that of the reference (experimental) structure. Our previous findings, regarding the folding model and conformations found at the folding-transition temperature for protein A (Maisuradze et al., J. Am. Chem. Soc. 132, 9444, 2010), were confirmed by the conformational space network analysis. In the methodology and in the analysis of the results, the shortest path identified by using the shortest-path algorithm corresponds to the most probable folding pathway in the conformational space network.

A study of the α-helical intermediate preceding the aggregation of the amino-terminal fragment of the β amyloid peptide (Aβ(1-28))

The β amyloid (Aβ) peptide aggregates to form β-rich structures that are known to trigger Alzheimer's disease. Experiments suggest that an α-helical intermediate precedes the formation of these aggregates. However, a description at the molecular level of the α-to-β transition has not been obtained. Because it has been proposed that the transition might be initiated in the amino-terminal region of Aβ, we studied the aggregation of the 28-residue amino-terminal fragment of Aβ (Aβ(1-28)) using molecular dynamics and a coarse-grained force field. Simulations starting from extended and helical conformations showed that oligomerization is initiated by the formation of intermolecular β-sheets between the residues in the N-terminal regions. In simulations starting from the α-helical conformation, forcing residues 17-21 to remain in the initial (helical) conformation prevents aggregation but allows for the formation of dimers, indicating that oligomerization, initiated along the nonhelical N-terminal regions, cannot progress without the α-to-β transition propagating along the chains.

Coarse-grained force field: general folding theory

We review the coarse-grained UNited RESidue (UNRES) force field for the simulations of protein structure and dynamics, which is being developed in our laboratory over the last several years. UNRES is a physics-based force field, the prototype of which is defined as a potential of mean force of polypeptide chains in water, where all the degrees of freedom except the coordinates of α-carbon atoms and side-chain centers have been integrated out. We describe the initial implementation of UNRES to protein-structure prediction formulated as a search for the global minimum of the potential-energy function and its subsequent molecular dynamics and extensions of molecular-dynamics implementation, which enabled us to study protein-folding pathways and thermodynamics, as well as to reformulate the protein-structure prediction problem as a search for the conformational ensemble with the lowest free energy at temperatures below the folding-transition temperature. Applications of UNRES to study biological problems are also described.

Simple physics-based analytical formulas for the potentials of mean force of the interaction of amino-acid side chains in water. VI. Oppositely charged side chains

The two-site coarse-grained model for the interactions of charged side chains, to be used with our coarse-grained UNRES force field for protein simulations proposed in the accompanying paper, has been extended to pairs of oppositely charged side chains. The potentials of mean force of four pairs of molecules modeling charged amino-acid side chains, i.e., propionate-n-pentylamine cation (for aspartic acid-lysine), butyrate-n-pentylamine cation (for glutamic acid-lysine), propionate-1-butylguanidine (for aspartic acid-arginine), and butyrate-1-butylguanidine (for glutamic acid-arginine) pairs were determined by umbrella-sampling molecular dynamics simulations in explicit water as functions of distance and orientation, and the analytical expression was fitted to the potentials of mean force. Compared to pairs of like-charged side chains discussed in the accompanying paper, an average quadrupole-quadrupole interaction term had to be introduced to reproduce the Coulombic interactions, and a multistate model of charge distribution had to be introduced to fit the potentials of mean force of all oppositely charged pairs well. The model reproduces all salt-bridge minima and, consequently, is likely to improve the performance of the UNRES force field.

Simple physics-based analytical formulas for the potentials of mean force of the interaction of amino-acid side chains in water. V. Like-charged side chains

A new model of side-chain-side-chain interactions for charged side-chains of amino acids, to be used in the UNRES force-field, has been developed, in which a side chain consists of a nonpolar and a charged site. The interaction energy between the nonpolar sites is composed of a Gay-Berne and a cavity term; the interaction energy between the charged sites consists of a Lennard-Jones term, a Coulombic term, a generalized-Born term, and a cavity term, while the interaction energy between the nonpolar and charged sites is composed of a Gay-Berne and a polarization term. We parametrized the energy function for the models of all six pairs of natural like-charged amino-acid side chains, namely propionate-propionate (for the aspartic acid-aspartic acid pair), butyrate-butyrate (for the glutamic acid-glutamic acid pair), propionate-butyrate (for the aspartic acid-glutamic acid pair), pentylamine cation-pentylamine cation (for the lysine-lysine pair), 1-butylguanidine cation-1-butylguanidine cation (for the arginine-arginine pair), and pentylamine cation-1-butylguanidine cation (for the lysine-arginine pair). By using umbrella-sampling molecular dynamics simulations in explicit TIP3P water, we determined the potentials of mean force of the above-mentioned pairs as functions of distance and orientation and fitted analytical expressions to them. The positions and depths of the contact minima and the positions and heights of the desolvation maxima, including their dependence on the orientation of the molecules were well represented by analytical expressions for all systems. The values of the parameters of all the energy components are physically reasonable, which justifies use of such potentials in coarse-grain protein-folding simulations.

Mechanism of fiber assembly: treatment of Aβ peptide aggregation with a coarse-grained united-residue force field

The growth mechanism of β-amyloid (Aβ) peptide fibrils was studied by a physics-based coarse-grained united-residue model and molecular dynamics (MD) simulations. To identify the mechanism of monomer addition to an Aβ(1-40) fibril, we placed an unstructured monomer at a distance of 20 Å from a fibril template and allowed it to interact freely with the latter. The monomer was not biased towards fibril conformation by either the force field or the MD algorithm. With the use of a coarse-grained model with replica-exchange molecular dynamics, a longer timescale was accessible, making it possible to observe how the monomers probe different binding modes during their search for the fibril conformation. Although different assembly pathways were seen, they all follow a dock-lock mechanism with two distinct locking stages, consistent with experimental data on fibril elongation. Whereas these experiments have not been able to characterize the conformations populating the different stages, we have been able to describe these different stages explicitly by following free monomers as they dock onto a fibril template and to adopt the fibril conformation (i.e., we describe fibril elongation step by step at the molecular level). During the first stage of the assembly ("docking"), the monomer tries different conformations. After docking, the monomer is locked into the fibril through two different locking stages. In the first stage, the monomer forms hydrogen bonds with the fibril template along one of the strands in a two-stranded β-hairpin; in the second stage, hydrogen bonds are formed along the second strand, locking the monomer into the fibril structure. The data reveal a free-energy barrier separating the two locking stages. The importance of hydrophobic interactions and hydrogen bonds in the stability of the Aβ fibril structure was examined by carrying out additional canonical MD simulations of oligomers with different numbers of chains (4-16 chains), with the fibril structure as the initial conformation. The data confirm that the structures are stabilized largely by hydrophobic interactions and show that intermolecular hydrogen bonds are highly stable and contribute to the stability of the oligomers as well.

PDZ binding to the BAR domain of PICK1 is elucidated by coarse-grained molecular dynamics

A key regulator of α-amino-3-hydroxy-5-methylisoxazole-4-propionic acid (AMPA) receptor traffic, PICK1 is known to interact with over 40 other proteins, including receptors, transporters and ionic channels, and to be active mostly as a homodimer. The current lack of a complete PICK1 structure determined at atomic resolution hinders the elucidation of its functional mechanisms. Here, we identify interactions between the component PDZ and BAR domains of PICK1 by calculating possible binding sites for the PDZ domain of PICK1 (PICK1-PDZ) to the homology-modeled, crescent-shaped dimer of the PICK1-BAR domain using multiplexed replica-exchange molecular dynamics (MREMD) and canonical molecular dynamics simulations with the coarse-grained UNRES force field. The MREMD results show that the preferred binding site for the single PDZ domain is the concave cavity of the BAR dimer. A second possible binding site is near the N-terminus of the BAR domain that is linked directly to the PDZ domain. Subsequent short canonical molecular dynamics simulations used to determine how the PICK1-PDZ domain moves to the preferred binding site on the BAR domain of PICK1 revealed that initial hydrophobic interactions drive the progress of the simulated binding. Thus, the concave face of the BAR dimer accommodates the PDZ domain first by weak hydrophobic interactions and then the PDZ domain slides to the center of the concave face, where more favorable hydrophobic interactions take over.

Evidence, from simulations, of a single state with residual native structure at the thermal denaturation midpoint of a small globular protein

The folding of the B-domain of staphylococcal protein A has been studied by coarse-grained canonical and multiplexed replica-exchange molecular dynamics simulations with the UNRES force field in a broad range of temperatures (270 K < or = T < or = 350 K). In canonical simulations, the folding was found to occur either directly to the native state or through kinetic traps, mainly the topological mirror image of the native three-helix bundle. The latter folding scenario was observed more frequently at low temperatures. With increase of temperature, the frequency of the transitions between the folded and misfolded/unfolded states increased and the folded state became more diffuse with conformations exhibiting increased root-mean-square deviations from the experimental structure (from about 4 A at T = 300 K to 8.7 A at T = 325 K). An analysis of the equilibrium conformational ensemble determined from multiplexed replica exchange simulations at the folding-transition temperature (T(f) = 325 K) showed that the conformational ensemble at this temperature is a collection of conformations with residual secondary structures, which possess native or near-native clusters of nonpolar residues in place, and not a 50-50% mixture of fully folded and fully unfolded conformations. These findings contradict the quasi-chemical picture of two- or multistate protein folding, which assumes an equilibrium between the folded, unfolded, and intermediate states, with equilibrium shifting with temperature but with the native conformations remaining essentially unchanged. Our results also suggest that long-range hydrophobic contacts are the essential factor to keep the structure of a protein thermally stable.

Coarse-grained model of nucleic acid bases

Atomistic simulations of nucleic acids are prohibitively expensive and, consequently, reduced models of these compounds are of great interest in the field. In this work, we propose a physics-based coarse-grained model of nucleic-acid bases in which each base is represented by several (3-5) interaction centers. van der Waals interactions are modeled by Lennard-Jones spheres with a 12-6 potential energy function. The charge distribution is modeled by a set of electric dipole moments located at the centers of the Lennard-Jones spheres. The method for computing the Lennard-Jones parameters, electric dipole moments (their magnitude and orientation) and positions of the interaction centers is described. Several models with different numbers of interaction centers were tested. The model with three-center cytosine, four-center guanine, four-center thymine, and five-center adenine satisfactorily reproduces the canonical Watson-Crick hydrogen bonding and stacking interaction energies of the all-atom AMBER model. The computation time with the coarse-grained model is reduced seven times compared with that of the all-atom model.

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.

beta-hairpin-forming peptides; models of early stages of protein folding

Formation of beta-hairpins is considered the initial step of folding of many proteins and, consequently, peptides constituting the beta-hairpin sequence of proteins (the beta-hairpin-forming peptides) are considered as models of early stages of protein folding. In this article, we discuss the results of experimental studies (circular-dichroism, infrared and nuclear magnetic resonance spectroscopy, and differential scanning calorimetry) of the structure of beta-hairpin-forming peptides excised from the B1 domain of protein G, which are known to fold on their own. We demonstrate that local interactions at the turn sequence and hydrophobic interactions between nonpolar residues are the dominant structure-determining factors, while there is no convincing evidence that stable backbone hydrogen bonds are formed in these peptides in aqueous solution. Consequently, the most plausible mechanism for folding of the beta-hairpin sequence appears to be the broken-zipper mechanism consisting of the following three steps: (i) bending the chain at the turn sequence owing to favorable local interactions, (ii) formation of loose hydrophobic contacts between nonpolar residues, which occur close to the contacts in the native structure of the protein but not exactly in the same position and, finally, (iii) formation of backbone hydrogen bonds and locking the hydrophobic contacts in the native positions as a hydrophobic core develops, sufficient to dehydrate the backbone peptide groups. This mechanism provides sufficient uniqueness (contacts form between residues that become close together because the chain is bent at the turn position) and robustness (contacts need not occur at once in the native positions) for folding a beta-hairpin sequence.