Implementation of a Serial Replica Exchange Method in a Physics-Based United-Residue (UNRES) Force Field

The kinetic-trapping problem in simulating protein folding can be overcome by using a Replica Exchange Method (REM). However, in implementing REM in molecular dynamics simulations, synchronization between processors on parallel computers is required, and communication between processors limits its ability to sample conformational space in a complex system efficiently. To minimize communication between processors during the simulation, a Serial Replica Exchange Method (SREM) has been proposed recently by Hagan et al. (J. Phys. Chem. B2007, 111, 1416-1423). Here, we report the implementation of this new SREM algorithm with our physics-based united-residue (UNRES) force field. The method has been tested on the protein 1E0L with a temperature-independent UNRES force field and on terminally blocked deca-alanine (Ala(10)) and 1GAB with the recently introduced temperature-dependent UNRES force field. With the temperature-independent force field, SREM reproduces the results of REM but is more efficient in terms of wall-clock time and scales better on distributed-memory machines. However, exact application of SREM to the temperature-dependent UNRES algorithm requires the determination of a four-dimensional distribution of UNRES energy components instead of a one-dimensional energy distribution for each temperature, which is prohibitively expensive. Hence, we assumed that the temperature dependence of the force field can be ignored for neighboring temperatures. This version of SREM worked for Ala(10) which is a simple system but failed to reproduce the thermodynamic results as well as regular REM on the more complex 1GAB protein. Hence, SREM can be applied to the temperature-independent but not to the temperature-dependent UNRES force field.

Computational techniques for efficient conformational sampling of proteins

In this review, we summarize the computational methods for sampling the conformational space of biomacromolecules. We discuss the methods applicable to find only lowest energy conformations (global minimization of the potential-energy function) and to generate canonical ensembles (canonical Monte Carlo method and canonical molecular dynamics method and their extensions). Special attention is devoted to the use of coarse-grained models that enable simulations to be enhanced by several orders of magnitude.

Potential of Mean Force of Hydrophobic Association: Dependence on Solute Size

The potentials of mean force (PMFs) were determined for systems involving formation of nonpolar dimers composed of methane, ethane, propane, isobutane, and neopentane, respectively, in water, using the TIP3P water model, and in vacuo. A series of umbrella-sampling molecular dynamics simulations with the AMBER force field was carried out for each pair in either water or in vacuo. The PMFs were calculated by using the weighted histogram analysis method (WHAM). The shape of the PMFs for dimers of all five nonpolar molecules is characteristic of hydrophobic interactions with contact and solvent-separated minima and desolvation maxima. The positions of all these minima and maxima change with the size of the nonpolar molecule, that is, for larger molecules they shift toward larger distances. The PMF of the neopentane dimer is similar to those of other small nonpolar molecules studied in this work, and hence the neopentane dimer is too small to be treated as a nanoscale hydrophobic object. The solvent contribution to the PMF was also computed by subtracting the PMF determined in vacuo from the PMF in explicit solvent. The molecular surface area model correctly describes the solvent contribution to the PMF together with the changes of the height and positions of the desolvation barrier for all dimers investigated. The water molecules in the first solvation sphere of the dimer are more ordered compared to bulk water, with their dipole moments pointing away from the surface of the dimer. The average number of hydrogen bonds per water molecule in this first hydration shell is smaller compared to that in bulk water, which can be explained by coordination of water molecules to the hydrocarbon surface. In the second hydration shell, the average number of hydrogen bonds is greater compared to bulk water, which can be explained by increased ordering of water from the first hydration shell; the net effect is more efficient hydrogen bonding between the water molecules in the first and second hydration shells.

Molecular dynamics with the United-residue force field: ab initio folding simulations of multichain proteins

The implementation of molecular dynamics with the united-residue (UNRES) force field is extended to treat multichain proteins. Constant temperature was maintained in the simulations with Berendsen or Langevin thermostats. The method was tested on three alpha-helical proteins (1G6U and GCN4-p1, each with two chains, and 1C94, with four chains). Simulations were carried out for both the isolated single chains and the multichain complexes. The proteins were folded by starting from the extended conformation with random initial velocities and with the chains parallel to each other. No symmetry constraints or structure information were included for the single chains or the multichain complexes. In the case of single-chain simulations, a high percentage of the trajectories (100% for 1G6U, 90% for GCN4-p1, and 80% for 1C94) converged to nativelike structures (assumed as the experimental structure of a monomer in the multichain complex), showing that, for the proteins studied in this work with the UNRES force field, the interactions between chains are not critical for stabilization of the individual chains. In the case of multichain simulations, the native structures of the 1G6U and GCN4-p1 complexes, but not that of 1C94, are predicted successfully. The association of the subunits does not follow a unique mechanism; the monomers were observed to fold both before and simultaneously with their association.

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.

Topology of Type II REases revisited; structural classes and the common conserved core

Type II restriction endonucleases (REases) are deoxyribonucleases that cleave DNA sequences with remarkable specificity. Type II REases are highly divergent in sequence as well as in topology, i.e. the connectivity of secondary structure elements. A widely held assumption is that a structural core of five β-strands flanked by two α-helices is common to these enzymes. We introduce a systematic procedure to enumerate secondary structure elements in an unambiguous and reproducible way, and use it to analyze the currently available X-ray structures of Type II REases. Based on this analysis, we propose an alternative definition of the core, which we term the αβα-core. The αβα-core includes the most frequently observed secondary structure elements and is not a sandwich, as it consists of a five-strand β-sheet and two α-helices on the same face of the β-sheet. We use the αβα-core connectivity as a basis for grouping the Type II REases into distinct structural classes. In these new structural classes, the connectivity correlates with the angles between the secondary structure elements and with the cleavage patterns of the REases. We show that there exists a substructure of the αβα-core, namely a common conserved core, ccc, defined here as one α-helix and four β-strands common to all Type II REase of known structure.

Theoretical study of the energetics of the reactions of triplet dioxygen with hydroquinone, semiquinone, and their protonated forms: relation to the mechanism of superoxide generation in the respiratory chain

One-electron reduction of the dioxygen molecule by the reduced form of mitochondrial ubiquinones (Q) of the NADH dehydrogenase (complex I) and mitochondrial cytochrome bc1 (complex III) is believed to be the main source of the superoxide anion radical O2*- and the hydroperoxide radical OOH*. In this work, we modeled the energetics of four possible reactions of the triplet ((3)Sigma(g)) dioxygen-molecule reduction by fully reduced and protonated ubiquinone (QH2; reaction 1), its deprotonated form (QH-; reaction 2), the semiquinone radical (QH*; reaction 3), and the semiquinone anion radical (Q*-; reaction 4), by means of ab initio calculations with the 6-31G(d) and 6-31+G(d) basis set in the restricted open-shell Hartree-Fock (ROHF), unrestricted Hartree-Fock (UHF), and complete active space self-consistent field (CASSCF) with dynamic correlation [at the second-order Møller-Plesset (MP2) or multiple reference Møller-Plesset (MRMP), respectively] schemes and the basis set superposition error (BSSE) correction included, as well as semiempirical AM1 and PM3 calculations in the UHF and ROHF schemes. 2-Butene-1,4-dione and p-benzoquinone were selected as model compounds. For the reduced forms of both compounds, reaction 1 turned out to be energetically unfavorable at all levels of theory, this agreeing with the experimentally observed diminished reductive properties of hydroquinone derivatives at low pH. For 2-butene-1,4-dione treated at the most advanced MRMP/CASSCF/6-31+G(d) level, the energies of reactions 1-4 are 4.7, -34.3, -15.0, and -4.1 kcal/mol, respectively. This finding suggests that reactions 2 and 3 are the most likely mechanisms of electron transfer to molecular oxygen in aprotic environments and that proton transfer is involved in this process. Nearly the same energies of reactions 2 and 3 were calculated at the MRMP/CASSCF/6-31+G(d) level for reduced forms of p-benzoquinone. Inclusion of diffuse functions in the basis set and dynamic correlation at the CASSCF level appears essential. Because deprotonated ubiquinol is unlikely to exist in physiological environments, reaction 3 appears to be the most likely mechanism of one-electron reduction of oxygen; however, if oxygen can penetrate cytochrome bc1 as far as the Q(o) center where ubiquinol can be deprotonated, reaction 2 can also come into play. The energies of reactions 2 and 3 calculated at the MRMP/CASSCF/6-31+G(d) level are most closely reproduced in the ab initio and semiempirical UHF PM3 calculations. Additional semiempirical calculations on more realistic models of ubiquinone, 2,3-dimethoxy-6-methyl-p-benzoquinone and 2,3-dimethoxy-5-isoprenyl-6-methyl-p-benzoquinone, gave qualitatively the same relations between the energies of reactions 2 and 3 as those carried out for p-benzoquinone species, thereby suggesting that this method could be used in studying electron-transfer reactions from reduced quinone derivatives to molecular oxygen in more complex systems, such as a model of the Q(o) site of cytochrome bc1, where applying ab initio methods is unfeasible.

Simple Physics-Based Analytical Formulas for the Potentials of Mean Force for the Interaction of Amino Acid Side Chains in Water. 2. Tests with Simple Spherical Systems

Simple analytical functions consisting of electrostatic, polarization, Lennard-Jones or modified Lennard-Jones, and cavity terms are proposed to express the potentials of mean force analytically for spherical particles interacting in water. The cavity term was expressed either through the molecular-surface area of the solute or by using the Gaussian-overlap model of hydrophobic hydration developed in paper 1 of this series. The analytical expressions were fitted to the potentials of mean force of a methane homodimer, heterodimers composed of a methane molecule, and an ammonium cation or a chloride anion, respectively, and dimers consisting of two chloride anions, two ammonium cations, or a chloride ion and an ammonium cation. The potentials of mean force for these dimers were determined by umbrella-sampling molecular dynamics simulations with the AMBER 7.0 force field with TIP3P water either in our earlier work or in this work. For all systems, the analytical formulas fitted the potentials of mean force very well. However, using the molecular-surface area to express the cavity term provided a good fit only when the nonbonded interactions were expressed by an all-repulsive modified Lennard-Jones potential but also resulted in non-physical values of some of the parameters. Conversely, the use of our new Gaussian-overlap-based expression for the cavity term provided a good fit to the potentials of mean force (PMFs) with Lennard-Jones nonbonded potential, and the values of all parameters were physically reasonable.

Simple Physics-Based Analytical Formulas for the Potentials of Mean Force for the Interaction of Amino Acid Side Chains in Water. 1. Approximate Expression for the Free Energy of Hydrophobic Association Based on a Gaussian-Overlap Model

A physics-based model is proposed to derive approximate analytical expressions for the cavity component of the free energy of hydrophobic association of spherical and spheroidal solutes in water. The model is based on the difference between the number and context of the water molecules in the hydration sphere of a hydrophobic dimer and of two isolated hydrophobic solutes. It is assumed that the water molecules touching the convex part of the molecular surface of the dimer and those in the hydration spheres of the monomers contribute equally to the free energy of solvation, and those touching the saddle part of the molecular surface of the dimer result in a more pronounced increase in free energy because of their more restricted mobility (entropy loss) and fewer favorable electrostatic interactions with other water molecules. The density of water in the hydration sphere around a single solute particle is approximated by the derivative of a Gaussian centered on the solute molecule with respect to its standard deviation. On the basis of this approximation, the number of water molecules in different parts of the hydration sphere of the dimer is expressed in terms of the first and the second mixed derivatives of the two Gaussians centered on the first and second solute molecules, respectively, with respect to the standard deviations of these Gaussians, and plausible analytical expressions for the cavity component of the hydrophobic-association energy of spherical and spheroidal solutes are introduced. As opposed to earlier hydration-shell models, our expressions reproduce the desolvation maxima in the potentials of mean force of pairs of nonpolar solutes in water, and their advantage over the models based on molecular-surface area is that they have continuous gradients in the coordinates of solute centers.

Simple physics-based analytical formulas for the potentials of mean force for the interaction of amino acid side chains in water. 3. Calculation and parameterization of the potentials of mean force of pairs of identical hydrophobic side chains

The potentials of mean force of homodimers of the molecules modeling hydrophobic amino acid side chains (ethane (for alanine), propane (for proline), isobutane (for valine), isopentane (for leucine and isoleucine), ethylbenzene (for phenylalanine), and methyl propyl sulfide (for methionine)) were determined by umbrella-sampling molecular dynamics simulations in explicit water as functions of distance and orientation. Analytical expressions consisting of the Gay-Berne term to represent effective van der Waals interactions and the cavity term derived in paper 1 of this series were fitted to the potentials of mean force. 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 the analytical expressions for all systems, which justifies use of such potentials in coarse-grain protein-folding simulations.

Further evidence for the absence of polyproline II stretch in the XAO peptide

It has been suggested that the alanine-based peptide with sequence Ac-XX-[A](7)-OO-NH(2), termed XAO where X denotes diaminobutyric acid and O denotes ornithine, exists in a predominantly polyproline-helix (P(II)) conformation in aqueous solution. In our recent work, we demonstrated that this "polyproline conformation" should be regarded as a set of local conformational states rather than as the overall conformation of the molecule. In this work, we present further evidence to support this statement. Differential scanning calorimetry measurements showed only a very small peak in the heat capacity of an aqueous solution of XAO at 57 degrees C, whereas the suggested transition to the P(II) structure should occur at approximately 30 degrees C. We also demonstrate that the temperature dependence of the (3)J(HNHalpha) coupling constants of the alanine residues can be explained qualitatively in terms of Boltzmann averaging over all local conformational states; therefore, this temperature dependence proves that a conformational transition does not occur. Canonical MD simulations with the solvent represented by the generalized Born model, and with time-averaged NMR-derived restraints, demonstrate the presence of an ensemble of structures with a substantial amount of local P(II) conformational states but not with an overall P(II) conformation.

Implementation of a symplectic multiple-time-step molecular dynamics algorithm, based on the united-residue mesoscopic potential energy function

A symplectic multiple-time-step (MTS) algorithm has been developed for the united-residue (UNRES) force field. In this algorithm, the slow-varying forces (which contain most of the long-range interactions and are, therefore, expensive to compute) are integrated with a larger time step, termed the basic time step, and the fast-varying forces are integrated with a shorter time step, which is an integral fraction of the basic time step. Based on the split operator formalism, the equations of motion were derived. Separation of the fast- and slow-varying forces leads to stable molecular dynamics with longer time steps. The algorithms were tested with the Ala(10) polypeptide chain and two versions of the UNRES force field: the current one in which the energy components accounting for the energetics of side-chain rotamers (U(rot)) can lead to numerically unstable forces and a modified one in which the the present U(rot) was replaced by a numerically stable expression which, at present, is parametrized only for polyalanine chains. With the modified UNRES potential, stable trajectories were obtained even when extending the basic time step to 15 fs and, with the original UNRES potentials, the basic time step is 1 fs. An adaptive multiple-time-step (A-MTS) algorithm is proposed to handle instabilities in the forces; in this method, the number of substeps in the basic time step varies depending on the change of the magnitude of the acceleration. With this algorithm, the basic time step is 1 fs but the number of substeps and, consequently, the computational cost are reduced with respect to the MTS algorithm. The use of the UNRES mesoscopic energy function and the algorithms derived in this work enables one to increase the simulation time period by several orders of magnitude compared to conventional atomic-resolution molecular dynamics approaches and, consequently, such an approach appears applicable to simulating protein-folding pathways, protein functional dynamics in a real molecular environment, and dynamical molecular recognition processes.

Modification and optimization of the united-residue (UNRES) potential energy function for canonical simulations. I. Temperature dependence of the effective energy function and tests of the optimization method with single training proteins

We report the modification and parametrization of the united-residue (UNRES) force field for energy-based protein structure prediction and protein folding simulations. We tested the approach on three training proteins separately: 1E0L (beta), 1GAB (alpha), and 1E0G (alpha + beta). Heretofore, the UNRES force field had been designed and parametrized to locate native-like structures of proteins as global minima of their effective potential energy surfaces, which largely neglected the conformational entropy because decoys composed of only lowest-energy conformations were used to optimize the force field. Recently, we developed a mesoscopic dynamics procedure for UNRES and applied it with success to simulate protein folding pathways. However, the force field turned out to be largely biased toward -helical structures in canonical simulations because the conformational entropy had been neglected in the parametrization. We applied the hierarchical optimization method, developed in our earlier work, to optimize the force field; in this method, the conformational space of a training protein is divided into levels, each corresponding to a certain degree of native-likeness. The levels are ordered according to increasing native-likeness; level 0 corresponds to structures with no native-like elements, and the highest level corresponds to the fully native-like structures. The aim of optimization is to achieve the order of the free energies of levels, decreasing as their native-likeness increases. The procedure is iterative, and decoys of the training protein(s) generated with the energy function parameters of the preceding iteration are used to optimize the force field in a current iteration. We applied the multiplexing replica-exchange molecular dynamics (MREMD) method, recently implemented in UNRES, to generate decoys; with this modification, conformational entropy is taken into account. Moreover, we optimized the free-energy gaps between levels at temperatures corresponding to a predominance of folded or unfolded structures, as well as to structures at the putative folding-transition temperature, changing the sign of the gaps at the transition temperature. This enabled us to obtain force fields characterized by a single peak in the heat capacity at the transition temperature. Furthermore, we introduced temperature dependence to the UNRES force field; this is consistent with the fact that it is a free-energy and not a potential energy function. beta

Resolution of the excitation-emission spectra of FMN in rigid poly(vinyl alcohol) matrices

The excitation-emission spectra of flavin mononucleotide (FMN) were measured in rigid PVA films for concentrations ranging from 6.92 x 10(-4)M to 1.03 M. The theoretical three-linear decomposition of the excitation-emission spectra indicated the presence of two absorption and emission centers corresponding to FMN monomer and dimer, respectively. The component of the fluorescence profile corresponding to the FMN monomer has a large negative part which is the mirror image of the emission band profile of the dimer. The elimination of this part by taking a linear combination of the emission components of the monomer and of the dimer resulted in emission spectrum, which is in a very good agreement with the monomer spectrum measured directly. The appearance of a negative part of the monomer emission profile obtained by trilinear decomposition of the emission-absorption spectra of FMN can be explained in terms of the non-radiative reverse energy transfer from the FMN dimers to the FMN monomers. The presented results confirm that the FMN molecules in rigid PVA form dimers but not higher order aggregates. Moreover, they enable to obtain fluorescence spectra of dimers and suggest that FMN dimers may take part in the process of non-radiative energy transfer occurring in photoreception phenomena.

Assessment of two theoretical methods to estimate potentiometric titration curves of peptides: comparison with experiment

We compared the ability of two theoretical methods of pH-dependent conformational calculations to reproduce experimental potentiometric titration curves of two models of peptides: Ac-K5-NHMe in 95% methanol (MeOH)/5% water mixture and Ac-XX(A)7OO-NH2 (XAO) (where X is diaminobutyric acid, A is alanine, and O is ornithine) in water, methanol (MeOH), and dimethyl sulfoxide (DMSO), respectively. The titration curve of the former was taken from the literature, and the curve of the latter was determined in this work. The first theoretical method involves a conformational search using the electrostatically driven Monte Carlo (EDMC) method with a low-cost energy function (ECEPP/3 plus the SRFOPT surface-solvation model, assumming that all titratable groups are uncharged) and subsequent reevaluation of the free energy at a given pH with the Poisson-Boltzmann equation, considering variable protonation states. In the second procedure, molecular dynamics (MD) simulations are run with the AMBER force field and the generalized Born model of electrostatic solvation, and the protonation states are sampled during constant-pH MD runs. In all three solvents, the first pKa of XAO is strongly downshifted compared to the value for the reference compounds (ethylamine and propylamine, respectively); the water and methanol curves have one, and the DMSO curve has two jumps characteristic of remarkable differences in the dissociation constants of acidic groups. The predicted titration curves of Ac-K5-NHMe are in good agreement with the experimental ones; better agreement is achieved with the MD-based method. The titration curves of XAO in methanol and DMSO, calculated using the MD-based approach, trace the shape of the experimental curves, reproducing the pH jump, while those calculated with the EDMC-based approach and the titration curve in water calculated using the MD-based approach have smooth shapes characteristic of the titration of weak multifunctional acids with small differences between the dissociation constants. Nevertheless, quantitative agreement between theoretically predicted and experimental titration curves is not achieved in all three solvents even with the MD-based approach, which is manifested by a smaller pH range of the calculated titration curves with respect to the experimental curves. The poorer agreement obtained for water than for the nonaqueous solvents suggests a significant role of specific solvation in water, which cannot be accounted for by the mean-field solvation models.

Polyproline II conformation is one of many local conformational states and is not an overall conformation of unfolded peptides and proteins

The alanine-based peptide Ac-XX(A)7OO-NH2, referred to as XAO (where X, A, and O denote diaminobutyric acid, alanine, and ornithine, respectively), has recently been proposed to possess a well defined polyproline II (P(II)) conformation at low temperatures. Based on the results of extensive NMR and CD investigations combined with theoretical calculations, reported here, we present evidence that, on the contrary, this peptide does not have any significant amount of organized P(II) structure but exists in an ensemble of conformations with a distorted bend in the N- and C-terminal regions. The conformational ensemble was obtained by molecular dynamics/simulated annealing calculations using the amber suite of programs with time-averaged distance and dihedral-angle restraints obtained from rotating-frame nuclear Overhauser effect (ROE) volumes and vicinal coupling constants 3J(HN Eta alpha), respectively. The computed ensemble-averaged radius of gyration Rg (7.4 +/- 1.0) A is in excellent agreement with that measured by small-angle x-ray scattering (SAXS) whereas, if the XAO peptide were in the P(II) conformation, Rg would be 11.6 A. Depending on the pH, peptide concentration, and temperature, the CD spectra of XAO do or do not possess the maximum with positive ellipticity in the 217-nm region, which is characteristic of the P(II) structure, reflecting a shifting conformational equilibrium rather than an all-or-none transition. The "P(II) conformation" should, therefore, be considered as one of the accessible conformational states of individual amino acid residues in peptides and proteins rather than as a structure of most of the chain in the early stage of folding.

Molecular dynamics study of amyloid formation of two Abl-SH3 domain peptides

Molecular dynamics (MD) simulations were carried out for two-strand and ten-strand beta-sheets constructed from two peptides corresponding to the diverging turn of two homologous Abl-SH3 domains, DLSFMKGE (MK; from Drosophila) and DLSFKKGE (KK; from man), in explicit water at the temperatures of 30, 170/190 and 300 K. It was found that the 2 x MK beta-sheet is more stable than the 2 x KK beta-sheet, and that the 10 x MK beta-sheet is more stable than the 10 x KK beta-sheet; this suggests that the MK systems are fibril-creating and the KK systems are not. These results might explain why most SH3 domains possess two conserved basic residues at the diverging turn, which may act as gatekeepers in order to avoid aggregation.

Kinetic Studies of Folding of the B-domain of Staphylococcal Protein A with Molecular Dynamics and a United-residue (UNRES) Model of Polypeptide Chains

Langevin dynamics is used with our physics-based united-residue (UNRES) force field to study the folding pathways of the B-domain of staphylococcal protein A (1BDD (alpha; 46 residues)). With 400 trajectories of protein A started from the extended state (to gather meaningful statistics), and simulated for more than 35 ns each, 380 of them folded to the native structure. The simulations were carried out at the optimal folding temperature of protein A with this force field. To the best of our knowledge, this is the first simulation study of protein-folding kinetics with a physics-based force field in which reliable statistics can be gathered. In all the simulations, the C-terminal alpha-helix forms first. The ensemble of the native basin has an average RMSD value of 4 A from the native structure. There is a stable intermediate along the folding pathway, in which the N-terminal alpha-helix is unfolded; this intermediate appears on the way to the native structure in less than one-fourth of the folding pathways, while the remaining ones proceed directly to the native state. Non-native structures persist until the end of the simulations, but the native-like structures dominate. To express the kinetics of protein A folding quantitatively, two observables were used: (i) the average alpha-helix content (averaged over all trajectories within a given time window); and (ii) the fraction of conformations (averaged over all trajectories within a given time window) with Calpha RMSD values from the native structure less than 5 A (fraction of completely folded structures). The alpha-helix content grows quickly with time, and its variation fits well to a single-exponential term, suggesting fast two-state kinetics. On the other hand, the fraction of folded structures changes more slowly with time and fits to a sum of two exponentials, in agreement with the appearance of the intermediate, found when analyzing the folding pathways. This observation demonstrates that different qualitative and quantitative conclusions about folding kinetics can be drawn depending on which observable is monitored.

Molecular dynamics with the united-residue model of polypeptide chains. I. Lagrange equations of motion and tests of numerical stability in the microcanonical mode

The Lagrange formalism was implemented to derive the equations of motion for the physics-based united-residue (UNRES) force field developed in our laboratory. The C(alpha)...C(alpha) and C(alpha)...SC (SC denoting a side-chain center) virtual-bond vectors were chosen as variables. The velocity Verlet algorithm was adopted to integrate the equations of motion. Tests on the unblocked Ala(10) polypeptide showed that the algorithm is stable in short periods of time up to the time step of 1.467 fs; however, even with the shorter time step of 0.489 fs, some drift of the total energy occurs because of momentary jumps of the acceleration. These jumps are caused by numerical instability of the forces arising from the U(rot) component of UNRES that describes the energetics of side-chain-rotameric states. Test runs on the Gly(10) sequence (in which U(rot) is not present) and on the Ala(10) sequence with U(rot) replaced by a simple numerically stable harmonic potential confirmed this observation; oscillations of the total energy were observed only up to the time step of 7.335 fs, and some drift in the total energy or instability of the trajectories started to appear in long-time (2 ns and longer) trajectories only for the time step of 9.78 fs. These results demonstrate that the present U(rot) components (which are statistical potentials derived from the Protein Data Bank) must be replaced with more numerically stable functions; this work is under way in our laboratory. For the purpose of our present work, a nonsymplectic variable-time-step algorithm was introduced to reduce the energy drift for regular polypeptide sequences. The algorithm scales down the time step at a given point of a trajectory if the maximum change of acceleration exceeds a selected cutoff value. With this algorithm, the total energy is reasonably conserved up to a time step of 2.445 fs, as tested on the unblocked Ala(10) polypeptide. We also tried a symplectic multiple-time-step reversible RESPA algorithm and achieved satisfactory energy conservation for time steps up to 7.335 fs. However, at present, it appears that the reversible RESPA algorithm is several times more expensive than the variable-time-step algorithm because of the necessity to perform additional matrix multiplications. We also observed that, because Ala(10) folds and unfolds within picoseconds in the microcanonical mode, this suggests that the effective (event-based) time unit in UNRES dynamics is much larger than that of all-atom dynamics because of averaging over the fast-moving degrees of freedom in deriving the UNRES potential.

Molecular dynamics with the united-residue model of polypeptide chains. II. Langevin and Berendsen-bath dynamics and tests on model alpha-helical systems

The implementation of molecular dynamics (MD) with our physics-based protein united-residue (UNRES) force field, described in the accompanying paper, was extended to Langevin dynamics. The equations of motion are integrated by using a simplified stochastic velocity Verlet algorithm. To compare the results to those with all-atom simulations with implicit solvent in which no explicit stochastic and friction forces are present, we alternatively introduced the Berendsen thermostat. Test simulations on the Ala(10) polypeptide demonstrated that the average kinetic energy is stable with about a 5 fs time step. To determine the correspondence between the UNRES time step and the time step of all-atom molecular dynamics, all-atom simulations with the AMBER 99 force field and explicit solvent and also with implicit solvent taken into account within the framework of the generalized Born/surface area (GBSA) model were carried out on the unblocked Ala(10) polypeptide. We found that the UNRES time scale is 4 times longer than that of all-atom MD simulations because the degrees of freedom corresponding to the fastest motions in UNRES are averaged out. When the reduction of the computational cost for evaluation of the UNRES energy function is also taken into account, UNRES (with hydration included implicitly in the side chain-side chain interaction potential) offers about at least a 4000-fold speed up of computations relative to all-atom simulations with explicit solvent and at least a 65-fold speed up relative to all-atom simulations with implicit solvent. To carry out an initial full-blown test of the UNRES/MD approach, we ran Berendsen-bath and Langevin dynamics simulations of the 46-residue B-domain of staphylococcal protein A. We were able to determine the folding temperature at which all trajectories converged to nativelike structures with both approaches. For comparison, we carried out ab initio folding simulations of this protein at the AMBER 99/GBSA level. The average CPU time for folding protein A by UNRES molecular dynamics was 30 min with a single Alpha processor, compared to about 152 h for all-atom simulations with implicit solvent. It can be concluded that the UNRES/MD approach will enable us to carry out microsecond and, possibly, millisecond simulations of protein folding and, consequently, of the folding process of proteins in real time.