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The research presents various simple physics-based analytical formulas consisting of electrostatic, polarization and cavity terms to explain the potentials of mean force for the spherical particles of amino acid side chains interacting in water. The results show that the newly developed Gaussian-overlap-based expression for the cavity term fits the potentials of mean force (PMFs) perfectly and all the parameters are reasonable.

Physics-based analytical forms are proposed to express the cavity component of the free energy of hydrophobic association of spherical and spheroidal solute particles. These expressions are derived on the basis of an analysis of the change of the number and free energy of water molecules upon the formation of a hydrophobic dimmer and a Gaussian model of the density of the solute.

The study employs the umbrella-sampling molecular dynamics simulations in explicit water as functions of distance and orientation to calculate and parameterize the potentials of mean force of identical hydrophobic side chains of amino acid. The analysis of the positions and depths of the contact minima and the positions and heights of the desolvation maxima proves and confirms the application of such potentials in coarse-grain protein-folding simulations.

The application of hierarchical optimization of protein potential-energy landscapes to optimize the UNRES force field using multiple training proteins is reported. The resulting force field trained on a set of four proteins gives good medium-resolution predictions of the structures of most of the alpha- and the alpha + beta-proteins, but still could not handle nontrivial beta-motifs sufficiently well.

The application of the proposed method of hierarchical optimization of the protein energy landscape to optimize the off-lattice united-residue (UNRES) force field using single training proteins is presented. Technical details of the method, including the maintenance of proper secondary structure and a method to classify structure, are described.

The development of proposed method for optimizing potential-energy functions for protein-structure prediction and folding simulations aimed at obtaining hierarchical energy landscapes of the protein chosen to calibrate a force field is presented. The structure of each benchmark protein is described in terms of structural levels, the number of native structural elements increasing in a given order with level number.

The potentials of mean force are determined for ionic heterocomplexes. The resulting heteroconjugation equilibrium constants are calculated.

Multibody terms pertaining to the correlation between electrostatic interaction in the UNRES force field were reparameterized on relevant model systems. The resultant force field predicted the structures of proteins with alpha and beta structure.

The potentials of mean force (PMF) of the interactions between models of like-charged and between models of charged and nonpolar amino acid side chains in water to design improved side chain-side chain interaction potentials for our united residue UNRES force field for protein-structure prediction. It is seen that the PMF curves for pairs of simple ions in water are quite complex and possess minima even for pairs of positively or negatively charged ions.

Three energy functions were developed to simulate the free energy of proteins from the alpha, beta and alpha/beta structural classes is presented. The study finally uses a simple model system to show that the VMC protocol does not require exhaustive sampling of medium- and high-energy structures in order to optimize the parameters of the potential energy adequately.

The potentials of mean force (PMFs) are determined for systems that have involved the formation of nonpolar dimers composed of methane, ethane, propane, isobutane and neopentane, respectively, in water, by using the TIP3P water model and in vacuo. It is shown that the neopentane dimer is too small to be treated as a macroscopic hydrophobic object.

Study has shown that ubiquinone-mediated one-electron reduction of the triplet-oxygen molecule in non-polar environment is likely to proceed via electron transfer from the ubiquinol monoanion or from neutral ubisemiquinone with preference to the first process. These results agree with the theoretical and experiment findings that proton-donor groups are required for anthraquinone-mediated oxygen reduction.

Molecular dynamics is implemented with united-residue (UNRES) force field to treat multichain proteins. Simulations are carried out for both isolated single chains and the multichain complexes.

Study is presented on the modification and parametrization of the united-residue (UNRES) force field for energy-based protein structure prediction and protein folding simulations. Result demonstrates that it is possible to optimize the UNRES force field for canonical simulations using a protein of any type of secondary structure.

The clarifications for the inconsistency in the results obtained for the molecular origin on anticooperativity in hydrophobic association is presented.

A study introduces the Langevin dynamics with the united-residue (UNRES) force field to take into account the dynamic effects of the solvent. It is found that the UNRES time scale is 4 times longer than that of all-atom molecular dynamics simulations because the degrees of freedom corresponding to the fastest motions in UNRES are averaged out.

The equations of motion for the physics-based united-residue (UNRES) force field are derived by implementing the Lagrange formalism. The results suggest that the effective time unit in UNRES dynamics is much larger than that of all-atom dynamics.

The potential of mean force for the formation of a dimer and trimers of methane molecules in three specific configurations in explicit water to explain multibody effects in hydrophobic association on a molecular level. Cooperativity was observed which was manifested as a reduction of the height of the desolvation barrier for the trimer in isosceles triangle geometry, however anticooperativity was observed for linear trimers.

A study was conducted on the photophysical properties of tyrosine and its derivatives with free and blocked functional groups in water by steady-state and time-resolved fluorescence and global analysis. Tyrosine fluorescence intensity decays in water at pH = 5.5 in the short-wavelength region (290-320 nm) were monoexponential, and at longer wavelengths they were biexponential.

The physics-based potentials of side-chain-side-chain interactions corresponding to pairs composed of charged and polar, polar and polar, charged and hydrophobic, and hydrophobic and hydrophobic side chains have been determined. A total of 144 four-dimensional potentials of mean force (PMFs) of all possible pairs of molecules modeling these pairs were determined by umbrella-sampling molecular dynamics simulations in explicit water as functions of distance and orientation, and the analytical expressions were then fitted to the PMFs. Depending on the type of interacting sites, the analytical approximation to the PMF is a sum of terms corresponding to van der Waals interactions and cavity-creation involving the nonpolar sections of the side chains and van der Waals, cavity-creation, and electrostatic (charge-dipole or dipole-dipole) interaction energies and polarization energies involving the charged or polar sections of the side chains. The model used in this work reproduces all features of the interacting pairs. The UNited RESidue force field with the new side-chain-side-chain interaction potentials was preliminarily tested with the N-terminal part of the B-domain of staphylococcal protein A (PDBL 1BDD; a three-α-helix bundle) and UPF0291 protein YnzC from Bacillus subtilis (PDB: 2HEP; an α-helical hairpin). [ABSTRACT FROM AUTHOR]

A method for surmounting a protein folding problem involving the multiple-minima was proposed. The location of a group of large basins containing low-energy minima in the original energy surface was the underlying principle of the method. This was achieved by coupling the superbasins in the original surface to basins in a highly deformed energy surface. To carry out the deformation, the distance scaling method and the diffusion equation method were implemented.

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 (3g) 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 initiocalculations with the 6-31G(d) and 6-31G(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 1turned 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/6-31G(d) level, the energies of reactions 1−4are 4.7, −34.3, −15.0, and −4.1 kcal/mol, respectively. This finding suggests that reactions 2and 3are 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 2and 3were calculated at the MRMP/CASSCF/6-31G(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 3appears to be the most likely mechanism of one-electron reduction of oxygen; however, if oxygen can penetrate cytochrome bc1as far as the Qocenter where ubiquinol can be deprotonated, reaction 2can also come into play. The energies of reactions 2and 3calculated at the MRMP/CASSCF/6-31G(d) level are most closely reproduced in the ab initioand 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 2and 3as 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 Qosite of cytochrome bc1, where applying ab initiomethods is unfeasible. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 -helical proteins (1G6Uand 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 1G6Uand 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. [ABSTRACT FROM AUTHOR]

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(), 1GAB(), and 1E0G( ). 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. [ABSTRACT FROM AUTHOR]

The implementation of molecular dynamics (MD) with our physics-based protein united-residue (UNRES) force field, described in the accompanying paper (Khalili et al. J. Phys. Chem. B2005, 109, 13785), 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 Ala10 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 Ala10 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. [ABSTRACT FROM AUTHOR]

Phosphorylation is a common post-translational modification of the amino-acid side chains (serine, tyrosine, and threonine) that contain hydroxyl groups. The transfer of the negatively charged phosphate group from an ATP molecule to such amino-acid side chains leads to changes in the local conformations of proteins and the pattern of interactions with other amino-acid side-chains. A convenient characteristic of the side chain-side chain interactions in the context of an aqueous environment is the potential of mean force (PMF) in water. A series of umbrella-sampling molecular dynamic (MD) simulations with the AMBER force field were carried out for pairs of O-phosphorylated serine (pSer), threonine (pThr), and tyrosine, (pTyr) with natural amino acids in a TIP3P water model as a solvent at 298 K. The weighted-histogram analysis method was used to calculate the four-dimensional potentials of mean force. The results demonstrate that the positions and depths of the contact minima and the positions and heights of the desolvation maxima, including their dependence on the relative orientation depend on the character of the interacting pairs. More distinct minima are observed for oppositely charged pairs such as, e.g., O-phosphorylated side-chains and positively charged ones, such as the side-chains of lysine and arginine.

The folding temperature of the trp-cage mini-protein was determined to be in the range 311-317 K depending on the method used. Our study is focused on determining the structure and dynamics of the polypeptide chain close to its unfolding or melting temperature. At T = 305 K, Trp6-Arg16 and Trp6-Pro12 long-range interactions are observed, and at T = 313 K, only the Trp6-Arg16 interactions remain, while all of mentioned interactions are observed in the native state of the protein. Partial (at T = 305 K) and complete (at T = 313 K) melting of the N-terminal α-helix is observed, manifested by the appearance of minor sets of signals in NMR spectra. Our key findings are: (i) conformational phase transition (melting point) could be described as a cooperative breaking of the Trp6-Pro12 long-range hydrophobic interaction and the melting of the N-terminal α-helix; (ii) many ROE signals corresponding to local or short-range interactions vanish rapidly with temperature increase; however, long-range interaction such as Trp6-Arg16 remains until 313 K. The presence of the native long-range interaction at 313 K makes that conformational ensemble resemble a very diffuse native state structure, but it is not a simple mixture of the folded and unfolded states, as could be expected on the basis of the common two-state folding mechanism.

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.

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.

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., (© 2011 American Chemical Society)

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., (© 2011 American Chemical Society)