Simple Physics-Based Analytical Formulas for the Potentials of Mean Force of the Interaction of Amino Acid Side Chains in Water. VII. Charged–Hydrophobic/Polar and Polar–Hydrophobic/Polar Side Chains

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)

Dependence of the Formation of Tau and Aβ Peptide Mixed Aggregates on the Secondary Structure of the N‑Terminal Region of Aβ

One of the hallmarks of Alzheimer’s disease is the formation of aggregates of the tau protein, a process that can be facilitated by the presence of fibrils formed by the amyloid β peptide (Aβ). However, the mechanism that triggers tau aggregation is still a matter of debate. The effect of Aβ₄₀ fibrils on the aggregation of the repeat domain of tau (TauRD) is investigated here by employing coarse-grained molecular dynamics simulations. The results indicate that the repeat domain of tau has a high affinity for Aβ₄₀ fibrils, with the ₂₆₁GSTENLK₂₆₇ fragment of tau driving TauRD toward the ₁₆KLVFFA₂₁ fragment in Aβ₄₀. Monomeric Aβ₄₀, in which the ₁₆KLVFFA₂₁ fragment is rarely found in an extended conformation (as in the fibril), has a low affinity for the TauRD, indicating that the ability of Aβ₄₀ fibrils to bind to the TauRD depends on the ₁₆KLVFFA₂₁ fragment of Aβ adopting an extended conformation

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 nonfolding 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 <i>θ</i> and backbone virtual-bond-dihedral angle <i>γ</i> 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

DNA Duplex Formation with a Coarse-Grained Model

A middle-resolution coarse-grained model of DNA is proposed. The DNA chain is built of spherical and planar rigid bodies connected by elastic virtual bonds. The bonded part of the potential energy function is fit to potentials of mean force of model systems. The rigid bodies are sets of neutral, charged, and dipolar beads. Electrostatic and van der Waals interactions are parametrized by our recently developed procedure [Maciejczyk, M.; Spasic, A.; Liwo, A.; Scheraga, H.A. <i>J. Comp. Chem.</i> <b>2010</b>, <i>31</i>, 1644]. Interactions with the solvent and an ionic cloud are approximated by a multipole–multipole Debye–Hückel model. A very efficient <i>R</i>-RATTLE algorithm, for integrating the movement of rigid bodies, is implemented. It is the first coarse-grained model, in which both bonded and nonbonded interactions were parametrized ab initio and which folds stable double helices from separated complementary strands, with the final conformation close to the geometry of experimentally determined structures

New Insights into Protein (Un)Folding Dynamics

A fundamental open problem in biophysics is how the folded structure of the main chain (MC) of a protein is determined by the physics of the interactions between the side chains (SCs). All-atom molecular dynamics simulations of a model protein (Trp-cage) revealed that strong correlations between the motions of the SCs and the MC occur transiently at 380 K in unfolded segments of the protein and during the simulations of the whole amino-acid sequence at 450 K. The high correlation between the SC and MC fluctuations is a fundamental property of the unfolded state and is also relevant to unstructured proteins as intrinsically disordered proteins (IDPs), for which new reaction coordinates are introduced. The presented findings may open a new door as to how functions of IDPs are related to conformations, which play a crucial role in neurodegenerative diseases

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 parametrized 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 <i>ab initio</i> 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 C_<i>i</i>–1^α···C_<i>i</i>^α (λ⁽¹⁾) and C_<i>i</i>^α···C_<i>i</i>+1^α (λ⁽²⁾) 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 nonequivalent 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 C_<i>i</i>–1^α···C_<i>i</i>^α···C_<i>i</i>+1^α plane to the respective potentials for the l-amino acid residues

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 parametrized 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 <i>ab initio</i> 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 C_<i>i</i>–1^α···C_<i>i</i>^α (λ⁽¹⁾) and C_<i>i</i>^α···C_<i>i</i>+1^α (λ⁽²⁾) 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 nonequivalent 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 C_<i>i</i>–1^α···C_<i>i</i>^α···C_<i>i</i>+1^α plane to the respective potentials for the l-amino acid residues

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 parametrized 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 <i>ab initio</i> 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 C_<i>i</i>–1^α···C_<i>i</i>^α (λ⁽¹⁾) and C_<i>i</i>^α···C_<i>i</i>+1^α (λ⁽²⁾) 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 nonequivalent 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 C_<i>i</i>–1^α···C_<i>i</i>^α···C_<i>i</i>+1^α plane to the respective potentials for the l-amino acid residues

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 parametrized 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 <i>ab initio</i> 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 C_<i>i</i>–1^α···C_<i>i</i>^α (λ⁽¹⁾) and C_<i>i</i>^α···C_<i>i</i>+1^α (λ⁽²⁾) 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 nonequivalent 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 C_<i>i</i>–1^α···C_<i>i</i>^α···C_<i>i</i>+1^α plane to the respective potentials for the l-amino acid residues

Physics-Based Potentials for the Coupling between Backbone- and Side-Chain-Local Conformational States in the United Residue (UNRES) Force Field for Protein Simulations

The UNited RESidue (UNRES) model of polypeptide chains is a coarse-grained model in which each amino-acid residue is reduced to two interaction sites, namely, a united peptide group (p) located halfway between the two neighboring α-carbon atoms (C^αs), which serve only as geometrical points, and a united side chain (SC) attached to the respective C^α. Owing to this simplification, millisecond molecular dynamics simulations of large systems can be performed. While UNRES predicts overall folds well, it reproduces the details of local chain conformation with lower accuracy. Recently, we implemented new knowledge-based torsional potentials (Krupa et al. <i>J. Chem. Theory Comput.</i> <b>2013</b>, <i>9</i>, 4620–4632) that depend on the virtual-bond dihedral angles involving side chains: C^α···C^α···C^α···SC (τ⁽¹⁾), SC···C^α···C^α···C^α (τ⁽²⁾), and SC···C^α···C^α···SC (τ⁽³⁾) in the UNRES force field. These potentials resulted in significant improvement of the simulated structures, especially in the loop regions. In this work, we introduce the physics-based counterparts of these potentials, which we derived from the all-atom energy surfaces of terminally blocked amino-acid residues by Boltzmann integration over the angles λ⁽¹⁾ and λ⁽²⁾ for rotation about the C^α···C^α virtual-bond angles and over the side-chain angles χ. The energy surfaces were, in turn, calculated by using the semiempirical AM1 method of molecular quantum mechanics. Entropy contribution was evaluated with use of the harmonic approximation from Hessian matrices. One-dimensional Fourier series in the respective virtual-bond-dihedral angles were fitted to the calculated potentials, and these expressions have been implemented in the UNRES force field. Basic calibration of the UNRES force field with the new potentials was carried out with eight training proteins, by selecting the optimal weight of the new energy terms and reducing the weight of the regular torsional terms. The force field was subsequently benchmarked with a set of 22 proteins not used in the calibration. The new potentials result in a decrease of the root-mean-square deviation of the average conformation from the respective experimental structure by 0.86 Å on average; however, improvement of up to 5 Å was observed for some proteins