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

Scale-consistent approach to the derivation of coarse-grained force fields for simulating structure, dynamics, and thermodynamics of biopolymers

In this chapter the scale-consistent approach to the derivation of coarse-grained force fields developed in our laboratory is presented, in which the effective energy function originates from the potential of mean force of the system under consideration and embeds atomistically detailed interactions in the resulting energy terms through use of Kubo's cluster-cumulant expansion, appropriate selection of the major degrees of freedom to be averaged out in the derivation of analytical approximations to the energy terms, and appropriate expression of the interaction energies at the all-atom level in these degrees of freedom. Our approach enables the developers to find correct functional forms of the effective coarse-grained energy terms, without having to import them from all-atom force fields or deriving them on a heuristic basis. In particular, the energy terms derived in such a way exhibit correct dependence on coarse-grained geometry, in particular on site orientation. Moreover, analytical formulas for the multibody (correlation) terms, which appear to be crucial for coarse-grained modeling of many of the regular structures such as, e.g., protein α-helices and β-sheets, can be derived in a systematic way. Implementation of the developed theory to the UNIfied COarse-gRaiNed (UNICORN) model of biological macromolecules, which consists of the UNRES (for proteins), NARES-2P (for nucleic acids), and SUGRES-1P (for polysaccharides) components, and is being developed in our laboratory is described. Successful applications of UNICORN to the prediction of protein structure, simulating the folding and stability of proteins and nucleic acids, and solving biological problems are discussed.

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).

Perspective: Coarse-grained models for biomolecular systems

By focusing on essential features, while averaging over less important details, coarse-grained (CG) models provide significant computational and conceptual advantages with respect to more detailed models. Consequently, despite dramatic advances in computational methodologies and resources, CG models enjoy surging popularity and are becoming increasingly equal partners to atomically detailed models. This perspective surveys the rapidly developing landscape of CG models for biomolecular systems. In particular, this review seeks to provide a balanced, coherent, and unified presentation of several distinct approaches for developing CG models, including top-down, network-based, native-centric, knowledge-based, and bottom-up modeling strategies. The review summarizes their basic philosophies, theoretical foundations, typical applications, and recent developments. Additionally, the review identifies fundamental inter-relationships among the diverse approaches and discusses outstanding challenges in the field. When carefully applied and assessed, current CG models provide highly efficient means for investigating the biological consequences of basic physicochemical principles. Moreover, rigorous bottom-up approaches hold great promise for further improving the accuracy and scope of CG models for biomolecular 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.

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.

Direct Measurement of the Kinetics and Thermodynamics of Association of Hydrophobic Molecules from Molecular Dynamics Simulations

A detailed study of the kinetics and thermodynamics of associations of model hydrophobic molecules is likely to be valuable for understanding the fundamental driving forces for processes such as protein folding and protein-protein association. To this end, we present results from a series of 500 ns long molecular dynamics (MD) simulations examining associations of 13 types of different alkane pairs in explicit water. In addition to providing accurate measurements of the association thermodynamics, the unbiased nature of the configurational sampling in the MD simulations allows the association and dissociation kinetics to be directly quantified. We show that by choosing a suitable reaction coordinate, the computed free energies of all of the alkane-alkane complexes can be linearly related to their buried molecular surface areas, that their dissociation kinetics can be reliably estimated from the height of the barrier on the computed free energy surfaces, and that their association kinetics are effectively diffusion-limited.

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.

Computing free energy hypersurfaces for anisotropic intermolecular associations

We previously used an adaptive reaction coordinate force biasing method for calculating the free energy of conformation (Naidoo and Brady, J Am Chem Soc 1999, 121, 2244) and chemical reactions (Rajamani et al., J Comput Chem 2003, 24, 1775) amongst others. Here, we describe a generalized version able to produce free energies in multiple dimensions, descriptively named the free energies from adaptive reaction coordinate forces method. To illustrate it, we describe how we calculate a multidimensional intermolecular orientational free energy, which can be used to investigate complex systems such as protein conformation and liquids. This multidimensional intermolecular free energy W(r, theta(1), theta(2), phi) provides a measure of orientationally dependent interactions that are appropriate for applications in systems that inherently have molecular anisotropic features. It is a highly informative free energy volume, which can be used to parameterize key terms such as the Gay-Berne intermolecular potential in coarse grain simulations. To demonstrate the value of the information gained from the W(r, theta(1), theta(2), phi) hypersurfaces we calculated them for TIP3P, TIP4P, and TIP5P dimer water models in vacuum. A comparison with a commonly used one-dimensional distance free energy profile is made to illustrate the significant increase in configurational information. The W(r) plots show little difference between the three models while the W(r, theta(1), theta(2), phi) hypersurfaces reveal the underlying energetic reasons why these potentials reproduce tetrahedrality in the condensed phase so differently from each.

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

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

Simple physics-based analytical formulas for the potentials of mean force for the interaction of amino acid side chains in water. IV. Pairs of different hydrophobic side chains

The potentials of mean force of 21 heterodimers 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), methyl propyl sulfide (for methionine), and indole (for tryptophane) 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 proposed in our earlier work 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, are well represented by the analytical expressions for all systems; large deviations between the MD-determined PMF and the analytical approximations are observed for pairs involving the least spheroidal solutes: ethylbenzene, indole, and methyl propyl sulfide at short distances at which the PMF is high and, consequently, these regions are rarely visited. When data from the PMF within only 10 kcal/mol above the global minimum are considered, the standard deviation between the MD-determined and the fitted PMF is from 0.25 to 0.55 kcal/mol (the relative standard deviation being from 4% to 8%); it is larger for pairs involving nonspherical solute molecules. The free energies of contact computed from the PMF surfaces are well correlated with those determined from protein-crystal data with a slope close to that relating the free energies of transfer of amino acids (from water to n-octanol) to the average contact free energies determined from protein-crystal data. These observations justify future use of the determined potentials in coarse-grained protein-folding simulations.

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. 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.