Four-body Potentials Research Articles

Accessing the applicability of the MBE approach for constructing potential energy surfaces of nitrogen clusters

The many-body expansion approach (MBE) is employed to build potential energy surfaces (PESs) for nitrogen clusters, which serve as a possible candidate of high energy density materials. The ground-state global PES of N 4 developed by Truhlar’s group is utilized to construct the two-body, three-body, and four-body interaction potentials. The energy of nitrogen clusters is thus obtained by summing the many-body interaction energies up to four-body term. This work found that such a many-body expansion could give reasonable energy predictions for loose chain nitrogen molecules while produce bad predictions for compact structures. The failure is caused by the neglect of high-order ( n ≥ 5) n -body interactions that are significant in the bonding area. The difficulties in applying the MBE method to construct the potential energy surface of nitrogen clusters are analyzed.

Atomistic two-, three- and four-body potentials. Spatial and material settings

In molecular dynamics or molecular statics (MD/MS) multi-body potentials empirically capture the energetic interactions in atomistic systems enabling the computation of the corresponding atomistic forces as energetic conjugates to the atomistic positions. We distinguish here between spatial and material atomistic positions and consequently between the corresponding spatial and material atomistic forces. In quasi-statics, i.e. MS, the former, also denoted as deformational atomistic forces, contribute to the classical deformational mechanics (i.e., equilibrium) problem that seeks to minimise the total potential energy of an atomistic system with respect to the atomistic positions relative to the ambient space. The latter, also denoted as configurational atomistic forces, contribute to the configurational mechanics (i.e., non-equilibrium) problem that determines the release of total potential energy of an atomistic system upon variation of the atomistic positions relative to the ambient material, i.e., due to perturbations of the material (initial) atomistic configuration. The importance of material atomistic forces is that they drive energetically favourable re-organisations of the material atomistic configuration, thereby characterising the tendency of generic atomistic defects to propagate. In this contribution we focus on two-, three-, and four-body potentials, whereby we distinguish between novel stretch- and classical angle-based potentials for the two latter cases. Taken together, as the main contribution, we derive expressions for the corresponding spatial and, for the first time, material atomistic forces and highlight their striking formal similarity. The derivations are detailed but the final expression compact and well-suited for numerical implementation.

Multiparticle potentials from lightlike Wilson lines in quark-gluon plasmas: A generalized relation of in-medium splitting rates to jet-quenching parameters q^

A powerful historical insight about the theory of in-medium showering in QCD backgrounds was that splitting rates can be related to a parameter $\stackrel{^}{q}$ that characterizes the rate of transverse-momentum kicks to a high-energy particle from the medium. Another powerful insight was that $\stackrel{^}{q}$ can be defined (with caveats) even when the medium is strongly coupled, using long, narrow Wilson loops whose two long edges are lightlike Wilson lines. The medium effects for the original calculations of in-medium splitting rates can be formulated in terms of three-body imaginary-valued ``potentials'' that are defined with three long, lightlike Wilson lines. Corrections due to the overlap of two consecutive splittings can be calculated using similarly defined four-body potentials. I give a simple argument for how such $N$-body potentials can be determined in the appropriate limit just from the knowledge of the values of $\stackrel{^}{q}$ for different color representations. For $N>3$, the $N$-body potentials have a nontrivial color structure, which will complicate calculations of overlap corrections outside of the large-${N}_{\mathrm{c}}$ or soft bremsstrahlung limits.

All-Atom Four-Body Knowledge-Based Statistical Potentials to Distinguish Native Protein Structures from Nonnative Folds.

Recent advances in understanding protein folding have benefitted from coarse-grained representations of protein structures. Empirical energy functions derived from these techniques occasionally succeed in distinguishing native structures from their corresponding ensembles of nonnative folds or decoys which display varying degrees of structural dissimilarity to the native proteins. Here we utilized atomic coordinates of single protein chains, comprising a large diverse training set, to develop and evaluate twelve all-atom four-body statistical potentials obtained by exploring alternative values for a pair of inherent parameters. Delaunay tessellation was performed on the atomic coordinates of each protein to objectively identify all quadruplets of interacting atoms, and atomic potentials were generated via statistical analysis of the data and implementation of the inverted Boltzmann principle. Our potentials were evaluated using benchmarking datasets from Decoys-‘R'-Us, and comparisons were made with twelve other physics- and knowledge-based potentials. Ranking 3rd, our best potential tied CHARMM19 and surpassed AMBER force field potentials. We illustrate how a generalized version of our potential can be used to empirically calculate binding energies for target-ligand complexes, using HIV-1 protease-inhibitor complexes for a practical application. The combined results suggest an accurate and efficient atomic four-body statistical potential for protein structure prediction and assessment.

Extending the Four-Body Problem of Wolfes to Non-Translationally Invariant Interactions

We propose and solve exactly the Schr\"odinger equation of a bound quantum system consisting in four particles moving on a real line with both translationally invariant four particles interactions of Wolfes type \cite{Wolf74} and additional non translationally invariant four-body potentials. We also generalize and solve exactly this problem in any $D$-dimensional space by providing full eigensolutions and the corresponding energy spectrum. We discuss the domain of the coupling constant where the irregular solutions becomes physically acceptable

Potentials 'R'Us web-server for protein energy estimations with coarse-grained knowledge-based potentials

BackgroundKnowledge-based potentials have been widely used in the last 20 years for fold recognition, protein structure prediction from amino acid sequence, ligand binding, protein design, and many other purposes. However generally these are not readily accessible online.ResultsOur new knowledge-based potential server makes available many of these potentials for easy use to automatically compute the energies of protein structures or models supplied. Our web server for protein energy estimation uses four-body potentials, short-range potentials, and 23 different two-body potentials. Users can select potentials according to their needs and preferences. Files containing the coordinates of protein atoms in the PDB format can be uploaded as input. The results will be returned to the user's email address.ConclusionsOur Potentials 'R'Us server is an easily accessible, freely available tool with a web interface that collects all existing and future protein coarse-grained potentials and computes energies of multiple structural models.

Multi-Body Knowledge-Based Potentials for Protein Structure Prediction Evaluation

Knowledge-based potentials have been widely used in the last 20 years for fold recognition, evaluation of protein structure predictions from amino acid sequences, ligand binding, protein design, and many other purposes. The most commonly known are two-body residue-level contact potentials, especially those first introduced by Miyazawa and Jernigan in 1985, and then rederived using an updated, larger protein dataset in 1996. Dense packing of residues in globular proteins is one of their characteristic features. Because of such dense packing cooperative multi-body interactions, especially in protein cores are important. The four-body contact potentials and short-range interaction potentials have been derived by considering different aspects of protein structures than those used to derive pair-contact potentials. The four-body contact potentials are appropriate for representing the cooperative parts of the protein folding process, and we have shown that they are quite successful for recognizing the native structures among hundreds or even thousands of decoys from the Decoys‘R’Us database. Short-range interaction energies allow us to estimate free energies from the statistical distribution of local conformational descriptors. We developed two types of four-body potentials: sequential and non-sequential ones. We have found that combining the former ones with short-range interactions yields excellent results for threadings, that significantly outperforms all other methods for coarse-grained models of proteins. We have developed also our knowledge-based potential server http://gor.bb.iastate.edu/potential for coarse-grained protein energy estimations that uses two types of four-body potentials, short-range potentials, and 23 different two-body potentials.

AatA Is a Novel Autotransporter and Virulence Factor of Avian Pathogenic Escherichia coli

Autotransporters (AT) are widespread in Gram-negative bacteria, and many of them are involved in virulence. An open reading frame (APECO1_O1CoBM96) encoding a novel AT was located in the pathogenicity island of avian pathogenic Escherichia coli (APEC) O1's virulence plasmid, pAPEC-O1-ColBM. This 3.5-kb APEC autotransporter gene (aatA) is predicted to encode a 123.7-kDa protein with a 25-amino-acid signal peptide, an 857-amino-acid passenger domain, and a 284-amino-acid beta domain. The three-dimensional structure of AatA was also predicted by the threading method using the I-TASSER online server and then was refined using four-body contact potentials. Molecular analysis of AatA revealed that it is translocated to the cell surface, where it elicits antibody production in infected chickens. Gene prevalence analysis indicated that aatA is strongly associated with E. coli from avian sources but not with E. coli isolated from human hosts. Also, AatA was shown to enhance adhesion of APEC to chicken embryo fibroblast cells and to contribute to APEC virulence.

Four‐body contact potentials derived from two protein datasets to discriminate native structures from decoys

Two-body inter-residue contact potentials for proteins have often been extracted and extensively used for threading. Here, we have developed a new scheme to derive four-body contact potentials as a way to consider protein interactions in a more cooperative model. We use several datasets of protein native structures to demonstrate that around 500 chains are sufficient to provide a good estimate of these four-body contact potentials by obtaining convergent threading results. We also have deliberately chosen two sets of protein native structures differing in resolution, one with all chains' resolution better than 1.5 A and the other with 94.2% of the structures having a resolution worse than 1.5 A to investigate whether potentials from well-refined protein datasets perform better in threading. However, potentials from well-refined proteins did not generate statistically significant better threading results. Our four-body contact potentials can discriminate well between native structures and partially unfolded or deliberately misfolded structures. Compared with another set of four-body contact potentials derived by using a Delaunay tessellation algorithm, our four-body contact potentials appear to offer a better characterization of the interactions between backbones and side chains and provide better threading results, somewhat complementary to those found using other potentials.

NONADDITIVE EFFECTS IN H2 AND HF TETRAMERS

Nonadditive three- and four-body interaction energies have been calculated for HF tetramer at the MP2/aug-cc-pVTZ level and for H 2 tetramer at the MP4(SDTQ)/aug-cc-pVTZ level using the so-called fifteen-point method. Calculated results show that with intermolecular distances decreasing from 3.0 to 1.7 Å the nonadditive three- and four-body interactions may be: (a) more and more attractive; (b) more and more repulsive; or (c) extremely weak. Strangely the minimum point of nonadditive three- and four-body interaction potentials has not been found up to now. For both H 2 and HF tetramers the nonadditive four-body interaction energy makes a negligible contribution to total binding energy when intermolecular distances are compressed from 3.0 to 1.7 Å.

Four-body potentials reveal protein-specific correlations to stability changes caused by hydrophobic core mutations

Mutational experiments show how changes in the hydrophobic cores of proteins affect their stabilities. Here, we estimate these effects computationally, using four-body likelihood potentials obtained by simplicial neighborhood analysis of protein packing (SNAPP). In this procedure, the volume of a known protein structure is tiled with tetrahedra having the center of mass of one amino acid side-chain at each vertex. Log-likelihoods are computed for the 8855 possible tetrahedra with equivalent compositions from structural databases and amino acid frequencies. The sum of these four-body potentials for tetrahedra present in a given protein yields the SNAPP score. Mutations change this sum by changing the compositions of tetrahedra containing the mutated residue and their related potentials. Linear correlation coefficients between experimental mutational stability changes, Δ(Δ G unfold), and those based on SNAPP scoring range from 0.70 to 0.94 for hydrophobic core mutations in five different proteins. Accurate predictions for the effects of hydrophobic core mutations can therefore be obtained by virtual mutagenesis, based on changes to the total SNAPP likelihood potential. Significantly, slopes of the relation between Δ(Δ G unfold) and Δ SNAPP for different proteins are statistically distinct, and we show that these protein-specific effects can be estimated using the average SNAPP score per residue, which is readily derived from the analysis itself. This result enhances the predictive value of statistical potentials and supports previous suggestions that “comparable” mutations in different proteins may lead to different Δ(Δ G unfold) values because of differences in their flexibility and/or conformational entropy.

Lattice protein folding with two and four-body statistical potentials.

The cooperative folding of proteins implies a description by multibody potentials. Such multibody potentials can be generalized from common two-body statistical potentials through a relation to probability distributions of residue clusters via the Boltzmann condition. In this exploratory study, we compare a four-body statistical potential, defined by the Delaunay tessellation of protein structures, to the Miyazawa-Jernigan (MJ) potential for protein structure prediction, using a lattice chain growth algorithm. We use the four-body potential as a discriminatory function for conformational ensembles generated with the MJ potential and examine performance on a set of 22 proteins of 30-76 residues in length. We find that the four-body potential yields comparable results to the two-body MJ potential, namely, an average coordinate root-mean-square deviation (cRMSD) value of 8 A for the lowest energy configurations of all-alpha proteins, and somewhat poorer cRMSD values for other protein classes. For both two and four-body potentials, superpositions of some predicted and native structures show a rough overall agreement. Formulating the four-body potential using larger data sets and direct, but costly, generation of conformational ensembles with multibody potentials may offer further improvements. Proteins 2001;43:161-174.

Rigorous Proof of a Liquid-Vapor Phase Transition in a Continuum Particle System

We consider particles in ${\Bbb R}^d, d \geq 2$, interacting via attractive pair and repulsive four-body potentials of the Kac type. Perturbing about mean field theory, valid when the interaction range becomes infinite, we prove rigorously the existence of a liquid-gas phase transition when the interaction range is finite but long compared to the interparticle spacing.

A domain decomposition parallel processing algorithm for molecular dynamics simulations of systems of arbitrary connectivity

We describe in this paper methods for applying domain decomposition to a general purpose molecular dynamics program. The algorithm is suitable for either distributed memory parallel computers or shared memory machines with message passing libraries. A method is discussed in detail which allows molecules of arbitrary connectivity to be simulated within the domain decomposition approach. The algorithm also contains techniques to handle both rigid bond constraints and special CH 2 constraints which involve five atoms. Examples are given for molecular systems containing both types of constraints plus two, three- and four-body intramolecular potentials as well as short-range and long-range non-bonded potentials. The algorithm has been implemented on a Silicon Graphics Power Challenge machine using the MPI message passing library.

Monte Carlo and mean-field studies of phase evolution in concentrated surfactant solutions

A two-dimensional lattice model, originally introduced by Granek et al. [J. Chem. Phys. 101, 4331 (1994)], is used to demonstrate the intricate coupling between the intramicellar interactions that determine the optimal aggregation geometry of surfactant molecules in dilute solution, and the intermicellar interactions that govern the phase behavior at higher concentrations. Three very different scenarios of self-assembly and phase evolution are analyzed in detail, based on Monte Carlo studies and theoretical interpretations involving mean-field, Landau–Ginzburg, Bethe–Peierls, and virial expansion schemes. The basic particles in the model are ‘‘unit micelles’’ which, due to spontaneous self-assembly or because of excluded area interactions, can fuse to form larger aggregates. These aggregates are envisaged as flat micelles composed of a bilayerlike body surrounded by a curved semitoroidal rim. The system’s Hamiltonian involves one- through four-body potentials between the unit micelles, which account for their tendency to form aggregates of different shapes, e.g., elongated vs disklike micelles. Equivalently, the configurational energy of the system is a sum of micellar self-energies involving the packing free energies of the constituent molecules in the bilayer body and in rim segments of different local curvature. The rim energy is a sum of a line tension term and a 1D curvature energy which depends on the rim spontaneous curvature and bending rigidity. Different combinations of these molecular parameters imply different optimal packing geometries and hence different self-assembly and phase behaviors. The emphasis in this paper is on systems of ‘‘curvature loving’’ amphiphiles which, in our model, are characterized by negative line tension. The three systems studied are: (i) A dilute solution of stable disklike micelles which, upon increasing the concentration, undergoes a first-order phase transition to a continuous bilayer with isolated hole defects. An intermediate modulated ‘‘checkerboard’’ phase appears under certain conditions at low temperatures. (ii) A system of unit micelles which in dilute solution tend to associate into linear micelles. These micelles are rodlike at low temperatures, becoming increasingly more flexible as the temperature increases. Upon increasing the concentration the micelles grow and undergo (in 2D) a continuous transition into nematic and ‘‘stripe’’ phases of long rods. At still higher concentrations the micellar stripes fuse into continuous sheets with line defects. (iii) A system in which, already in dilute solution, the micelles favor the formation of branched aggregates, analogous to the branched cylindrical micelles recently observed in certain surfactant solutions. As the concentration increases the micelles associate into networks (‘‘gels’’) composed of a mesh of linear micelles linked by ‘‘T-like’’ intermicellar junctions. The network may span the entire system or phase separate and coexist with a dilute micellar phase, depending on the details of the molecular packing parameters.

Using the PHONON program for interpreting inelastic neutron scattering data

The PHONON program was developed into a generally useful form for lattice dynamic calculations using the quasi-harmonic approximation and made available to other users interested in the interpretation of INS data. By diagonalizing the dynamical matrix, the actual values of the ω( q) for each wave vector q are determined. The dispersion curves can, therefore, be obtained and plotted using NAG graphic routines on a suitable computer. The weighted phonon density of states can also be obtained by integration over all the possible values of q lying in the first Brillouin zone of the crystal. There is a wide range of two-body potentials programmed into the code which are used for the short-range repulsive parts of the potential. Ionic polarisation is modelled using the shell model. There are also a number of many-body potentials for molecular systems and lattice structures. These include angle bending three-body potentials and four-body (torsional) potentials.

A domain decomposition parallel processing algorithm for molecular dynamics simulations of polymers

We describe in this paper a domain decomposition molecular dynamics algorithm for use on distributed memory parallel computers which is capable of handling systems containing rigid bond constraints and three- and four-body potentials as well as non-bonded potentials. The algorithm has been successfully implemented on the Fujitsu 1024 processor element AP1000 machine. The performance has been compared with and benchmarked against the alternative cloning method of parallel processing [D. Brown, J.H.R. Clarke, M. Okuda and T. Yamazaki, J. Chem. Phys., 100 (1994) 1684] and results obtained using other scalar and vector machines. Two parallel versions of the SHAKE algorithm, which solves the bond length constraints problem, have been compared with regard to optimising the performance of this procedure.

Reduced coordinates on the configuration space of three and four atoms.

A framework for constructing three- and four-body interatomic potentials is proposed. The study is motivated by a need for an understanding of the morphology of these potentials and consequently the morphology of the Born-Oppenheimer energy surface (i.e., the location of the surface minimum and saddle points). Invariant coordinate systems on the configuration space of three and four atoms are therefore developed which allow the morphology of three- and four-body terms to be chosen. Direct applications of these coordinates include the construction of interatomic potentials and an analysis of the local geometry around an atom in a crystal.

Approximate equations of state for a finite lattice system with two-, three-, and four-body potentials: Models for the binding of oxygen to hemoglobin.

Biochemical binding systems present many-body problems, even after integrating over solvent and polymer configurational degrees of freedom, because biological molecules are dense in cellular compartments. Furthermore, the resulting effective potentials are likely to include substantial n-body contributions with n>2. The binding of oxygen to hemoglobin is studied as a prototype problem of this kind. (i) The Hill coefficient (simply related to the change in the fractional binding with a change in the ligand chemical potential) has proven to be a useful, model-independent descriptor of ligand binding behavior. An obvious generalization of this approach is given and the resulting coefficients are shown to describe stably the binding of oxygen to hemoglobin over a broad range of free-ligand concentrations. (ii) An expansion of the standard Helmholtz free energy in the local binding density is used to find the excess standard Helmholtz free energy for saturation of hemoglobin with oxygen. This requires equations to change from local potential field variables to density variables that are equivalent to the two-, three-, and four-point Ornstein-Zernike equations. The method permits exact inclusion of two-, three-, and four-body potentials in the limit of weak interaction potentials, but is very poor for strong interactions. (iii) An expansion of the two-body direct correlation function with r\ensuremath{\ne}0 and the core condition are used in the compressibility equation to find the excess pressure as a function of the fractional saturation, and thus also to find the standard free-energy change upon saturating the protein with ligand.At first order, this procedure provides a model for the binding of oxygen to hemoglobin that does not require the assumption of pair decomposability of the interaction potentials or explicit calculation of a partition function, and yet is competitive with models that require both. The model is accurate for a wide range of interaction potentials. The result is a one-parameter equation of state that does not exhibit a phase transition for a finite system, but may exhibit a phase transition in the limit of infinite system size (depending on the magnitude of the cooperativeness parameter).

Multibody quark forces in quantum chromodynamics.

A general exposition of multibody quark forces in quantum chromodynamics is given. The low-energy Hamiltonian involves two-, three-, and four-body quark potentials. We compute the short-range three- and four-body potentials to lowest-order terms in v/c and show that the former does not contribute to the mass of baryons.