Limited conformational space for early-stage protein folding simulation

MOTIVATION: The problem of early-stage protein folding is critical for protein structure prediction. The model presented introduces a common definition of protein structures which may be treated as the possible in silico early-stage form of the polypeptide chain. Limitation of the conformational space to the ellipse path on the Ramachandran map was tested as a possible sub-space to represent the early-stage structure for simulation of protein folding. The proposed conformational sub-space was developed on the basis of the backbone conformation, with side-chain interactions excluded. RESULTS: The ellipse-path-limited conformation of BPTI was created using the criterion of shortest distance between Phi, Psi angles in native form of protein and the Phi, Psi angles belonging to the ellipse. No knots were observed in the structure created according to ellipse-path conformational sub-space. The energy minimization procedure applied to ellipse-path derived conformation directed structural changes toward the native form of the protein with SS-bonds system introduced to the procedure. AVAILABILITY: Program 'Ellipse' to create the ellipse-path derived structure available on request: myroterm@cyf-kr.edu.pl     

Consistent force field calculations on 2,5-diketopiperazine and its 3.6-dimethyl derivatives

The minimum energy conformations are calculated for 2, 5-diketopiperazine (DKP) and its 3,6-dimethyl derivatives (DL-DMDKP and LL-DMDKP), using a consistent force field approach developed previously. The energy function parameters that were not required in earlier calculations on alkanes, amides, mid lactams are fitted to spectral and conformational data on the diketopiperazines. Vibrational assignments are suggested for DKP. Conformational energies are also determined over a range of selected values for ring dihedral angles, and the shape of the potential energy functions is examined over deviations from planarity. DKP and LL-DMDKP are found to have non-planar minimum energy conformations, separated from planar by less than a kcal/mole. DL-DMKP exhibits a nearly flat trough about the planar conformation. Calculations of minimum energies with one dihedral angle coordinate constrainted show a coupling between bond angles and dihedral angles in agreement with recent suggestions of Benedetti.     

Molecular dynamics simulations of peptides and proteins with amplified collective motions

We present a novel method that uses the collective modes obtained with a coarse-grained model/anisotropic network model to guide the atomic-level simulations. Based on this model, local collective modes can be calculated according to a single configuration in the conformational space of the protein. In the molecular dynamics simulations, the motions along the slowest few modes are coupled to a higher temperature by the weak coupling method to amplify the collective motions. This amplified-collective-motion (ACM) method is applied to two test systems. One is an S-peptide analog. We realized the refolding of the denatured peptide in eight simulations out of 10 using the method. The other system is bacteriophage T4 lysozyme. Much more extensive domain motions between the N-terminal and C-terminal domain of T4 lysozyme are observed in the ACM simulation compared to a conventional simulation. The ACM method allows for extensive sampling in conformational space while still restricting the sampled configurations within low energy areas. The method can be applied in both explicit and implicit solvent simulations, and may be further applied to important biological problems, such as long timescale functional motions, protein folding/unfolding, and structure prediction.     

Continuous metadynamics in essential coordinates as a tool for free energy modelling of conformational changes

Modelling of conformational changes in biopolymers is one of the greatest challenges of molecular biophysics. Metadynamics is a recently introduced free energy modelling technique that enhances sampling of configurational (e.g. conformational) space within a molecular dynamics simulation. This enhancement is achieved by the addition of a history-dependent bias potential, which drives the system from previously visited regions. Discontinuous metadynamics in the space of essential dynamics eigenvectors (collective motions) has been proposed and tested in conformational change modelling. Here, we present an implementation of two continuous formulations of metadynamics in the essential subspace. The method was performed in a modified version of the molecular dynamics package GROMACS. These implementations were tested on conformational changes in cyclohexane, alanine dipeptide (terminally blocked alanine, Ace-Ala-Nme) and SH3 domain. The results illustrate that metadynamics in the space of essential coordinates can accurately model free energy surfaces associated with conformational changes. Figure The conformational free energy surface of cyclohexane in the space of the two most intensive collective motions.

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Shape of the conformational energy surface near the global minimum and low-frequency vibrations in the native conformation of globular proteins

Based on the assumption that the conformational energy surface of a protein molecule can be approximated near the global minimum point by a multidimensional parabola, conformational fluctuations in the native state are discussed. In this approximation the conformational fluctuations can be viewed as excitations of coupled harmonic oscillations of dihedral angles. For the purpose of estimating the range of frequencies vibrations, globular proteins are assumed to made of homogeneous continuous elastic material. The number of vibrational modes in such an elastic body, with the wavelength no less than the characteristic length of an amino acid residue, are estimated roughly to be three times the number of amino acid residues in a protein, which is slightly less than the number of variable dihedral angles in a protein. Their frequencies, when converted to the wavenumber of corresponding light, are found to range from 1.8 × 10 cm^?1 to 2.1 × 10²cm^?1 for a protein with the diameter d = 40 Å, when Young's E = 10¹¹ dyne/cm² is assumed. A significant fraction of the coupled vibrations of dihedral angles in real globular proteins are collective ones, i.e., those involving the whole protein molecules. Based on these results, it concluded that the depth of the global minimum s at least 150 Kcal/mol.     

Synthesis, X-ray crystallographic structures of thio substituted N-acetyl N'-methylamide alanine and evaluation of sp sulfur parameters of the CFF91 force field.

Acetyl thioalanine N-methyl (Ac-Alat-NHMe) and thioacetyl alanine N-methyl (Act-Ala-NHMe) were synthesized, crystallized and their X-ray diffraction structures determined for the first time. Both molecules adopted beta-sheet conformations and showed similar hydrogen bonding patterns with one molecular surface forming two oxo hydrogen bonds and the other forming two thio hydrogen bonds. The crystal structure data for the two thioamides provided a validation of the thioamide parameters for the newly derived CFF91 force field because the observed crystal (phi, psi) angles were situated in the global minimum regions of the theoretical (phi, psi) map predicted using the parameters. In addition, the parameters were further validated because conformational energy minimization of the crystal structure produced low deviations in unit cell dimensions, bond lengths, bond angles and torsional angles, and a 120-ps molecular dynamics simulation also gave a low deviation for the most probable N-H...S=C bond distance.     

Conformational and helicoidal analysis of the molecular dynamics of proteins: "curves," dials and windows for a 50 psec dynamic trajectory of BPTI

A new procedure for the graphic analysis of molecular dynamics (MD) simulations on proteins is introduced, in which comprehensive visualization of results and pattern recognition is greatly facilitated. The method involves determining the conformational and helicoidal parameters for each structure entering the analysis via the method "Curves," developed for proteins by Sklenar, Etchebest, and Lavery (Proteins: Structure, Function Genet. 6:46-60, 1989) followed by a novel computer graphic display of the results. The graphic display is organized systematically using conformation wheels ("dials") for each torsional parameter and "windows" on the range values assumed by the linear and angular helicoidal parameters, and is present in a form isomorphous with the primary structure per se. The complete time evolution of dynamic structure can then be depicted in a set of four composite figures. Dynamic aspects of secondary and tertiary structure are also provided. The procedure is illustrated with an analysis of a 50 psec in vacuo simulation on the 58 residue protein, bovine pancreatic trypsin inhibitor (BPTI), in the vicinity of the local minimum on the energy surface corresponding to a high resolution crystal structure. The time evolution of 272 conformational and 788 helicoidal parameters for BPTI is analyzed. A number of interesting features can be discerned in the analysis, including the dynamic range of conformational and helicoidal motions, the dynamic extent of 2 degrees structure motifs, and the calculated fluctuations in the helix axis. This approach is expected to be useful for a critical analysis of the effects of various assumptions about force field parameters, truncation of potentials, solvation, and electrostatic effects, and can thus contribute to the development of more reliable simulation protocols for proteins. Extensions of the analysis to present differential changes in conformational and helicoidal parameters is expected to be valuable in MD studies of protein complexes with substrates, inhibitors, and effectors and in determining the nature of structural changes in protein-protein interactions.     

A stereochemical model for phospholipids. II: Conformational analysis and molecular packing of phosphatidylethanolamine (PE)

A partition energy method procedure was applied to select the energetically favoured conformations of phosphatidylethanolamine (PE) as polar constituents of phospholipid molecules. The result indicated a large degree of freedom for the two torsion angles of the ester bond of the phosphate and a gauche, gauche star conformation for the ethane bond.A packing process of the molecule was carried out through a potential energy calculation by considering the conformers selected above, using previously published procedure and conventions. All the arrangements which possess the best packing energy values were characterised by an orientation of the PN dipolar segment parallel to the lattice plain. Rotation of the internal torsion angles and rotation in the eulerian space of the molecule produced differences in the charged groups that interact. An additional minimum was present in the energy packing process of those conformers which have the first torsion angle of the phosphate in a trans conformation. This minimum, which corresponds to an orientation of the molecule orthogonal to the lattice plane, requires a complete neutralisation of the point charges on the system.The results of the calculation underline the importance of changes in the behaviour of the polar group of the phospholipids in the packing process.     

Modeling mutations in protein structures

We describe an automated method for the modeling of point mutations in protein structures. The protein is represented by all non-hydrogen atoms. The scoring function consists of several types of physical potential energy terms and homology-derived restraints. The optimization method implements a combination of conjugate gradient minimization and molecular dynamics with simulated annealing. The testing set consists of 717 pairs of known protein structures differing by a single mutation. Twelve variations of the scoring function were tested in three different environments of the mutated residue. The best-performing protocol optimizes all the atoms of the mutated residue, with respect to a scoring function that includes molecular mechanics energy terms for bond distances, angles, dihedral angles, peptide bond planarity, and non-bonded atomic contacts represented by Lennard-Jones potential, dihedral angle restraints derived from the aligned homologous structure, and a statistical potential for non-bonded atomic interactions extracted from a large set of known protein structures. The current method compares favorably with other tested approaches, especially when predicting long and flexible side-chains. In addition to the thoroughness of the conformational search, sampled degrees of freedom, and the scoring function type, the accuracy of the method was also evaluated as a function of the flexibility of the mutated side-chain, the relative volume change of the mutated residue, and its residue type. The results suggest that further improvement is likely to be achieved by concentrating on the improvement of the scoring function, in addition to or instead of increasing the variety of sampled conformations.     

Ribonuclease A. Analysis of the hydrogen bond geometry, and spatial accessibility at the active site

The hydrogen bonding of bovine ribonuclease A derived from the high resolution X-ray structure has been studied in detail. Correlations have been examined for main-chain-main-chain hydrogen bond angles, torsion angles and distances, respectively. Differences are found consistently for correlations associated with alpha-helix and beta-sheet, respectively. Ten of the 124 side-chains have four or more hydrogen bond contacts; two, including Glu-101, have five or more. Three potential C = O---H, three N---X and three potential side-chain H-bonds fail to form. A search for highly inaccessible buried residues resulted in nine outstanding examples, all of which are conserved across 38 known mammalian ribonuclease A sequences, indicating the importance of these residues for structural stability. Of the two histidines in the active site, His-12 has five hydrogen bonds and His-119 three. The conformational space accessible to these two catalytically important residues studied by means of simple non-bonded contact energy calculations confirms the existence of two alternative, interchangeable locations for His-119, while His-12 is locked in a local energy minimum.     

Efficient sampling in collective coordinate space

Collective motions in biological macromolecules have been shown to be important for function. The most important collective motions occur on slow time scales, which poses a sampling problem in dynamic simulation of biomolecules. We present a novel method for efficient conformational sampling. The method combines the simulation of an ensemble of concurrent trajectories with restraints acting on the ensemble of structures as a whole. Two properties of the ensemble may be restrained: (i) the variance of the ensemble and (ii) the average position of the ensemble. Both properties are defined in a subspace of collective coordinate space spanned by an arbitrary number of modes. We show that weak restraints on the ensemble variance suffice for an increase in sampling efficiency along soft modes by two orders of magnitudes. The resulting trajectories exhibit virtually the same structural quality as trajectories generated by restraint-free-molecular dynamics simulation, as judged by standard structure validation tools. The method is used to probe the resistance of a structure against conformational changes along collective modes and clearly distinguishes soft from stiff modes. Further applications are discussed. Proteins 2000;39:82-88.     

Polarization Force Fields for Peptides Implemented in ECEPP2 and MM2

Abstract

The empirical conformational energy program for peptides (ECEPP2) and molecular mechanics (MM2) have been used for the simulation of the For-Gly-NH₂ backbone. I propose two different methods for the calculation of the polarization energy term: the polarization procedure by non-interacting induced dipoles (NID) which assumes scalar isotropic point polarizabilities and the polarization scheme by interacting induced dipoles (ID) which calculates tensor effective anisotropic point polarizabilities (method of Applequist). I present a comparative study of ECEPP2 and MM2 + polarization. I discuss molecular mechanics results including the total energy differences, partitional analyses of the total steric energies and torsion dihedral angles. The γ global and the α, β and Δ local minima are stabilized by intramolecular hydrogen bonds. Although ECEPP2-based calculations rather under or over-estimate the relative energy of some local minima, the ID polarization energy term represents a significant correction to the total relative energy.     

Monte Carlo minimization with thermalization for global optimization of polypeptide conformations in cartesian coordinate space.

A new minimization procedure for the global optimization in cartesian coordinate space of the conformational energy of a polypeptide chain is presented. The Metropolis Monte Carlo minimization is thereby supplemented by a thermalization process, which is initiated whenever a structure becomes trapped in an area containing closely located local minima in the conformational space. The method has been applied to the endogenous opioid pentapeptide methionine enkephalin. Five among 13 different starting conformations led to the same apparent global minimum of an in-house developed energy function, a type II' reverse turn, the central residues of which are Gly-3-Phe-4. A comparison between the ECEPP/2 global minimum conformation of methionine enkephalin and the apparent one achieved by the present method shows that minimum-energy conformations having a certain similarity can be generated by relatively different force fields.     

A Monte Carlo method for generating structures of short single-stranded DNA sequences

A Monte Carlo method has been developed for generating the conformations of short single-stranded DNAs from arbitrary starting states. The chain conformers are constructed from energetically favorable arrangements of the constituent mononucleotides. Minimum energy states of individual dinucleotide monophosphate molecules are identified using a torsion angle minimizer. The glycosyl and acyclic backbone torsions of the dimers are allowed to vary, while the sugar rings are held fixed in one of the two preferred puckered forms. A total of 108 conformationally distinct states per dimer are considered in this first stage of minimization. The torsion angles within 5 kcal/mole of the global minimum in the resulting optimized states are then allowed to vary by ±10° in an effort to estimate the breadth of the different local minima. The energies of a total of 2187 (3⁷) angle combinations are examined per local conformational minimum. Finally, the energies of all dinucleotide conformers are scaled so that the populations of differently puckered sugar rings in the theoretical sample match those found in nmr solution studies. This last step is necessitated by limitations in the theoretical methods to predict DNA sugar puckering accurately. The conformer populations of the individual acyclic torsion angles in the composite dimer ensembles are found to be in good agreement with the distributions of backbone conformations deduced from nmr coupling constants and the frequencies of glycosyl conformations in x-ray crystal structures, suggesting that the low energy states are reasonable. The low energy dimer forms (consisting of 150–325 conformational states per dimer step) are next used as variables in a Monte Carlo algorithm, which generates the conformations of single-stranded d(CX_nG) chains, where X = A, T and n = 3, 4, 5. The oligonucleotides are built sequentially from the 5′ end of the chain using random numbers to select the conformations of overlapping dimer units. The simulations are very fast, involving a total of 10⁶ conformations per chain sequence. The potential errors in the buildup procedure are minimized by taking advantage of known rotational interdependences in the sugar–phosphate backbone. The distributions of oligonucleotide conformations are examined in terms of the magnitudes, positions, and orientations of the end-to-end vectors of the chains. The differences in overall flexibility and extension of the oligomers are discussed in terms of the conformations of the constituent dinucleotide steps, while the general methodology is discussed and compared with other nucleic acid model building techniques. © 1993 John Wiley & Sons, Inc.     

Theoretical conformational analysis of doulble-stranded polynucleotides]

Calculations of intramolecular interaction energy of two-stranded helical homopolynucleotide in the function of nine conformational variables have been carried out by the method of atom-atom potential functions. Four of these variables determine mutual position of base pairs, other four--deoxiribose ring conformation and other one--orientation of this ring with respect to the base. For this purpose an algorythm connecting dependent variables with independent ones has been developed. The investigation of energy function has shown that in the space of conformational parameters there are two valleys, which correspond to A-and B-families of conformations. Experimentaly determined conformations of two-stranded helical polynucleotides are located along the bottoms of these valleys. Along the bottom of each valley the intramolecular interaction energy changes rather little when conformational parameters change within a wide range. The valleys are separated by an energetical barrier.     

New methodology for computer-aided modelling of biomolecular structure and dynamics. 1. Non-cyclic structures

A general methodology is proposed for the conformational modelling of biomolecular systems. The approach allows one: (i) to describe the system under investigation by an arbitrary set of internal variables, i.e., torsion angles, bond angles, and bond lengths; it offers a possibility to pass from the free structure to a completely fixed one with the number of variables from 3N to zero, respectively, where N is the number of atoms; (ii) to consider both, a single molecule and a complex of many molecules, (e.g., proteins, water, ligands, etc.) in terms of one universal model; (iii) to study the dynamics of the system using explicit analytical Lagrangian equations of motion, thus opening up possibilities for investigations of slow concerted motions such as domain oscillations in proteins etc.; (iv) to calculate the partial derivatives of various functions of conformation, e.g., the conformational energy or external constraints imposed, using a standard efficient procedure regardless of the variables and the structure of the system. The approach is meant to be used in various investigations concerning the conformations and dynamics of biomacromolecules.     

Simulation of conformational possibilities of DNA via calculation of nonbonded interactions of complementary dinucleoside phosphate complexes

Using classical potential functions, we carried out potential-energy calculations on the complementary deoxydinucleoside phosphate complexes dApdA:dUpdU, dUpdA:dUpdA, and dApdU:dApdU. All dihedral and bond angles, except those of the nitrogen bases, were varied. The resulting minimum-energy conformations of the complexes are close to DNA A- and B-family conformations, with a typical arrangement of the nitrogen bases. The dihedral and bond angles of one of the molecules forming the complex can thereby differ by several degrees from those of the other molecule. For different base sequences, some dihedral and bond angles may vary over a range of several degrees without appreciably changing the total energy of the complex. Some low-energy conformations of the complexes corresponding to other regions of the conformational space are also found. The biological consequences of possible changes in dihedral and bond angles, occurring on interaction with other molecules, are discussed.     

Stereochemical effects of non-ionic methylphosphonates on nucleotide conformations.

The preferred conformations of deoxyribo and ribonucleoside 3'-methylphosphonates are analysed by minimizing the conformational energy as a function of all the major parameters including the sugar ring for both the S- and R-isomers. The results show that neither the substitution nor the nature of the diastereomer affects significantly the preferred conformations compared to the naturally occurring nucleoside 3'-phosphates. The preferred range of C3'-O3' bond torsions or the phase angles of pseudorotation (P) of the sugar are unaffected. The chiral substitution on the phosphate always adopts a conformation distal to the secondary C3' carbon atom in the minimum energy conformational state. Further, it introduces certain restrictions on the preferred range of P-O3' torsions depending on the methylphosphonate configuration. Methylphosphonate, especially the S-isomer, renders the normal gauche- range of P-O3' bond torsions responsible for the stacked helical duplexes to be energetically unfavourable besides introducing a high energy barrier between trans and gauche conformations. Therefore it is suggested that duplexes with S-methylphosphonate may favour extended phosphodiester conformations. These factors explain the observed lower melting temperature as well as the downfield shifts in the 31P signals in duplexes containing the S-isomer.