The Lagrange multiplier method for manipulating geometries. Implementation and applications in molecular mechanics

It is shown that a Lagrange multiplier method to constrain one or several internal coordinates, or averages and combinations of these, is easily implemented in a molecular mechanics computer program that uses Newton–Raphson (NR ) minimization. Results are given for constraints on nonbonded distances and torsion angles. When a potential energy surface is to be explored, it is much better to constrain the average of three torsion angles around a bond than to constrain a single torsion angle. Certain conversions can only be achieved when averages of torsion angles around different bonds are constrained. Combinations of constraints have been applied to evaluate differences between calculated and observed geometries and to obtain transition states for relatively large molecules from results for smaller molecules at relatively low costs. The efficieny of the combination of the Lagrange multiplier method and NR minimization in terms of computing time can be rated as good.     

Conformational sampling by a general linearized embedding algorithm

Linearized embedding is a variant on the usual distance geometry methods for finding atomic Cartesian coordinates given constraints on interatomic distances. Instead of dealing primarily with the matrix of interatomic distances, linearized embedding concentrates on properties of the metric matrix, the matrix of inner products between pairs of vectors defining local coordinate systems within the molecule. We developed a pair of general computer programs that first convert a given arbitrary conformation of any covalent molecule from atomic Cartesian coordinates representation to internal local coordinate systems enforcing rigid valence geometry and then generate a random sampling of conformers in terms of atomic Cartesian coordinates that satisfy the rigid local geometry and a given list of interatomic distance constraints. We studied the sampling properties of this linearized embedding algorithm vs. a standard metric matrix embedding program, DGEOM, on cyclohexane, cycloheptane, and a cyclic pentapeptide. Linearized embedding always produces exactly correct bond lengths, bond angles, planarities, and chiralities; it runs at least two times faster per structure generated, and is successful as much as four times as often at refining these structures to full agreement with the constraints. It samples the full range of allowed conformations broadly, although not perfectly uniformly. Because local geometry is rigid, linearized embedding's sampling in terms of torsion angles is more restricted than that of DGEOM, but it finds in some instances conformations missed by DGEOM. © 1992 by John Wiley & Sons, Inc.     

Systematic drug receptor mapping: A new approach to the analysis of conformational energy calculations of flexible molecules with application to dopaminergic and adrenergic agonists

The computer program PRODIS is used to find low energy conformations of flexible molecules by searching the potential energy surface(s) of one or more torsion angles via rigid rotation. The n-dimensional grid of energy versus torsion angles is then converted to a Boltzman probability distribution, with the probability being represented not as a function of torsion angle, but rather a distance between two atoms. These atoms are chosen by comparison with a known, active analogue in which certain atoms have previously been determined as requirements for drug activity. PRODIS produces a list of low energy conformations, their corresponding interatomic distances and the Boltzman probability for each distance ±0.125, as well as the total probability for each conformation. The user also specifies a target interatomic distance and range (usually derived from a more rigid analogue) for which PRODIS lists all conformations and their Boltzman probability that meet this distance.     

Structure distribution in an elastin-mimetic peptide (VPGVG)3 investigated by solid-state NMR

Elastin is an extracellular-matrix protein that imparts elasticity to tissues. We have used solid-state NMR to determine a number of distances and torsion angles in an elastin-mimetic peptide, (VPGVG)3, to understand the structural basis of elasticity. C-H and C-N distances between the V6 carbonyl and the V9 amide segment were measured using 13C-15N and 13C-1H rotational-echo double-resonance experiments. The results indicate the coexistence of two types of intramolecular distances: a third of the molecules have short C-H and C-N distances of 3.3 +/- 0.2 and 4.3 +/- 0.2 A, respectively, while the rest have longer distances of about 7 A. Complementing the distance constraints, we measured the (phi, psi ) torsion angles of the central pentameric unit using dipolar correlation NMR. The -angles of P7 and G8 are predominantly ~150, thus restricting the majority of the peptide to be extended. Combining all torsion angles measured for the five residues, the G8 C chemical shift, and the V6-V9 distances, we obtained a bimodal structure distribution for the PG residues in VPGVG. The minor form is a compact structure with a V6-V9 C=O-HN hydrogen bond and can be either a type II -turn or a previously unidentified turn with Pro (phi = -70, psi= 20 +/- 20) and Gly ( phi= -100 +/- 20, psi = -20 +/- 20). The major form is an extended and distorted beta-strand without a V6-V9 hydrogen bond and differs from the ideal parallel and antiparallel beta-strands. The other three residues in the VPGVG unit mainly adopt antiparallel beta-sheet torsion angles. Since (VPGVG)3 has the same 13C and 15N isotropic and anisotropic chemical shifts as the elastin-mimetic protein (VPGXG)n (X = V and K, n = 195), the observed conformational distribution around Pro and Gly sheds light on the molecular mechanism of elastin elasticity.     

A fast and efficient method to generate biologically relevant conformations

Summary Mutual binding between a ligand of low molecular weight and its macromolecular receptor demands structural complementarity of both species at the recognition site. To predict binding properties of new molecules before synthesis, information about possible conformations of drug molecules at the active site is required, especially if the 3D structure of the receptor is not known. The statistical analysis of small-molecule crystal data allows one to elucidate conformational preferences of molecular fragments and accordingly to compile libraries of putative ligand conformations. A comparison of geometries adopted by corresponding fragments in ligands bound to proteins shows similar distributions in conformation space. We have developed an automatic procedure that generates different conformers of a given ligand. The entire molecule is decomposed into its individual ring and open-chain torsional fragments, each used in a variety of favorable conformations. The latter ones are produced according to the library information about conformational preferences. During this building process, an extensive energy ranking is applied. Conformers ranked as energetically favorable are subjected to an optimization in torsion angle space. During minimization, unfavorable van der Waals interactions are removed while keeping the open-chain torsion angles as close as possible to the experimentally most frequently observed values. In order to assess how well the generated conformers map conformation space, a comparison with experimental data has been performed. This comparison gives some confidence in the efficiency and completeness of this approach. For some ligands that had been structurally characterized by protein crystallography, the program was used to generate sets of some 10 to 100 conformers. Among these, geometries are found that fall convincingly close to the conformations actually adopted by these ligands at the binding site.     

蒙特卡洛模拟退火与距离限制相结合的方法在多肽溶液构象分析中的应用

多维核磁共振技术的飞速发展议得其在生物大分子结构测定方面的应用已经达到可以与【射线晶体学并驾齐驱的地步．蛋白质结构堆积紧密，较适合于用核磁共振方法给出确定的结构．与蛋白质不同的是多肽的柔性较大，在溶液中可能存在多种构象，核磁共振实验给出的只是平均信息＊．利用核磁，（振数据构建分子结构模型常用的方法有距离几何、分子动力学等，在由核磁共振NOESZ得到的距离信息足够多时可以给出较好的结果问．由于多肽本身的特点：柔性较大，由核磁共振得到的距离信息较少等，利用距离几何、分子动力学方法进行构象搜索时容易陷入…     

Searching for conformers of nine- to twelve-ring hydrocarbons on the MM2 and MM3 energy surfaces: Stochastic search for interconversion pathways

The stochastic search method was employed to find as many conformers on the MM2 and MM3 energy surfaces as possible for cyclic saturated hydrocarbons with ring sizes from 9 through 12. The number found was 8 MM2 (8 MM3) for 9 rings, 18 MM2 (16 MM3) for 10 rings, 40 MM2 (29 MM3) for 11 rings, and 111 MM2 (90 MM3) for 12 rings. A measure of similarity between pairs of conformers of a compound, called conformational distance, is described. It was used to correlate similar MM2 and MM3 conformers. It was discovered that some conformers on each energy surface are not close to minima on the other surface in rings larger than 9. On refinement with the other optimizer, they changed considerably—going downhill to other previously found minima on the other energy surface or (in a few cases) going to minima which had not been found by direct searches. Conformational distance was also employed as an indication of which pairs of MM2 (or MM3) conformers are likely to interconvert rapidly. A new stochastic procedure of using small kicks was used to search for the most likely interconversion processes among the conformers. There is fairly good agreement between the most facile pathways located by it and unusually short conformational distances. Several additional 12-ring conformers (not found with previous methods) were located through application of this small kick procedure.     

A semiempirical conformational analysis of the 3-phenyl-1,2,3-oxathiazolidine 2-oxide

An extensive conformational search for the heterocyclic compound 3-phenyl-1,2,3-oxathiazolidine 2-oxide has been carried out using the semiempirical quantum-mechanical method PM3. All torsion angles were varied in the geometry optimization procedure which led to the prediction of four distinct conformers. Barriers to internal rotation leading to conformational interconversions were also calculated. The results of the present study indicate that this five-member ring substituted heterocyclic system may be expected to exist as a mixture of stable conformers rather than a unique molecular entity.     

蒙特卡洛模拟退火与距离限制相结合的方法——在多肽溶液构象分析中的应用

Distance geometry and molecular dynamics are currently employed in determining molecular structures with interatomic distances from NMR NOESY experiment. Because of the flexibility of peptide, distances obtained from NMR are usually not sufficient to confine its structure. Both distance geometry and molecular dynamics will bias in the conformational space at this circumstance. Constraint Monte Carlo simulated annealing was established to solve this problem. Distance constraints were included into the ECEPP/2 force field by introducing a harmonic energy term. Conformational analysis of a pentapeptide with eight interatomic distances from NMR was carried out as a test. By comparison of the 100 conformers obtained from constraint simulated annealing and the 100 conformers from distance geometry calculation, it was found that constraint simulated annealing can cover the outcomes of distance geometry and at the same time give more con-formers fitting to the experimental data. The result shows that constraint Monte-Carlo simulated annealing is more valid in constructing peptide structures from NMR distances than currently employed methods when no sufficient distances from NMR are available.     

Development of a quasi-dynamic pharmacophore model for anti-complement peptide analogues

Three quasi-dynamic pharmacophore models have been constructed for the complement inhibitor peptide compstatin, using first principles. Uniform sampling along 5-ns molecular dynamics trajectories provided dynamic conformers that are thought to represent the entire conformational space for nine training set molecules, compstatin, four active analogues, and four inactive analogues. The pharmacophore models were built using mixed physicochemical and structural properties of residues indispensable for structural stability and activity. Owing to the size and flexibility of compstatin, one-dimensional probability distributions of intrapharmacophore point distances, angles, and dihedral angles of different analogues spread over wide and overlapping ranges. More robust two-dimensional distance-angle probability distributions for two pharmacophore models discriminated individual analogues in terms of specific distance-angle pairs, but overall failed to identify the active and the inactive analogues as two distinct groups. Two-dimensional distance-dihedral angle probability distributions in a third pharmacophore model allowed discrimination of the groups of active and inactive analogues more effectively, with the highest-activity analogue having distinct behavior. The present study indicates that more stringent structural constraints should be used for a set of structurally similar but flexible peptides, as opposed to organic molecules, to convert dynamic conformers into pharmacophore models. Flexibility is a general aspect of the structure and function of peptides and should be taken into account in ligand-based pharmacophore design. However, the discrimination of activity using multidimensional probability surfaces depends on the peptide system, the selection of the training set, the molecular dynamics protocol, and the selection of the type and number of pharmacophore points.     

On the optical activity of paraffinic hydrocarbons

An expression for potential energy as a function of torsion angle about a saturated CC bond is developed analytically. This equation is used in the development of two methods for the conformational analysis of paraffins-one based on consideration only of exactly staggered conformers and one giving consideration to energy at 1° intervals of torsion angle. Two methods of assigning molecular rotations to hydrocarbon conformers are considered. Various combinations of results so obtained are compared as ways to calculate rotations for paraffins.     

Conformational variety for the ansa chain of rifamycins: Comparison of observed crystal structures and molecular dynamics simulations

The antibiotic activity (via inhibition of DNA-dependent RNA polymerase, DDRP) of rifamycins has been correlated to the conformation of the ansa chain, which can be described by means of 17 torsion angles defined along the ansa backbone. It has been shown that favourable or unfavourable conformations of the ansa chain in rifamycin crystals are generally diagnostic of activity or inactivity against isolated DDRP. The principles of structure correlation suggest that the torsional variety observed in rifamycin crystals should mimic the dynamic flexibility of the ansa chain in solution. Twenty-six crystal structures of rifamycins are grouped into two classes (active and non-active). For each class the variance of the 17 ansa backbone torsion angles is analysed. Active compounds show a well-defined common pattern, while non-active molecules are more scattered, mainly due to steric constraints forcing the molecules into unfavourable conformations. The experimental distributions of torsion angles are compared to the torsional freedom of the ansa chain simulated by molecular dynamics calculations performed at different temperatures and conditions on rifamycin S and rifamycin O, which represent a typical active and a typical sterically constrained molecule, respectively. It is shown that the torsional variety found in the crystalline state samples the dynamic behaviour of the ansa chain for active compounds. The methods of circular statistics are illustrated to describe torsion angle distributions.     

Application of standard DFT theory for nonbonded interactions in soft matter: prototype study of poly-para-phenylene

We present a detailed analysis of the application of density functional theory (DFT) methods to the study of structural properties of molecular and supramolecular systems, using as a paradigmatic example three para-phenylene-based systems: isolated biphenyl, single chain poly-para-phenylene, and crystalline biphenyl. We use different functionals for the exchange correlation potential, the local density (LDA), and generalized gradient approximations (GGA), and also different basis sets expansions, localized, plane waves (PW), and mixed (localized plus PW), within the reciprocal space formulation for the hamiltonian. We find that regardless of the choice of basis functions, the GGA calculations yield larger interring distances and torsion angles than LDA. For the same XC approximation, the agreement between calculations with different basis functions lies within 1% (LDA) or 0.5% (GGA) for distances, and while PW and mixed basis calculations agree within 1 degrees for torsion angles, the localized basis results show larger angles by approximately 8 degrees and a nonmonotonic dependence on basis size, with differences within 6 degrees. The most prominent features, namely the torsion between rings for isolated molecule and infinite chain, and planarity for the molecule in crystalline environment, are well reproduced by all DFT calculations.     

Electron diffraction study of the molecular structure and conformation of gaseous 2-furoyl chloride

The molecular structure of 2-furoyl chloride has been investigated by gas-phase electron diffraction at 86°C. Two distinct conformers were identified, a more stable planar form with the furan oxygen and the carbonyl oxygen syn and a less stable planar (or nearly planar) anti form. Assuming that the two forms differ in their geometries only in the O=C---C---O torsion angles and assuming the furan ring to have C_2v symmetry, the results for some of the distances (r_a) and angles (_a) are: r(C---H) = 1.110(20) Å, r(C=O) = 1.207(6) Å, r(C---O) = 1.378(10) Å, r(C??? = 1.465(13) A, (r(C---C)) (average carbon—carbon distance in the furan ring) = 1.392(8) A Δr(C---C) (difference between single and double carbon—carbon distances in the furan ring) = 0.069 A (assumed), r(C---Cl) = 1.787(6) A, C=C---COCl = 131.6(9)°, C=C---O = 110.9(4)°, C=C---H = 127.7(13.4)°, C---C=O = 125.8(8)° and C---C---Cl = 111.8(6)°. At 359 K the observed amount of the conformer with the oxygen atoms syn was 69.8(14.2)%.     

Conformational distribution of heptaalanine: Analysis using a new Monte Carlo chain growth method

A distribution of conformations of heptaalanine is obtained using a new Monte Carlo (MC) method that grows the chain atom by atom. Resulting configurations are energy minimized and a detailed analysis is performed of the minimum-energy conformers using a method of classification that partitions ?ψ space. The MC-generated configurations are compared with those generated from high-temperature molecular dynamics (MD) simulations. It is found that the new method generates a wide distribution of low-energy conformers at least 10 times more quickly than the MD. An analysis of the generated energy minima demonstrates that they can be divided into clusters in the space defined by the five pairs of ?—ψ angles of the inner residues. The space occupied by the minima populating each cluster is restricted. © 1992 by John Wiley & Sons, Inc.     

Solution-phase structural characterization of supramolecular assemblies by molecular diffraction

Structures of four molecular squares based on rhenium coordination chemistry have been characterized in the solution phase using pair distribution function (PDF) analyses of wide-angle X-ray scattering measured to better than 1 A spatial resolution. In this report we have focused, in particular, on a comparison of structures for pyrazine- and bipyridine-edged squares measured in solution with structures determined for these molecules in the solid state using X-ray crystallography and models derived from geometry optimization and molecular dynamics simulations using a classical force field. The wide-angle scattering for these assemblies is dominated by pair correlations involving one or more rhenium atoms, with both edge and diagonal Re-Re interactions appearing prominently in PDF plots. The pyrazine square is characterized by a relatively rigid structure in solution, with PDF peak positions and linewidths corresponding closely to those calculated from crystal structure data. For the bipyridine-edged square, the experimental PDF peaks measured along the molecular sides match the positions and linewidths of the PDF peaks calculated from static models. In contrast, PDF peaks measured across the diagonal distances of the molecular square deviate significantly from those calculated from the static crystallographic and energy minimized models. The experimental data are instead indicative of configurational broadening of the diagonal distances. In this respect, molecular dynamics simulations point to the importance of butterfly type motions that modulate the Re-Re diagonal distance. Indeed, the experimental data are reasonably well fit by assuming a bimodal distribution of butterfly conformers differing by approximately 25 degrees in the Re-Re-Re-Re torsion angle. Additionally, the measurements provide evidence for solvent ordering by the supramolecular assemblies detected as regions of solvent association and exclusion.     

Similarity searching in databases of flexible 3D structures using smoothed bounded distance matrices

This paper describes a method for calculating the similarity between pairs of chemical structures represented by 3D molecular graphs. The method is based on a graph matching procedure that accommodates conformational flexibility by using distance ranges between pairs of atoms, rather than fixing the atom pair distances. These distance ranges are generated using triangle and tetrangle bound smoothing techniques from distance geometry. The effectiveness of the proposed method in retrieving other compounds of like biological activity is evaluated, and the results are compared with those obtained from other, 2D-based methods for similarity searching.     

A conformational study of the trisaccharide beta-D-Glcp-(1-->2)[beta-D-Glcp-(1-->3)]alpha-D-Glcp-OMe by NMR NOESY and TROESY experiments, computer simulations, and X-ray crystal structure analysis

Proton-proton cross-relaxation rates have been measured for the trisaccharide beta-D-Glcp-(l --> 2)[beta-D-Glcp-(1 --> 3)]alpha-D-Glcp-OMe in D2O as well as in D2O/[D6]DMSO 7:3 solution at 30 degrees C by means of one-dimensional NMR pulsed field gradient 1H,1H NOESY and TROESY experiments. Interatomic distances for the trisaccharide in D2O were calculated from the cross-relaxation rates for two intraresidue and three interglycosidic proton pairs, using the isolated spin-pair approximation. In the solvent mixture one intraresidue and three interglycosidic distances were derived without the use of a specific molecular model. In this case the distances were calculated from the cross-relaxation rates in combination with "model-free" motional parameters previously derived from 13C relaxation measurements. The proton-proton distances for interglycosidic pairs were compared with those averaged from Metropolis Monte Carlo and Langevin Dynamics simulations with the HSEA, PARM22, and CHEAT95 force fields. The crystal structure of the trisaccharide was solved by analysis of X-ray data. Interresidue proton pairs from the crystal structure and those observed by NMR experiments were similar. However, the corresponding proton-proton distances generated by computer simulations were longer. For the (1 --> 2) linkage the glycosidic torsion angles of the crystal structure were found in a region of conformational space populated by all three force fields, whereas for the (1 --> 3) linkage they occupied a region of low population density, as seen from the simulations.     

Crystal and molecular structure of an (S)-(+)-enantiomer of modafinil, a novel wake-promoting agent

The (+)-enantiomer of modafinil [(RS)-2-(diphenylmethylsulfinyl)acetamide], a novel wake-promoting agent, was clarified to be S-configuration by X-ray crystal structure analysis. The crystal consists of two crystallographically independent conformers that are different at the torsion angles around the sulfinylacetamide moiety, and this results from the molecular packing requirement to form a two-dimensional hydrogen-bonding network via neighboring amide groups in the crystal. The crystal structure is characterized by the formation of alternative hydrophobic and hydrophilic layers, which are formed among the symmetry-translated assemblies of diphenylmethyl and sulfinylacetamide moieties, respectively. The spatial orientation between the diphenyl and amide groups is believed to be important for the activity of modafinil.