Carbon-13 nuclear magnetic resonance for organic chemists | George C. Levy and Gordon L. Nelson,Wiley-Interscience,New York, 1966, pp. xiii + 222, price 4.50

A Lagrange multipliers method is shown to be a convenient way of applying constraints to a Newton-Raphson minimization algorithm in a conformational analysis program. To illustrate this method two applications are dealt with in some detail. The Eckart conditions are used as a set of constraints to avoid mathematical difficulties associated with the empirical valence-force potential in a conformational analysis problem. For the molecules considered the use of constraints speeds up the Convergence. The second application illustrates how the method can be used to study selected sections through potential energy surfaces.     

Application of some methods of constrained optimization to the calculation of the molecular strain energy

The Lagrange multipliers method is applied to the minimization algorithm of the molecular potential energy proposed by Boyd in order to keep the bond lengths constant during the optimization. The results obtained for a series of molecules and the approximations supposed in the new algorithm are analysed.The first steps to include the penalty methods in the minimization of the molecular potential energy with constraints in the valence coordinates are given.     

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.     

Lagrange multiplier based transport theory for quantum wires

We discuss how the Lagrange multiplier method of nonequilibrium steady state statistical mechanics can be applied to describe the electronic transport in a quantum wire. We describe the theoretical scheme using a tight-binding model. The Hamiltonian of the wire is extended via a Lagrange multiplier to "open" the quantum system and to drive current through it. The diagonalization of the extended Hamiltonian yields the transport properties of wire. We show that the Lagrange multiplier method is equivalent to the Landauer approach within the considered model.     

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.     

Constrained optimization in delocalized internal coordinates

Using the recently introduced delocalized internal coordinates, in conjunction with the classical method of Lagrange multipliers, an algorithm for constrained optimization is presented in which the desired constraints do not have to be satisfied in the starting geometry. The method used is related to a previous algorithm by the same author for constrained optimization in Cartesian coordinates [J. Comput. Chem., 13 , 240 (1992)], but is simpler and far more efficient. Any internal (distance or angle/torsion) constraint can be imposed between any atoms in the system whether or not the atoms involved are formally bonded. Imposed constraints can be satisfied exactly. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 :1079–1095, 1997     

A method for constrained energy minimization of macromolecules

Two algorithms for the local energy minimization of the structure of macromolecules in the presence of constraints are proposed. They are a combination of the method of steepest descents and the method of conjugate gradients with the procedure SHAKE, by which distance constraints can be satisfied. The two algorithms are tested by applying them to a small protein, the bovine pancreatic trypsin inhibitor (BPTI), and compared with the penalty function method for conserving constraints. The efficiency of the proposed methods depends on the level of interdependence of the constraints. For bond-length constraints, the use of SHAKE is superior to the penalty function method. However, when bond-angle constraints are included, SHAKE is more efficient only if the curvature of the penalty function is considerably greater than that of the potential function being minimized. The results indicate that with bond-length constraints the minimization behavior is similar to that without constraints. However, the simultaneous application of bond-length and bond-angle constraints appears to confine the molecule to a very limited part of configuration space, very different from the part covered by an unconstrained minimization. This conclusion calls into question energy minimizations of protein systems in which only the dihedral angles are allowed to vary.     

Electron trajectories in molecular orbitals

The time-dependent Schrödinger equation can be rewritten so that its interpretation is no longer probabilistic. Two well-known and related reformulations are Bohmian mechanics and quantum hydrodynamics. In these formulations, quantum particles follow real, deterministic trajectories influenced by a quantum force. Generally, trajectory methods are not applied to electronic structure calculations as they predict that the electrons in a ground-state, real, molecular wavefunction are motionless. However, a spin-dependent momentum can be recovered from the nonrelativistic limit of the Dirac equation. Therefore, we developed new, spin-dependent equations of motion for the quantum hydrodynamics of electrons in molecular orbitals. The equations are based on a Lagrange multiplier, which constrains each electron to an isosurface of its molecular orbital, as required by the spin-dependent momentum. Both the momentum and the Lagrange multiplier provide a unique perspective on the properties of electrons in molecules.     

Revealing the configuration and crystal packing of organic compounds by solid-state NMR spectroscopy: methoxycarbonylurea, a case study

The molecular configuration and intermolecular arrangement of polycrystalline methoxycarbonylurea (MCU) has been studied by a combination of chemical editing, rotational echo double resonance (REDOR) spectroscopy and ab initio calculations. From the multispin IS(n) REDOR experiments several dipolar couplings were determined and converted into distance constraints. Intra- and intermolecular dipolar couplings were distinguished by isotope dilution. The configuration of the MCU molecule can be determined from three torsion angles Psi1, Psi2, and Psi3. Ab initio calculations showed that these angles are either 0 degrees or 180 degrees (Z or E). From the REDOR experiments, the E configuration was found for Psi1 and Psi2 and the Z configuration for Psi3. Thus the configuration of MCU in the solid state was determined to be EEZ. Distance constraints for the intermolecular arrangement of MCU were obtained by performing REDOR experiments on 13C15N2 MCU with different degrees of isotope dilution and on a cocrystallized 1:1 mixture of 13C(urea) MCU and 15N(amide) MCU. By combining these distance constraints with molecular modeling, three different possible packing motifs for MCU molecules were found. The molecules in these motifs are arranged as linear chains with methoxy groups at the borders of the chains. All the intermolecular hydrogen bond donors and acceptors in the interior of the chain are saturated.     

A kinematic view of loop closure

We consider the problem of loop closure, i.e., of finding the ensemble of possible backbone structures of a chain segment of a protein molecule that is geometrically consistent with preceding and following parts of the chain whose structures are given. We reduce this problem of determining the loop conformations of six torsions to finding the real roots of a 16th degree polynomial in one variable, based on the robotics literature on the kinematics of the equivalent rotator linkage in the most general case of oblique rotators. We provide a simple intuitive view and derivation of the polynomial for the case in which each of the three pair of torsional axes has a common point. Our method generalizes previous work on analytical loop closure in that the torsion angles need not be consecutive, and any rigid intervening segments are allowed between the free torsions. Our approach also allows for a small degree of flexibility in the bond angles and the peptide torsion angles; this substantially enlarges the space of solvable configurations as is demonstrated by an application of the method to the modeling of cyclic pentapeptides. We give further applications to two important problems. First, we show that this analytical loop closure algorithm can be efficiently combined with an existing loop-construction algorithm to sample loops longer than three residues. Second, we show that Monte Carlo minimization is made severalfold more efficient by employing the local moves generated by the loop closure algorithm, when applied to the global minimization of an eight-residue loop. Our loop closure algorithm is freely available at http://dillgroup. ucsf.edu/loop_closure/.     

Polymer dynamics in torsion space

The large scale motion of proteins, or covalently bonded polymers in general, is governed by the dynamics of the torsion angles, with bond lengths and bond angles kept approximately constant. In the present work, the Lagrangian equations of torsion motion are derived for a general macromolecule. The dynamics is implemented numerically for a test protein, using the velocity Verlet method as the integrator. The results indicate time steps of up to about 30 fs can be used for short time (up to at least 20 ps) simulations, before the dynamics and energy start to differ significantly from results obtained with smaller time steps. For longer time simulations, up to 1000 ps, a time step of 10 fs is relatively safe. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

A self-organizing algorithm for molecular alignment and pharmacophore development

We present a method for simultaneous three-dimensional (3D) structure generation and pharmacophore-based alignment using a self-organizing algorithm called Stochastic Proximity Embedding (SPE). Current flexible molecular alignment methods either start from a single low-energy structure for each molecule and tweak bonds or torsion angles, or choose from multiple conformations of each molecule. Methods that generate structures and align them iteratively (e.g., genetic algorithms) are often slow. In earlier work, we used SPE to generate good-quality 3D conformations by iteratively adjusting pairwise distances between atoms based on a set of geometric rules, and showed that it samples conformational space better and runs faster than earlier programs. In this work, we run SPE on the entire ensemble of molecules to be aligned. Additional information about which atoms or groups of atoms in each molecule correspond to points in the pharmacophore can come from an automatically generated hypothesis or be specified manually. We add distance terms to SPE to bring pharmacophore points from different molecules closer in space, and also to line up normal/direction vectors associated with these points. We also permit pharmacophore points to be constrained to lie near external coordinates from a binding site. The aligned 3D molecular structures are nearly correct if the pharmacophore hypothesis is chemically feasible; postprocessing by minimization of suitable distance and energy functions further improves the structures and weeds out infeasible hypotheses. The method can be used to test 3D pharmacophores for a diverse set of active ligands, starting from only a hypothesis about corresponding atoms or groups.     

Conformational energy analysis of substituted diphenylethanes: Part II. Empirical energy calculations on stilbene dihalogenides

The geometry and energy of the stable conformations of the isomeric forms of 1,2-halogeno-1,2-diphenylethanes have been obtained by means of empirical energy functions. A minimization of the conformational energy with respect to the torsion angles and the valence angles around the asymmetrically substituted carbon atoms has been carried out. The evaluated populations of the stable conformations showed good agreement with available experimental data. CNDO/2 calculations on the low-energy conformations of the isomeric forms of 1,2-difluoro-, 1,2-fluorochloro-, and 1,2-dichlorodiphenylethane have been carried out. These yielded improved estimates of the dipole moments for the dichloro isomers.     

Quantum-Chemical and semiclassical calculations of intermolecular interactions of phospholipids

By use of empirical 0–1–6–12 atom–atom potential functions and the PCILOCC method intra- and intermolecular interactions of glycero–phosphoryl–ethanolamine model head groups in a planar layer crystal were calculated. Starting from investigations of the two-dimensional energy-contour diagrams the minima of energy as a function of all head group torsion angles were calculated using a gradient procedure. Within an interval of 15 kcal/mol above the energy of the global minimum we obtained about 30 local minima. These results demonstrate a high flexibility of the investigated phosphorylethanolamine head group in agreement with experiment. The ethanolamine moiety exists in enantiomeric conformations. With the torsion angles of the 0–1–6–12 energy minimization procedure PCILOCC calculations were carried out. These calculations yield the x-ray conformation as the most stable one (unit-cell stabilization energy = ?36.3 kcal/mol). The PCILOCC as well as the potential function calculations show that the conformation of phospholipid head groups in layer crystals is determined by intramolecular as well as by intermolecular interactions with neighboring phospholipid molecules.     

The ellipsoid algorithm as a method for the determination of polypeptide conformations from experimental distance constraints and energy minimization

A new method for constrained nonlinear optimization known as the ellipsoid algorithm is evaluated as a means of determining and refining the conformations of peptides. Advantages of the ellipsoid algorithm over conventional optimization methods include that it avoids many local minima that other methods would be trapped by, and that it is sometimes able to find optimum solutions in which the constraints are satisfied exactly. The dihedral angles about single bonds were used as variables to keep the dimensionality low (the rate of convergence decreases rapidly with increasing dimensionality of the problem). The method is evaluated on problems involving distance constraints, and for minimization of conformational energy functions. In an initial application, conformations consistent with an experimental set of NMR distance constraints were obtained in a problem involving 48 variable dihedral angles.     

On the use of constraints in molecular mechanics. II. The Lagrange multiplier method and non-full-matrix Newton-Raphson minimization

It is shown how the Lagrange Multiplier method for constrained minimization can be implemented in a molecular mechanics program using the common approximations to the full-matrix Newton-Raphson minimization. The method reduces the number of cycles to achieve convergence, and also stabilizes the refinement process. Increases in computer memory requirements are small. As an application, the conformational surface of cycloheptane is calculated.     

Molecular parameters for the prediction of polyurethane structures

Recent efforts to determine the structures of poly(MDI/diol) hard segments in polyurethane elastomers have relied on the structures determined by single-crystal x-ray methods for diphenylmethane urethane model compounds. We have surveyed the structure of six model compounds, and have derived average values for the bond lengths, bond angles, and bond torsion angles for use in future analyses. The applicability of these averages to polymer structures is discussed, and the data are used to derive models for the poly(MDI-butanediol) chain which are found to be consistent with the fiber repeat determine by x-ray methods.     

Gold nanorod-seeded growth of silver nanostructures: from homogeneous coating to anisotropic coating

Single crystalline gold nanorods (Au NRs) dominated by {110} side facets were employed as seeds to tailor the deposition of Ag. Apart from homogeneous coating, anisotropic coating of Ag was observed and resulted in an orange slice-like shape for the Au@Ag nanocrystal. Different growth rates for the {110} side facets were responsible for this shape: among the four {110} facets, two of the neighboring {110} facets grew more quickly and another two grew more slowly, thus inducing the anisotropic deposition of Ag around the Au NR. This growth behavior is believed to be a consequence of competition between the strong stabilization of cetyltrimethylammomium bromide (CTAB) molecules to the {110} facets of Ag and minimization of the overall surface energy. Although the reason for the anisotropic coating remains to be clarified, our results lead to one important conclusion: The interaction of CTAB and metal can be utilized to tune the shapes of bimetallic structures.