Shapes and Sizes of Linear and Circular Multiple-ring Macromolecules

The shapes and sizes of linear and circular multiple-ring macromolecules in the framework of the Gaussian model have been numerically investigated in terms of shape factors, asphericity and prolateness factors and parameters, and shrinking factors. Simple analytic expressions for the eigenpolynomials of the Kirchhoff or architecture matrices for both linear and circular multi-rings in the limit of an infinitely large individual ring have been obtained via a new recursion method. It is found that for both types of multiple rings, shape asymmetry increases while size decreases as the number of rings increases, and that asphericity and prolateness parameters for a circular 99-ring macromolecule or a doubly stranded closed random walk have stronger dependence on dimensionality of the space in which the molecule is embedded than those of its linear counterpart. © 1997 John Wiley & Sons, Ltd.     

Shape anisotropy of polymers in disordered environment

We study the influence of structural obstacles in a disordered environment on the size and shape characteristics of long flexible polymer macromolecules. We use the model of self-avoiding random walks on diluted regular lattices at the percolation threshold in space dimensions d=2 and d=3. Applying the pruned-enriched Rosenbluth method, we numerically estimate rotationally invariant universal quantities such as the averaged asphericity and prolateness of polymer chain configurations. Our results quantitatively reveal the extent of anisotropy of macromolecules due to the presence of structural defects.     

Low-dimensional manifolds in reaction-diffusion equations. 1. Fundamental aspects

The approach to equilibrium for systems of reaction-diffusion equations on bounded domains is studied geometrically. It is shown that equilibrium is approached via low-dimensional manifolds in the infinite-dimensional function space for these dissipative, parabolic systems. The fundamental aspects of this process are mapped out in some detail for single species cases and for two-species cases where there is an exact solution. It is shown how the manifolds reduce the dimensionality of the system from infinite dimensions to only a few dimensions.     

The dimensionality of chromatographic separations

The box-counting or capacity dimension algorithm, known from the fractal mathematics literature, is used to measure the dimensionality D of chromatographic separation techniques for any number of dimensions. It is shown that D has limit properties that match Giddings' sample dimensionality s. D values are shown to be sensitive to the uniformity of peak spacing. A number of examples are given where D is calculated for various limits in one- and two-dimensional separations and for heart-cutting separations. The use of D as a quantitative measure of multidimensional orthogonality is suggested as D, due to the scale-free nature, is not dependent on the effective separation area. The connection to statistical peak overlap theory is discussed.     

DYNAMIC MONTE CARLO STUDY ON THE CORRELATION BETWEEN SHAPE AND SIZE OF LINEAR POLYMER CHAINS

The correlation between shape and size of linear chains on the simple cubic lattice is investigated using a dynamicMonte Carlo technique. A positive correlation between the asphericity parameter A and the square of the end-to-end distanceR~2, as well as that between A and the square of the radius of gyration S~2, is found for both RW and SAW chains, indicatingthat a chain conformation of small size is usually more spherical than one of large size. The result can explain why the shapeof the SAW chain deviate much more from a sphere than that of the RW chain, and can also explain the similar dependenceof size and shape on chain stiffness and on the distance of the first bead of a chain from an infinitely large flat surface.     

Some model inhomogeneous electron liquid in D dimensions: relation between energy and chemical potential and a spatial generalisation of Kato's nuclear cusp theorem

After a brief survey on the application of the Thomas–Fermi statistical method to bare Coulomb potential satisfying Laplace equation in D dimensions, which then relates energy and chemical potential, we next focus attention on model atomic ions with merely one and two electrons in D dimensions. In particular, for a bare Coulomb field we use the nodeless radial eigenfunctions given by Herschbach in D dimensions to derive the spatial generalization of Kato's nuclear cusp theorem in D dimensions for D?>?1. The unbinding of H ^? as a function of dimensionality D is also briefly referred.     

Projection of multidimensional GC data into alternative dimensions—exploiting sample dimensionality and structured retention patterns

Comprehensive multidimensional gas chromatography (GC×GC) is a powerful separation technique. One of the features of this technique is that it offers separations with more apparent structure than that offered by conventional one-dimensional GC (1-D GC). While some previous studies have alluded to this structure, and used structured retention patterns for some simple classifications, the topic of structured retention in GC×GC has not been studied in any great detail. Using the separation of fatty acid methyl esters (FAME) on both nonpolar/polar and polar/nonpolar column sets, the interaction between the separation dimensions and the sample dimensions is explored here. The GC×GC separation of a series of compounds is presented as a projection of the sample from sample space, a p-dimensional space with dimensions defined by the dimensionality of the sample, into separation space: for GC×GC, a two-dimensional plane passing through the sample space in an orientation defined by the separation conditions. Using this conceptual model and some a priori knowledge of the sample, it is shown how the image of the sample in the separation space can be used to construct an image of the sample in alternate dimensions, such as second dimension retention factor (²k) vs. chain length in the case of FAME. These projections into alternate dimensions should facilitate the interpretation of the complex patterns found within the GC×GC chromatogram for the identification and classification of compounds.     

Reappraisal of four different approaches for finding the mean reaction time in the multi-trap variant of the Adam-Delbrück problem

Adam and Delbrück argued that the dimensionality of the diffusion space determines the average lifetime of a diffusing particle confined to a region with a central trap. Doubts have often been aired as to whether their calculation is relevant to real biological systems, where the number of traps is usually much larger than unity; or whether the rate enhancement is merely a manifestation of an increase in the concentration of the traps; or whether the diverse multi-trap versions of their expression for the mean lifetime in two dimensions are trustworthy. These issues are addressed, and the long-standing problem of finding the low-density limit of trapping time in two dimensions solved, by examining previous treatments of the problem, and by carrying out simulations of two-dimensional systems in which the particles undergo a Pearsonian random walk, and the traps are distributed randomly or on a square lattice. The mean lifetimes are found to be different in the two situations, and it is concluded that the neglect of this aspect lies at the root of the conflict between some of the existing expressions for the mean lifetime. Relations expressing the mean lifetime as a function of the concentration of the traps are presented together with a discussion of their applicability.     

Metadynamics in the conformational space nonlinearly dimensionally reduced by Isomap

Atomic motions in molecules are not linear. This infers that nonlinear dimensionality reduction methods can outperform linear ones in analysis of collective atomic motions. In addition, nonlinear collective motions can be used as potentially efficient guides for biased simulation techniques. Here we present a simulation with a bias potential acting in the directions of collective motions determined by a nonlinear dimensionality reduction method. Ad hoc generated conformations of trans,trans-1,2,4-trifluorocyclooctane were analyzed by Isomap method to map these 72-dimensional coordinates to three dimensions, as described by Brown and co-workers [J. Chem. Phys. 129, 064118 (2008)]. Metadynamics employing the three-dimensional embeddings as collective variables was applied to explore all relevant conformations of the studied system and to calculate its conformational free energy surface. The method sampled all relevant conformations (boat, boat-chair, and crown) and corresponding transition structures inaccessible by an unbiased simulation. This scheme allows to use essentially any parameter of the system as a collective variable in biased simulations. Moreover, the scheme we used for mapping out-of-sample conformations from the 72D to 3D space can be used as a general purpose mapping for dimensionality reduction, beyond the context of molecular modeling.     

Dimensional effects on the LO-TO splitting in CF4: first-principles and infrared absorption studies

The development of longitudinal optical-transverse optical (LO-TO) modes in CF(4) has been studied experimentally and theoretically as a function of dimensionality. Infrared absorption experiments for CF(4) adsorbed on single-walled carbon nanotubes indicate a lack of LO-TO splitting at low coverage and a gradual appearance of LO-TO modes as the coverage of CF(4) on the nanotubes is increased. We have performed density functional perturbation theory calculations for the vibrational frequencies, IR absorption spectra, and phonon density of states for CF(4) in one, two, and three dimensions. The calculations demonstrate that LO-TO splitting in 1D is qualitatively different from that computed for 2D or the bulk. The magnitude of the splitting in 1D is about one-half that computed for the bulk, and the LO mode is very weakly blue-shifted in 1D. We predict that the phonon density of states changes dramatically as the dimensionality of the crystal is changed. This prediction can be tested experimentally via inelastic neutron scattering. We conclude that LO-TO splitting can be used as a probe to identify 1D states of matter.     

Lattice Monte Carlo investigations on copolymer systems, 1. Diblock copolymers

Symmetric diblock copolymers in dilute solution were examined by means of Monte Carlo simulations on a cubic lattice with respect to chain- and block dimensions, shape, local structure and number of contacts. The solvent was either a common good one, a common θ-solvent or a selective one for the two blocks. In all cases, repulsive interactions are operative between the blocks. In addition, the underlying homopolymers (athermal and θ) were divided into two parts (and treated as a block copolymer) for comparison. Chain-length was varied from 40 to 1280 segments leading to the expected values for the critical exponent 2v ≈ 1.2 for good solvent quality and 2v ≈ 1.0 for θ-solvent. Copolymers in a selective solvent scale with an intermediate exponent, 2v ≈ 1.13. The deviation of the mean squared dimensions of the copolymers from the sum of those of two homopolymers of the same length and for the same solvent quality as the blocks is largest for block copolymers in a common θ-solvent (where it exceeds 20%), while the blocks themselves have mostly the same dimensions as their underlying homopolymers of equal length. The shape of the copolymers, expressed by the parameter δ (asphericity) becomes more rod-like with increasing chain-length if there are (compact) θ-blocks in the molecule which are subject to mutual repulsive interaction. In these cases, θ exceeds the value of the homopolymers in the limit of infinite chain-length. The number of contacts per segment approaches a limiting value with increasing chain-length which is ≈0.20 for athermal chains and athermal blocks. For θ-chains and θ-blocks, a limiting value is not yet reached within the range of chainlengths investigated. The number of contacts per segment between two different blocks quickly tends to zero with increasing chain-length.     

The Effects of Chemical Bonding at Subatomic Resolution: A Case Study on α-Boron

Similar to classical asphericity shifts, aspherical deformations of the electron density in the atomic core region can result in core asphericity shifts in refinements using a Hansen-Coppens multipolar model (HCM), especially when highly precise experimental datasets with resolutions far beyond sin(θ)/λ ≤ 1.0 Å⁻¹ are employed. These shifts are about two orders of magnitude smaller than their counterparts caused by valence shell deformations, and their underlying deformations are mainly of dipolar character for 1st row atoms. Here, we analyze the resolution dependence of core asphericity shifts in α-boron. Based on theoretical structure factors, an appropriate Extended HCM (EHCM) is developed, which is tested against experimental high-resolution (sin(θ)/λ ≤ 1.6 Å⁻¹) single-crystal diffraction data. Bond length deviations due to core asphericity shifts of α-boron in the order of 4–6·10⁻⁴ Å are small but significant at this resolution and can be effectively compensated by an EHCM, although the correlation of the additional model parameters with positional parameters prevented a free refinement of all core model parameters. For high quality, high resolution data, a proper treatment with an EHCM or other equivalent methods is therefore highly recommended.     

Ring-polymer instanton method for calculating tunneling splittings

The semiclassical instanton expression for the tunneling splitting between two symmetric wells is rederived, starting from the ring-polymer representation of the quantum partition function. This leads to simpler mathematics by replacing functional determinants with matrix determinants. By exploiting the simple Hu?ckel-like structure of the matrices, we derive an expression for the instanton tunneling splitting in terms of a minimum on the potential surface of a linear polymer. The latter is a section cut out of a ring polymer, consisting of an infinite number of beads, which describes a periodic orbit on the inverted potential surface. The approach is straightforward to generalize to multiple dimensions, and we demonstrate that it is computationally practical by carrying out instanton calculations of tunneling splittings in HO(2) and malonaldehyde in full dimensionality.     

Construction of the free energy landscape of biomolecules via dihedral angle principal component analysis

A systematic approach to construct a low-dimensional free energy landscape from a classical molecular dynamics (MD) simulation is presented. The approach is based on the recently proposed dihedral angle principal component analysis (dPCA), which avoids artifacts due to the mixing of internal and overall motions in Cartesian coordinates and circumvents problems associated with the circularity of angular variables. Requiring that the energy landscape reproduces the correct number, energy, and location of the system's metastable states and barriers, the dimensionality of the free energy landscape (i.e., the number of essential components) is obtained. This dimensionality can be determined from the distribution and autocorrelation of the principal components. By performing an 800 ns MD simulation of the folding of hepta-alanine in explicit water and using geometric and kinetic clustering techniques, it is shown that a five-dimensional dPCA energy landscape is a suitable and accurate representation of the full-dimensional landscape. In the second step, the dPCA energy landscape can be employed (e.g., in a Langevin simulation) to facilitate a detailed investigation of biomolecular dynamics in low dimensions. Finally, several ways to visualize the multidimensional energy landscape are discussed.     

Application of pattern recognition and feature extraction techniques to volatile constituent metabolic profiles obtained by capillary gas chromatography

The applicability of threshold logic units, a form of nonparametric pattern recognition, to the processing of metabolic profile data obtained by high-efficiency glass capillary column gas chromatography has been investigated. The test data included profiles of the volatile constituents of urine from normal individuals and from individuals with diabetes mellitus. A feature extraction algorithm allowed for dimensionality reduction and indicated the constituents most important in the normal versus pathological distinction. With an optimum number of dimensions, a normal versus pathological prediction rate of 93.75% was achieved. Gas chromatography-mass spectrometry was utilized to identify important profile constituents.     

Instantaneous normal mode analysis of orientational motions in liquid water: local structural effects

We have investigated the effects of local structures on the orientational motions in liquid water in terms of the instantaneous normal mode (INM) analysis. The local structures of a molecule in liquid water are characterized by two different kinds of index: the asphericity parameter of its Voronoi polyhedron and the numbers of the H bonds donated and accepted by the molecule. According to the two kinds of index, the molecules in the simulated water are classified into subensembles, for which the rotational contributions to the INM spectrum are calculated. Our results indicate that by increasing the asphericity, the rotational contribution has a shift toward the high-frequency end in the real spectrum and a decrease in the fraction of the imaginary modes. Furthermore, we find that this shift essentially relies on the number of the donated H bonds of a molecule, but has almost nothing to do with that of the accepted H bonds. The local structural effects resulting from the geometry of water molecule are also discussed.     

高分子链形状与尺寸关联的Monte Carlo模拟

运用MonteCarlo方法对线型高分子链格点模型的构型进行了模拟,研究了构型的尺寸(采用平方末端距R²,平方回转半径S²来表征)和形状(由非球形因子A表征)之间的关联.对任何长度的高分子链,其关联系数CA,R²和CA,S²均为正值,表明高分子链的形状与尺寸之间存在正关联,即尺寸小的构型其非球形因子A一般也小,反之尺寸大的构型其非球形因子A一般也大.关联系数CA,R²和CA,S²均随链长的增大而减小,近似地与链长的倒数(n^-1)成正比.研究还表明,关联系数的极限值(链长n很大时)与格点的类型无关,与链样本产生的方式也无关,但与链是否考虑排斥体积有关,考虑了排斥体积后,关联系数增大.     

X-ray crystallographic study of ranaconitine

The crystal structure of natural diterpenoid alkaloid ranaconitine isolated from Aconitum sinomontanum Nakai has been determined by single crystal X-ray diffraction analysis. The crystal presents a monoclinic system, space group C2 with Z = 4, unit cell dimensions a = 30.972(19) angstrom, b = 7.688(5) angstrom, and c = 19.632(12) angstrom. Moreover, the intermolecular O-H...O hydrogen bonds and weak pi-pi interactions play a critical role in expanding the dimensionality.