Supercollisions and energy transfer of highly vibrationally excited molecules

Collisional energy-transfer probability distribution functions of highly vibrationally excited molecules and the existence of supercollisions remain as the outstanding questions in the field of intermolecular energy transfer. In this investigation, collisional interactions between ground state Kr atoms and highly vibrationally excited azulene molecules (4.66 eV internal energy) were examined at a collision energy of 410 cm-1 using a crossed molecular beam apparatus and time-sliced ion imaging techniques. A large amount of energy transfer (1000-5000 cm-1) in the backward direction was observed. We report the experimental measurement for the shape of the energy-transfer probability distribution function along with a direct observation of supercollisions.     

Adsorption‐Induced Structural Phase Transformation in Nanopores

We report a new type of structural transformation occurring in methane adsorbed in micropores. The observed methane structures are defined by probability distributions of molecular positions. The mechanism of the transformation has been modeled using Monte Carlo method. The transformation is totally determined by a reconstruction of the probability distribution functions of adsorbed molecules. The methane molecules have some freedom to move in the pore but most of the time they are confined to the positions around the high probability adsorption sites. The observed high‐probability structures evolve as a function of temperature and pressure. The transformation is strongly discontinuous at low temperature and becomes continuous at high temperature. The mechanism of the transformation is influenced by a competition between different components of the interaction and the thermal energy. The methane structure represents a new state of matter, intermediate between solid and liquid.     

A new derivation of average molecular weights of nonlinear polymers.

A new method for calculating average molecular weights is presented for nonlinear polymers. In contrast to the previous methods of Flory and Stockmayer which first calculate the distribution of all species and then use the distributions to calculate average properties, the new method calculates these properties directly. In contrast to the method of Gordon, probability generating functions are not required. Starting with elementary probability and utilizing the recursive nature of network polymers property relations can be developed more simply. We illustrate the method for calculations of Mw Mz, and the gel point for a wide variety of polyfunctional polymerizations.     

A Method for Following Changes in Molecular Weight Distributions of Polymers on Degradation: Development and Comparison with Ultrasonic Degradation Experiments

When polymers are degraded by submitting them to shearing action, neither the probability of scission of a macromolecule, as a function of molecular weight (MW), nor the probability distribution of the lengths of the resulting fragments is known a priori. Information about these probabilities can be obtained by comparing observed changes in the molecular weight distributions (MWD) of degrading polymer with changes calculated from models. These models are based on assumed functions relating the probabilities mentioned to the MW of the polymer and that of the fragments. In order to make this comparison, equations are derived for various moments of the MWDs in terms of arbitrary initial MWDs and probability functions. Some simplified forms of these equations arising from chosen simple functions for the probabilities are also given. Data are presented to show that the precision of the weight-average MW calculated from gel-permeation chromatography (GPC) is about twice that of the number-average and z-average MW. On the basis of this observation, a particular index is recommended to indicate the shape of MWDs derived from GPC. This index, calculated from the experimental MWDs of degrading polystyrene as a function of degree of degradation, is compared with index curves computed from models with specified forms for the two probability functions. These comparisons suggest that this polymer has a minimum degradable MW of about 40,000. Finally experimental MWDs are shown to agree well with those calculated.     

Normalizing molecular docking rankings using virtually generated decoys

Drug discovery research often relies on the use of virtual screening via molecular docking to identify active hits in compound libraries. An area for improvement among many state-of-the-art docking methods is the accuracy of the scoring functions used to differentiate active from nonactive ligands. Many contemporary scoring functions are influenced by the physical properties of the docked molecule. This bias can cause molecules with certain physical properties to incorrectly score better than others. Since variation in physical properties is inevitable in large screening libraries, it is desirable to account for this bias. In this paper, we present a method of normalizing docking scores using virtually generated decoy sets with matched physical properties. First, our method generates a set of property-matched decoys for every molecule in the screening library. Each library molecule and its decoy set are docked using a state-of-the-art method, producing a set of raw docking scores. Next, the raw docking score of each library molecule is normalized against the scores of its decoys. The normalized score represents the probability that the raw docking score was drawn from the background distribution of nonactive property-matched decoys. Assuming that the distribution of scores of active molecules differs from the nonactive score distribution, we expect that the score of an active compound will have a low probability of having been drawn from the nonactive score distribution. In addition to the use of decoys in normalizing docking scores, we suggest that decoy sets may be a useful tool to evaluate, improve, or develop scoring functions. We show that by analyzing docking scores of library molecules with respect to the docking scores of their virtually generated property-matched decoys, one can gain insight into the advantages, limitations, and reliability of scoring functions.     

On the calculation of orientational autocorrelation functions for simple models of molecular reorientation

It is shown how a theorem about characteristic functions of random variables having a gaussian probability distribution may be used to obtain all the orientation autocorrelation functions for various two dimensional models of molecular reorientation. It is also shown how a simple modification of the Smoluchkowski equation may be used to obtain approximate expressions for these functions for a sphere undergoing rotational brownian motion when inertial effects are included.     

Structure of microemulsion-ABA triblock copolymer networks

Structural equilibrium properties of transient networks formed by microemulsion droplets and ABA triblock copolymers in solution have been studied by Monte Carlo simulation. The droplets were represented by soft spheres, and the polymers were represented by junctions connected by harmonic bonds with an angular potential regulating the intrinsic chain stiffness. The interaction parameters were selected such that the end A-blocks were localized inside the droplets and the middle B-block in the continuous phase. The influence of (i) the polymer concentration, (ii) the polymer stiffness, and (iii) the contour length of the middle B-block on the formation and the structure of the microemulsion-polymer network were investigated using polymer end-to-end separation probability distribution functions, droplet radial distribution functions, droplet-droplet nearest-neighbor probability distribution functions, and network connectivity indicators. An increase of the polymer-droplet number ratio had a strong impact on the network formation. Under typical conditions and at an intermediate polymer-droplet number ratio, (i) the fraction of polymers forming bridges between droplets increased from essentially zero to unity and (ii) the fraction of polymers that were forming loops decreased as the ratio of the polymer end-to-end separation and the surface-to-surface separation between neighboring droplets for a hypothetical homogeneous droplet distribution was increased from 0.5 to 2. For long and flexible polymers, a mesoscopic segregation triggered by a depletion attraction between droplets appeared, and, furthermore, for sufficiently stiff chains, only bridge conformations occurred. The percolation probability could be represented as a function of the average droplet cluster size only, across all systems.     

交联高分子的分子量分布

从概率母函数出发,利用Lagrange展开,导出了起始分子为Flory和Poisson分布的交联高聚物的分子量分布函数,并用以描述交联分子结构的溶胶分数。     

From secondary structure to three-dimensional structure: Improved dihedral angle probability distribution function for use with energy searches for native structures of polypeptides and proteins

An improved scheme to help in the prediction of protein structure is presented. This procedure generates improved starting conformations of a protein suitable for energy minimization. Trivariate gaussian distribution functions for the π, ψ, and χ¹ dihedral angles have been derived, using conformational data from high resolution protein structures selected from the Protein Data Bank (PDB). These trivariate probability functions generate initial values for the π, ψ, and χ¹ dihedral angles which reflect the experimental values found in the PDB. These starting structures speed the search of the conformational space by focusing the search mainly in the regions of native proteins. The efficiency of the new trivariate probability distributions is demonstrated by comparing the results for the α-class polypeptide fragment, the mutant Antennapedia (C39 → S) homeodomain (2HOA), with those from two reference probability functions. The first reference probability function is a uniform or flat probability function and the second is a bivariate probability function for π and ψ. The trivariate gaussian probability functions are shown to search the conformational space more efficiently than the other two probability functions. The trivariate gaussian probability functions are also tested on the binding domain of Streptococcal protein G (2GB1), an α/β class protein. Since presently available energy functions are not accurate enough to identify the most native-like energy-minimized structures, three selection criteria were used to identify a native-like structure with a 1.90-Å rmsd from the NMR structure as the best structure for the Antennapedia fragment. Each individual selection criterion (ECEPP/3 energy, ECEPP/3 energy-plus-free energy of hydration, or a knowledge-based mean field method) was unable to identify a native-like structure, but simultaneous application of more than one selection criterion resulted in a successful identification of a native-like structure for the Antennapedia fragment. In addition to these tests, structure predictions are made for the Antennapedia polypeptide, using a Pattern Recognition-based Importance-Sampling Minimization (PRISM) procedure to predict the backbone conformational state of the mutant Antennapedia homeodomain. The ten most probable backbone conformational state predictions were used with the trivariate and bivariate gaussian dihedral angle probability distributions to generate starting structures (i.e., dihedral angles) suitable for energy minimization. The final energy-minimized structures show that neither the trivariate nor the bivariate gaussian probability distributions are able to overcome the inaccuracies in the backbone conformational state predictions to produce a native-like structure. Until highly accurate predictions of the backbone conformational states become available, application of these dihedral angle probability distributions must be limited to problems, such as homology modeling, in which only a limited portion of the backbone (e.g., surface loops) must be explored. © 1996 John Wiley & Sons, Inc.     

Electron localization function as information measure

The conditional two-electron probability function, which defines the electron localization function (ELF) of Becke and Edgecombe in the Kohn-Sham theory, is interpreted as the nonadditive (interorbital) Fisher information contained in the electron distribution. The probability normalization considerations suggest a use of the related information measure defined in terms of the unity-normalized probability distributions (shape factors of the electron densities), as the key ingredient of the modified information-theoretic ELF. This modified Fisher information density is validated by a comparison with the original two-electron probability function. Illustrative applications to typical molecular systems demonstrate the adequacy of the modified information-theoretic ELF in extracting the key features of the electron distributions in molecules. The overall Fisher information itself and the associated information-distance quantities are also proposed as complementary localization functions.     

Carbohydrate clustering in aqueous solutions and the dynamics of confined water

We use molecular dynamics simulations to investigate structure and dynamics of fructose aqueous solutions in the 1-5 M concentration range at ambient conditions. We analyze hydration structures, H-bond statistics, and size distribution of H-bonded carbohydrate clusters as functions of concentration. We find that the local tetrahedral order of water is reasonably well-preserved and that the solute tends to appear as scattered "isolated" molecules at low concentrations and as H-bonded clusters for less diluted solutions. The sugar cluster size distribution exhibits a sharp transition to a percolated cluster between 3.5 and 3.8 M. The percolated cluster forms an intertwined network of H-bonded saccharides that imprisons water. For the dynamics, we find good agreement between simulation and available experimental results for the self-diffusion coefficients. Water librational dynamics is little affected by sugar concentration, whereas reorientational relaxation is described by a concentration-independent bulk-like component attributed to noninterfacial water molecules and a slower component (strongly concentration dependent) that arises from interfacial solvent molecules and, hence, depends on the dynamics of the cluster structure itself. Analysis of H-bonding survival probability functions indicates that the formation of carbohydrate clusters upon increasing concentration enhances the H-bond relaxation time and slows down the entire system dynamics. We find that multiexponential or stretched-exponential fits alone cannot describe the H-bond survival probabilities for the entire postlibrational time span of our data (0.1-100 ps), as opposed to a combined stretched-plus-biexponential function, which provides excellent fits. Our results suggest that water dynamics in concentrated fructose solutions resembles in many ways that of protein hydration water.     

Dependence of the cyclization behavior of multifunctional network molecules on molecular size

The tendency for cyclization of nonlinear network molecules is quantitatively expressed by the cyclization probability, defined as the ratio of cyclization to total reaction rate. In contrast to branching, the rate of cyclization depends on the configurational statistical mechanics of segments joining functional groups. Direct integration of the joint configurational probability density in the subspace of encounters yields the distribution of the number of configurationally formed intramolecular functional group encounters (nearest neighbors). Statistically independent network segments are assumed to obey the Gaussian statistics. A recursive relationship for the distribution of chain lengths is developed, and it is shown that for large molecular sizes this relationship tends to a limiting distribution. Corresponding average and standard deviation follow power law dependence on degree of polymerization (DP). With these results, cyclization probabilities are explicitly expressed as functions of DP. When segmental diffusion is the rate controlling factor, cyclization is similar to short range (or primary) cyclization. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 861–870, 2005     

Analytical HDMR formulas for functions expressed as quadratic polynomials with a multivariate normal distribution

High Dimensional Model Representation (HDMR) is a general set of quantitative model assessment and analysis tools for systems with many variables. A general formulation for the HDMR component functions with independent and correlated variables was obtained previously. Since the HDMR component functions generally are coupled to one another and involve multi-dimensional integrals, explicit formulas for the component functions are not available for an arbitrary function with an arbitrary probability distribution amongst their variables. This paper presents analytical formulas for the HDMR component functions and the corresponding sensitivity indexes for the common case of a function expressed as a quadratic polynomial with a multivariate normal distribution over its variables. This advance is important for practical applications of HDMR with correlated variables.     

Configurational statistics of macromolecules in solution from correlations of liquid molecules

We address the relevant quest for a simple formalism describing the microstructure of liquid solutions of polymer chains. On the basis of a recent relativistic-type picture of self-diffusion in (simple) liquids named Brownian relativity (BWR), a covariant van Hove's distribution function in a Vineyard-like convolution approximation is proposed to relate the statistical features of liquid and chain molecules forming a dilute polymer solution. It provides an extension of the Gaussian statistics of ideal chains to correlated systems, allowing an analysis of macromolecular configurations in solution by the only statistical properties of the liquid units (and vice versa). However, the mathematical solution to this issue is not straightforward because, when the liquid and polymer van Hove's functions are equated, an inverse problem takes place. It presents some conceptual analogies with a scattering experiment in which the correlation of the liquid molecules acts as the radiation source and the macromolecule as the scatterer. After inverting the equation by a theorem coming from the Tikhonov's approach, it turns out that the probability distribution function of a real polymer can be expressed from a static Ornstein-Uhlenbeck process, modified by correlations. This result is used to show that the probability distribution of a true self-avoiding walk polymer (TSWP) can be modeled as a universal Percus-Yevick hard-sphere solution for the total correlation function of the liquid units. This method suits in particular the configurational analysis of single macromolecules. The analytical study of arbitrary many-polymer systems may require further mathematical investigation.     

Photonic superdiffusive motion in resonance radiation trapping

In this work we consider the relation between the jump length probability density function and the line shape function in resonance radiation trapping in atomic vapors. The two-sided jump length probability density function suitable for a unidimensional formulation of radiative transfer is also derived. As a side result, a procedure to obtain the Maxwell distribution of velocities from the Maxwell-Boltzmann distribution of speeds was obtained. General relations that give the asymptotic jump length behavior and the Levy flight parameter mu for any line shape are obtained. The results are applied to generalized Doppler, generalized Lorentz, and Voigt line shape functions. It is concluded that the lighter the tail of the line shape function, the less heavy the tail of the jump length probability density function, although this tail is always heavy, with mu < or =1.     

Enthalpy distributions and densities of states in linear polymers

Using polypropylene as an example, we applied a method we have recently developed to calculate the probability distribution of enthalpy from the temperature dependence of the heat capacity. The method involves the use of local temperature expansions of the heat capacity to calculate a set of moments of the enthalpy distribution. Using the maximum‐entropy method, one can then construct the enthalpy distribution for the system. The method is completely model free. The enthalpy distribution so obtained is the analogue of the Maxwell–Boltzmann distribution of kinetic energies for a gas, and like that function, tells one the distribution of enthalpies that an average unit in the polymer chain can have, a quantity that is crucial to understanding the chemical and physical properties of a polymer. Given the enthalpy distribution, one can then calculate the Gibbs free energy and the density of states that correspond to a particular value of enthalpy, thus giving one an expanded thermodynamics of the system in addition to the usual average quantities. We illustrate the fact that the Gibbs free‐energy distribution for this system scales as a simple function of temperature and that the density‐of ‐states function yields a simple empirical partition function for the system giving both the average thermodynamics and the distribution functions. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 1513–1518, 2001     

Convergence of the distribution functions for wormlike chains

The Daniels-type distribution functions of the end-to-end distance of three-dimensional and two-dimensional wormlike chains are obtained to terms of order t^?10, by an operational method with use of a digital computer, where t is the ratio of the total chain contour length to the Kuhn segment length in three-dimensional cases and of the contour length to the persistence length in two-dimensional cases. The convergence of the ring-closure probability and the mean reciprocal distance is examined on the basis of these distribution functions. A similar study of the moment-based distribution functions is also made.     

Adaptive‐numerical‐bias metadynamics

A metadynamics scheme is presented in which the free energy surface is filled with progressively adding adaptive biasing potentials, obtained from the accumulated probability distribution of the collective variables. Instead of adding Gaussians with assigned height and width in conventional metadynamics method, here we add a more realistic adaptive biasing potential to the Hamiltonian of the system. The shape of the adaptive biasing potential is adjusted on the fly by sampling over the visited states. As the top of the barrier is approached, the biasing potentials become wider. This decreases the problem of trapping the system in the niches, introduced by the addition of Gaussians of fixed height in metadynamics. Our results for the free energy profiles of three test systems show that this method is more accurate and converges more quickly than the conventional metadynamics, and is quite comparable (in accuracy and convergence rate) with the well‐tempered metadynamics method. © 2017 Wiley Periodicals, Inc.